![Preface
This book is based on a graduate course taught four times, once in French at
the Université de Montréal and then three times in English at the Institut
für Theoretische Physik, in Innsbruck, Austria, at the Center for Quantum
Spacetime, Department of Physics, Sogang University, Seoul, Korea, and most
recently, a part of it at the African Institute for Mathematical Sciences (AIMS),
Cape Town, South Africa.
The course covered the contents of the magnificent Erice lectures of Coleman
[31], “The Uses of Instantons”, in addition to several chapters based on
independent research papers. However, it might be more properly entitled, “The
Uses of Instantons for Dummies”. I met Sidney Coleman a few times, more
than 30 years ago, and although I am sure that he was less impressed with the
meetings than I was and probably relegated them to the dustbin of the memory,
my debt to him is enormous. Without his lecture notes I cannot imagine how I
would ever have been able to understand what the uses of instantons actually
were. However, in his lecture notes, one finds that he also thanks and expresses
gratitude to a multitude of eminent and great theoretical physicists of the era,
indeed thanking them for “patiently explaining large portions of the subject” to
him. Unfortunately, we cannot all be so lucky. Coleman’s lecture notes are a work
of art; it is clear when one reads them that one is enjoying a master impressionist
painter’s review of a subject, a review that transmits, as he says, the “awe and
joy” of the beauty of the “wonderful things brought back from far places”. But
then the hard work begins.
Hence, through diligent, fastidious and brute force work, I have been able, I
hope, to produce what I believe is a well-rounded, detailed monograph, essentially
explaining in a manner accessible to first- and second-year graduate students the
beauty and the depth of what is contained in Coleman’s lectures and in some
elaborations of the whole field itself.
I am indebted to many, but I will thank explicitly Luc Vinet for impelling me to
first give this course when I started out at the Université de Montréal; Gebhard
Grübl for the opportunity to teach the course at the Universität Innsbruck in
Innsbruck, Austria; Bum-Hoon Lee for the same honour at Sogang University
in Seoul, Korea; and Fritz Hahne for the opportunity to give the lectures at
the African Institute for Mathematical Sciences, Cape Town, South Africa. I
thank the many students who took my course and suggested corrections to my](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-17-320.jpg)
![2 Introduction
Unstable
Stable
Figure 1.1. A system trapped in the false vacuum will tunnel through the
barrier to the state of lower energy
space. In this case we are, generally speaking, considering tunnelling through
a barrier, as depicted in Figure 1.1. Classically the particle is not allowed to
enter the space where the potential energy is greater than the total energy of the
particle. Indeed, if the energy of a particle is given by
E = T + V =
q̇2
2
+ V (q) (1.2)
then for a classically fixed energy, regions where E V (q) require that T = q̇2
2 0,
which means that the kinetic energy has to be negative, and such regions are
classically forbidden. Then what takes the role of the dominant contribution in
the limit → 0, since no classical path can interpolate between the initial and
final states?
Heuristically such a region is attainable if t becomes imaginary. Indeed, if t →
−iτ then
dq
dt
2
→
i
dq
dτ
2
= −
dq
dτ
2
, T becomes negative and then perhaps
such regions are accessible. Hence it could be interesting to see what happens if we
analytically continue to imaginary time, equivalent to continuing from Minkowski
spacetime to Euclidean space, which is exactly what we will do. In fact, we
will be able to obtain many results of the usual semi-classical WKB (Wentzel,
Kramers and Brillouin) approximation [119, 77, 22], using the Euclidean space
path integral. The amplitudes that we can calculate, although valid for the small
limit, are not normally attainable in any order in perturbation theory; they
behave like ∼ Ke−S0/
(1 + o()). Such a behaviour actually corresponds to an
essential singularity at = 0.
The importance of being able to do this is manifold. Indeed, it is interesting
to be able to reproduce the results that can be obtained by the standard
WKB method for quantum mechanics using a technique that seems to have
absolutely no connection with that method. Additionally, the methods that we
will enunciate here can be generalized rather easily to essentially any quantum
system, especially to the case of quantum field theory. Tunnelling phenomena](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-20-320.jpg)
![2
Quantum Mechanics and the Path Integral
2.1 Schrödinger Equation and Probability
Our starting point will be single-particle quantum mechanics as defined by the
Schrödinger equation
i
d
dt
Ψ(x,t) = ĥ
x,−i
d
dx
Ψ(x,t). (2.1)
Here ĥ(x,−i d
dx ) is a self-adjoint operator, the Hamiltonian on the space of wave-
functions Ψ(x,t), where x stands for any number of spatial degrees of freedom.
The connection to physics of Ψ(x,t) comes from the interpretation of Ψ(x,t) as
the amplitude of the probability to find the particle between x and x + dx at
time t; hence, the probability density is given by
P[x,x + dx] = Ψ∗
(x,t)Ψ(x,t). (2.2)
Correspondingly, the probability of finding the particle in a volume V is given by
P[V ] =
V
dxΨ∗
(x,t)Ψ(x,t). (2.3)
The state of the system is completely described by the wave function Ψ(x,t). It
is the content of a standard course on quantum mechanics to find Ψ(x,t) for a
given ĥ(x,−i d
dx ).
2.2 Position and Momentum Eigenstates
For our purposes, we introduce the set of (improper) states |x which diagonalize
the position operator X̂, with
X̂|x = x|x (2.4)
and
dx|xx| = I. (2.5)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-22-320.jpg)
![2.2 Position and Momentum Eigenstates 5
We are in principle working in d dimensions, but we suppress the explicit
dependence on the number of coordinates. The states are improper in the sense
that the normalization is
x|y = δ(x − y), (2.6)
where δ(x−y) is the Dirac delta function. We also introduce the set of (improper)
states |p which diagonalize the momentum operator P̂
P̂|p = p|p (2.7)
with
dp|pp| = 1 (2.8)
but as with the position eigenstates
p|p
= δ(p − p
), (2.9)
where δ(p − p
) is the Dirac delta function in momentum space. The improper
states |x and |p are not vectors in the Hilbert space of states, they have infinite
norm. They actually define vector-valued distributions, linear maps from the
space of the square integrable functions of x or p or some suitable set of test
functions usually taken to be of compact support, to actual vectors in the Hilbert
space,
|x : f(x) → |f ∼
dxf(x)|x, (2.10)
where the ∼ should be interpreted as “loosely defined by”. For a more rigorous
definition, see the book by Reed and Simon [107] or Glimm and Jaffe [55].
The operators X̂ and P̂ must satisfy the canonical commutation relation
[X̂,P̂] = i. (2.11)
The algebraic relation Equation (2.11) is not adequate to determine P̂
completely; there are infinitely many representations of the commutator
Equation (2.11) in which X̂ is diagonal. Taking the matrix element of Equation
(2.11) between position eigenstates gives
(x − y)x|P̂|y = x|[X̂,P̂]|y = ix|y = iδ(x − y). (2.12)
For the more mathematically inclined, this expression does not make good sense,
since the position and momentum operators are unbounded, though self-adjoint
operators. They may only act on their respective domains and, correspondingly,
the product of two unbounded operators requires proper analysis of the domains
and ranges of the operators concerned and similar other difficulties can exist. We
leave these subtleties out in what follows, and refer the interested reader to the](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-23-320.jpg)
![6 Quantum Mechanics and the Path Integral
book on functional analysis by Reed and Simon [107]. We find the solution for
x|P̂|y as
x|P̂|y = −i
d
dx
δ(x − y) + cδ(x − y)
= −i
d
dx
x|y + cδ(x − y), (2.13)
where c is an arbitrary constant, using the property of the δ function that (x −
y)δ(x−y) ≡ 0. We will call the x representation the one in which the momentum
operator is represented by a simple derivative, i.e. c = 0,
x|P̂|y = −i
d
dx
x|y. (2.14)
In this representation,
x|P̂|p =
dyx|P̂|yy|p =
dy
−i
d
dx
x|y
y|p
= −i
d
dx
x|p. (2.15)
Acting to the right directly in the left-hand side of Equation (2.15) gives
x|P̂|p = px|p = −i
d
dx
x|p. (2.16)
The appropriately normalized solution of the resulting differential equation is
x|p =
1
(2π)
d
2
ei p·x
, (2.17)
where d is the number of spatial dimensions.
2.3 Energy Eigenstates and Semi-Classical States
We can write the eigenstate of the Hamiltonian in the form |ΨE,
ĥ(X̂,P̂)|ΨE = E|ΨE, (2.18)
where ĥ(X̂,P̂) is defined such that
x|ĥ(X̂,P̂)|f = ĥ
x,−i
d
dx
x|f (2.19)
for any vector |f in the Hilbert space. Then
x|ĥ(X̂,P̂)|ΨE = ĥ
x,−i
d
dx
x|ΨE = Ex|ΨE, (2.20)
which implies the energy eigenfunctions are given by
ΨE(x) = x|ΨE. (2.21)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-24-320.jpg)
![2.4 Time Evolution and Transition Amplitudes 7
Correspondingly,
|ΨE =
dx|xx|ΨE =
dxΨE(x)|x (2.22)
and
ĥ(x,−i
d
dx
)ΨE(x) = EΨE(x). (2.23)
A particle described by ΨE(x) is most likely to be found in the region where
ΨE(x) is peaked. The time-dependent solution of the Schrödinger equation for
static Hamiltonians is given by ΨE(x,t) = ΨE(x)e− i
Et
, and the most general
state of the system is a linear superposition
Ψ(x,t) =
E
AEΨE(x)e− i
Et
(2.24)
with
E
A∗
EAE = 1. (2.25)
Suppose the Hamiltonian can be modified by adjusting the potential, say, such
that ΨE(x) approaches a delta function:
ΨE(x) → δ(x − x0). (2.26)
We would then say that a particle in the energy level E is localized at the point
x0. But in the limit of Equation (2.26) we clearly have
|ΨE → |x0 (2.27)
from Equation (2.22). Thus the states |x describe particles localized at the
spatial point x. This is conceptually important for the semi-classical limit. Semi-
classically we think of particles as localized at points in the configuration space.
Thus the states |x and their generalizations are useful in the description of
quantum systems in the semi-classical limit.
2.4 Time Evolution and Transition Amplitudes
Given a particle in a state |Ψ;0 = |Ψ at t = 0, the Schrödinger equation,
Equation (2.1), governs the time evolution of the state. The state at t = T is
given by
|Ψ;T = e−i T
ĥ(X̂,P̂ )
|Ψ, (2.28)
which satisfies the Schrödinger equation. The exponential of a self-adjoint
operator, which accurs on the right-hand side of Equation (2.28), is rigorously
defined via the spectral representation [107]. The probability amplitude for
finding the particle in a state |Φ at t = T is then given by
Φ|Ψ;T = Φ|e−i
T ĥ(X̂,P̂ )
|Ψ. (2.29)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-25-320.jpg)
![8 Quantum Mechanics and the Path Integral
We could derive an expression for this matrix element in terms of a “path
integral”. Such an integral would be defined as an integral over the space of
all classical paths starting from the initial state and ending at the final state,
and we would find that the function that we would integrate is the exponential
of −i times the classical action for each path. This is the standard Feynman path
integral [45, 46], which was actually suggested by Dirac [40].
2.5 The Euclidean Path Integral
Rather than the matrix element Equation (2.29), we are more interested in a
path-integral representation of the matrix element
Φ|e−
βĥ(X̂,P̂ )
|Ψ, (2.30)
where β can be thought of as imaginary time
T → −iβ. (2.31)
The derivation of the path-integral representation of Equation (2.30) is more
rigorous than that for Equation (2.29); however, the derivation which follows can
be almost identically taken over to the case of real time. This can be completed
by the reader. It is the matrix element of Equation (2.30) that will interest us
in future chapters.
First of all, due to the linearity of quantum mechanics, it is sufficient to
consider the matrix element
y|e− β
ĥ(X̂,P̂ )
|x. (2.32)
To obtain Equation (2.30) we just integrate over x and y with appropriate
smearing functions as in Equation (2.10). Now we write
e−
βĥ(X̂,P̂ )
= e−
ĥ(X̂,P̂ )
· e−
ĥ(X̂,P̂ )
···e−
ĥ(X̂,P̂ )
N+1 factors
, (2.33)
where we mean N + 1 factors on the right-hand side and (N + 1) = β. Next we
insert complete sets of position eigenstates
dzi|zizi| = I, (2.34)
where I is the identity operator. Between the evolution operators appearing on
the right-hand side of Equation (2.33), there will be N such insertions, i.e. i :
1 → N. Consider one of the matrix elements
zi|e−
ĥ(X̂,P̂ )
|zi−1 (2.35)
between position eigenstates |ziand |zi−1 for Hamiltonians of the form
ĥ(X̂,P̂) =
P̂2
2
+ V (X̂). (2.36)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-26-320.jpg)
![2.5 The Euclidean Path Integral 9
Then
zi|e−
ĥ(X̂,P̂ )
|zi−1 = zi|1 −
P̂2
/2 + V (X̂) |zi−1 + o(2
)
=
dpizi|1 −
p2
i /2 + V (zi−1) |pipi|zi−1 + o(2
)
=
dpi 1 −
p2
i /2 + V (zi−1) zi|pipi|zi−1 + o(2
)
=
dpi
(2π)d
e
−
p2
i
2 +V (zi−1)−ipi
(zi−zi−1)
+o(2
)
=
⎛
⎝
dpi
(2π)d
e
−
p2
i
2 −i
pi(zi−zi−1)
− 1
2
zi−zi−1
2
⎞
⎠×
×e
−
1
2
zi−zi−1
2
+V (zi−1)
+o(2
)
, (2.37)
where in the second step, we have inserted a complete set of momentum
eigenstates after letting V (X̂) act on the position eigenstate |zi−1. The first
factor in the last equality is just a (shifted) Gaussian integral, and can be easily
evaluated to give
N =
dpi
(2π)d
e
−
2 (pi−i
(zi−zi−1)
)2
=
1
√
2π
d
. (2.38)
Now we use Equations (2.37) and (2.38) in Equation (2.33), inserting an
independent complete set of position eigenstates between each of the factors
to yield
y|e− β
ĥ(X̂,P̂ )
|x =
dz1 ···dzN
(2π)
Nd
2
N+1
i=1
e
−
1
2
zi−zi−1
2
+V (zi−1)
+o(2
)
=
dz1 ···dzN
(2π)
Nd
2
e
−
N+1
i=1
1
2
zi−zi−1
2
+V (zi−1)
+o(2
)
,
(2.39)
where we define z0 = x and zN+1 = y. Equation (2.39) is actually as far as
one can go rigorously. It expresses the matrix element as a path integral over
piecewise straight (N pieces), continuous paths weighted by the exponential of a
discretized approximation to the negative Euclidean action. In the limit N → ∞,
the o(2
) terms are expected to be negligible. Additionally, in the limit that the
path becomes differentiable, which is actually almost never the case,
N
i=1
1
2
zi − zi−1
2
+ V (zi−1)
→
dτ
V (z(τ)) +
1
2
ż(τ))2
, (2.40)
where τ ∈ [0,β] parametrizes the path such that z(0) = x and z(β) = y. Hence
the matrix element Equation (2.32) can be formally written as the integral over](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-27-320.jpg)
![10 Quantum Mechanics and the Path Integral
classical paths,
y|e− β
ĥ(X̂,P̂ )
|x = N
Dz(τ)e− 1
β
0 dτ(1
2 (ż(τ))2
+V (z(τ)))
= N
Dz(τ)e−
SE[z(τ)]
, (2.41)
where SE[z(τ)] is the classical Euclidean action for each path z(τ), which starts
at x and ends at y. Dz(τ) is the formal integration measure over the space of
all such paths and N is a formally infinite or ill-defined normalization constant,
the limit of 1
(2π)
Nd
2
as N → ∞.
There exists a celebrated measure defined on the space of paths, the so-called
Wiener measure [121], which was defined in the rigorous study of Brownian
motion. One can use it to define the Euclidean path integral rigorously and
unambiguously, certainly for quantum mechanics, but also in many instances for
quantum field theory. We are not interested in these mathematical details, and
we will use and manipulate the path integral as if it were an ordinary integral.
We will have to define what we mean by this measure and normalization more
carefully, later. The measure actually only makes sense, in any rigorous way,
for the discretized version Equation (2.39) including the limit N → ∞; however,
strictly speaking the path integral for smooth paths, Equation (2.41), is just a
formal expression.
We will record here the corresponding formula in Minkowski time:
y|e− iT
ĥ(X̂,P̂ )
|x = N
Dz(t)e
i
T
0 dt(1
2 (ż(t))2
−V (z(t)))
= N
Dz(t)e
i
SM [z(t)]
. (2.42)
This formula can be proved formally by following each of the steps that we have
done for the case of the Euclidean path integral; we leave the details to the
reader. However, the Gaussian integral that we encountered at Equation (2.38)
becomes
N =
dpi
(2π)d
e
−i
2
pi−i
(zi−zi−1)
2
. (2.43)
This expression is ill-defined, but it only contributes to an irrelevant normaliza-
tion constant. We can make it well-defined by adding a small negative imaginary
part to the Hamiltonian, which then yields
N =
i
2π
d
. (2.44)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-28-320.jpg)
![2.5 The Euclidean Path Integral 11
Adding the imaginary part to the Hamiltonian is known in other words as
the “i-epsilon” prescription (note this “epsilon” has nothing to do with the
appearing in our formulas above). Such a deformation can be effected in the case
at hand by changing the pi → (1−iξ)pi in the exponent of Equation (2.43) with
infinitesimal ξ (instead of using the usual “epsilon”). It is tantamount to defining
the Minkowski path integral by starting with the Euclidean path integral and
continuing this back to Minkowski space.
For the remainder of this book, we will be interested in the path-integral
representation, Equation (2.39), of the matrix element Equation (2.32). We will
apply methods that are standard for ordinary integrals to obtain approximations
for the matrix element. We will use the saddle point method for evaluation of the
path integral. This involves finding the critical points of the Euclidean action and
then expanding about the critical point in a (functional) Taylor expansion. The
value of the action at the critical point is a constant as far as the integration is
concerned and just comes out of the integral. This term alone already gives much
novel information about the matrix element. It is usually non-perturbative in the
coupling constant. The first variation of the action vanishes by definition at the
critical point. The first non-trivial term, the second-order term in the Taylor
expansion, yields a Gaussian path integral. The remaining higher-order terms
in the Taylor expansion give perturbative corrections to the Gaussian integral.
The Gaussian integral can sometimes be done explicitly, although this too can
be prohibitively complicated.
We will work with the formal path integral, Equation (2.41), rather than the
exact discretized version, Equation (2.39). First of all it is much easier to find
the critical points of the classical Euclidean action rather than its discretized
analogue. Secondly, in the limit that N → ∞, the critical points for the discrete
action should approach those of the classical action. The actual path integral to
be done always remains defined by the discretized version. The critical point of
the classical action is only to be used as a centre point about which to perform
the path integral Equation (2.39) in the Gaussian approximation and in further
perturbative expansion. As stressed by Coleman [31], the set of smooth paths is a
negligible fraction of the set of all paths. However, this does not dissuade us from
using a particular smooth path, that which is a solution of the classical equations
of motion, as a centre point about which to perform the functional integration in a
Gaussian approximation. The Gaussian path integral corresponds to integration
over all paths, especially including those which are arbitrarily non-smooth, but
which are centred on the particular smooth path corresponding to the solution of
the equations of motion, with a quadratic approximation to the action (or what
is called Gaussian since it leads to an (infinite) product of Gaussian integrals).
It actually receives most of its contribution from extremely non-smooth paths.
However, the Gaussian path integral can be evaluated in some cases exactly,](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-29-320.jpg)
![3
The Symmetric Double Well
In this chapter we will consider in detail a simple quantum mechanical system
where “instantons”, critical points of the classical Euclidean action, can be used
to uncover non-perturbative information about the energy levels and matrix
elements. We will also explicitly see the use of the particular matrix element
(2.27) that we consider. The model we will consider has the classical Euclidean
action
SE[z(τ)] =
β
2
− β
2
dτ
1
2
(ż(τ))2
+ V (z(τ))
. (3.1)
We choose for convenience the domain [−β
2 , β
2 ] and we will choose the potential
explicitly later. We will always have in mind that β → ∞, thus if β is considered
finite, it is to be treated as arbitrarily large. The potential, for now, is simply
required to be a symmetric double well potential, adjusted so that the energy is
equal to zero at the bottom of each well, located at ±a, as depicted in Figure 3.1.
3.1 Classical Critical Points
The critical points of the action, Equation (3.1), are achieved at solutions of the
equations of motion
δSE[z(τ)]
δz(τ)
z(τ)=z̄(τ)
= −¨
z̄(τ
) + V
(z̄(τ
)) = 0. (3.2)
We assume z̄(τ) satisfies Equation (3.2). Then writing z(τ) = z̄(τ) + δz(τ) and
expanding in a Taylor series, we find
SE[z(τ)] = SE[z̄(τ)] +
1
2
dτ
dτ δ2
SE[z(τ)]
δz(τ)δz(τ)
z(τ)=z̄(τ)
δz(τ
)δz(τ
) + ··· ,
(3.3)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-31-320.jpg)
![14 The Symmetric Double Well
V(z)
–a a
Figure 3.1. A symmetric double well potential with minima at ±a
where we note that the first-order variation is absent as the equations of motion,
Equation (3.2), are satisfied. The second-order variation is given by
δ2
SE[z(τ)]
δz(τ)δz(τ)
z(τ)=z̄(τ)
=
−
d2
dτ2 + V
(z̄(τ
)
δ(τ
− τ
). (3.4)
Then we have
SE[z(τ)] = SE[z̄(τ)] +
1
2
β
2
− β
2
dτδz(τ)
−
d2
dτ2
+ V
(z̄(τ))
δz(τ) + ··· . (3.5)
We can expand δz(τ) in terms of the complete orthonormal set of eigenfunctions
zn(τ) of the hermitean operator entering in the second-order term
−
d2
dτ2
+ V
(z̄(τ))
zn(τ) = λnzn(τ), n = 0,1,2,3,··· ,∞ (3.6)
supplied with the boundary conditions
zn(−
β
2
) = zn(
β
2
) = 0. (3.7)
Completeness implies
∞
n=0
zn(τ)zn(τ
) = δ(τ − τ
) (3.8)
while orthonormality gives
β
2
− β
2
dτzn(τ)zm(τ) = δnm. (3.9)
Thus expanding
δz(τ) =
∞
n=0
cnzn(τ) (3.10)
we find
SE[z(τ)] = SE[z̄(τ)] +
1
2
∞
n=0
λnc2
n + o(c3
n) (3.11)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-32-320.jpg)
![3.2 Analysis of the Euclidean Path Integral 15
using the orthonormality Equation (3.9) of the zn(τ)’s.
3.2 Analysis of the Euclidean Path Integral
The original matrix element that we wish to study, Equation (2.32), is given by
y|e− β
ĥ(X̂,P̂ )
|x = z̄(β/2)|e− β
ĥ(X̂,P̂ )
|z̄(−β/2), (3.12)
as we have not yet picked the boundary conditions on z̄(±β/2). Then we get
z̄(β/2)|e− β
ĥ(X̂,P̂ )
|z̄(−β/2) = N
Dz(τ)e
− 1
SE[z̄(τ)]+ 1
2
∞
n=0 λnc2
n+o(c3
n)
= e−
SE[z̄(τ)]
N
Dz(τ)e
− 1
∞
n=0
1
2
λnc2
n+o(c3
n)
.
(3.13)
Now we will begin to define the path integration measure as
Dz(τ) →
∞
n=0
dcn
√
2π
, (3.14)
integrating over all possible values of the cn’s as a reasonable way of integrating
over all paths. The factor of
√
2π in the denominator is purely a convention
and is done for convenience as we shall see; any difference in the normalization
obtained this way can be absorbed into the still undefined normalization
constant, N. Then the expression for the matrix element in Equation (3.13)
becomes
z̄(β/2)|e− β
ĥ(X̂,P̂ )
|z̄(−β/2) = e−
SE[z̄(τ)]
N
∞
n=0
dcn
√
2π
e
− 1
∞
n=0
1
2
λnc2
n+o(c3
n)
.
(3.15)
Scaling cn = c̃n
√
gives for the right-hand side
= e−
SE[z̄(τ)]
N
∞
n=0
dc̃n
√
2π
e
−
1
2
λnc̃2
n+o()
= e−
SE[z̄(τ)]
N
∞
n=0
1
√
λn
(1 + o())
. (3.16)
This infinite product of eigenvalues for the operators which arise typically does
not converge. We will address and resolve this difficulty later and, assuming that
it is so done, we formally write “det” for the product of all the eigenvalues of the
operator. This yields the formula
z̄(β/2)|e− β
ĥ(X̂,P̂ )
|z̄(−β/2)= e−
SE[z̄(τ)]
Ndet
− 1
2
−
d2
dτ2
+ V
(z̄(τ))
(1 + o())
.
(3.17)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-33-320.jpg)
![16 The Symmetric Double Well
Thus we see the matrix element has a non-perturbative contribution in
coming from the exponential of the value of the classical action at the critical
point divided by , e−
SE[z̄(τ)]
, multiplying the yet undefined normalization and
determinant and an expression which has a perturbative expansion in positive
powers of .
3.3 Tunnelling Amplitudes and the Instanton
To proceed further we have to be more specific. We shall consider the following
matrix elements
±a|e− β
ĥ(X̂,P̂ )
|a = ∓a|e− β
ĥ(X̂,P̂ )
| − a. (3.18)
The equality of these matrix elements is easily obtained here by using the
assumed parity reflection symmetry of the Hamiltonian,
x|e− β
ĥ(X̂,P̂ )
|y = x|PPe− β
ĥ(X̂,P̂ )
PP|y
= −x|Pe− β
ĥ(X̂,P̂ )
P| − y
= −x|e− β
ĥ(X̂,P̂ )
| − y, (3.19)
where P is the parity operator which satisfies P2
= 1, P|x = | − x and
[P,ĥ(X̂,P̂)] = 0.
The equation which z̄(τ) satisfies is
− ¨
z̄(τ) + V
(z̄(τ)) = 0, (3.20)
which is exactly the equation of motion for a particle in real time moving in the
reversed potential −V (z), as in Figure 3.2. Because of the matrix elements that
we are interested in, Equation (3.18), the corresponding classical solutions are
those which start at and return to either ±a or those that interpolate between
–V(z)
–a a
Figure 3.2. Inverted double well potential for z̄(τ)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-34-320.jpg)
![18 The Symmetric Double Well
a
–a
z[τ]
–
Figure 3.3. Interpolating kink instanton for the symmetric double well
The action for the constant solutions, Equation (3.21), is evidently zero. For
the interpolating solution implicitly determined by Equation (3.24), it is
SE[z̄(τ)] =
β
2
− β
2
dτ
1
2
˙
z̄2
(τ) + V (z̄(τ))
=
β
2
− β
2
dτ ˙
z̄2
(τ) − c2
=
β
2
− β
2
2V (z̄(τ)) + c2
dz̄
dτ
dτ
− βc2
=
a
−a
dz̄
2V (z̄) + c2
− βc2
. (3.26)
For large β, we neglect c in the integral for SE[z̄(τ)] ≡ S0, and the term −βc2
,
yielding
S0 =
a
−a
dz̄
2V (z̄). (3.27)
This is exactly the action corresponding to the classical solution for β = ∞
depicted in Figure 3.3. Such Euclidean time classical solutions are called
“instantons”.
For large τ the approximate equation satisfied by z̄(τ) is
dz̄
dτ
= ω(a − z̄), (3.28)
obtained by expanding Equation (3.23) as z̄ → a−
from below and where ω2
is the
second derivative of the potential at z̄ = a. There is a corresponding, equivalent
analysis for τ → −∞. These have the solution
|z(τ)| = a − Ce−ω|τ|
. (3.29)
Thus the instanton is exponentially close to ±a for |τ| 1
ω . Its size is 1
ω which
is of order 1, compared with and β. For large |τ| , the solution is essentially
equal to ±a, which is just the trivial solution. The solution is “on” only for an](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-36-320.jpg)
![3.5 Evaluation of the Determinant 27
V˝(z[τ ])
ω2
–
Figure 3.6. The behaviour of V
(z(τ) between τ = ±∞
We will express the matrix element in terms of a Euclidean time evolution
operator U β
2 ,−β
2 as
N
Dz(τ)e
− 1
β
2
−
β
2
dτ 1
2
(˙
z̄2
(τ)+V
(z̄(τ−τ1))z2
(τ))
≡
z = 0
U β
2
,− β
2
z = 0
(3.57)
with explicitly,
U β
2
,− β
2
= T
e
− 1
β
2
−
β
2
dτ
1
2
P̂ 2
+
V (z̄(τ−τ1))
2
X̂2
. (3.58)
Now
U β
2
,− β
2
= U β
2
,τ1+ 1
2ω U (τ1+ 1
2ω ,τ1− 1
2ω )U τ1− 1
2ω ,− β
2
≈ U0 β
2
,τ1+ 1
2ω U (τ1+ 1
2ω ,τ1− 1
2ω )U0
τ1− 1
2ω ,− β
2
, (3.59)
where on the intervals
τ1 + 1
2ω , β
2 and
−β
2 ,τ1 − 1
2ω we can replace the full
evolution operator with the free evolution operator
U0
(τ,τ
) = T
e
− 1
τ
τ dτ 1
2
−2 d2
dz2 +ω2
z2
= e−
(τ−τ)
ĥ0
(X̂,P̂ )
(3.60)
as V
(z̄(τ) is essentially constant and equal to ω2
on these intervals. Then
inserting complete sets of free eigenstates, which are just simple harmonic
oscillator states |En for an oscillator of frequency ω , we obtain
U β
2
,− β
2
=
n,m
e
−
β
2
−τ1− 1
2ω
En
|EnEn|U (τ1+ 1
2ω ,τ1− 1
2ω )|Em
× Em|e
−
τ1− 1
2ω + β
2
Em
(3.61)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-45-320.jpg)
![3.5 Evaluation of the Determinant 29
all giving
N
det
−
d2
dτ2
+ V
(z̄2n+1(τ))
− 1
2
= N
⎛
⎝
det
− d2
dτ2 + V
(z̄2n+1(τ))
λ2n+1
0
⎞
⎠
− 1
2
= N
det
−
d2
dτ2
+ ω2
− 1
2
κλ
1
2
0
2n+1
.
(3.65)
Hence
K = κλ
1
2
0 . (3.66)
It only remains to calculate two things, the free determinant and the correction
factor K.
3.5.1 Calculation of the Free Determinant
To calculate the free determinant, we will use the method of Affleck and Coleman
[31, 114, 36]. Consider the more general case
det
−
d2
dτ2
+ W(τ)
, (3.67)
where the operator acts on the space of functions which vanish at ±β
2 . Formally
we want to compute the infinite product of the eigenvalues of the eigenvalue
problem
−
d2
dτ2
+ W(τ)
ψλn (τ) = λnψλn (τ), ψλn
±
β
2
= 0. (3.68)
The eigenvalues generally increase unboundedly, hence the infinite product is
actually ill-defined. Consider, nevertheless, an ancillary problem
−
d2
dτ2
+ W(τ)
ψλ(τ) = λψλ(τ), ψλ
−
β
2
= 0,
d
dτ
ψλ (τ)
− β
2
= 1. (3.69)
There exists, in general, a solution for each λ; the second boundary condition
can always be satisfied by adjusting the normalization. On the other hand, the
equation in λ
ψλ
β
2
= 0 (3.70)
has solutions exactly at the eigenvalues λ = λn. Affleck and Coleman [31, 114, 36]
propose to define the ratio of the determinant for two different potentials as
det
− d2
dτ2 + W1(τ) − λ
det
− d2
dτ2 + W2(τ) − λ
=
ψ1
λ
β
2
ψ2
λ
β
2
. (3.71)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-47-320.jpg)
![32 The Symmetric Double Well
from Equations (3.64) and (3.66) for n = 0. Thus
1
K2
=
⎛
⎝
ψ0
β
2 /λ0
ψ0
0
β
2
⎞
⎠, (3.87)
where λ0 is the smallest eigenvalue in the presence of an instanton. To calculate
ψ0
β
2 and λ0 approximately we describe again the procedure given in Coleman
[31]. First we need to solve
−∂2
τ + V
(z̄(τ)) ψ0(τ) = 0 (3.88)
with the boundary conditions ψ0(−β/2) = 0 and ∂τ ψ0(−β/2) = 1. We already
know one solution of Equation (3.88), albeit one that does not satisfy the
boundary conditions: the zero mode of the operator in Equation (3.30) due to
time translation invariance, we will call it here x1(τ):
x1(τ) =
1
√
S0
dz̄
dτ
. (3.89)
x1(τ) → Ae−ω|τ|
as τ → ±∞. A is determined by the equation of motion,
Equation (3.30), which integrated once corresponds to
˙
z̄(τ) =
2V (z̄(τ)). (3.90)
Once we have A we can compute ψ(β
2 ) and λ0.
We know that there must exist a second independent solution of the differential
Equation (3.88), y1(τ) which we normalize so that the Wronskian
x1
dy1
dτ
− y1
dx1
dτ
= 2A2
. (3.91)
We remind the reader that the Wronskian between two linearly independent
solutions of a linear second-order differential equation is non-zero, and with no
first derivative term, as in Equation (3.88), is a constant. Then as τ → ±∞ we
have
ẏ1(τ) ± ωy1(τ) = 2Aωeω|τ|
(3.92)
using the known behaviour of x1(τ). The general solution of Equation (3.92) is
any particular solution plus an arbitrary factor times the homogeneous solution
y1(τ) = ±Aeω|τ|
+ Be∓ω|τ|
, (3.93)
where B is an arbitrary constant. Evidently the homogenous solution is a
negligible perturbation on the particular solution, and y1(τ) → ±Aeω|τ|
as
τ → ±∞. Then we construct ψ0(τ) as
ψ0(τ) =
1
2ωA
eωβ/2
x1(τ) + e−ωβ/2
y1(τ) , (3.94)](https://image.slidesharecdn.com/3400189-250606090352-d8a05511/85/The-Theory-And-Applications-Of-Instanton-Calculations-Manu-Paranjape-50-320.jpg)
The Theory And Applications Of Instanton Calculations Manu Paranjape
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- 5. THE THEORY AND APPLICATIONS OF INSTANTON CALCULATIONS Instantons, or pseudoparticles, are solutions to the equations of motion in classical field theories on a Euclidean spacetime. Instantons are found everywhere in quantum theories as they have many applications in quantum tunnelling. Diverse physical phenomena may be described through quantum tunnelling, for example: the Josephson effect, the decay of meta-stable nuclear states, band formation in tight binding models of crystalline solids, the structure of the gauge theory vacuum, confinement in 2+1 dimensions, and the decay of superheated or supercooled phases. Drawing inspiration from Sidney Coleman’s Erice lectures, this volume provides an accessible, detailed introduction to instanton methods, with many applications, making it a valuable resource for graduate students in many areas of physics, from condensed matter, particle and nuclear physics, to string theory. Manu Paranjape has been a professor at the Université de Montréal for the past 30 years. In this time he has worked on quantum field theory, the Skyrme model, non-commutative geometry, quantum spin tunnelling and conformal gravity. Whilst working on induced fermion numbers, he discovered induced angular momentum on flux tube solitons, and more recently he discovered the existence of negative-mass bubbles in de Sitter space, which merited a prize in the Gravity Research Foundation essay competition.
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- 7. S. W. Hawking and G. F. R. Ellis The Large Scale Structure of Space-Time† F. Iachello and A. Arima The Interacting Boson Model† F. Iachello and P. van Isacker The Interacting Boson-Fermion Model† C. Itzykson and J. M. Drouffe Statistical Field Theory Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory† C. Itzykson and J. M. Drouffe Statistical Field Theory Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems† G. Jaroszkiewicz Principles of Discrete Time Mechanics G. Jaroszkiewicz Quantized Detector Networks C. V. Johnson D-Branes† P. S. Joshi Gravitational Collapse and Spacetime Singularities† J. I. Kapusta and C. Gale Finite-Temperature Field Theory: Principles and Applications, 2nd edition† V. E. Korepin, N. M. Bogoliubov and A. G. Izergin Quantum Inverse Scattering Method and Correlation Functions† J. Kroon Conformal Methods in General Relativity M. Le Bellac Thermal Field Theory† Y. Makeenko Methods of Contemporary Gauge Theory† S. Mallik and S. Sarkar Hadrons at Finite Temperature N. Manton and P. Sutcliffe Topological Solitons† N. H. March Liquid Metals: Concepts and Theory† I. Montvay and G. Münster Quantum Fields on a Lattice† P. Nath Supersymmetry, Supergravity, and Unification L. O’Raifeartaigh Group Structure of Gauge Theories† T. Ortín Gravity and Strings, 2nd edition A. M. Ozorio de Almeida Hamiltonian Systems: Chaos and Quantization† M. Paranjape The Theory and Applications of Instanton Calculations L. Parker and D. Toms Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity R. Penrose and W. Rindler Spinors and Space-Time Volume 1: Two-Spinor Calculus and Relativistic Fields† R. Penrose and W. Rindler Spinors and Space-Time Volume 2: Spinor and Twistor Methods in Space-Time Geometry† S. Pokorski Gauge Field Theories, 2nd edition† J. Polchinski String Theory Volume 1: An Introduction to the Bosonic String† J. Polchinski String Theory Volume 2: Superstring Theory and Beyond† J. C. Polkinghorne Models of High Energy Processes† V. N. Popov Functional Integrals and Collective Excitations† L. V. Prokhorov and S. V. Shabanov Hamiltonian Mechanics of Gauge Systems S. Raychaudhuri and K. Sridhar Particle Physics of Brane Worlds and Extra Dimensions A. Recknagel and V. Schiomerus Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes R. J. Rivers Path Integral Methods in Quantum Field Theory† R. G. Roberts The Structure of the Proton: Deep Inelastic Scattering† C. Rovelli Quantum Gravity† W. C. Saslaw Gravitational Physics of Stellar and Galactic Systems† R. N. Sen Causality, Measurement Theory and the Differentiable Structure of Space-Time M. Shifman and A. Yung Supersymmetric Solitons H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt Exact Solutions of Einstein’s Field Equations, 2nd edition† J. Stewart Advanced General Relativity† J. C. Taylor Gauge Theories of Weak Interactions† T. Thiemann Modern Canonical Quantum General Relativity† D. J. Toms The Schwinger Action Principle and Effective Action† A. Vilenkin and E. P. S. Shellard Cosmic Strings and Other Topological Defects† R. S. Ward and R. O. Wells, Jr Twistor Geometry and Field Theory† E. J. Weinberg Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics J. R. Wilson and G. J. Mathews Relativistic Numerical Hydrodynamics† † Available in paperback
- 9. The Theory and Applications of Instanton Calculations MANU PARANJAPE Université de Montréal
- 10. University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06-04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107155473 DOI: 10.1017/9781316658741 c Manu Paranjape 2018 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2018 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library. Library of Congress Cataloguing in Publication Data Names: Paranjape, M. B., author. Title: The theory and applications of instanton calculations / Manu Paranjape (Universite de Montreal). Other titles: Cambridge monographs on mathematical physics. Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2017. | Series: Cambridge monographs on mathematical physics | Includes bibliographical references and index. Identifiers: LCCN 2017033141| ISBN 9781107155473 (hardback ; alk. paper) | Subjects: LCSH: Quantum field theory–Mathematics. | Instantons. Classification: LCC QC174.17.M35 P36 2017 | DDC 530.12–dc23 LC record available at https://lccn.loc.gov/2017033141 ISBN 978-1-107-15547-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. Contents Preface page xiii 1 Introduction 1 1.1 A Note on Notation 3 2 Quantum Mechanics and the Path Integral 4 2.1 Schrödinger Equation and Probability 4 2.2 Position and Momentum Eigenstates 4 2.3 Energy Eigenstates and Semi-Classical States 6 2.4 Time Evolution and Transition Amplitudes 7 2.5 The Euclidean Path Integral 8 3 The Symmetric Double Well 13 3.1 Classical Critical Points 13 3.2 Analysis of the Euclidean Path Integral 15 3.3 Tunnelling Amplitudes and the Instanton 16 3.4 The Instanton Contribution to the Path Integral 19 3.4.1 Translational Invariance Zero Mode 19 3.4.2 Multi-instanton Contribution 21 3.4.3 Two-dimensional Integral Paradigm 24 3.5 Evaluation of the Determinant 25 3.5.1 Calculation of the Free Determinant 29 3.5.2 Evaluation of K 31 3.6 Extracting the Lowest Energy Levels 35 3.7 Tunnelling in Periodic Potentials 36 4 Decay of a Meta-stable State 41 4.1 Decay Amplitude and Bounce Instantons 41 4.2 Calculating the Determinant 44 4.3 Negative Mode 44 4.4 Defining the Analytic Continuation 46 4.4.1 An Explicit Example 46
- 12. viii Contents 4.5 Extracting the Imaginary Part 50 4.5.1 A Little Complex Analysis 50 4.6 Analysis for the General Case 54 5 Quantum Field Theory and the Path Integral 59 5.1 Preliminaries 59 5.2 Canonical Quantization 61 5.2.1 Canonical Quantization of Particle Mechanics 61 5.2.2 Canonical Quantization of Fields 61 5.3 Quantization via the Path Integral 63 5.3.1 The Gaussian Functional Integral 64 5.3.2 The Propagator 67 5.3.3 Analytic Continuation to Euclidean Time 68 6 Decay of the False Vacuum 71 6.1 The Bounce Instanton Solution 72 6.2 The Thin-Wall Approximation 75 6.3 The Fluctuation Determinant 77 6.4 The Fate of the False Vacuum Continued 79 6.4.1 Minkowski Evolution After the Tunnelling 80 6.4.2 Energetics 83 6.5 Technical Details 84 6.5.1 Exactly One Negative Mode 84 6.5.2 Fluctuation Determinant and Renormalization 86 6.6 Gravitational Corrections: Coleman–De Luccia 90 6.6.1 Gravitational Bounce 92 6.7 Induced Vacuum Decay 100 6.7.1 Cosmic String Decay 100 6.7.2 Energetics and Dynamics of the Thin, False String 102 6.7.3 Instantons and the Bulge 104 6.7.4 Tunnelling Amplitude 107 7 Large Orders in Perturbation Theory 111 7.1 Generalities 111 7.2 Particle Mechanics 112 7.3 Generalization to Field Theory 117 7.4 Instantons and Quantum Spin Tunnelling 118 7.5 Spin-Coherent States and the Path Integral for Spin Systems 118 7.6 Coordinate-Independent Formalism 121 7.6.1 Coordinate-Dependent Analysis 121 7.6.2 Coordinate-Independent Analysis 124
- 13. Contents ix 7.7 Instantons in the Spin Exchange Model 128 7.8 The Haldane-like Spin Chain and Instantons 135 7.8.1 Even Number of Sites and Spin-Coherent State Path Integral 137 7.8.2 Odd Spin Chain, Frustration and Solitons 139 8 Quantum Electrodynamics in 1+1 Dimensions 143 8.1 The Abelian Higgs Model 143 8.2 The Euclidean Theory and Finite Action 145 8.2.1 Topological Homotopy Classes 146 8.2.2 Nielsen–Olesen Vortices 147 8.3 Tunnelling Transitions 151 8.4 The Wilson Loop 152 8.4.1 Expectation Value of the Wilson Loop Operator 154 9 The Polyakov Proof of Confinement 157 9.1 Georgi–Glashow model 157 9.2 Euclidean Theory 159 9.2.1 Topological Homotopy Classes 161 9.2.2 Magnetic Monopole Solutions 162 9.3 Monopole Ansatz with Maximal Symmetry 166 9.3.1 Monopole Equations 167 9.4 Non-Abelian Gauge Field Theories 167 9.4.1 Classical Non-Abelian Gauge Invariance 168 9.4.2 The Field Strength 169 9.5 Quantizing Gauge Field Configurations 173 9.5.1 The Faddeev–Popov Determinant 174 9.6 Monopoles in the Functional Integral 177 9.6.1 The Classical Action 177 9.6.2 Monopole Contribution: Zero Modes 178 9.6.3 Defining the Integration Measure 180 9.7 Coulomb Gas and Debye Screening 183 10 Monopole Pair Production 185 10.1 ’t Hooft–Polyakov Magnetic Monopoles 185 10.2 The Euclidean Equations of Motion 185 10.3 The Point Monopole Approximation 187 10.4 The Euclidean Action 188 10.5 The Coulomb Energy 190 10.6 The Fluctuation Determinant 193 10.7 The Final Amplitude for Decay 199
- 14. x Contents 11 Quantum Chromodynamics (QCD) 201 11.1 Definition of QCD 201 11.1.1 The Quark Model and Chiral Symmetry 202 11.1.2 Problems with Chiral Symmetry 204 11.1.3 The Lagrangian of QCD 207 11.2 Topology of the Gauge Fields 209 11.2.1 Topological Winding Number 211 11.3 The Yang–Mills Functional Integral 213 11.3.1 Finite Action Gauge Fields in a Box 214 11.3.2 The Theta Vacua 220 11.3.3 The Yang–Mills Instantons 222 11.4 Theta Vacua in QCD 224 11.4.1 Instantons: Specifics 227 11.4.2 Transitions Between Vacua 230 11.5 Instantons and Confinement 231 11.6 Quarks in QCD 237 11.6.1 Quantum Fermi Fields 240 11.6.2 Fermionic Functional Integral 241 11.6.3 The Axial Anomaly 246 11.6.4 The U(1) Problem 249 11.6.5 Why is there no Goldstone Boson? 253 12 Instantons, Supersymmetry and Morse Theory 259 12.1 A Little Differential Geometry 259 12.1.1 Riemannian Manifolds 259 12.1.2 The Tangent Space, Cotangent Space and Tensors 260 12.2 The de Rham Cohomology 261 12.2.1 The Exterior Algebra 261 12.2.2 Exterior Derivative 262 12.2.3 Integration 263 12.2.4 The Laplacian and the Hodge Decomposition 264 12.2.5 Homology 265 12.2.6 De Rham Cohomology 266 12.3 Supersymmetric Quantum Mechanics 267 12.3.1 The Supersymmetry Algebra 267 12.3.2 Supersymmetric Cohomology 269 12.3.3 1-d Supersymmetric Quantum Mechanics 271 12.3.4 A Useful Deformation 274 12.4 Morse Theory 277 12.4.1 Supersymmetry and the Exterior Algebra 279 12.4.2 The Witten Deformation 280
- 15. Contents xi 12.4.3 The Weak Morse Inequalities 282 12.4.4 Polynomial Morse Inequalities 285 12.4.5 Witten’s Coboundary Operator 288 12.4.6 Supersymmetric Sigma Model 289 12.4.7 The Instanton Calculation 294 Appendix A An Aside on O(4) 297 Appendix B Asymptotic Analysis 299 Bibliography 301 Index 308
- 17. Preface This book is based on a graduate course taught four times, once in French at the Université de Montréal and then three times in English at the Institut für Theoretische Physik, in Innsbruck, Austria, at the Center for Quantum Spacetime, Department of Physics, Sogang University, Seoul, Korea, and most recently, a part of it at the African Institute for Mathematical Sciences (AIMS), Cape Town, South Africa. The course covered the contents of the magnificent Erice lectures of Coleman [31], “The Uses of Instantons”, in addition to several chapters based on independent research papers. However, it might be more properly entitled, “The Uses of Instantons for Dummies”. I met Sidney Coleman a few times, more than 30 years ago, and although I am sure that he was less impressed with the meetings than I was and probably relegated them to the dustbin of the memory, my debt to him is enormous. Without his lecture notes I cannot imagine how I would ever have been able to understand what the uses of instantons actually were. However, in his lecture notes, one finds that he also thanks and expresses gratitude to a multitude of eminent and great theoretical physicists of the era, indeed thanking them for “patiently explaining large portions of the subject” to him. Unfortunately, we cannot all be so lucky. Coleman’s lecture notes are a work of art; it is clear when one reads them that one is enjoying a master impressionist painter’s review of a subject, a review that transmits, as he says, the “awe and joy” of the beauty of the “wonderful things brought back from far places”. But then the hard work begins. Hence, through diligent, fastidious and brute force work, I have been able, I hope, to produce what I believe is a well-rounded, detailed monograph, essentially explaining in a manner accessible to first- and second-year graduate students the beauty and the depth of what is contained in Coleman’s lectures and in some elaborations of the whole field itself. I am indebted to many, but I will thank explicitly Luc Vinet for impelling me to first give this course when I started out at the Université de Montréal; Gebhard Grübl for the opportunity to teach the course at the Universität Innsbruck in Innsbruck, Austria; Bum-Hoon Lee for the same honour at Sogang University in Seoul, Korea; and Fritz Hahne for the opportunity to give the lectures at the African Institute for Mathematical Sciences, Cape Town, South Africa. I thank the many students who took my course and suggested corrections to my
- 18. xiv Preface lectures. I thank Nick Manton, Chris Dobson, and Duncan Dormor, respectively, Fellow, Master and President of St John’s College, University of Cambridge in 2015, for making available to me the many assets of the College that made it possible to work uninterrupted and in a pleasant ambiance on this book, during my stay as an Overseas Visiting Scholar. I also thank my many colleagues and friends who have helped me through discussions and advice; these include Ian Affleck, Richard MacKenzie, Éric Dupuis, Jacques Hurtubise, Keshav Dasgupta and Gordon Semenoff. I especially thank my wife Suneeti Phadke, who started the typing of my lectures in TeX and effectively typed more than half the book while caring for a six-month-old baby. This was no easy feat for someone with a background in Russian literature, devoid of the intricacies of mathematical typesetting. This book would not have come to fruition had it not been for her monumental efforts. I also thank my children Kiran and Meghana, whose very existence makes it a joy and a wonder to be alive.
- 19. 1 Introduction This book covers the methods by which we can use instantons. What is an instanton? A straightforward definition is the following. Given a quantum system, an instanton is a solution of the equations of motion of the corresponding classical system; however, not for ordinary time, but for the analytically continued classical system in imaginary time. This means that we replace t with −iτ in the classical equations of motion. Such solutions are alternatively called the solutions of the Euclidean equations of motion. This type of classical solution can be important in the semi-classical limit → 0. The Feynman path integral, which we will study in its Euclideanized form in great detail in this book, gives the matrix element corresponding to the amplitude for an initial state at t = ti to be found in a final state at t = tf as a “path integral” final,tf |initial,tf = final,tf |e− i (tf −ti)ĥ(q̂,p̂) |initial,ti = final,tf initial,ti DpDqe i dt(pq̇−h(p,q)) (1.1) where ĥ(q̂,p̂) is the quantum Hamiltonian and h(q,p) is the corresponding classical Hamiltonian of the dynamical system. The “path integral” and integration measure DpDq defines an integration over all classical “paths” which satisfy the boundary conditions corresponding to the initial state at ti and to the final state at tf . It is intuitively evident, or certainly from the approximation method of stationary phase, that the dominant contribution, as → 0, should come from the neighbourhood of the classical path which corresponds to a stationary (critical) point of the exponent, since the contributions from non- stationary points of the exponent become suppressed as the regions around them cause wild, self-annihilating variations of the exponential. However, the situation can occur where the particle (or quantum system in general) is classically forbidden from entering some parts of the configuration
- 20. 2 Introduction Unstable Stable Figure 1.1. A system trapped in the false vacuum will tunnel through the barrier to the state of lower energy space. In this case we are, generally speaking, considering tunnelling through a barrier, as depicted in Figure 1.1. Classically the particle is not allowed to enter the space where the potential energy is greater than the total energy of the particle. Indeed, if the energy of a particle is given by E = T + V = q̇2 2 + V (q) (1.2) then for a classically fixed energy, regions where E V (q) require that T = q̇2 2 0, which means that the kinetic energy has to be negative, and such regions are classically forbidden. Then what takes the role of the dominant contribution in the limit → 0, since no classical path can interpolate between the initial and final states? Heuristically such a region is attainable if t becomes imaginary. Indeed, if t → −iτ then dq dt 2 → i dq dτ 2 = − dq dτ 2 , T becomes negative and then perhaps such regions are accessible. Hence it could be interesting to see what happens if we analytically continue to imaginary time, equivalent to continuing from Minkowski spacetime to Euclidean space, which is exactly what we will do. In fact, we will be able to obtain many results of the usual semi-classical WKB (Wentzel, Kramers and Brillouin) approximation [119, 77, 22], using the Euclidean space path integral. The amplitudes that we can calculate, although valid for the small limit, are not normally attainable in any order in perturbation theory; they behave like ∼ Ke−S0/ (1 + o()). Such a behaviour actually corresponds to an essential singularity at = 0. The importance of being able to do this is manifold. Indeed, it is interesting to be able to reproduce the results that can be obtained by the standard WKB method for quantum mechanics using a technique that seems to have absolutely no connection with that method. Additionally, the methods that we will enunciate here can be generalized rather easily to essentially any quantum system, especially to the case of quantum field theory. Tunnelling phenomena
- 21. 1.1 A Note on Notation 3 in quantum field theory are extremely important. The structure of the quantum chromodynamics (QCD) vacuum and its low-energy excitations is intimately connected to tunnelling. Various properties of the phases of quantum field theories are dramatically altered by the existence of tunelling. The decay of the false vacuum and the escape from inflation is also a tunnelling effect that is of paramount importance to cosmology, especially the early universe. The method of instantons lets us study all of these phenomena in one general framework. 1.1 A Note on Notation We will use the following notation throughout this book: metric ημν = (1,−1,−1,−1) (1.3) Minkowsi time t (1.4) Euclidean time τ (1.5) Analytic continuation of time t → −iτ (1.6)
- 22. 2 Quantum Mechanics and the Path Integral 2.1 Schrödinger Equation and Probability Our starting point will be single-particle quantum mechanics as defined by the Schrödinger equation i d dt Ψ(x,t) = ĥ x,−i d dx Ψ(x,t). (2.1) Here ĥ(x,−i d dx ) is a self-adjoint operator, the Hamiltonian on the space of wave- functions Ψ(x,t), where x stands for any number of spatial degrees of freedom. The connection to physics of Ψ(x,t) comes from the interpretation of Ψ(x,t) as the amplitude of the probability to find the particle between x and x + dx at time t; hence, the probability density is given by P[x,x + dx] = Ψ∗ (x,t)Ψ(x,t). (2.2) Correspondingly, the probability of finding the particle in a volume V is given by P[V ] = V dxΨ∗ (x,t)Ψ(x,t). (2.3) The state of the system is completely described by the wave function Ψ(x,t). It is the content of a standard course on quantum mechanics to find Ψ(x,t) for a given ĥ(x,−i d dx ). 2.2 Position and Momentum Eigenstates For our purposes, we introduce the set of (improper) states |x which diagonalize the position operator X̂, with X̂|x = x|x (2.4) and dx|xx| = I. (2.5)
- 23. 2.2 Position and Momentum Eigenstates 5 We are in principle working in d dimensions, but we suppress the explicit dependence on the number of coordinates. The states are improper in the sense that the normalization is x|y = δ(x − y), (2.6) where δ(x−y) is the Dirac delta function. We also introduce the set of (improper) states |p which diagonalize the momentum operator P̂ P̂|p = p|p (2.7) with dp|pp| = 1 (2.8) but as with the position eigenstates p|p = δ(p − p ), (2.9) where δ(p − p ) is the Dirac delta function in momentum space. The improper states |x and |p are not vectors in the Hilbert space of states, they have infinite norm. They actually define vector-valued distributions, linear maps from the space of the square integrable functions of x or p or some suitable set of test functions usually taken to be of compact support, to actual vectors in the Hilbert space, |x : f(x) → |f ∼ dxf(x)|x, (2.10) where the ∼ should be interpreted as “loosely defined by”. For a more rigorous definition, see the book by Reed and Simon [107] or Glimm and Jaffe [55]. The operators X̂ and P̂ must satisfy the canonical commutation relation [X̂,P̂] = i. (2.11) The algebraic relation Equation (2.11) is not adequate to determine P̂ completely; there are infinitely many representations of the commutator Equation (2.11) in which X̂ is diagonal. Taking the matrix element of Equation (2.11) between position eigenstates gives (x − y)x|P̂|y = x|[X̂,P̂]|y = ix|y = iδ(x − y). (2.12) For the more mathematically inclined, this expression does not make good sense, since the position and momentum operators are unbounded, though self-adjoint operators. They may only act on their respective domains and, correspondingly, the product of two unbounded operators requires proper analysis of the domains and ranges of the operators concerned and similar other difficulties can exist. We leave these subtleties out in what follows, and refer the interested reader to the
- 24. 6 Quantum Mechanics and the Path Integral book on functional analysis by Reed and Simon [107]. We find the solution for x|P̂|y as x|P̂|y = −i d dx δ(x − y) + cδ(x − y) = −i d dx x|y + cδ(x − y), (2.13) where c is an arbitrary constant, using the property of the δ function that (x − y)δ(x−y) ≡ 0. We will call the x representation the one in which the momentum operator is represented by a simple derivative, i.e. c = 0, x|P̂|y = −i d dx x|y. (2.14) In this representation, x|P̂|p = dyx|P̂|yy|p = dy −i d dx x|y y|p = −i d dx x|p. (2.15) Acting to the right directly in the left-hand side of Equation (2.15) gives x|P̂|p = px|p = −i d dx x|p. (2.16) The appropriately normalized solution of the resulting differential equation is x|p = 1 (2π) d 2 ei p·x , (2.17) where d is the number of spatial dimensions. 2.3 Energy Eigenstates and Semi-Classical States We can write the eigenstate of the Hamiltonian in the form |ΨE, ĥ(X̂,P̂)|ΨE = E|ΨE, (2.18) where ĥ(X̂,P̂) is defined such that x|ĥ(X̂,P̂)|f = ĥ x,−i d dx x|f (2.19) for any vector |f in the Hilbert space. Then x|ĥ(X̂,P̂)|ΨE = ĥ x,−i d dx x|ΨE = Ex|ΨE, (2.20) which implies the energy eigenfunctions are given by ΨE(x) = x|ΨE. (2.21)
- 25. 2.4 Time Evolution and Transition Amplitudes 7 Correspondingly, |ΨE = dx|xx|ΨE = dxΨE(x)|x (2.22) and ĥ(x,−i d dx )ΨE(x) = EΨE(x). (2.23) A particle described by ΨE(x) is most likely to be found in the region where ΨE(x) is peaked. The time-dependent solution of the Schrödinger equation for static Hamiltonians is given by ΨE(x,t) = ΨE(x)e− i Et , and the most general state of the system is a linear superposition Ψ(x,t) = E AEΨE(x)e− i Et (2.24) with E A∗ EAE = 1. (2.25) Suppose the Hamiltonian can be modified by adjusting the potential, say, such that ΨE(x) approaches a delta function: ΨE(x) → δ(x − x0). (2.26) We would then say that a particle in the energy level E is localized at the point x0. But in the limit of Equation (2.26) we clearly have |ΨE → |x0 (2.27) from Equation (2.22). Thus the states |x describe particles localized at the spatial point x. This is conceptually important for the semi-classical limit. Semi- classically we think of particles as localized at points in the configuration space. Thus the states |x and their generalizations are useful in the description of quantum systems in the semi-classical limit. 2.4 Time Evolution and Transition Amplitudes Given a particle in a state |Ψ;0 = |Ψ at t = 0, the Schrödinger equation, Equation (2.1), governs the time evolution of the state. The state at t = T is given by |Ψ;T = e−i T ĥ(X̂,P̂ ) |Ψ, (2.28) which satisfies the Schrödinger equation. The exponential of a self-adjoint operator, which accurs on the right-hand side of Equation (2.28), is rigorously defined via the spectral representation [107]. The probability amplitude for finding the particle in a state |Φ at t = T is then given by Φ|Ψ;T = Φ|e−i T ĥ(X̂,P̂ ) |Ψ. (2.29)
- 26. 8 Quantum Mechanics and the Path Integral We could derive an expression for this matrix element in terms of a “path integral”. Such an integral would be defined as an integral over the space of all classical paths starting from the initial state and ending at the final state, and we would find that the function that we would integrate is the exponential of −i times the classical action for each path. This is the standard Feynman path integral [45, 46], which was actually suggested by Dirac [40]. 2.5 The Euclidean Path Integral Rather than the matrix element Equation (2.29), we are more interested in a path-integral representation of the matrix element Φ|e− βĥ(X̂,P̂ ) |Ψ, (2.30) where β can be thought of as imaginary time T → −iβ. (2.31) The derivation of the path-integral representation of Equation (2.30) is more rigorous than that for Equation (2.29); however, the derivation which follows can be almost identically taken over to the case of real time. This can be completed by the reader. It is the matrix element of Equation (2.30) that will interest us in future chapters. First of all, due to the linearity of quantum mechanics, it is sufficient to consider the matrix element y|e− β ĥ(X̂,P̂ ) |x. (2.32) To obtain Equation (2.30) we just integrate over x and y with appropriate smearing functions as in Equation (2.10). Now we write e− βĥ(X̂,P̂ ) = e− ĥ(X̂,P̂ ) · e− ĥ(X̂,P̂ ) ···e− ĥ(X̂,P̂ ) N+1 factors , (2.33) where we mean N + 1 factors on the right-hand side and (N + 1) = β. Next we insert complete sets of position eigenstates dzi|zizi| = I, (2.34) where I is the identity operator. Between the evolution operators appearing on the right-hand side of Equation (2.33), there will be N such insertions, i.e. i : 1 → N. Consider one of the matrix elements zi|e− ĥ(X̂,P̂ ) |zi−1 (2.35) between position eigenstates |ziand |zi−1 for Hamiltonians of the form ĥ(X̂,P̂) = P̂2 2 + V (X̂). (2.36)
- 27. 2.5 The Euclidean Path Integral 9 Then zi|e− ĥ(X̂,P̂ ) |zi−1 = zi|1 − P̂2 /2 + V (X̂) |zi−1 + o(2 ) = dpizi|1 − p2 i /2 + V (zi−1) |pipi|zi−1 + o(2 ) = dpi 1 − p2 i /2 + V (zi−1) zi|pipi|zi−1 + o(2 ) = dpi (2π)d e − p2 i 2 +V (zi−1)−ipi (zi−zi−1) +o(2 ) = ⎛ ⎝ dpi (2π)d e − p2 i 2 −i pi(zi−zi−1) − 1 2 zi−zi−1 2 ⎞ ⎠× ×e − 1 2 zi−zi−1 2 +V (zi−1) +o(2 ) , (2.37) where in the second step, we have inserted a complete set of momentum eigenstates after letting V (X̂) act on the position eigenstate |zi−1. The first factor in the last equality is just a (shifted) Gaussian integral, and can be easily evaluated to give N = dpi (2π)d e − 2 (pi−i (zi−zi−1) )2 = 1 √ 2π d . (2.38) Now we use Equations (2.37) and (2.38) in Equation (2.33), inserting an independent complete set of position eigenstates between each of the factors to yield y|e− β ĥ(X̂,P̂ ) |x = dz1 ···dzN (2π) Nd 2 N+1 i=1 e − 1 2 zi−zi−1 2 +V (zi−1) +o(2 ) = dz1 ···dzN (2π) Nd 2 e − N+1 i=1 1 2 zi−zi−1 2 +V (zi−1) +o(2 ) , (2.39) where we define z0 = x and zN+1 = y. Equation (2.39) is actually as far as one can go rigorously. It expresses the matrix element as a path integral over piecewise straight (N pieces), continuous paths weighted by the exponential of a discretized approximation to the negative Euclidean action. In the limit N → ∞, the o(2 ) terms are expected to be negligible. Additionally, in the limit that the path becomes differentiable, which is actually almost never the case, N i=1 1 2 zi − zi−1 2 + V (zi−1) → dτ V (z(τ)) + 1 2 ż(τ))2 , (2.40) where τ ∈ [0,β] parametrizes the path such that z(0) = x and z(β) = y. Hence the matrix element Equation (2.32) can be formally written as the integral over
- 28. 10 Quantum Mechanics and the Path Integral classical paths, y|e− β ĥ(X̂,P̂ ) |x = N Dz(τ)e− 1 β 0 dτ(1 2 (ż(τ))2 +V (z(τ))) = N Dz(τ)e− SE[z(τ)] , (2.41) where SE[z(τ)] is the classical Euclidean action for each path z(τ), which starts at x and ends at y. Dz(τ) is the formal integration measure over the space of all such paths and N is a formally infinite or ill-defined normalization constant, the limit of 1 (2π) Nd 2 as N → ∞. There exists a celebrated measure defined on the space of paths, the so-called Wiener measure [121], which was defined in the rigorous study of Brownian motion. One can use it to define the Euclidean path integral rigorously and unambiguously, certainly for quantum mechanics, but also in many instances for quantum field theory. We are not interested in these mathematical details, and we will use and manipulate the path integral as if it were an ordinary integral. We will have to define what we mean by this measure and normalization more carefully, later. The measure actually only makes sense, in any rigorous way, for the discretized version Equation (2.39) including the limit N → ∞; however, strictly speaking the path integral for smooth paths, Equation (2.41), is just a formal expression. We will record here the corresponding formula in Minkowski time: y|e− iT ĥ(X̂,P̂ ) |x = N Dz(t)e i T 0 dt(1 2 (ż(t))2 −V (z(t))) = N Dz(t)e i SM [z(t)] . (2.42) This formula can be proved formally by following each of the steps that we have done for the case of the Euclidean path integral; we leave the details to the reader. However, the Gaussian integral that we encountered at Equation (2.38) becomes N = dpi (2π)d e −i 2 pi−i (zi−zi−1) 2 . (2.43) This expression is ill-defined, but it only contributes to an irrelevant normaliza- tion constant. We can make it well-defined by adding a small negative imaginary part to the Hamiltonian, which then yields N = i 2π d . (2.44)
- 29. 2.5 The Euclidean Path Integral 11 Adding the imaginary part to the Hamiltonian is known in other words as the “i-epsilon” prescription (note this “epsilon” has nothing to do with the appearing in our formulas above). Such a deformation can be effected in the case at hand by changing the pi → (1−iξ)pi in the exponent of Equation (2.43) with infinitesimal ξ (instead of using the usual “epsilon”). It is tantamount to defining the Minkowski path integral by starting with the Euclidean path integral and continuing this back to Minkowski space. For the remainder of this book, we will be interested in the path-integral representation, Equation (2.39), of the matrix element Equation (2.32). We will apply methods that are standard for ordinary integrals to obtain approximations for the matrix element. We will use the saddle point method for evaluation of the path integral. This involves finding the critical points of the Euclidean action and then expanding about the critical point in a (functional) Taylor expansion. The value of the action at the critical point is a constant as far as the integration is concerned and just comes out of the integral. This term alone already gives much novel information about the matrix element. It is usually non-perturbative in the coupling constant. The first variation of the action vanishes by definition at the critical point. The first non-trivial term, the second-order term in the Taylor expansion, yields a Gaussian path integral. The remaining higher-order terms in the Taylor expansion give perturbative corrections to the Gaussian integral. The Gaussian integral can sometimes be done explicitly, although this too can be prohibitively complicated. We will work with the formal path integral, Equation (2.41), rather than the exact discretized version, Equation (2.39). First of all it is much easier to find the critical points of the classical Euclidean action rather than its discretized analogue. Secondly, in the limit that N → ∞, the critical points for the discrete action should approach those of the classical action. The actual path integral to be done always remains defined by the discretized version. The critical point of the classical action is only to be used as a centre point about which to perform the path integral Equation (2.39) in the Gaussian approximation and in further perturbative expansion. As stressed by Coleman [31], the set of smooth paths is a negligible fraction of the set of all paths. However, this does not dissuade us from using a particular smooth path, that which is a solution of the classical equations of motion, as a centre point about which to perform the functional integration in a Gaussian approximation. The Gaussian path integral corresponds to integration over all paths, especially including those which are arbitrarily non-smooth, but which are centred on the particular smooth path corresponding to the solution of the equations of motion, with a quadratic approximation to the action (or what is called Gaussian since it leads to an (infinite) product of Gaussian integrals). It actually receives most of its contribution from extremely non-smooth paths. However, the Gaussian path integral can be evaluated in some cases exactly,
- 30. 12 Quantum Mechanics and the Path Integral and in other cases in a perturbative approximation. In this way the exact definition of the formal path integral, Equation (2.41), is not absolutely essential for our further considerations. We will, however, continue to frame our analysis in terms of it, content with the understanding that underlying it a more rigorous expression always exists.
- 31. 3 The Symmetric Double Well In this chapter we will consider in detail a simple quantum mechanical system where “instantons”, critical points of the classical Euclidean action, can be used to uncover non-perturbative information about the energy levels and matrix elements. We will also explicitly see the use of the particular matrix element (2.27) that we consider. The model we will consider has the classical Euclidean action SE[z(τ)] = β 2 − β 2 dτ 1 2 (ż(τ))2 + V (z(τ)) . (3.1) We choose for convenience the domain [−β 2 , β 2 ] and we will choose the potential explicitly later. We will always have in mind that β → ∞, thus if β is considered finite, it is to be treated as arbitrarily large. The potential, for now, is simply required to be a symmetric double well potential, adjusted so that the energy is equal to zero at the bottom of each well, located at ±a, as depicted in Figure 3.1. 3.1 Classical Critical Points The critical points of the action, Equation (3.1), are achieved at solutions of the equations of motion δSE[z(τ)] δz(τ) z(τ)=z̄(τ) = −¨ z̄(τ ) + V (z̄(τ )) = 0. (3.2) We assume z̄(τ) satisfies Equation (3.2). Then writing z(τ) = z̄(τ) + δz(τ) and expanding in a Taylor series, we find SE[z(τ)] = SE[z̄(τ)] + 1 2 dτ dτ δ2 SE[z(τ)] δz(τ)δz(τ) z(τ)=z̄(τ) δz(τ )δz(τ ) + ··· , (3.3)
- 32. 14 The Symmetric Double Well V(z) –a a Figure 3.1. A symmetric double well potential with minima at ±a where we note that the first-order variation is absent as the equations of motion, Equation (3.2), are satisfied. The second-order variation is given by δ2 SE[z(τ)] δz(τ)δz(τ) z(τ)=z̄(τ) = − d2 dτ2 + V (z̄(τ ) δ(τ − τ ). (3.4) Then we have SE[z(τ)] = SE[z̄(τ)] + 1 2 β 2 − β 2 dτδz(τ) − d2 dτ2 + V (z̄(τ)) δz(τ) + ··· . (3.5) We can expand δz(τ) in terms of the complete orthonormal set of eigenfunctions zn(τ) of the hermitean operator entering in the second-order term − d2 dτ2 + V (z̄(τ)) zn(τ) = λnzn(τ), n = 0,1,2,3,··· ,∞ (3.6) supplied with the boundary conditions zn(− β 2 ) = zn( β 2 ) = 0. (3.7) Completeness implies ∞ n=0 zn(τ)zn(τ ) = δ(τ − τ ) (3.8) while orthonormality gives β 2 − β 2 dτzn(τ)zm(τ) = δnm. (3.9) Thus expanding δz(τ) = ∞ n=0 cnzn(τ) (3.10) we find SE[z(τ)] = SE[z̄(τ)] + 1 2 ∞ n=0 λnc2 n + o(c3 n) (3.11)
- 33. 3.2 Analysis of the Euclidean Path Integral 15 using the orthonormality Equation (3.9) of the zn(τ)’s. 3.2 Analysis of the Euclidean Path Integral The original matrix element that we wish to study, Equation (2.32), is given by y|e− β ĥ(X̂,P̂ ) |x = z̄(β/2)|e− β ĥ(X̂,P̂ ) |z̄(−β/2), (3.12) as we have not yet picked the boundary conditions on z̄(±β/2). Then we get z̄(β/2)|e− β ĥ(X̂,P̂ ) |z̄(−β/2) = N Dz(τ)e − 1 SE[z̄(τ)]+ 1 2 ∞ n=0 λnc2 n+o(c3 n) = e− SE[z̄(τ)] N Dz(τ)e − 1 ∞ n=0 1 2 λnc2 n+o(c3 n) . (3.13) Now we will begin to define the path integration measure as Dz(τ) → ∞ n=0 dcn √ 2π , (3.14) integrating over all possible values of the cn’s as a reasonable way of integrating over all paths. The factor of √ 2π in the denominator is purely a convention and is done for convenience as we shall see; any difference in the normalization obtained this way can be absorbed into the still undefined normalization constant, N. Then the expression for the matrix element in Equation (3.13) becomes z̄(β/2)|e− β ĥ(X̂,P̂ ) |z̄(−β/2) = e− SE[z̄(τ)] N ∞ n=0 dcn √ 2π e − 1 ∞ n=0 1 2 λnc2 n+o(c3 n) . (3.15) Scaling cn = c̃n √ gives for the right-hand side = e− SE[z̄(τ)] N ∞ n=0 dc̃n √ 2π e − 1 2 λnc̃2 n+o() = e− SE[z̄(τ)] N ∞ n=0 1 √ λn (1 + o()) . (3.16) This infinite product of eigenvalues for the operators which arise typically does not converge. We will address and resolve this difficulty later and, assuming that it is so done, we formally write “det” for the product of all the eigenvalues of the operator. This yields the formula z̄(β/2)|e− β ĥ(X̂,P̂ ) |z̄(−β/2)= e− SE[z̄(τ)] Ndet − 1 2 − d2 dτ2 + V (z̄(τ)) (1 + o()) . (3.17)
- 34. 16 The Symmetric Double Well Thus we see the matrix element has a non-perturbative contribution in coming from the exponential of the value of the classical action at the critical point divided by , e− SE[z̄(τ)] , multiplying the yet undefined normalization and determinant and an expression which has a perturbative expansion in positive powers of . 3.3 Tunnelling Amplitudes and the Instanton To proceed further we have to be more specific. We shall consider the following matrix elements ±a|e− β ĥ(X̂,P̂ ) |a = ∓a|e− β ĥ(X̂,P̂ ) | − a. (3.18) The equality of these matrix elements is easily obtained here by using the assumed parity reflection symmetry of the Hamiltonian, x|e− β ĥ(X̂,P̂ ) |y = x|PPe− β ĥ(X̂,P̂ ) PP|y = −x|Pe− β ĥ(X̂,P̂ ) P| − y = −x|e− β ĥ(X̂,P̂ ) | − y, (3.19) where P is the parity operator which satisfies P2 = 1, P|x = | − x and [P,ĥ(X̂,P̂)] = 0. The equation which z̄(τ) satisfies is − ¨ z̄(τ) + V (z̄(τ)) = 0, (3.20) which is exactly the equation of motion for a particle in real time moving in the reversed potential −V (z), as in Figure 3.2. Because of the matrix elements that we are interested in, Equation (3.18), the corresponding classical solutions are those which start at and return to either ±a or those that interpolate between –V(z) –a a Figure 3.2. Inverted double well potential for z̄(τ)
- 35. 3.3 Tunnelling Amplitudes and the Instanton 17 ±a and ∓a, and each in time β. The trivial solutions z̄(τ) = ±a (3.21) satisfy the first condition while the second condition can be obtained by integrating Equation (3.20). Straightforwardly, ¨ z̄(τ) ˙ z̄(τ) = V (z̄(τ)) ˙ z̄(τ), (3.22) which implies ˙ z̄(τ) = 2V (z̄(τ)) + c2, (3.23) where c is an integration constant. Integrating one more time and choosing the solution that interpolates from −a to a, we get z̄(τ) −a dz̄ 2V (z̄) + c2 = τ − β 2 dτ = τ + β 2 (3.24) and c is determined by a −a dz̄ 2V (z̄) + c2 = β. (3.25) We note that this last Equation (3.25) does not depend on the details of the solution, but only on the fact that it must interpolate from −a to a. Obviously from Equation (3.23), c is the initial velocity. The initial velocity is not arbitrary, the solution must interpolate from −a to a in Euclidean time β, and Equation (3.25) implicitly gives c as a function of β. There is no solution that starts with vanishing initial velocity but interpolates between ±a in finite time β; vanishing initial velocity requires infinite time. As β → ∞, the only way for the integral in Equation (3.25) to diverge to give an infinite or very large β is for the denominator to vanish. This only occurs for V (z̄) → 0 and for c → 0. V (z̄) → 0 occurs as z̄ → ±a, which is near the start and end of the trajectory. Also, physically, if the particle is to interpolate from −a to a in a longer and longer time, β, then it must start out at −a with a smaller and smaller initial velocity, c. For larger and larger β, c must vanish in an appropriate fashion. Heuristically, for small c, the solution spends most of its time near z̄ = ±a and interpolates from one to the other relatively quickly. Then the major contribution to the integral comes from the region around z̄ = ±a. Since the integral diverges logarithmically when c = 0, for a typical potential V (which must vanish quadratically at z̄ = ±a as V has a double zero at ±a), the integral must behave as −lnc, i.e. β ∼ −lnc which is equivalent to c ∼ e−β , which means that c must vanish exponentially with large β. For sufficiently large β we may neglect c altogether.
- 36. 18 The Symmetric Double Well a –a z[τ] – Figure 3.3. Interpolating kink instanton for the symmetric double well The action for the constant solutions, Equation (3.21), is evidently zero. For the interpolating solution implicitly determined by Equation (3.24), it is SE[z̄(τ)] = β 2 − β 2 dτ 1 2 ˙ z̄2 (τ) + V (z̄(τ)) = β 2 − β 2 dτ ˙ z̄2 (τ) − c2 = β 2 − β 2 2V (z̄(τ)) + c2 dz̄ dτ dτ − βc2 = a −a dz̄ 2V (z̄) + c2 − βc2 . (3.26) For large β, we neglect c in the integral for SE[z̄(τ)] ≡ S0, and the term −βc2 , yielding S0 = a −a dz̄ 2V (z̄). (3.27) This is exactly the action corresponding to the classical solution for β = ∞ depicted in Figure 3.3. Such Euclidean time classical solutions are called “instantons”. For large τ the approximate equation satisfied by z̄(τ) is dz̄ dτ = ω(a − z̄), (3.28) obtained by expanding Equation (3.23) as z̄ → a− from below and where ω2 is the second derivative of the potential at z̄ = a. There is a corresponding, equivalent analysis for τ → −∞. These have the solution |z(τ)| = a − Ce−ω|τ| . (3.29) Thus the instanton is exponentially close to ±a for |τ| 1 ω . Its size is 1 ω which is of order 1, compared with and β. For large |τ| , the solution is essentially equal to ±a, which is just the trivial solution. The solution is “on” only for an
- 37. 3.4 The Instanton Contribution to the Path Integral 19 “instant”, the relatively short time compared with β, during which it interpolates between −a and +a. Hence the name instanton. Reversing the time direction gives another solution which starts at +a and interpolates to −a, aptly called an anti-instanton. It clearly has the same action as an instanton. 3.4 The Instanton Contribution to the Path Integral 3.4.1 Translational Invariance Zero Mode As we have seen, for very large β, the instanton corresponding to infinite β is an arbitrarily close and perfectly good approximation to the true instanton. Evidently with the infinite β instanton, we may choose the time arbitrarily at which the solution crosses over from −a to +a. The solution of z̄(τ) 0 dz 2V (z) = τ − τ0 (3.30) corresponds to an instanton which crosses over around τ = τ0 . Thus the position of the instanton τ0 gives a one-parameter family of solutions, each with the same classical action. The point is that for large enough β, there exists a one- parameter family of approximate critical points with action arbitrarily close to S0. The contribution to the path integral from the vicinity of these approximate critical points will be of a slightly modified form, since the first variation of the action about the approximate critical point does not quite vanish. Thus the contribution will be of the form, the exponential of the negative action at the approximate critical point, multiplied by a Gaussian integral with a linear shift, the shift coming from the non-vanishing first variation of the action. The shift will be proportional to some arbitrarily small function f(β) as β → ∞ . The higher-order terms give perturbative corrections in , as in Equation (3.16), and can be dropped. Then, considering a typical Gaussian integral with a small linear shift, as arises in the integration about an approximate critical point, we have ∞ −∞ dx √ 2π e− 1 (α2 x2 +2f(β)x) = e f2(β) α2 1 α . (3.31) We see that to be able to neglect the effects of the shift, f(β) must be so small that f2 (β) 1, given that α, being independent of and β, is of order 1. Typically, f(β) is exponentially small in β, just as earlier c was found to be. f(β) needs to be determined and depends of the details of the dynamics. In any case, β must be greater than a certain value determined by the value of . This is, however, no strong constraint other than imposing that we must consider the limit that β is arbitrarily large while all other constants (especially ) are held fixed. Hence, assuming β is sufficiently large, we can neglect the effect of the linear shift and we must include the contribution from these approximate critical points. To do so, we simply integrate over the position of the instanton and
- 38. 20 The Symmetric Double Well perform the Gaussian integral over directions in path space which are orthogonal to the direction corresponding to translations of the instanton. The easiest way to perform such a constrained Gaussian integral is to use the following observations. In the infinite β limit, the translated instantons become exact critical points and correspondingly the fluctuation directions about a given instanton contain a flat direction. This means that the action does not change to second order for variations along this direction. Precisely, this means that the eigenfrequencies, λn, contain a zero mode, λ0 = 0. We can explicitly construct this zero mode since − d2 dτ2 + V (z̄(τ − τ1)) dz̄(τ − τ1) dτ1 = − d dτ (−¨ z̄(τ − τ1) + V (z̄(τ − τ1))) = 0, (3.32) the second term vanishing by the equation of motion, Equation (3.20), which is clearly also valid for z̄(τ −τ1). This mode occurs because of the time translation invariance when β is infinite. The corresponding normalized zero mode is z0(τ) = 1 √ S0 d dτ1 z̄(τ − τ1). (3.33) Clearly ∞ −∞ dτ 1 √ S0 d dτ1 z̄(τ − τ1) 2 = 1 S0 ∞ −∞ dτ 1 2 ˙ z̄2 (τ − τ1) + V (z̄(τ − τ1)) = 1 (3.34) using the equation of motion, Equation (3.23), with c = 0 (infinite β). Integration in the path integral, Equation (3.15), over the coefficient of this mode yields a divergence as the frequency is zero dc0 √ 2π e− 1 λ0c2 0 = dc0 √ 2π 1 = ∞. (3.35) However, integrating over the position of the instanton is equivalent to integrating over c0. τ1 is called a collective coordinate of the instanton corresponding to its position in Euclidean time. Indeed, if z̄(τ −τ1) is an instanton at position τ1, the change in the path obtained by infinitesimally changing τ1 is δz(τ) = d dτ1 z̄(τ − τ1)dτ1 = S0z0(τ). (3.36) The change induced by varying c0 is, however, δz(τ) = z0(τ)dc0. (3.37) Thus dc0 √ 2π = S0 2π dτ1 (3.38) and when integrating over the position τ1 we should multiply by the normalizing factor S0 2π . Clearly for infinite β the integral over τ1 diverges, reflecting the equivalent infinity obtained when integrating over c0.
- 39. 3.4 The Instanton Contribution to the Path Integral 21 This divergence is not disturbing, since for a positive definite Hamiltonian the infinite β limit of the matrix element, Equation (2.32), is strictly zero, and for large β it is an expression which vanishes exponentially. Thus in the large β limit, the Gaussian integrals in the directions orthogonal to the flat direction must combine to give an expression which indeed vanishes exponentially with β, as we will see. For the time being, for finite β, the integration over the position then gives a factor that is linear in β S0 2π β. (3.39) Thus, so far the path integral has yielded a|e− β ĥ(X̂,P̂ ) | − a = e− S0 S0 2π 1 2 βN det − d2 dτ2 + V (z̄(τ) − 1 2 , (3.40) where det means the “determinant” excluding the zero eigenvalue. We will leave the evaluation of the determinant for a little later when will show that N det − d2 dτ2 + V (z̄(τ)) − 1 2 = KN det − d2 dτ2 + ω2 − 1 2 , (3.41) where ω was defined at Equation (3.28), and we will evaluate K, which is, most importantly, independent of and β. 3.4.2 Multi-instanton Contribution To proceed further, we must realize that there are also other approximate critical points which give significant contributions to the path integral. These correspond to classical configurations which have, for example, an instanton at τ1, an anti- instanton at τ2 and again an instanton at τ3. If τi are well separated within the interval β, these configurations are approximately critical, with an error of the same order as for the approximate critical points previously considered. More generally we can have a string of n pairs of an instanton followed by an anti- instanton, plus a final instanton completing the interpolation from −a to a. We denote such a configuration as z̄2n+1(τ). The positions are arbitrary except that the order of the instantons and the anti-instantons must be preserved and they must be well separated. The action for 2n + 1 such objects is just (2n + 1)S0 to the same degree of accuracy. One would, at first sight, conclude that this contribution, including the Gaussian integral about these approximate critical points, is exponentially suppressed relative to the contribution from the single instanton sector. Indeed, we would find that the contribution of the 2n+1-instantons and anti-instantons
- 40. 22 The Symmetric Double Well to the matrix element1 , a|e− β ĥ(X̂,P̂ ) | − a2n+1 = e− (2n+1)S0 N det − d2 dτ2 + V (z̄2n+1(τ) − 1 2 (3.42) is suppressed by e− 2nS0 relative to the one instanton contribution. This is true; however, we must analyse the effects of zero modes. For 2n + 1 instantons and anti-instantons there are 2n + 1 zero modes corresponding to the independent translation of each object. This is actually only true for infinitely separated objects with β infinite; however, for β large, it is an arbitrarily good approximation. Thus there exist 2n + 1 zero frequencies in the determinant which should not be included in the path integration and, correspondingly, we should integrate over the positions of the 2n + 1 instantons and anti-instantons. This integration is constrained by the condition that their order is preserved. Hence we get the factor β 2 − β 2 dτ1 β 2 τ1 dτ2 β 2 τ2 dτ3 ··· β 2 τ2n−1 dτ2n β 2 τ2n dτ2n+1 = β2n+1 (2n + 1)! . (3.43) Furthermore, from exactly the same analysis as the integration over the position of the single instanton, the integration is normalized correctly only when each factor is multiplied by S0 2π 1 2 . Thus we find a e− β ĥ(X̂,P̂ ) − a 2n+1 = e− S0 S0 2π 1 2 β 2n+1 N (2n + 1)! det − d2 dτ2 + V (z̄2n+1(τ)) − 1 2 , (3.44) where det again means the determinant with the 2n + 1 zero modes removed. We will show later that N det − d2 dτ2 + V (z̄2n+1(τ)) − 1 2 = K2n+1 N det − d2 dτ2 + ω2 − 1 2 (3.45) for the same K as in the case of one instanton, as in Equation (3.41). Now even if e− S0 is very small, our whole analysis is done at fixed with β → ∞; the relevant parameter, as can be seen from Equation (3.44), is δ = S0 2π 1 2 e− S0 Kβ, (3.46) which is arbitrarily large in this limit. Thus it seems that the contribution from the strings of instanton and anti-instanton pairs is proportional to δ2n+1 and 1 Here the subscript 2n + 1 signifies that we are calculating only the contribution to the matrix element from 2n + 1 instantons and anti-instantons.
- 41. 3.4 The Instanton Contribution to the Path Integral 23 seems to get larger and larger. However, the denominator contains (2n + 1)!, which must be taken into account. For large enough n, the denominator always dominates, δ2n+1 (2n + 1)!, and so renders the contribution small. We require, however, for the consistency of our approximations that when n is large enough so that this is true, the average space per instanton or anti- instanton, β 2n+1 , is still large compared to the size of these objects ∼ 1/ω, which is independent of both and β. This is satisfied as β → ∞. Hence we require n large enough such that δ2n+1 (2n + 1)! 1; (3.47) however, with β 2n + 1 1 ω . (3.48) Taking the logarithm of Equation (3.47) after multiplying by (2n + 1)! yields in the Stirling approximation (2n + 1)lnδ (2n + 1)ln(2n + 1) − (2n + 1). (3.49) Neglecting the second term on the right-hand side and combining with Equation (3.48) yields δ = S0 2π 1 2 e− S0 K β 2n + 1 ωβ. (3.50) That such an n can exist simply requires S0 2π 1 2 e− S0 K ≪ ω. We will evaluate K explicitly and find that it does not depend on or β. The inequality then is clearly satisfied for → 0, which brings into focus that underneath everything we are interested in the semi-classical limit. A tiny parenthetical remark is in order: in integrating over the positions of the instantons, we should always maintain the constraint that the instantons are well separated. Thus we should not integrate the position of one instanton exactly from that of the preceding one to that of the succeeding one, but we should leave a gap of the order of 1 ω which is the size of the instanton. Such a correction corresponds to a contribution which behaves to leading order as 1 ω βn−1 (n−1)! , which is negligible in comparison to βn n! if 1 ω β. When the density of instantons and anti-instantons becomes large, all of our approximations break down, and such configurations are no longer even approximately critical. Thus we do not expect any significant contribution to the path integral from the regions of the space of paths which include these configurations. Hence we should actually truncate the series in the number of instantons for some large enough n; however, this is not necessary. We will always assume that we work in the limit that β should be sufficiently large and sufficiently small so that the contribution from the terms in the series with
- 42. 24 The Symmetric Double Well f(x) • • • • • • 4S0 3S0 2S0 S0 Figure 3.4. A simple function analogous to the action n greater than some N is already negligible, while there is still a lot of room per instanton, i.e. β/N is still large. This should still correspond to a dilute “gas” of instantons and anti-instantons. Then the remaining terms in the series can be maintained, although they do not represent the contribution from any part of path space. It is simply easier to sum the series to infinity, knowing that the contribution added in from n greater than some N makes only a negligible change. The sum to infinity is straightforward. We find a e− β ĥ(X̂,P̂ ) − a = N det − d2 dτ2 + ω2 − 1 2 sinh S0 2π 1 2 e− S0 Kβ . (3.51) 3.4.3 Two-dimensional Integral Paradigm A simple two-dimensional, ordinary integral which serves as a paradigm exhibiting many of the features of the path integral just considered is given by I = dxdye− 1 (f(x)+ α2 2 y2 ) (3.52) where y corresponds to the transverse directions and plays no role. f(x) is a function of the form depicted in Figure 3.4 and increases sharply in steps of S0, and the length of each plateau is βn n! . In the limit that the steps become sharp, the integral can be done exactly and yields I = (2π) 1 2 α ∞ n=0 e− nS0 βn n! = (2π) 1 2 α e βe − S0 . (3.53)
- 43. 3.5 Evaluation of the Determinant 25 Obviously this is exactly analogous to the path integral just considered for β → ∞ and → 0 . The plateaux correspond to the critical points. Clearly we cannot consider just the lowest critical point since the volume associated with the higher critical points is sufficiently large that their contribution does not damp out until n becomes large enough. In terms of physically intuitive arguments, the volume is like the entropy factor associated with n instantons, βn n! , while the exponential, e− nS0 , is like the Boltzmann factor. In statistical mechanics, even though the Boltzmann factor is much smaller for higher energy levels, their contribution to the partition function can be significant due to a large enough entropy. We can further model the aspect of approximate critical points by giving the plateaux in Figure 3.4 a very small slope. Clearly the integral is only negligibly modified if the slope is taken to be exponentially small in β. 3.5 Evaluation of the Determinant Finally, we are left with the evaluation of the determinant. We wish to show for the case of 2n + 1 instantons and anti-instantons N det − d2 dτ2 + V (z̄2n+1(τ)) − 1 2 = K2n+1 N det − d2 dτ2 + ω2 − 1 2 (3.54) and to evaluate K. Physically this means that the effect of each instanton and anti-instanton is simply to multiply the free determinant by a factor of 1 K2 . Intuitively this is very reasonable, and we expect that for well-separated instantons their effect would be independent of each other. To obtain the det we will work in the finite large interval, β, with boundary conditions that the wave function must vanish at the end points. Consider first the case of just one instanton. Because of the finite interval, time translation will not be an exact symmetry and the operator − d2 dτ2 + V (z̄(τ)) will not have an exact zero mode. However, as β → ∞ one mode will approach zero. The det is then obtained by calculating the full determinant on the finite interval, β, and then dividing out by the smallest eigenvalue. There should be a rigorous theorem proving first that the operator in question has a positive definite spectrum on the finite interval, β, for any potential, V (z), of the type considered and the corresponding instanton, z̄(τ), and secondly as β → ∞, one bound state drops to exactly zero; this is reasonable and taken as a hypothesis. Thus we will study the full determinant on the interval β which has the path-integral representation N det − d2 dτ2 + V (z̄(τ − τ1)) − 1 2 = N Dz(τ)e − 1 β 2 − β 2 dτ 1 2 (ż2 (τ)+V (z̄(τ−τ1))z2 (τ)) (3.55)
- 44. 26 The Symmetric Double Well –a a V˝(z) ω2 Figure 3.5. The behaviour of V (z) between ±a with the boundary conditions that z(β 2 ) = z(−β 2 ) = 0 in the path integral. The path integral on the right-hand side is performed in exactly the same manner as in Equation (3.15). This determinant actually corresponds to the matrix element of the Euclidean time evolution operator with a time-dependent Hamiltonian, z = 0|T e − 1 β 2 − β 2 dτ 1 2 P̂ 2 + V (z̄(τ−τ1)) 2 X̂2 |z = 0, (3.56) where T denotes the operation of Euclidean time ordering. This time ordering is effectively described by the product representation of Equation (2.33), where the appropriate Hamiltonian is entered into each Euclidean time slice. This can be shown to give the path integral, Equation (3.55), adapting with minimal changes the demonstration in Chapter 2. We leave it to the reader to confirm the details. Consider first the behaviour of V (z) which controls the Euclidean time- dependent frequency in the path integral Equation (3.55). V (±a) = ω2 is the parabolic curvature at the bottom of each well. In between, at z = 0, V (0) will drop to some negative value giving the curvature at the top of the potential hill separating the two wells. We will have a function as depicted in Figure 3.5. Thus V (z̄(τ)) will start out at ω2 at τ = −∞, until z̄(τ) starts to cross over from −a to a, where it will trace out the potential well of Figure 3.5, and again it will regain the value ω2 for z̄(τ) = a at τ = ∞, corresponding to the function of τ as in Figure 3.6. Thus the path integral in Equation (3.55) is exactly equal to the matrix element or “Euclidean persistence amplitude” that a particle at position zero will remain at position zero in Euclidean time β in a quadratic potential with a time-dependent frequency given by V (z̄(τ)) depicted in Figure 3.6.
- 45. 3.5 Evaluation of the Determinant 27 V˝(z[τ ]) ω2 – Figure 3.6. The behaviour of V (z(τ) between τ = ±∞ We will express the matrix element in terms of a Euclidean time evolution operator U β 2 ,−β 2 as N Dz(τ)e − 1 β 2 − β 2 dτ 1 2 (˙ z̄2 (τ)+V (z̄(τ−τ1))z2 (τ)) ≡ z = 0 U β 2 ,− β 2 z = 0 (3.57) with explicitly, U β 2 ,− β 2 = T e − 1 β 2 − β 2 dτ 1 2 P̂ 2 + V (z̄(τ−τ1)) 2 X̂2 . (3.58) Now U β 2 ,− β 2 = U β 2 ,τ1+ 1 2ω U (τ1+ 1 2ω ,τ1− 1 2ω )U τ1− 1 2ω ,− β 2 ≈ U0 β 2 ,τ1+ 1 2ω U (τ1+ 1 2ω ,τ1− 1 2ω )U0 τ1− 1 2ω ,− β 2 , (3.59) where on the intervals τ1 + 1 2ω , β 2 and −β 2 ,τ1 − 1 2ω we can replace the full evolution operator with the free evolution operator U0 (τ,τ ) = T e − 1 τ τ dτ 1 2 −2 d2 dz2 +ω2 z2 = e− (τ−τ) ĥ0 (X̂,P̂ ) (3.60) as V (z̄(τ) is essentially constant and equal to ω2 on these intervals. Then inserting complete sets of free eigenstates, which are just simple harmonic oscillator states |En for an oscillator of frequency ω , we obtain U β 2 ,− β 2 = n,m e − β 2 −τ1− 1 2ω En |EnEn|U (τ1+ 1 2ω ,τ1− 1 2ω )|Em × Em|e − τ1− 1 2ω + β 2 Em (3.61)
- 46. 28 The Symmetric Double Well Now we use the “ground state saturation approximation”, i.e. when β is huge and the instanton is not near the boundaries, only the ground state contribution is important. Using this twice we obtain U β 2 ,− β 2 ≈ e β 2 −τ1− 1 2ω E0 |E0E0|U (τ1+ 1 2ω ,τ1− 1 2ω )|E0E0|e − τ1− 1 2ω + β 2 E0 = U0 β 2 ,τ1+ 1 2ω |E0E0|U0 (τ1+ 1 2ω ,τ1− 1 2ω )|E0E0|U0 τ1− 1 2ω ,− β 2 × × E0|U (τ1+ 1 2ω ,τ1− 1 2ω )|E0 E0|U0 (τ1+ 1 2ω ,τ1− 1 2ω )|E0 ≈ n,m U0 β 2 ,τ1+ 1 2ω |EnEn|U0 (τ1+ 1 2ω ,τ1− 1 2ω )|EmEm|U0 τ1− 1 2ω ,− β 2 × × E0|U (τ1+ 1 2ω ,τ1− 1 2ω )|E0 E0|U0 (τ1+ 1 2ω ,τ1− 1 2ω )|E0 = U0 β 2 ,− β 2 E0|U τ1 + 1 2ω ,τ1 − 1 2ω |E0 E0|U0 (τ1+ 1 2ω ,τ1− 1 2ω )|E0 ≡ U0 β 2 ,− β 2 κ, (3.62) where κ is the ratio of the two amplitudes over the short time period during which V (z̄(τ) is non-trivially time-dependent. κ is surely independent of the position τ1 of the instanton. The full evolution operator in fact simply does not depend on the position, nor does the denominator. Indeed, U (τ1+ 1 2ω ,τ1− 1 2ω ) = T ⎛ ⎝e − 1 τ1+ 1 2ω τ1− 1 2ω dτ 1 2 −2 d2 dz2 +V (z̄(τ−τ1))z2 ⎞ ⎠ = T ⎛ ⎝e − 1 1 2ω − 1 2ω dτ 1 2 −2 d2 dz2 +V (z̄(τ ))z2 ⎞ ⎠, (3.63) since the integration variable is a dummy, thus exhibiting manifest τ1 independence. Clearly for n well-separated instantons the result applies also, we simply apply an appropriately adapted version of the same arguments. We convert the determinant into a persistence amplitude for the related quadratic quantum mechanical process, which we then further break up into free evolution in the gaps between the instantons and full evolution during the instanton, use the ground state saturation approximation, giving the result, to leading approximation N det − d2 dτ2 + V (z̄2n+1(τ)) − 1 2 = N det − d2 dτ2 + ω2 − 1 2 κ2n+1 . (3.64) The relationship of κ to the K fixed by Equation (3.41) is obtained by dividing out by the lowest energy eigenvalue, call it λ0. We will show that this eigenvalue is exponentially small for large β. For 2n + 1 instantons there are 2n + 1 such eigenvalues which are all equal, in first approximation, and we must remove them
- 47. 3.5 Evaluation of the Determinant 29 all giving N det − d2 dτ2 + V (z̄2n+1(τ)) − 1 2 = N ⎛ ⎝ det − d2 dτ2 + V (z̄2n+1(τ)) λ2n+1 0 ⎞ ⎠ − 1 2 = N det − d2 dτ2 + ω2 − 1 2 κλ 1 2 0 2n+1 . (3.65) Hence K = κλ 1 2 0 . (3.66) It only remains to calculate two things, the free determinant and the correction factor K. 3.5.1 Calculation of the Free Determinant To calculate the free determinant, we will use the method of Affleck and Coleman [31, 114, 36]. Consider the more general case det − d2 dτ2 + W(τ) , (3.67) where the operator acts on the space of functions which vanish at ±β 2 . Formally we want to compute the infinite product of the eigenvalues of the eigenvalue problem − d2 dτ2 + W(τ) ψλn (τ) = λnψλn (τ), ψλn ± β 2 = 0. (3.68) The eigenvalues generally increase unboundedly, hence the infinite product is actually ill-defined. Consider, nevertheless, an ancillary problem − d2 dτ2 + W(τ) ψλ(τ) = λψλ(τ), ψλ − β 2 = 0, d dτ ψλ (τ) − β 2 = 1. (3.69) There exists, in general, a solution for each λ; the second boundary condition can always be satisfied by adjusting the normalization. On the other hand, the equation in λ ψλ β 2 = 0 (3.70) has solutions exactly at the eigenvalues λ = λn. Affleck and Coleman [31, 114, 36] propose to define the ratio of the determinant for two different potentials as det − d2 dτ2 + W1(τ) − λ det − d2 dτ2 + W2(τ) − λ = ψ1 λ β 2 ψ2 λ β 2 . (3.71)
- 48. 30 The Symmetric Double Well The left-hand side is defined as the infinite product ∞ n=1 (λ1 n − λ) (λ2 n − λ) , (3.72) where the potentials and the labelling of the eigenvalues are assumed to be such that as the eigenvalues become large, they approach each other sufficiently fast, lim n→∞ (λ1 n − λ2 n) = 0 (3.73) so that the infinite product in Equation (3.72) does conceivably converge. To prove Equation (3.71) we observe that the zeros, λ = λ1 n, and poles, λ = λ2 n, of the left-hand side are at the same place as those of the right-hand side, as evinced by the solutions of Equation (3.70). Thus the ratio of the two sides !∞ n=1 (λ1 n−λ) (λ2 n−λ) ψ1 λ β 2 /ψ2 λ β 2 ≡ g(λ) (3.74) defines an analytic function g(λ) without zeros or poles. Now as |λ| → ∞ in all directions except the real axis, the numerator in Equation (3.74) is equal to 1. For the denominator, as λ → ∞ the potentials W1 and W2 become negligible perturbations compared to the term on the right-hand side of Equation (3.69), which we can consider as a potential −λ. Neglecting the potentials, clearly ψ1 λ β 2 and ψ2 λ β 2 approach each other, and hence the denominator also approaches 1 in the same limit. Therefore, g(λ) defines an everywhere-analytic function of λ which approaches the constant 1 at infinity, and now in all directions including the real axis, as it does so infinitesimally close to the real axis. By a theorem of complex analysis, a meromorphic function that approaches 1 in all directions at infinity must be equal to 1 everywhere g(λ) = 1 (3.75) establishing Equation (3.71). Reorganizing the terms in Equation (3.71), formally we obtain det − d2 dτ2 + W1(τ) − λ ψ1 λ β 2 = det − d2 dτ2 + W2(τ) − λ ψ2 λ β 2 , (3.76) where both sides are constants independent of the potentials Wi. We now finally choose N by defining det − d2 dτ2 + W(τ) ψ0 β 2 ≡ 2πN2 (3.77)
- 49. 3.5 Evaluation of the Determinant 31 and we will show that this choice is appropriate. Then Ndet − 1 2 − d2 dτ2 + ω2 = 2πψ0 0 β 2 − 1 2 , (3.78) where ψ0 0(τ) is the solution of Equation (3.69) for the free theory. It is easy to see that this solution is given by ψ0 0(τ) = 1 ω sinhω τ + β 2 (3.79) giving Ndet −1 2 − d2 dτ2 + ω2 = 2π eωβ − e−ωβ 2ω − 1 2 ≈ ω π 1 2 e −ω β 2 . (3.80) We can compare this result with the direct calculation of the Euclidean persistence amplitude of the free harmonic oscillator. We find Ndet − 1 2 − d2 dτ2 + ω2 = x = 0 e − β − 2 2 d2 dx2 + 1 2 ω2 x2 x = 0 # = e− βE0 x = 0| E0E0| x = 0 + ··· , (3.81) where |E0 is the ground state. Clearly the normalized wave function is x|E0 = ω π 1 4 e− ω 2 x2 (3.82) while E0 = 1 2 ω (3.83) giving x = 0| E0 = ω π 1 4 . (3.84) Hence Equation (3.81) yields Ndet − 1 2 − d2 dτ2 + ω2 = ω π 1 2 e −ω β 2 (3.85) in agreement with Equation (3.80), and confirming the definition of the normalization N chosen in Equation (3.77). 3.5.2 Evaluation of K Finally we must evaluate the factor K. K is given by the ratio 1 K2 = det − d2 dτ2 + V (z̄(τ − τ1) det − d2 dτ2 + ω2 (3.86)
- 50. 32 The Symmetric Double Well from Equations (3.64) and (3.66) for n = 0. Thus 1 K2 = ⎛ ⎝ ψ0 β 2 /λ0 ψ0 0 β 2 ⎞ ⎠, (3.87) where λ0 is the smallest eigenvalue in the presence of an instanton. To calculate ψ0 β 2 and λ0 approximately we describe again the procedure given in Coleman [31]. First we need to solve −∂2 τ + V (z̄(τ)) ψ0(τ) = 0 (3.88) with the boundary conditions ψ0(−β/2) = 0 and ∂τ ψ0(−β/2) = 1. We already know one solution of Equation (3.88), albeit one that does not satisfy the boundary conditions: the zero mode of the operator in Equation (3.30) due to time translation invariance, we will call it here x1(τ): x1(τ) = 1 √ S0 dz̄ dτ . (3.89) x1(τ) → Ae−ω|τ| as τ → ±∞. A is determined by the equation of motion, Equation (3.30), which integrated once corresponds to ˙ z̄(τ) = 2V (z̄(τ)). (3.90) Once we have A we can compute ψ(β 2 ) and λ0. We know that there must exist a second independent solution of the differential Equation (3.88), y1(τ) which we normalize so that the Wronskian x1 dy1 dτ − y1 dx1 dτ = 2A2 . (3.91) We remind the reader that the Wronskian between two linearly independent solutions of a linear second-order differential equation is non-zero, and with no first derivative term, as in Equation (3.88), is a constant. Then as τ → ±∞ we have ẏ1(τ) ± ωy1(τ) = 2Aωeω|τ| (3.92) using the known behaviour of x1(τ). The general solution of Equation (3.92) is any particular solution plus an arbitrary factor times the homogeneous solution y1(τ) = ±Aeω|τ| + Be∓ω|τ| , (3.93) where B is an arbitrary constant. Evidently the homogenous solution is a negligible perturbation on the particular solution, and y1(τ) → ±Aeω|τ| as τ → ±∞. Then we construct ψ0(τ) as ψ0(τ) = 1 2ωA eωβ/2 x1(τ) + e−ωβ/2 y1(τ) , (3.94)
- 51. 3.5 Evaluation of the Determinant 33 verifying ψ0(−β/2) = 1 2ωA eωβ/2 x1 (−β/2) + e−ωβ/2 y1(−β/2) ≈ 1 2ωA eωβ/2 Ae−ωβ/2 + e−ωβ/2 (−A)eωβ/2 = 0 (3.95) while dψ0(−β/2) dτ −β 2 ≈ 1 2ωA eωβ/2 d dτ Aeωτ −β 2 + e−ωβ/2 d dτ (−A)e−ωτ −β 2 = 1. (3.96) Then it is also easy to see ψ0(β/2) = 1 ω , (3.97) which we will need later. We also need to calculate the smallest eigenvalue λ0 of Equation (3.69). To do this we convert the differential equation to an integral equation using the corresponding Green function. The Green function satisfying the appropriate boundary conditions is constructed from x1(τ) and y1(τ) using standard techniques and is given by G(τ,τ ) = $ 1 2A2 (−y1(τ )x1(τ) + x1(τ )y1(τ)) τ τ 0 τ τ . (3.98) Then the differential equation is converted to an integral equation ψλ(τ) = ψ0(τ) + λ 2A2 τ −β 2 dτ (x1(τ )y1(τ) − y1(τ )x1(τ))ψλ(τ ) ≈ ψ0(τ) + λ 2A2 τ −β 2 dτ (x1(τ )y1(τ) − y1(τ )x1(τ))ψ0(τ ). (3.99) This wave function vanishes for the lowest eigenvalue λ0 (and actually for all eigenvalues λn) at τ = β/2 by Equation (3.70), thus ψ0(β/2) + λ 2A2 β 2 −β 2 dτ (x1(τ )y1(β/2) − y1(τ )x1(β/2))ψ0(τ ) ≈ 1 ω − λ 2A2 β 2 −β 2 dτ (x1(τ )y1(β/2) − y1(τ )x1(β/2)) 1 2ωA eωβ/2 x1(τ ) + e−ωβ/2 y1(τ ) ≈ 1 ω − λ 2A2 β 2 −β 2 dτ (x1(τ )eωβ/2 − y1(τ )e−ωβ/2 ) 1 2ω eωβ/2 x1(τ ) + e−ωβ/2 y1(τ )
- 52. 34 The Symmetric Double Well ≈ 1 ω − λ 2A2ω β 2 −β 2 dτ (x2 1(τ )eωβ − y2 1(τ )e−ωβ ) ≈ 1 ω − λ 4A2ω β 2 −β 2 dτ x2 1(τ )eωβ = 1 ω − λ 4A2ω eωβ = 0. (3.100) In the penultimate equation, we can drop the second term because it behaves at most as ∼ β, since y1(τ) ∼ eβ/2 at the boundaries of the integration domain at ±β/2, while the first term behaves as ∼ eβ since % x2 1(τ)dτ is normalized to 1. This gives quite simply λ0 ≈ 4A2 e−ωβ . (3.101) Then finally we get K = ψ0 0 (β/2) ψ0 (β/2)/λ0 1 2 = eωβ /2ω (1/ω4A2e−ωβ) = 2A2 . (3.102) Thus we have found that the matrix element a|e−βĥ(X̂,P̂ )/ | − a = sinh S0 2π 1 2 e−S0/ 2A2 β ω π 1 2 e −ω β 2 . (3.103) To see explicitly see how to compute A, we can consider a convenient, completely integrable example, V (x) = (γ2 /2)(x2 − a2 )2 , which has ω2 = V (±a) = (2γa)2 . Then Equation (3.30) yields z̄(τ−τ1) 0 dz γ(z2 − a2) = τ − τ1 (3.104) with exact solution z̄(τ) = atanh(aγ(τ − τ1)). (3.105) Thus A is determined by x1(τ) = ˙ z̄(τ) √ S0 = a2 γ √ S0 cosh2 (aγ(τ − τ1)) , (3.106) which behaves as lim τ→±∞ x1(τ) = 4a2 γ √ S0 e−2aγ|τ| = 2aω √ S0 e−ω|τ| = Ae−ω|τ| . (3.107) √ S0 is calculated from Equation (3.27), giving S0 = a −a dzγ(z2 − a2 ) = 4 3 γa3 = 2 3 ωa2 . (3.108) Hence A = 2aω √ (2/3)ωa2 = 6 ω , for this example.
- 53. 3.6 Extracting the Lowest Energy Levels 35 3.6 Extracting the Lowest Energy Levels On the other hand, the matrix element of Equation (3.103) can be evaluated by inserting a complete set of energy eigenstates between the operator and the position eigenstates on the left-hand side, yielding a|e−βĥ(X̂,P̂ )/ | − a = e−βE0/ a|E0E0| − a + e−βE1/ a|E1E1| − a + ··· , (3.109) where we have explicitly written only the first two terms as we expect that the two classical states, |±a, are reorganized due to tunnelling into the two lowest-lying states, |E0 and |E1. Indeed, comparing Equation (3.103) and Equation (3.109) we find E0 = 2 ω − S0 2π 1 2 e−S0/ 2A2 (3.110) while E1 = 2 ω + S0 2π 1 2 e−S0/ 2A2 . (3.111) It should be stressed that our calculation is only valid for the energy difference, not for the corrections to the energies directly. Indeed, there are ordinary perturbative corrections to the energy levels which are normally far greater than the non-perturbative, exponentially suppressed correction that we have calculated. However, none of these perturbative corrections can see any tunnelling phenomena. Thus our calculation gives the leading term in the correction due to tunnelling. Thus, the energy splitting which relies on tunnelling is found only through our calculation, and not through perturbative calculations. We also find the relations a|E0E0| − a = ω π 1 2 (3.112) in addition to a|E1E1| − a = − ω π 1 2 (3.113) while a simple adaptation of our analysis yields a|E0E0|a = ω π 1 2 (3.114) in addition to a|E1E1|a = ω π 1 2 . (3.115) These yield E0| − a = E0|a while E1| − a = −E1|a which are consistent with |E0 being an even function, i.e. |E0 being an even superposition of the position eigenstates |a and | − a while |E1 being an odd function and hence an odd superposition of these two position eigenstates.
- 54. 36 The Symmetric Double Well Figure 3.7. A generic periodic potential with minima occurring at na with n ∈ Z, where a is the distance between neighbouring minima 3.7 Tunnelling in Periodic Potentials We will end this chapter with an application of the method to periodic potentials. Periodic potentials are very important in condensed matter physics, as crystal lattices are well-approximated by the theory of electrons in a periodic potential furnished by the atomic nuclei. The idea is easiest to enunciate in a one- dimensional example. Consider a potential of the form given in Figure 3.7. A particle in the presence of such a potential with minimal energy will classically, certainly, be localized in the bottom of the wells of the potential. If there is no tunnelling, there would be an infinite number of degenerate states corresponding to the state where the particle is localized in state labelled by integer n ∈ Z. This could also be a very large, finite number of minima. However, quantum tunnelling will completely change the spectrum. Just as in the case of the double well potential, the states will reorganize so that the most symmetric superposition will correspond to the true ground state, and various other superpositions will give rise to excited states, albeit with excitation energies proportional to the tunnelling amplitude. The tunnelling amplitude is expected to be exponentially small and non-perturbative in the coupling constant. As in the case of the double well potential, the instanton trajectories will correspond to solutions of the analogous dynamical problem in the inverted potential in Euclidean time (as depicted in Figure 3.8), where the trajectories commence at the top of a potential hill, stay there for a long time, then quickly fall through the minimum of the inverted potential, and then arrive at the top of the adjacent potential hill, and stay there for the remaining positive Euclidean time. For the simple, real-time Lagrangian L = 1 2 ẋ2 − V (x), (3.116)
- 55. Exploring the Variety of Random Documents with Different Content
- 56. as you may think proper, and with what alterations and arrangements you may think necessary. I do not know how to make a book, any more than a watch, but you have learned the trade completely; I therefore beg your assistance, for which I shall feel very grateful.” But even these anxieties could not engross his confidential correspondence. In the same letter we have pleasant mention of New Zealand and its missionaries:—“I have no doubt about New Zealand; we must pray much for them, and labour hard, and God will bless the labour of our hands.” Nor is science quite forgotten:—“I have sent you a small box of fossils and minerals, by Captain Dixon, of the Phœnix, from Point Dalrymple principally; the whole of them came from Van Diemen’s Land.” Mr. Wilberforce and other friends of religion were consulted; and under their advice his pamphlet was published in London, though not till the year 1826. It is entitled, “An Answer to certain Calumnies, etc., by the Rev. Samuel Marsden, principal Chaplain to the colony of New South Wales.” It contains a temperate, and at the same time a conclusive answer, to all the charges made against him. To some of these we have already had occasion to refer; others have lost their interest. The charge of hypocrisy was chiefly grounded on the fact that a windmill, on Mr. Marsden’s property, had been seen at work on Sunday. But “the mill,” he says, “was not in my possession at that time, nor was I in New South Wales. I never heard of the circumstance taking place but once; and the commissioner of inquiry was the person who told me of it after my return from New Zealand. I expressed my regret to the commissioner that anything should have taken place, in my absence, which had the appearance that I sanctioned the violation of the sabbath-day. As I was twelve hundred miles off at the time, it was out of my power to prevent what had happened; but I assured him it should not happen again, for the mill should be taken down, which was done.” How few, it is to be feared, would make such a sacrifice, simply to avoid the possibility of a return of the appearance of evil! The charge of bigotry arose out of his interference with Mr. Crook, a person in the colony who had formerly been intended for the South Sea mission. It was at the
- 57. request of the missionaries themselves, that Mr. Marsden, as agent of their Society, had been led to interfere; but he was represented, in consequence, as “a persecutor of dissenters.” Messrs. Bennett and Tyerman were then in Australia; and in answer to Mr. Marsden’s request that “they would do him the favour to communicate to him their impartial opinion, how far he had in any way merited such an accusation, either as it respects Mr. C. or any other missionary belonging to the London Missionary Society,” he received a grateful acknowledgment of his services, which we are happy to insert:— “Sydney, May 11, 1825. “Rev. and dear Sir,—We have to acknowledge the receipt of your letter of the 5th inst., requesting our opinion, as the representatives of the London Missionary Society, on one of the malicious charges against you in the outrageous publication lately come to the colony. It is with the utmost satisfaction we state, as our decided opinion, that the charge of intolerance or persecution towards Mr. Crook, or any other missionary connected with the London Society, or, indeed, connected with any other missionary society, is utterly untrue. We believe it to have originated in malice or culpable ignorance, and to be a gross libel. “We rejoice, sir, to take the opportunity to say that the South Sea mission, and all its missionaries, have been, and continue, to be, exceedingly indebted to your singular kindness and persevering zeal in their behalf. No temporal reward, we are persuaded, would have been equivalent to the most valuable services which you have so long and so faithfully rendered to this mission and its missionaries. After all your upright and perfectly disinterested kindness towards the missionaries, when they have been residing on the Islands,—when they have been residing in the colony, on their way from England to the Islands, —when they have voluntarily returned from the Islands to the colony,—and when, from dire necessity and cruel persecution, compelled to flee from the scenes of their missionary labours,
- 58. and take up their residence here; that you have met with so much calumny, and so few returns of grateful acknowledgment, for all you have done and borne on their behalf, is to us a matter of surprise and regret. “Allow us, dear sir, to conclude by expressing our hope, that the other envenomed shafts aimed at you in this infamous publication, will prove as impotent as that aimed at you through that Society, in whose name, and as whose representatives, we beg to renew its cordial thanks and unqualified acknowledgments. And desiring to present our own thanks in the amplest and most respectful manner, “We remain, rev. and dear sir, most faithfully, “Your obliged and obedient servants, “George Bennett. “Daniel Tyerman.” The case of James Ring, we cannot pass unnoticed. It shows the cruelty with which Mr. Marsden’s reputation was assailed on the one hand, and his own firm and resolute bearing on the other. Ring was a convict, who for his general good conduct had been assigned as a domestic servant to Mr. Marsden. He was permitted by the latter, in accordance with the usual custom, to work occasionally at his own trade—that of a painter and glazier, on his own account, and as a reward for his good conduct. He was frequently employed in this way by the residents at Paramatta; amongst others by the chief magistrate himself. This man having been ill-treated and severely beaten by another servant, applied, with Mrs. Marsden’s approbation, to the magistrates of Paramatta for redress; instead of receiving which, he was charged by them with being illegally at large, and committed to the common jail. Mr. Marsden was then absent on duty in the country: on appearing before the bench of magistrates upon his return home, he at once stated that he had given permission to Ring to work occasionally for himself, and that therefore if there was any blame it lay with him,
- 59. and not the prisoner. The magistrates not only ordered Mr. Marsden to be fined two shillings and sixpence per day for each day his servant had been thus at large, under the assumed plea of his transgressing a general government order, but also ordered Ring to be remanded to jail and ironed; and he was subsequently worked in irons in a penal gang. “At this conviction there was no informer, nor evidence,” (we are now quoting Mr. Marsden’s words, from a statement which he made before a court of inquiry instituted by Lord Bathurst, the colonial minister at home, to investigate the subject at Mr. Marsden’s request,) “but the bench convicted me on my own admission that I had granted indulgence to my servant to do jobs in the town. There were two convictions, the first was on the 17th of May, 1823. On the 23rd of the same month, without a hearing, or being present, without informer, evidence, or notice, on the same charge I was convicted in the penal sum of ten pounds. On the 7th of June, a convict constable entered my house with a warrant of execution, and levied the fine by distress and sale of my property.” These convictions took place under an obsolete colonial regulation of 1802, made in the first instance by Governor King, to meet a temporary emergency; but virtually set aside by a general order of Governor Macquarie’s, of a much later date, granting the indulgence under certain regulations, with which Mr. Marsden had complied. Mr. Marsden says, in his official defence, that he “was the only person in the colony who was ever fined under such circumstances, since the first establishment of the colony, to the present time.” And he adds a statement which, had it not come down to us thus accredited under his own hand, would have seemed incredible, namely that “the two magistrates by whom the fines were inflicted, Dr.—— and Lieut.——, were doing, on that very day, the same thing for which they fined me and punished my servant, and I pointed that out to them at the time they were sitting on the bench, and which they could not deny.” Denial indeed was out of the question, since, says Mr. Marsden, “one of Dr.——’s convict servants, Henry Buckingham, by trade a tailor, was working for me, and had been so for months. Lieut.—— at that
- 60. very time also had two convict servants belonging to Dr. Harris, working for him at his own house.” In vain did Mr. Marsden appeal to the governor; even he was afraid to breast the torrent, which for a time bore all before it. “He found no reason to interfere with the colonial law.” Mr. Marsden prayed him at least to bring the matter before a full bench of magistrates, in whose hands he would leave his character; this, too, the governor declined, whereupon as a last step, he laid the affair before the supreme court for its decision; prosecuting the magistrates, and obtaining a verdict for the amount of the fine so unjustly levied. They now affected to triumph in the small amount of the damages in which they were cast, “wishing,” he says, “to make the world believe that the injury I had sustained was proportionally small.” And thus even his forbearance and his Christian spirit in rendering good for evil, were turned against him; for he had instructed his solicitor expressly, not to insert in the indictment the count or charge of malice, but merely to sue for the recovery of the amount of the fine. He states the case thus in simple and forcible language. “I may here observe, the only error it appears I committed originally was in not prosecuting the magistrates for vindictive damages before the supreme court. Had I alleged malice, I must have obtained a verdict accordingly; but I sought for no vindictive damages; I sought redress no further than to set my character right with the public. To have done more than this would not have become me, according to my judgment, as a minister of the gospel, and I instructed my solicitor, Mr. Norton, merely to sue for the amount of the award which had been levied on my property by warrant and distress of sale. The court gave me the amount I prosecuted for, with costs of suit, and with this I was perfectly satisfied.” For two whole years this miserable affair lingered on. The unfortunate man Ring at length gave way to despondency, made his escape from the colony, and found his way to New Zealand, but was never heard of more. Mr. Marsden was much concerned for Ring’s misfortunes, and deplored his rashness in making his escape when all his sufferings were unmerited. “I knew,” he says, “if he should
- 61. return to England and be apprehended as a returned felon, his life would be forfeited.” Such even to a recent period was the severity of our penal code, an escaped felon was consigned to the gallows. With a view of preventing this additional calamity, he wrote to the Right Honourable Mr. Robert Peel, his Majesty’s secretary of state for the home department, under date of July 1824; and having stated the case, he says: “I feel exceedingly for Ring; should he return to England and fall a sacrifice to the law, I should never forgive myself unless I used every means in my power to save him. The above statement of facts might have some influence with the executive in saving his life, if the circumstances of the case could reach the throne of mercy.” The contents of this letter were transmitted by Mr. Peel to Lord Bathurst the colonial secretary, and his lordship ordered the governor of New South Wales to establish a formal inquiry into the case. A court was accordingly summoned at Sydney, consisting of the governor assisted by two assessors, the chief justice and the newly-appointed archdeacon Scott, before which Mr. Marsden was cited to appear. He did so, the whole affair was investigated, and the result was, as the reader will have anticipated, not only Mr. Marsden’s entire acquittal of the charges which wantonness and malice had preferred, but the establishment of his reputation as a man of high courage and pure integrity, and a Christian minister of spotless character. The Christian reader will probably ask what were the effects of these various trials upon Mr. Marsden’s mind and temper? Did he become selfish and morose? were his spiritual affections quickened? As a minister of Christ, did his light shine with a more resplendent ray, or was it disturbed and overcast with gloom? To suggest and answer such inquiries are the proper uses of biography, especially the biography of religious men. With regard, then, to his habitual temper and tone of mind nothing can be more cheering than a letter, which we now insert, written to a lady in solitude, when the storm of insult and misrepresentation was at its highest pitch. “Paramatta, December 26, 1824.
- 62. “Dear Mrs. F.,—I received your kind letter by Mr. Franklane, and was happy to learn that you and your little boy were well. The circumstance to which you allude is not worthy to be had in recollection for a single moment, and I hope you will blot it out of your remembrance for ever; we are so weak and foolish, and I may add sinful, that we allow real or imaginary trifles to vex and tease our minds, while subjects of eternal moment make little impression upon us. It is a matter of no moment to our great adversary, if he can only divert our minds from attending to the best things. He wishes at all times ‘a root of bitterness’ should ‘spring up’ in our minds, as this will eat like a canker every pious feeling, every Christian disposition. ‘Learn of me,’ says our blessed Lord, ‘for I am meek and lowly in heart, and ye shall find rest unto your souls.’ ‘The meek will he guide in judgment, and the meek will he teach his way.’ It is for want of this meekness, this humility of mind, that we are soon angry. The apostle exhorts us ‘to be kindly affectioned one towards another,’ and live in unity and godly love, and ‘bear ye one another’s burdens, and so fulfil the law of Christ.’ Situated as you are, remote from all Christian society, and from the public ordinances of religion, you will want, in a very especial manner, the consolations which can only be derived from the Holy Scriptures. You are in a barren and thirsty land where no water is; you have none to give you to drink of the waters of Bethlehem, and you must not be surprised if you grow weary and faint in your mind. Though God is everywhere, and his presence fills heaven and earth, yet all places are not equally favourable for the growth of religion in our souls. We want Christian society; we want the public ordinances; we want social worship. All these are needful to keep up the life of God in our souls. Without communion and fellowship with God, without our souls are going forth after him, we cannot be easy, we cannot be happy; we are dissatisfied with ourselves, and with all around us. A little matter puts us out of humour, Satan easily gains an advantage over us, we become a prey to discontent, to murmuring, and are prone to overlook all the great things the
- 63. Lord hath done for us. Under your peculiar circumstances you will require much prayer, and much watchfulness; religion is a very tender plant, it is soon injured, it requires much nourishing in the most favourable situations, but it calls for more attention, where it is more exposed to blights and storms. A plant removed from a rich cultivated soil, into a barren uncultivated spot soon droops and pines away. I hope this will not be the case with you, though you must expect to feel some change in your feelings of a religious nature. Without much care the sabbaths will be a weariness; instead of your soul being nourished and fed upon this day, it will sicken, languish, and pine. I most sincerely wish you had the gospel preached unto you; this would be the greatest blessing, but it cannot be at present. There is no man to care for your souls, you have no shepherd to watch over you, and must consider yourselves as sheep without a shepherd. You know how easily sheep are scattered, how they wander when left to themselves, how soon the wolves destroy them. It is impossible to calculate the loss you must suffer, for want of the public ordinances of religion. My people, says God, perish for lack of knowledge. You know it is true that there is a Saviour, you have your Bible to instruct you, and you have gained much knowledge of Divine things, but still you will want feeding on the bread of life, you will want Jesus to be set before your eyes continually as crucified. You will want eternal things to be impressed upon your minds from time to time. Though you know these things, yet you will require to have your minds stirred up, by being put in remembrance of these things. As you cannot enjoy the public ordinances, I would have you to have stated times for reading the Scriptures and private prayer; these means God may bless to your soul. Isaac lived in a retired situation, he had no public ordinances to attend, but we are told he planted a grove, and built an altar, and called upon the name of the Lord. This you have within your power to do. Imitate his example, labour to possess his precious faith, and then it will be a matter of little importance where you dwell. With the Saviour you will be happy, without
- 64. him you never can be. When you once believe on him, when he becomes precious to your soul, then you will seek all your happiness in him. May the Father of mercies give you a right judgment in all things, lead you to build your hopes of a blessed immortality upon that chief corner stone, which he hath laid in Zion; then you will never be ashamed through the countless ages of eternity. “Mrs. M. and my family unite in kind regards to you, wishing you every blessing that the upper and nether springs can afford. “In great haste. I remain, dear Mrs. F——, “Yours very faithfully, “Samuel Marsden.” Systematic theology, or indeed deep learning in any of its branches, sacred or profane, Mr. Marsden had never cultivated. His life had not been given to abstraction and close study, but to the most active pursuits. Activity, however, is not inconsistent with deep thoughtfulness, and it affords some aids to reflection and observation, which often lay the foundation for a breadth of mind and a solid wisdom to which the mere student or man of letters seldom attains. Mr. Marsden, too, was well acquainted with his Bible, and, above most men, with himself. Thus, without being in any sense a learned divine, he was an instructive minister, and often an original thinker. His early acquaintance with Dr. Mason Good had led him deeply to consider the question of the deity of Christ and the following letter upon this all-important doctrine proves how capable he was of standing forward in its defence, and how deeply alive he was to its importance. It was addressed to one who had begun to doubt upon the subject of our Lord’s Divine nature. “Paramatta, June 13, 1825. “My dear Sir,—I ought to have answered your letter long ago, but was prevented from one thing and another, which called away my attention when I was determined to write. I received the books you sent me. That respecting our Lord’s Divinity I
- 65. read with care and attention. I found nothing in it that would satisfy me; there was no food to the soul, no bread, no water of life. I found nothing that suited my ruined state. I know I have destroyed myself by my iniquities, that I am hopeless and helpless, and must be eternally undone unless I can find a Divine Saviour who is able and willing to answer all the demands of law and justice. If I were alone in the world, and no individual but myself believed that Jesus was God over all blessed for evermore, and that he had died for my sins, that the penalty due to them was laid upon him, I know and am persuaded unless I believed this I could not be saved. I find no difficulty in my mind in praying to him, because I believe he is able to save. The dying thief did this in the very face of death: ‘Lord, remember me when thou comest into thy kingdom.’ Jesus promised that he should be with him that very day in paradise. Stephen, we are told, was a man full of faith and the Holy Ghost; he was mighty in the Scriptures, so that none of the Jewish priests were able to withstand his arguments which he advanced in support of the doctrine that Jesus was the Son of God. When he was brought to the place of execution his only hope of eternal life was in Jesus. ‘Lord Jesus, receive my spirit,’ was his dying prayer. He fled to him as the Almighty God at this most awful period. No other foundation can any man lay than that is laid, says St. Paul, which is Christ Jesus. It is to no purpose to quote Scripture on this important doctrine, I mean any particular passage, for Jesus is the sum and substance of them all. I am fully convinced that no man can have a well- grounded hope of salvation unless he believes in the Divinity of our Lord and only Saviour. I would ask you, why should you not have as firm a hope as any other man in the world of eternal life, if you do not believe in the Divinity of our Lord? Admitting that you have the same view as the author of the work you sent me to read, of God and religion, I may put the question to you, Can you depend on the foundation your hope stands upon? Does it now give you full satisfaction? Are you sure that you are right? I believe Jesus to be a Divine person, I believe him to be
- 66. God over all; I have no doubt upon this point, and I believe that all will be saved by him who trust in him for salvation. This doctrine is as clear to me as the sun at noon-day, and while I believe this doctrine it administers comfort to my mind, and gives me hope of a better state. I envy none their views of religion. I am satisfied with my own, though I am not satisfied with the attainments I have made in it, because I have not made those advances in divine knowledge in all the fruits of the Spirit I might have done. This is matter of shame, and regret, and humiliation. Examine the Christian religion as it stands revealed, with prayer for Divine illumination, and that God who giveth wisdom to all who call upon him for it will impart it to you. I have never met with a Socinian who wished me to embrace his faith, which has surprised me. I feel very differently. I wish all to believe in our Lord, because I believe this is necessary to salvation, as far as I understand the Scriptures; and I would wish all men to be saved, and to come to the knowledge of the truth. I would not change my views of religion for ten thousand worlds. But I must drop this subject, and reply to your last note. “Our affectionate regards to Mrs. F.; accept the same from, “Dear sir, yours very sincerely, “Samuel Marsden.” He remembered with gratitude his early friends, and was now in a condition to repay their kindness, and in his turn to repeat the Christian liberality which had once been extended to himself. From a private letter to the Rev. J. Pratt, we venture to make the following interesting quotation: “I believe in the year 1786 I first turned my attention to the ministry, and from the year 1787 to 1793 I received pecuniary assistance, more or less, from the Elland Society, but to what amount I never knew. First I studied under the Rev. S. Stores,
- 67. near Leeds. In 1788, I went to the late Rev. Joseph Milner, and remained two years with him. From Hull I went to Cambridge, and in 1793 I left Cambridge, was ordained, and came out to New South Wales. I shall be much obliged to you to learn, if you can, the amount of my expenses to the Elland Society. I have always considered that a just debt, which I ought to pay. If you can send me the amount I shall be much obliged to you. I purpose to pay the amount from time to time, in sums not less than 50l. per annum. When I close the Society’s accounts on the 31st of December next, I will give your Society credit for 50l., and will thank you to pay the same to the Elland Society on my account. When I know the whole amount, I will then inform you how I purpose to liquidate it. Should the Elland Society not be in existence, I have to request that the Church Missionary Society will assist some pious young man with a loan, per annum, of not less than 50l., to get into the church as a missionary. In the midst of all my difficulties God has always blessed my basket and my store, and prospered me in all that I have set my hand unto. The greatest part of my property is in the charge of common felons, more than a hundred miles from my house, in the woods, and much of it I never saw, yet it has been taken care of, and will be. A kind providence has watched over all that I have had, and I can truly say I feel no more concern about my sheep and cattle than if they were under my own eye. I have never once visited the place where many of them are, having no time to do this. We may trust God with all we have. I wish to be thankful to him who has poured out his benefits upon me and mine.” The practical wisdom, the spirit of calm submission to the Divine will when danger appears, and the simple faith in Christ displayed in the following letter require no comment, nor will its affectionate and paternal tone pass unnoticed. It appears to have been written to a lady on the eve of a voyage to England. We could wish that a copy of it were placed in the hands of every lady who may be compelled to go to sea. “Paramatta, May 27, 1826.
- 68. “My Dear Mrs.——,—Should you sail to-morrow it will not be in my power to see you again. I feel much for your very trying situation; why and wherefore you are so severely exercised remains at present known to the only wise God. If time does not reveal the mystery, eternity will. Clouds and darkness are round about the paths of the Almighty, and his footsteps are not known. You must now cast yourself and your little ones upon the bosom of the great deep. Remember always that he who holdeth the waters in the hollow of his hand, will continually watch over you and yours; winds and seas are under his sovereign control. We are prone to imagine that we are in much more danger on the seas than on dry land, but this is not really the case; our times are all in his hands, and if we only reflected that the hairs of our heads are all numbered, we should often be relieved from unnecessary and anxious fears. As for myself, I am constrained to believe that I am as safe in a storm as in a calm from what I have seen and known. Should you meet with raging seas and stormy winds, let not these distress you; they can do no more to injure you than the breath of a fly, or the drop of a bucket, without Divine permission. The promise is, ‘When thou passest through the waters I will be with thee.’ This is sufficient for the Christian to rest upon. You must live near to God in prayer. Labour to get right views of the Redeemer, who gave his life as a ransom for you. Humble faith in the Saviour will enable you to overcome every trial and bear every burden. No doubt but that you will have many painful exercises before you see the shores of old England. Tribulations will meet us, and follow us, and attend us all our journey through, and it is through much tribulation we must enter the kingdom of God. Could you and I meet on your arrival in London, and could we put our trials in opposite sides, it is very probable that mine would overbalance yours during the period you were at sea. You are not to conclude when the storm blows hard, the waves roar, and seas run mountain high, that you are more tried and distressed than others.
- 69. “I hope the captain will be kind to you and the children; if he should not you will have no remedy but patience. Should the servant woman behave ill, you must submit to this also, because you can do no good in complaining. Should the woman leave you ... this is no more than what has happened to my own family. I should recommend you to give the children their dinner in your own cabin; never bring them to table but at the particular request of the captain. This precaution may prevent unpleasant disputes. You will soon see what the feelings of the captain and his wife are, and regulate your conduct accordingly. When I returned to England, when I entered the ship I resolved that I would not have any difference with any one during my passage; whatever provocations I might meet with, I would not notice them; and that resolution I kept to the last. “If you take no offence at anything, but go on quietly your own way, those who would wish to annoy you, will cease to do so, finding their labour in vain. Never appear to see or hear anything that you have not the power to remedy. If you should even know that the persons intended to vex you, never notice their conduct. There will be no occasions for these precautions if your companions on board be such as they ought to be. “Let your passage be pleasant or not, take your Bible for your constant companion. The comfort to be derived from the Divine promises will always be sweet and seasonable. ‘They that love thy law,’ says the Psalmist, ‘nothing shall offend them.’ If Jesus be precious to your soul, you will be able to bear every trial with Divine submission. To believe that Jesus is your Saviour, and that he is God over all blessed for evermore, will make you happy in the midst of the sea, as well as on dry land. Wishing you a safe and pleasant passage, and a happy meeting of your friends in England, and praying that the God of all grace may preserve you and yours in his everlasting kingdom, I subscribe myself,
- 70. “Yours respectfully, “Samuel Marsden.” More than two years had now passed since Mr. Marsden’s last visit to New Zealand. The close of the year 1826 found him preparing for another, his fifth voyage, of twelve hundred miles, to the scene of those missions he had so long regarded with all a parent’s fondness. A great change had just taken place in the conduct of several chiefs towards the missionaries in consequence of their fierce intestine wars. At Wangaroa the whole of the Wesleyan missionary premises had been destroyed; the property of all the missionaries was frequently plundered, and their lives were exposed to the greatest danger. The worst consequences were apprehended, and the missionaries, warned of their danger by the friendly natives, were in daily expectation of being at least stripped of everything they possessed, according to the New Zealand custom. For a time the Wesleyan mission was suspended, and their pious and zealous missionary, Mr. Turner, took refuge at Sydney, and found a home at the parsonage of Paramatta. The clergy of the church mission deeply sympathized with him. Mr. Henry Williams writes: “The return of Mr. Turner will be a convincing proof of our feelings on this point. In the present unsettled state of things we consider ourselves merely as tenants for the time being, who may receive our discharge at any hour.” His brother, the Rev. William Williams, in another communication says: “We are prepared to depart or stay according to the conduct of the natives; for it is, I believe, our united determination to remain until we are absolutely driven away. When the natives are in our houses, carrying away our property, it will then be time for us to take refuge in our boats.” As soon as the painful intelligence reached New South Wales, Mr. Marsden determined to proceed to the Bay of Islands, and use his utmost exertions to prevent the abandonment of the mission. He was under no apprehension of suffering injury from the natives; and his long acquaintance with their character and habits led him to anticipate that the storm would soon pass away. Accordingly, he
- 71. sailed for New Zealand in H.M.S. Rainbow, and arrived in the Bay of Islands on the 5th April, 1827. He had reached the period of life when even the most active crave for some repose, and feel themselves entitled to the luxury of rest; but his ardent zeal never seems to have wanted other refreshment than a change of duties and of scene. He found the state of things improved; peace had been restored; and the missionaries were once more out of danger. He conferred with them, and gave them spiritual counsel. As far as time would permit, he reasoned with the chiefs upon the baneful consequences of the late war, and, at the end of five days from his arrival, he was again upon the ocean, on his way back to Sydney. “He was not wanted in New Zealand;” in Australia, besides domestic cares, many circumstances combined to make his presence desirable. Thus he was instant in season, out of season; disinterested, nay indifferent and utterly regardless of the honours and preferments which even good men covet; and ever finding in the work itself, and in Him for the love of whom it was undertaken, an abundant recompense. Brief as the visit was, it confirmed his faith, and reassured his confidence in the speedy conversion of New Zealand. He found the missionaries living in unity and godly love, and devoting themselves to the work. “I trust,” he says, “that the Great Head of the church will bless their labours.” In consequence of his co-operation with the missionaries, the beneficial labours of the press now for the first time reached the Maori tribes. During a visit to Sydney, Mr. Davis had carried through the press a translation of the first three chapters of Genesis, the twentieth of Exodus, part of the fifth of Matthew, the first of John, and some hymns. These were small beginnings, but not to be despised; they prepared the way for the translation of the New Testament into Maori, which was printed a few years afterwards at the expense of the British and Foreign Bible Society. The importance of this work can scarcely be estimated, and it affords a striking example of the way in which that noble institution becomes the silent handmaid, preparing the rich repast which our various
- 72. missionary societies are ever more distributing abroad, with bounteous hand, to feed the starving myriads of the heathen world. Nor was the Polynesian mission forgotten by its old friend. The London Missionary Society now conducted its affairs on so wide a basis, and to so great an extent, that Mr. Marsden’s direct assistance was no longer wanted. But how much he loved the work, how much he revered the missionaries, those who shall read the extract with which this chapter concludes will be at no loss to judge. “Paramatta, February 4, 1826. “My dear Sir,—It is not long since I wrote to you, but as a friend of mine is returning, the Rev. Mr. Nott, who has been twenty- seven years a missionary in the Society Islands, I could not deny myself the pleasure of introducing him to you. Mr. Nott was one of the first missionaries who was sent out to the Islands. Like Caleb, he always said the missionaries were able to take the land. He remained a long time in Tahiti alone, labouring by himself when all his colleagues were gone, and lived with and as the natives, under the full persuasion that the mission would succeed. He remained breaking up the ground, sowing the gospel seed, until he saw it spring up, and waiting until part of the harvest was gathered in, until many of the poor heathen crossed the river Jordan, with the heavenly Canaan full in view. Such have been the fruits of his patient perseverance and faith. Should his life be spared, I shall expect to see him again in fourteen months returning to his labours, to die amongst his people, and to be buried with them. “I venerate the man more than you can conceive: in my estimation, he is a great man: his piety, his simplicity, his meekness, his apostolic appearance, all unite to make him great in my view, and more honourable than any of the famed heroes of ancient or modern times. I think Mrs. Good will like to see such a character return from a savage nation, whom God has so
- 73. honoured in his work. I shall leave Mr. Nott to tell his own story, while you listen to his report.... “I remain, my dear sir, “Your’s affectionately, “Samuel Marsden.” “To John Mason Good, M.D.”
- 74. CHAPTER XI. Death of Dr. Mason Good—Malicious Charges brought against Mr. Marsden and confuted—Sixth Voyage to New Zealand— Frightful state of the Island—Battle of the Maories—Their Cannibalism—Progress of the Mission—Mr. Marsden’s return— Death of Mrs. Marsden—Anticipation of his own decease. The shadows of evening now began to fall on him whose life had hitherto been full of energy, and to whom sickness appears to have been a stranger. He had arrived at the period when early friendships are almost extinct, and the few who survive are dropping into the grave. The year 1827 witnessed the death of Dr. Mason Good. Nearly twenty years had elapsed since he and Mr. Marsden had taken leave of one another; but their friendship had not cooled during that long term of absence; it seems rather to have gained strength with distance and declining years. Dr. Mason Good felt, and gratefully acknowledged, that to the conversations, and yet more to the high example of Mr. Marsden, he owed it, under God, that he was led to seek, through faith in Jesus, that holiness and peace which he found at last, and which shed so bright a lustre on his closing years. He had seen in his friend a living instance of disinterestedness, zeal, and humility combined, all springing from the love of God, and directed for Christ’s sake towards the welfare of man; such as he had never seen before—such as, he confessed, his own Socinian principles were incapable of producing. Far his superior as a scholar and a man of genius, he perceived and felt his inferiority in all that relates to the highest destinies of man; he sat, as a little child, a learner, in his presence; and God, who is rich in mercy, brought home the lessons to his soul. Nothing, on the other hand, could exceed the respect, almost amounting to reverence, mingled however with the warmest
- 75. affection, with which Mr. Marsden viewed his absent friend. In every difficulty he had recourse to him for advice; more than once he intrusted the defence of his character and reputation entirely to his discretion. A correspondence of nearly twenty years, a few specimens of which are in the reader’s hand, show the depth of his esteem. Upon his death a fuller tide of affection gushed out; while he wrote thus to the mourning widow:— “Paramatta, November 9, 1827. “My dear Mrs. Good,—A few days ago we received two letters from your daughter M—, informing us of the death of your much revered husband. I had seen his death noticed in one of the London papers, but had not received any other information. I feel for all your loss. He was a blessing to the Christian world, and to mankind at large. No one I esteemed more, and his memory will always be dear to me. When I was with you, he and I had many serious conversations on the subject of religion. “His great talents, united with his child-like simplicity, interested me much. I always experienced the greatest pleasure in his company, as well as advantage; in knowledge I found myself an infant in his presence, but yet at perfect ease. His gentle manners, his mild address, often made me forget to whom I was speaking; and after retiring from his presence I, on reflecting, have been ashamed that I should presume to talk to him as I had done, as if he were my equal. I never could account for the ease and freedom I felt in his company, in giving my opinion upon the various subjects we were wont to converse upon. He was a very learned man, and knew a thousand times more of men and things than I did, excepting on the subject of religion; here I always felt myself at home; and he would attend to what I said with the sweetest simplicity and the greatest openness of mind. In our various conversations on the most important doctrines of the gospel, he manifested a humble desire to know the truth, though he proceeded with great caution. I experienced no difficulty in my own mind in urging
- 76. the truths of religion upon him, by every argument in my power. I always saw, or thought I saw, the Day-star from on high dawning upon his mind; and my own soul was animated and refreshed whenever the subjects of the gospel engaged our conversation. Perhaps our mutual friend, Dr. Gregory, may remember the observations I made to him, on what passed between your dear husband and myself, respecting religion, and what were my views of the state of his mind at that time; the period to which I allude was when he joined the Church Missionary Society, or intended to join it. I had the firmest conviction in my mind that he would embrace the gospel, and cordially believe to the salvation of his soul. I could never account for that love which I have continued to have for Dr. Good, even here at the ends of the earth, but from the communion of saints. Though the affliction of yourself and your dear daughters must be severe, having lost such a husband and father, yet you cannot sorrow as those without hope; you must be satisfied that the Lord has taken him away from the evil to come; and as he cannot now return to you, comfort one another with the hope that you shall go to him. He finished his course with joy, and the work that had been given him to do; and came to the grave like a shock of corn that was fully ripe. This consideration should reconcile you to the Divine dispensation, and constrain you to say, ‘Not my will, but Thine be done.’ You and your dear husband had travelled long together; few in this miserable world were so happy and blessed as you were for so long a period. Remember all the way the Lord hath led you in this wilderness; recall to mind his mercies of old, and bless his name. I have long wished to see you face to face; but that wish will never be gratified. The day may come when, in another and a better world, we may recount all our travels here below. We are sure that we are fast approaching to the end of our journey, and shall soon arrive at the banks of Jordan. Let us labour, my dear madam, to keep the promised land in view. You have the consolation of your two amiable daughters’ company. I have never thought of Mrs. N. but with
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