The Collective Coordinate Fix

Arindam Bhattacharya,1 Jordan Cotler,1,2 Aurélien Dersy,1 Matthew D. Schwartz1

1 Department of Physics, Harvard University, Cambridge, MA 02138, USA
2 Society of Fellows, Harvard University, Cambridge, MA 02138, USA

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Abstract

Collective coordinates are frequently employed in path integrals to manage
divergences caused by fluctuations around saddle points that align with
classical symmetries. These coordinates parameterize a manifold of zero modes
and more broadly provide judicious coordinates on the space of fields.
However, changing from local coordinates around a saddle point to more global
collective coordinates is remarkably subtle. The main complication is that the
mapping from local coordinates to collective coordinates is generically multi-
valued. Consequently one is forced to either restrict the domain of path
integral in a delicate way, or otherwise correct for the multi-valuedness by
dividing the path integral by certain intersection numbers. We provide a
careful treatment of how to fix collective coordinates while accounting for
these intersection numbers, and then demonstrate the importance of the fix for
free theories. We also provide a detailed study of the fix for interacting
theories and show that the contributions of higher intersections to the path
integral can be non-perturbatively suppressed. Using a variety of examples
ranging from single-particle quantum mechanics to quantum field theory, we
explain and resolve various pitfalls in the implementation of collective
coordinates.

###### Contents

  1. 1 Introduction
  2. 2 Collective coordinates: general results
  3. 3 Free theory
  4. 4 Double well potential
    1. 4.1 Normal modes at finite $T$
    2. 4.2 Collective coordinates
    3. 4.3 Energy splitting
    4. 4.4 Free twisted partition function
  5. 5 Quantum field theory
    1. 5.1 Basis and cross check
    2. 5.2 Collective coordinates for dilatations
    3. 5.3 Collective coordinates for translations
  6. 6 Conclusions
  7. A Metric approach to Jacobians in the path integral
  8. B Mapping QFT to the double well

##  1 Introduction

One of the great advantages of the path integral for a quantum theory is that
it allows for expansions around saddle points other than the free theory,
providing access to non-perturbative physics. For example, the lifetime of the
Standard Model can be computed by expanding the path integral around a non-
trivial configuration of the Higgs field [1, 2, 3, 4, 5, 6, 7]. In many field
theories (including the Standard Model) there are families of equivalent non-
trivial saddles related by symmetries, such as translations, dilatations, or
internal symmetries. Expanding around any one of these saddles leads to a
divergent 1-loop saddle point calculation because the zero eigenvalues from
the flat directions make the functional determinant vanish. Such infinities
are usually handled by introducing collective coordinates along the manifold
of zero modes, effectively trading local coordinates around a single saddle
for more global coordinates around the entire manifold of zero modes [8, 9,
10]. Then the integral over the zero mode manifold can be handled non-
perturbatively, giving a factor of the volume of the symmetry group. The
remaining path integral can then be evaluated using a saddle-point
approximation, and can be regularized to be finite. While this procedure gives
sensible answers in some situations where it can be checked (for example, the
energy splitting in a symmetric double-well potential can be alternatively
computed with the WKB approximation [11, 12, 13, 14, 15, 16]), the procedure
also happens to give very wrong answers in other situations, such as in free
theories.

To see the problem clearly, consider in more detail the decay rate calculation
in a quantum field theory with action $S=\int
d^{4}x\left(\frac{1}{2}\,\partial_{\mu}\phi\,\partial^{\mu}\phi+\lambda\phi^{4}\right)$
with $\lambda<0$. As we will review in Section 5, the decay rate involves an
expansion around a non-trivial instanton background $\phi_{b}(x)$ normalized
by an expansion around the trivial vacuum $\phi(x)=0$. The decay rate $\Gamma$
can be written as [6, 5, 7]

$\Gamma=\lim_{T\rightarrow\infty}\frac{1}{T}\text{Im}\frac{\int_{C_{b}}\mathcal{D}\phi\,e^{-S[\phi]}}{\int_{C_{\text{FV}}}\mathcal{D}\phi\,e^{-S_{0}[\phi]}}\,,$
(1.1)

where $S_{0}[\phi]=\int
d^{4}x\left(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\right)$ is the
free action. Here $C_{b}$ is a path of steepest descent in the interacting
theory passing through $\phi_{b}(x)$ and $C_{\text{FV}}$ is the path of
steepest descent in the free theory passing though the free vacuum
$\phi(x)=0$. To evaluate the numerator in the saddle point approximation, we
expand

$\phi(x)=\phi_{b}(x)+\sum_{m=0}^{\infty}a_{m}\phi_{m}(x)\,,$ (1.2)

where $\\{\phi_{m}(x)\\}$ forms a basis of normal modes around the bounce
$\phi_{b}(x)$. Five of these modes, corresponding to four translations and one
dilatation mode, have zero eigenvalue with respect to the operator
$L^{\prime\prime}[\phi_{b}]$, where $L$ is the Lagrangian, and so a naïve
saddle point approach gives infinite contributions to the numerator of Eq.
(1.1). A more careful treatment of steepest descent is required to resolve
these infinities. To do so, we can change to a collective coordinate basis

$\phi(x)=\phi_{b}(x+\xi,R)+\sum_{m\geq 5}\alpha_{m}\phi_{m}(x+\xi,R)\,,$ (1.3)

where the five zero-mode coordinates $\alpha_{m}$ for $m=0,1,2,3,4$ are traded
for five collective coordinates $\xi^{\mu}$ and $R$. The integral over
$\xi^{\mu}$ then gives a factor of the spacetime volume $T\mathcal{V}$ for the
numerator of Eq. (1.1), as expected for a decay rate in spacetime. However, a
complication arises if we use the same exact basis of Eq. (1.3), which happens
to also diagonalize normal modes in the free theory, for the denominator of
Eq. (1.1). In particular, the same $T\mathcal{V}$ factor apparently arises,
which is unexpected. In this paper, we identify the cause of this unexpected
factor, and show that the appropriate treatment of collective coordinates
yields sensible results for both the numerator and denominator of Eq. (1.1).
We more broadly study general coordinates on field space in which similar
subtleties arise.

To understand the source of the subtleties, let us recall how
$\delta$-functions can be used to change variables in integrals. In a finite-
dimensional integral $\int_{\Omega}d^{d}x\,A(x)$ over
$x\in\Omega\subseteq\mathbb{R}^{d}$, we can implement the change of variables
$x=f(y)$ using the resolution of the identity

$\displaystyle 1=\int_{f^{-1}(\Omega)}\\!d^{d}y\,\left|\frac{\partial
f(y)}{\partial y}\right|\,\delta^{d}(f(y)-x)\,,$ (1.4)

so that

$\displaystyle\int_{\Omega}d^{d}x\,A(x)$
$\displaystyle=\int_{\Omega}d^{d}x\int_{f^{-1}(\Omega)}\\!d^{d}y\,\left|\frac{\partial
f(y)}{\partial y}\right|\,\delta^{d}(f(y)-x)\,A(x)$ (1.5)
$\displaystyle=\int_{f^{-1}(\Omega)}\\!d^{d}y\,\left|\frac{\partial
f(y)}{\partial y}\right|\,A(f(y))\,.$ (1.6)

The above can be regarded as a trick to find the right Jacobian factor and
domain of integration. Crucially, we notice that the above assumes that $f$ is
an invertible function on the domain of integration. But what if $f$ multiply-
covers the domain of integration? A simple example is the 1-dimensional
integral $\int_{0}^{\infty}dx\,A(x)$ with the change of variables $x=y^{2}$.
In this setting the salient resolution of the identity is

$\displaystyle
1=\frac{1}{2}\int_{-\infty}^{\infty}dy\,|2y|\,\delta(y^{2}-x)\,.$ (1.7)

Here, we need the factor of $\frac{1}{2}$ out front because the map
$f(y)=y^{2}$ taking $\mathbb{R}\to\mathbb{R}_{\geq 0}$ doubly covers the
positive real line. Going back to Eq. (1.4), if $f(y)=x$ has $N_{f}[x]$
solutions for fixed $x$, then we need to correct Eq. (1.4) by

$\displaystyle
1=\frac{1}{N_{f}[x]}\int_{f^{-1}(\Omega)}\\!d^{d}y\,\left|\frac{\partial
f(y)}{\partial y}\right|\,\delta^{d}(f(y)-x)\,,$ (1.8)

where we call $N_{f}[x]$ the intersection number. A similar subtlety arises in
gauge theories where there can be multiple solutions to a local gauge-fixing
condition, and so we must introduce a multiplicative factor to correct for the
so-called Gribov copies [17].

To see how the intersection number appears in a path integral, consider non-
relativistic quantum mechanics where the path integral is over paths $x(t)$.
For a time-independent Lagrangian, there will always be a flat direction:
$S[x(t)]=S[x(t+t_{0})]$ for any path $x(t)$ and any $t_{0}$. To change from
orthonormal coordinates on path space to collective coordinates we can employ
the resolution of the identity111We have used
$\Big{|}\langle\dot{X}(t+t_{0})|x(t)\rangle\Big{|}=\Big{|}\langle
X(t+t_{0})|\dot{x}(t)\rangle\Big{|}$ obtained via integration by parts in $t$.

$1=\frac{1}{N_{X}[x]}\int dt_{0}\,\delta\Big{[}\langle
X(t+t_{0})|x(t)\rangle\Big{]}\,\Big{|}\langle
X(t+t_{0})|\dot{x}(t)\rangle\Big{|}\,.$ (1.9)

Here $\langle x|y\rangle=\int dt\,x(t)y(t)$ denotes the $L^{2}$ inner product,
$x(t)$ is a trajectory in the domain of the path integral,
$\dot{x}(t)=\partial_{t}x(t)$ is the time derivative, $X(t+t_{0})$ is the path
whose coordinate we want to exchange for $t_{0}$, and $N_{X}[x]$ is the
intersection number. That is, solving $\langle X(t+t_{0})|x(t)\rangle=0$ will
replace the time-translation zero mode with the collective coordinate $t_{0}$
for any path $x(t)$. Then the intersection number $N_{X}[x]$ counts the number
of values of $t_{0}$ for which $\langle X(t+t_{0})|x(t)\rangle=0$ for a given
path $x(t)$. The intersection number is a complicated integer-valued
functional of the path which must be carefully taken into account.

The textbook narrative is that a collective coordinate can be simply pulled
out of the path integral giving a volume factor [18, 19, 20, 21, 22].
Unfortunately, this standard treatment does not properly account for the
multi-valuedness of the coordinate change associated with going from local
coordinates to collective coordinates. In this paper, we explore through
examples how and when collective coordinates can be used. Our key observations
are that (1) the transition to collective coordinates does not involve only an
innocuous coordinate change; (2) in interacting theories, non-Gaussian terms
in the action are important to suppress high-multiplicity intersections; and
(3) the Jacobian factor incurred by changing to collective coordinates plays
an important role in assuring the consistency of the implementation.

Our paper begins in Section 2 with an overview of how to change from local
coordinates to collective coordinates. We then proceed through a set of
examples of increasing complexity. In Section 3, we discuss a free particle
with time compactified on a circle. For this system time-translation
invariance is exact. We explicitly compute the intersection number associated
with the collective coordinates and illustrate the proper treatment of the
Jacobian. In Section 4, we consider the symmetric double-well potential in
quantum mechanics, focusing on the computation of the energy splitting between
the two lowest levels. This theory has a non-trivial instanton and a basis of
excitations around the instanton whose analytic form is explicitly known. We
further show that the intersection number apropos to the collective
coordinates counts the number of times a given path $x(t)$ crosses $x=0$. We
then demonstrate that paths with $N_{X}[x]>1$ give exponentially small
contributions to the path integral in the interacting theory, but give
$\mathcal{O}(1)$ contributions in the free theory. In Section 5 we consider
$\lambda\phi^{4}$ quantum field theory and discuss collective coordinates for
dilatation and translations. In Section 6 we conclude with a discussion.

##  2 Collective coordinates: general results

We begin by discussing how collective coordinates are used in quantum
mechanics. In evaluating the path integral, we often expand around some saddle
point $x_{I}(t)$. This saddle can be the free solution $x=0$, an instanton, or
simply some classical path satisfying the boundary conditions of the path
integral for the problem of interest. For concreteness, suppose we work in
Euclidean signature with a time-independent Lagrangian of the form
$L[x]=\frac{1}{2}\dot{x}^{2}+V(x)$ with action $S[x]=\int dt\,L[x]$. Then we
can expand a generic path in an orthonormal basis of eigenstates of
$\frac{\delta^{2}S}{\delta x^{2}}\big{|}_{x=x_{I}}$. That is, we write

$x(t)=x_{I}(t)+a_{0}x_{0}(t)+\sum_{n=1}^{\infty}a_{n}x_{n}(t)$ (2.1)

where

$[-\partial_{t}^{2}+V^{\prime\prime}(x_{I})]\,x_{n}=\lambda_{n}x_{n}\,.$ (2.2)

In (2.1) we have separated out the zero mode $x_{0}(t)\propto\dot{x}_{I}(t)$.
We can see that $\lambda_{0}=0$ using $\frac{\delta S}{\delta
x}\big{|}_{x=x_{I}}=0$ so that

$0=\partial_{t}\left.\frac{\delta S}{\delta x(t)}\right|_{x=x_{I}}=\int
dt^{\prime}\,\left.\frac{\delta^{2}S}{\delta x(t)\delta
x(t^{\prime})}\right|_{x=x_{I}}\dot{x}_{I}(t^{\prime})\,.$ (2.3)

Then the path integral becomes, to 1-loop order in the saddle-point
approximation,

$Z=\mathcal{N}\int\mathcal{D}x\
e^{-S}=\mathcal{N}e^{-S[x_{I}]}\int\frac{da_{0}}{\sqrt{2\pi}}\
\prod_{n=1}^{\infty}\int\frac{da_{n}}{\sqrt{2\pi}}\
e^{-\lambda_{n}a_{n}^{2}}=\mathcal{N}e^{-S[x_{I}]}\sqrt{\frac{1}{\lambda_{0}}}\prod_{n=1}^{\infty}\sqrt{\frac{1}{\lambda_{n}}}\,.$
(2.4)

Because of the zero eigenvalue, Eq. (2.4) is infinite. The existence of a zero
mode is because the action $S[x(t+t_{0})]$ is independent of $t_{0}$. We
therefore would like to trade $a_{0}$ for $t_{0}$ and integrate over $t_{0}$
giving a factor of the volume of time $T$.

To trade $a_{0}$ for $t_{0}$ we transform from coordinates $\\{a_{0},a_{n}\\}$
to collective coordinates $\\{t_{0},\alpha_{n}\\}$ of the form

$x(t)=x_{I}(t+t_{0})+\sum_{n=1}^{\infty}\alpha_{n}x_{n}(t+t_{0})\,.$ (2.5)

Since $\\{\alpha_{0},\alpha_{n}\\}$ are global coordinates, for which each
path in the integration domain is represented exactly once, the zero mode can
be projected out with

$\big{\langle}x(t)-x_{I}(t+t_{0})\big{|}x_{0}(t+t_{0})\big{\rangle}=0$ (2.6)

where $\langle x|y\rangle$ is the inner product under which the $x_{n}$ are
orthonormal. Eq. (2.6) can be solved to find a coordinate $t_{0}$ for a given
path $x(t)$. Importantly, however, there can be multiple solutions and we
denote by $N_{x_{0}}[x]$ the number of solutions for $t_{0}$ given $x$. The
Jacobian for the coordinate change is determined by the partial derivatives

$\frac{\partial a_{m}}{\partial
t_{0}}=\big{\langle}x_{m}(t)\big{|}\partial_{t_{0}}x_{I}(t+t_{0})+\sum_{n=1}^{\infty}\alpha_{n}\partial_{t_{0}}x_{n}(t+t_{0})\big{\rangle},\quad\frac{\partial
a_{m}}{\partial\alpha_{n}}=\big{\langle}x_{m}(t)\big{|}x_{n}(t+t_{0})\big{\rangle}\,.$
(2.7)

After some algebra (see Appendix B of [6]), this Jacobian can be simplified to

$\displaystyle\mathcal{J}$
$\displaystyle=\big{\langle}x_{0}(t+t_{0})\big{|}\partial_{t_{0}}{x}_{I}(t+t_{0})+\sum_{n=1}^{\infty}\alpha_{n}\partial_{t_{0}}{x}_{n}(t+t_{0})\big{\rangle}$
(2.8)
$\displaystyle=\big{\langle}x_{0}(t)\big{|}\dot{x}_{I}(t)+\sum_{n=1}^{\infty}\alpha_{n}\dot{x}_{n}(t)\big{\rangle}\,.$
(2.9)

Alternatively, the functional determinant can be computed as $\sqrt{\det G}$
where $G$ is the Riemannian metric on the space of fields in collective
coordinates (see Appendix A). That the Jacobian is independent of $t_{0}$ is
non-trivial. Thus we have

$\boxed{Z=\mathcal{N}e^{-S[x_{I}]}\int\frac{dt_{0}}{\sqrt{2\pi}}\
\int\prod_{m=1}^{\infty}\frac{d\alpha_{m}}{\sqrt{2\pi}}\
e^{-\lambda_{m}\alpha_{m}^{2}}\frac{1}{N_{x_{0}}[x]}\Big{|}\big{\langle}x_{0}(t)\big{|}\dot{x}_{I}(t)+\sum_{n=1}^{\infty}\alpha_{n}\dot{x}_{n}(t)\big{\rangle}\Big{|}\\\
}$ (2.10)

This is the formulation we should use if we want to implement the
$\\{t_{0},\alpha_{n}\\}$ coordinates as in Eq. (2.5).

From Eq. (2.10) we can integrate in $\alpha_{0}$ by including $\int
d\alpha_{0}\,\delta(\alpha_{0})$. If we then relabel $\alpha_{0}\rightarrow
a_{0}$ and $\alpha_{n}\rightarrow a_{n}$, the resulting expression would be
exactly the same as if we had expanded

$Z=\mathcal{N}\int dt_{0}\left\\{\int\mathcal{D}x\
e^{-S[x]}\frac{1}{N_{x_{0}}[x]}\delta\Big{[}\big{\langle}x_{0}(t)\big{|}x(t)\big{\rangle}\Big{]}\,\Big{|}\big{\langle}x_{0}(t)\big{|}\dot{x}(t)\big{\rangle}\Big{|}\right\\}$
(2.11)

using Eq. (2.1). In the above equation we can no longer use $t_{0}$ as a
coordinate: it is a constant from the point of view of the path integral (the
part of the expression in the curly brackets). That is, we cannot evaluate the
path integral expression using the coordinates in Eq. (2.5). If we want to use
$t_{0}$ as a coordinate, we should instead use Eq. (2.10).

There was nothing essential about $x_{I}(t)$ or $x_{0}(t)$ in our derivation.
Indeed, $x_{I}(t)$ does not even appear in Eq. (2.11). We can replace
$x_{0}(t)$ by any path $X(t)$ leading to

$\boxed{Z=\mathcal{N}\int dt_{0}\left\\{\int\mathcal{D}x\
e^{-S[x]}\frac{1}{N_{X}[x]}\,\delta\Big{[}\big{\langle}X(t)\big{|}x(t)\big{\rangle}\Big{]}\,\Big{|}\big{\langle}X(t)\big{|}\dot{x}(t)\big{\rangle}\Big{|}\right\\}\\\
}$ (2.12)

where $N_{X}[x]$ counts the number of times $\langle X(t+t_{0})|x(t)\rangle$
vanishes over the integration range of $t_{0}$ for fixed $x(t)$. Note that the
above equation can also be derived inserting the identity of Eq. (1.9) inside
the path integral and using $S[x(t+t_{0})]=S[x(t)]$.

To summarize, the key new element in our analysis is the intersection number
$N_{X}[x]$. If we write a generic path $x(t)$ in an orthonormal basis, then
$N_{X}[x]$ is the number of solutions to

$\langle X(t+t_{0})|x(t)\rangle=0$ (2.13)

over the range of $t_{0}$. Eq. (2.13) allows us to solve for $t_{0}$ in terms
of the coordinates labeling an orthonormal basis, and there may be multiple
solutions. When the collective coordinate $t_{0}$ is one of the coordinates,
Eq. (2.13) cannot be used. Instead, in collective coordinates, $N_{X}[x]$
counts the number of ways of constructing the same path $x(t)$ using different
values of $t_{0}$. That is, the same function would be overcounted in the path
integral in the collective coordinates if it can be described using multiple
values of $t_{0}$, and so we must divide by a $N_{X}[x]$ factor to compensate.
We will see examples of the use of $N_{X}[x]$ in the following.

##  3 Free theory

As a first example, we consider a free non-relativistic quantum theory for a
particle with mass $m=1$ and Lagrangian $L=\frac{1}{2}\dot{x}^{2}$. The
propagator going from $a$ to $b$ in time $T$ is

$D(a,b)=\langle
b|e^{-HT}|a\rangle=\int_{-\infty}^{\infty}\frac{dp}{2\pi}\,e^{ip(b-a)}e^{-\frac{p^{2}}{2}T}=\sqrt{\frac{1}{2\pi
T}}\,e^{-\frac{(b-a)^{2}}{2T}}$ (3.1)

and the partition function is

$Z_{0}=\text{Tr}(e^{-HT})=\int_{-\mathcal{V}/2}^{\mathcal{V}/2}dx_{0}\,D(x_{0},x_{0})=\sqrt{\frac{1}{2\pi
T}}\,\mathcal{V}\,,$ (3.2)

where $\mathcal{V}$ is the volume of space used the regulate this integral. To
compute the partition function with the path integral, we parameterize paths
with periodic boundary conditions in time as

$x(t)=x_{0}+\sum_{n\,\text{even}}\left[a_{n}s_{n}(t)+b_{n}c_{n}(t)\right]\,,$
(3.3)

where

$s_{n}(t)=\sqrt{\frac{2}{T}}\sin\\!\left(\frac{n\pi\left(t+\frac{T}{2}\right)}{T}\right),\quad
c_{n}(t)=\sqrt{\frac{2}{T}}\cos\\!\left(\frac{n\pi\left(t+\frac{T}{2}\right)}{T}\right).$
(3.4)

Then the partition function is given by

$Z_{0}=\mathcal{N}\int\mathcal{D}x\
e^{-S[x]}=\mathcal{N}\int_{-\mathcal{V}/2}^{\mathcal{V}/2}dx_{0}\prod_{n\
\text{even}}\left[\int_{-\infty}^{\infty}\frac{da_{n}}{\sqrt{2\pi}}e^{-\frac{1}{2}\lambda_{n}a_{n}^{2}}\int_{-\infty}^{\infty}\frac{db_{n}}{\sqrt{2\pi}}e^{-\frac{1}{2}\lambda_{n}b_{n}^{2}}\right]$
(3.5)
$=\mathcal{N}\mathcal{V}\left[\prod_{n\,\text{even}}\sqrt{\frac{1}{\lambda_{n}}}\right]^{2}$
(3.6)

where

$\lambda_{n}=\left(\frac{n\pi}{T}\right)^{2}$ (3.7)

are the eigenvalues of the fluctuations under $\left.\frac{\delta^{2}S}{\delta
x(t)\,\delta
x(t^{\prime})}\right|_{x=0}=-\delta(t-t^{\prime})\,\partial_{t}^{2}$. We then
match on to Eq. (3.2) with

$\mathcal{N}=\sqrt{\frac{1}{2\pi
T}}\left[\prod_{n\,\text{even}}\frac{T}{n\pi}\right]^{-2}$ (3.8)

which fixes the normalization of the path integral. Note that $\mathcal{V}$ is
only needed to regulate the $x_{0}$ integral: we can take the integrals over
$a_{n}$ and $b_{n}$ to go to from $-\infty$ to $\infty$ because the path
integral is exponentially suppressed at large $a_{n},b_{n}$.

Now we want to reproduce the partition function $Z_{0}$ with collective
coordinates, using Eq. (2.12). We first need to choose which path $X(t)$ to
project onto, i.e. which coordinate to swap for the collective coordinate
$t_{0}$. The coordinate to swap should be one from a parameterization of paths
where time is shifted by $t_{0}$, i.e. from

$x(t)=x_{0}+\sum_{n\,\text{even}}\left(\alpha_{n}s_{n}(t+t_{0})+\beta_{n}c_{n}(t+t_{0})\right).$
(3.9)

In the simplest case, we would swap a single coordinate, say $\beta_{m}$, for
$t_{0}$. That would mean choosing $X(t)=c_{m}(t)$. To change to coordinates
involving $t_{0}$ we have to start from coordinates not involving $t_{0}$,
such as the original coordinates in Eq. (3.3). The condition $\beta_{m}=0$ is
$\langle c_{m}(t+t_{0})|x(t)\rangle=0$ as in Eq. (2.13), which in the original
coordinates amounts to

$\tan\\!\left(\frac{m\pi t_{0}}{T}\right)=-\frac{b_{m}}{a_{m}}\,.$ (3.10)

This equation allows us to solve for $t_{0}$ and to transform from the
$\\{a_{n},b_{n}\\}$ coordinates to the $\\{t_{0},\alpha_{n},\beta_{n}\\}$
coordinates with no $\beta_{m}$ component. Note that over the range
$0<t_{0}\leqslant T$, Eq. (3.10) has $m$ possible solutions so that222In this
case, we can alternatively restrict the range $0<t_{0}\leqslant\frac{T}{m}$
and take $N_{X}[x]=1$, but restricting the range is untenable in more
complicated examples. $N_{X}[x]=m$.

The Jacobian arising from Eq. (2.12) is

$\mathcal{J}=\big{\langle}x(t)\big{|}\dot{c}_{m}(t)\big{\rangle}=\sqrt{\lambda_{m}}\,a_{m}\,,$
(3.11)

and thus in collective coordinates the partition function is

$Z_{\text{coll}}=\mathcal{N}\frac{1}{m}\int_{0}^{T}dt_{0}\int_{0}^{\mathcal{V}}dx_{0}\prod_{\begin{subarray}{c}n\,\text{even}\\\
n\neq m\end{subarray}}\left(\int_{-\infty}^{\infty}\frac{da_{n}}{\sqrt{2\pi}}\
e^{-\frac{1}{2}\lambda_{n}a_{n}^{2}}\int_{-\infty}^{\infty}\frac{db_{n}}{\sqrt{2\pi}}e^{-\frac{1}{2}\lambda_{n}b_{n}^{2}}\right)\\\
\times\int\frac{da_{m}}{\sqrt{2\pi}}\sqrt{\lambda_{m}}|a_{m}|e^{-\frac{1}{2}a_{m}^{2}\lambda_{m}}\,.$
(3.12)

Using Eq. (3.8) the above agrees precisely with $Z_{0}$ in Eq. (3.2).

The next simplest case is to choose $X(t)$ to be a sum of two modes instead of
one mode. Suppose we choose $X(t)=A\,c_{m}(t)+B\,c_{k}(t)$ which is a sum of
cosine modes. Then the coordinate $t_{0}$ would be determined from the
coordinates in Eq. (3.3) using the condition

$\langle
X(t+t_{0})|x(t)\rangle=A\,b_{m}c_{m}(t_{0})+A\,a_{m}s_{m}(t_{0})+B\,b_{k}c_{m}(t_{0})+B\,a_{k}s_{m}(t_{0})=0\,.$
(3.13)

The number of solutions $N_{X}[x]$ for this equation is a complicated function
of $a_{m},b_{m},a_{k}$, and $b_{k}$. It turns out to be somewhat easier to use
polar coordinates. Let us write an arbitrary linear combination of the $m$ and
$k$ sines and cosine modes as

$X(t)=R_{m}s_{m}(t+\phi_{m})+R_{k}s_{k}(t+\phi_{k})\,.$ (3.14)

Changing to the polar coordinates $a_{n}=\rho_{n}s_{n}(\theta_{n})$ and
$b_{n}=\rho_{n}c_{n}(\theta_{n})$ for convenience, we find the overlap

$\langle
X(t+t_{0})|x(t)\rangle=R_{m}\rho_{m}s_{m}(t_{0}+\theta_{m}+\phi_{m})+R_{k}\rho_{k}s_{k}(t_{0}+\theta_{k}+\phi_{k})$
(3.15)

which only involves two frequencies, as expected.

We now want to replace the $a_{m},b_{m},a_{k},b_{k}$ integrals from the
original partition function, which gave rise to a
$\frac{1}{\lambda_{m}}\frac{1}{\lambda_{k}}$ term, with new integrals in our
polar coordinates. As a cross check, we see that before introducing collective
coordinates we have

$Z_{\text{polar}}=\left(\frac{\lambda_{m}}{2\pi}\frac{\lambda_{k}}{2\pi}Z_{0}\right)\frac{m\pi}{T}\frac{k\pi}{T}\int_{0}^{\infty}d\rho_{m}\rho_{m}e^{-\frac{1}{2}\lambda_{m}\rho^{2}_{m}}\int_{0}^{\infty}d\rho_{k}\rho_{k}e^{-\frac{1}{2}\lambda_{m}\rho^{2}_{m}}\int_{0}^{\frac{2T}{m}}d\theta_{m}\int_{0}^{\frac{2T}{k}}d\theta_{k}=Z_{0}\,,$
(3.16)

as expected. Here $\frac{\lambda_{m}}{2\pi}\frac{\lambda_{k}}{2\pi}Z_{0}$
represents the partition function where we have performed all integrals except
for those over $a_{m},b_{m},a_{k},b_{k}$. The $\frac{m\pi}{T}\frac{k\pi}{T}$
factors arise from the Jacobian induced by our polar coordinates.

Figure 1: For the free quantum mechanical theory where the coefficient of a
linear combination of two normal modes is traded for a collective coordinate,
the intersection number $N_{X}$ cannot be neglected. $N_{X}$ is integer-
valued. After integrating over the modulus $\rho_{m}$, we find that $N_{X}$
depends on the modulus $\rho_{k}$ and the relative angle
$\theta_{m}-\theta_{k}$. Shown are the value of $N_{X}$ for $m=26$ and $k=4$
for a few choices of $\theta_{m}-\theta_{k}$.

Now we move to collective coordinates by inserting into the partially-
integrated partition function our Eq. (2.12) which includes an integral over
$t_{0}$, a $\delta$-function, and a Jacobian. Doing so and then shifting
$\theta_{m}\rightarrow\theta_{m}-\phi_{m}$,
$\theta_{k}\rightarrow\theta_{k}-\phi_{k}$, rescaling
$\rho_{m}\rightarrow\frac{\rho_{m}}{R_{m}},\rho_{k}\rightarrow\frac{\rho_{k}}{R_{k}}$,
taking $\rho_{k}\rightarrow\rho_{k}\rho_{m}$, and performing the $\rho_{m}$
integral leads to

$Z_{\text{coll}}=Z_{0}\times\left(\frac{\lambda_{m}}{2\pi}\frac{\lambda_{k}}{2\pi}\right)\frac{m\pi}{T}\frac{k\pi}{T}\int_{0}^{\infty}d\rho_{k}\frac{2\rho_{k}R_{k}^{2}R_{m}^{2}}{(R_{k}^{2}\lambda_{m}+R_{m}^{2}\lambda_{k}\rho_{k}^{2})^{2}}\int_{0}^{\frac{2T}{m}}d\theta_{m}\int_{0}^{\frac{2T}{k}}d\theta_{k}\\\
\times\frac{1}{N_{X}(\rho_{k},\theta_{k},\theta_{m})}\int_{0}^{T}dt_{0}\,\delta[s_{m}(\theta_{m})+\rho_{k}s_{k}(\theta_{k})]\left|\frac{m\pi}{T}c_{m}(\theta_{m})+\rho_{k}\frac{k\pi}{T}c_{k}(\theta_{k})\right|.$
(3.17)

One can check that without the second line, we still have
$Z_{\text{coll}}=Z_{0}$ . The intersection number $N_{X}$ counts the number of
times the $\delta$-function fires for a given path. We can compute the
intersection number by counting the roots of $\sin(\pi
m(t_{0}+\theta_{m})/T)+\rho_{k}\sin(\pi k(t_{0}+\theta_{k})/T)$ on the
interval $0<t_{0}<T$. Due to periodicity, the intersection number depends only
on $\theta_{k}-\theta_{m}$. The function for $m=26$ and $k=4$ with three
choices of $\theta_{k}-\theta_{m}$ is shown in Fig. 1.

To proceed, one can first observe that the integral is periodic in
$\theta_{m}$ and $\theta_{k}$, so we can extend their limits of integration to
$0<\theta_{m},\theta_{k}<T$ and pull out an overall factor of $\frac{4}{mk}$.
Then we change variables from $(\theta_{m},\theta_{k})$ to
$\theta_{+}=\frac{1}{2}(\theta_{m}+\theta_{k})$ and
$\theta_{-}=\theta_{m}-\theta_{k}$ and integrate over $\theta_{+}$. In this
integral, the $\delta$-function fires the number of times given by the number
of solutions of
$s_{m}(\theta_{+}+\frac{1}{2}\theta_{-})+\rho_{k}s_{k}(\theta_{+}-\frac{1}{2}\theta_{-})$
for $\theta_{+}$. But this number is identical to
$N_{X}(\rho_{k},\theta_{-})$, so that again $Z_{\text{coll}}=Z_{0}$ as
expected. One can of course integrate over $\theta_{m}$ or $\theta_{k}$ first
(rather than $\theta_{+}$) to get the same result, but the integrals are much
more difficult.

In the general case where $X(t)$ has components of arbitrary frequency, the
intersection number $N_{X}(\rho_{n},\theta_{n})$ would be a function of the
entire configuration space. Thus the intersection number is generally a
horrible mess, but accounting for the intersection number in the path integral
is nonetheless required for collective coordinates to work. Luckily, for some
choices of $X(t)$ the intersection number has a more physical interpretation
and can be managed either analytically or approximately. A representative
example is the double-well potential in quantum mechanics to which we now
turn.

##  4 Double well potential

For the next example, we consider the symmetric double well potential in
quantum mechanics whose Euclidean action is

$S[x]=\int_{-T/2}^{T/2}dt\left[\frac{m}{2}\dot{x}^{2}+V(x)\right],\quad
V(x)=\frac{m\omega^{2}}{8a^{2}}(x^{2}-a^{2})^{2}\,.$ (4.1)

There are two dimensionless scales in the problem: ${ma^{2}\omega}/{\hbar}$
which indicates the size of the barrier and $\omega T/\hbar$ which indicates
time. We will set $m=\hbar=1$ but keep $a,\omega$ and $T$ to assist with the
physical interpretation of various limits.

The main quantity of interest in this theory to compute with path integral
methods is the splitting of the two lowest energy eigenstates. This splitting,
and the entire spectrum, have been studied in great detail both analytically
and numerically [23, 24, 25, 26, 27, 28, 29, 13]. It is nevertheless difficult
to find a rigorous analysis of the path integral computation of the energy
splitting. Typically treatments are not only cavalier with regard to the
introduction of collective coordinates, but also with regard to the associated
Jacobian and finite-time regulation. We will attempt to clarify all of these
issues in this section.

Since the potential $V(x)$ is symmetric, the parity operator $P$ which takes
$x\rightarrow-x$ commutes with the Hamiltonian. Thus energy eigenstates are
parity even or parity odd. In the limit $a^{2}\omega\rightarrow\infty$, the
barrier becomes enormous and the spectrum is that of two harmonic oscillators
with frequency $\omega$. That is, $E_{0}^{+}=E_{0}^{-}=\frac{\omega}{2}$ and
the higher modes are spaced by $\omega$. Expanding the potential around a
minimum and working perturbatively in the limit of large but finite
$a^{2}\omega$, the energies receive corrections but the even and odd modes
remain degenerate to all orders. The WKB approximation can be used to compute
the energy splitting with the leading order result [13, 29, 30, 14]

$\Delta_{n}^{\text{WKB}}=E_{n}^{-}-E_{n}^{+}=\omega\frac{2}{\sqrt{2\pi}}\left(8\omega
a^{2}\right)^{n+1/2}e^{-\frac{2\omega a^{2}}{3}}\,.$ (4.2)

In particular, the ground state splitting can be written as

$\Delta_{0}^{\mathrm{WKB}}=4\omega\sqrt{\frac{\omega
a^{2}}{\pi}}e^{-\frac{2\omega a^{2}}{3}}\,.$ (4.3)

In fact, the energies can be determined for this system by solving the Ricatti
equation iteratively though the Exact WKB method relatively easily to
arbitrary order [31, 12, 32, 28, 33, 34]. However, our focus here is not on
precision but on the role of collective coordinates in the path integral
version of the calculation.

To compute the energy splitting with the path integral, it is helpful to set
up the problem so that the leading result as $T\rightarrow\infty$ is directly
proportional to $\Delta_{0}$. Defining
$E_{n}^{\pm}=E_{n}\mp\frac{\Delta_{n}}{2}$ the partition function is

$Z=\text{Tr}[e^{-HT}]=\sum_{n}(e^{-E_{n}^{+}T}+e^{-E_{n}^{-}T})\ ,$ (4.4)

and the twisted partition function is

$Z_{P}=\text{Tr}[Pe^{-HT}]=\sum_{n}(e^{-E_{n}^{+}T}-e^{-E_{n}^{-}T})\,.$ (4.5)

$Z_{P}$ is useful for extracting the energy since $E_{n}^{+}=E_{n}^{-}$ to all
orders in perturbation theory; that is, the splittings are non-perturbative.
Since $Z-Z_{P}$ involves only odd modes and $Z+Z_{P}$ involves only even modes
we see that

$\lim_{T\rightarrow\infty}-\frac{1}{T}\ln\left(\frac{Z-Z_{P}}{Z+Z_{P}}\right)=E_{0}^{-}-E_{0}^{+}=\Delta_{0}\,.$
(4.6)

The above formula makes no assumption about $\Delta_{0}$ being small, but is
useful only if $Z$ and $Z_{P}$ are known non-perturbatively in $\Delta_{0}$.
If we only know $Z_{P}$ in the small $\Delta_{n}$ limit, then we can use

$\lim_{T\rightarrow\infty}\frac{2}{T}\Delta_{n}\lim_{\Delta_{n}\rightarrow
0}\frac{1}{\Delta_{n}}\frac{Z_{P}}{Z}=\lim_{T\rightarrow\infty}\frac{2}{T}\frac{\sum
e^{-E^{+}_{n}T}(\Delta_{n}T)}{\sum e^{-E^{+}_{n}T}}=\Delta_{0}\,.$ (4.7)

This formula will allow us to compute the splitting of the lowest energy
states directly with the path integral: if we compute $Z_{P}$ and $Z$ in the
saddle-point approximation and find at small splitting and large $T$ that
$Z_{P}\sim ZAT$, then the energy splitting is $\Delta_{0}=2A$.

The twisted partition function can be computed with the path integral using
twisted boundary conditions:

$Z_{P}=\text{Tr}(Pe^{-HT})=\mathcal{N}\int_{-\infty}^{\infty}db\
\langle-b|e^{-HT}|b\rangle=\mathcal{N}\int_{-\infty}^{\infty}db\int_{x\left(-\frac{T}{2}\right)=-b}^{x\left(\frac{T}{2}\right)=b}\mathcal{D}x\
e^{-S[x]}\,.$ (4.8)

All paths involved go from $-b$ to $b$ in time $T$ and we integrate over $b$
to take the trace. For the normalization $\mathcal{N}$, we can match to the
partition function of the harmonic oscillator.

Figure 2: The twisted partition function sums over paths starting and ending
at any $b$. At finite $T$, the double well admits exact instanton solutions
for any $b$. For any $b$ and large values of $T$, these solutions spend most
of the time in one of the two wells.

For any $b>0$, the classical path from $-b$ to $b$ for a large time $T$ is
exponentially close to the $T=\infty$ instanton

$x_{I}(t)=a\tanh\frac{\omega t}{2}\,.$ (4.9)

For $b<0$, the classical path from $-b$ to $b$ for a large time $T$ is closer
to the anti-instanton solution $-x_{I}(t)$. Both the instanton and anti-
instanton have an action given by

$S_{I}\equiv S[x_{I}]=a^{2}\omega\tanh\frac{\omega
T}{4}-\frac{a^{2}\omega}{3}\tanh^{3}\frac{T\omega}{4}=\frac{2a^{2}\omega}{3}+\mathcal{O}(e^{-T\omega})\,.$
(4.10)

In particular for $b\neq a$ these (anti-)instanton paths start at $-b$, shoot
up to the top of hill of the inverted potential, stay there for almost the
entire time, roll across, and then stay at the other hill, ending at $b$ after
a total time $T$. Examples (found numerically with the shooting method) that
have $b$ slightly lower or larger than $a$ are shown in Fig. 2. In particular,
the part of the motion between $-b$ and $-a$ is given by a solution to the
Euclidean equations of motion

$\displaystyle\tilde{x}_{b}(t)=-a\coth{\frac{\omega t}{2}}\,,$ (4.11)

and provides the corresponding action

$\displaystyle S[\tilde{x}_{b}(t)]=\int_{a}^{b}dx\
\sqrt{2V(x)}=\frac{\omega}{6a}(a-b)^{2}(2a+b)\,.$ (4.12)

Thus in the large $T$ limit, we find that the classical path from $-b$ to $b$
($b>0$) has action

$\displaystyle
S[x_{b}(t)]=S[x_{I}]+2S[\tilde{x}_{b}(t)]=S_{I}+\frac{\omega}{3a}(a-b)^{2}(2a+b)\,.$
(4.13)

Note that when $\left|\frac{b}{a}\right|\gg 1$ then
$S[x]\approx\frac{2a^{2}\omega}{3}+\frac{\omega|b|^{3}}{3a}$ and so the part
of the action arising from the path approaching the hills is significant. The
effect is that the contributions to the twisted partition function with
$|b|\gg a$ will be small, however we still need to include the $b$ dependence
to get the numerical prefactor correct.

For the fluctuations, the tail regions before and after $|x|\approx a$ give
exponentially small contributions, so we can expand paths as $x(t)\rightarrow
x_{I}(t)+x(t)$. Then the twisted partition function in the leading saddle
point approximation becomes

$\displaystyle Z_{P}$ $\displaystyle\approx 2\mathcal{N}\int_{0}^{\infty}db\
e^{-S_{I}}e^{-2S[\tilde{x}_{b}(t)]}\int_{x\left(-\frac{T}{2}\right)=0}^{x\left(\frac{T}{2}\right)=0}\mathcal{D}x\,\exp\left[-\frac{1}{2}\int_{-\frac{T}{2}}^{\frac{T}{2}}dt\,x(t)[-\partial_{t}^{2}+V^{\prime\prime}(x_{I})]x(t)\right]$
(4.14) $\displaystyle\approx\mathcal{N}\sqrt{\frac{4\pi}{\omega}}e^{-S_{I}}\
\int_{x\left(-\frac{T}{2}\right)=0}^{x\left(\frac{T}{2}\right)=0}\mathcal{D}x\,\exp\left[-\frac{1}{2}\int_{-\frac{T}{2}}^{\frac{T}{2}}dt\,x(t)[-\partial_{t}^{2}+V^{\prime\prime}(x_{I})]x(t)\right]$
(4.15)

where we have performed the integral over $b$ in the large $S_{I}$ limit. The
factor of $2$ in front accounts for the anti-instanton solutions, which go
from $-b$ to $b$ where $b<0$, and give the same contribution as the instanton
solutions. Note that the instantons saturate the twisted boundary conditions
and the fluctuations are Dirichlet. Next, we want to compute the path integral
in the leading saddle-point approximation, carefully treating the collective
coordinate and Gaussian integration over the fluctuations.

###  4.1 Normal modes at finite $T$

Fluctuations around the instanton are described by normal modes satisfying

$\big{[}-\partial_{t}^{2}+V^{\prime\prime}(x_{I})\big{]}\,x_{n}=\lambda_{n}x_{n}$
(4.16)

where
$V^{\prime\prime}(x_{I})=\omega^{2}\left(1-\frac{3}{2}\operatorname{sech}^{2}\frac{\omega
t}{2}\right)$. Upon rescaling $t\rightarrow t/2$, we recognize that Eq. (4.16)
is equivalent to the time-independent Schrödinger equation with a Pöschl-
Teller potential [35] having $\lambda=2$. The spectrum of the differential
operator in Eq. (4.16) has bound states and a continuum. The continuum becomes
discrete at finite $T$ and we can write a trajectory as

$x(t)=x_{I}(t)+a_{0}x_{0}(t)+a_{1}x_{1}(t)+\sum_{k}(a_{k}^{+}x_{k}^{+}(t)+a_{k}^{-}x_{k}^{-}(t))$
(4.17)

with modes defined as follows.

The first bound state is333This function as written has eigenvalue exactly
$0$, but does not satisfy the Dirichlet boundary conditions. If we impose the
Dirichlet boundary conditions at $t=\pm\frac{T}{2}$, the eigenvalue gets
lifted to $\lambda_{0}\approx e^{-\omega T}$ (see [24, App. E] or [30, Eq.
23.55b]). While an exponentially small eigenvalue renders finite the
functional determinant in the path integral, it does not produce the correct
answer for the twisted partition function because higher order terms in
fluctuations in the $x_{0}$ direction are important. Collective coordinates
allow for inclusion of fluctuations due to time translations to all orders.

$x_{0}(t)=\sqrt{\frac{3\omega}{8}}\frac{1}{\cosh^{2}\frac{\omega
t}{2}}=\frac{\sqrt{6}}{2}\,\dot{x}_{I}(t),\quad\lambda_{0}\approx 0\,.$ (4.18)

Note that $x_{0}(t)$ is proportional to the velocity of the instanton.

The second bound state has odd parity and finite eigenvalue:

$x_{1}(t)=\frac{\sqrt{3\omega}}{2}\frac{\sinh\frac{\omega
t}{2}}{\cosh^{2}\frac{\omega t}{2}},\quad\lambda_{1}=\frac{3}{4}\omega^{2}\,.$
(4.19)

For $T\rightarrow\infty$ the remaining modes form a continuum. However, if we
consistently maintain boundary conditions at finite $T$, these remaining modes
are discrete. The eigenvalue equation is solved by

$x_{k}(t)=\frac{1}{4}\left(1+4\frac{k^{2}}{\omega^{2}}+6i\frac{k}{\omega}\tanh\frac{\omega
t}{2}-3\tanh^{2}\frac{\omega
t}{2}\right)e^{ikt},\quad\lambda_{k}=\omega^{2}+k^{2}\,.$ (4.20)

Paths $x_{k}(-t)$ are also eigenstates with the same eigenvalues. Since the
Pöschl-Teller potential is symmetric, it is sensible to use parity
eigenstates. Explicitly, these modes are

$\displaystyle x_{k}^{+}(t)$ $\displaystyle=N_{k}[x_{k}(t)+x_{k}(-t)]$ (4.21)
$\displaystyle=\frac{N_{k}}{2}\left[\left(1+\frac{4k^{2}}{\omega^{2}}\right)\cos(kt)-\frac{6k}{\omega}\sin(kt)\tanh\left(\frac{\omega
t}{2}\right)-3\cos(kt)\tanh^{2}\left(\frac{\omega t}{2}\right)\right]$

with $N_{k}=\sqrt{\frac{2\omega^{4}}{T(4k^{4}+5k^{2}\omega^{2}+\omega^{4})}}$
, and

$\displaystyle x_{k}^{-}(t)$ $\displaystyle=-iN_{k}[x_{k}(t)-x_{k}(-t)]$
(4.22)
$\displaystyle=\frac{N_{k}}{2}\left[\left(1+\frac{4k^{2}}{\omega^{2}}\right)\sin(kt)+\frac{6k}{\omega}\cos(kt)\tanh\left(\frac{\omega
t}{2}\right)-3\sin(kt)\tanh^{2}\left(\frac{\omega t}{2}\right)\right]\,.$

Note that the modes $x_{k}^{\pm}(t)$ are manifestly real. As written, the
modes are normalized at large $T$ to $\langle
x_{k}^{\pm}|x_{k}^{\pm}\rangle=1+\mathcal{O}\left(\frac{1}{T}\right)$. The
eigenvalues are quantized by the boundary conditions
$x_{k}^{\pm}\left(\frac{T}{2}\right)=0$ which leads at large $T$ to

$k^{-}(n)\approx\frac{2n\pi}{T},\quad k^{+}(n)\approx\frac{(2n-1)\pi}{T},\quad
n=1,2,3,...\,.$ (4.23)

As $T\rightarrow\infty$, the modes approach

$x_{k}^{\pm}(t)\rightarrow
s_{n}(t),\quad\lambda_{n}\approx\omega^{2}+\left(\frac{n\pi}{T}\right)^{2},\quad
n=1,2,3,...$ (4.24)

with $s_{n}$ as in Eq. (3.4). As $T\to\infty$ the $x_{k}^{+}$ become the
even-$n$ modes and $x^{-}_{k}$ become the odd-$n$ modes. The modes
$x_{k}^{\pm}$ and $s_{n}$ differ only close to $t=0$, as shown in Fig. 3. For
small $t$, it is enough to know the values of the modes and their derivatives
at $t=0$:

$x_{k}^{+}(0)=\frac{1}{\sqrt{2T}},\quad
x_{k}^{-}(0)=0,\quad\dot{x}_{k}^{+}(0)=0,\quad\dot{x}_{k}^{-}(0)=\frac{4\sqrt{2}n\pi}{T^{3/2}}\,.$
(4.25)

When $t$ is not small, the $s_{n}$ are excellent approximations to the
$x_{k}^{\pm}$.

Figure 3: Normal modes of the double well at finite $T$ (plotted with
$T=200$), with Dirichlet boundary conditions at $t=\pm\frac{T}{2}$. Modes have
even or odd parity. The two bound state modes $x_{0}$ and $x_{1}$ are shown,
as are two of the continuum modes $x_{k}^{\pm}$ with $k=5$. These continuum
modes are similar to sine functions except near $t=0$.

###  4.2 Collective coordinates

Now we would like to change to collective coordinates. To do so we need to
consider the intersection number $N_{X}[x]$. This is the number of times a
given path $x(t)$ has vanishing projection against $x_{0}(t+t_{0})$ over the
integration range of $t_{0}$. Parameterizing $x(t)$ as in Eq. (4.17), we need
to find the number of zeros of

$\big{\langle}x_{0}(t+t_{0})\big{|}x(t)\big{\rangle}=\big{\langle}x_{0}(t+t_{0})\big{|}x_{I}(t)\big{\rangle}+a_{0}\big{\langle}x_{0}(t+t_{0})\big{|}x_{0}(t)\big{\rangle}+a_{1}\big{\langle}x_{0}(t+t_{0})\big{|}x_{1}(t)\big{\rangle}\\\\[5.69054pt]
\qquad\qquad+\sum_{k}\Big{[}a_{k}^{+}\big{\langle}x_{0}(t+t_{0})\big{|}x_{k}^{+}(t)\big{\rangle}+a_{k}^{-}\big{\langle}x_{0}(t+t_{0})\big{|}x_{k}^{-}(t)\big{\rangle}\Big{]}$
(4.26)

for fixed values of the coordinates $\\{a_{0},a_{1},a_{k}^{\pm}\\}$. To
evaluate the above expression we note that there are three relevant time
scales: the overall time $T$, the collective coordinate time $t_{0}$, and the
inverse frequency $\omega^{-1}$ associated with the curvature of the
potential. The two bound states $x_{0}$ and $x_{1}$ have support of order
$\omega^{-1}$ while the continuum modes have support over all $t$. For
$t_{0}\ll\omega^{-1}\ll T$, we have

$\langle
x_{0}(t+t_{0})|x(t)\rangle=-\sqrt{\frac{2a^{2}\omega}{3}}\,t_{0}+a_{0}-\frac{3\pi}{16\sqrt{2}}\,a_{1}\omega
t_{0}+\cdots\,.$ (4.27)

If we are permitted to drop the $a_{1}$ term and the additional terms
thereafter, the above reduces to $t_{0}\approx a_{0}\,S^{-1/2}_{I}$. This
fiducial trade-off between $a_{0}$ and $t_{0}$ is the change to collective
coordinates one normally anticipates: motion along the direction of the zero
mode $x_{0}$ is exchanged for motion along $t_{0}$. However, the relationship
between $a_{0}$ and $t_{0}$ only holds for $t_{0}\ll\omega^{-1}\ll T$. For
larger $t_{0}$ the other terms in Eq. (4.27) are important.

In the regime $\omega^{-1}\ll t_{0}\ll T$, the detailed structure of
$x_{0}(t+t_{0})$ of order $\omega^{-1}$ cannot be resolved. In fact, as
$\omega t_{0}\rightarrow\infty$, the mode
$x_{0}(t-t_{0})\sim\frac{1}{\cosh^{2}\frac{\omega(t-t_{0})}{2}}$ approximates
a $\delta$-function:

$x_{0}(t-t_{0})\approx\sqrt{\frac{6}{\omega}}\,\delta(t-t_{0})\,,\quad\omega^{-1}\ll
t_{0}\ll T\,,$ (4.28)

so that

$\big{\langle}x_{0}(t-t_{0})\big{|}x(t)\big{\rangle}\approx\sqrt{\frac{6}{\omega}}\int
dt\,\delta(t-t_{0})x(t)=\sqrt{\frac{6}{\omega}}\,x(t_{0})\,.$ (4.29)

The equation above leads to a nice interpretation of $N_{x_{0}}[x]$: it is the
number of times $x(t_{0})=0$, i.e. the number of times a given path $x(t)$
crosses $x=0$ on its way from $-b$ to $b$. Similarly, in the same limit the
Jacobian in Eq. (2.12)
$\mathcal{J}=\langle\dot{x}_{0}(t)|x(t)\rangle\approx-\dot{x}(0)$ becomes the
velocity of the path at $t_{0}$, i.e. as the path crosses the origin.

###  4.3 Energy splitting

Now let us use the collective coordinates to compute the energy splitting. To
do so, it is easiest to parameterize paths by

$x(t)=x_{I}(t+t_{0})+\alpha_{1}x_{1}(t+t_{0})+\sum_{k}\Big{[}\alpha_{k}^{+}x_{k}^{+}(t+t_{0})+\alpha_{k}^{-}x_{k}^{-}(t+t_{0})\Big{]}$
(4.30)

and to use Eq. (2.10). The Jacobian evaluates to

$\displaystyle\mathcal{J}$
$\displaystyle=\big{\langle}\dot{x}_{0}(t)\big{|}x_{I}(t)+\alpha_{1}x_{1}(t)+\sum_{k}\Big{[}\alpha_{k}^{+}x_{k}^{+}(t)+\alpha_{k}^{-}x_{k}^{-}(t)\Big{]}\,\big{\rangle}$
(4.31) $\displaystyle\approx-\sqrt{\frac{2\omega
a^{2}}{3}}-\frac{3\pi\omega}{16\sqrt{2}}\alpha_{1}+\sum_{k}\alpha_{k}^{+}\frac{4\sqrt{2}n\pi}{T^{3/2}}$
(4.32)

where $n$ refers to the approximate wavenumber from Eq. (4.23).

The action of a path $x(t)$ parameterized in collective coordinates is

$S[x]=\frac{1}{2}\alpha_{1}^{2}\lambda_{1}+\frac{1}{2}\sum_{k}\Big{[}(\alpha_{k}^{+})^{2}\lambda_{k}^{+}+(\alpha_{k}^{-})^{2}\lambda_{k}^{-}\Big{]}\,.$
(4.33)

Since $\lambda_{1}=\frac{3}{4}\omega^{2}$ and
$\lambda_{k}\approx\omega^{2}+\left(\frac{n\pi}{T}\right)^{2}$, the above
action implies that the path integral is dominated by the regime
$\alpha_{1}\lesssim\omega^{-1}$ and $\alpha_{k}^{\pm}\lesssim\omega^{-1}$ (and
even for $n>T\omega$ the scaling is
$\alpha^{\pm}_{k}\lesssim\frac{T}{n}\lesssim\omega^{-1}$ as well). Thus, for
purposes of computing the energy splitting where we are interested in the
limit of large $a^{2}\omega$ and large $T$, the Jacobian factor in Eq. (4.32)
reduces to

$\mathcal{J}\approx-\sqrt{\frac{2\omega a^{2}}{3}}=-\sqrt{S_{I}}\,.$ (4.34)

This is the result usually found in the literature.

In collective coordinate, the twisted partition function is then

$Z_{P}=\mathcal{N}\sqrt{\frac{4\pi}{\omega}}\sqrt{S_{I}}\,e^{-S_{I}}\int_{-\frac{T}{2}}^{\frac{T}{2}}\frac{dt_{0}}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\frac{d\alpha_{1}}{\sqrt{2\pi}}\,\prod_{k}\int_{-\infty}^{\infty}\frac{d\alpha_{k}^{\pm}}{\sqrt{2\pi}}\
e^{-\frac{1}{2}\lambda_{1}\alpha_{1}^{2}-\frac{1}{2}\lambda_{k}(\alpha_{k}^{\pm})^{2}}\frac{1}{N_{x_{0}}[x]}$
(4.35)

where, as discussed, $N_{x_{0}}[x]$ counts the number of times a path $x(t)$
crosses $x=0$. Although $N_{x_{0}}[x]$ is an incredibly complicated functional
of paths, it can be simplified for the calculation of the twisted partition
function in the large $T$ limit.

We reiterate that according to Eq. (4.29), $N_{x_{0}}[x]$ counts the number of
times the path crosses the origin. Since all paths integrated over in the
twisted partition function go from $-b$ to $b$ they must all cross the origin
at least once – this is essential, otherwise the change to collective
coordinate $t_{0}$ would not be valid. Because the potential $V(x)$ is largest
at $x=0$, larger values of $N_{x_{0}}[x]$ will give exponentially suppressed
contributions to the path integral compared with the contributions at
$N_{x_{0}}[x]=1$. Consequently $N_{x_{0}}[x]\geqslant 1$ and paths with
$N_{x_{0}}[x]>1$ are negligible. Thus we can approximate $N_{x_{0}}[x]=1$. So
we have

$\displaystyle Z_{P}$
$\displaystyle\approx\mathcal{N}\sqrt{\frac{4\pi}{\omega}}T\sqrt{\frac{S_{I}}{2\pi}}\,e^{-S_{I}}\int_{-\infty}^{\infty}\frac{d\alpha_{1}}{\sqrt{2\pi}}\prod_{k}\int_{-\infty}^{\infty}\frac{d\alpha_{k}^{\pm}}{\sqrt{2\pi}}\
e^{-\frac{1}{2}\lambda_{1}\alpha_{1}^{2}-\frac{1}{2}\lambda_{k}(\alpha_{k}^{\pm})^{2}}$
(4.36)
$\displaystyle=\mathcal{N}\sqrt{\frac{4\pi}{\omega}}T\sqrt{\frac{S_{I}}{2\pi}}\,e^{-S_{I}}\sqrt{\frac{1}{\lambda_{1}}}\prod_{k}\sqrt{\frac{1}{\lambda_{k}^{\pm}}}$
(4.37)
$\displaystyle=\mathcal{N}\sqrt{\frac{4\pi}{\omega}}T\sqrt{\frac{S_{I}}{2\pi}}\,e^{-S_{I}}\left[\text{det}^{\prime}(-\partial_{t}^{2}+V^{\prime\prime}(x_{I}))\right]^{-1/2}\,.$
(4.38)

As such we have reduced our calculation to the textbook treatment.

We can also understand the $N_{x_{0}}[x]\approx 1$ replacement from a
different perspective, which allows a more straightforward generalization to
the QFT case in Section 5. If we solve the condition $\langle
x_{0}(t+t_{0})|x(t)\rangle=0$ for $t_{0}$ using Eq. (4.27) and expand at large
$S_{I}$, we find

$t_{0}=\frac{a_{0}}{\sqrt{S_{I}}}-\frac{3\pi\omega}{16\sqrt{2}S_{I}}\,a_{0}\,a_{1}+\cdots\,.$
(4.39)

Here $a_{n}\lesssim\omega^{-1}$ just as $\alpha_{n}\lesssim\omega^{-1}$.
However, the mode with zero eigenvalue with coordinate $a_{0}$ is
unsuppressed. Considering only the zero mode, the condition determining
$t_{0}$ is

$\displaystyle 0$ $\displaystyle=\langle
x_{0}(t+t_{0})|x_{I}(t)\rangle+a_{0}\langle x_{0}(t+t_{0})|x_{0}(t)\rangle$
(4.40) $\displaystyle=\frac{3}{\omega}\sqrt{S_{I}}\,\frac{(\omega
t_{0}-\sinh(\omega t_{0}))}{\cosh\omega t_{0}-1}+a_{0}\,\frac{3(-2+\omega
t_{0}\coth\frac{\omega t_{0}}{2})}{\cosh\omega t_{0}-1}\,.$ (4.41)

The first function on the second line is a monotonically decreasing odd
function of $t_{0}$ and the second function is a positive monotonically
decreasing function of $|t_{0}|$ with a maximum at $t_{0}=0$; these conditions
imply that there is exactly one solution to (4.40) for a given $a_{0}$. There
are more solutions for $t_{0}$ if we turn on some of the other $a_{n}$’s,
analogous to the non-linear terms in Eq. (4.39). Such additional solutions for
$t_{0}$ require large values of $a_{n}\gtrsim\sqrt{S_{I}}$. However, when
$a_{n}\gtrsim\omega^{-1}$, the path integral becomes exponentially suppressed.
Thus any path with $N_{x_{0}}[x]>1$ is exponentially suppressed by the
contributions of large $a_{n}$ and so we can simply set $N_{x_{0}}[x]=1$.

The above argument can be stated more physically. We expand around a
particular instanton which crosses the origin at $t_{0}=0$. We can shift the
path using the zero mode, changing the time the path crosses the origin. But
to get an additional crossing, the quantum corrections must be enormous: as
large as the classical contribution. Such large corrections are exponentially
suppressed.

There is a close analogy between this calculation, where the energy splitting
is given by paths that cross $x=0$ exactly once, and the calculation of the
tunneling rate in an asymmetric double well. In the latter, one way to derive
the formula for the decay rate is to restrict to paths that cross the origin
only once [5, 6]. This restriction can be understood physically: if the paths
do not cross the origin at all, they do not mediate transitions from the false
vacuum to the true one; to find a well-defined decay rate one must be in a
regime where the back-scattering off the far end of the potential is
irrelevant, which can be enforced by demanding at most one crossing. Thus the
decay rate is determined by paths that cross the origin exactly once, akin to
the energy splitting in the symmetric double well.

For completeness, we finish the energy-splitting calculation. The product over
continuum eigenvalues is nearly the same as for a simple harmonic oscillator.
For the harmonic oscillator, one expands around the classical solution
$x_{\text{cl}}(t)=b\cosh(\omega t)/\cosh(\omega T/2)$ where the fluctuations
have eigenvalues $\lambda_{n}^{\text{SHO}}=\omega^{2}+({n\pi}/{T})^{2}$, and
so the partition function is

$\displaystyle Z_{\text{SHO}}$
$\displaystyle=\mathcal{N}\int_{-\infty}^{\infty}db\ e^{-\omega
b^{2}}\,\prod_{n}\int_{-\infty}^{\infty}\frac{da_{n}}{\sqrt{2\pi}}\,e^{-\frac{1}{2}\lambda_{n}a_{n}^{2}}$
(4.42)
$\displaystyle=\mathcal{N}\sqrt{\frac{\pi}{\omega}}\prod_{n}\sqrt{\frac{1}{\lambda_{n}^{\text{SHO}}}}=\mathcal{N}\sqrt{\frac{\pi}{\omega}}\
\text{det}^{-1/2}(-\partial_{t}^{2}+\omega^{2})$ (4.43)

which defines for us the normalization $\mathcal{N}$ in terms of
$Z_{\text{SHO}}$. For the double well, we can then plug $\mathcal{N}$ into Eq.
(4.38), obtaining

$\displaystyle Z_{P}=Z_{\text{SHO}}\
2T\sqrt{\frac{S_{I}}{2\pi}}\,e^{-S_{I}}\left(\frac{\det(-\partial_{t}^{2}+\omega^{2})}{\text{det}^{\prime}(-\partial_{t}^{2}+V^{\prime\prime}(x_{I}))}\right)^{1/2}\,.$
(4.44)

The ratio of determinants can be computed using the Gelfand-Yanglom method
[36, 37] and yields $12\omega^{2}$. Using Eq. (4.7), we find

$\Delta_{0}=2\sqrt{\frac{6S_{I}}{\pi}}\,e^{-S_{I}}\omega$ (4.45)

Recalling $S_{I}=\frac{2\omega a^{2}}{3}$, the above exactly matches the WKB
result in Eq. (4.3). We have used that $Z\approx 2Z_{\text{SHO}}$ in the
small-energy-splitting limit.

###  4.4 Free twisted partition function

In the interacting theory, integration over the collective coordinate $t_{0}$
gave a factor of $T$ in the twisted partition function, as expected for the
computation of the energy splitting. The free theory is also time-translation
invariant, and so using collective coordinates would seem to give a factor of
$T$. However, the free twisted partition function is just $Z_{P}=\frac{1}{2}$
(as we review shortly). So what is the fate of the putative factor of $T$? To
find out, we need to properly compute the twisted partition function with
action $S_{0}[x]=\int dt\,\frac{1}{2}\dot{x}^{2}$ using collective
coordinates.

#### Expected results

First we recall the expected result. The twisted partition function in the
free theory can be computed as the zero frequency limit of the twisted
partition function for the harmonic oscillator, namely

$Z_{P}^{0}=\text{Tr}(Pe^{-HT})=\lim_{\omega\rightarrow
0}\,\sum_{n=0}^{\infty}e^{-\omega\left(n+\frac{1}{2}\right)T}(-1)^{n}=\frac{1}{2}\,.$
(4.46)

Alternatively, we can recall that the position-space propagator in quantum
mechanics is

$D(a,b)=\langle
b|e^{-HT}|a\rangle=\int_{-\infty}^{\infty}\frac{dp}{2\pi}\langle
b|e^{-\frac{1}{2}p^{2}T}|p\rangle\langle
p|a\rangle=\int_{-\infty}^{\infty}\frac{dp}{2\pi}\,e^{ip(b-a)}e^{-\frac{p^{2}}{2}T}=\sqrt{\frac{1}{2\pi
T}}\,e^{-\frac{(b-a)^{2}}{2T}}\,.$ (4.47)

So the twisted partition function is

$Z_{P}^{0}=\int_{-\infty}^{\infty}db\
D(b,-b)=\int_{-\infty}^{\infty}db\,\sqrt{\frac{1}{2\pi
T}}\,e^{-\frac{2b^{2}}{T}}=\frac{1}{2}\,.$ (4.48)

In the path integral approach, we compute

$Z_{P}^{0}=\mathcal{N}\int_{-\infty}^{\infty}db\int_{x\left(-\frac{T}{2}\right)=-b}^{x\left(\frac{T}{2}\right)=b}\mathcal{D}x\
e^{-S[x]}\,.$ (4.49)

Here the classical path going from $-b$ to $b$ in time $T$ is
$x_{c}(t)=\frac{2t}{T}b$ with action $S_{0}[x_{c}]=\frac{2b^{2}}{T}$, as
expected from the exponent of $D(b,-b)$. A general path is then parameterized
by

$x(t)=x_{c}(t)+c_{n}s_{n}(t),\quad n=1,2,3,\ldots\,,$ (4.50)

with $s_{n}(t)$ as in Eq. (3.4). These $s_{n}(t)$ modes satisfy
$\partial_{t}^{2}s_{n}=-\lambda_{n}s_{n}$ with
$\lambda_{n}=\left(\frac{n\pi}{T}\right)^{2}$. Then

$Z_{P}^{0}=\mathcal{N}\int_{-\infty}^{\infty}db\
e^{-\frac{2b^{2}}{T}}\prod_{n=1}^{\infty}\int_{-\infty}^{\infty}\frac{dc_{n}}{\sqrt{2\pi}}\
e^{-\frac{1}{2}\lambda_{n}c_{n}^{2}}=\mathcal{N}\frac{\sqrt{2\pi
T}}{2}\prod_{n}\frac{T}{n\pi}$ (4.51)

so that to have $Z_{P}^{0}=\frac{1}{2}$ we need

$\mathcal{N}=\frac{1}{\sqrt{2\pi
T}}\left[\prod_{n}\frac{T}{n\pi}\right]^{-1}\,.$ (4.52)

#### Collective coordinates

Now we attempt to compute $Z_{P}^{0}$ using collective coordinates in the
double-well basis. Noting that the formula in Eq. (2.12) is basis-independent,
the only feature from the double well basis we carry over is that the function
$X(t)=x_{0}(t)$ is the one used to project out the collective coordinate.
Indeed, if we had tried to use the parametrization in Eq. (4.17) for the paths
in the free theory case we would have run into problems because $x_{I}(t)$ is
not a saddle point of the free action (in fact, it has much larger action than
the classical path; for paths from $-a$ to $a$,
$S_{0}[x_{I}]=\frac{a^{2}\omega}{3}\gg S_{0}[x_{c}]=\frac{2a^{2}}{T})$. Since
$x_{I}(t)$ is not a saddle point, in attempting to use Eq. (4.17) in the path
integral, we would find linear fluctuation terms which would require
completing the square in the action, effectively reducing the computation to
expanding around the classical path. The result would be the same as using the
parameterization in Eq. (4.50) to begin with.

Marching forward, we want to evaluate

$Z_{P}^{\text{coll},0}=\mathcal{N}\int_{-\infty}^{\infty}db\
e^{-\frac{2b^{2}}{T}}\int_{-\frac{T}{2}}^{\frac{T}{2}}dt_{0}\
\prod_{n}\int_{-\infty}^{\infty}\frac{dc_{n}}{\sqrt{2\pi}}\
e^{-\frac{1}{2}\lambda_{n}c_{n}^{2}}\frac{1}{N_{x_{0}}[x]}\,\delta\Big{[}\big{\langle}x_{0}(t)\big{|}x(t)\big{\rangle}\Big{]}\,\Big{|}\big{\langle}x_{0}(t)\big{|}\dot{x}(t)\big{\rangle}\Big{|}\,.$
(4.53)

As discussed in the double-well case, $\delta[\langle x_{0}(t)|x(t)\rangle]$
forces the path to cross $x=0$ at $t=0$ and $N_{x_{0}}[x]$ counts the number
of times a given path crosses $x=0$. To compute the argument of the $\delta$
function we need the projections

$\displaystyle\langle x_{0}(t)|s_{n}(t)\rangle$
$\displaystyle=\sqrt{\frac{2}{T}}\int_{-\infty}^{\infty}dt\
x_{0}(t)\sin\frac{n\pi\left(t+\frac{T}{2}\right)}{T}=\sqrt{\frac{12}{T\omega}}\sin\frac{n\pi}{2}\,Q_{n}$
(4.54)
$\displaystyle=(-1)^{\frac{n-1}{2}}\sqrt{\frac{12}{T\omega}}\,\delta_{n,\,\text{odd}}\,Q_{n}$
(4.55)

which we calculate in the $T\rightarrow\infty$ limit, where

$Q_{n}=\frac{\frac{n\pi^{2}}{T\omega}}{\sinh\frac{n\pi^{2}}{T\omega}}\approx
1\,.$ (4.56)

Thus the $\delta$ function involves only the odd modes, and sets
$c_{1}-c_{3}+c_{4}-c_{5}+\cdots=0$ up to corrections in $\frac{1}{T}$. The
Jacobian is

$\langle x_{0}(t)|\dot{x}(t)\rangle=\langle
x_{0}(t)|\dot{x_{c}}(t)\rangle+\sum_{n}c_{n}\langle
x_{0}(t)|\dot{s}_{n}(t)\rangle=\frac{2b}{T}\sqrt{\frac{6}{\omega}}+\sum_{n}(-1)^{\frac{n}{2}}\sqrt{\frac{12}{\omega
T}}\sqrt{\lambda_{n}}\,c_{n}\,Q_{n}\,\delta_{n,\,\text{even}}$ (4.57)

which involves only the even modes. Since the $\delta$ function only involves
the odd modes and the Jacobian only involves the even modes, the integration
over the even and odd modes factorizes.

With the factor of $N_{x_{0}}[x]$, even though the path integral factorizes,
the integrals are challenging. To show that $N_{x_{0}}[x]$ is necessary, it
suffices to ignore it by setting it to 1 and show that the wrong answer
results. Let us do so and behold the consequences of such ignorance.

The formulas

$\int d^{N}c\ e^{-\pi\vec{c}\cdot
A\cdot\vec{c}}|\vec{w}\cdot\vec{c}|=\frac{1}{\pi}\frac{\sqrt{\vec{w}\cdot
A^{-1}\cdot\vec{w}}}{\sqrt{\det(A)}}$ (4.58)

and

$\int d^{N}c\
e^{-\pi\vec{c}^{\,2}}\delta(\vec{w}\cdot\vec{c})=\frac{1}{\sqrt{\vec{w}^{2}}}$
(4.59)

are helpful. For the odd modes we find

$I_{\text{odd}}=\prod_{n\,\text{odd}}\int\frac{dc_{n}}{\sqrt{2\pi}}\,e^{-\frac{1}{2}c_{n}^{2}\lambda_{n}}\delta\left(c_{n}\sqrt{\frac{12}{\omega
T}}\sin\frac{n\pi}{2}Q_{n}\right)=\left(\prod_{n\,\text{odd}}\sqrt{\frac{1}{\lambda_{n}}}\right)\sqrt{\frac{\omega}{3\pi
T}}$ (4.60)

and for the even modes we have

$\displaystyle I_{\text{even}}$ $\displaystyle=\int db\
e^{-\frac{2b^{2}}{T}}\prod_{n\,\text{even}}\int\frac{dc_{n}}{\sqrt{2\pi}}\,e^{-\frac{1}{2}\lambda_{n}c_{n}^{2}}\left|\frac{2b}{T}\sqrt{\frac{6}{\omega}}+\sum_{n}(-1)^{\frac{n}{2}}\sqrt{\frac{12}{\omega
T}}\sqrt{\lambda_{n}}\,c_{n}\,Q_{n}\,\delta_{n,\text{even}}\right|$ (4.61)
$\displaystyle=\left(\prod_{n\,\text{even}}\sqrt{\frac{1}{\lambda_{n}}}\right)\left(\sqrt{T}+\frac{3}{\omega\sqrt{T}}+\cdots\right)\,.$
(4.62)

So all together we obtain

$\displaystyle Z_{P}^{\text{coll},0}$ $\displaystyle=\frac{1}{\sqrt{2\pi
T}}\left[\prod_{n}\sqrt{\frac{1}{\lambda_{n}}}\right]^{-1}\left(\prod_{n\,\text{odd}}\sqrt{\frac{1}{\lambda_{n}}}\right)\sqrt{\frac{\omega}{3\pi
T}}\left(\prod_{n\,\text{even}}\sqrt{\frac{1}{\lambda_{n}}}\right)\sqrt{T}\int_{-T/2}^{T/2}dt_{0}$
(4.63) $\displaystyle=\sqrt{\frac{\omega T}{6\pi^{2}}}\,.$ (4.64)

This does not agree with the correct answer $Z_{P}^{0}=\frac{1}{2}$, as in Eq.
(4.48). Indeed Eq. (4.64) is too large by a factor of $\sqrt{\omega T}$. Thus
we confirm that using collective coordinates without the factor of
$N_{x_{0}}[x]$ is incorrect.

To see that the right answer does emerge if $N_{x_{0}}[x]$ is included is
challenging because of the extreme complexity of $N_{x_{0}}[x]$. However,
incorporating $N_{x_{0}}[x]$ is essentially the same challenging problem
discussed in Section 3. It does not seem that working in the double-well basis
rather than a Fourier basis provides any additional insight, so we simply
refer the reader back to Section 3 and move on to quantum field theory.

##  5 Quantum field theory

For our final example, we consider the $\lambda\phi^{4}$ quantum field theory
in 4 Euclidean dimensions. The Lagrangian density is
$\mathcal{L}[\phi]=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+V(\phi)$
with $V(\phi)=\frac{1}{4}\lambda\phi^{4}$. For $\lambda<0$ this theory has a
5-parameter family of Fubini-Lipatov instanton saddle points [38, 39] called
“bounces” given by

$\phi_{b}^{R,\xi}(x)=\sqrt{\frac{8}{-\lambda}}\frac{R}{R^{2}+(x^{\mu}+\xi^{\mu})^{2}}\,,$
(5.1)

with action

$S_{b}=S[\phi_{b}]=-\frac{8\pi^{2}}{3\lambda}\,.$ (5.2)

We will generally pick a fiducial member of this family to expand around and
denote it by $\phi_{b}(r)$ with $R$ fixed and $\xi=0$. Here
$r=\sqrt{\sum_{\mu}x_{\mu}^{2}}$ is the Euclidean radial coordinate.

With $\lambda<0$ the vacuum at $\phi=0$ is unstable. The corresponding decay
rate can be computed perturbatively as

$\Gamma=\lim_{T\rightarrow\infty}\frac{1}{T}\text{Im}\frac{\int_{C_{b}}\mathcal{D}\phi\,e^{-S[\phi_{b}]+\frac{1}{2}S^{\prime\prime}[\phi_{b}]\phi^{2}+\cdots}}{\int_{C_{\text{FV}}}\mathcal{D}\phi\,e^{-\frac{1}{2}\int
d^{4}x(\partial_{\mu}\phi)^{2}}}$ (5.3)

where $C_{b}$ is a path of steepest descent passing though the bounce
$\phi_{b}$ and $C_{\text{FV}}$ is the expansion around the false vacuum
$\phi=0$. A derivation of this formula can be found in [7]. The numerator of
this expression is expected to have a factor of $T\mathcal{V}$ from the
translation invariance of the action (the $\xi^{\mu}$ moduli). Such a factor
makes $\Gamma\propto\mathcal{V}$, i.e. the decay is a rate per unit volume, as
expected in a translationally invariant theory. The denominator, which
comprises an expansion around the free theory, is not expected to have a
factor of $T\mathcal{V}$. The goal of this section is to understand why the
numerator is proportional to $T\mathcal{V}$ but the denominator is not,
despite the fact that both actions are translation and scale invariant.

###  5.1 Basis and cross check

Remarkably, the eigenstates for expanding around the bounce are known in
closed form [7]. Even more remarkably, these same eigenstates also diagonalize
fluctuations around the false vacuum. The modes are

$\hat{\phi}_{ns\ell
m}(r,\alpha,\theta,\phi)=\frac{\sqrt{2n+3}}{2\pi}\frac{1}{r}P_{n+1}^{-s-1}\left(\frac{R^{2}-r^{2}}{R^{2}+r^{2}}\right)\sqrt{\frac{\pi}{2}}\,Y^{s\ell
m}(\alpha,\theta,\phi)$ (5.4)

where $P_{\ell}^{m}(x)$ are associated Legendre polynomials and $Y^{s\ell
m}(\alpha,\theta,\phi)$ are 3D spherical harmonics (which are also Legendre
polynomials). We will use the “box” inner product

$\langle\phi_{i}|\phi_{j}\rangle=-\int d^{4}x\,\phi_{i}(x)\Box\phi_{j}(x)$
(5.5)

under which the modes are orthonormal, namely

$\langle\hat{\phi}_{ns\ell
m}|\hat{\phi}_{n^{\prime}s^{\prime}\ell^{\prime}m^{\prime}}\rangle=-\int
d^{4}x\,\hat{\phi}_{ns\ell
m}\Box\hat{\phi}_{n^{\prime}s^{\prime}\ell^{\prime}m^{\prime}}=\delta_{nn^{\prime}}\delta_{ss^{\prime}}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,.$
(5.6)

The modes satisfy

$\Box\,\hat{\phi}_{ns\ell
m}=\lambda_{n}V^{\prime\prime}[\phi_{b}]\hat{\phi}_{ns\ell
m},\quad\lambda_{n}=\frac{(n+1)(n+2)}{6}=\frac{1}{3},1,2,\frac{10}{3},...\,,$
(5.7)

and there are $d_{n}=\frac{(n+1)(n+2)(2n+3)}{6}=1,5,14,...$ modes with
eigenvalue $\lambda_{n}$.

The $n=0$ mode with multiplicity $d_{0}=1$ is proportional to the bounce

$\phi_{0000}=\frac{\sqrt{3}}{2\pi}\frac{R}{r^{2}+R^{2}}=\frac{1}{2\pi}\sqrt{\frac{-3\lambda}{8}}\,\phi_{b}=\frac{1}{2\sqrt{S_{b}}}\,\phi_{b}\,.$
(5.8)

Normalizing with respect to our box inner product, the mode is independent of
$\lambda$. Of the five $n=1$ modes the one with $s=0$ is the dilatation mode
associated with changes in $R$, namely

$\phi_{d}\equiv\phi_{1000}=\frac{\sqrt{5}R}{2\pi}\frac{R^{2}-r^{2}}{(R^{2}+r^{2})^{2}}=-\sqrt{\frac{5}{12S_{b}}}R\,\partial_{R}\phi_{b}\,.$
(5.9)

The other four $n=1$ modes, with $s=1$, correspond to translations. They are
linear combinations of the $\partial_{\mu}\phi_{b}$.

Field configurations around the bounce can be parameterized as

$\phi(x)=\phi_{b}(x)+\sum_{n}c_{n}\hat{\phi}_{n}(x)$ (5.10)

where we are using $n$ to abbreviate all the indices. The quadratic action
around the bounce is

$\displaystyle S[\phi]$ $\displaystyle\approx
S[\phi_{b}]+\sum_{n,\,m}\frac{1}{2}\int
d^{4}x\,c_{n}\hat{\phi}_{n}[-\Box+V^{\prime\prime}(\phi_{b})]c_{m}\hat{\phi}_{m}$
(5.11)
$\displaystyle=S_{b}+\sum_{n}\frac{1}{2}c_{n}^{2}\cdot\left(\frac{\lambda_{n}-1}{\lambda_{n}}\right)$
(5.12)
$\displaystyle=S_{b}+\frac{1}{2}c_{0}^{2}\cdot(-2)+\frac{1}{2}c_{1}^{2}\cdot(0)+\frac{1}{2}c_{2}^{2}\cdot\left(\frac{1}{2}\right)+\cdots\,.$
(5.13)

The unusual factor $(\lambda_{n}-1)/\lambda_{n}$ is a result of the box inner
product. If we had used the “$V$” inner product of Ref. [7], namely
$\langle\phi_{i}|\phi_{j}\rangle_{V}=-\int
d^{4}x\,V^{\prime\prime}[\phi_{b}]\phi_{i}\phi_{j}$, we would have found the
action $S=S_{b}+\frac{1}{2}c_{n}^{2}(\lambda_{n}-1)+\cdots$. For present
purposes, the box inner product is preferable because it is well-defined even
for $V=0$ so we can use it in the free theory. Importantly, there is one mode
($n=0$) with a negative eigenvalue and five modes ($n=1$) with zero eigenvalue
in either inner product.

Writing $\phi(x)=\sum_{n}b_{n}\hat{\phi}_{n}(x)$, the free action in these
coordinates is

$S[0]+\frac{1}{2}\int
d^{4}x\,S^{\prime\prime}[0]\phi(x)^{2}=\sum_{n,\,m}\frac{1}{2}\int
d^{4}x(b_{n}\hat{\phi}_{n})[-\Box](b_{m}\hat{\phi}_{m})=\sum_{n}\frac{1}{2}b_{n}^{2}\,.$
(5.14)

The reason that the same basis diagonalizes the expansion around the bounce
and around the free theory is that the bounce is itself one of the the normal
modes (the $n=0$ one). Thus we can move from the bounce to the false vacuum by
varying a single coordinate.

For a cross check, let us compute the propagator in the free theory. We can
write the propagator in terms of the path integral as

$\displaystyle D(x,0)$
$\displaystyle=\frac{\int\mathcal{D}\phi\,e^{-S[\phi]}\phi(x)\phi(0)}{\int\mathcal{D}\phi
e^{-S[\phi]}}=\frac{\int\prod_{n}db_{n}\,e^{-\frac{1}{2}b_{n}^{2}}\sum_{k,j}[b_{k}\phi_{k}(x)][b_{j}\phi_{j}(0)]}{\int\prod_{n}dc_{n}\,e^{-\frac{1}{2}c_{n}^{2}}}$
(5.15) $\displaystyle=\sum_{n}\phi_{n}(x)\phi_{n}(0)$ (5.16)
$\displaystyle=\frac{1}{4\pi^{2}Rr}\sum_{n}(2n+3)P_{n+1}^{-1}\\!\left(\frac{R^{2}-r^{2}}{R^{2}+r^{2}}\right)\,.$
(5.17)

Using the integral representation

$P_{n+1}^{-1}(x)=\frac{1}{i\pi(n+1)}\int_{0}^{\pi}dt\cos(t)\left(\sqrt{x^{2}-1}\cos
t+x\right)^{n+1}$ (5.18)

we can perform the sum, leading to

$D(x,0)=\frac{1}{4\pi^{2}r^{2}}$ (5.19)

which is the correct answer and as expected is independent of $R$.

###  5.2 Collective coordinates for dilatations

To go to collective coordinates, we consider first the $R$ coordinate
associated with dilatations. The $\lambda\phi^{4}$ theory is classically scale
invariant, including if we expand around the bounce or the false vacuum.
Ultimately, the divergence associated with an integration over $R$ will be
regulated by the scale anomaly, which shows up at higher orders in the saddle
point approximation. We are not interested in this regulation here. Rather, we
are interested in showing why a divergent integral over the collective
coordinate $R$ should show up in the expansion around the bounce but not
around the free theory.

Let us denote the dilatation mode as $\phi_{d}\equiv\phi_{1000}$. This is the
mode we denoted by $X(t)=x_{0}(t)$ in the quantum mechanical case, and is the
mode we would like to remove. While $R$ is already incorporated into our basis
expansion, let us make the $R$-dependence explicit by fixing $R=\overline{R}$
and writing

$\phi(x)=\phi_{b}(\overline{R},x)+c_{d}\phi_{d}(\overline{R},x)+\sum_{n\geq
1}c_{n}\phi_{n}(\overline{R},x)\,.$ (5.20)

Then the collective coordinate parametrization where $R$ is a coordinate and
$\phi_{d}$ is removed is

$\phi(x)=\phi_{b}(R,x)+\sum_{n\geq 1}\alpha_{n}\phi_{n}(R,x)\,.$ (5.21)

In these collective coordinates, the Jacobian factor from Eq. (2.10) is

$\mathcal{J}=\langle\phi_{d}|\partial_{R}\phi\rangle=-\sqrt{\frac{3}{5}}\frac{1}{R}\left(2\sqrt{S_{b}}+\alpha_{0}\right)+\frac{8}{\sqrt{35}}\frac{1}{R}\alpha_{2}$
(5.22)

where $\alpha_{2}$ is the coefficient of $\phi_{2000}$. Only the radial modes
with $\ell=0$ (and only two of them, with $n=0,2$) contribute. Accordingly the
expansion of the path integral around the bounce is

$e^{-S_{b}}\int\frac{dR}{R}\int\prod_{n}d\alpha_{n}\,e^{-\frac{1}{2}\alpha_{n}^{2}\frac{\lambda_{n}-1}{\lambda_{n}}}\delta(\alpha_{1})\left|\sqrt{\frac{3}{5}}\left(2\sqrt{S_{b}}+\alpha_{0}\right)-\frac{8}{\sqrt{35}}\alpha_{2}\right|\frac{1}{N_{d}[\phi]}$
(5.23)

where $N_{d}[\phi]$ counts the number of solutions to
$\langle\phi_{d}(R,x)|\phi(\overline{R},x)\rangle=0$. For the free theory, the
path integral in the collective coordinate basis is

$\int\frac{dR}{R}\int\prod_{n}d\alpha_{n}\,e^{-\frac{1}{2}\alpha_{n}^{2}}\delta(\alpha_{1})\left|\sqrt{\frac{3}{5}}\alpha_{0}-\frac{8}{\sqrt{35}}\alpha_{2}\right|\frac{1}{N_{d}[\phi]}\,.$
(5.24)

The integrands of both Eqs. (5.23) and (5.24) have a dependence on $R$ which
enters through $N_{d}[\phi]$.

In the interacting theory, $S_{b}\sim\frac{1}{-\lambda}\gg 1$ while
$\lambda_{n}\sim 1$ so that the part of the path integral domain with
$\alpha_{n}\lesssim 1$ dominates. Then we can approximate the Jacobian as
$\sqrt{\frac{3}{5}}\left(2\sqrt{S_{b}}+\alpha_{0}\right)-\frac{8}{\sqrt{35}}\alpha_{2}\approx\sqrt{\frac{12S_{b}}{5}}$.
If we can justify ignoring the $N_{d}[\phi]$ factor, the path integral reduces
to the usual expression used to compute the decay rate in $\lambda\phi^{4}$
QFT.

In the free theory, we also have $\alpha_{n}\lesssim 1$. There $S_{b}$ does
not appear and we can simply perform the integrals over $\alpha_{0}$ and
$\alpha_{2}$ and compare to what we would have gotten without collective
coordinates, where the $R$ integral is replaced by $\alpha_{1}$. The ratio of
‘collective coordinate integral’ and the ‘non-collective coordinate integral’
if we set $N_{d}[\phi]=1$ is

$\frac{\int\frac{dR}{R}\int d\alpha_{0}\int
d\alpha_{2}e^{-\frac{1}{2}\alpha_{0}^{2}-\frac{1}{2}\alpha_{2}^{2}}\left|\sqrt{\frac{3}{5}}\alpha_{0}-\frac{8}{\sqrt{35}}\alpha_{2}\right|}{\int
d\alpha_{0}d\alpha_{1}d\alpha_{2}e^{-\frac{1}{2}\alpha_{0}^{2}-\frac{1}{2}\alpha_{1}^{2}-\frac{1}{2}\alpha_{2}^{2}}}=\int\frac{dR}{R}\sqrt{\frac{17}{7\pi^{2}}}\neq
1\,.$ (5.25)

Since this ratio is not 1, we cannot expect $N_{d}[\phi]=1$ to be a good
approximation in the free theory. So if the standard computation for the
interacting theory is correct, why can we drop $N_{d}[\phi]$ when expanding
around the bounce but not around the false vacuum?

To proceed, we have to understand the intersection number $N_{d}[\phi]$. By
analogy to the non-relativistic quantum mechanics case, $N_{d}[\phi]$ counts
the number of solutions to $\langle\phi_{d}(x,R)|\phi\rangle=0$ for a given
$\phi$ over the range of $R$. As in the quantum mechanics case, we need to
work in a basis other than the collective coordinate one to evaluate
$N_{\delta}[\phi]$. In the basis with fixed $\overline{R}$ as in Eq. (5.20),
we find

$\big{\langle}\phi_{d}(x,R)\big{|}\phi(x)\big{\rangle}=2\sqrt{S_{b}}f_{0}(R,\overline{R})+\sum_{n}c_{n}f_{n}(R,\overline{R})$
(5.26)

where

$f_{n}(R,\overline{R})=-\int
d^{4}x\,\phi_{d}(x,R)\Box\phi_{n}(x,\overline{R})\,.$ (5.27)

For example, with $z=R/\overline{R}$ we have

$\displaystyle f_{0}(z)$
$\displaystyle=\frac{\sqrt{15}z\left(-1-9z^{2}+9z^{4}+z^{6}-12\left(z^{4}+z^{2}\right)\ln
z\right)}{\left(z^{2}-1\right)^{4}}$ (5.28) $\displaystyle f_{1}(z)$
$\displaystyle=-\frac{5z\left(-1-46(z^{2}-z^{6})+z^{8}-12z^{2}\left(3+z^{2}\right)\left(1+3z^{2}\right)\ln
z\right)}{\left(z^{2}-1\right)^{5}}\,.$ (5.29)

Figure 4: We depict the functions
$f_{n}(R/\overline{R})=\langle\phi_{d}(x,R)|\phi_{n}(x,\overline{R})\rangle$.
The collective coordinate $R$ is determined by the condition
$2\sqrt{S_{b}}f_{0}(R/\overline{R})+c_{n}f_{n}(R/\overline{R})=0$ which may
have multiple solutions.

For the interacting theory, we can examine whether $N_{d}[\phi]=1$ is a good
approximation using the same approach as for the double-well potential in
Section 4.3. Close to $R=\overline{R}$ we have

$\big{\langle}\phi_{d}(x,R)\big{|}\phi(x)\big{\rangle}=c_{1}+\left(\sqrt{\frac{12S_{b}}{5}}+\sqrt{\frac{3}{5}}c_{0}-\frac{8}{\sqrt{35}}c_{2}\right)\left(\frac{R-\overline{R}}{\overline{R}}\right)\\\
+\frac{-7\sqrt{15}c_{0}-85c_{1}+8\sqrt{35}c_{2}+40\sqrt{5}c_{3}-14\sqrt{15S_{b}}}{70}\left(\frac{R-\overline{R}}{\overline{R}}\right)^{2}+\cdots\,.$
(5.30)

Since $\sqrt{S_{b}}\gg 1$ we can solve $\langle\phi_{d}|\phi\rangle=0$ as a
series in $\frac{1}{\sqrt{S_{b}}}$. The leading terms are

$R=\overline{R}\left(1-\sqrt{\frac{5}{12S_{b}}}c_{1}+\frac{14\sqrt{15}c_{0}c_{1}+35c_{1}^{2}-16\sqrt{35}c_{1}c_{2}}{168S_{b}}+\cdots\right)\,.$
(5.31)

Since $\lambda_{n}\approx 1$, the path integral becomes suppressed if any
$c_{n}\gtrsim 1$ on account of the $e^{-c_{n}^{2}\lambda_{n}}$ terms. The only
exception is the zero mode: excursions in the $c_{1}$ direction by arbitrary
amounts only affect the integrand by an order 1 amount. From Eq. (5.31) we see
that there is always at least one solution with nonzero $c_{1}$. To have
multiple solutions the nonlinear terms need to become important. If we just
consider the $c_{n}$ contributions, we require $c_{n}\gtrsim\sqrt{S_{b}}\gg 1$
and so the path integral is affected by an exponentially small amount (of
order $e^{-c_{1}^{2}\lambda_{R}}\approx e^{-S_{b}}$). The situation is akin to
our quantum mechanical examples: the quantum fluctuations are too weak to
significantly alter the classical trajectory. It remains to check the
possibility of additional solutions due to large values of $c_{1}$.

For large $c_{1}$, meaning $c_{1}\gtrsim\sqrt{S_{b}}$, the expansion in Eq.
(5.31) is no longer useful. If we consider only paths of the form
$\phi(x)=\phi_{b}(x,\overline{R})+c_{1}\phi_{1}(x,\overline{R})$, then a
solution to $\langle\phi_{d}(x,R)|\phi(x)\rangle=0$ requires

$\frac{c_{1}}{\sqrt{S_{b}}}=-2\,\frac{f_{0}(R,\overline{R})}{f_{1}(R,\overline{R})}=\vbox{\hbox{
\leavevmode\hbox to178.6pt{\vbox
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For a given $c_{1}$ there is always at least one solution to the above
equation. More precisely, since $f_{1}(R,\overline{R})=0$ at
$R/\overline{R}=x_{c}$ and $R/\overline{R}=1/x_{c}$ where $x_{c}\approx
0.303$, there will be one solution with
$x_{c}\overline{R}<R<\frac{1}{x_{c}}\overline{R}$. However, if
$|c_{1}/{\sqrt{S_{b}}}|>\sqrt{12/5}$, there will be a second solution. So here
$1\leq N_{d}[\phi]\leq 2$, but we would like to determine whether the second
solution is exponentially suppressed or not.

Although the action is exactly independent of the collective coordinate $R$,
it is only independent of the coordinate $c_{1}$ to quadratic order. Expanding
to higher orders there is a quartic term

$S[\phi_{b}+c_{1}\phi_{1}]=S_{b}-\frac{5}{36S_{b}}c_{1}^{4}\,.$ (5.33)

The sign is negative since $\lambda<0$; when expanding around the bounce to
quadratic order, there is only one direction in which the potential decreases,
but to quartic order it decreases in every direction (except the flat ones)!
So we do not want to include terms to quartic order in $c_{1}$. We do,
however, want to include terms to quartic order in the collective coordinate
$R$. If we did not include these terms, the action would not be independent of
$R$. Large $R$ leads to a large factor of the volume of the dilatation group,
but not an exponentially large factor from fluctuations in a direction of
instability.

We are primarily interested in whether a field configuration has large
quadratic action in our collective coordinates and not in the coordinates at
fixed $\overline{R}$. The $\alpha_{n}$ coordinates are given by

$\displaystyle\alpha_{n}$
$\displaystyle=\big{\langle}\phi_{b}(x,\overline{R})+c_{1}\phi_{1}(x,\overline{R})-\phi_{b}(x,R)\big{|}\phi_{n}(x,R)\big{\rangle}$
(5.34)
$\displaystyle=g_{n}(R,\overline{R})+c_{1}f_{n}(R,\overline{R})-2\sqrt{S_{b}}\,\delta_{n0}$
(5.35)

where
$g_{n}(R,\overline{R})=\langle\phi_{b}(x,\overline{R})|\phi_{n}(x,R)\rangle$
and we note that $g_{1}(R,\overline{R})=f_{0}(R,\overline{R})$. When
$c_{1}\ll\sqrt{S_{b}}$, we have
$c_{1}\approx\sqrt{\frac{12S_{b}}{5}}(R-\overline{R})/\overline{R}$. In this
regime $R$ is very close to $\overline{R}$ and all the $\alpha_{n}$ are small,
namely $\alpha_{n}\ll\sqrt{S_{b}}$. When $c_{1}\gtrsim\sqrt{S_{b}}$ then all
$\alpha_{n}\gtrsim\sqrt{S_{b}}$ as well. The corresponding field
configurations, which include all of the $N_{d}[\phi]=2$ field configurations
with only $c_{1}$ turned on, are all suppressed. Other field configurations
with $N_{d}[\phi]>1$ would have various $c_{n}\gtrsim\sqrt{S_{b}}$ leading to
$\alpha_{n}\gtrsim\sqrt{S_{b}}$ as well. Thus we can approximate $N_{d}[\phi]$
as equal to $1$ up to exponentially small corrections in the path integral.

The free field theory behaves very differently, akin to the free quantum
mechanics example. In the free field theory, the
$\sqrt{\frac{12S_{b}}{5}}(\frac{R-\overline{R}}{\overline{R}})$ term in Eq.
(5.30) is not present. Then the leading solution in the regime
$\frac{R-\overline{R}}{\overline{R}}\ll 1$ is

$R=\overline{R}\left(1-\frac{7\sqrt{5}c_{1}}{7\sqrt{3}c_{0}-8\sqrt{7}c_{2}}+\cdots\right)\,.$
(5.36)

There is no small parameter in the free theory and no zero mode. So even with
$c_{n}\lesssim 1$ there are multiple solutions to
$\langle\phi_{d}|\phi\rangle=0$. That is, field configurations with
$N_{d}[\phi]\gtrsim 1$ are not suppressed and in fact $N_{d}[\phi]$ can be
arbitrarily large. Including these large values of $N_{d}[\phi]$ must
compensate for the unregulated $\int\frac{dR}{R}$ integral, akin to the free
quantum mechanics case. Showing the cancellation explicitly seems
extraordinarily challenging: collective coordinates around the bounce are very
cumbersome to use for the free theory.

An alternative approach to understand the exponential suppression when
$N_{d}[\phi]>1$ in the interacting theory, but not in the free theory, is to
change coordinates to map directly to the double well. This approach is
considered in Appendix B.

###  5.3 Collective coordinates for translations

In addition to the dilatation zero mode proportional to
$\partial_{R}\phi_{b}$, the $\lambda\phi^{4}$ theory also has zero modes
associated with translations which are linear combinations of the
$\partial_{\mu}\phi_{b}$. When introducing collective coordinates for
translations, one must also justify why the approximation $N_{\delta}=1$ is
valid in the interacting theory but not for the free theory. The argument is
essentially the same as for the double-well potential and for the dilatation
mode: in the interacting theory paths with $N_{\delta}>1$ give an
exponentially suppressed contribution to the action. Because
$\partial_{\mu}\phi_{b}$ breaks Lorentz invariance, or rather $SO(4)$
invariance in Euclidean signature, the modes which contribute to the
projections
$\langle\partial_{\mu}\phi_{b}(x)|\phi(x^{\mu}+x_{0}^{\mu})\rangle$ are not so
easy to identify.

To prove that the analysis for the translation modes proceeds as with the
dilatation mode, the easiest approach is to make a coordinate transformation
to map any particular direction in spacetime to $r$. This can be done because
the theory is scale invariant and the classical theory has an exact $SO(5)$
symmetry as part of a larger conformal symmetry. To make the map explicit, we
employ the conformal mapping with coordinates

$\eta^{\mu}=\frac{2x^{\mu}R}{R^{2}+r^{2}},\quad\eta^{5}=\frac{R^{2}-r^{2}}{R^{2}+r^{2}}\,.$
(5.37)

These new coordinates satisfy $\sum_{\mu=1}^{4}\eta_{\mu}^{2}+\eta_{5}^{2}=1$,
i.e. they parameterize the 4-sphere. The inverse of this conformal mapping is
the stereographic projection to Cartesian coordinates.

On the 4-sphere, the integration volume is

$d^{4}x=\left(\frac{2R}{R^{2}+r^{2}}\right)^{-4}d\Omega_{4}\,.$ (5.38)

If we also rescale the fields as

$\Phi(\eta)=\frac{r^{2}+R^{2}}{2R}\phi(x)=\frac{R}{1+\eta_{5}}\phi(x)\,,$
(5.39)

the action becomes

$\int
d^{4}x\left[-\frac{1}{2}\phi\Box\phi+\frac{\lambda}{4}\phi^{4}\right]=\int
d\Omega_{4}\left[\frac{1}{2}\Phi(-\vec{L}^{2}+2)\Phi+\frac{\lambda}{4}\Phi^{4}\right],$
(5.40)

where $\vec{L}^{2}$ is the angular momentum operator. The equations of motion
are now easily solved by a mode with $\vec{L}^{2}\Phi=0$, namely
$\Phi_{b}=\sqrt{\frac{2}{-\lambda}}$. Using Eq. (5.39) we find
$\Phi_{b}=\frac{r^{2}+R^{2}}{2R}\phi_{b}$, as expected.

The normal modes from expanding around $\Phi_{b}$ are spherical harmonics in 5
dimensions, which are identical to the conformally mapped fluctuations in Eq.
(5.4). Indeed, it is through this conformal mapping that the exact
eigenfunctions were first derived [7, 40, 41]. In these coordinates there is
an exact $SO(5)$ symmetry among the 5 zero modes. We can therefore rotate any
of the translation modes to the dilatation mode. Mapping back to flat space,
the analysis of the path integral and $N_{\delta}$ proceeds identically to the
discussion in Section 5.2.

##  6 Conclusions

In this paper we have provided a resolution to a long-standing puzzle
involving collective coordinates. Collective coordinates are a way to factor
out an integral over the volume of a symmetry group from the path integral and
still exploit a saddle point approximation in the non-symmetry directions. The
puzzle is how collective coordinates can be consistently introduced through a
change of coordinates, independent of the action. Doing so appears to produce
correct results in interacting theories where the symmetry volume is expected,
but not in free theories, where it is unexpected. The resolution to this
puzzle is that the change of coordinates is allowed and valid, but it is
multi-valued: a given path $x(t)$ in quantum mechanics may be described by
multiple values of the collective coordinates $t_{0}$. We quantified this
multi-valuedness with the intersection number $N_{X}[x]$: it counts the number
of times two paths $X(t)$ and $x(t)$ intersect as one is shifted by the
collective coordinate over the symmetry volume. The correct way to introduce a
collective coordinate is therefore with a resolution of the identity of the
form

$1=\int dt_{0}\frac{1}{N_{X}[x]}\,\delta\Big{[}\langle
X(t+t_{0})|x(t)\rangle\Big{]}\,\Big{|}\langle
X(t+t_{0})|\dot{x}(t)\rangle\Big{|}\,.$ (6.1)

Without the factor of $N_{X}[x]$, the path integral can give the wrong answer.

To call out and rectify the mistreatment of collective coordinates we have
explored examples in quantum mechanics and quantum field theory. For the
splitting between the two lowest energy levels in the symmetric double well
potential in quantum mechanics, the $N_{X}[x]$ factor counts the number of
times a path crosses the potential barrier. Because each crossing gives
additional suppression by $e^{-S_{I}}$ where $S_{I}$ is the instanton action,
we argued that one can approximate $N_{X}[x]=1$ up to exponentially small
corrections. We made special effort to regulate the theory in finite time and
carefully take the limit of a combination of the twisted and untwisted
partition function to extract the energy splitting. While the final result is
well-known through the WKB approximation, most path integral treatments
involve unjustified approximations. We have attempted to provide a complete
path integral treatment of the splitting where each step is under explicit
control.

In the interacting theories we have explored here, namely the double well in
quantum mechanics and interacting scalar quantum field theory, we found that
the intersection number could be approximated as 1. In the free theories, the
intersection number cannot be approximated as 1 and in fact it can be
arbitrarily large, as needed to compensate for the volume factor which arises
in collective coordinates.

It would be interesting to explore cases intermediate between these two
extremes. An example is the instanton-anti-instanton pair in QCD as a function
of separation. This configuration interpolates between the vacuum at zero
separation and a saddle point of the action at infinite separation. The
separation distance is a quasi-collective coordinate [42, 43, 44], becoming a
flat direction at infinite separation and a unexceptional normal mode
fluctuation at zero separation. In the dilute instanton gas approximation, the
dependence of the action on the separation distance is neglected, so that the
separation is treated as a collective coordinate. Thus the techniques
developed here could provide insights into the validity of the dilute
instanton gas approximation. Although QCD is a significant step up in
complexity from the examples considered here, it is of direct phenomenological
relevance and is certainly worth examining in more detail.

### Acknowledgements

We would like to thank G. Dunne, M. Mariño, M. Serone, and J. Stout for
valuable discussions. M.D.S. and A.B. are supported in part by the U.S.
Department of Energy under contract DE-SC0013607. J.C. is supported by a
Junior Fellowship from the Harvard Society of Fellows.

##  Appendix A Metric approach to Jacobians in the path integral

In this Appendix we explain an approach for computing Jacobians arising from
changes of coordinates in path integrals. To this end, we review how to use
Riemannian geometry to manipulate the path integral measure. For concreteness
we consider a quantum mechanical path integral for a particle in $d$ spatial
dimensions, although our arguments here readily generalize to field theory.

The path integral measure and its normalization depend on a metric on the
space of paths. If $K$ is a positive-definite operator, we can begin by
defining an inner product

$\big{\langle}x(t)\big{|}y(t)\big{\rangle}_{K}=\int dt\,x(t)\cdot K\cdot
y(t)\,.$ (A.1)

It is most common to then choose a basis of functions $x_{n}(t)$ that are
orthonormal with respect to this inner product. If we write
$x(t)=\sum_{n}c_{n}x_{n}(t)$ where $\langle x_{m}|x_{n}\rangle=\delta_{mn}$
then the path integral measure is

${\mathcal{D}}x=\prod_{n}dc_{n}\,.$ (A.2)

Factors of $2\pi$ or an overall normalization ${\mathcal{N}}$ can be
optionally absorbed into the measure; these will drop out of physical
quantities calculated by ratios of path integrals.

Now we consider a set of coordinates which are not orthonormal. If we have any
set of coordinates $\\{a_{n}\\}$ which can parametrize a general path $x(t)$
then we can define a distance element between paths as

$ds^{2}=\sum_{m,n}G_{mn}(a)da_{m}da_{n}=\sum_{m,n}\Bigg{\langle}\frac{\partial
x}{\partial a_{m}}\Bigg{|}\frac{\partial x}{\partial
a_{n}}\Bigg{\rangle}\,da_{m}\,da_{n}\,.$ (A.3)

In this case the path integral measure becomes

${\mathcal{D}}x=\sqrt{\det G}\prod_{n}da_{n}\,,$ (A.4)

which is the analog of the measure $\sqrt{\det g}\,d^{d}x$ in Riemannian
geometry.

When we compute correlation functions, we divide by a partition function
normalization computed using the same measure in Eq. (A.4). We observe that
correlation functions are independent of the choice of positive-definite $K$
in Eq. (A.1), since changing $K$ to $K^{\prime}$ affects the square root
determinant multiplicatively by $\sqrt{\frac{\det K^{\prime}}{\det K}}$ which
cancels out the same factor in the partition function normalization.

Let us show how these more abstract ideas manifest in two examples.

Example 1: First, suppose we consider paths $x:[0,T]\to\mathbb{R}$ with
$x(0)=x_{i}$ and $x(T)=x_{f}$. The space of such paths can be parameterized by

$\displaystyle
x(t)=x_{i}+\frac{t}{T}(x_{f}-x_{i})+\sum_{n=1}^{\infty}c_{n}\,\sqrt{\frac{2}{T}}\sin\\!\left(\frac{\pi
n}{T}\,t\right)$ (A.5)

with coordinates $\\{c_{n}\\}$. Taking $K=1$, corresponding to the ordinary
‘flat’ $L^{2}$ inner product, the metric is

$G_{mn}=\Big{\langle}\frac{\partial x}{\partial c_{m}}\Big{|}\frac{\partial
x}{\partial
c_{n}}\Big{\rangle}=\frac{2}{T}\int_{0}^{T}dt\,\sin\\!\left(\frac{\pi
m}{T}\,t\right)\,\sin\\!\left(\frac{\pi n}{T}\,t\right)=\delta_{mn}$ (A.6)

and so

$\displaystyle ds^{2}=\sum_{n}dc_{n}^{2}\,.$ (A.7)

For this metric $\sqrt{\det G}=1$ and the path integral measure is simply

$\displaystyle\mathcal{D}x=\prod_{n}\,dc_{n}\,.$ (A.8)

Example 2: Now consider coordinates for paths around some instanton
$x_{I}(t)$, namely $x(t)=x_{I}(t)+\sum_{n=0}^{\infty}a_{n}\,x_{n}(t)$. If
$\langle x_{m}(t)|x_{n}(t)\rangle=\delta_{mn}$, then, by the same arguments as
in the first example above, the path integral measure can be expressed in
coordinates $\\{a_{n}\\}$ as $\prod_{n=0}^{\infty}da_{n}$. If $x_{0}(t)$ is a
time translation zero mode of the action of our theory, then it is instead
convenient to work with an expansion
$x(t)=x_{I}(t+t_{0})+\sum_{n=1}^{\infty}\alpha_{n}\,x_{n}(t+t_{0})$ in
coordinates $\\{t_{0},\alpha_{n>0}\\}$. In these coordinates, the metric is

$ds^{2}=\langle\dot{x}(t)|\dot{x}(t)\rangle\,dt_{0}^{2}+2\sum_{n=1}^{\infty}\langle\dot{x}(t)|x_{n}(t+t_{0})\rangle\,dt_{0}\,d\alpha_{n}+\sum_{n=1}^{\infty}d\alpha_{n}^{2}$
(A.9)

and so

$\sqrt{\det
G}=\left(\langle\dot{x}(t)|\dot{x}(t)\rangle-\sum_{n=1}^{\infty}\langle\dot{x}(t)|x_{n}(t+t_{0})\rangle\langle
x_{n}(t+t_{0})|\dot{x}(t)\rangle\right)^{1/2}\,.$ (A.10)

Since $\sum_{n=0}^{\infty}\langle a(t)|x_{n}(t+t_{0})\rangle\langle
x_{n}(t+t_{0})|b(t)\rangle=\langle a(t)|b(t)\rangle$ by completeness of
$\\{x_{n\geq 0}(t+t_{0})\\}$ for fixed $t_{0}$, the above simplifies to

$\sqrt{\det G}=\left|\langle
x_{0}(t+t_{0})|\dot{x}(t)\rangle\right|=\left|\langle
x_{0}(t)|\dot{x}_{I}(t)+\sum_{n=1}^{\infty}\alpha_{n}\dot{x}_{n}(t)\rangle\right|\,.$
(A.11)

Thus in the $\\{t_{0},\alpha_{n>0}\\}$ coordinates the path integral measure
is

$\mathcal{D}x=dt_{0}\prod_{m=1}^{\infty}d\alpha_{m}\,\left|\langle
x_{0}(t)|\dot{x}_{I}(t)\rangle+\sum_{n=1}^{\infty}\alpha_{n}\langle
x_{0}(t)|\dot{x}_{n}(t)\rangle\right|$ (A.12)

which agrees with Eq. (2.9). Of course, we also have to be careful about the
change in the domain of path integration coming from our change in
coordinates. The modification of the domain and subtleties with multi-valued
coordinate changes are more readily understood using the methods in the main
text.

##  Appendix B Mapping QFT to the double well

In the quantum mechanics case, we saw that $N_{\delta}$ counted the number of
times a particular path crossed $x=0$. This led to a simple physical argument
for why paths with $N_{\delta}>1$ were exponentially suppressed in the
interacting theory where the potential is high at $x=0$, but unsuppressed in
the free theory where $x=0$ is not special in any sense. In the scalar quantum
field theory we justified why $N_{\delta}=1$ is appropriate for the
interacting theory, but did not provide a physical interpretation of
$N_{\delta}$. We can do so by mapping the theory to quantum mechanics. This
mapping provides an alternative way of thinking about the bounces in QFT.

To map QFT to QM we follow [44, 45] and use a conformal mapping to turn the
dilatation direction into time. Let us take $r=Re^{\omega\tau}$ with $\omega$
an arbitrary frequency, and consider spherically-symmetric field
configurations of the form

$\phi(r)=e^{-\omega\tau}h(\tau)\,.$ (B.1)

Then the action of $\lambda\phi^{4}$ theory evaluated on such field
configurations is

$S[\phi]=\frac{2\pi^{2}R^{2}}{\omega}\int
d\tau\left[\frac{1}{2}(\partial_{\tau}h)^{2}+\frac{1}{2}\omega^{2}h^{2}+R^{2}\omega^{2}\frac{\lambda}{4}h^{4}\right]\,.$
(B.2)

The equations of motion are then $h^{\prime\prime}(\tau)-V^{\prime}(h)=0$
where $V(h)=\frac{1}{2}\omega^{2}h^{2}+R^{2}\omega^{2}\frac{\lambda}{4}h^{4}$.
Recalling that $\lambda<0$, the above is an inverted double-well potential.
The bounce is written as

$h_{b}(\tau)=\sqrt{\frac{2}{-\lambda}}\frac{1}{R\cosh(\omega\tau)}\,.$ (B.3)

The above satisfies the equation of motion of a particle in an inverted
inverted double well, starting at the metastable local maximum at $h=0$. The
particle rolls slowly off of this hump starting at $\tau=-\infty$, bounces off
a wall, and ends up back on top of the hump at $\tau=\infty$. The flat
direction is associated with the time at the center of the bounce. The normal
modes which had satisfied
$\Box\phi_{n}=\lambda_{n}V^{\prime\prime}(\phi_{b})\phi_{n}$ now satisfy

$[\partial_{\tau}^{2}-\omega^{2}]h_{n}=-\lambda_{n}\frac{6\omega^{2}}{\cosh^{2}(\omega\tau)}h_{n}\,.$
(B.4)

However, there is a better basis we can use. Expanding around $h_{b}(\tau)$
the quadratic fluctuations satisfy

$\left[\partial_{\tau}^{2}-\omega^{2}+\frac{6\omega^{2}}{\cosh^{2}(\omega\tau)}\right]\eta_{n}=\hat{\lambda}_{n}\eta_{n}\,,$
(B.5)

which reproduces the Pöschl-Teller potential with $\lambda=2$. There are thus
two bound states and a continuum. The $h_{n}$ and $\eta_{h}$ bases are
different, except for the dilatation zero mode

$h_{d}(\tau)=\eta_{d}(\tau)=\frac{\sqrt{5}}{4\pi
R}\frac{\sinh(\omega\tau)}{\cosh^{2}(\omega\tau)}$ (B.6)

which has the same form in both bases (it has $\hat{\lambda}_{1}=0$ and
$\lambda_{1}=1$ so that $\frac{\lambda_{1}-1}{\lambda_{1}}=0$ as in Eq.
(5.13)).

The zero condition $\langle\phi_{d}(x,R)|\phi(x)\rangle=0$ becomes
$\langle\eta_{1}(\tau-\tau_{0})|h(\tau)\rangle=0$ after the mapping to the
double well, and can be studied using $\eta$ modes quite similarly to the way
we studied the double well in Section 4. We write

$h(\tau)=b_{0}\eta_{0}(\tau)+b_{1}\eta_{1}(\tau)+\sum_{n\geq
2}b_{n}(\tau)\eta_{n}(\tau)$ (B.7)

where $\eta_{n}$ are the continuum modes and $\eta_{0},\eta_{1}$ are the bound
states. As with the double well, the bound states have support for
$\tau\sim\omega^{-1}$ while the continuum modes must be regulated with
$-T<\tau<T$, or equivalently $Re^{-\omega T}<r<Re^{\omega T}$. The projection
that needs to be studied now involves the collective coordinate $\tau_{0}$ and
is

$\langle\eta_{1}(\tau-\tau_{0})|h(\tau)\rangle=0\,.$ (B.8)

In the regime $\tau_{0}\ll\omega^{-1}$ we reduce to the case where we can
trade $\tau_{0}$ for $b_{1}$. In the regime $\omega^{-1}\ll\tau_{0}\ll T$ the
two bound states have support only near $\tau=0$ and in particular

$\eta_{1}(\tau-\tau_{0})\approx\delta^{\prime}(\tau-\tau_{0}),\quad\omega^{-1}\ll\tau_{0}\ll
T\,,$ (B.9)

so that the zero overlap condition becomes $h^{\prime}(\tau_{0})=0$. Thus
$\tau_{0}$ is the time when $h(\tau)$ stops and $N_{\delta}[h]$ counts the
number of times $h$ stops.

Plugging (B.1) into the action of the free theory, the potential is
$V_{0}(h)=\frac{1}{2}\omega^{2}h^{2}$ which is just a stable quadratic
potential. Thus the $h$ actions for the free and interacting theories are
similar, and the action of a given path in the interacting theory is always
strictly lower than the action in the free theory. Nevertheless, since the
saddle point we expand around has large action in the interacting theory,
perturbations around this saddle must be large to slow the particle down so
that it stops multiple times. Such large quantum fluctuations are
exponentially suppressed. In the free theory, the saddle point is just $h=0$
and the particle can stop multiple times without much penalty. Hence in the
full quantum field theory setting, $N_{d}[\phi]=1$ is a good approximation for
the interacting theory but not for the free theory.

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