Statistical approach to quantum field theory

Chapter 2
Path Integrals in Quantum and Statistical
Mechanics
There exist three apparently different formulations of quantum mechanics: HEISEN-
BERG’s matrix mechanics, SCHRÖDINGER’s wave mechanics and FEYNMAN’s path
integral approach. In contrast to matrix and wave mechanics, which are based on the
Hamiltonian approach the latter is based on the Lagrangian approach.
2.1 Summing Over All Paths
Already back in 1933 DIRAC asked himself, whether the classical Lagrangian and
action are as significant in quantum mechanics as they are in classical mechanics
[1, 2]. He observed that the probability amplitude
K t,q ,q = q e−i ˆHt/
|q (2.1)
for the propagation of a system from a point with coordinate q to another point with
coordinate q in time t is given by
K t,q ,p ∝ eiS[qcl]/
, (2.2)
where qcl denotes the classical trajectory from q to q . In the exponent the action of
this trajectory enters as a multiple of Planck’s reduced constant . For a free particle
with Lagrangian
L0 =
m
2
˙q2
(2.3)
the formula (2.2) is verified easily: A free particle moves with constant velocity
(q − q)/t from q to q and the action of the classical trajectory is
S[qcl] =
t
0
dsL0 qcl(s) =
m
2t
q − q
2
.
The factor of proportionality in (2.2) is then uniquely fixed by the condition
e−i ˆHt/ → 1 for t → 0, which in position space reads
lim
t→0
K t,q ,q = δ q ,q . (2.4)
A. Wipf, Statistical Approach to Quantum Field Theory, Lecture Notes in Physics 864,
DOI 10.1007/978-3-642-33105-3_2, © Springer-Verlag Berlin Heidelberg 2013
5

6 2 Path Integrals in Quantum and Statistical Mechanics
Alternatively, it is fixed by the property e−i ˆHt/ e−i ˆHs/ = e−i ˆH(t+s)/ that takes the
form
duK t,q ,u K(s,u,q) = K t + s,q ,q (2.5)
in position space. Thus, the correct free-particle propagator on a line is given by
K0 t,q ,q =
m
2πi t
1/2
eim(q −q)2/2 t
. (2.6)
Similar results hold for the harmonic oscillator or systems for which ˆq(t) fulfills
the classical equation of motion. For such systems V (ˆq) = V ( ˆq ) holds true.
However, for general systems the simple formula (2.2) must be extended and it was
FEYNMAN who discovered this extension back in 1948. He realized that all paths
from q to q (and not only the classical path) contribute to the propagator. This
means that in quantum mechanics a particle can potentially move on any path q(s)
from the initial to the final destination,
q(0) = q and q(t) = q . (2.7)
The probability amplitude emerges as the superposition of contributions from all
trajectories,
K t,q ,q ∼
all paths
eiS[path]/
, (2.8)
where a single path contributes a term ∼exp(iS[path]/ ).
In passing we note that already in 1923 WIENER introduced the sum over all
paths in his studies of stochastic processes [3]. Thereby a single path was weighted
with a real and positive probability and not with a complex amplitude as in (2.8).
Wiener’s path integral corresponds to Feynman’s path integral for imaginary time
and describes quantum systems in thermal equilibrium with a heat bath at fixed
temperature. In this book we will explain this extraordinary result and apply it to in-
teresting physical systems. Moreover, the path integral method allows for a uniform
treatment of quantum mechanics, quantum field theory and statistical mechanics and
can be regarded as a basic tool in modern theoretical physics. It represents an alter-
native approach to the canonical quantization of classical systems and earned its first
success in the 1950s. The path integral method is very beautifully and intelligibly
presented in Feynman’s original work [4] as well as in his book with HIBBS [5].
The latter reference contains many applications and is still recognized as a standard
reference. Functional integrals have been developed further by outstanding mathe-
maticians and physicists, especially by KAC [9]. An adequate reference for these
developments is contained in the review article by GELFAND and YAGLOM [10].
In the present chapter we can only give a short introduction to path integrals. For a
deeper understanding the reader should consult more specialized books and review
articles. Some of them are listed in the bibliography at the end of this chapter.

2.2 Recalling Quantum Mechanics 7
2.2 Recalling Quantum Mechanics
There are two well-established ways to quantize a classical system: canonical quan-
tization and path integral quantization. For completeness and later use we recall the
main steps of canonical quantization both in Schrödinger’s wave mechanics and
Heisenberg’s matrix mechanics.
A classical system is described by its coordinates {qi} and momenta {pi} on
phase space Γ . An observable O is a real-valued function on Γ . Examples are the
coordinates on phase space and the energy H(q,p) of the system under considera-
tion. We assume that phase space comes along with a symplectic structure and has
local coordinates with Poisson brackets
qi
,pj = δi
j . (2.9)
The brackets are extended to observables through antisymmetry and the derivation
rule {OP,Q} = O{P,Q} + {O,Q}P . The evolution in time of an observable is
determined by
˙O = {O,H}, e.g. ˙qi
= qi
,H and ˙pi = {pi,H}. (2.10)
In the canonical quantization the function on phase space are mapped to operators
and the Poisson brackets of two functions become commutators of the associated
operators:
O(q,p) → Ô(ˆq, ˆp) and {O,P} →
1
i
[ Ô, ˆP]. (2.11)
The time evolution of an (not explicitly time-dependent) observable is determined
by Heisenberg’s equation
d Ô
dt
=
i
[ ˆH, Ô]. (2.12)
In particular the phase space coordinates (qi,pi) become operators with commuta-
tion relations [ˆqi, ˆpj ] = i δi
j and time evolution given by
dˆqi
dt
=
i ˆH, ˆqi
and
d ˆpi
dt
=
i
[ ˆH, ˆpi].
For a system of non-relativistic and spinless particles the Hamiltonian reads
ˆH = ˆH0 + ˆV with ˆH0 =
1
2m
ˆp2
i , (2.13)
and one arrives at Heisenberg’s equations of motion,
dˆqi
dt
=
ˆpi
2m
and
d ˆpi
dt
= − ˆV ,i . (2.14)
Observables are represented by hermitian operators on a Hilbert space H , whose
elements characterize the states of the system:
Ô(ˆq, ˆp) : H → H . (2.15)

Consider a particle confined to an endless wire. Its Hilbert space is H = L2(R) and
its position and momentum operator are represented in position space as
(ˆqψ)(q) = qψ(q) and ( ˆpψ)(q) =
i
∂qψ(q). (2.16)
In experiments we can measure matrix elements of observables, represented by her-
mitian operators, and in particular expectation values of hermitian operators in a
state of the system. The time dependence of an expectation value ψ| Ô(t)|ψ is
determined by the Heisenberg equation (2.12).
The transition from the Heisenberg picture to the Schrödinger picture involves a
time-dependent similarity transformation,
Ôs = e−it ˆH/ Ôeit ˆH/
and |ψs = e−it ˆH/
|ψ , (2.17)
and leads to time-independent observables in the Schrödinger picture,
d
dt
Ôs = e−it ˆH/
−
i
[ ˆH, Ô] +
d
dt
Ô eit ˆH/
= 0.
Note that the Hamiltonian operator is the same in both pictures, ˆHs = ˆH and that
all expectation values are left invariant by the similarity transformation,
ψ| Ô(t)|ψ = ψs(t) Ôs ψs(t) . (2.18)
A state vector in the Schrödinger picture |ψs(t) fulfills the Schrödinger equation
i
d
dt
|ψs = ˆH|ψs ⇐⇒ ψs(t) = e−it ˆH/
ψs(0) . (2.19)
In position space this formal solution of the evolution equation has the form
ψs t,q ≡ q |ψs(t) = q e−it ˆH/
|q q|ψs(0) dq
≡ K t,q ,q ψs(0,q)dq, (2.20)
where we inserted the resolution of the identity with ˆq-eigenstates,
dq|q q| = 1, (2.21)
and introduced the kernel of the unitary time evolution operator
K t,q ,q = q ˆK(t)|q , ˆK(t) = e−it ˆH/
. (2.22)
The propagator K(t,q ,q) is interpreted as the probability amplitude for the prop-
agation from q at time 0 to q at time t. This is emphasized by the notation
K t,q ,q ≡ q ,t|q,0 . (2.23)
The amplitude solves the time-dependent Schrödinger equation
i
d
dt
K t,q ,q = ˆHK t,q ,q , (2.24)

2.3 Feynman–Kac Formula 9
where ˆH acts on q , and fulfills the initial condition
lim
t→0
K t,q ,q = δ q − q . (2.25)
The conditions (2.24) and (2.25) uniquely define the propagator. In particular for a
non-relativistic particle with Hamiltonian (2.13) in d dimensions the solution reads
K0 t,q ,q = q e−it ˆH0/
|q =
m
2πi t
d/2
eim(q −q)2/2 t
, q,q ∈ Rd
. (2.26)
In one dimension we recover the result (2.6). After this preliminaries we now turn
to the path integral representation of the propagator.
2.3 Feynman–Kac Formula
We shall derive Feynman’s path integral representation for the unitary time evolution
operator exp(−i ˆHt) as well as Kac’s path integral representation for the positive
operator exp(− ˆHτ). Thereby we shall utilize the product formula of TROTTER. In
case of matrices this formula was already verified by LIE and has the form:
Theorem 2.1 (Lie’s theorem) For two matrices A and B
eA+B
= lim
n→∞
eA/n
eB/n n
.
To prove this theorem we define for each n the two matrices Sn := exp(A/n +
B/n) and Tn := exp(A/n)exp(B/n) and telescope the difference of their nth powers,
Sn
n − Tn
n = Sn−1
n (Sn − Tn) + Sn−2
n (Sn − Tn)Tn + ··· + (Sn − Tn)Tn−1
n .
Since the norm of a product is less or equal than the product of the norms we have
exp(X) ≤ exp( X ). Using the triangle inequality we have
Sn , Tn ≤ a1/n
with a = e A + B
and therefore
Sn
n − Tn
n ≡ eA+B
− eA/n
eB/n n
≤ n × a(n−1)/n
Sn − Tn .
Finally, using Sn − Tn = −[A,B]/2n2 + O(1/n3), the product formula is verified
for matrices. But the theorem also holds for unbounded self-adjoint operators.
Theorem 2.2 (Trotter’s theorem) If Â and ˆB are self-adjoint operators and Â + ˆB
is essentially self-adjoint on the intersection D of their domains, then
e−it( Â+ ˆB)
= s- lim
n→∞
e−it Â/n
e−it ˆB/n n
. (2.27)
If in addition Â and ˆB are bounded from below, then
e−τ( Â+ ˆB)
= s- lim
n→∞
e−τ Â/n
e−τ ˆB/n n
. (2.28)

The convergence here is in the sense of the strong operator topology. For opera-
tors Ân and Â on a common domain D in the Hilbert space we have s-limn→∞ Ân =
Â iff Ânψ − Âψ → 0 for all ψ ∈ D. Formula (2.27) is used in quantum mechan-
ics and formula (2.28) finds its application in statistical physics and the Euclidean
formulation of quantum mechanics [7, 8].
Let us assume that ˆH can be written as ˆH = ˆH0 + ˆV and apply the product
formula to the evolution kernel in (2.22). With ε = t/n and = 1 we obtain
K t,q ,q = lim
n→∞
q e−iε ˆH0 e−iε ˆV n
|q
= lim
n→∞
dq1 ···dqn−1
j=n−1
j=0
qj+1|e−iε ˆH0 e−iε ˆV
|qj , (2.29)
where we repeatedly inserted the resolution of the identity (2.21) and denoted the
initial and final point by q0 = q and qn = q , respectively. The potential ˆV is diago-
nal in position space such that
qj+1|e−iε ˆH0 e−iε ˆV
|qj = qj+1|e−iε ˆH0 |qj e−iεV (qj )
. (2.30)
Here we insert the result (2.26) for the propagator of the free particle with Hamilto-
nian ˆH0 and obtain
K t,q ,q = lim
n→∞
dq1 ···dqn−1
m
2πiε
n/2
× exp iε
j=n−1
j=0
m
2
qj+1 − qj
ε
2
− V (qj ) . (2.31)
This is the celebrated Feynman–Kac formula, which provides the path integral rep-
resentation for the propagator. To make clear why it is called path integral, we divide
the time interval [0,t] into n subintervals of equal length ε = t/n and identify qk
with q(s = kε). Now we connect the points
(0,q0),(ε,q1),...,(t − ε,qn−1),(t,qn)
by straight line segments, which give rise to a broken-line path as depicted in
Fig. 2.1. The exponent in (2.31) is just the Riemann integral for the action of a
particle moving along the broken-line path,
j=n−1
j=0
ε
m
2
qj+1 − qj
ε
2
− V (qj ) =
t
0
ds
m
2
dq
ds
2
− V q(s) . (2.32)
The integral dq1 ···dqn−1 represents the sum over all broken-line paths from q
to q . Every continuous path can be approximated by a broken-line path if only ε is
small enough. Next we perform the so-called continuum limit ε → 0 or equivalently
n → ∞. In this limit the finite-dimensional integral (2.31) turns into an infinite-
dimensional (formal) integral over all paths from q to q . With the definition
m
2πiε
n/2
=: C (2.33)

2.4 Euclidean Path Integral 11
Fig. 2.1 Broken-line path
entering the discretized path
integral (2.31)
we arrive at the formal result
K t,q ,q = C
q(t)=q
q(0)=q
DqeiS[q]/
. (2.34)
The ‘measure’ Dq is defined via the limiting process n → ∞ in (2.31). Since the
infinite product of Lebesgue measures does not exist, D has no precise mathematical
meaning. Only after a continuation to imaginary time a measure on all paths can be
rigorously defined.
The formula (2.34) holds true for more general systems, for example interacting
particles moving in more than one dimension and in the presence of external fields.
It also applies to mechanical systems with generalized coordinates q1,...,qN . The
formula is also correct in quantum field theories where one integrates over all fields
instead of all paths. Further properties of the path integral as well as many examples
and applications can be found in the reference given at the end of this chapter.
2.4 Euclidean Path Integral
The oscillating integrand exp(iS) entering the path integral (2.34) leads to distri-
butions. If only we could suppress these oscillations, then it may be possible to
construct a well-defined path integral. This may explain why most rigorous work on
path integrals is based on imaginary time. For imaginary time it is indeed possible
to construct a measure on all paths: the Wiener measure. The continuation from real
to imaginary time is achieved by a Wick rotation and the continuation from imagi-
nary time back to real time by an inverse Wick rotation. In practice, one replaces t
by −iτ in the path integral (2.34), works with the resulting Euclidean path integral,
and replaces τ by it in the final expressions.
2.4.1 Quantum Mechanics in Imaginary Time
The unitary time evolution operator has the spectral representation
ˆK(t) = e−i ˆHt
= e−iEt
d ˆPE, (2.35)

where ˆPE is the spectral family of the Hamiltonian operator ˆH. If ˆH has discrete
spectrum then ˆPE is the orthogonal projector onto the subspace of H spanned by
all eigenfunctions with energies less than E. In the following we assume that the
Hamiltonian operator is bounded from below. Then we can subtract its ground state
energy to obtain a non-negative ˆH for which the integration limits in (2.35) are 0
and ∞. With the substitution t → t − iτ we obtain
e−(τ+it) ˆH
=
∞
0
e−E(τ+it)
d ˆPE. (2.36)
This defines a holomorphic semigroup in the lower complex half-plane
{t − iτ ∈ C,τ ≥ 0}. (2.37)
If the operator (2.36) is known on the lower imaginary axis (t = 0,τ ≥ 0), then one
can perform an analytic continuation to the real axis (t,τ = 0). The analytic con-
tinuation to complex time t → −iτ corresponds to a transition from the Minkowski
metric ds2 = dt2 − dx2 − dy2 − dz2 to a metric with Euclidean signature. Hence a
theory with imaginary time is called Euclidean theory.
The time evolution operator ˆK(t) exists for real time and defines a one-
parametric unitary group. It fulfills the Schrödinger equation
i
d
dt
ˆK(t) = ˆH ˆK(t)
with a complex and oscillating kernel K(t,q ,q) = q | ˆK(t)|q . For imaginary time
we have a hermitian (and not unitary) evolution operator
ˆK(τ) = e−τ ˆH
(2.38)
with positive spectrum. ˆK(τ) exists for positive τ and form a semigroup only. For
almost all initial data evolution back into the ‘imaginary past’ is impossible.
The evolution operator for imaginary time satisfies the heat equation
d
dτ
ˆK(τ) = − ˆH ˆK(τ), (2.39)
instead of the Schrödinger equation and has kernel
K τ,q ,q = q e−τ ˆH
|q , K 0,q ,q = δ q ,q . (2.40)
This kernel is real1 for a real Hamiltonian. Furthermore it is strictly positive:
Theorem 2.3 Let the potential V be continuous and bounded from below and ˆH =
−Δ + ˆV be an essentially self-adjoint operator. Then
q e−τ ˆH
|q > 0. (2.41)
1If we couple the system to a magnetic field, ˆH and ˆK(τ) become complex quantities.

2.4 Euclidean Path Integral 13
The reader may consult the textbook [6] for a proof of this theorem. As examples
we consider the kernel of the free particle with mass m,
K0 τ,q ,q =
m
2πτ
d/2
e−m(q −q)2/2τ
, (2.42)
and of the harmonic oscillator with frequency ω,
Kω τ,q ,q =
mω/(2π)
sinhωτ
d/2
exp −
mω
2
q
2
+ q2
cothωτ −
2q q
sinhωτ
,
(2.43)
both for imaginary time and in d dimensions. Both kernels are strictly positive. This
positivity is essential for the far-reaching relation of Euclidean quantum theory and
probability theory: The quantity
Pτ (q) ≡ CK(τ,q,0) (2.44)
can be interpreted as probability for the transition from point 0 to point q during the
time interval τ.2 The probability of ending somewhere should be 1,
dqPτ (q) = 1, (2.45)
and this requirement determines the constant C. For a free particle we obtain
Pτ (q) =
m
2πτ
d/2
e−mq2/2τ
.
It represents the probability density for Brownian motion with diffusion coefficient
inversely proportional to the mass, D = 1/2m.
In quantum field theory vacuum expectation values of products of field operators
at different spacetime points encode all information about the theory. They deter-
mine scattering amplitudes and spectral properties of the particles and hence play a
distinguished role. In quantum mechanics these expectation values are given by
W(n)
(t1,...,tn) = 0|ˆq(t1)··· ˆq(tn)|0 , ˆq(t) = eit ˆH
ˆqe−it ˆH
. (2.46)
These Wightman functions are not symmetric in their arguments t1,...,tn since
the position operators at different times do not commute. Again we normalize the
Hamiltonian such that the energy of the ground state |0 vanishes and perform an
analytic continuation of the Wightman functions to complex times zi = ti − iτi:
W(n)
(z1,...,zn) = 0|ˆqe−i(z1−z2) ˆH
ˆqe−i(z2−z3) ˆH
ˆq ··· ˆqe−i(zn−1−zn) ˆH
ˆq|0 . (2.47)
We used that ˆH annihilates the ground state or that exp(iζ ˆH)|0 = |0 . The func-
tions W(n) are well-defined if the imaginary parts of their arguments zk are ordered
according to
(zk − zk+1) ≤ 0.
2To keep the notation simple, we use q as the final point.

With zi = ti − iτi one ends up with analytic functions W(n) in the region
τ1 > τ2 > ··· > τn. (2.48)
The Wightman distributions for real time represent boundary values of the analytic
Wightman functions with complex arguments:
W(n)
(t1,...,tn) = lim
zi→0
(zk+1−zk)>0
W(n)
(z1,...,zn). (2.49)
On the other hand, if the arguments are purely imaginary then we obtain the
Schwinger functions. For τ1 > τ2 > ··· > τn they are given by
S(n)
(τ1,...,τn) = W(n)
(−iτ1,...,−iτn)
= 0|ˆqe−(τ1−τ2) ˆH
ˆqe−(τ2−τ3) ˆH
ˆq ··· ˆqe−(τn−1−τn) ˆH
ˆq|0 . (2.50)
As an example we consider the harmonic oscillator with Hamiltonian
ˆH = ωâ†
â,
expressed in terms of the step operators â, â†, which obey the commutation relation
[â, â†] = 1. The ground state |0 in annihilated by â and hence has zero energy.
The first excited state |1 = â†|0 has energy ω. The two-point Wightman function
depends on the time difference only,
W(2)
(t1 − t2) = 0|ˆq(t1)ˆq(t2)|0 =
1
2mω
0| â + â†
e−i(t1−t2) ˆH
â + â†
|0
=
1
2mω
1|e−itωâ† â
|1 =
e−iω(t1−t2)
2mω
.
The corresponding Schwinger function is given by
S(2)
(τ1 − τ2) =
e−ω(τ1−τ2)
2mω
(τ1 > τ2). (2.51)
In a relativistic quantum field theory the Schwinger functions S(n)(x1,...,xn) are
invariant under Euclidean Lorentz transformation from the group SO(4). This in-
variance together with locality imply that the S(n) are symmetric functions of their
arguments xi ∈ R4. This is not necessarily true for the Schwinger functions in quan-
tum mechanics.
2.4.2 Imaginary-Time Path Integral
To formulate the path integral for imaginary time we employ the product formula
(2.28), which follows from the product formula (2.27) through the substitution of it

2.5 Path Integral in Quantum Statistics 15
by τ. For such systems the analogue of (2.31) for Euclidean time τ is obtained by
the substitution of iε by ε. Thus we find
K τ,q ,q = ˆq e−τ ˆH/
|ˆq
= lim
n→∞
dq1 ···dqn−1
m
2π ε
n/2
e−SE(q0,q1,...,qn)/
,
SE(...) = ε
n−1
j=0
m
2
qj+1 − qj
ε
2
+ V (qj ) ,
(2.52)
where q0 = q and qn = q . The multi-dimensional integral represents the sum over
all broken-line paths from q to q . Interpreting SE as Hamiltonian of a classical
lattice model and as temperature, it is (up to the fixed endpoints) the partition
function of a one-dimensional lattice model on a lattice with n + 1 sites. The real-
valued variable qj defined on site j enters the action SE, which contains interactions
between the variables qj and qj+1 at neighboring sites. The values of the lattice field
{0,1,...,n − 1,n} → {q0,q1,...,qn−1,qn}
are prescribed at the end points q0 = q and qn = q . Note that the classical limit
→ 0 corresponds to the low-temperature limit of the lattice-system.
The multi-dimensional integral (2.52) corresponds to the summation over all lat-
tice fields. What happens to the finite-dimensional integral when we take the con-
tinuum limit n → ∞? Then we obtain the Euclidean path integral representation for
the positive kernel
K τ,q ,q = q e−τ ˆH/
|q = C
q(τ)=q
q(0)=q
Dqe−SE[q]/
. (2.53)
The integrand contains the Euclidean action
SE[q] =
τ
0
dσ
m
2
˙q2
+ V q(σ) , (2.54)
which for many physical systems is bounded from below.
2.5 Path Integral in Quantum Statistics
The Euclidean path integral formulation immediately leads to an interesting connec-
tion between quantum statistical mechanics and classical statistical physics. Indeed,
if we set τ/ ≡ β and integrate over q = q in (2.53), then we end up with the path
integral representation for the canonical partition function of a quantum system with
Hamiltonian ˆH at inverse temperature β = 1/kBT . More precisely, setting q = q
and τ = β in the left-hand side of this formula, then the integral over q yields the
trace of exp(−β ˆH), which is just the canonical partition function,
dqK( β,q,q) = tre−β ˆH
= Z(β) = e−βEn with β =
1
kBT
. (2.55)

Setting q = q in the Euclidean path integral in (2.53) means that we integrate over
paths beginning and ending at q during the imaginary-time interval [0, β]. The
ﬁnal integral over q leads to the path integral over all periodic paths with period β,
Z(β) = C Dqe−SE[q]/
, q( β) = q(0). (2.56)
For example, the kernel of the harmonic oscillator in (2.43) on the diagonal is
Kω(β,q,q) =
mω
2π sinh(ωβ)
exp −mω tanh(ωβ/2)q2
, (2.57)
where we used units with = 1. The integral over q yields the partition function
Z(β) =
mω
2π sinh(ωβ)
dq exp −mω tanh(ωβ/2)q2
=
1
2sinh(ωβ/2)
=
e−ωβ/2
1 − e−ωβ
= e−ωβ/2
∞
n=0
e−nωβ
, (2.58)
where we used sinhx = 2sinhx/2coshx/2. A comparison with the spectral sum
over all energies in (2.55) yields the energies of the oscillator with (angular) fre-
quency ω,
En = ω n +
1
2
, n = 0,1,2,.... (2.59)
For large values of ωβ, i.e. for very low temperature, the spectral sum is dominated
by the contribution of the ground state energy. Thus for cold systems the free energy
converges to the ground state energy
F(β) ≡ −
1
β
logZ(β)
ωβ→∞
→ E0. (2.60)
One often is interested in the energies and wave functions of excited states. We now
discuss an elegant method to extract this information from the path integral.
2.5.1 Thermal Correlation Functions
The energies of excited states are encoded in the thermal correlation functions.
These functions are expectation values of products of the position operator
ˆqE(τ) = eτ ˆH/
ˆqe−τ ˆH/
, ˆqE(0) = ˆq(0), (2.61)
at different imaginary times in the canonical ensemble,
ˆqE(τ1)··· ˆqE(τn) β
≡
1
Z(β)
tr e−β ˆH
ˆqE(τ1)··· ˆqE(τn) . (2.62)
The normalizing function Z(β) is the partition function (2.56). From the thermal
two-point function

2.5 Path Integral in Quantum Statistics 17
ˆqE(τ1)ˆqE(τ2) β
=
1
Z(β)
tr e−β ˆH
ˆqE(τ1)ˆqE(τ2)
=
1
Z(β)
tr e−(β−τ1) ˆH
ˆqe−(τ1−τ2) ˆH
ˆqe−τ2 ˆH
(2.63)
we can extract the energy gap between the ground state and the first excited state.
For this purpose we use orthonormal energy eigenstates |n to calculate the trace
and in addition insert the resolution of the identity-operator 1 = |m m|. This
yields
... β =
1
Z(β) n,m
e−(β−τ1+τ2)En e−(τ1−τ2)Em n|ˆq|m m|ˆq|n . (2.64)
Note that in the sum over n the contributions from the excited states are exponen-
tially suppressed at low temperatures β → ∞, implying that the thermal two-point
function converges to the Schwinger function in this limit:
β→∞
→
m≥0
e−(τ1−τ2)(Em−E0)
0|ˆq|m
2
= 0|ˆqE(τ1)ˆqE(τ2)|0 .
(2.65)
In the first step we used that for low temperature the partition function tends to
exp(−βE0). Likewise, we find for the one-point function the result
lim
β→∞
ˆqE(τ) β
= 0|ˆq|0 . (2.66)
In the connected two-point function
ˆqE(τ1)ˆqE(τ2) c,β
≡ ˆqE(τ1)ˆqE(τ2) β
− ˆqE(τ1) β
ˆqE(τ2) β
(2.67)
the term with m = 0 in the sum (2.65) is absent and this leads to an exponential
decaying function for large time-differences,
lim
β→∞
ˆqE(τ1)ˆqE(τ2) c,β
=
m>0
e−(τ1−τ2)(Em−E0)
0|ˆq|m
2
. (2.68)
For large time-differences τ1 − τ2 the term with m = 1 dominates the sum such that
ˆqE(τ1)ˆqE(τ2) c,β→∞
→ e−(E1−E0)(τ1−τ2)
0|ˆq|1
2
, τ1 − τ2 → ∞. (2.69)
It follows that we can read off the energy gap E1 − E0 as well as the transition
probability | 0|q|1 |2 from the asymptotics of the connected two-point function.
To arrive at the path integral representation for the thermal two-point correlation
function we consider the matrix elements
q ˆK(β)ˆqE(τ1)ˆqE(τ2)|q , with ˆqE(τ) = ˆK(−τ)ˆq ˆK(τ). (2.70)
Here ˆK(τ) = exp(−τ ˆH) denotes the evolution operator for imaginary time with
path integral representation given in (2.53). Now we insert twice the resolution of
the identity and obtain
... = dv du q ˆK(β − τ1)|v v v| ˆK(τ1 − τ2)|u u u| ˆK(τ2)|q .

In this result we make use of the path integral representations for the three propa-
gators to ﬁnd the path integral representation: Firstly, we sum over all paths from
q → u in the time interval τ2 and multiply the result with the coordinate u at time τ2.
Next we sum over all paths u → v in the time interval τ1 − τ2 and multiply with the
coordinate v at time τ1. The last step includes the summation over all paths v → q
in the time interval β − τ1. The integration over the intermediate positions u and v
means that the summation extends over all paths q → q and not only over paths
going through u at time τ2 and v at time τ1. Besides exp(−SE), the integrand in-
cludes the multiplicative factor vu = q(τ1)q(τ2). Since the entire propagation time
is β we end up with
q e−β ˆH
ˆqE(τ1)ˆqE(τ2)|q = C
q(β)=q
q(0)=q
Dqe−SE[q]
q(τ1)q(τ2), τ1 > τ2. (2.71)
The thermal expectation value is given by the trace. Thus we set q = q , integrate
over q and divide the result by the partition function Z(β). Integrating over q is
equivalent to summing over all periodic paths with period β. Hence, we obtain
=
1
Z(β)
Dqe−SE[q]
q(τ1)q(τ2) (2.72)
with partition function given in (2.56). In the derivation we assumed the time-order
τ1 > τ2 when applying the Trotter formula.
The path integral representation of higher time-ordered correlation functions are
obtained in a similar fashion. They are all generated by the kernel
Z β,j,q ,q = C
q(β)=q
q(0)=q
Dqe−SE[q]+ dτj(τ)q(τ)
, (2.73)
in which one integrates over all paths from q to q , or by the partition function in
presence of an external source,
Z(β,j) = dqZ(β,j,q,q) = C
q(0)=q(β)
Dqe−SE[q]+ dτj(τ)q(τ)
. (2.74)
The object in (2.73) generates matrix elements similarly as in (2.71) but with an
arbitrary number of insertions of position operators. The function Z(β,j) generates
all time-ordered thermal correlation functions. For example, the thermal two-point
function follows by differentiating the generating function (2.74) twice:
T ˆqE(τ1)ˆqE(τ2) β
=
1
Z(β,0)
δ2
δj(τ1)δj(τ2)
Z(β,j)
j=0
, (2.75)
wherein T indicates the time ordering. Since the right-hand side is symmetric in its
arguments τ1,τ2 and both sides are identical for τ1 > τ2, we must include the time-
ordering on the left-hand side. The ordering also results from a repeated calculation
for τ2 > τ1.
The connected correlation functions are generated by the logarithm of the parti-
tion function, called Schwinger functional
W(β,j) ≡ logZ(β,j), (2.76)

2.6 The Harmonic Oscillator 19
by repeated differentiations with respect to the external source,
T ˆqE(τ1)ˆqE(τ2)··· ˆqE(τn) c,β
=
δn
δj(τ1)···δj(τn)
W(β,j)
j=0
. (2.77)
If we consider conservative systems and a time-independent source j, then the
Schwinger functional is proportional to the free energy in the presence of the source.
2.6 The Harmonic Oscillator
We wish to study the path integral for the Euclidean oscillator with discretized time.
The results are instructive particularly with regard to lattice ﬁeld theories consid-
ered later in this book. So let us discretize the Euclidean time interval [0,τ] with n
sampling points separated by a lattice constant ε = τ/n. For the Lagrangian
L =
m
2
˙q2
+ μq2
(2.78)
the discretized path integral over periodic paths reads
Z = dq1 ···dqn
m
2πε
n/2
exp −ε
n−1
j=0
m
2
qj+1 − qj
ε
2
+ μq2
j
=
m
2πε
n/2
dq1 ···dqn exp −
1
2
(q,Aq) , (2.79)
where we assumed q0 = qn and introduced the symmetric matrix
A =
m
ε
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
α −1 0 ··· 0 −1
−1 α −1 ··· 0 0
...
...
0 0 ··· −1 α −1
−1 0 ··· 0 −1 α
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
, α = 2 1 +
μ
m
ε2
. (2.80)
This is a Toeplitz matrix, in which each descending diagonal from left to right is
constant. This property results from the invariance of the action under lattice trans-
lations. For the explicit calculation of Z we consider the generating function
Z[j] =
m
2πε
n/2
dn
q exp −
1
2
(q,Aq) + (j,q)
=
(m/ε)n/2
√
detA
exp
1
2
j,A−1
j . (2.81)
Here we applied the known result for Gaussian integrals. The n eigenvalues of A are
λk =
m
ε
α − 2cos
2π
n
k =
2
ε
με2
+ 2msin2 πk
n
, k = 1,...,n (2.82)

and the corresponding orthonormal eigenvectors have the form
ψ(k) =
1
√
n
zk
,z2k
,...,znk T
with z = e2πi/n
. (2.83)
With the spectral resolution for the inverse matrix A−1 = k λ−1
k ψ†(k)ψ(k) we
obtain
A−1
pq
=
ε
2n
n
k=1
e2πik(p−q)/n
με2 + 2msin2 πk
n
. (2.84)
Note that the connected correlation function
qi1 ···qim =
∂m
∂ji1 ···∂qim
logZ[j]
j=0
(2.85)
of the harmonic oscillator vanishes for m > 2. This means that all correlation func-
tions are given in terms of the two-point function
qiqj c = qiqj =
∂2
∂ji∂jj
j,A−1
j = A−1
ij
. (2.86)
As a consequence of time-translation invariance the expectation value
q2
i =
ε
2n
n
k=1
1
με2 + 2msin2 πk
n
(2.87)
is independent of i. This and similar expectation values, together with the virial the-
orem, yield the ground state energies of Hamiltonians discretized on ﬁnite lattices.
More details and numerical results are found in the chapter on simulations.
2.7 Problems
2.1 (Gaussian integral) Show that
dz1 d¯z1 ···dzn d¯zn exp −
ij
¯ziAij zj = πn
(detA)−1
with A being a positive Hermitian n×n matrix and zi complex integration variables.
2.2 (Harmonic oscillator) In (2.43) we quoted the result for the kernel Kω(τ,q ,q)
of the d-dimensional harmonic oscillator with Hamiltonian
ˆH =
1
2m
ˆp2
+
mω2
2
ˆq2
at imaginary time τ. Derive this formula.

2.7 Problems 21
Hint: Express the kernel in terms of the eigenfunctions of ˆH, which for = m =
ω = 1 are given by
exp −ξ2
− η2
∞
n=0
ζn
2nn!
Hn(ξ)Hn(η) =
1
1 − ζ2
exp
−(ξ2 + η2 − 2ξηζ)
1 − ζ2
.
The functions Hn denote the Hermite polynomials.
Comment This result also follows from the direct evaluation of the path integral.
2.3 (Free particle on a circle) A free particle moves on an interval and obeys peri-
odic boundary conditions. Compute the time evolution kernel K(tb − ta,qb,qa) =
qb,tb|qa,ta . Use the familiar formula for the kernel of the free particle (2.26) and
enforce the periodic boundary conditions by a suitable sum over the evolution kernel
for the particle on R.
2.4 (Connected and unconnected correlation function) The unconnected thermal
correlation functions are given by
T ˆqE(τ1)··· ˆqE(τn) β
=
1
Z(β)
δn
δj(τ1)···δj(τn)
Z(β,j)
j=0
with generating functional
Z(β,j) = Dq exp −SE[q] +
β
0
j(τ)q(τ) ,
wherein one integrates over all β-periodic paths. Assume that the Euclidean La-
grangian density
LE(q, ˙q) =
1
2
˙q2
+ V (q)
contains an even potential, i.e. V (−q) = V (q).
(a) Show that ˆqE(τ) β = 0.
(b) Express the unconnected 4-point function T ˆqE(τ1)··· ˆqE(τ4) β via connected
correlation functions.
2.5 (Semi-classical expansion of the partition function) In Chap. 2.5 we discussed
the path integral representation of the thermal partition function, given by
Z(β) = C dq
q( β)=q
q(0)=q
Dqe−SE[q]/
.
We rescale the imaginary time and the amplitude according to
τ → τ and q(.) → q(.).
After rescaling the ‘time interval’ is of length β instead of β and
Z(β) = C dq
q(β)=q/
q(0)=q/
Dq exp −
β
0
1
2
m˙q2
+ V q(.) dτ .

For a moving particle the kinetic energy dominates the potential energy for small .
Thus we decompose each path into its constant part and the ﬂuctuations about the
constant part: q(.) = q/ + ξ(.). Show that
Z(β) =
C
dq
ξ(β)=0
ξ(0)=0
Dξ exp −
β
0
1
2
m˙ξ2
+ V (q + ξ) dτ .
Determine the constant C by considering the limiting case V = 0 with the well-
known result Z(β,q,q) = (m/2πβ 2)1/2. Then expand the integrand in powers of
and prove the intermediate result
Z =
C
dqe−βV (q)
ξ(β)=0
ξ(0)=0
Dξe− 1
2 m dτ ˙ξ2
1 − V (q) ξ(τ)
−
1
2
2
V (q) ξ2
(τ) − V 2
(q) ξ(τ) ξ(s) + ··· .
Conditional expectation values as
ξ(τ1)ξ(τ2) = ξ(τ2)ξ(τ1) = C
ξ(β)=0
ξ(0)=0
Dξe− 1
2 m dτ ˙ξ2
ξ(τ1)ξ(τ2)
are computed by differentiating the generating functional
C
ξ(β)=0
ξ(0)=0
Dξe− 1
2 m dτ ˙ξ2+ dτjξ
=
m
2πβ
exp
1
mβ
β
0
dτ
τ
0
dτ (β − τ)τ j(τ)j τ .
Prove this formula for the generating functional and compute the leading and sub-
leading contributions in the semi-classical expansion.
2.6 (High-temperature expansion of the partition function) Analyze the temperature
dependence of the partition function (set = 1). Repeat the calculation in prob-
lem 2.5 but this time with the rescalings
τ → βτ and ξ → βξ,
and show that
Z(β) =
C
√
β
dq
ξ(1)=0
ξ(0)=0
Dξ exp −
1
0
m
2
˙ξ2
+ βV (q + βξ) dτ .
Expand Z(β) in powers of the inverse temperature and use the generating functional
in problem 2.5 (with β = 1) to compute the correlation functions. The remaining in-
tegrals over correlation functions are easily calculated. Determine the contributions
of order T 1/2,T −1/2 and T −3/2 in the high-temperature expansion of Z(β).

References 23
References
1. P.A.M. Dirac, The Lagrangian in quantum mechanics. Phys. Z. Sowjetunion 3, 64 (1933)
2. P.A.M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, London,
1947)
3. N. Wiener, Differential space. J. Math. Phys. Sci. 2, 132 (1923)
4. R. Feynman, Spacetime approach to non-relativistic quantum mechanic. Rev. Mod. Phys. 20,
267 (1948)
5. R. Feynman, A. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York,
1965)
6. J. Glimm, A. Jaffe, Quantum Physics: A Functional Integral Point of View (Springer, Berlin,
1981)
7. P.R. Chernoff, Note on product formulas for operator semigroups. J. Funct. Anal. 2, 238
(1968)
8. M. Reed, B. Simon, Methods of Modern Mathematical Physics I (Academic Press, New York,
1972)
9. M. Kac, Random walk and the theory of Brownian motion. Am. Math. Mon. 54, 369 (1947)
10. I.M. Gel’fand, A.M. Yaglom, Integration in functional spaces and its applications in quantum
physics. J. Math. Phys. 1, 48 (1960)
11. G. Roepstorff, Path Integral Approach to Quantum Physics (Springer, Berlin, 1996)
12. L.S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981)
13. E. Nelson, Feynman integrals and the Schrödinger equation. J. Math. Phys. 5, 332 (1964)
14. S.G. Brush, Functional integrals and statistical physics. Rev. Mod. Phys. 33, 79 (1961)
15. J. Zinn-Justin, Path Integrals in Quantum Mechanics (Oxford University Press, London,
2004)
16. H. Kleinert, Path Integral, in Quantum Mechanics, Statistics, Polymer Physics and Financial
Markets (World Scientiﬁc, Singapore, 2006)
17. U. Mosel, Path Integrals in Field Theory: An Introduction (Springer, Berlin, 2004)

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