![Postulates, statements, theorems
Theorem 1.1 Wick theorem (page 10)
Lemma 1.2 Commutator of renorm operators (page 20)
Lemma 1.3 Product of regular operators (page 23)
Theorem 1.4 Properties of phys operators (page 23)
Lemma 1.5 Product of phys operators (page 24)
Lemma 1.6 Commutator [phys,renorm] (page 24)
Lemma 1.7 Commutator [unphys,renorm] (page 24)
Lemma 1.8 Commutator [phys,unphys] (page 25)
Lemma 1.9 t-derivative of operator (page 25)
Lemma 2.1 Disconnected part of product (page 47)
Theorem 2.2 Multiple commutator (page 48)
Lemma 2.3 Number of independent loops (page 49)
Theorem 2.4 Convergence of loop integrals (page 51)
Theorem 2.5 Cluster separability of smooth potentials (page 54)
Theorem 2.6 Smoothness of products of potentials (page 54)
Theorem 2.7 Smoothness of commutators of potentials (page 55)
Statement 4.1 Absence of self-scattering (page 77)
Postulate 4.2 Charge renormalization condition (page 78)
https://doi.org/10.1515/9783110493207-203](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-19-320.jpg)
![XVIII | Preface
cluster separability. Our first two chapters have a mostly technical character. They de-
fine our terminology and notation and prepare us for a more in-depth study of QED in
the two following chapters.
In Chapter 3, Quantum electrodynamics, we introduce the important concept of
the quantum field. This idea will be applied to systems of charged particles and pho-
tons in the formalism of QED. Here we will obtain an interacting theory, which satisfies
the principles of relativistic invariance and cluster separability, where the number of
particles is not fixed. However, the “naïve” version of QED presented here is unsat-
isfactory, since it cannot calculate scattering amplitudes beyond the lowest orders of
perturbation theory.
Chapter 4, Renormalization, completes the second volume of the book. We will dis-
cussthe plagueof ultravioletdivergencesin the “naïve”QED and explain how they can
be eliminated by adding counterterms to the Hamiltonian. As a result, we will get the
traditional “renormalized” QED, which has proven itself in precision calculations of
scattering cross sections and energy levels in systems of charged particles. However,
this theory failed to provide a well-defined interacting Hamiltonian and the interact-
ing time evolution (= dynamics). We will address these issues in the third volume of
our book.
As in the first volume, here we refrain from criticism and unconventional interpre-
tations, trying to keep in line with generally accepted approaches. The main purpose
of this volume is to explain the basic concepts and terminology of QFT. For the most
part, we will adhere to the logic of QFT formulated by Weinberg in the series of articles
[19, 18, 20] and in the excellent textbook [21]. A critical discussion of the traditional
approaches and a new look at the theory of relativity will be presented in Volume 3
[17].
References to Volume 1 [16] of this book will be prefixed with “1-”. For example,
(1-7.14) is formula (7.14) from Volume 1.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-24-320.jpg)
![1.1 Creation and annihilation operators | 5
in sectors with n electrons. We should distinguish two alternatives. In the first case,
the created one-particle state (psz) is among the states present in (1.14), for example,
(psz) = (pisiz). Since electrons are fermions and two fermions cannot occupy the same
state due to the Pauli principle, this action leads to the zero result, i. e.,
a†
psz
|p1s1z, . . . , pi−1s(i−1)z, pisiz, pi+1s(i+1)z, . . . , pnsnz⟩ = 0. (1.15)
In the second case, the created state (psz) is not among the single-particle states form-
ing (1.14). Then, the creation operator a†
psz
simply adds one electron to the beginning
of the particle list, so
a†
psz
|p1s1z, p2s2z, . . . , pn, snz⟩ ≡ |psz, p1s1z, p2s2z, . . . , pnsnz⟩. (1.16)
In this case, the operator a†
psz
converts a state with n electrons to a state with n+1 elec-
trons. By repeatedly applying creation operators to the vacuum vector |vac⟩, we can
construct all basis vectors in the purely electronic part of the Fock space. For example,
a†
psz
|vac⟩ = |psz⟩, (1.17)
a†
p1s1z
a†
p2s2z
|vac⟩ = |p1s1z, p2s2z⟩
are basis vectors in the one-electron and two-electron sectors.
We define the electron annihilation operator apsz
as a Hermitian conjugate to the
creation operator a†
psz
. One can prove [21] that the action of apsz
on the n-electron state
(1.14) is as follows. If the one-electron state with parameters (psz) is already occupied,
for example, (psz) = (pisiz), then this state is “annihilated” and the number of parti-
cles in the system decreases by one, i. e.,
apsz
|p1s1z, . . . , pi−1s(i−1)z, pisiz, pi+1s(i+1)z, . . . , pnsnz⟩
= (−1)𝒫
|p1s1z, . . . , pi−1s(i−1)z, pi+1s(i+1)z, . . . , pnsnz⟩. (1.18)
Here 𝒫 is the number of permutations of neighboring particles, which is necessary to
move the annihilated one-particle state i to the first place in the list. If the state (psz)
is absent, i. e., (psz) ̸
= (pisiz) for all i, then
apsz
|p1s1z, p2s2z, . . . , pnsnz⟩ = 0. (1.19)
Acting on the vacuum state, annihilation operators always yield zero, i. e.,
apsz
|vac⟩ = 0. (1.20)
The above formulas define the actions of creation and annihilation operators on
the basis vectors in purely electronic sectors. These rules do not change in the pres-
ence of other particles, and they extend to any linear combinations of basis vectors by](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-29-320.jpg)
![1.1 Creation and annihilation operators | 7
1.1.6 Creation and annihilation operators for photons
For photons that are bosons, the properties of creation and annihilation operators
differ slightly from the fermion operators described above. Two or more photons can
coexist in the same quantum state. Therefore, we determine the action of the photon
creation operator c†
pτ
7
on a multi-photon state as
c†
pτ|p1τ1, p2τ2, . . . , pnτn⟩ = |pτ, p1τ1, p2τ2, . . . , pnτn⟩,
regardless of whether there was a particle (pτ) in the initial state or not. As in the case
of fermions, boson annihilation operators cpτ are defined as Hermitian conjugates of
the creation operators. The photon annihilation operator cpτ completely destroys a
multi-photon state, so
cpτ|p1τ1, p2τ2, . . . , pnτn⟩ = 0
if the annihilated one-photon state (pτ) was absent there. If the photon (pτ) was
present, then the annihilation operator cp,τ simply removes this component, thus
generating an (n − 1)-photon state,
cpiτi
|p1τ1, . . . , pi−1τi−1, piτi, pi+1τi+1, . . . , pnτn⟩
= |p1τ1, . . . , pi−1τi−1, pi+1τi+1, . . . , pnτn⟩.
The above formulas can be extended without change to states where, in addition
to photons, other particles are also present. Also, the action of operators extends by
linearity to superpositions of basis vectors. From these rules, proceeding in analogy
with Subsection 1.1.5, we obtain the following commutation relations for the photon
annihilation and creation operators:
[cpτ, c†
pτ ] = δp,p δττ ,
[cpτ, cpτ ] = [c†
pτ, c†
pτ ] = 0.
1.1.7 Particle number operators
With the help of creation and annihilation operators, we can build explicit expressions
for various useful observables in the Fock space. Consider, for example, the product
of two photon operators,
Npτ = c†
pτcpτ. (1.22)
7 The photon’s momentum is p and τ is its helicity.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-31-320.jpg)
![1.1 Creation and annihilation operators | 9
The same arguments can be applied to the operators of positrons (bpsz
and b†
psz
),
protons (dpsz
and d†
psz
), antiprotons (fpsz
and f†
psz
) and photons (cpτ and c†
pτ). So, finally,
we get the following set of anticommutation and commutation relations relevant to
QED:
{apsz
, a†
ps
z
} = {bpsz
, b†
ps
z
} = {dpsz
, d†
ps
z
} = {fpsz
, f †
ps
z
} = δ(p − p
)δszs
z
, (1.24)
{apsz
, aps
z
} = {bpsz
, bps
z
} = {dpsz
, dps
z
} = {fpsz
, fps
z
}
= {a†
psz
, a†
ps
z
} = {b†
psz
, b†
ps
z
} = {d†
psz
, d†
ps
z
}
= {f†
psz
, f †
ps
z
} = 0, (1.25)
[cpτ, c†
pτ ] = δ(p − p
)δττ , (1.26)
[c†
pτ, c†
pτ ] = [cpτ, cpτ ] = 0. (1.27)
Commutators of operators related to different particles are always zero.
In the limit of continuous momentum, the counterpart of the particle counter
(1.22) is the operator
ρpτ = c†
pτcpτ, (1.28)
which can be interpreted as the density of photons with helicity τ and momentum p.
Having summed the density (1.28) by the photon polarizations and integrating it over
the entire momentum space, we obtain the operator of the total number of photons in
the system
Nph = ∑
τ
∫ dpc†
pτcpτ. (1.29)
We can also write similar expressions for the numbers of other particles. For example,
Nel = ∑
sz
∫ dpa†
psz
apsz
(1.30)
is the electron number operator. Then the operator
N = Nel + Npo + Npr + Nan + Nph (1.31)
expresses the total number of particles in the system.
1.1.9 Normal ordering
It is necessary to note the important property of operators (1.29) and (1.30). Being ex-
pressed through particle creation and annihilation operators, they are applicable in](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-33-320.jpg)
![1.1 Creation and annihilation operators | 11
(i) put the contracted operators next to each other (i. e., in the αα†
configuration) and
(ii) rearrange in the normal order the operators left after all contractions.
The proof of this theorem can be found in many textbooks on quantum field the-
ory, for instance in [1]. Here we simply illustrate this result by the example of the prod-
uct of electron operators aq ap a†
pa†
q. According to Wick’s theorem, in a normally or-
dered form, this operator is the sum of the fully ordered product and six contractions13
:
: aq ap a†
pa†
q : ≡ a†
pa†
qaq ap ,
aq ap a†
pa†
q ≡ −a†
qaq δ(p − p
),
aq ap a†
pa†
q ≡ −a†
pap δ(q − q
),
aq ap a†
pa†
q ≡ a†
qap δ(p − q
),
aq ap a†
pa†
q ≡ a†
paq δ(q − p
),
aq ap a†
pa†
q ≡ δ(p − p
)δ(q − q
),
aq ap a†
pa†
q ≡ −δ(p − q
)δ(q − p
).
1.1.10 Noninteracting energy and momentum
Now we can fully appreciate the benefits of introducing creation and annihilation op-
erators. In particular, with their help it is easy to obtain a compact expression for the
noninteracting Hamiltonian H0. It is obtained simply from the particle number oper-
ator (1.31), multiplying the integrands (particle densities in the momentum space) by
the energies of free particles, i. e.,
H0 = Hel+po
0 + Hpr+an
0 + Hph
0 , (1.32)
Hel+po
0 = ∫ dpωp ∑
sz=±1/2
[a†
psz
apsz
+ b†
psz
bpsz
],
Hpr+an
0 = ∫ dpΩp ∑
sz=±1/2
[d†
psz
dpsz
+ f †
psz
fpsz
],
Hph
0 = c ∫ dpp ∑
τ=±1
c†
pτcpτ. (1.33)
Here ωp = √m2
ec4 + p2c2 are energies of free electrons and positrons, Ωp =
√m2
pc4 + p2c2 are energies of free protons and antiprotons and cp are energies of
13 Contracted pairs of operators are marked with overline signs.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-35-320.jpg)
![12 | 1 Fock space
free photons. It is not difficult to verify that H0 in (1.32) acts on states from the
sector H (1, 0, 0, 0, 0) in the same way as equation (1.12) and H0 acts in the sector
H (2, 0, 0, 0, 2) exactly as (1.13). So, we got a single expression for the energy that
works equally well in all sectors of the Fock space.14
Similar arguments show that the
operator
P0 = Pel+po
0 + Ppr+an
0 + Pph
0 , (1.34)
Pel+po
0 = ∫ dpp ∑
sz=±1/2
[a†
psz
apsz
+ b†
psz
bpsz
],
Ppr+an
0 = ∫ dpp ∑
sz=±1/2
[d†
psz
dpsz
+ f†
psz
fpsz
],
Pph
0 = ∫ dpp ∑
τ=±1
c†
pτcpτ (1.35)
has the meaning of the total momentum.
1.1.11 Noninteracting angular momentum and boost
Expressions for the generators J0 and K0 in the Fock space are more complicated,
since they requirethe participation of derivativesof particle operators. For illustration,
consider the example of a massive spinless particle with creation and annihilation
operators α†
p and αp, respectively. The effect of the rotation e− i
ℏ
J0zφ
on the one-particle
state |p⟩ (see (1-5.10))
e− i
ℏ
J0zφ
|p⟩ = |px cos φ + py sin φ, py cos φ − px sin φ, pz⟩ ≡ |φp⟩
can be imagined as annihilation of the initial state |p⟩ = |px, py, pz⟩ followed by cre-
ation of the rotated state |φp⟩, i. e.,
e− i
ℏ
J0zφ
|p⟩ = α†
φpαp|px, py, pz⟩.
Therefore, for an arbitrary one-particle state the operator of finite rotation has the form
e− i
ℏ
J0zφ
= ∫ dpα†
φpαp. (1.36)
It is not difficult to see that the same form is valid in the entire Fock space. Then the
explicit expression for the generator J0z is obtained by taking the derivative of (1.36)
14 Note that our expression for the energy does not contain the problematic infinite term (so-called
vacuum energy) that is typical for approaches based on quantum fields; see, for example, formula
(2.31) in [10].](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-36-320.jpg)
![14 | 1 Fock space
of the Poincaré group in the Fock space. By construction, this representation induces
transformations (1-5.8)–(1-5.10), (1-5.30) of one-particle states. From this, it is not dif-
ficult to find out how creation and annihilation operators transform under the action
of (1.40).
As an example, consider the boost transformation. For the electron creation op-
erators, we get15
e− ic
ℏ
K0⋅θ
a†
psz
e
ic
ℏ
K0⋅θ
|vac⟩ = e− ic
ℏ
K0⋅θ
a†
psz
|vac⟩ = e− ic
ℏ
K0⋅θ
|psz⟩
= √
ωθp
ωp
∑
s
z
D
1/2
s
zsz
(φW (p, θ))|(θp)s
z⟩
= √
ωθp
ωp
∑
s
z
D
1/2
s
zsz
(φW (p, θ))a†
(θp)s
z
|vac⟩.
Therefore16
e− ic
ℏ
K0⋅θ
a†
psz
e
ic
ℏ
K0⋅θ
= √
ωθp
ωp
∑
s
z
D
1/2
s
zsz
(φW (p, θ))a†
(θp)s
z
= √
ωθp
ωp
∑
s
z
D
1/2∗
szs
z
(−φW (p, θ))a†
(θp)s
z
. (1.41)
The transformation law for annihilation operators is obtained by the Hermitian con-
jugation of (1.41),
e− ic
ℏ
K0⋅θ
apsz
e
ic
ℏ
K0⋅θ
= √
ωθp
ωp
∑
s
z
D
1/2
szs
z
(−φW (p, θ))a(θp)s
z
. (1.42)
Actions of rotations and translations are derived in a similar way. We have
e− i
ℏ
J0⋅φ
a†
psz
e
i
ℏ
J0⋅φ
= ∑
s
z
D
1/2∗
szs
z
(−φ)a†
(φp)s
z
, (1.43)
e− i
ℏ
J0⋅φ
apsz
e
i
ℏ
J0⋅φ
= ∑
s
z
D
1/2
szs
z
(−φ)a(φp)s
z
, (1.44)
e− i
ℏ
P0⋅r
e
i
ℏ
H0t
a†
psz
e− i
ℏ
H0t
e
i
ℏ
P0⋅r
= e− i
ℏ
p⋅r
e
i
ℏ
ωpt
a†
psz
, (1.45)
e− i
ℏ
P0⋅r
e
i
ℏ
H0t
apsz
e− i
ℏ
H0t
e
i
ℏ
P0⋅r
= e
i
ℏ
p⋅r
e− i
ℏ
ωpt
apsz
. (1.46)
15 Here we took into account the fact that the vacuum vector is invariant with respect to U0 and used
equation (1-5.30), where the Wigner angle φW (p, θ) is defined by formula (1-5.18).
16 We took into account that for unitary representatives of rotations D1/2 T∗
(−φ) ≡ D1/2†
(−φ) =
[D1/2
(−φ)]−1
= D1/2
(φ).](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-38-320.jpg)
![16 | 1 Fock space
It then follows that operators of conserved observables commute with the Hamilto-
nian [F, H] = [F, H0 + V] = 0, which imposes some restrictions on the interaction
operator V. For example, in the instant form of dynamics adopted in our book, the
conservation of the total momentum and the total angular momentum means that18
[V, P0] = 0, (1.49)
[V, J0] = 0. (1.50)
It is also known that electromagnetic interactions conserve the lepton charge.19
There-
fore, H = H0 + V must commute with the lepton number operator
NL = Nel − Npo = ∑
sz
∫ dp(a†
psz
apsz
− b†
psz
bpsz
). (1.51)
Since H0 already commutes with NL, we get
[V, NL] = 0. (1.52)
In addition, all known interactions preserve the baryon charge,20
i. e.,
NB = Npr − Nan = ∑
sz
∫ dp(d†
psz
dpsz
− f †
psz
fpsz
). (1.53)
Hence, V must commute with the baryon number operator, i. e.,
[V, NB] = 0. (1.54)
Taking into account that the electrons have a charge of −e, that the protons have a
charge of +e and that the charge of antiparticles is opposite to the charge of particles,
we can introduce the electric charge operator
Q = e(NB − NL)
= e ∑
sz
∫ dp(b†
psz
bpsz
− a†
psz
apsz
+ d†
psz
dpsz
− f†
psz
fpsz
) (1.55)
and obtain the law of its conservation,
[H, Q] = [V, Q] = e[V, NB − NL] = 0, (1.56)
from equations (1.52) and (1.54).
18 The conservation of energy is a consequence of the trivial equality [H, H] = 0.
19 In our case this is the number of electrons minus the number of positrons.
20 In our case this is the number of protons minus the number of antiprotons.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-40-320.jpg)
![1.2 Interaction potentials | 17
As we have just found out, in QED both operators H0 and V commute with total
momentum P0, total angular momentum J0, lepton charge NL, baryon charge NB and
electric charge Q. Then, from the formulas in Section 1-7.1 it follows that the scattering
operators F, Σ and S also commute with P0, J0, NL, NB and Q. This means that the
corresponding observables are conserved in collisions.
Although separately the numbers of particles of a certain type (for example, elec-
trons or protons) may not be conserved, the conservation laws require that charged
particles be born and destroyed only together with their antiparticles, i. e., in pairs.
The pair production does not occur in low-energy reactions, because such processes
require additional energy of 2mec2
= 2 × 0.51 MeV = 1.02 MeV for an electron–positron
pair and 2mpc2
= 1876.6 MeV for a proton–antiproton pair. Such high-energy pro-
cesses can be ignored in classical electrodynamics. However, even in the low-energy
limit, it is necessary to take into account the emission of photons. Photons have zero
mass, and the energy threshold for their creation is zero. Moreover, photons have zero
charges (lepton, baryon and electric), so no conservation laws can limit their creation
and destruction. Photons can be created (radiated) and annihilated (absorbed) in any
quantities.
1.2.2 General form of interaction operators
The well-known theorem21
claims that in the Fock space any operator V satisfying the
conservation laws (1.49)–(1.50) can be written in the form of a polynomial in creation
and annihilation operators,22
i. e.,
V =
∞
∑
N=0
∞
∑
M=0
VNM, (1.57)
VNM = ∑
{η,η}
∫ dq
1 ⋅ ⋅ ⋅ dq
N dq1 ⋅ ⋅ ⋅ dqM
× DNM(q
1η
1, . . . , q
N η
N ; q1η1, . . . , qMηM)
× δ(
N
∑
i=1
q
i −
M
∑
j=1
qj)α†
q
1η
1
⋅ ⋅ ⋅ α†
q
N η
N
αq1η1
⋅ ⋅ ⋅ αqM ηM
, (1.58)
where the summation is over all spin/helicity indices η, η
and the integration is car-
ried out over all particle momenta. The individual terms (monomials) VNM in the ex-
pansion (1.57) will be called potentials. Each potential is a normally ordered product of
21 See p. 175 in [21].
22 Here symbols α†
, α refer to generic creation–annihilation operators without specifying the particle
type.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-41-320.jpg)
![18 | 1 Fock space
N creation operators α†
and M annihilation operators α. The pair of nonnegative inte-
gers [N : M] will be called the index of the potential VNM. A potential is called bosonic
if it has an even number of fermion particle operators Nf + Mf . The conservation laws
(1.52), (1.54) and (1.56),
[V, NL] = [V, NB] = [V, Q] = 0, (1.59)
require that all interaction potentials in QED are bosonic. We are only interested in
Hermitian operators V.
In (1.58) DNM is a numerical coefficient function, which depends on the momenta
and spin projections (or helicities) of all particles being created and destroyed. To
satisfy the requirement [V, J0] = 0, this function must be rotationally invariant. The
translational invariance ([V, P0] = 0) of (1.57)–(1.58) is guaranteed by the momentum
delta function
δ(
N
∑
i=1
q
i −
M
∑
j=1
qj).
This delta function also expresses the momentum conservation: the sum of the mo-
menta of annihilated particles is equal to the sum of the momenta of created particles.
The potential energy operator V enters formulas for the S-operator (1-7.14), (1-7.17)
and (1-7.18) in a t-dependent form, i. e.,
V(t) = e
i
ℏ
H0t
Ve− i
ℏ
H0t
. (1.60)
We shall call regular those operators that satisfy conservation laws (1.49), (1.50) and
(1.59) and whose t-dependence is determined by the free Hamiltonian H0, as in equa-
tion (1.60). Equivalently, a t-dependent regular operator V(t) satisfies the following
differential equation:
d
dt
V(t) =
d
dt
e
i
ℏ
H0t
Ve− i
ℏ
H0t
=
i
ℏ
e
i
ℏ
H0t
[H0, V]e− i
ℏ
H0t
=
i
ℏ
[H0, V(t)]. (1.61)
In our convention, if a regular operator V is written without its t-argument, then either
this operator is t-independent (i. e., it commutes with H0), or its value is taken at t = 0.
One final remark on notation. If the coefficient function of the potential VNM is
DNM, then we will use the symbol VNM ∘ ζ for an operator whose coefficient function
D
NM is the product of DNM and a numeric function ζ of the same arguments, i. e.,
D
NM(q
1η
1, . . . , q
N η
N ; q1η1, . . . , qMηM)
= DNM(q
1η
1, . . . , q
N η
N ; q1η1, . . . , qMηM)ζ (q
1η
1, . . . , q
N η
N ; q1η1, . . . , qMηM).
Then, inserting (1.58) in (1.60) and using (1.45)–(1.48), we conclude that any regular
potential VNM(t) takes the form
VNM(t) = e
i
ℏ
H0t
VNMe− i
ℏ
H0t
= VNM ∘ e
i
ℏ
ℰNM t
, (1.62)](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-42-320.jpg)
![20 | 1 Fock space
Figure 1.1: Locations of different types of operators in the “index
space” [N : M]. R = renorm, O = oscillation.
1.2.3 Five types of regular potentials
In this subsection we are going to introduce a classification of regular potentials (1.58),
by dividing them into five groups depending on the index [N : M]. We call these types
of operators renorm, oscillation, decay, phys and unphys.24
This classification will help
in our study of renormalization in Chapter 4 and also in Volume 3, where we will for-
mulate the “dressing” approach to QFT.
Renorm potentials have either index [0 : 0] or index [1 : 1]. In the former case,
the operator simply multiplies states by a numerical constant C. In the latter case, it
is assumed that the particles that are produced and destroyed have the same type. In
QED, the most general form of a renorm potential is25
R ∝ a†
a + b†
b + d†
d + f †
f + c†
c + C. (1.69)
Renorm potentials are characterized by the property that their energy functions (1.63)
are identically zero. This means that such potentials always have an energy shell,
where they do not vanish.
Lemma 1.2. Any two renorm operators commute with each other.
Proof. A general renorm operator is the sum (1.69). The summands referring to differ-
ent particles commute, because particle operators of different particles always com-
mute. It is not difficult to verify that two renorm operators, corresponding to the same
particle, commute as well:
[∫ dpf(p)α†
pαp, ∫ dqg(q)α†
qαq] = 0.
The free Hamiltonian (1.32) and the total momentum (1.34) are examples of renorm
operators. In particular, this implies that renorm potentials commute with H0, so reg-
ular renorm operators are independent of t.
24 The correlation between potential’s index [N : M] and its type is shown in Figure 1.1.
25 For brevity, here we write only the operator structure of R, omitting numerical multipliers, indices,
summation and integration signs.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-44-320.jpg)
![1.2 Interaction potentials | 21
Oscillation potentials have index [1 : 1]. In contrast to renorm potentials with the
same index, oscillation potentials create and destroy different types of particles having
different masses. For this reason, their energy functions (1.63) never turn to zero, so
they do not have energy shells. In nature, oscillation potentials act on particles such
as kaons and neutrinos. A vivid experimental manifestation of such interactions are
time-dependent oscillations between different types of particles [6].
In QED there cannot be oscillation interactions, because they would violate the
lepton and/or baryon conservation laws.
Decay potentials satisfy two conditions:
(1) their indices are either [1 : N] or [N : 1] with N ≥ 2;
(2) they have a nonempty energy shell, where their coefficient functions do not van-
ish.
These potentials describe decay processes 1 → N,26
in which one particle decays into
N products. Moreover, we require that the laws of conservation of energy and momen-
tum are fulfilled in the decay, i. e., there is a nontrivial energy shell, where the coeffi-
cient function does not vanish. Decay terms are not present in the QED Hamiltonian
and in the corresponding S-matrix, because decays of electrons, protons or photons
would be against conservation laws.27
Nevertheless, decays of elementary particles
play a huge role in other branches of high-energy physics, and we will discuss them
in the third volume.
Phys potentials have at least two creation operators and at least two annihila-
tion operators (they have indices [N : M] where N ≥ 2 and M ≥ 2). For phys poten-
tials, the energy shell always exists. For example, in the case of the phys operator28
d†
(p+k)ρ
f†
(q−k)σ
apτbqη the energy shell is the set of solutions of the equation Ωp+k+Ωq−k =
ωp + ωq in the nine-dimensional momentum space {p, q, k}. This equation has non-
trivial solutions, so the energy shell is not empty.
All regular operators that do not belong to any of the four above classes will be
called unphys potentials. They can be divided into two subclasses with the following
indices:
(1) [0 : N] or [N : 0], where N ≥ 1. Obviously, in this case the energy shell is absent.
26 And also inverse processes N → 1.
27 In principle, one photon can decay into an odd number of other photons without violat-
ing the conservation laws. For example, such a process could be described by the potential
c†
k1τ1
c†
k2τ2
c†
k3τ3
c(k1+k2+k3)τ4
, which formally satisfies all conservation laws if the momenta of all in-
volved photons are collinear and k1 + k2 + k3 − |k1 + k2 + k3| = 0. However, as shown in [5], such
terms in the S-operator are zero on the energy shell, so photon decays are forbidden in QED.
28 This operator describes the conversion reaction electron + positron → proton + antiproton. In the
arguments of particle operators, we have already taken into account the momentum conservation law.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-45-320.jpg)
![22 | 1 Fock space
(2) [1 : N] or [N : 1], where N ≥ 2. These are the same indices as for decay poten-
tials, but here we demand that either the energy shell does not exist or that the
coefficient function vanishes on the energy shell.
Here is an example of an unphys potential with condition (2):
a†
(p−k)σc†
kτapρ. (1.70)
The energy shell equation ωp−k + ck = ωp has only one solution, k = 0. However,
the zero vector is excluded from the photon’s momentum spectrum (see Subsec-
tion 1-5.4.1), so the potential (1.70) has an empty energy shell. This means that a free
electron cannot emit a photon without violating the energy–momentum conservation
law.
Table 1.1: Types of regular potentials in the Fock space.
Potential Index [N : M] Energy shell Example
renorm [0 : 0],[1 : 1] yes a†
pap
oscillation [1 : 1] no forbidden in QED
unphys [0 : M ≥ 1],[N ≥ 1 : 0] no a†
pb†
−p−k
c†
k
unphys [1 : M ≥ 2],[N ≥ 2 : 1] no a†
pap−kck
decay [1 : M ≥ 2],[N ≥ 2 : 1] yes forbidden in QED
phys [N ≥ 2 : M ≥ 2] yes d†
q+k
a†
p−k
dqap
The properties of potentials considered above are summarized in Table 1.1. These five
types of interactions exhaust all possibilities; therefore any regular operator V must
have a unique expansion
V = Vren
+ Vunp
+ Vdec
+ Vphys
+ Vosc
.
As mentioned above, there are no oscillation and decay interactions in QED, so every-
where in this volume we will assume that the most general potential is equal to the
sum of renorm, unphys and phys parts:
VQED
= Vren
+ Vunp
+ Vphys
.
Now we need to figure out how to perform various manipulations with these three
classes of potentials. In particular, we want to learn how to calculate products, com-
mutators and t-integrals that are necessary for computing scattering operators from
Section 1-7.1.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-46-320.jpg)
![1.2 Interaction potentials | 23
1.2.4 Products and commutators of regular potentials
Let us first prove a few simple results.
Lemma 1.3. The product of two or more regular operators is regular.
Proof. By definition, if operators A(t) and B(t) are regular, then
A(t) = e
i
ℏ
H0t
Ae− i
ℏ
H0t
,
B(t) = e
i
ℏ
H0t
Be− i
ℏ
H0t
.
Hence, their product C(t) = A(t)B(t) has the t-dependence
C(t) = e
i
ℏ
H0t
Ae− i
ℏ
H0t
e
i
ℏ
H0t
Be− i
ℏ
H0t
= e
i
ℏ
H0t
ABe− i
ℏ
H0t
characteristic of regular operators. The conservation laws (1.49), (1.50) and (1.59) are
valid for the product AB, just as they are valid for A and B separately. Therefore, C(t)
is regular.
Theorem 1.4. A Hermitian operator P is phys if and only if it yields zero when acting on
both the vacuum vector |vac⟩ and one-particle states |1⟩ ≡ α†
|vac⟩29
:
P|vac⟩ = 0, (1.71)
P|1⟩ ≡ Pα†
|vac⟩ = 0. (1.72)
Proof. By definition, normally ordered phys potentials have (at least) two annihila-
tion operators on the right. So, they yield zero when applied to the vacuum or any
one-particle state. Therefore, equations (1.71) and (1.72) are satisfied for any phys
operator P.
Letusnowprovetheconverse. Renormoperatorscannotsatisfyrequirements(1.71)
and (1.72), because they preserve the number of particles. Unphys operators [1 : M]
can satisfy these requirements. For example,
α†
1 α2α3|vac⟩ = 0,
α†
1 α2α3|1⟩ = 0.
However, in order to be Hermitian, such operators must always be present in pairs with
[M : 1] operators, like α†
2α†
3α1. Then there exists at least one single-particle state |1⟩ for
which equation (1.72) is not valid, that is,
α†
3α†
2α1|1⟩ = α†
3α†
2|vac⟩ ̸
= 0.
29 Here α means any of the five particle operators (a, b, d, f , c) related to QED. Momentum and spin
labels are omitted for brevity.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-47-320.jpg)
![24 | 1 Fock space
Similar arguments apply to unphys operators with indices [0 : M] and [N : 0]. Hence,
the only remaining possibility for the potential P is to be phys.
Lemma 1.5. The product and commutator of any phys operators A and B are also phys.
Proof. By Theorem 1.4, if A and B are phys, then
A|vac⟩ = B|vac⟩ = A|1⟩ = B|1⟩ = 0.
The same properties are valid for the Hermitian combinations i(AB−BA) and AB+BA.
Hence, both the commutator [A, B] and the anticommutator {A, B} are phys. The same
conclusion is true for the product, which can be expressed as the sum
AB =
1
2
{A, B} +
1
2
[A, B].
Lemma 1.6. If R is a renorm operator, P is a phys operator and [P, R] ̸
= 0, then operator
[P, R] is of the phys type.
Proof. Let us first check how this commutator acts on the vacuum and single-particle
states.30
We have
i(PR − RP)|vac⟩ = iPR|vac⟩ = iPC0|vac⟩ = 0,
i(PR − RP)|1⟩ = iPR|1⟩ = iPC1|1
⟩ = 0.
This means that the Hermitian commutator i[P, R] turns vectors |vac⟩ and |1⟩ to zero.
By Lemma 1.4 this operator is phys.
Lemma 1.7. If R is a renorm operator, U is an unphys operator and [U, R] ̸
= 0, then
operator [U, R] has the unphys type.
Idea of the proof. Let us first calculate the commutator of the renorm operator R =
∫ dpf(p)α†
pαp with a particle creation operator31
We have
[α†
q, R] = α†
q(∫ dpf(p)α†
pαp) − (∫ dpf (p)α†
pαp)α†
q
= ± ∫ dpf(p)α†
pα†
qαp − ∫ dpf (p)α†
pαpα†
q
= ∫ dpf(p)α†
pαpα†
q − ∫ dpf (p)α†
pδ(p − q) − ∫ dpf (p)α†
pαpα†
q
= −f (q)α†
q.
30 Here we took into account that renorm operators preserve the number of particles: R|vac⟩ = const×
|vac⟩, R|1⟩ = |1
⟩ and phys operators turn the states |vac⟩ and |1⟩ to zero.
31 The upper sign is for bosons, the lower sign is for fermions.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-48-320.jpg)
![1.2 Interaction potentials | 25
Similarly, we obtain the commutator with an annihilation operator:
[αq, R] = f(q)αq.
Now, as an example of an unphys operator, we take a potential with index [2 : 1],
U = ∫ dq1dq2dpD(q1, q2; p)δ(q1 + q2 − p)α†
q1
α†
q2
αp.
The index of the commutator is also [2 : 1]. We have
[U, R] = ∫ dq1dq2dpD(q1, q2; p)δ(q1 + q2 − p)α†
q1
α†
q2
[αp, R]
+ ∫ dq1dq2dpD(q1, q2; p)δ(q1 + q2 − p)α†
q1
[α†
q2
, R]αp
+ ∫ dq1dq2dpD(q1, q2; p)δ(q1 + q2 − p)[α†
q1
, R]α†
q2
αp
= ∫ dq1dq2dpD(q1, q2; p)f(p)δ(q1 + q2 − p)α†
q1
α†
q2
αp
− ∫ dq1dq2dpD(q1, q2; p)f(q2)δ(q1 + q2 − p)α†
q1
α†
q2
αp
− ∫ dq1dq2dpD(q1, q2; p)f(q1)δ(q1 + q2 − p)α†
q1
α†
q2
αp.
Moreover, if the operator U does not have an energy shell, then [U, R] also does not
have it, i. e., its type is unphys. If U has an energy shell where the coefficient function
D(q1, q2; p) is zero, then [U, R] also has this property.
Lemma 1.8. The commutator [P, U] of an Hermitian phys operator P and an Hermitian
unphys operator U cannot contain renorm terms.
Proof. Applying the operator [P, U] to a single-particle state |1⟩ and using (1.72), we
obtain
[P, U]|1⟩ = (PU − UP)|1⟩ = PU|1⟩. (1.73)
If the commutator [P, U] contained renorm terms, then the right-hand side of (1.73)
would have a nonzero one-particle component. However, the range of any phys P does
not include the one-particle sector. This implies [P, U]ren
= 0.
Finally, it is easy to verify that there are no restrictions on the type of the commu-
tator of two unphys operators [U, U
]. It can contain unphys, phys and renorm parts.
The above results are summarized in Table 1.2.
1.2.5 More about t-integrals
Lemma 1.9. The t-derivative of a regular operator A(t) is regular, and its renorm part
vanishes.](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-49-320.jpg)
![26 | 1 Fock space
Table 1.2: Commutators, t-derivatives and t-integrals with regular operator A in the Fock space. (No-
tation: P = phys, U = unphys, R = renorm, NR = nonregular.)
Type of A [A, P] [A, U] [A, R] dA
dt
A A
⏟⏟
⏟⏟⏟
⏟⏟
P P P+U P P P P
U P+U P+U+R U U U 0
R P U 0 0 NR ∞
Proof. According to (1.61), the derivative of A(t) is equal to the commutator with regu-
lar H0. Then by Lemma 1.3 this derivative is regular.
Suppose, by contradiction, that d
dt
A(t) has a nonzero renorm part R. Then R does
not depend on t, because it is regular. It follows that the most general form of A(t) is
A(t) = Rt + S, where S is any operator independent of t. From the condition that the
renorm part of the regular operator A(t) cannot depend on t, we obtain R = 0.
From equation (1.65) we conclude that t-integrals of regular phys and unphys oper-
ators are regular. However, this property does not hold for t-integrals of renorm opera-
tors. As we know, renorm operators are independent of t. Hence, when the interaction
is adiabatically switched on, as in (1-7.26), we obtain
Vren
(t) = lim
ϵ→+0
(−
i
ℏ
0
∫
−∞
Vren
eϵt
dt
−
i
ℏ
t
∫
0
Vren
e−ϵt
dt
)
= −(
i
ℏ
)Vren
∘ lim
ϵ→+0
(
eϵt
ϵ
t=0
t=−∞
−
e−ϵt
ϵ
t=t
t=0
)
= −(
i
ℏ
)Vren
∘ lim
ϵ→+0
(
1
ϵ
−
e−ϵt
ϵ
+
1
ϵ
)
= −(
i
ℏ
)Vren
∘ lim
ϵ→+0
(
1
ϵ
+ t + ⋅ ⋅ ⋅), (1.74)
Vren
⏟⏟⏟⏟⏟⏟⏟ = lim
t→∞
Vren
(t) = ∞. (1.75)
Hence, renorm operators differ from all others in that their t-integrals (1.74)–(1.75) are
infinite and nonregular.32
By definition, an unphys operator Vunp
either does not have an energy shell, or its
coefficient function vanishes on the energy shell. Then, from equation (1.66) it follows
that for any unphys operator
Vunp
⏟⏟⏟⏟⏟⏟⏟⏟⏟ = 0. (1.76)
Results obtained in this subsection are shown in the last three columns of Table 1.2.
32 As we shall see in Subsection 4.1.1, correctly renormalized expressions for scattering operators
should not contain renorm terms and pathological constructs like (1.74)–(1.75).](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-50-320.jpg)
![59
60
61
62
63
Liṅg Eklingakadiwán (Ditto 222, Ráj. Gaz. III. 53). The blend is natural. The fierce noon-
tide sun is Mahákála the Destroyer. Like Śiva the Sun is lord of the Moon. And marshalled
by Somanátha the great Soul Home the souls of the dead pass heavenwards along the rays
of the setting sun. [Compare Sachau’s Alberuni, II. 168.] It is the common sun element in
Śaivism and in Vaishnavism that gives their holiness to the sunset shrines of Somanátha
and Dwárka. For (Ditto, 169) the setting sun is the door whence men march forth into the
world of existence Westwards, heavenwards. ↑
This explanation is hardly satisfactory. The name Gehlot seems to be Guhila-putra
from Gobhila-putra an ancient Bráhman gotra, one of the not uncommon cases of Rájputs
with a Bráhman gotra. The Rájput use of a Bráhman gotra is generally considered a
technical affiliation, a mark of respect for some Bráhman teacher. It seems doubtful
whether the practice is not a reminiscence of an ancestral Bráhman strain. This view finds
confirmation in the Aitpur inscription (Tod’s Annals, I. 802) which states that Guhadit the
founder of the Gohil tribe was of Bráhman race Vipra kula. Compare the legend (Rás Málá,
I. 13) that makes the first Śíláditya of Valabhi (a.d. 590–609) the son of a Bráhman
woman. Compare (Elliot, I. 411) the Bráhman Chách (a.d. 630–670) marrying the widow
of the Sháhi king of Alor in Sindh who is written of as a Rájput though like the later
(a.d. 850–1060) Shahiyas of Kábul (Alberuni, Sachau II. 13) the dynasty may possibly
have been Bráhmans.60 The following passage from Hodgson’s Essays (J. A. Soc. Bl. II.
218) throws light on the subject: Among the Khás or Rájputs of Nepál the sons of
Bráhmans by Khás women take their fathers’ gotras. Compare Ibbetson’s Panjáb Census
1881 page 236. ↑
In support of a Bráhman origin is Prinsep’s conjecture (J. A. S. Bl. LXXIV. [Feb.
1838] page 93) that Divaij the name of the first recorded king may be Dvija or Twice-born.
But Divaij for Deváditya, like Silaij for Śíláditya, seems simpler and the care with which
the writer speaks of Chach as the Bráhman almost implies that his predecessors were not
Bráhmans. According to Elliot (II. 426) the Páls of Kábul were Rájputs, perhaps Bhattias. ↑
Tod’s Annals, I. 229–231. ↑
Annals, I. 229. ↑
Gladwin’s Áin-i-Akbari, II. 81; Tod’s Annals, I. 235 and note *. Tod’s dates are
confused. The Aitpur inscription (Ditto, page 230) gives Śakti Kumára’s date a.d. 968 (S.
1024) while the authorities which Tod accepts (Ditto, 231) give a.d. 1068 (S. 1125). That
the Moris were not driven out of Chitor as early as a.d. 728 is proved by the Navsárí](https://image.slidesharecdn.com/49769-250225045129-fd89d3d5/85/Elementary-Particle-Theory-Volume-2-Quantum-Electrodynamics-1st-Edition-Eugene-Stefanovich-76-320.jpg)
Elementary Particle Theory Volume 2 Quantum Electrodynamics 1st Edition Eugene Stefanovich
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- 5. Elementary Particle Theory Volume 2 Quantum Electrodynamics 1st Edition Eugene Stefanovich Digital Instant Download Author(s): Eugene Stefanovich ISBN(s): 9783110493207, 3110493209 Edition: 1 File Details: PDF, 2.92 MB Year: 2019 Language: english
- 7. Eugene Stefanovich Elementary Particle Theory
- 8. De Gruyter Studies in Mathematical Physics | Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia Volume 46
- 9. Eugene Stefanovich Elementary Particle Theory | Volume 2: Quantum Electrodynamics
- 10. Mathematics Subject Classification 2010 Primary: 81-02, 81V10, 81T15; Secondary: 47A40, 81T18 Author Dr Eugene Stefanovich San Jose, California USA eugene_stefanovich@usa.net ISBN 978-3-11-049089-3 e-ISBN (PDF) 978-3-11-049320-7 e-ISBN (EPUB) 978-3-11-049143-2 ISSN 2194-3532 Library of Congress Control Number: 2018016481 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com
- 11. Contents List of figures | IX List of tables | XI Postulates, statements, theorems | XIII Conventional notation | XV Preface | XVII 1 Fock space | 1 1.1 Creation and annihilation operators | 1 1.1.1 Sectors with fixed numbers of particles | 1 1.1.2 Particle observables in Fock space | 3 1.1.3 Noninteracting representation of Poincaré group | 3 1.1.4 Creation and annihilation operators for fermions | 4 1.1.5 Anticommutators of particle operators | 6 1.1.6 Creation and annihilation operators for photons | 7 1.1.7 Particle number operators | 7 1.1.8 Continuous spectrum of momentum | 8 1.1.9 Normal ordering | 9 1.1.10 Noninteracting energy and momentum | 11 1.1.11 Noninteracting angular momentum and boost | 12 1.1.12 Poincaré transformations of particle operators | 13 1.2 Interaction potentials | 15 1.2.1 Conservation laws | 15 1.2.2 General form of interaction operators | 17 1.2.3 Five types of regular potentials | 20 1.2.4 Products and commutators of regular potentials | 23 1.2.5 More about t-integrals | 25 1.2.6 Solution of one commutator equation | 27 1.2.7 Two-particle potentials | 28 1.2.8 Momentum-dependent potentials | 31 2 Scattering in Fock space | 33 2.1 Toy model theory | 33 2.1.1 Fock space and Hamiltonian | 33 2.1.2 S-operator in second order | 35 2.1.3 Drawing diagrams in toy model | 36
- 12. VI | Contents 2.1.4 Reading diagrams in toy model | 39 2.1.5 Scattering in second order | 40 2.2 Renormalization in toy model | 41 2.2.1 Renormalization of electron self-scattering in second order | 41 2.2.2 Renormalization of electron self-scattering in fourth order | 43 2.3 Diagrams in general theory | 46 2.3.1 Products of diagrams | 46 2.3.2 Connected and disconnected diagrams | 47 2.3.3 Divergence of loop integrals | 50 2.4 Cluster separability | 52 2.4.1 Cluster separability of interaction | 52 2.4.2 Cluster separability of S-operator | 54 3 Quantum electrodynamics | 57 3.1 Interaction in QED | 57 3.1.1 Why do we need quantum fields? | 58 3.1.2 Simple quantum field theories | 58 3.1.3 Interaction operators in QED | 60 3.2 S-operator in QED | 62 3.2.1 S-operator in second order | 62 3.2.2 Covariant form of S-operator | 66 3.2.3 Feynman gauge | 68 3.2.4 Feynman diagrams | 70 3.2.5 Compton scattering | 72 3.2.6 Virtual particles? | 73 4 Renormalization | 75 4.1 Two renormalization conditions | 75 4.1.1 No self-scattering condition | 75 4.1.2 Charge renormalization | 78 4.1.3 Renormalization by counterterms | 78 4.1.4 Diagrams of electron–proton scattering | 79 4.1.5 Regularization | 80 4.2 Counterterms | 81 4.2.1 Electron’s self-scattering | 81 4.2.2 Electron self-scattering counterterm | 83 4.2.3 Fitting coefficient (δm)2 | 84 4.2.4 Fitting coefficient (Z2 − 1)2 | 85 4.2.5 Photon’s self-scattering | 86 4.2.6 Photon self-energy counterterm | 87 4.2.7 Applying charge renormalization condition | 89 4.2.8 Vertex renormalization | 90
- 13. Contents | VII 4.3 Renormalized S-matrix | 93 4.3.1 “Vacuum polarization” diagrams | 93 4.3.2 Vertex diagram | 93 4.3.3 Ladder diagram | 95 4.3.4 Cross ladder diagram | 98 4.3.5 Renormalizability | 101 A Useful integrals | 103 B Quantum fields of fermions | 107 B.1 Pauli matrices | 107 B.2 Dirac gamma matrices | 108 B.3 Dirac representation of Lorentz group | 109 B.4 Construction of Dirac field | 112 B.5 Properties of functions u and v | 114 B.6 Explicit formulas for u and v | 115 B.7 Useful notation | 118 B.8 Poincaré transformations of fields | 119 B.9 Approximation (v/c)2 | 120 B.10 Anticommutation relations | 122 B.11 Dirac equation | 123 B.12 Fermion propagator | 125 C Quantum field of photons | 129 C.1 Construction of photon quantum field | 129 C.2 Properties of function eμ(p, τ) | 130 C.3 Useful commutator | 131 C.4 Commutator of photon fields | 133 C.5 Photon propagator | 133 C.6 Poincaré transformations of photon field | 135 D QED interaction in terms of particle operators | 139 D.1 Current density | 139 D.2 First-order interaction in QED | 142 D.3 Second-order interaction in QED | 142 E Relativistic invariance of QFT | 155 E.1 Relativistic invariance of simple QFT | 155 E.2 Relativistic invariance of QED | 156 F Loop integrals in QED | 163 F.1 Schwinger–Feynman integration trick | 163
- 14. VIII | Contents F.2 Some basic four-dimensional integrals | 164 F.3 Electron self-energy integral | 167 F.4 Vertex integral | 170 F.4.1 Calculation of M | 172 F.4.2 Calculation of Mσ | 173 F.4.3 Calculation of Mστ | 174 F.4.4 Complete integral | 175 F.5 Integral for ladder diagram | 178 F.5.1 Calculation of LI | 179 F.5.2 Calculation of LII | 181 F.5.3 Calculation of LIII | 182 F.5.4 Complete integral | 184 G Scattering matrix in (v/c)2 approximation | 185 G.1 Second perturbation order | 185 G.2 Vertex contribution in fourth order | 187 H Checks of physical dimensions | 191 Bibliography | 193 Index | 195
- 15. List of figures Figure 1.1 Operators in “index space” (page 20) Figure 2.1 Diagrams for operators V1 and V1(t) (page 37) Figure 2.2 Normal ordering of the product of two diagrams (page 38) Figure 2.3 Renorm diagrams in Vc Vc Vc Vc (page 44) Figure 2.4 Diagram of the counterterm Q2 (page 44) Figure 2.5 Renorm diagrams in Vc Vc + Vc Vc Vc (page 45) Figure 2.6 To the proof of Lemma 2.3 (page 49) Figure 2.7 Generic diagram in a hypothetical theory (page 49) Figure 3.1 Second-order diagram for e− + p+ scattering (page 72) Figure 3.2 e− + γ scattering diagrams (page 73) Figure 4.1 e− + p+ scattering diagrams up to the fourth order (page 80) Figure 4.2 Electron self-scattering diagrams (page 81) Figure 4.3 Photon self-scattering diagrams (page 86) Figure 4.4 “Vacuum polarization” diagrams (page 89) Figure 4.5 Vertex diagrams (page 91) Figure 4.6 Ladder diagram (page 95) Figure 4.7 Cross-ladder diagram (page 98) Figure A.1 To the calculation of integral (A.10) (page 104) Figure F.1 Wick rotation in the integral (F.6) (page 164) Figure F.2 Integration area in (F.45) (page 179) https://doi.org/10.1515/9783110493207-201
- 17. List of tables Table 1.1 Types of potentials in the Fock space of QED (page 22) Table 1.2 Commutators, t-derivatives and t-integrals (page 26) Table 3.1 Components of Feynman diagrams (page 71) https://doi.org/10.1515/9783110493207-202
- 19. Postulates, statements, theorems Theorem 1.1 Wick theorem (page 10) Lemma 1.2 Commutator of renorm operators (page 20) Lemma 1.3 Product of regular operators (page 23) Theorem 1.4 Properties of phys operators (page 23) Lemma 1.5 Product of phys operators (page 24) Lemma 1.6 Commutator [phys,renorm] (page 24) Lemma 1.7 Commutator [unphys,renorm] (page 24) Lemma 1.8 Commutator [phys,unphys] (page 25) Lemma 1.9 t-derivative of operator (page 25) Lemma 2.1 Disconnected part of product (page 47) Theorem 2.2 Multiple commutator (page 48) Lemma 2.3 Number of independent loops (page 49) Theorem 2.4 Convergence of loop integrals (page 51) Theorem 2.5 Cluster separability of smooth potentials (page 54) Theorem 2.6 Smoothness of products of potentials (page 54) Theorem 2.7 Smoothness of commutators of potentials (page 55) Statement 4.1 Absence of self-scattering (page 77) Postulate 4.2 Charge renormalization condition (page 78) https://doi.org/10.1515/9783110493207-203
- 21. Conventional notation See also Conventional notation of Volume 1. Fock space |vac⟩ vacuum state vector (page 2) a† , a electron creation/annihilation operators (page 4) b† , b positron creation/annihilation operators (page 9) c† , c photon creation/annihilation operators (page 7) d† , d proton creation/annihilation operators (page 9) f† , f antiproton creation/annihilation operators (page 9) α† , α creation/annihilation operators of generic particles (page 10) VNM ∘ ζ product of coefficient functions (page 18) ℰA energy function of operator A (page 19) H Fock space (page 1) H (i, j, k, l, m) sector in the Fock space (page 1) : abcd : normally ordered product (page 10) Quantum fields ψ( ̃ x) electron–positron quantum field (page 112) ψ( ̃ x) Dirac-conjugated field (page 113) Ψ( ̃ x) proton–antiproton quantum field (page 113) 𝒜μ( ̃ x) photon quantum field (page 129) 𝒰μ (p s z, psz) ≡ u (p , s z) γμ u(p, sz) (page 118) 𝒲μ (p s z, psz) ≡ w (p , s z) γμ w(p, sz) (page 118) Hn naïve Hamiltonian of QED (page 61) Hc QED Hamiltonian with counterterms (page 42) j μ ep( ̃ x) electron–positron current density operator (page 139) j μ pa( ̃ x) proton–antiproton current density operator (page 139) jμ ( ̃ x) total current density operator (page 139) λ infrared cutoff (page 80) Λ ultraviolet cutoff (page 80) Miscellaneous γμ Dirac gamma matrices (page 108) a, b, c, . . . Dirac indices (page 112) https://doi.org/10.1515/9783110493207-204
- 22. XVI | Conventional notation 𝒟(Λ) Dirac representation of the Lorentz group (page B.27) J, K generators of the Dirac representation (page 109) / k ≡ kμγμ (page 109) σ, σ0 Pauli matrices (page 107) α = e2 /(4πℏc) fine structure constant (page 66) 𝜕μ 4-gradient (− 1 c 𝜕 𝜕t , 𝜕 𝜕x , 𝜕 𝜕y , 𝜕 𝜕z ) (page 83) ̃ p ≡ (ωp, cpx, cpy, cpz) energy-momentum 4-vector (page 112) ̃ x ≡ (t, x/c, y/c, z/c) time-position 4-vector (page 112) ̃ p ⋅ ̃ x ≡ ωpt − p ⋅ x (page 112) k ≡ q − q = p − p transferred 3-momentum (page 66) ̃ k ≡ ̃ q − ̃ q = ̃ p − ̃ p transferred 4-momentum (page 66)
- 23. Preface In a successful theory of elementary particles, at least three important conditions must be fulfilled: (1) relativistic invariance in the instant form of dynamics; (2) cluster separability of the interaction; (3) description of processes involving creation and destruction of particles. In the first volume of our book we discussed interacting quantum theories in Hilbert spaces with a fixed set of particles. We showed how it is possible to satisfy the first two requirements (relativistic invariance and cluster separability).1 However, these theories were fundamentally incomplete, due to their inability to describe physical processes that change the types and/or number of particles in the system. Thus, con- dition 3 from our list was not fulfilled. Familiar examples of the creation and annihilation processes are emission and ab- sorption of light (photons), decays, neutrino oscillations, etc. Particles are produced especially intensively at high energies. This is due to the famous Einstein formula E = mc2 , which says, in particular, that if the system has sufficient energy E of rel- ative motion, then this energy can be transformed into the mass m of newly created particles. Even in the simplest two-particle case, the energy of the relative motion of these reactants is unlimited. Therefore, there is no limit to the number of new particles that can be created in a collision. To advance in the study of such processes, the first thing to do is to build a Hilbert space of states H , which is capable of describing particle creation and annihilation. Such a space must include states with arbitrary numbers (from zero to infinity) of par- ticles of all types. It is called the Fock space. This construction is rather simple. How- ever, the next step – the definition of realistic interaction operators in the Fock space – is highly nontrivial. A big part of our third volume will be devoted to the solution of this problem. Here we will prepare ourselves to this task by starting with a more traditional approach, which is known as the renormalized relativistic quantum field theory (QFT). Our discussions in this book are limited to electromagnetic phenomena, so we will be interested in the simplest and most successful type of QFT – quantum electrodynamics (QED). In Chapter 1, Fock space, we will describe the basic mathematical machinery of Fock spaces, including creation and annihilation operators, normal ordering and clas- sification of interaction potentials. A simple toy model with variable number of particles will be presented in Chap- ter 2, Scattering in Fock space. In this example, we will discuss such important in- gredients of QFT as the S-matrix formalism, renormalization, diagram technique and 1 See, e. g., Subsection 1-6.4.6. https://doi.org/10.1515/9783110493207-205
- 24. XVIII | Preface cluster separability. Our first two chapters have a mostly technical character. They de- fine our terminology and notation and prepare us for a more in-depth study of QED in the two following chapters. In Chapter 3, Quantum electrodynamics, we introduce the important concept of the quantum field. This idea will be applied to systems of charged particles and pho- tons in the formalism of QED. Here we will obtain an interacting theory, which satisfies the principles of relativistic invariance and cluster separability, where the number of particles is not fixed. However, the “naïve” version of QED presented here is unsat- isfactory, since it cannot calculate scattering amplitudes beyond the lowest orders of perturbation theory. Chapter 4, Renormalization, completes the second volume of the book. We will dis- cussthe plagueof ultravioletdivergencesin the “naïve”QED and explain how they can be eliminated by adding counterterms to the Hamiltonian. As a result, we will get the traditional “renormalized” QED, which has proven itself in precision calculations of scattering cross sections and energy levels in systems of charged particles. However, this theory failed to provide a well-defined interacting Hamiltonian and the interact- ing time evolution (= dynamics). We will address these issues in the third volume of our book. As in the first volume, here we refrain from criticism and unconventional interpre- tations, trying to keep in line with generally accepted approaches. The main purpose of this volume is to explain the basic concepts and terminology of QFT. For the most part, we will adhere to the logic of QFT formulated by Weinberg in the series of articles [19, 18, 20] and in the excellent textbook [21]. A critical discussion of the traditional approaches and a new look at the theory of relativity will be presented in Volume 3 [17]. References to Volume 1 [16] of this book will be prefixed with “1-”. For example, (1-7.14) is formula (7.14) from Volume 1.
- 25. 1 Fock space There are more things in Heaven and on earth, dear Horacio, than are dreamed of in your philosophy. Hamlet In this chapter, we construct the Fock space H populated by particles of five types: electrons e− , positrons e+ , protons p+ , antiprotons1 p− and photons γ. We will practice constructions of simple interaction operators and study their properties. In compar- ison with Volume 1, the main novelty is in working with operators that change the number of particles. This will prepare us for mastering a more realistic theory – quan- tum electrodynamics (QED) – in Chapters 3 and 4. 1.1 Creation and annihilation operators Here we introduce the concepts of creation and annihilation operators. Though lack- ing autonomous physical meaning, these operators greatly simplify calculations in H . 1.1.1 Sectors with fixed numbers of particles The numbers of particles of each type are easily measurable in experiments, so we have the right to introduce in our theory five new observables, namely, the numbers of electrons (Nel), positrons (Npo), protons (Npr), antiprotons (Nan) and photons (Nph). Unlike in ordinary quantum mechanics from Volume 1, here we will not assume that the numbers of particles are fixed. We would like to treat these quantities on the same footing as other quantum observables. In particular, we will also take into account their quantum uncertainty. Then, in accordance with general quantum rules, these observables should be represented in the Hilbert space (= Fock space) H by five Her- mitian operators. Obviously, their allowed values (spectra) are nonnegative integers (0, 1, 2, . . .). From part (II) of Postulate 1-6.1, it follows that these observables are mea- surable simultaneously, so that the particle number operators commute with each other and have common eigensubspaces. Hence, the Fock space H splits into a di- rect sum of orthogonal subspaces, or sectors, H (i, j, k, l, m) containing i electrons, 1 In this book, protons and antiprotons are regarded as simple point charges. Their internal structure is ignored, as well as their participation in strong nuclear interactions. https://doi.org/10.1515/9783110493207-001
- 26. 2 | 1 Fock space j positrons, k protons, l antiprotons and m photons, so H = ∞ ⨁ ijklm=0 H (i, j, k, l, m), (1.1) where NelH (i, j, k, l, m) = iH (i, j, k, l, m), NpoH (i, j, k, l, m) = jH (i, j, k, l, m), NprH (i, j, k, l, m) = kH (i, j, k, l, m), NanH (i, j, k, l, m) = lH (i, j, k, l, m), NphH (i, j, k, l, m) = mH (i, j, k, l, m). The one-dimensional subspace without particles H (0, 0, 0, 0, 0) is called the vac- uum subspace. The vacuum vector |vac⟩ is defined in this subspace up to an unimpor- tant phase factor. Single-particle sectors are built according to the recipes from Chapter 1-5. The sub- spaces H (1, 0, 0, 0, 0) and H (0, 1, 0, 0, 0) contain one electron and one positron, re- spectively. These subspaces carry unitary irreducible representations of the Poincaré group with mass me = 0.511 MeV/c2 and spin 1/2 (see Table 1-5.1). The subspaces H (0, 0, 1, 0, 0) and H (0, 0, 0, 1, 0) contain one proton and one antiproton, respec- tively. These particles have mass mp = 938.3 MeV/c2 and spin 1/2. The subspace H (0, 0, 0, 0, 1) contains one photon with zero mass. This subspace is the direct sum of two irreducible massless subspaces with helicities 1 and −1 (see Subsection 1-5.4.4). Sectors with two or more particles are constructed as (anti)symmetrized products of single-particle sectors.2 For example, if Hel is the one-electron sector and Hph is the one-photon sector, then sectors having only electrons and photons can be written as H (0, 0, 0, 0, 0) = |vac⟩, (1.2) H (1, 0, 0, 0, 0) = Hel, (1.3) H (0, 0, 0, 0, 1) = Hph, (1.4) H (1, 0, 0, 0, 1) = Hel ⊗ Hph, (1.5) H (2, 0, 0, 0, 0) = Hel ⊗asym Hel, (1.6) H (0, 0, 0, 0, 2) = Hph ⊗sym Hph, (1.7) H (1, 0, 0, 0, 2) = Hel ⊗ (Hph ⊗sym Hph), (1.8) H (2, 0, 0, 0, 1) = (Hel ⊗asym Hel) ⊗ Hph, (1.9) H (2, 0, 0, 0, 2) = (Hel ⊗asym Hel) ⊗ (Hph ⊗sym Hph). (1.10) . . . 2 See Subsection 1-6.1.3. Note that electrons and protons are fermions, while photons are bosons.
- 27. 1.1 Creation and annihilation operators | 3 1.1.2 Particle observables in Fock space As explained in Subsection 1-6.1.2, in each sector of the Fock space we can define ob- servables of individual particles populating this sector, i. e., their positions, momenta, spins, etc. For example, in each (massive) one-particle subspace there is a Newton– Wigner operator describing measurements of the particle’s position r. In n-particle sectors, in addition to the center-of-energy position R, positions ri of individual parti- cles are defined as well. In each sector, we can choose a basis of common eigenvectors of a complete set of commuting one-particle observables. For further discussions it will be convenient to use the basis which diagonalizes momenta p and spin components sz of massive particles or helicities τ of massless particles. For example, basis vectors in the two- electron sector H (2, 0, 0, 0, 0) = Hel ⊗asym Hel will be denoted as |p1s1z, p2s2z⟩. Thus, in each sector one can define many-particle wave functions in the momentum–spin representation. The arbitrary state |Ψ⟩ in the Fock space can have components in many or all sec- tors.3 So the number of particles in the state |Ψ⟩ can be undefined, and a complete de- scription of such a state requires the introduction of multi-sector state vectors, which can be expanded in the basis described above. 1.1.3 Noninteracting representation of Poincaré group The construction given above gives us the Fock space H , where many-particle states and observables of our theory live and where a convenient orthonormal basis is de- fined. To complete this formalism, we need to construct a realistic interacting rep- resentation Ug of the Poincaré group in H . Let us first solve a simpler problem and define a noninteracting representation U0 g there. From Subsection 1-6.2.1 we already know how to build noninteracting representa- tions of the Poincaré group in each separate sector of H . This is done with the help of the tensor product4 of one-particle irreducible representations corresponding to elec- trons Uel g , photons Uph g , etc. Then, the noninteracting representation of the Poincaré group in the entire Fock space is formed as the direct sum of such sector representa- tions. In accordance with the sector decomposition (1.2)–(1.10), we can write U0 g = 1 ⊕ Uel g ⊕ Uph g ⊕ (Uel g ⊗ Uph g ) ⊕ (Uel g ⊗asym Uel g ) ⊕ ⋅ ⋅ ⋅ . (1.11) 3 Superselection rules forbid creating linear combinations of states with different charges. We will not discuss these rules here. 4 With the appropriate (anti)symmetrization.
- 28. 4 | 1 Fock space The generators of this representation will be denoted by {H0, P0, J0, K0}. In each sector, these generators are simply sums of single-particle generators.5 As usual, we assume that the operators H0, P0 and J0 represent the total energy, momentum and angular momentum, respectively. Here we immediately notice a serious problem, which was not present in quantum mechanics with fixed particle content. For example, according to (1.11), a free Hamil- tonian should be represented as a direct sum of sector components, i. e., H0 = 0 ⊕ H0(1, 0, 0, 0, 0) ⊕ H0(0, 0, 0, 0, 1) ⊕ H0(1, 0, 0, 0, 1) ⊕ ⋅ ⋅ ⋅ . It is tempting to use the notation from Section 1-6.2 and express Hamiltonians in each sector through observables of individual particles there: p1, p2, etc. For example, in the one-electron sector H (1, 0, 0, 0, 0), the free Hamiltonian is equal to H0(1, 0, 0, 0, 0) = √m2 ec4 + p2c2 (1.12) and the Hamiltonian in the sector H (2, 0, 0, 0, 2) is6 H0(2, 0, 0, 0, 2) = p1c + p2c + √m2 ec4 + p2 3c2 + √m2 ec4 + p2 4c2. (1.13) Obviously, such a notation is very cumbersome, because it does not give a single ex- pression for the operator H0 in the entire Fock space. Moreover, it is completely unclear how to use the single-particle observables for constructing operators of interactions that change the number of particles, i. e., moving state vectors across sector bound- aries. We need to find a simple and universal method for writing operators in the Fock space. This problem is solved by introducing creation and annihilation operators. 1.1.4 Creation and annihilation operators for fermions To begin with, it will be useful to consider the simpler case of a discrete spectrum of momentum. In theory, such a spectrum can be produced by standard methods of placing the system in an impenetrable box or using periodic boundary conditions. Then the eigenvalues of the momentum operator form a discrete three-dimensional lattice pi. In the limit of infinite box size, the usual continuous momentum spectrum is restored. First turn to creation and annihilation operators for electrons. We define a (linear) creation operator a† psz of an electron with momentum p and spin projection sz by its action on basis vectors |p1s1z, p2s2z, . . . , pnsnz⟩. (1.14) 5 For example, equations (1-6.10)–(1-6.13) are valid in each two-particle sector. 6 Two photons are labeled by indices 1 and 2, two electrons by indices 3 and 4.
- 29. 1.1 Creation and annihilation operators | 5 in sectors with n electrons. We should distinguish two alternatives. In the first case, the created one-particle state (psz) is among the states present in (1.14), for example, (psz) = (pisiz). Since electrons are fermions and two fermions cannot occupy the same state due to the Pauli principle, this action leads to the zero result, i. e., a† psz |p1s1z, . . . , pi−1s(i−1)z, pisiz, pi+1s(i+1)z, . . . , pnsnz⟩ = 0. (1.15) In the second case, the created state (psz) is not among the single-particle states form- ing (1.14). Then, the creation operator a† psz simply adds one electron to the beginning of the particle list, so a† psz |p1s1z, p2s2z, . . . , pn, snz⟩ ≡ |psz, p1s1z, p2s2z, . . . , pnsnz⟩. (1.16) In this case, the operator a† psz converts a state with n electrons to a state with n+1 elec- trons. By repeatedly applying creation operators to the vacuum vector |vac⟩, we can construct all basis vectors in the purely electronic part of the Fock space. For example, a† psz |vac⟩ = |psz⟩, (1.17) a† p1s1z a† p2s2z |vac⟩ = |p1s1z, p2s2z⟩ are basis vectors in the one-electron and two-electron sectors. We define the electron annihilation operator apsz as a Hermitian conjugate to the creation operator a† psz . One can prove [21] that the action of apsz on the n-electron state (1.14) is as follows. If the one-electron state with parameters (psz) is already occupied, for example, (psz) = (pisiz), then this state is “annihilated” and the number of parti- cles in the system decreases by one, i. e., apsz |p1s1z, . . . , pi−1s(i−1)z, pisiz, pi+1s(i+1)z, . . . , pnsnz⟩ = (−1)𝒫 |p1s1z, . . . , pi−1s(i−1)z, pi+1s(i+1)z, . . . , pnsnz⟩. (1.18) Here 𝒫 is the number of permutations of neighboring particles, which is necessary to move the annihilated one-particle state i to the first place in the list. If the state (psz) is absent, i. e., (psz) ̸ = (pisiz) for all i, then apsz |p1s1z, p2s2z, . . . , pnsnz⟩ = 0. (1.19) Acting on the vacuum state, annihilation operators always yield zero, i. e., apsz |vac⟩ = 0. (1.20) The above formulas define the actions of creation and annihilation operators on the basis vectors in purely electronic sectors. These rules do not change in the pres- ence of other particles, and they extend to any linear combinations of basis vectors by
- 30. 6 | 1 Fock space linearity. Creation and annihilation operators for other fermions – positrons, protons and antiprotons – are defined similarly. For brevity, we will call the creation and annihilation operators jointly particle operators. In this way we will distinguish them from particle observables, such as mo- mentum pi, position ri, energy hi, etc. It should be emphasized that the (creation and annihilation) particle operators are not intended to directly describe any physical pro- cess or quantity. They are only formal mathematical objects intended to simplify the notation for working with other operators having direct physical meanings. Some ex- amples will be provided in Subsection 1.1.10. 1.1.5 Anticommutators of particle operators In practical calculations, we often encounter anticommutators of fermion particle op- erators. First we consider the case of annihilation/creation of unequal states of parti- cles, such as (psz) ̸ = (p s z). In this case, the anticommutator is {aps z , a† psz } ≡ a† psz aps z + aps z a† psz . Acting by this operator on a one-particle state |p s z ⟩, which differs from both |psz⟩ and |p s z⟩, we get (a† psz aps z + aps z a† psz )|p s z ⟩ = aps z |psz, p s z ⟩ = 0. Similarly we obtain (a† psz aps z + aps z a† psz )|psz⟩ = 0, (a† psz aps z + aps z a† psz )|p s z⟩ = a† psz |vac⟩ + aps z |psz, p s z⟩ = |psz⟩ − |psz⟩ = 0. It is not difficult to show that the result remains zero when acting on any n-particle state and also on their linear combinations. Thus, we conclude that in the entire Fock space {aps z , a† psz } = 0, if (psz) ̸ = (p s z). In the case (psz) = (p s z) a similar calculation yields {a† psz , apsz } = 1. Therefore, for all values of p, p , sz and s z we can write {a† psz , aps z } = δp,p δszs z . (1.21) Using similar arguments, one can show that {a† psz , a† ps z } = {apsz , aps z } = 0.
- 31. 1.1 Creation and annihilation operators | 7 1.1.6 Creation and annihilation operators for photons For photons that are bosons, the properties of creation and annihilation operators differ slightly from the fermion operators described above. Two or more photons can coexist in the same quantum state. Therefore, we determine the action of the photon creation operator c† pτ 7 on a multi-photon state as c† pτ|p1τ1, p2τ2, . . . , pnτn⟩ = |pτ, p1τ1, p2τ2, . . . , pnτn⟩, regardless of whether there was a particle (pτ) in the initial state or not. As in the case of fermions, boson annihilation operators cpτ are defined as Hermitian conjugates of the creation operators. The photon annihilation operator cpτ completely destroys a multi-photon state, so cpτ|p1τ1, p2τ2, . . . , pnτn⟩ = 0 if the annihilated one-photon state (pτ) was absent there. If the photon (pτ) was present, then the annihilation operator cp,τ simply removes this component, thus generating an (n − 1)-photon state, cpiτi |p1τ1, . . . , pi−1τi−1, piτi, pi+1τi+1, . . . , pnτn⟩ = |p1τ1, . . . , pi−1τi−1, pi+1τi+1, . . . , pnτn⟩. The above formulas can be extended without change to states where, in addition to photons, other particles are also present. Also, the action of operators extends by linearity to superpositions of basis vectors. From these rules, proceeding in analogy with Subsection 1.1.5, we obtain the following commutation relations for the photon annihilation and creation operators: [cpτ, c† pτ ] = δp,p δττ , [cpτ, cpτ ] = [c† pτ, c† pτ ] = 0. 1.1.7 Particle number operators With the help of creation and annihilation operators, we can build explicit expressions for various useful observables in the Fock space. Consider, for example, the product of two photon operators, Npτ = c† pτcpτ. (1.22) 7 The photon’s momentum is p and τ is its helicity.
- 32. 8 | 1 Fock space Acting on the state of two photons with quantum numbers (pτ), this operator gives Npτ|pτ, pτ⟩ = Npτc† pτc† pτ|vac⟩ = c† pτcpτc† pτc† pτ|vac⟩ = c† pτc† pτcpτc† pτ|vac⟩ + c† pτc† pτ|vac⟩ = c† pτc† pτc† pτcpτ|vac⟩ + 2c† pτc† pτ|vac⟩ = 2|pτ, pτ⟩, but acting on the state |pτ, p τ ⟩, we get Npτ|pτ, p τ ⟩ = Npτc† pτc† pτ |vac⟩ = c† pτcpτc† pτc† pτ |vac⟩ = c† pτc† pτcpτc† pτ |vac⟩ + c† pτc† pτ |vac⟩ = c† pτc† pτc† pτ cpτ|vac⟩ + c† pτc† pτ |vac⟩ = |pτ, p τ ⟩. These examples should convince us that the operator Npτ acts as a counter of photons with quantum numbers (pτ). 1.1.8 Continuous spectrum of momentum The properties of creation and annihilation operators presented in the previous sub- sections were derived for the case of discrete momentum spectrum. In reality, the mo- mentum spectrum is continuous, and these results must be modified by taking the limit of a “very large box.” It is not difficult to guess that in this limit equation (1.21) goes into {aps z , a† psz } = δszs z δ(p − p ). (1.23) The sequence of formulas8 δszs z δ(p − p ) = ⟨psz p s z⟩ = ⟨vac|apsz a† ps z |vac⟩ = −⟨vac|a† ps z apsz |vac⟩ + δszs z δ(p − p ) = δszs z δ(p − p ) confirms the consistency of our choice (1.23). 8 The first equality is obtained from the normalization of momentum eigenvectors (1-5.21); the second equality follows from the definition of the creation operator (1.17); the third one from formula (1.23); and the fourth one from (1.20).
- 33. 1.1 Creation and annihilation operators | 9 The same arguments can be applied to the operators of positrons (bpsz and b† psz ), protons (dpsz and d† psz ), antiprotons (fpsz and f† psz ) and photons (cpτ and c† pτ). So, finally, we get the following set of anticommutation and commutation relations relevant to QED: {apsz , a† ps z } = {bpsz , b† ps z } = {dpsz , d† ps z } = {fpsz , f † ps z } = δ(p − p )δszs z , (1.24) {apsz , aps z } = {bpsz , bps z } = {dpsz , dps z } = {fpsz , fps z } = {a† psz , a† ps z } = {b† psz , b† ps z } = {d† psz , d† ps z } = {f† psz , f † ps z } = 0, (1.25) [cpτ, c† pτ ] = δ(p − p )δττ , (1.26) [c† pτ, c† pτ ] = [cpτ, cpτ ] = 0. (1.27) Commutators of operators related to different particles are always zero. In the limit of continuous momentum, the counterpart of the particle counter (1.22) is the operator ρpτ = c† pτcpτ, (1.28) which can be interpreted as the density of photons with helicity τ and momentum p. Having summed the density (1.28) by the photon polarizations and integrating it over the entire momentum space, we obtain the operator of the total number of photons in the system Nph = ∑ τ ∫ dpc† pτcpτ. (1.29) We can also write similar expressions for the numbers of other particles. For example, Nel = ∑ sz ∫ dpa† psz apsz (1.30) is the electron number operator. Then the operator N = Nel + Npo + Npr + Nan + Nph (1.31) expresses the total number of particles in the system. 1.1.9 Normal ordering It is necessary to note the important property of operators (1.29) and (1.30). Being ex- pressed through particle creation and annihilation operators, they are applicable in
- 34. 10 | 1 Fock space the entire Fock space. We will follow this principle in our construction of other observ- ables as well. Thus, we intend to express operators in the Fock space in the form of polyno- mials in creation and annihilation operators. But for this, we need to overcome one notational problem related to the noncommutativity of particle operators: two differ- ent polynomials can represent the same operator. In order to have unified polynomial representatives, we will always agree to write the products of particle operators in the normal order, i. e., creation operators to the left and annihilation operators to the right. Using (anti)commutation relations (1.24)–(1.27), we can always convert any product of particle operators into a normally ordered form. Let us illustrate the above with one example. We have aps z cqτ a† psz c† qτ = aps z a† psz cqτ c† qτ = (a† psz aps z + δ(p − p )δszs z )(−c† qτcqτ + δ(q − q )δττ ) = −a† psz c† qτaps z cqτ + a† psz aps z δ(q − q )δττ − c† qτcqτ δ(p − p )δszs z + δ(p − p )δszs z δ(q − q )δττ , where the right-hand side is in a normally ordered form. As can be seen from this example, the transition to the normal order is accom- plished by moving all creation operators9 α† p to the leftmost positions. Permutations of operators of different particles have no additional effect. When on its way to the left a creation operator α† p meets an annihilation operator of the same particle αq, two terms appear10 instead of one (αqα† p). In the first term, the creation operator simply “jumps over” the annihilation operator, leading to the product ±(α† pαq). In the second term, the two operators contract, producing the delta function δ(p − q). The normal ordering in complex products of particle operators can be very labo- rious. Here, the celebrated Wick theorem comes to the rescue. Theorem 1.1 (Wick). When transformed to the normally ordered form, an arbitrary prod- uct abc ⋅ ⋅ ⋅ of particle operators becomes equal to the fully ordered term : abc ⋅ ⋅ ⋅ :11 plus the sum of terms with all possible contractions.12 Each term in this sum includes the fac- tor (−1)𝒫 , where 𝒫 is the number of permutations of the fermionic operators needed in order to 9 Here, for brevity, we drop the spin/polarization labels and use symbols α† , α to denote generic par- ticles operators (bosons and fermions). 10 They come from the (anti)commutation relation αqα† p = ±α† pαq + δ(p − q), where the minus (plus) sign refers to fermions (bosons). 11 The : abc ⋅ ⋅ ⋅ : symbol means that (i) particle operators are rearranged in the normal order and (ii) the resulting operator is multiplied by (−1)𝒫 , where 𝒫 is the number of permutations of fermionic factors. For example, : ap cq a† pc† q : = −a† pc† qap cq . 12 That is, contractions should be written for all pairs appearing in the “wrong” order α . . . α† .
- 35. 1.1 Creation and annihilation operators | 11 (i) put the contracted operators next to each other (i. e., in the αα† configuration) and (ii) rearrange in the normal order the operators left after all contractions. The proof of this theorem can be found in many textbooks on quantum field the- ory, for instance in [1]. Here we simply illustrate this result by the example of the prod- uct of electron operators aq ap a† pa† q. According to Wick’s theorem, in a normally or- dered form, this operator is the sum of the fully ordered product and six contractions13 : : aq ap a† pa† q : ≡ a† pa† qaq ap , aq ap a† pa† q ≡ −a† qaq δ(p − p ), aq ap a† pa† q ≡ −a† pap δ(q − q ), aq ap a† pa† q ≡ a† qap δ(p − q ), aq ap a† pa† q ≡ a† paq δ(q − p ), aq ap a† pa† q ≡ δ(p − p )δ(q − q ), aq ap a† pa† q ≡ −δ(p − q )δ(q − p ). 1.1.10 Noninteracting energy and momentum Now we can fully appreciate the benefits of introducing creation and annihilation op- erators. In particular, with their help it is easy to obtain a compact expression for the noninteracting Hamiltonian H0. It is obtained simply from the particle number oper- ator (1.31), multiplying the integrands (particle densities in the momentum space) by the energies of free particles, i. e., H0 = Hel+po 0 + Hpr+an 0 + Hph 0 , (1.32) Hel+po 0 = ∫ dpωp ∑ sz=±1/2 [a† psz apsz + b† psz bpsz ], Hpr+an 0 = ∫ dpΩp ∑ sz=±1/2 [d† psz dpsz + f † psz fpsz ], Hph 0 = c ∫ dpp ∑ τ=±1 c† pτcpτ. (1.33) Here ωp = √m2 ec4 + p2c2 are energies of free electrons and positrons, Ωp = √m2 pc4 + p2c2 are energies of free protons and antiprotons and cp are energies of 13 Contracted pairs of operators are marked with overline signs.
- 36. 12 | 1 Fock space free photons. It is not difficult to verify that H0 in (1.32) acts on states from the sector H (1, 0, 0, 0, 0) in the same way as equation (1.12) and H0 acts in the sector H (2, 0, 0, 0, 2) exactly as (1.13). So, we got a single expression for the energy that works equally well in all sectors of the Fock space.14 Similar arguments show that the operator P0 = Pel+po 0 + Ppr+an 0 + Pph 0 , (1.34) Pel+po 0 = ∫ dpp ∑ sz=±1/2 [a† psz apsz + b† psz bpsz ], Ppr+an 0 = ∫ dpp ∑ sz=±1/2 [d† psz dpsz + f† psz fpsz ], Pph 0 = ∫ dpp ∑ τ=±1 c† pτcpτ (1.35) has the meaning of the total momentum. 1.1.11 Noninteracting angular momentum and boost Expressions for the generators J0 and K0 in the Fock space are more complicated, since they requirethe participation of derivativesof particle operators. For illustration, consider the example of a massive spinless particle with creation and annihilation operators α† p and αp, respectively. The effect of the rotation e− i ℏ J0zφ on the one-particle state |p⟩ (see (1-5.10)) e− i ℏ J0zφ |p⟩ = |px cos φ + py sin φ, py cos φ − px sin φ, pz⟩ ≡ |φp⟩ can be imagined as annihilation of the initial state |p⟩ = |px, py, pz⟩ followed by cre- ation of the rotated state |φp⟩, i. e., e− i ℏ J0zφ |p⟩ = α† φpαp|px, py, pz⟩. Therefore, for an arbitrary one-particle state the operator of finite rotation has the form e− i ℏ J0zφ = ∫ dpα† φpαp. (1.36) It is not difficult to see that the same form is valid in the entire Fock space. Then the explicit expression for the generator J0z is obtained by taking the derivative of (1.36) 14 Note that our expression for the energy does not contain the problematic infinite term (so-called vacuum energy) that is typical for approaches based on quantum fields; see, for example, formula (2.31) in [10].
- 37. 1.1 Creation and annihilation operators | 13 with respect to φ, J0z = iℏ lim φ→0 d dφ e− i ℏ J0zφ = iℏ lim φ→0 d dφ ∫ dpα† px cos φ+py sin φ,py cos φ−px sin φ,pz αp = iℏ ∫ dp(py 𝜕α† p 𝜕px − px 𝜕α† p 𝜕py )αp. (1.37) The action of a boost along the z-axis is obtained from (1-5.30) and (1-5.11). We have e− ic ℏ K0zθ |p⟩ = √ ωθp ωp |θp⟩, (1.38) where the rapidity vector is θ = (0, 0, θ). This transformation can be represented as annihilation of the state |p⟩ = |px, py, pz⟩ and then creation of the state (1.38): e− ic ℏ K0zθ |p⟩ = √ ωθp ωp α† θpαp|p⟩. Thus, for all states in the Fock space the finite boost operator is e− ic ℏ K0zθ = ∫ dp√ ωθp ωp α† θpαp. The explicit formula for K0z is obtained by taking the derivative of this expression with respect to θ, K0z = iℏ c lim θ→0 d dθ e− ic ℏ K0zθ = iℏ c lim θ→0 d dθ ∫ dp√ ωp cosh θ + cpz sinh θ ωp α† px,py,pz cosh θ+ωp cosh θαp = iℏ ∫ dp( pz 2ωp α† pαp + ωp c2 𝜕α† p 𝜕pz αp). (1.39) Similar derivations can be done for other components of J0 and K0. 1.1.12 Poincaré transformations of particle operators Having defined all ten generators {H0, P0, J0, K0} we secured the noninteracting rep- resentation U0(θ; φ; r; t) ≡ e− ic ℏ K0⋅θ e− i ℏ J0⋅φ e− i ℏ P0⋅r e i ℏ H0t (1.40)
- 38. 14 | 1 Fock space of the Poincaré group in the Fock space. By construction, this representation induces transformations (1-5.8)–(1-5.10), (1-5.30) of one-particle states. From this, it is not dif- ficult to find out how creation and annihilation operators transform under the action of (1.40). As an example, consider the boost transformation. For the electron creation op- erators, we get15 e− ic ℏ K0⋅θ a† psz e ic ℏ K0⋅θ |vac⟩ = e− ic ℏ K0⋅θ a† psz |vac⟩ = e− ic ℏ K0⋅θ |psz⟩ = √ ωθp ωp ∑ s z D 1/2 s zsz (φW (p, θ))|(θp)s z⟩ = √ ωθp ωp ∑ s z D 1/2 s zsz (φW (p, θ))a† (θp)s z |vac⟩. Therefore16 e− ic ℏ K0⋅θ a† psz e ic ℏ K0⋅θ = √ ωθp ωp ∑ s z D 1/2 s zsz (φW (p, θ))a† (θp)s z = √ ωθp ωp ∑ s z D 1/2∗ szs z (−φW (p, θ))a† (θp)s z . (1.41) The transformation law for annihilation operators is obtained by the Hermitian con- jugation of (1.41), e− ic ℏ K0⋅θ apsz e ic ℏ K0⋅θ = √ ωθp ωp ∑ s z D 1/2 szs z (−φW (p, θ))a(θp)s z . (1.42) Actions of rotations and translations are derived in a similar way. We have e− i ℏ J0⋅φ a† psz e i ℏ J0⋅φ = ∑ s z D 1/2∗ szs z (−φ)a† (φp)s z , (1.43) e− i ℏ J0⋅φ apsz e i ℏ J0⋅φ = ∑ s z D 1/2 szs z (−φ)a(φp)s z , (1.44) e− i ℏ P0⋅r e i ℏ H0t a† psz e− i ℏ H0t e i ℏ P0⋅r = e− i ℏ p⋅r e i ℏ ωpt a† psz , (1.45) e− i ℏ P0⋅r e i ℏ H0t apsz e− i ℏ H0t e i ℏ P0⋅r = e i ℏ p⋅r e− i ℏ ωpt apsz . (1.46) 15 Here we took into account the fact that the vacuum vector is invariant with respect to U0 and used equation (1-5.30), where the Wigner angle φW (p, θ) is defined by formula (1-5.18). 16 We took into account that for unitary representatives of rotations D1/2 T∗ (−φ) ≡ D1/2† (−φ) = [D1/2 (−φ)]−1 = D1/2 (φ).
- 39. 1.2 Interaction potentials | 15 Transformations of photon operators are obtained from equation (1-5.69); we have U0(Λ; r; t)c† pτU−1 0 (Λ; r; t) = √ |Λp| p e− i ℏ (p⋅r)+ ic ℏ pt eiτφW (p,Λ) c† (Λp)τ, (1.47) U0(Λ; r; t)cpτU−1 0 (Λ; r; t) = √ |Λp| p e i ℏ (p⋅r)− ic ℏ pt e−iτφW (p,Λ) c(Λp)τ. (1.48) 1.2 Interaction potentials We would like to learn how to calculate the S-operator in QED, that is, the quantity most directly comparable with the experiment. Formulas derived in Section 1-7.1 tell us that in order to achieve this goal, we need to know the interacting part V of the total Hamiltonian H = H0 + V. The potential energy V in QED will be explicitly formulated only in Section 3.1. In the meantime, we will be interested in general properties of interactions and S-operators in the Fock space. In particular, we will try to find the limitations imposed on the choice of the operator V by a number of physical principles, such as conservation laws and cluster separability. Note that in our approach we postulate that the interaction V has no effect on the structure of the state space (Fock space). All the properties of this space17 defined in the noninteracting case remain true also in the presence of interactions. In Chap- ter 4 we will explain that even the necessity of renormalization will not force us to change the parameters (e. g., masses) of the particles from which the Fock space is constructed. In this respect, our approach differs from the axiomatic or constructive quantum field theory, where the Hilbert space of states has a non-Fock structure that depends on interactions. For more discussions see Volume 3. 1.2.1 Conservation laws From the experiment, we know that electromagnetic interactions obey certain impor- tant constraints, which are called conservation laws. An observable F is referred to as conserved if it remains unchanged during the time evolution, i. e., F(t) ≡ e i ℏ Ht F(0)e− i ℏ Ht = F(0). 17 The inner product, the mutual orthogonality of n-particle sectors, the form of the particle number operators, etc.
- 40. 16 | 1 Fock space It then follows that operators of conserved observables commute with the Hamilto- nian [F, H] = [F, H0 + V] = 0, which imposes some restrictions on the interaction operator V. For example, in the instant form of dynamics adopted in our book, the conservation of the total momentum and the total angular momentum means that18 [V, P0] = 0, (1.49) [V, J0] = 0. (1.50) It is also known that electromagnetic interactions conserve the lepton charge.19 There- fore, H = H0 + V must commute with the lepton number operator NL = Nel − Npo = ∑ sz ∫ dp(a† psz apsz − b† psz bpsz ). (1.51) Since H0 already commutes with NL, we get [V, NL] = 0. (1.52) In addition, all known interactions preserve the baryon charge,20 i. e., NB = Npr − Nan = ∑ sz ∫ dp(d† psz dpsz − f † psz fpsz ). (1.53) Hence, V must commute with the baryon number operator, i. e., [V, NB] = 0. (1.54) Taking into account that the electrons have a charge of −e, that the protons have a charge of +e and that the charge of antiparticles is opposite to the charge of particles, we can introduce the electric charge operator Q = e(NB − NL) = e ∑ sz ∫ dp(b† psz bpsz − a† psz apsz + d† psz dpsz − f† psz fpsz ) (1.55) and obtain the law of its conservation, [H, Q] = [V, Q] = e[V, NB − NL] = 0, (1.56) from equations (1.52) and (1.54). 18 The conservation of energy is a consequence of the trivial equality [H, H] = 0. 19 In our case this is the number of electrons minus the number of positrons. 20 In our case this is the number of protons minus the number of antiprotons.
- 41. 1.2 Interaction potentials | 17 As we have just found out, in QED both operators H0 and V commute with total momentum P0, total angular momentum J0, lepton charge NL, baryon charge NB and electric charge Q. Then, from the formulas in Section 1-7.1 it follows that the scattering operators F, Σ and S also commute with P0, J0, NL, NB and Q. This means that the corresponding observables are conserved in collisions. Although separately the numbers of particles of a certain type (for example, elec- trons or protons) may not be conserved, the conservation laws require that charged particles be born and destroyed only together with their antiparticles, i. e., in pairs. The pair production does not occur in low-energy reactions, because such processes require additional energy of 2mec2 = 2 × 0.51 MeV = 1.02 MeV for an electron–positron pair and 2mpc2 = 1876.6 MeV for a proton–antiproton pair. Such high-energy pro- cesses can be ignored in classical electrodynamics. However, even in the low-energy limit, it is necessary to take into account the emission of photons. Photons have zero mass, and the energy threshold for their creation is zero. Moreover, photons have zero charges (lepton, baryon and electric), so no conservation laws can limit their creation and destruction. Photons can be created (radiated) and annihilated (absorbed) in any quantities. 1.2.2 General form of interaction operators The well-known theorem21 claims that in the Fock space any operator V satisfying the conservation laws (1.49)–(1.50) can be written in the form of a polynomial in creation and annihilation operators,22 i. e., V = ∞ ∑ N=0 ∞ ∑ M=0 VNM, (1.57) VNM = ∑ {η,η} ∫ dq 1 ⋅ ⋅ ⋅ dq N dq1 ⋅ ⋅ ⋅ dqM × DNM(q 1η 1, . . . , q N η N ; q1η1, . . . , qMηM) × δ( N ∑ i=1 q i − M ∑ j=1 qj)α† q 1η 1 ⋅ ⋅ ⋅ α† q N η N αq1η1 ⋅ ⋅ ⋅ αqM ηM , (1.58) where the summation is over all spin/helicity indices η, η and the integration is car- ried out over all particle momenta. The individual terms (monomials) VNM in the ex- pansion (1.57) will be called potentials. Each potential is a normally ordered product of 21 See p. 175 in [21]. 22 Here symbols α† , α refer to generic creation–annihilation operators without specifying the particle type.
- 42. 18 | 1 Fock space N creation operators α† and M annihilation operators α. The pair of nonnegative inte- gers [N : M] will be called the index of the potential VNM. A potential is called bosonic if it has an even number of fermion particle operators Nf + Mf . The conservation laws (1.52), (1.54) and (1.56), [V, NL] = [V, NB] = [V, Q] = 0, (1.59) require that all interaction potentials in QED are bosonic. We are only interested in Hermitian operators V. In (1.58) DNM is a numerical coefficient function, which depends on the momenta and spin projections (or helicities) of all particles being created and destroyed. To satisfy the requirement [V, J0] = 0, this function must be rotationally invariant. The translational invariance ([V, P0] = 0) of (1.57)–(1.58) is guaranteed by the momentum delta function δ( N ∑ i=1 q i − M ∑ j=1 qj). This delta function also expresses the momentum conservation: the sum of the mo- menta of annihilated particles is equal to the sum of the momenta of created particles. The potential energy operator V enters formulas for the S-operator (1-7.14), (1-7.17) and (1-7.18) in a t-dependent form, i. e., V(t) = e i ℏ H0t Ve− i ℏ H0t . (1.60) We shall call regular those operators that satisfy conservation laws (1.49), (1.50) and (1.59) and whose t-dependence is determined by the free Hamiltonian H0, as in equa- tion (1.60). Equivalently, a t-dependent regular operator V(t) satisfies the following differential equation: d dt V(t) = d dt e i ℏ H0t Ve− i ℏ H0t = i ℏ e i ℏ H0t [H0, V]e− i ℏ H0t = i ℏ [H0, V(t)]. (1.61) In our convention, if a regular operator V is written without its t-argument, then either this operator is t-independent (i. e., it commutes with H0), or its value is taken at t = 0. One final remark on notation. If the coefficient function of the potential VNM is DNM, then we will use the symbol VNM ∘ ζ for an operator whose coefficient function D NM is the product of DNM and a numeric function ζ of the same arguments, i. e., D NM(q 1η 1, . . . , q N η N ; q1η1, . . . , qMηM) = DNM(q 1η 1, . . . , q N η N ; q1η1, . . . , qMηM)ζ (q 1η 1, . . . , q N η N ; q1η1, . . . , qMηM). Then, inserting (1.58) in (1.60) and using (1.45)–(1.48), we conclude that any regular potential VNM(t) takes the form VNM(t) = e i ℏ H0t VNMe− i ℏ H0t = VNM ∘ e i ℏ ℰNM t , (1.62)
- 43. 1.2 Interaction potentials | 19 where ℰNM(q 1, . . . , q N , q1, . . . , qM) ≡ N ∑ i=1 √m2 i c4 + q2 i c2 − M ∑ j=1 √m2 j c4 + q2 j c2 (1.63) is the difference between the energies of particles created and destroyed by the mono- mial VNM. This difference is called the energy function of the potential VNM. We can also extend this notation to general sums of potentials VNM and write V(t) = e i ℏ H0t Ve− i ℏ H0t = V ∘ e i ℏ ℰV t , (1.64) where ℰV formally denotes energy functions of the monomials in V. In this economical notation we obtain23 d dt V(t) = V(t) ∘ ( i ℏ ℰV ), V(t) = − i ℏ t ∫ −∞ V(t )dt = V(t) ∘ ( −1 ℰV ), (1.65) V ⏟⏟ ⏟⏟⏟ ⏟⏟ ≡ − i ℏ ∞ ∫ −∞ V(t)dt = −2πiV ∘ δ(ℰV ). (1.66) For example, formula (1.66) means that each monomial in V ⏟⏟ ⏟⏟⏟ ⏟⏟ is different from zero only on the surface that is a solution of the equation ℰNM(q 1, . . . , q N , q1, . . . , qM) = 0 (1.67) (if such a solution exists). This surface in the momentum space is called the energy surface or the energy shell of the potential VNM. We will also say that the operator V ⏟⏟ ⏟⏟⏟ ⏟⏟ in equation (1.66) is zero outside its energy shell ℰV = 0. Note that the scattering operator (1-7.14) S = 1 + Σ ⏟⏟ ⏟⏟⏟ ⏟⏟ is different from 1 only on the energy shell, i. e., where the energy conservation condition (1.67) is fulfilled. It is easy to verify that the energy function of the product of two regular operators is equal to the sum of their energy functions, i. e., ℰAB = ℰA + ℰB. This implies the following equality: AB ⏟⏟ ⏟⏟⏟ ⏟⏟ = − AB ∘ (ℰB)−1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ = −AB ∘ (ℰB)−1 δ(ℰA + ℰB) = AB ∘ (ℰA)−1 δ(ℰA + ℰB) = − AB ⏟⏟ ⏟⏟⏟ ⏟⏟, (1.68) which we will find useful in the third volume. 23 Here we tacitly assume the adiabatic switching of the interaction (1-7.26) and use formulas (1-7.12), (1-7.13) and (1-7.27).
- 44. 20 | 1 Fock space Figure 1.1: Locations of different types of operators in the “index space” [N : M]. R = renorm, O = oscillation. 1.2.3 Five types of regular potentials In this subsection we are going to introduce a classification of regular potentials (1.58), by dividing them into five groups depending on the index [N : M]. We call these types of operators renorm, oscillation, decay, phys and unphys.24 This classification will help in our study of renormalization in Chapter 4 and also in Volume 3, where we will for- mulate the “dressing” approach to QFT. Renorm potentials have either index [0 : 0] or index [1 : 1]. In the former case, the operator simply multiplies states by a numerical constant C. In the latter case, it is assumed that the particles that are produced and destroyed have the same type. In QED, the most general form of a renorm potential is25 R ∝ a† a + b† b + d† d + f † f + c† c + C. (1.69) Renorm potentials are characterized by the property that their energy functions (1.63) are identically zero. This means that such potentials always have an energy shell, where they do not vanish. Lemma 1.2. Any two renorm operators commute with each other. Proof. A general renorm operator is the sum (1.69). The summands referring to differ- ent particles commute, because particle operators of different particles always com- mute. It is not difficult to verify that two renorm operators, corresponding to the same particle, commute as well: [∫ dpf(p)α† pαp, ∫ dqg(q)α† qαq] = 0. The free Hamiltonian (1.32) and the total momentum (1.34) are examples of renorm operators. In particular, this implies that renorm potentials commute with H0, so reg- ular renorm operators are independent of t. 24 The correlation between potential’s index [N : M] and its type is shown in Figure 1.1. 25 For brevity, here we write only the operator structure of R, omitting numerical multipliers, indices, summation and integration signs.
- 45. 1.2 Interaction potentials | 21 Oscillation potentials have index [1 : 1]. In contrast to renorm potentials with the same index, oscillation potentials create and destroy different types of particles having different masses. For this reason, their energy functions (1.63) never turn to zero, so they do not have energy shells. In nature, oscillation potentials act on particles such as kaons and neutrinos. A vivid experimental manifestation of such interactions are time-dependent oscillations between different types of particles [6]. In QED there cannot be oscillation interactions, because they would violate the lepton and/or baryon conservation laws. Decay potentials satisfy two conditions: (1) their indices are either [1 : N] or [N : 1] with N ≥ 2; (2) they have a nonempty energy shell, where their coefficient functions do not van- ish. These potentials describe decay processes 1 → N,26 in which one particle decays into N products. Moreover, we require that the laws of conservation of energy and momen- tum are fulfilled in the decay, i. e., there is a nontrivial energy shell, where the coeffi- cient function does not vanish. Decay terms are not present in the QED Hamiltonian and in the corresponding S-matrix, because decays of electrons, protons or photons would be against conservation laws.27 Nevertheless, decays of elementary particles play a huge role in other branches of high-energy physics, and we will discuss them in the third volume. Phys potentials have at least two creation operators and at least two annihila- tion operators (they have indices [N : M] where N ≥ 2 and M ≥ 2). For phys poten- tials, the energy shell always exists. For example, in the case of the phys operator28 d† (p+k)ρ f† (q−k)σ apτbqη the energy shell is the set of solutions of the equation Ωp+k+Ωq−k = ωp + ωq in the nine-dimensional momentum space {p, q, k}. This equation has non- trivial solutions, so the energy shell is not empty. All regular operators that do not belong to any of the four above classes will be called unphys potentials. They can be divided into two subclasses with the following indices: (1) [0 : N] or [N : 0], where N ≥ 1. Obviously, in this case the energy shell is absent. 26 And also inverse processes N → 1. 27 In principle, one photon can decay into an odd number of other photons without violat- ing the conservation laws. For example, such a process could be described by the potential c† k1τ1 c† k2τ2 c† k3τ3 c(k1+k2+k3)τ4 , which formally satisfies all conservation laws if the momenta of all in- volved photons are collinear and k1 + k2 + k3 − |k1 + k2 + k3| = 0. However, as shown in [5], such terms in the S-operator are zero on the energy shell, so photon decays are forbidden in QED. 28 This operator describes the conversion reaction electron + positron → proton + antiproton. In the arguments of particle operators, we have already taken into account the momentum conservation law.
- 46. 22 | 1 Fock space (2) [1 : N] or [N : 1], where N ≥ 2. These are the same indices as for decay poten- tials, but here we demand that either the energy shell does not exist or that the coefficient function vanishes on the energy shell. Here is an example of an unphys potential with condition (2): a† (p−k)σc† kτapρ. (1.70) The energy shell equation ωp−k + ck = ωp has only one solution, k = 0. However, the zero vector is excluded from the photon’s momentum spectrum (see Subsec- tion 1-5.4.1), so the potential (1.70) has an empty energy shell. This means that a free electron cannot emit a photon without violating the energy–momentum conservation law. Table 1.1: Types of regular potentials in the Fock space. Potential Index [N : M] Energy shell Example renorm [0 : 0],[1 : 1] yes a† pap oscillation [1 : 1] no forbidden in QED unphys [0 : M ≥ 1],[N ≥ 1 : 0] no a† pb† −p−k c† k unphys [1 : M ≥ 2],[N ≥ 2 : 1] no a† pap−kck decay [1 : M ≥ 2],[N ≥ 2 : 1] yes forbidden in QED phys [N ≥ 2 : M ≥ 2] yes d† q+k a† p−k dqap The properties of potentials considered above are summarized in Table 1.1. These five types of interactions exhaust all possibilities; therefore any regular operator V must have a unique expansion V = Vren + Vunp + Vdec + Vphys + Vosc . As mentioned above, there are no oscillation and decay interactions in QED, so every- where in this volume we will assume that the most general potential is equal to the sum of renorm, unphys and phys parts: VQED = Vren + Vunp + Vphys . Now we need to figure out how to perform various manipulations with these three classes of potentials. In particular, we want to learn how to calculate products, com- mutators and t-integrals that are necessary for computing scattering operators from Section 1-7.1.
- 47. 1.2 Interaction potentials | 23 1.2.4 Products and commutators of regular potentials Let us first prove a few simple results. Lemma 1.3. The product of two or more regular operators is regular. Proof. By definition, if operators A(t) and B(t) are regular, then A(t) = e i ℏ H0t Ae− i ℏ H0t , B(t) = e i ℏ H0t Be− i ℏ H0t . Hence, their product C(t) = A(t)B(t) has the t-dependence C(t) = e i ℏ H0t Ae− i ℏ H0t e i ℏ H0t Be− i ℏ H0t = e i ℏ H0t ABe− i ℏ H0t characteristic of regular operators. The conservation laws (1.49), (1.50) and (1.59) are valid for the product AB, just as they are valid for A and B separately. Therefore, C(t) is regular. Theorem 1.4. A Hermitian operator P is phys if and only if it yields zero when acting on both the vacuum vector |vac⟩ and one-particle states |1⟩ ≡ α† |vac⟩29 : P|vac⟩ = 0, (1.71) P|1⟩ ≡ Pα† |vac⟩ = 0. (1.72) Proof. By definition, normally ordered phys potentials have (at least) two annihila- tion operators on the right. So, they yield zero when applied to the vacuum or any one-particle state. Therefore, equations (1.71) and (1.72) are satisfied for any phys operator P. Letusnowprovetheconverse. Renormoperatorscannotsatisfyrequirements(1.71) and (1.72), because they preserve the number of particles. Unphys operators [1 : M] can satisfy these requirements. For example, α† 1 α2α3|vac⟩ = 0, α† 1 α2α3|1⟩ = 0. However, in order to be Hermitian, such operators must always be present in pairs with [M : 1] operators, like α† 2α† 3α1. Then there exists at least one single-particle state |1⟩ for which equation (1.72) is not valid, that is, α† 3α† 2α1|1⟩ = α† 3α† 2|vac⟩ ̸ = 0. 29 Here α means any of the five particle operators (a, b, d, f , c) related to QED. Momentum and spin labels are omitted for brevity.
- 48. 24 | 1 Fock space Similar arguments apply to unphys operators with indices [0 : M] and [N : 0]. Hence, the only remaining possibility for the potential P is to be phys. Lemma 1.5. The product and commutator of any phys operators A and B are also phys. Proof. By Theorem 1.4, if A and B are phys, then A|vac⟩ = B|vac⟩ = A|1⟩ = B|1⟩ = 0. The same properties are valid for the Hermitian combinations i(AB−BA) and AB+BA. Hence, both the commutator [A, B] and the anticommutator {A, B} are phys. The same conclusion is true for the product, which can be expressed as the sum AB = 1 2 {A, B} + 1 2 [A, B]. Lemma 1.6. If R is a renorm operator, P is a phys operator and [P, R] ̸ = 0, then operator [P, R] is of the phys type. Proof. Let us first check how this commutator acts on the vacuum and single-particle states.30 We have i(PR − RP)|vac⟩ = iPR|vac⟩ = iPC0|vac⟩ = 0, i(PR − RP)|1⟩ = iPR|1⟩ = iPC1|1 ⟩ = 0. This means that the Hermitian commutator i[P, R] turns vectors |vac⟩ and |1⟩ to zero. By Lemma 1.4 this operator is phys. Lemma 1.7. If R is a renorm operator, U is an unphys operator and [U, R] ̸ = 0, then operator [U, R] has the unphys type. Idea of the proof. Let us first calculate the commutator of the renorm operator R = ∫ dpf(p)α† pαp with a particle creation operator31 We have [α† q, R] = α† q(∫ dpf(p)α† pαp) − (∫ dpf (p)α† pαp)α† q = ± ∫ dpf(p)α† pα† qαp − ∫ dpf (p)α† pαpα† q = ∫ dpf(p)α† pαpα† q − ∫ dpf (p)α† pδ(p − q) − ∫ dpf (p)α† pαpα† q = −f (q)α† q. 30 Here we took into account that renorm operators preserve the number of particles: R|vac⟩ = const× |vac⟩, R|1⟩ = |1 ⟩ and phys operators turn the states |vac⟩ and |1⟩ to zero. 31 The upper sign is for bosons, the lower sign is for fermions.
- 49. 1.2 Interaction potentials | 25 Similarly, we obtain the commutator with an annihilation operator: [αq, R] = f(q)αq. Now, as an example of an unphys operator, we take a potential with index [2 : 1], U = ∫ dq1dq2dpD(q1, q2; p)δ(q1 + q2 − p)α† q1 α† q2 αp. The index of the commutator is also [2 : 1]. We have [U, R] = ∫ dq1dq2dpD(q1, q2; p)δ(q1 + q2 − p)α† q1 α† q2 [αp, R] + ∫ dq1dq2dpD(q1, q2; p)δ(q1 + q2 − p)α† q1 [α† q2 , R]αp + ∫ dq1dq2dpD(q1, q2; p)δ(q1 + q2 − p)[α† q1 , R]α† q2 αp = ∫ dq1dq2dpD(q1, q2; p)f(p)δ(q1 + q2 − p)α† q1 α† q2 αp − ∫ dq1dq2dpD(q1, q2; p)f(q2)δ(q1 + q2 − p)α† q1 α† q2 αp − ∫ dq1dq2dpD(q1, q2; p)f(q1)δ(q1 + q2 − p)α† q1 α† q2 αp. Moreover, if the operator U does not have an energy shell, then [U, R] also does not have it, i. e., its type is unphys. If U has an energy shell where the coefficient function D(q1, q2; p) is zero, then [U, R] also has this property. Lemma 1.8. The commutator [P, U] of an Hermitian phys operator P and an Hermitian unphys operator U cannot contain renorm terms. Proof. Applying the operator [P, U] to a single-particle state |1⟩ and using (1.72), we obtain [P, U]|1⟩ = (PU − UP)|1⟩ = PU|1⟩. (1.73) If the commutator [P, U] contained renorm terms, then the right-hand side of (1.73) would have a nonzero one-particle component. However, the range of any phys P does not include the one-particle sector. This implies [P, U]ren = 0. Finally, it is easy to verify that there are no restrictions on the type of the commu- tator of two unphys operators [U, U ]. It can contain unphys, phys and renorm parts. The above results are summarized in Table 1.2. 1.2.5 More about t-integrals Lemma 1.9. The t-derivative of a regular operator A(t) is regular, and its renorm part vanishes.
- 50. 26 | 1 Fock space Table 1.2: Commutators, t-derivatives and t-integrals with regular operator A in the Fock space. (No- tation: P = phys, U = unphys, R = renorm, NR = nonregular.) Type of A [A, P] [A, U] [A, R] dA dt A A ⏟⏟ ⏟⏟⏟ ⏟⏟ P P P+U P P P P U P+U P+U+R U U U 0 R P U 0 0 NR ∞ Proof. According to (1.61), the derivative of A(t) is equal to the commutator with regu- lar H0. Then by Lemma 1.3 this derivative is regular. Suppose, by contradiction, that d dt A(t) has a nonzero renorm part R. Then R does not depend on t, because it is regular. It follows that the most general form of A(t) is A(t) = Rt + S, where S is any operator independent of t. From the condition that the renorm part of the regular operator A(t) cannot depend on t, we obtain R = 0. From equation (1.65) we conclude that t-integrals of regular phys and unphys oper- ators are regular. However, this property does not hold for t-integrals of renorm opera- tors. As we know, renorm operators are independent of t. Hence, when the interaction is adiabatically switched on, as in (1-7.26), we obtain Vren (t) = lim ϵ→+0 (− i ℏ 0 ∫ −∞ Vren eϵt dt − i ℏ t ∫ 0 Vren e−ϵt dt ) = −( i ℏ )Vren ∘ lim ϵ→+0 ( eϵt ϵ t=0 t=−∞ − e−ϵt ϵ t=t t=0 ) = −( i ℏ )Vren ∘ lim ϵ→+0 ( 1 ϵ − e−ϵt ϵ + 1 ϵ ) = −( i ℏ )Vren ∘ lim ϵ→+0 ( 1 ϵ + t + ⋅ ⋅ ⋅), (1.74) Vren ⏟⏟⏟⏟⏟⏟⏟ = lim t→∞ Vren (t) = ∞. (1.75) Hence, renorm operators differ from all others in that their t-integrals (1.74)–(1.75) are infinite and nonregular.32 By definition, an unphys operator Vunp either does not have an energy shell, or its coefficient function vanishes on the energy shell. Then, from equation (1.66) it follows that for any unphys operator Vunp ⏟⏟⏟⏟⏟⏟⏟⏟⏟ = 0. (1.76) Results obtained in this subsection are shown in the last three columns of Table 1.2. 32 As we shall see in Subsection 4.1.1, correctly renormalized expressions for scattering operators should not contain renorm terms and pathological constructs like (1.74)–(1.75).
- 51. Another Random Scribd Document with Unrelated Content
- 52. Dhruvasena dated a.d. 634 (G. 315) and an unpublished copperplate in the possession of the chief of Morbí belonging to his successor Dhruvasena III. dated a.d. 651 (G. 332) prove that Dharasena’s reign did not last more than seventeen years. The well known Sanskrit poem Bhaṭṭikávya seems to have been composed in the reign of this king as at the end of his work the author says it was written at Valabhi protected (governed) by the king the illustrious Dharasena.34 The author’s application to Dharasena of the title Narendra Lord of Men is a further proof of his great power. Dhruvasena III. a.d. 650–656.Dharasena IV. was not succeeded by his son but by Dhruvasena the son of Derabhaṭa the son of Dharasena IV.’s paternal grand- uncle. Derabhaṭa appears not to have been ruler of Valabhi itself but of some district in the south of the Valabhi territory. His epithets describe him as like the royal sage Agastya spreading to the south, and as the lord of the earth which has for its two breasts the Sahya and Vindhya hills. This description may apply to part of the province south of Kaira where the Sahyádri and Vindhya mountains may be said to unite. In the absence of a male heir in the direct line, Derabhaṭa’s son Dhruvasena appears to have succeeded to the throne of Valabhi. The only known copperplate of Dhruvasena III.’s, dated a.d. 651 (G. 332), records the grant of the village of Peḍhapadra in Vanthali, the modern Vanthali in the Navánagar State of North Káthiáváḍa. A copperplate of his elder brother and successor Kharagraha dated a.d. 656 (G. 337) shows that Dhruvasena’s reign cannot have lasted over six years. Kharagraha, a.d. 656–665.The less than usually complimentary and respectful reference to Dhruvasena III. in the attributes of Kharagraha suggests that Kharagraha took the kingdom by force from his younger brother as the rightful successor of his father. At all events the succession of Kharagraha to Dhruvasena was not in the usual peaceful manner. Kharagraha’s grant dated a.d. 656 (G. 337) is written by the Divirapati or Chief Secretary and minister of peace and war Anahilla son of Skandabhaṭa.35 The Dútaka or causer of the gift was the Pramátṛi or survey officer Śríná. Śíláditya III. a.d. 666–675.Kharagraha was succeeded by Śíláditya III. son of Kharagraha’s elder brother Śíláditya II. Śíláditya II. seems not to have ruled
- 53. at Valabhi but like Derabhaṭa to have been governor of Southern Valabhi, as he is mentioned out of the order of succession and with the title Lord of the Earth containing the Vindhya mountain. Three grants of Śíláditya III. remain, two dated a.d. 666 (G. 346)36 and the third dated a.d. 671 (G. 352).37 He is called Parama-bhaṭṭáraka Great Lord, Mahárájádhirája Chief King among Great Kings, and Parameśvara Great Ruler. These titles continue to be applied to all Chapter VIII. The Valabhis, a.d. 509–766. Śíláditya IV. a.d. 691. subsequent Valabhi kings. Even the name Śíláditya is repeated though each king must have had some personal name. Śíláditya IV. a.d. 691.Śíláditya III. was succeeded by his son Śíláditya IV. of whom one grant dated a.d. 691 (G. 372) remains. The officer who prepared the grant is mentioned as the general Divirapati Śrí Haragaṇa the son of Bappa Bhogika. The Dútaka or gift-causer is the prince Kharagraha, which may perhaps be the personal name of the next king Śíláditya V. Śíláditya V. a.d. 722.Of Śíláditya V. the son and successor of Śíláditya IV. two grants dated a.d. 722 (G. 403) both from Gondal remain. Both record grants to the same person. The writer of both was general Gillaka son of Buddhabhaṭṭa, and the gift-causer of both prince Śíláditya. Śíláditya VI. a.d. 760.Of Śíláditya VI. the son and successor of the last, one grant dated a.d. 760 (G. 441) remains. The grantee is an Atharvavedi Bráhman. The writer is Sasyagupta son of Emapatha and the gift-causer is Gánjaśáti Śrí Jajjar (or Jajjir). Śíláditya VII. a.d. 766.Of Śíláditya VII. the son and successor of the last, who is also called Dhrúbhaṭa (Sk. Dhruvabhaṭa), one grant dated a.d. 766 (G. 447) remains. Valabhi Family Tree.The following is the genealogy of the Valabhi Dynasty: VALABHI FAMILY TREE, a.d. 509–766.
- 54. Bhaṭárka a.d. 509. (Gupta 190?). Dharasena I. Droṇasiṃha. Dhruvasena I. a.d. 526. (Gupta 207). Dharapaṭṭa. Guhasena a.d. 559, 565, 567, (Gupta 240, 246, 248). Dharasena II. a.d. 571, 588, 589 (Gupta 252, 269, 270). Śíláditya I. or Dharmáditya I. a.d. 605, 609 (Gupta 286, 290). Kharagraha I. Dharasena III. Dhruvasena II. or Báláditya, a.d. 629 (Gupta 310). Derabhaṭa. Śíláditya II. Kharagraha II. or Dharmáditya II. a.d. 656 (Gupta 337). Dhruvasena III. a.d. 651 (Gupta 332). Dharasena IV. a.d. 645, 649, (Gupta 326, 330). Śíláditya III. a.d. 671 (Gupta 352). Śíláditya IV. a.d. 691, 698 (Gupta 372 & 379). Śíláditya V. a.d. 722 (Gupta 403).
- 55. Śíláditya VI. a.d. 760 (Gupta 441). Śíláditya VII. or Dhrúbhaṭa, a.d. 766 (Gupta 447). Chapter VIII. The Valabhis, a.d. 509–766. The Fall of Valabhi, a.d. 750–770. The Fall of Valabhi, a.d. 750–770.Of the overthrow of Valabhi many explanations have been offered.38 The only explanation in agreement with the copperplate evidence that a Śíláditya was ruling at Valabhi as late as a.d. 766 (Val. Saṃ. 447)40 is the Hindu account preserved by Alberuni (a.d. 1030)41 that soon after the Sindh capital Mansúra was founded, say a.d. 750–770, Ranka a disaffected subject of the era-making Valabhi, with presents of money persuaded the Arab lord of Mansúra to send a naval expedition against the king of Valabhi. In a night attack king Valabha was killed and his people and town were destroyed. Alberuni adds: Men say that still in our time such traces are left in Chapter VIII. The Valabhis, a.d. 509–766. The Fall of Valabhi, a.d. 750–770. that country as are found in places wasted by an unexpected attack.42 For this expedition against Valabhi Alberuni gives no date. But as Mansúra was not founded till a.d. 75043 and as the latest Valabhi copperplate is a.d. 766 the expedition must have taken place between a.d. 750 and 770. In support of the Hindu tradition of an expedition from Mansúra against Valabhi between a.d. 750 and 770 it is to be noted that the Arab historians of Sindh record that in a.d. 758 (H. 140) the Khalif Mansúr sent Amru bin Jamal with a fleet of barks to the coast of Barada.44 Twenty years later a.d. 776 (H. 160) a second expedition succeeded in taking the town, but, as sickness broke out, they had to return. The question remains should the word, which in these extracts Elliot reads Barada, be read Balaba. The lax rules of Arab cursive writing would cause little difficulty in adopting the reading Balaba.45 Further it is hard to believe
- 56. that Valabhi, though to some extent sheltered by its distance from the coast and probably a place of less importance than its chroniclers describe, should be unknown to the Arab raiders of the seventh and eighth centuries and after its fall be known to Alberuni in the eleventh century. At the same time, as during the eighth century there was, or at least as there may have been,46 a town Barada on the south-west coast of Káthiáváḍa the identification Chapter VIII. The Valabhis, a.d. 509–766. The Fall of Valabhi, a.d. 750–770. of the raids against Barada with the traditional expedition against Balaba though perhaps probable cannot be considered certain. Further the statement of the Sindh historians47 that at this time the Sindh Arabs also made a naval expedition against Kandahár seems in agreement with the traditional account in Tod that after the destruction of Valabhi the rulers retired to a fort near Cambay from which after a few years they were driven.48 If this fort is the Kandahár of the Sindh writers and Gandhár on the Broach coast about twenty miles south of Cambay, identifications which are in agreement with other passages, the Arab and Rájput accounts would fairly agree.49 The Importance of Valabhi.The discovery of its lost site; the natural but mistaken identification of its rulers with the famous eighth and ninth century (a.d. 753–972) Balharas of Málkhet in the East Dakhan;50 the tracing to Valabhi of the Rána of Udepur in Mewáḍ the head of the Sesodias or Gohils the most exalted of Hindu families51; and in later times the wealth of Valabhi copperplates have combined to make the Valabhis one of the best known of Gujarát dynasties. Except the complete genealogy, covering the 250 years from the beginning of the sixth to the middle of the eighth century, little is known of Valabhi or its chiefs. The Chapter VIII. The Valabhis, a.d. 509–766. The Importance of Valabhi, a.d. 750–770. origin of the city and of its rulers, the extent of their sway, and the cause and date of their overthrow are all uncertain. The unfitness of the site, the want of reservoirs or other stone remains, the uncertainty when its rulers gained an independent position, the fact that only one of them claimed the title Chakravarti or All Ruler are hardly
- 57. consistent with any far-reaching authority. Add to this the continuance of Maitraka or Mer power in North Káthiáváḍa, the separateness though perhaps dependence of Sauráshṭra even in the time of Valabhi’s greatest power,52 the rare mention of Valabhi in contemporary Gujarát grants,53 and the absence of trustworthy reference in the accounts of the Arab raids of the seventh or eighth centuries tend to raise a doubt whether, except perhaps during the ten years ending 650, Valabhi was ever of more than local importance. Valabhi and the Gehlots.In connection with the pride of the Sesodias or Gohils of Mewáḍ in their Valabhi origin54 the question who were the Valabhis has a special interest. The text shows that Pandit Bhagvánlál was of opinion the Valabhis were Gurjjaras. The text also notes that the Pandit believed they reached south-east Káthiáváḍa by sea from near Broach and that if they did not come to Broach from Málwa at least the early rulers obtained (a.d. 520 and 526) investiture from the Málwa kings. Apart from the doubtful evidence of an early second to fifth century Bála or Valabhi three considerations weigh against the theory that the Valabhis entered Gujarát from Málwa in the sixth century. First their acceptance of the Gupta era and of the Gupta currency raises the presumption that the Valabhis were in Káthiáváḍa during Gupta ascendancy (a.d. 440–480): Second that the Sesodias trace their pedigree through Valabhi to an earlier settlement at Dhánk in south-west Káthiáváḍa and that the Válas of Dhánk still hold the place of heads of the Válas of Káthiáváḍa: And Third that both Sesodias and Válas trace their origin to Kanaksen a second century North Indian immigrant into Káthiáváḍa combine to raise the presumption that the Válas were in Káthiáváḍa before the historical founding of Valabhi in a.d. 52655 and that the city took its name from its founders the Válas or Bálas. Whether or not the ancestors of the Gohils and Válas were settled in Káthiáváḍa before the establishment of Valabhi about a.d. 526 Chapter VIII. The Valabhis, a.d. 509–766. Valabhi and the Gehlots. several considerations bear out the correctness of the Rájput traditions and the Jain records that the Gohils or Sesodias of Mewáḍ came from Bála or Valabhi in Káthiáváḍa. Such a withdrawal from the
- 58. coast, the result of the terror of Arab raids, is in agreement with the fact that from about the middle of the eighth century the rulers of Gujarát established an inland capital at Aṇahilaváḍa (a.d. 746).56 It is further in agreement with the establishment by the Gohil refugees of a town Balli in Mewáḍ; with the continuance as late as a.d. 968 (S. 1024) by the Sesodia chief of the Valabhi title Śíláditya or Sail57; and with the peculiar Valabhi blend of Sun and Śiva worship still to be found in Udepur.58 The question remains how far can the half-poetic accounts of the Sesodias be reconciled with a date for the fall of Valabhi so late as a.d. 766. The mythical wanderings, the caveborn Guha, and his rule at Idar can be easily spared. The name Gehlot which the Sesodias trace to the caveborn Guha may as the Bhávnagar Gehlots hold have its origin in Guhasena (a.d. 559–567) perhaps the first Valabhi chief of more than local distinction.59 Tod61 fixes the first historical date in the Sesodia family history at a.d. 720 or 728 the ousting of the Mori or Maurya of Chitor by Bappa or Sail. An inscription near Chitor shows the Mori in power in Chitor as late as a.d. 714 (S. 770).62 By counting back nine generations from Śakti Kumára the tenth from Bappa whose date is a.d. 1038 Tod fixes a.d. 720–728 as the date when the Gohils succeeded the Moris. But Chapter VIII. The Valabhis, a.d. 509–766. Valabhi and the Gehlots. the sufficient average allowance of twenty years for each reign would bring Bappa to a.d. 770 or 780 a date in agreement with a fall of Valabhi between a.d. 760 and 770, as well as with the statement of Abul Fazl, who, writing in a.d. 1590, says the Rána’s family had been in Mewáḍ for about 800 years.63 The Válas of Káthiáváḍa.The Arab accounts of the surprise-attack and of the failure of the invaders to make a settlement agree with the local and Rájputána traditions that a branch of the Valabhi family continued to rule at Vaḷeh until its conquest by Múla Rája Solaṇkhi in a.d. 950.64 Though their bards favour the explanation of Vála from the Gujaráti valvu return or the Persian válah65 noble the family claim to be of the old Valabhi stock. They still have the tradition they were driven out by the Musalmáns, they still keep up the family name of Selait or Śíláditya.66
- 59. The local tradition regarding the settlement of the Válas in the Balakshetra south of Valabhi is that it took place after the capture of Valabhi by Múla Rája Solaṇkhi (a.d. 950).67 If, as may perhaps be accepted, the present Válas represent the rulers of Valabhi it seems to follow the Válas were the overlords of Balakshetra at least from the time of the historical prosperity of Valabhi (a.d. 526–680). The traditions of the Bábriás who held the east of Sorath show that when they arrived (a.d. 1200–1250) the Vála Rájputs were in possession and suggest that the lands of the Válas originally stretched as far west as Diu.68 That the Válas held central Káthiáváḍa is shown by their possession of the old capital Vanthali nine miles south-west of Junágaḍh and by (about a.d. 850) their transfer of that town to the Chúḍásamás.69 Dhánk, about twenty-five miles north-west of Junágaḍh, was apparently held by the Válas under the Jetwas when (a.d. 800–1200?) Ghumli or Bhumli was the capital of south-west Káthiáváḍa. According to Jetwa accounts the Válas were newcomers whom the Jetwas allowed to settle at Dhánk.70 But as the Jetwas are not among the earliest settlers in Káthiáváḍa it seems more probable that, like the Chúḍásamás at Vanthali, the Jetwas found the Válas in possession. The close connection of the Válas with the earlier waves of Káthis is admitted.71 Considering that the present Chapter VIII. The Valabhis, a.d. 509–766. The Válas of Káthiáváḍa. (1881) total of Káthiáváḍa Vála Rájputs is about 900 against about 9000 Vála Káthis, the Válas,72 since their loss of power, seem either to have passed into unnoticeable subdivisions of other Rájput tribes or to have fallen to the position of Káthis. The Válas and Káthis.If from the first and not solely since the fall of Valabhi the Válas have been associated with the Káthis it seems best to suppose they held to the Káthis a position like that of the Jetwas to their followers the Mers. According to Tod73 both Válas and Káthis claim the title Tata Multánka Rai Lords of Tata and Multán. The accounts of the different sackings of Valabhi are too confused and the traces of an earlier settlement too scanty and doubtful to justify any attempt to carry back Valabhi and the Válas beyond the Maitraka overthrow of Gupta power in Káthiáváḍa (a.d. 470–480). The boast that Bhaṭárka, the reputed founder of the house
- 60. of Valabhi (a.d. 509), had obtained glory by dealing hundreds of blows on the large and very mighty armies of the Maitrakas who by force had subdued their enemies, together with the fact that the Valabhis did and the Maitrakas did not adopt the Gupta era and currency seem to show the Válas were settled in Káthiáváḍa at an earlier date than the Mers and Jetwas. That is, if the identification is correct, the Válas and Káthis were in Káthiáváḍa before the first wave of the White Huns approached. It has been noticed above under Skandagupta that the enemies, or some of the enemies, with whom, in the early years of his reign a.d. 452–454, Skandagupta had so fierce a struggle were still in a.d. 456 a source of anxiety and required the control of a specially able viceroy at Junágaḍh. Since no trace of the Káthis appears in Káthiáváḍa legends or traditions before the fifth century the suggestion may be offered that under Vála or Bála leadership the Káthis were among the enemies who on the death of Kumáragupta (a.d. 454) seized the Gupta possessions in Káthiáváḍa. Both Válas and Káthis would then be northerners driven south from Multán and South Chapter VIII. The Valabhis, a.d. 509–766. The Válas and Káthis. Sindh by the movements of tribes displaced by the advance of the Ephthalites or White Huns (a.d. 440–450) upon the earlier North Indian and border settlements of the Yuan-Yuan or Avars.74 Descent from Kanaksen, a.d. 150.The Sesodia or Gohil tradition is that the founder of the Válas was Kanaksen, who, in the second century after Christ, from North India established his power at Virát or Dholka in North Gujarát and at Dhánk in Káthiáváḍa.75 This tradition, which according to Tod76 is supported by at least ten genealogical lists derived from distinct sources, seems a reminiscence of some connection between the early Válas and the Kshatrapas of Junágaḍh with the family of the great Kushán emperor Kanishka (a.d. 78–98). Whether this high ancestry belongs of right to the Válas and Gohils or whether it has been won for them by their bards nothing in the records of Káthiáváḍa is likely to be able to prove. Besides by the Válas Kanaksen is claimed as an ancestor by the Chávaḍás of Okhámandal as the founder of Kanakapurí and as reigning in Kṛishṇa’s throne in Dwárká.77. In support of the form Kanaka for Kanishka is the
- 61. doubtful Kanaka-Śakas or Kanishka-Śakas of Varáhamihira (a.d. 580).78 The form Kanik is also used by Alberuni79 for the famous Vihára or monastery at Pesháwar of whose founder Kanak Alberuni retails many widespread legends. Tod80 says; ‘If the traditional date (a.d. 144) of Kanaksen’s arrival in Káthiáváḍa had been only a little earlier it would have fitted well with Wilson’s Kanishka of the Rája Tarangini.’ Information brought to light since Tod’s time shows that hardly any date could fit better than a.d. 144 for some member of the Kushán family, possibly a grandson of the great Kanishka, to make a settlement in Gujarát and Káthiáváḍa. The date agrees closely with the revolt against Vasudeva (a.d. 123–150), the second in succession from Kanishka, raised by the Panjáb Yaudheyas, whom the great Gujarát Kshatrapa Rudradáman (a.d. 143–158), the introducer of Kanishka’s (a.d. 78) era into Gujarát, humbled. The tradition calls Kanaksen Kośalaputra and brings him from Lohkot in North India.81 Kośala has been explained as Oudh and Lohkot as Lahore, but as Kanak came from the north not from the north-east an original Kushána-putra or Son of the Kushán may be the true form. Similarly Lohkot cannot be Lahore. It may be Alberuni’s Lauhavar or Lahur in the Káshmir uplands one of the main centres of Kushán power.82 Chapter VIII. The Valabhis, a.d. 509–766. Mewáḍ and the Persians. Mewáḍ and the Persians.One further point requires notice, the traditional connection between Valabhi and the Ránás of Mewáḍ with the Sassanian kings of Persia (a.d. 250–650). In support of the tradition Abul Fazl (a.d. 1590) says the Ránás of Mewáḍ consider themselves descendants of the Sassanian Naushirván (a.d. 531–579) and Tod quotes fuller details from the Persian history Maaser-al-Umra.83 No evidence seems to support a direct connection with Naushirván.84 At the same time marriage between the Valabhi chief and Maha Banu the fugitive daughter of Yezdigerd the last Sassanian (a.d. 651) is not impossible.85 And the remaining suggestion that the link may be Naushirván’s son Naushizád who fled from his father in a.d. 570 receives support in the statement of Procopius86 that Naushizád found shelter at Belapatan in Khuzistán perhaps
- 62. Balapatan in Gurjaristán. As these suggestions are unsupported by direct evidence, it seems best to look for the source of the legend in the fire symbols in use on Káthiáváḍa and Mewáḍ coins. These fire symbols, though in the main Indo-Skythian, betray from about the sixth century a more direct Sassanian influence. The use of similar coins coupled with their common sun worship seems sufficient to explain how the Agnikulas and other Káthiáváḍa and Mewáḍ Rájputs came to believe in some family connection between their chiefs and the fireworshipping kings of Persia.87 Válas.Can the Vála traditions of previous northern settlements be supported either by early Hindu inscriptions or from living traces in the present population of Northern India? The convenient and elaborate tribe and surname lists in the Census Report of the Panjáb, and vaguer information from Rájputána, show traces of Bálas and Válas among the Musalmán as well as among the Hindu population of Northern India.88 Among the tribes mentioned in Varáha-Mihira’s sixth century (a.d. 580)89 lists the Váhlikas appear along with the dwellers on Sindhu’s banks. An inscription of a king Chandra, probably Chandragupta and if so about a.d. 380–400,90 boasts of crossing the seven mouths of the Indus to attack the Váhlikas. These references suggest that the Bálas or Válas are the Válhikas and that the Bálhikas of the Harivaṃśa (a.d. 350–500 ?) are not as Langlois supposed people then ruling Chapter VIII. The Valabhis, a.d. 509–766. Válas. in Balkh but people then established in India.91 Does it follow that the Válhikas of the inscriptions and the Bálhikas of the Harivaṃśa are the Panjáb tribe referred to in the Mahábhárata as the Báhikas or Bálhikas, a people held to scorn as keeping no Bráhman rites, their Bráhmans degraded, their women abandoned?92 Of the two Mahábhárata forms Báhika and Bálhika recent scholars have preferred Bálhika with the sense of people of Balkh or Baktria.93 The name Bálhika might belong to more than one of the Central Asian invaders of Northern India during the centuries before and after Christ, whose manner of life might be expected to strike an Áryávarta Bráhman with horror. The date of the settlement of these northern tribes (b.c. 180–a.d. 300) does not conflict with the comparatively modern date
- 63. (a.d. 150–250) now generally received for the final revision of the Mahábhárata.94 This explanation does not remove the difficulty caused by references to Báhikas and Bálhikas95 in Páṇini and other writers earlier than the first of the after-Alexander Skythian invasions. At the same time as shown in the footnote there seems reason to hold that the change from the Bákhtri of Darius (b.c. 510) and Alexander the Great (b.c. 330) to the modern Balkh did not take place before the first century after Christ. If this view is correct it follows that Chapter VIII. The Valabhis, a.d. 509–766. Válas. if the form Bahlika occurs in Páṇini or other earlier writers it is a mistaken form due to some copyist’s confusion with the later name Bahlika. As used by Páṇini the name Báhika applied to certain Panjáb tribes seems a general term meaning Outsider a view which is supported by Brian Hodgson’s identification of the Mahábhárata Báhikas with the Bahings one of the outcaste or broken tribes of Nepál.97 The use of Báhika in the Mahábhárata would then be due either to the wish to identify new tribes with old or to the temptation to use a word which had a suitable meaning in Sanskrit. If then there is fair ground for holding that the correct form of the name in the Mahábhárata is Bálhika and that Bálhika means men of Balkh the question remains which of the different waves of Central Asian invaders in the centuries before and after Christ are most likely to have adopted or to have received the title of Baktrians. Between the second century before and the third century after Christ two sets of northerners might justly have claimed or have received the title of Baktrians. These northerners are the Baktrian Greeks about b.c. 180 and the Yuechi between b.c. 20 and a.d. 300. Yavana is so favourite a name among Indian writers that it may be accepted that whatever other northern tribes the name Yavana includes no name but Yavana passed into use for the Baktrian Greeks. Their long peaceful and civilised rule (b.c. 130–a.d. 300 ?) from their capital at Balkh entitles the Yuechi to the name Baktrians or Báhlikas. That the Yuechi were known in India as Baktrians is proved by the writer of the Periplus (a.d. 247), who, when Baktria was still under Yuechi rule, speaks of the Baktrianoi as a most warlike race governed by their own sovereign.98 It is known that in certain cases the Yuechi tribal names were of local origin.
- 64. Kushán the name of the leading tribe is according to some authorities a place-name.99 Chapter VIII. The Valabhis, a.d. 509–766. Válas. And it is established that the names of more than one of the tribes who about b.c. 50 joined under the head of the Kusháns were taken from the lands where they had settled. It is therefore in agreement both with the movements and with the practice of the Yuechi, that, on reaching India, a portion of them should be known as Báhlikas or Bálhikas. Though the evidence falls short of proof there seems fair reason to suggest that the present Rájput and Káthi Válas or Bálas of Gujarát and Rájputána, through a Sanskritised Váhlika, may be traced to some section of the Yuechi, who, as they passed south from Baktria, between the first century before and the fourth century after Christ, assumed or received the title of men of Balkh. One collateral point seems to deserve notice. St. Martin100 says: ‘The Greek historians do not show the least trace of the name Báhlika.’ Accepting Báhika, with the general sense of Outsider, as the form used by Indian writers before the Christian era and remembering101 Páṇini’s description of the Málavas and Kshudrakas as two Báhika tribes of the North-West the fact that Páṇini lived very shortly before or after the time of Alexander and was specially acquainted with the Panjáb leaves little doubt that when (a.d. 326) Alexander conquered their country the Malloi and Oxydrakai, that is the Málavas and Kshudrakas, were known as Báhikas. Seeing that Alexander’s writers were specially interested in and acquainted with the Malloi and Oxydrakai it is strange if St. Martin is correct in stating that Greek writings show no trace of the name Báhika. In explanation of this difficulty the following suggestion may be offered.102 As the Greeks sounded their kh (χ) as a spirant, the Indian Báhika would strike them as almost the exact equivalent of their own word βακχικος. More than one of Alexander’s writers has curious references to a Bacchic element in the Panjáb tribes. Arrian103 notices that, as Alexander’s fleet passed down the Jhelum, the people lined the banks chanting songs taught them by Dionysus and the Bacchantes. According to Quintus Curtius104 the name of Father Bacchus was famous among the people to the south of the Malloi. These
- 65. references are vague. But Strabo is definite.105 The Malloi and Oxydrakai are reported to be the descendants of Bacchus. This passage is the more important since Strabo’s use of the writings of Aristobulus Alexander’s historian and of Onesikritos Alexander’s pilot and Bráhman-interviewer gives his details a special value.106 It may be said Strabo explains why the Malloi and Oxydrakai were called Bacchic and Strabo’s explanation is not in agreement with the proposed Báhika origin. The answer is that Strabo’s explanation can be proved to be in part, if not altogether, fictitious. Strabo107 gives two reasons why the Oxydrakai Chapter VIII. The Valabhis, a.d. 509–766. Válas. were called Bacchic. First because the vine grew among them and second because their kings marched forth Bakkhikôs that is after the Bacchic manner. It is difficult to prove that in the time of Alexander the vine did not grow in the Panjáb. Still the fact that the vines of Nysa near Jalálábád and of the hill Meros are mentioned by several writers and that no vines are referred to in the Greek accounts of the Panjáb suggests that the vine theory is an after-thought.108 Strabo’s second explanation, the Bacchic pomp of their kings, can be more completely disproved. The evidence that neither the Malloi nor the Oxydrakai had a king is abundant.109 That the Greeks knew the Malloi and Oxydrakai were called Bakkhikoi and that they did not know why they had received that name favours the view that the explanation lies in the Indian name Báhika. One point remains. Does any trace of the original Báhikas or Outsiders survive? In Cutch Káthiáváḍa and North Gujarát are two tribes of half settled cattle-breeders and shepherds whose names Rahbáris as if Rahábaher and Bharváds as if Baherváda seem like Báhika to mean Outsider. Though in other respects both classes appear to have adopted ordinary Hindu practices the conduct of the Bharvád women of Káthiáváḍa during their special marriage seasons bears a curiously close resemblance to certain of the details in the Mahábhárata account of the Báhika women. Colonel Barton writes:110 ‘The great marriage festival of the Káthiáváḍa Bharváds which is held once in ten or twelve years is called the Milkdrinking, Dudhpíno, from the lavish use of milk or clarified butter. Under the exciting influence of the butter the women become frantic singing obscene songs breaking down hedges and
- 66. 1 2 3 spoiling the surrounding crops.’ Though the Bharváds are so long settled in Káthiáváḍa as to be considered aboriginals their own tradition preserves the memory of a former settlement in Márwár.111 This tradition is supported by the fact that the shrine of the family goddess of the Cutch Rabáris is in Jodhpur,112 and by the claim of the Cutch Bharváds that their home is in the North-West Provinces.113 Mr. Vajeshankar Gavrishankar, Náib Diván of Bhávnagar, has made a collection of articles found in Valabhi. The collection includes clay seals of four varieties and of about the seventh century with the Buddhist formula Ye Dhárma hetu Prabhavá: a small earthen tope with the same formula imprinted on its base with a seal; beads and ring stones nangs of several varieties of akik or carnelian and sphatik or coral some finished others half finished showing that as in modern Cambay the polishing of carnelians was a leading industry in early Valabhi. One circular figure of the size of a half rupee carved in black stone has engraved upon it the letters ma ro in characters of about the second century.2 A royal seal found by Colonel Watson in Vaḷeh bears on it an imperfect inscription of four lines in characters as old as Dhruvasena I. (a.d. 526). This seal contains the names of three generations of kings, two of which the grandfather and grandson read Ahivarmman and Pushyáṇa all three being called Mahárája or great king. The dynastic name is lost. The names on these moveable objects need not belong to Valabhi history. Still that seals of the second and fifth centuries have been discovered in Valabhi shows the place was in existence before the founding of the historical Valabhi kingdom. A further proof of the age of the city is the mention of it in the Kathásarit-ságara a comparatively modern work but of very old materials. To this evidence of age, with much hesitation, may be added Balai Ptolemy’s name for Gopnáth point which suggests that as early as the second century Vaḷeh or Baleh (compare Alberuni’s era of Balah) was known by its present name. Badly minted coins of the Gupta ruler Kumáragupta (a.d. 417–453) are so common as to suggest that they were the currency of Valabhi. ↑ The ma and ra are of the old style and the side and upper strokes, that is the káno and mátra of ro are horizontal. ↑ As suggested by Dr. Bühler (Ind. Ant. VI. 10), this is probably the Vihára called Śrí Bappapádiyavihára which is described as having been constructed by Áchárya Bhadanta Sthiramati who is mentioned as the grantee in a copperplate of Dharasena II. bearing date
- 67. 4 5 6 7 8 9 10 11 12 Gupta 269 (a.d. 588). The Sthiramati mentioned with titles of religious veneration in the copperplate is probably the same as that referred to by Hiuen Tsiang. (Ditto). ↑ Burgess’ Káthiáwár and Kutch, 187. ↑ Stories on record about two temples one at Śatruñjaya the other at Somanátha support this view. As regards the Śatruñjaya temple the tradition is that while the minister of Kumárapála (a.d. 1143–1174) of Aṇahilaváḍa was on a visit to Śatruñjaya to worship and meditate in the temple of Ádinátha, the wick of the lamp in the shrine was removed by mice and set on fire and almost destroyed the temple which was wholly of wood. The minister seeing the danger of wooden buildings determined to erect a stone edifice (Kumárapála Charita). The story about Somanátha is given in an inscription of the time of Kumárapála in the temple of Bhadrakáli which shows that before the stone temple was built by Bhímadeva I. (a.d. 1022–1072) the structure was of wood which was traditionally believed to be as old as the time of Kṛishṇa. Compare the Bhadrakáli inscription at Somanátha. ↑ The correctness of this inference seems open to question. The descent of the Valabhi plate character seems traceable from its natural local source the Skandagupta (a.d. 450) and the Rudradáman (a.d. 150) Girnár Inscriptions.—(A. M. T. J.) ↑ The era has been exhaustively discussed by Mr. Fleet in Corp. Ins. Ind. III. Introduction. ↑ Nepaul Inscriptions. The phrase acháṭa-bhaṭa is not uncommon. Mr. Fleet (Corp. Ins. Ind. III. page 98 note 2) explains acháṭa-bhaṭa-praveśya as “not to be entered either by regular (bhaṭa) or by irregular (cháṭa) troops.” ↑ Bühler in Ind. Ant. V. 205. ↑ Ind. Ant. VII. 68. ↑ Ind. Ant. VII. 68. ↑ Of the different territorial divisions the following examples occur: Of Vishaya or main division Svabhágapuravishaye and Súryapuravishaye: of Áhára or collectorate Kheṭaka- áhára the Kaira district and Hastavapra-áhára or Hastavapráharaṇí the Háthab district near Bhávnagar: of Pathaka or sub-division Nagar-panthaka Porbandar-panthaka (Pársis still talk of Navsári panthaka): of Sthali or petty division Vaṭasthalí, Loṇápadrakasthalí, and others. ↑
- 68. 13 14 Kárván seems to have suffered great desecration at the hands of the Musalmáns. All round the village chiefly under pipal trees, images and pieces of sculpture and large liṅgas lie scattered. To the north and east of the village on the banks of a large built pond called Káśíkuṇḍa are numerous sculptures and liṅgas. Partly embedded in the ground a pillar in style of about the eleventh century has a writing over it of latter times. The inscription contains the name of the place Sanskritised as Káyávarohana, and mentions an ascetic named Vírabahadraráśi who remained mute for twelve years. Near the pillar, at the steps leading to the water, is a carved doorway of about the tenth or eleventh century with some well-proportioned figures. The left doorpost has at the top a figure of Śiva, below the Śiva a figure of Súrya, below the Súrya a male and female, and under them attendants or gaṇas of Śiva. The right doorpost has at the top a figure of Vishṇu seated on Garuḍa, below the seated Vishṇu a standing Vishṇu with four hands, and below that two sitting male and female figures, the male with hands folded in worship the female holding a purse. These figures probably represent a married pair who paid for this gateway. Further below are figures of gaṇas of Śiva. In 1884 in repairing the south bank of the pond a number of carved stones were brought from the north of the town. About half a mile north-west of the town on the bank of a dry brook, is a temple of Chámundá Deví of about the tenth century. It contains a mutilated life-size image of Chámundá. Facing the temple lie mutilated figures of the seven Mátrikás and of Bhairava, probably the remains of a separate altar facing the temple with the mátri-maṇḍala or Mother-Meeting upon it. The village has a large modern temple of Śiva called Nakleśvara, on the site of some old temple and mostly built of old carved temple stones. In the temple close by are a number of old images of the sun and the boar incarnation of Vishṇu all of about the tenth or eleventh century. The name Nakleśvara would seem to have been derived from Nakuliśa the founder of the Páśupata sect and the temple may originally have had an image of Nakuliśa himself or a liṅga representing Nakulíśa. Close to the west of the village near a small dry reservoir called the Kuṇḍa of Rájarájeśvara lies a well-preserved black stone seated figure of Chaṇḍa one of the most respected of Śiva’s attendants, without whose worship all worship of Śiva is imperfect, and to whom all that remains after making oblations to Śiva is offered. A number of other sculptures lie on the bank of the pond. About a mile to the south of Kárván is a village called Lingthali the place of liṅgas. ↑ Compare Beal Buddhist Records, II. 268 note 76 and Ind. Ant. VI. 9. The meaning and reference of the title Bappa have been much discussed. The question is treated at length by Mr. Fleet (Corp. Ins. Ind. III. 186 note 1) with the result that the title is applied not to a religious teacher but to the father and predecessor of the king who makes the grant.
- 69. 15 16 17 18 19 20 According to Mr. Fleet bappa would be used in reference to a father, báva in reference to an uncle. ↑ Whether the Valabhis were or were not Gurjjaras the following facts favour the view that they entered Gujarát from Málwa. It has been shown (Fleet Ind. Ant. XX. 376) that while the Guptas used the so-called Northern year beginning with Chaitra, the Valabhi year began with Kártika (see Ind. Ant. XX. 376). And further Kielhorn in his examination of questions connected with the Vikrama era (Ind. Ant. XIX. and XX.) has given reasons for believing that the original Vikrama year began with Kártika and took its rise in Málwa. It seems therefore that when they settled in Gujarát, while they adopted the Gupta era the Valabhis still adhered to the old arrangement of the year to which they had been accustomed in their home in Málwa. The arrangement of the year entered into every detail of their lives, and was therefore much more difficult to change than the starting point of their era, which was important only for official acts.—(A. M. T. J.) ↑ Montfauçon’s Edition in Priaulx’s Indian Travels, 222–223. It seems doubtful if Cosmas meant that Gollas’ overlordship spread as far south as Kalyán. Compare Migne’s Patrologiæ Cursus, lxxxviii. 466; Yule’s Cathay, I. clxx. ↑ The Mehrs seem to have remained in power also in north-east Káthiáváḍa till the thirteenth century. Mokheráji Gohil the famous chief of Piram was the son of a daughter of Dhan Mehr or Mair of Dhanduka, Rás Mála, I. 316. ↑ All the silver and copper coins found in Valabhi and in the neighbouring town of Sihor are poor imitations of Kumáragupta’s (a.d. 417–453) and of Skandagupta’s (a.d. 454–470) coins, smaller lighter and of bad almost rude workmanship. The only traces of an independent currency are two copper coins of Dharasena, apparently Dharasena IV., the most powerful of the dynasty who was called Chakravartin or Emperor. The question of the Gupta-Valabhi coins is discussed in Jour. Royal As. Socy. for Jan. 1893 pages 133– 143. Dr. Bühler (page 138) holds the view put forward in this note of Dr. Bhagvánlál’s namely that the coins are Valabhi copies of Gupta currency. Mr. Smith (Ditto, 142–143) thinks they should be considered the coins of the kings whose names they bear. ↑ The three types of coins still current at Ujjain, Bhilsa, and Gwálior in the territories of His Highness Sindhia are imitations of the previous local Muhammadan coinage. ↑ As the date of Droṇasiṃha’s investiture is about a.d. 520 it is necessary to consider what kings at this period claimed the title of supreme lord and could boast of ruling the whole earth. The rulers of this period whom we know of are Mihirakula, Yaśodharman
- 70. 21 22 23 24 25 26 27 28 29 30 31 Vishṇuvardhana, the descendants of Kumáragupta’s son Puragupta, and the Gupta chiefs of Eastern Málwa. Neither Toramáṇa nor Mihirakula appears to have borne the paramount title of Parameśvara though the former is called Mahárájádhirája in the Eraṇ inscription and Avanipati or Lord of the Earth (= simply king) on his coins: in the Gwálior inscription Mihirakula is simply called Lord of the Earth. He was a powerful prince but he could hardly claim to be ruler of “the whole circumference of the earth.” He therefore cannot be the installer of Droṇasiṃha. Taking next the Guptas of Magadha we find on the Bhitári seal the title of Mahárájádhirája given to each of them, but there is considerable reason to believe that their power had long since shrunk to Magadha and Eastern Málwa, and if Hiuen Tsiang’s Báláditya is Narasiṃhagupta, he must have been about a.d. 520 a feudatory of Mihirakula, and could not be spoken of as supreme lord, nor as ruler of the whole earth. The Guptas of Málwa have even less claim to these titles, as Bhánugupta was a mere Mahárája, and all that is known of him is that he won a battle at Eraṇ in Eastern Málwa in a.d. 510–11. Last of all comes Vishṇuvardhana or Yaśodharman of Mandasor. In one of the Mandasor inscriptions he has the titles of Rájádhirája and Parameśvara (a.d. 532–33); in another he boasts of having carried his conquests from the Lauhitya (Brahmaputra) to the western ocean and from the Himálaya to mount Mahendra. It seems obvious that Yaśodharman is the Paramasvámi of the Valabhi plate, and that the reference to the western ocean relates to Bhaṭárka’s successes against the Maitrakas.—(A.M.T.J.) ↑ Ind. Ant. V. 204. ↑ Ind. Ant. IV. 104. ↑ In a commentary on the Kalpasútra Daṇḍanáyaka is described as meaning Tantrapâla that is head of a district. ↑ Ind. Ant. VII. 66; IV. 174. ↑ Ind. Ant. V. 206. ↑ Ind. Ant. XIV. 75. ↑ Kumárápála-Charita, Abu Inscriptions. ↑ Ind. Ant. VIII. 302, VII. 68, XIII. 160. ↑ Ind. Ant. VI. 9. ↑ Ind. Ant. VII. 90. ↑ This change of title was probably connected with the increase of Gurjara power, which resulted in the founding of the Gurjara kingdom of Broach about a.d. 580. See
- 71. 32 33 34 35 36 37 38 Chapter X. below. ↑ Ind. Ant. XI. 306. ↑ Ind. Ant. VI. 13. ↑ Kávyamidam rachitam mayá Valabhyám, Śrí Dharasena-narendra pálitáyám. ↑ Ind. Ant. VII. 76. ↑ Journ. Beng. A. S. IV. and an unpublished grant in the museum of the B. B. R. A. Soc. ↑ Ind. Ant. XI. 305. ↑ Since his authorities mention the destroyers of Valabhi under the vague term mlechchhas or barbarians and since the era in which they date the overthrow may be either the Vikrama b.c. 57, the Śaka a.d. 78, or the Valabhi a.d. 319, Tod is forced to offer many suggestions. His proposed dates are a.d. 244 Vik. Saṃ. 300 (Western India, 269), a.d. 424 Val. Saṃ. 105 (Ditto, 51 and 214), a.d. 524 Val. Saṃ. 205 (Annals of Rájasthán, I. 83 and 217–220), and a.d. 619 Val. Saṃ. 300 (Western India, 352). Tod identifies the barbarian destroyers of Valabhi either with the descendants of the second century Parthians, or with the White Huns Getes or Káthis, or with a mixture of these who in the beginning of the sixth century supplanted the Parthians (An. of Ráj. I. 83 and 217–220; Western India, 214, 352). Elliot (History, I. 408) accepting Tod’s date a.d. 524 refers the overthrow to Skythian barbarians from Sindh. Elphinstone, also accepting a.d. 524 as an approximate date, suggested (History, 3rd Edition, 212) as the destroyer the Sassanian Naushirván or Chosroes the Great (a.d. 531–579) citing in support of a Sassanian inroad Malcolm’s Persia, I. 141 and Pottinger’s Travels, 386. Forbes (Rás Málá, I. 22) notes that the Jain accounts give the date of the overthrow Vik. Saṃ. 375 that is a.d. 319 apparently in confusion with the epoch of the Gupta era which the Valabhi kings adopted.39 Forbes says (Ditto, 24): If the destroyers had not been called mlechchhas I might have supposed them to be the Dakhan Chálukyas. Genl. Cunningham (Anc. Geog. 318) holds that the date of the destruction was a.d. 658 and the destroyer the Ráshṭrakúṭa Rája Govind who restored the ancient family of Sauráshṭra. Thomas (Prinsep’s Useful Tables, 158) fixes the destruction of Valabhi at a.d. 745 (S. 802). In the Káthiáwár Gazetteer Col. Watson in one passage (page 671) says the destroyers may have been the early Muhammadans who retired as quickly as they came. In another passage (page 274), accepting Mr. Burgess’ (Arch. Sur. Rep. IV. 75) Gupta era of a.d. 195 and an overthrow date of a.d. 642, and citing a Wadhwán couplet telling how Ebhal Valabhi withstood the Iranians, Col. Watson
- 72. 39 40 41 42 suggests the destroyers may have been Iranians. If the Pársis came in a.d. 642 they must have come not as raiders but as refugees. If they could they would not have destroyed Valabhi. If the Pársis destroyed Valabhi where next did they flee to. ↑ Similarly S. 205 the date given by some of Col. Tod’s authorities (An. of Ráj. I. 82 and 217–220) represents a.d. 524 the practical establishment of the Valabhi dynasty. The mistake of ascribing an era to the overthrow not to the founding of a state occurs (compare Sachau’s Alberuni, II. 6) in the case both of the Vikrama era b.c. 57 and of the Śáliváhana era a.d. 78. In both these cases the error was intentional. It was devised with the aim of hiding the supremacy of foreigners in early Hindu history. So also, according to Alberuni’s information (Sachau, II. 7) the Guptakála a.d. 319 marks the ceasing not the beginning of the wicked and powerful Guptas. This device is not confined to India. His Mede informant told Herodotus (b.c. 450 Rawlinson’s Herodotus, I. 407) that b.c. 708 was the founding of the Median monarchy. The date really marked the overthrow of the Medes by the Assyrian Sargon. ↑ Tod (An. of Ráj. I. 231) notices what is perhaps a reminiscence of this date (a.d. 766). It is the story that Bappa, who according to Mewáḍ tradition is the founder of Gehlot power at Chitor, abandoned his country for Irán in a.d. 764 (S. 820). It seems probable that this Bappa or Saila is not the founder of Gehlot power at Chitor, but, according to the Valabhi use of Bappa, is the founder’s father and that this retreat to Irán refers to his being carried captive to Mansúra on the fall either of Valabhi or of Gandhár. ↑ Reinaud’s Fragments, 143 note 1; Mémoire Sur l’Inde, 105; Sachau’s Alberuni, I. 193. The treachery of the magician Ranka is the same cause as that assigned by Forbes (Rás Málá, I. 12–18) from Jain sources. The local legend (Ditto, 18) points the inevitable Tower of Siloam moral, a moral which (compare Rás Málá, I. 18) is probably at the root of the antique tale of Lot and the Cities of the Plain, that men whose city was so completely destroyed must have been sinners beyond others. Dr. Nicholson (J. R. A. S. Ser. I. Vol. XIII. page 153) in 1851 thought the site of Valabhi bore many traces of destruction by water. ↑ Lassen (Ind. Alt. III. 533) puts aside Alberuni’s Arab expedition from Mansúra as without historical support and inadmissible. Lassen held that Valabhi flourished long after its alleged destruction from Mansúra. Lassen’s statement (see Ind. Alt. III. 533) is based on the mistaken idea that as the Valabhis were the Balharas the Balharas’ capital Mánkir must be Valabhi. So far as is known, except Alberuni himself (see below) none of the Arab geographers of the ninth, tenth or eleventh centuries mentions Valabhi. It is true that
- 73. 43 44 45 46 according to Lassen (Ind. Alt. 536) Masudi a.d. 915, Istakhri a.d. 951, and Ibn Háukal a.d. 976 all attest the existence of Valabhi up to their own time. This remark is due either to the mistake regarding Malkhet or to the identification of Bálwi or Balzi in Sindh (Elliot’s History, I. 27–34) with Valabhi. The only known Musalmán reference to Valabhi later than a.d. 750 is Alberuni’s statement (Sachau, II. 7) that the Valabhi of the era is 30 yojanas or 200 miles south of Aṇahilaváḍa. That after its overthrow Valabhi remained, as it still continues, a local town has been shown in the text. Such an after-life is in no way inconsistent with its destruction as a leading capital in a.d. 767. ↑ According to Alberuni (Sachau, I. 21) Al Mansúra, which was close to Bráhmanábád about 47 miles north-east of Haidarábád (Elliot’s Musalmán Historians, I. 372–374) was built by the great Muhammad Kásim about a.d. 713. Apparently Alberuni wrote Muhammad Kásim by mistake for his grandson Amru Muhammad (Elliot, I. 372 note 1 and 442–3), who built the city a little before a.d. 750. Reinaud (Fragments, 210) makes Amru the son of Muhammad Kásim. Masudi (a.d. 915) gives the same date (a.d. 750), but (Elliot, I. 24) makes the builder the Ummayide governor Mansúr bin Jamhur. Idrísi (a.d. 1137 Elliot, I. 78) says Mansúra was built and named in honour of the Khalif Abu Jáfar-al-Mansur. If so its building would be later than a.d. 754. On such a point Idrísi’s authority carries little weight. ↑ Elliot, I. 244. ↑ That the word read Barada by Elliot is in the lax pointless shikasta writing is shown by the different proposed readings (Elliot, I. 444 note 1) Nárand, Barand, and Barid. So far as the original goes Balaba is probably as likely a rendering as Barada. Reinaud (Fragments, 212) says he cannot restore the name. ↑ Though, except as applied to the Porbandar range of hills, the name Barada is almost unknown, and though Ghumli not Barada was the early (eighth-twelfth century) capital of Porbandar some place named Barada seems to have existed on the Porbandar coast. As early as the second century a.d., Ptolemy (McCrindle, 37) has a town Barda-xema on the coast west of the village Kome (probably the road or kom) of Sauráshṭra; and St. Martin (Geographie Grecque et Latine de l’Inde, 203) identifies Pliny’s (a.d. 77) Varetatæ next the Odomberæ or people of Kachh with the Varadas according to Hemachandra (a.d. 1150) a class of foreigners or mlechchhas. A somewhat tempting identification of Barada is with Beruni’s Bárwi (Sachau, I. 208) or Baraoua (Reinaud’s Fragments, 121) 84 miles (14 parasangs) west of Somanátha. But an examination of Beruni’s text shows that Bárwi is not the name of a place but of a product of Kachh the bára or bezoar stone. ↑
- 74. 47 48 49 50 51 Elliot, I. 445. ↑ Compare Tod (Annals, I. 83 and 217). Gajni or Gayni another capital whence the last prince Śíláditya was expelled by Parthian invaders in the sixth century. ↑ Compare Reinaud (Fragments, 212 note 4) who identifies it with the Áin-i-Akbari Kandahár that is Gandhár in Broach. The identification is doubtful. Tod (Annals, I. 217) names the fort Gajni or Gayni and there was a fort Gajni close to Cambay. Elliot (I. 445) would identify the Arab Kandahár with Khandadár in north-west Káthiáváḍa. Even after a.d. 770 Valabhi seems to have been attacked by the Arabs. Dr. Bhagvánlál notices that two Jain dates for the destruction of the city 826 and 886 are in the Vira era and that this means not the Mahávira era of b.c. 526 but the Vikram era of b.c. 57. The corresponding dates are therefore a.d. 769 and 829. Evidence in support of the a.d. 769 and 770 defeat is given in the text. On behalf of Dr. Bhagvánlál’s second date a.d. 829 it is remarkable that in or about a.d. 830 (Elliot, I. 447) Músa the Arab governor of Sindh captured Bála the ruler of As Sharqi. As there seems no reason to identify this As Sharqi with the Sindh lake of As Sharqi mentioned in a raid in a.d. 750 (Elliot, I. 441: J. R. A. S. (1893) page 76) the phrase would mean Bála king of the east. The Arab record of the defeat of Bála would thus be in close agreement with the Jain date for the latest foreign attack on Valabhi. ↑ The identification of the Balharas of the Arab writers with the Chálukyas (a.d. 500– 753) and Ráshṭrakúṭas (a.d. 753–972) of Málkhet in the East Dakhan has been accepted. The vagueness of the early (a.d. 850–900) Arab geographers still more the inaccuracy of Idrísi (a.d. 1137) in placing the Balharas capital in Gujarát (Elliot, I. 87) suggested a connection between Balhara and Valabhi. The suitableness of this identification was increased by the use among Rájput writers of the title Balakarai for the Valabhi chief (Tod An. of Ráj. I. 83) and the absence among either the Chálukyas (a.d. 500–753) or the Ráshṭrakúṭas (a.d. 753–972) of Málkhet of any title resembling Balhara. Prof. Bhandárkar’s (Deccan History, 56–57) discovery that several of the early Chálukyas and Ráshṭrakúṭas had the personal name Vallabha Beloved settled the question and established the accuracy of all Masudi’s (a.d. 915) statements (Elliot, I. 19–21) regarding the Balhara who ruled the Kamkar, that is Kamrakara or Karnáṭak (Sachau’s Beruni, I. 202; II. 318) and had their Kánarese (Kiriya) capital at Mankir (Málkhet) 640 miles from the coast. ↑ After their withdrawal from Valabhi to Mewáḍ the Válas took the name of Gehlot (see below page 98), then of Aharya from a temporary capital near Udepur (Tod’s An. of Ráj. I. 215), next of Sesodia in the west of Mewáḍ (Tod’s An. of Raj. I. 216; Western India, 57).
- 75. 52 53 54 55 56 57 58 Since 1568 the Rána’s head-quarters have been at Udepur. Ráj. Gaz. III. 18. After the establishment of their power in Chitor (a.d. 780), a branch of the Gehlot or Gohil family withdrew to Kheir in south-west Márwár. These driven south by the Ráthoḍs in the end of the twelfth century are the Gohils of Piram, Bhávnagar, and Rájpipla in Káthiáváḍa and Gujarát. Tod’s Annals of Ráj. I. 114, 228. ↑ The somewhat doubtful Jáikadeva plates (above page 87 and Káthiáváḍa Gazetteer, 275) seem to show the continuance of Maitraka power in North Káthiáváḍa. This is supported by the expedition of the Arab chief of Sandhán in Kachch (a.d. 840) against the Medhs of Hind which ended in the capture of Mália in North Káthiáváḍa. Elliot, I. 450. Hiuen Tsiang (a.d. 630) (Beal’s Buddhist Records, II. 69) describes Sauráshṭra as a separate state but at the same time notes its dependence on Valabhi. Its rulers seem to have been Mehrs. In a.d. 713 (Elliot, I. 123) Muhammad Kasim made peace with the men of Surasht, Medhs, seafarers, and pirates. ↑ The only contemporary rulers in whose grants a reference to Valabhi has been traced are the Gurjjaras of Broach (a.d. 580–808) one of whom, Dadda II. (a.d. 633), is said (Ind. Ant. XIII. 79) to have gained renown by protecting the lord of Valabhi who had been defeated by the illustrious Śrí Harshadeva (a.d. 608–649), and another Jayabhaṭa in a.d. 706 (Ind. Ant. V. 115) claims to have quieted with the sword the impetuosity of the lord of Valabhi. ↑ Tod An. of Raj. I. 217: Western India, 269. ↑ Tod An. of Raj. I. 112 and Western India, 148: Rás Málá, I. 21. It is not clear whether these passages prove that the Sesodias or only the Válas claim an early settlement at Dhánk. In any case (see below page 101) both clans trace their origin to Kanaksen. ↑ Tod’s Western India, 51. ↑ Tod’s An. of Raj. I. 230. ↑ The cherished title of the later Valabhis, Śíláditya Sun of Virtue, confirms the special sun worship at Valabhi, which the mention of Dharapaṭṭa (a.d. 550) as a devotee of the supreme sun supports, and which the legends of Valabhi’s sun-horse and sun-fountain keep fresh (Rás Málá, I. 14–18). So the great one-stone liṅgas, the most notable trace of Valabhi city (J. R. A. S. Ser. I. Vol. XIII. 149 and XVII. 271), bear out the Valabhi copperplate claim that its rulers were great worshippers of Śiva. Similarly the Rána of Udepur, while enjoying the title of Sun of the Hindus, prospering under the sun banner, and specially worshipping the sun (Tod’s Annals, I. 565) is at the same time the Minister of Śiva the One
- 76. 59 60 61 62 63 Liṅg Eklingakadiwán (Ditto 222, Ráj. Gaz. III. 53). The blend is natural. The fierce noon- tide sun is Mahákála the Destroyer. Like Śiva the Sun is lord of the Moon. And marshalled by Somanátha the great Soul Home the souls of the dead pass heavenwards along the rays of the setting sun. [Compare Sachau’s Alberuni, II. 168.] It is the common sun element in Śaivism and in Vaishnavism that gives their holiness to the sunset shrines of Somanátha and Dwárka. For (Ditto, 169) the setting sun is the door whence men march forth into the world of existence Westwards, heavenwards. ↑ This explanation is hardly satisfactory. The name Gehlot seems to be Guhila-putra from Gobhila-putra an ancient Bráhman gotra, one of the not uncommon cases of Rájputs with a Bráhman gotra. The Rájput use of a Bráhman gotra is generally considered a technical affiliation, a mark of respect for some Bráhman teacher. It seems doubtful whether the practice is not a reminiscence of an ancestral Bráhman strain. This view finds confirmation in the Aitpur inscription (Tod’s Annals, I. 802) which states that Guhadit the founder of the Gohil tribe was of Bráhman race Vipra kula. Compare the legend (Rás Málá, I. 13) that makes the first Śíláditya of Valabhi (a.d. 590–609) the son of a Bráhman woman. Compare (Elliot, I. 411) the Bráhman Chách (a.d. 630–670) marrying the widow of the Sháhi king of Alor in Sindh who is written of as a Rájput though like the later (a.d. 850–1060) Shahiyas of Kábul (Alberuni, Sachau II. 13) the dynasty may possibly have been Bráhmans.60 The following passage from Hodgson’s Essays (J. A. Soc. Bl. II. 218) throws light on the subject: Among the Khás or Rájputs of Nepál the sons of Bráhmans by Khás women take their fathers’ gotras. Compare Ibbetson’s Panjáb Census 1881 page 236. ↑ In support of a Bráhman origin is Prinsep’s conjecture (J. A. S. Bl. LXXIV. [Feb. 1838] page 93) that Divaij the name of the first recorded king may be Dvija or Twice-born. But Divaij for Deváditya, like Silaij for Śíláditya, seems simpler and the care with which the writer speaks of Chach as the Bráhman almost implies that his predecessors were not Bráhmans. According to Elliot (II. 426) the Páls of Kábul were Rájputs, perhaps Bhattias. ↑ Tod’s Annals, I. 229–231. ↑ Annals, I. 229. ↑ Gladwin’s Áin-i-Akbari, II. 81; Tod’s Annals, I. 235 and note *. Tod’s dates are confused. The Aitpur inscription (Ditto, page 230) gives Śakti Kumára’s date a.d. 968 (S. 1024) while the authorities which Tod accepts (Ditto, 231) give a.d. 1068 (S. 1125). That the Moris were not driven out of Chitor as early as a.d. 728 is proved by the Navsárí
- 77. 64 65 66 67 68 69 70 71 inscription which mentions the Arabs defeating the Mauryas as late as a.d. 738–9 (Saṃ. 490). See above page 56. ↑ Tod Western India 268 says Siddha Rája (a.d. 1094–1143): Múla Rája (a.d. 942–997) seems correct. See Rás Málá, I. 65. ↑ Káthiáwár Gazetteer, 672. ↑ The chronicles of Bhadrod, fifty-one miles south-west of Bhávnagar, have (Káth. Gaz. 380) a Selait Vála as late as a.d. 1554. ↑ Káthiáwár Gazetteer, 672. Another account places the movement south after the arrival of the Gohils a.d. 1250. According to local traditions the Válas did not pass to Bhadrod near Mahuva till a.d. 1554 (Káth. Gaz. 380) and from Bhadrod (Káth. Gaz. 660) retired to Dholarva. ↑ Káth. Gaz. 111 and 132. According to the Áin-i-Akbari (Gladwin, II. 60) the inhabitants of the ports of Mahua and Tulája were of the Vála tribe. ↑ Káth. Gaz. 680. ↑ Káth. Gaz. 414. ↑ The Vála connection with the Káthis complicates their history. Col. Watson (Káth. Gaz. 130) seems to favour the view that the Válas were the earliest wave of Káthis who came into Káthiáváḍa from Málwa apparently with the Guptas (a.d. 450) (Ditto, 671). Col. Watson seems to have been led to this conclusion in consequence of the existence of the petty state of Kátti in west Khándesh. But the people of the Kátti state in west Khándesh are Bhils or Kolis. Neither the people nor the position of the country seems to show connection with the Káthis of Káthiáváḍa. Col. Watson (Káth. Gaz. 130) inclines to hold that the Válas are an example of the rising of a lower class to be Rájputs. That both Válas and Káthis are northerners admitted into Hinduism may be accepted. Still it seems probable that on arrival in Káthiáváḍa the Válas were the leaders of the Káthis and that it is mainly since the fall of Valabhi that a large branch of the Válas have sunk to be Káthis. The Káthi traditions admit the superiority of the Válas. According to Tod (Western India, 270: Annals, I. 112–113) the Káthis claim to be a branch or descendants of the Válas. In Káthiáváḍa the Válas, the highest division of Káthis (Rás Málá, I. 296; Káth. Gaz. 122, 123, 131, 139), admit that their founder was a Vála Rájput who lost caste by marrying a Káthi woman. Another tradition (Rás Málá, I. 296; Káth. Gaz. 122 note 1) records that the Káthis flying from Sindh took refuge with the Válas and became their followers. Col. Watson (Káth. Gaz. 130) considers the practice in Porbandar and Navánagar of styling any
- 78. 72 73 74 75 76 77 78 79 80 81 82 lady of the Dhánk Vála family who marries into their house Káthiáníbái the Káthi lady proves that the Válas are Káthis. But as this name must be used with respect it may be a trace that the Válas claim to be lords of the Káthis as the Jetwas claim to be lords of the Mers. That the position of the Válas and Káthis as Rájputs is doubtful in Káthiáváḍa and is assured (Tod’s Annals, I. 111) in Rájputána is strange. The explanation may perhaps be that aloofness from Muhammadans is the practical test of honour among Rájputána Hindus, and that in the troubled times between the thirteenth and the seventeenth centuries, like the Jhálás, the Válas and Káthis may have refused Moghal alliances, and so won the approval of the Ránás of Mewáḍ. ↑ Káth. Gaz. 110–129. ↑ Western India, 207; Annals, I. 112–113. ↑ It is worthy of note that Bálas and Káthiás are returned from neighbouring Panjáb districts. Bálas from Dehra Ismail Khán (Panjáb Census Report 1891 Part III. 310), Káthiá Rájputs from Montgomery (Ditto, 318), and Káthiá Játs from Jhang and Dera Ismail Khán (Ditto, 143). Compare Ibbetson’s (1881) Panjáb Census, I. 259, where the Káthias are identified with the Kathaioi who fought Alexander the Great (b.c. 325) and also with the Káthis of Káthiáváḍa. According to this report (page 240) the Válas are said to have come from Málwa and are returned in East Panjáb. ↑ Tod’s Annals, I. 83 and 215; Elliot, II. 410; Jour. B. Br. A. S. XXIII. ↑ Annals, I. 215. ↑ Kath. Gaz. 589. ↑ Bṛihat-Saṃhitá, XIV. 21. The usual explanation (compare Fleet Ind. Ant. XXII. 180) Gold-Śakas seems meaningless. ↑ Sachau, II. 11. Among the legends are the much-applied tales of the foot-stamped cloth and the self-sacrificing minister. ↑ Western India, 213. ↑ Tod’s Annals, I. 83, 215; Western India, 270–352. ↑ Sachau, I. 208, II. 341. For the alleged descent of the Sesodiás and Válas from Ráma of the Sun race the explanation may be offered that the greatness of Kanishka, whose power was spread from the Ganges to the Oxus, in accordance with the Hindu doctrine (compare Beal’s Buddhist Records, I. 99 & 152; Rás Málá, I. 320; Fryer’s New Account, 190) that a conqueror’s success is the fruit of transcendent merit in a former birth, led to
- 79. 83 84 85 86 87 88 89 90 91 92 Kanishka being considered an incarnation of Ráma. A connection between Kanishka and the race of the Sun would be made easy by the intentional confusing of the names Kshatrapa and Kshatriya and by the fact that during part at least of his life fire and the sun were Kanishka’s favourite deities. ↑ Gladwin’s Áin-i-Akbari, II. 81: Tod’s Annals, I. 235. ↑ The invasion of Sindh formerly (Reinaud’s Fragments, 29) supposed to be by Naushirván in person according to fuller accounts seems to have been a raid by the ruler of Seistán (Elliot, I. 407). Still Reinaud (Mémoire Sur l’Inde, 127) holds that in sign of vassalage the Sindh king added a Persian type to his coins. ↑ Compare Tod’s Annals, I. 235–239 and Rawlinson’s Seventh Monarchy, 576. ↑ Rawlinson Seventh Monarchy, 452 note 3. ↑ Compare Tod’s Annals, I. 63; Thomas’ Prinsep, I. 413; Cunningham’s Arch. Survey, VI. 201. According to their own accounts (Rás Málá, I. 296) the Káthis learned sun- worship from the Vála of Dhánk by whom the famous temple of the sun at Thán in Káthiáváḍa was built. ↑ Válas Musalmán Játs in Lahor and Gurdaspur: Váls in Gujarát and Gujranwálá: Váls in Mozafarnagar and Dhera Ismael Khan. Also Válahs Hindus in Kángra. Panjáb Census of 1891, III. 162. ↑ Bṛihaṭ Saṃhitá, V. 80. ↑ Corp. Ins. Ind. III. 140–141. ↑ The references are; Langlois’ Harivaṃśa, I. 388–420, II. 178. That in a.d. 247 Balkh or Báktria was free from Indian overlordship (McCrindle’s Periplus, 121), and that no more distant tribe than the Gandháras finds a place in the Harivaṃśa lists combine to make it almost certain that, at the time the Harivaṃśa was written, whatever their origin may have been, the Báhlikas were settled not in Báktria but in India. ↑ The passage from the Karṇa Parva or Eighth Book of the Mahábhárata is quoted in Muir’s Sanskrit Texts, II. 482, and in greater fullness in St. Martin’s Geog. Greque et Latine de l’Inde, 402–410. The Báhikas or Bálhikas are classed with the Madras, Gandháras, Araṭṭas, and other Panjáb tribes. In their Bráhman families it is said the eldest son alone is a Bráhman. The younger brothers are without restraint Kshatriyas, Vaiśyas, Śudras, even Barbers. A Bráhman may sink to be a Barber and a barber may rise to be a Bráhman. The Báhikas eat flesh even the flesh of the cow and drink liquor. Their women
- 80. 93 94 95 know no restraint. They dance in public places unclad save with garlands. In the Harivaṃśa (Langlois, I. 493 and II. 178, 388, 420) the Bahlikas occur in lists of kings and peoples. ↑ Kern in Muir’s Sanskrit Texts, II. 446. St. Martin (Geog. Greque et Latine de l’Inde, 149) takes Báhika to be a contraction of Báhlika. Reasons are given below for considering the Mahábhárata form Báhika a confusion with the earlier tribes of that name rather than a contraction of Báhlika or Bálhika. The form Báhika was also favoured by the writer in the Mahábhárata because it fitted with his punning derivation from their two fiend ancestors Vahi and Hika. St. Martin, 408. ↑ St. Martin Geog. Greque et Latine de l’Inde, 403, puts the probable date at b.c. 380 or about fifty years before Alexander. St. Martin held that the passage belonged to the final revision of the poem. Since St. Martin’s time the tendency has been to lower the date of the final revision by at least 500 years. The fact noted by St. Martin (Ditto, page 404) that Jartika which the Mahábhárata writer gives as another name for Báhika is a Sanskritised form of Jat further supports the later date. It is now generally accepted that the Jats are one of the leading tribes who about the beginning of the Christian era passed from Central Asia into India. ↑ The name Valabhi, as we learn from the Jain historians, is a Sanskritised form of Valahi, which can be easily traced back to one of the many forms (Bálhíka, Bálhika, Balhika, Bahlíka, Báhlika, Váhlíka, Vahlíka, Válhíka, Válhika, Valhika) of a tribal name which is of common occurrence in the Epics. This name is, no doubt rightly, traced back to the city of Balkh, and originally denoted merely the people of Baktria. There is, however, evidence that the name also denoted a tribe doubtless of Baktrian origin, but settled in India: the Emperor Chandra speaks of defeating the Váhlikas after crossing the seven mouths of the Indus: Varáha-Mihira speaks of the Válhikas along with the people who dwell on Sindhu’s banks (Bṛ. Saṃ. V. 80): and, most decisive of all, the Káśiká Vṛitti on Páṇ. VIII. iv. 9 (a.d. 650) gives Bahlíka as the name of the people of the Sauvíra country, which, as Alberuni tells us, corresponded to the modern Multán, the very country to which the traditions of the modern Válas point. If the usual derivation of the name Bálhika be accepted,96 it is possible to go a step further and fix a probable limit before which the tribe did not enter India. The name of Balkh in the sixth century b.c. was, as we learn from Darius’ inscriptions, Bákhtri, and the Greeks also knew it as Baktra: the Avesta form is Bakhdhi, which according to the laws of sound- change established by Prof. Darmsteter for the Arachosian language as represented by the modern Pushtu, would become Bahli (see Chants Populaires des Afghans, Introd. page
- 81. 96 97 98 99 100 101 102 103 104 105 xxvii). This reduction of the hard aspirates to spirants seems to have taken place about the first century a.d.: parallel cases are the change from Parthava to Palhava, and Mithra to Mihira. It would seem therefore that the Bahlikas did not enter India before the first century a.d.: and if we may identify their subduer Chandra with Chandragupta I., we should have the fourth century a.d. as a lower limit for dating their invasion. Unfortunately, however, these limits cannot at present be regarded as more than plausible: for the name Balhika or Valhika appears to occur in works that can hardly be as modern as the first century a.d. The Atharvaveda-pariśishtas might be put aside, as they show strong traces of Greek influence and are therefore of late date: and the supposed occurrences in Páṇini belong to the commentators and to the Gaṇapáṭha only and are of more or less uncertain age. But the name occurs, in the form Balhika, in one hymn of the Atharvaveda itself (Book V. 22) which there is no reason to suppose is of late date. The lower limit is also uncertain as the identification of Chandra of the inscription with the Gupta king is purely conjectural.—(A. M. T. J.) ↑ There is a very close parallel in the modern Panjáb, where (see Census Report of 1881) the national name Baluch has become a tribal name in the same way as Bálhika. ↑ Hodgson’s Essays on Indian Subjects, I. 405 Note. ↑ McCrindle’s Periplus, 121. Compare Rawlinson’s Seventh Monarchy, 79. The absence of Indian reference to the Yuechi supports the view that in India the Yuechi were known by some other name. ↑ According to Reinaud (Mémoire Sur l’Inde, 82 note 3) probably the modern Kochanya or Kashania sixty or seventy miles west of Samarkand. This is Hiuen Tsiang’s (a.d. 620) Ki’uh-shwangi-ni-kia or Kushánika. See Beal’s Buddhist Records, I. 34. ↑ Etude sur la Geographie Grecque et Latine de l’Inde, 147. ↑ McCrindle’s Alexander in India, 350. ↑ The suggestion is made by Mr. A. M. T. Jackson. ↑ McCrindle’s Alexander, 136. ↑ McCrindle’s Alexander, 252. ↑ Compare Strabo, XV. I. 8. The Oxydrakai are the descendants of Dionysus. Again, XV. I. 24: The Malloi and the Oxydrakai who as we have already said are fabled to be related to Dionysus. ↑
- 82. 106 107 108 109 110 111 112 113 See McCrindle’s Alexander, 157, 369, 378, 398. Compare St. Martin Geog. Grecque et Latine de l’Inde, 102. ↑ Strabo, XV. I. 8 and 24, Hamilton’s Translation, III. 76, 95. ↑ References to the vines of Nysa and Meros occur in Strabo, Pliny, Quintus Curtius, Philostratus, and Justin: McCrindle’s Alexander in India, 193 note 1, 321, and 339. Strabo (Hamilton’s Translation, III. 86) refers to a vine in the country of Musikanus or Upper Sindh. At the same time (Ditto, 108) Strabo accepts Megasthenês’ statement that in India the wild vine grows only in the hills. ↑ The Kathaioi Malloi and Oxydrakai are (Arrian in McCrindle’s Alexander, 115, 137, 140, 149) called independent in the sense of kingless: they (Ditto, 154) sent leading men not ambassadors: (compare also Diodorus Siculus and Plutarch, Ditto 287, 311): the Malloi had to chose a leader (Q. Curtius, Ditto 236). ↑ Káthiáwár Gazetteer, 138. ↑ Káthiáwár Gazetteer, 137. ↑ Cutch Gazetteer, 80. ↑ Cutch Gazetteer, 81. ↑
- 84. CHAPTER IX.
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