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\secteqno
\title{ \vspace*{-1.0in}
\bf Field Redefinition Invariance in Quantum Field Theory}
\author{{\sc Karyn M. Apfeldorf }$^\sharp \:\:$%
%
\thanks{apfel@utpapa.ph.utexas.edu}
%
{\sc , Carlos Ord\'o\~nez }$^\flat\:\:$%
%
\thanks{cordonez@utaphy.ph.utexas.edu $\:\:\:\:$ Current address:
University of Texas, Austin}
%
\\
\llap%
\small{\it Theory Group, Department of Physics }$^{\sharp \:\: \flat}$ \\
\small{\it RLM\,5.208 University of Texas at Austin} \\
\small{\it Austin, TEXAS 78712} \\
%
\\
\llap%
\small{\it Department of Physics and Astronomy }$^\flat$ \\
\small{\it Box 1807 Station B, Vanderbilt University } \\
\small{\it Nashville, TN 37235} \\
%
}
\date{}
\begin{document}
\maketitle
\thispagestyle{empty}
\begin{abstract}
\noindent
We investigate the consequences of field redefinition invariance
in quantum field theory by carefully performing nonlinear transformations
in the path integral. We first present a ``paradox'' whereby
a $1+1$ free massless scalar theory on a Minkowskian cylinder
is reduced to an effectively quantum mechanical theory.
We perform field redefinitions both before and after reduction
to suggest that one should not ignore operator ordering
issues in quantum field theory.
We next employ a discretized version of the path integral for a
free massless scalar quantum field in $d$ dimensions to
show that beyond the usual jacobian term,
an infinite series of divergent ``extra'' terms
arises in the action whenever a nonlinear field redefinition is made.
The explicit forms for the first couple of these terms are derived.
We evaluate Feynman diagrams to illustrate the importance of
retaining the extra terms, and conjecture that
these extra terms are the exact counterterms
necessary to render all physical quantities invariant under field
redefinitions. As a byproduct, we
see explicitly how these extra terms are essential to understanding why
the unphysical practice of dimensional regularization works at all.
We indicate how the extra counterterms
cancel out unwanted divergent contributions to physical quantities so that
the result is {\em consistent} with simply setting divergences
such as $\delta^{(d)}(0)$ equal to zero in evaluations of Feynman diagrams.
An exciting possibility is that these extra terms could
allow for the presence of anomalies of higher orders in $\hbar$
in quantum field theories with nonlinear symmetries.
\end{abstract}
\vfill\hfill
\vbox{
\hfill August 1994 \null\par
\hfill UTTG-29-93}
\newpage
\section{Introduction}
Perhaps the most primitive of all properties of a
quantum field theory is field redefinition invariance.
Indeed, experiments yield measurements and these numbers
relative to some scale have meaning, but the numbers in no
way imply a particular choice of fields.
One could go further to argue that in fact although
theoretical calculations using an effective quantum field theory
yield viable results, there is no proof that another framework
(perhaps not even a quantum field theory) could not yield similar
or equivalent results.
Under field redefinitions, physically meaningful quantities
must remain unchanged. For example, one expects the
poles of renormalized
propagators to remain the same under a field redefinition.
On the other hand, quantities related to choice of fields, such as
wave function renormalization factors, are physically insignificant
and may change.
In this paper we investigate the consequences of field redefinition
invariance in the context of quantum field theories, using as our basic
method of attack the consideration of
nonlinear point canonical transformations in the Feynman path integral.
Our analysis requires a careful handling of the path integral under
nonlinear change of variables, and we contend that in spite of the
path integral's ubiquitous employment,
manipulations involving the path integral must be
reevaluated~\footnote{We will avoid
completely mathematically rigorous questions of existence of path
integrals in continuum quantum field theory, and imagine an underlying
spacetime lattice with arbitrarily small spacings.}.
In the case of quantum mechanics, it has been known for some time that
${\cal O}(\hbar^2)$ terms arise in the
path integral action upon performing nonlinear point canonical
transformations of the coordinates. From the lagrangian point of
view this phenomenon is a manifestation of the stochastic nature
of the path integral, and is studied from a discretization of the
quantum mechanical path integral.
It is easy to understand why these so-called ``extra''
terms are generated since upon
quantization, the classical coordinates become quantum operators,
and therefore there are nontrivial issues of operator ordering.
In fact, it may be seen directly from the hamiltonian point of view
that these extra terms are generated by operator ordering~\footnote{
A given prescription for handling the discretized path integral
corresponds to a particular operator ordering of the associated
hamiltonian. In particular, the midpoint prescription corresponds
to Weyl ordering, and in the coordinate representation supports
the interpretation of Brownian motion. For a nice review of the Feynman
path integral in quantum mechanics, see the work of Grosche~\cite{grosche}.}.
In contrast to the case of quantum mechanics,
the standard lore in quantum field theory dictates
that under a field redefinition, the action changes by direct
substitution of the change of variables in the action together with
inclusion of the jacobian determinant of the transformation.
Furthermore, when dimensional regularization is employed, the
jacobian determinant does not contribute since upon exponentiation
the formally infinite spacetime delta function $\delta^{(d)} (0)$ generated by
the trace is set equal to zero. Similarly, there is also a standard
argument that any extra terms
generated in the path integral, since they are manifestations of
operator ordering (i.e. from $ [ \pi(x) ,\phi (x) ] =
-i \hbar \delta^{(d-1)}(0)$), would involve delta functions at zero argument
and therefore would vanish by dimensional regularization.
The usual justification for ignoring these terms is loosely as follows.
One presupposes the existence and necessity of local counterterms in the
action, so that jacobians and any other terms resulting from operator
ordering, which are local quantities in the action, have the effect of
just changing the coefficients of these local counterterms.
The argument presumes that
one is not interested in the initial values of the counterterms,
and therefore there is no need to worry about keeping track of
extra terms that could conceivably arise.
Even if the validity of dimensional regularization is not questioned, a
solid justification for setting infinite quantities equal to zero is lacking.
Certainly from the lattice point of view, the jacobian term, as well as any
extra term, is very real and does not vanish since the spacetime delta
function at zero argument
is just a power of the inverse lattice spacing i.e.,
$\delta^{(d)}(0) \sim a^{-d}.$
Dimensional regularization does in fact fail in cases where the action
has some feature depending on the dimensionality of spacetime.
Examples are theories
with chiral fermions, conformal symmetries, and theories with
nontrivial spacetime topology.
Our investigation reveals that upon making nonlinear field
redefinitions in the quantum field theory path integral extra terms
do appear. In the generic case, an infinite number of divergent ``extra''
terms appear. We give a detailed account of the method of derivation of
these terms and give the explicit form for the first couple of terms.
We conjecture that this series of ``extra'' terms gives the precise
counterterms necessary to ensure that physically significant
quantities remain unchanged under field redefinitions.
Furthermore, inclusion of these terms allows one to understand what
is really going on in the procedure of setting
$\delta^{(d)}(0) $ and other divergent quantities to zero in
dimensional regularization.
Especially when dimensional regularization is not applicable, one must
take care to include any divergent extra terms that may arise when
manipulating fields in the path integral. An important application
of this work could possibly be the existence of quantum anomalies of
higher orders in $\hbar$ in theories with nonlinear symmetries.
This paper is organized as follows.
In section 2, we present a simple ``paradox'' where quantum field
theory meets quantum mechanics, which suggests the
existence of extra terms in the quantum field theory action
upon change of field variables. We consider
the dimensional reduction of a free, massless $1+1$ dimensional real scalar
quantum field theory to an effectively quantum mechanical theory,
and perform field redefinitions both before and after reduction.
In section 3, we investigate the problem of ``extra'' terms in
quantum field theory more directly by
making general field redefinitions in a
discretized version of the path integral in a massless noninteracting
scalar quantum field theory in flat $d$-dimensional spacetime.
We show that an infinite series of extra terms arise upon
performing a nonlinear field transformation in the path
integral, and give explicit expressions for the first couple of
extra terms. In section 4, we turn to Feynman diagrams
to calculate corrections to the propagator
to the lowest couple of orders in the field redefinition
parameter. We discuss the important role of the
extra terms derived in the previous section.
We conjecture that the extra terms ensure that the
pole of the renormalized scalar
propagator remains unchanged to all orders, while only the wave function
gets renormalized. These calculations enable one to see explicitly
that the extra terms are essential to understanding why
the unphysical practice of dimensional regularization works at all.
Conclusions and further research questions are given in the final
section.
\section{Kaluza-Klein ``Paradox''}
An effective laboratory in which to learn about nonlinear point
canonical transformations in quantum field theory is that of a
1+1 dimensional real scalar field on a Minkowskian cylinder.
By integrating out the angular coordinate one arrives at an
effectively quantum mechanical problem, and then the
well-established results for nonlinear point canonical transformations
in quantum mechanics may be employed.
On the other hand, one may use the standard lore to make a
nonlinear field redefinition directly in the 1+1 dimensional
quantum field theory, and then afterwards integrate out the
angular coordinate to arrive again to a quantum mechanics action.
One expects that the two approaches should
give the same result, so if it is true that no new terms appear
in the latter scenario when one performs a field redefinition in the
quantum field theory, one must be able to explain
what happens to the extra terms that arise along the way in
the former scenario.
In the following we consider a real massless scalar field $\phi(x,t)$
in a flat $1+1$ Minkowskian spacetime with a periodic spatial coordinate
\beq
S = -\half \int d^2 x \:\: \eta^{\mu \nu} \partial_\mu \phi \partial_\nu \phi
\eeq
where
$\eta_{tt} = 1,$ $\eta_{x x} =-1$ and $x \cong x + 2 \pi R . $
\subsection{Method 1}
Using an angular coordinate $\theta = x/R,$ the action is
\beq
S = R \int d\theta dt \left[ \half \dot{\phi}^2
- \frac{1}{2 R^2} (\partial_\theta \phi)^2 \right] .
\eeq
One may convert this quantum field theory into an effectively
quantum mechanical problem by expanding the scalar field in a
Kaluza-Klein-like decomposition
\beq
\phi(\theta,t) = \sum_{p \in Z} a^{(p)}(t) e^{i p \theta}
\eeq
and integrating out the $\theta$ dependence in the action.
The resulting action is that of a sum of an infinite number of
quantum mechanical complex oscillators $a^{(m)}(t)$ with frequencies
$\omega_m = \frac{m}{R}$
dependent on mode number $m$ and radius $R$ of the internal dimension.
Explicitly,
\beq
S_0[a] = \sum_{p \in Z} \int dt \left[ \sum_{m \in Z}
\half g_{mp}[a] \dot{a}^{(m)} \dot{a}^{(p)}
- \half \frac{p^2}{R^2} g_{mp}[a] a^{(m)} a^{(p)} \right]
\label{eq:free}
\eeq
where the
metric
$$g_{mp}[a] = 2 \pi R \delta_{m,-p} = 2 \pi R \delta_{m+p=0}$$
is an infinite-dimensional unit matrix multiplied by $2 \pi R$.
We may consider a ``cutoff'' version of this theory, wherein
only modes with $|n| \leq N_{\rm max}$ are included. In this case,
$g_{mp}[a]$ is an ordinary $2N_{\rm max} +1$ by $2N_{\rm max} +1$
metric defining a Euclidean space of oscillators $a(t).$
Let us perform a change of variables in this quantum mechanical model.
In particular, we wish to make the change of variables corresponding to
\beq
\phi = F[\varphi] \equiv \varphi + \alpha \varphi^N .
\label{eq:Ntran}
\eeq
We expand the new field $\varphi (\theta , t)$ in modes
\beq
\varphi(\theta,t) = \sum_{p \in Z} b^{(p)}(t) e^{i p \theta} ,
\eeq
and define the following useful objects
\beqar
E^{(\ell)}_L [b] &\equiv& \sum_{p_1 \in Z} \ldots \sum_{p_L \in Z}
b^{(p_1)} \ldots b^{(p_L)} \delta^{\ell - \sum_{i=1}^L p_i =0} \\
B_L [ b ] &\equiv& E^{(0)}_L [ b ] = \int \frac{d\theta}{2 \pi}
\varphi(\theta, t)^L ,
\label{eq:defB}
\eeqar
where $\delta^{\ell=0}$ is one when $\ell=0$ and is zero if $\ell \neq 0.$
Note $E^{(\ell)}_L [b] $ involves the product of $L$ $b$'s whose mode
numbers add up to $\ell,$ and $B_L [b] $ involves $L$ $b$'s whose
overall sum of mode numbers vanishes.
A number of identities, the simplest being
\beqar
\sum_{p \in Z} E^{(\ell + p)}_M [b] E^{(-p+m)}_L [b] &=&
E^{(\ell+m)}_{M+L} [b]
\label{eq:ident1} \\
\frac{\partial \:\:}{\partial b^{(m)} } E^{(\ell)}_M [b] &=&
M E^{(\ell-m)}_{M-1} [b]
\eeqar
greatly facilitate subsequent calculations. Appendix A contains
useful identities and their proofs.
The transformation of the modes corresponding to the field redefinition
above in equation~\ref{eq:Ntran} is
\beq
a^{(p)} \equiv f^{(p)} [b] = b^{(p)} + \alpha E^{(p)}_N [b] .
\label{eq:atob}
\eeq
To implement this change of coordinates in the quantum mechanical
path integral, we must include not only the naive substitution
of change of variables in the action and the jacobian determinant, but also
the extra term due to the stochastic nature of the path integral.
Specifically, after
exponentiating the usual jacobian factor, the action becomes
\beq
S[b]=S_0[b] + S_{\rm jacobian}[b] + S_{\rm extra} [b]
\eeq
where $S_0 [b ] $ is the action obtained by direct substitution into
the original free action given in equation~\ref{eq:free},
and where the extra term of ${\cal O}(\hbar^2)$ is
\beq
S_{\rm extra} [b] = - \hbar^2 \int dt \sum_{k,\ell, p, n \in Z}
\frac{1}{8} g^{\ell n} [b] \Gamma^k_{\ell p}[b] \Gamma^p_{nk}[b] .
\eeq
A modified version of the
classic derivation of this extra term, which is written in terms of the
metric and connection in the new coordinates, is reviewed in section 3.1.
It is possible to compute all three contributions to the
action $S[b]$ exactly to all orders in the parameter $\alpha$.
The metric in the new coordinates is given by
%\beqar
%g_{mn}[b] &=& \sum_{\ell ,p \in Z}
%\frac{\partial f^{(\ell) }}{\partial b^{(m)}}
%\frac{\partial f^{(p)}}{\partial b^{(n)} } g^{(a)}_{\ell p} \nonumber \\
%&=& 2 \pi R
%\left( \delta_{m+n=0} + 2 \alpha N E^{(-m-n)}_{N-1}[b] + N^2 \alpha^2
%E^{(-m-n)}_{2N-2}[b] \right) ,
%\eeqar
\beq
g_{mn}[b] = 2 \pi R \left(
\delta_{m+n=0} + 2 \alpha N E^{(-m-n)}_{N-1}[b] + N^2 \alpha^2
E^{(-m-n)}_{2N-2}[b] \right) ,
\eeq
and the inverse metric, which may be obtained via a recursion equation
(see appendix B), is given by
\beq
g^{km}[b] = \frac{1}{2 \pi R} \sum_{j=0}^\infty (-\alpha N)^j (j+1)
E^{(k+m)}_{j(N-1)}[b].
\eeq
%The connection coefficients are computed in the usual manner
%\beq
%\Gamma^k_{\ell p}[b] = - \sum_{j=0}^\infty (-\alpha N)^{j+1} (N-1)
%E^{(k-p-\ell)}_{j(N-1)+N-2}[b].
%\eeq
In terms of this metric, direct substitution of the nonlinear change
of variables into the original action yields
\beq
S_0[b] = \sum_{p \in Z} \int dt \left[
\half \sum_{m \in Z} g_{mp}[b] \dot{b}^{(m)} \dot{b}^{(p)}
- \half \frac{p^2}{R^2} 2\pi R f^{(p)}[b] f^{(-p)}[b] \right] ,
\label{eq:m10}
\eeq
where $f^{(p)}[b]$ was given in equation~\ref{eq:atob}.
The jacobian factor may be evaluated (see appendix B) to give a
contribution to the action of
\beqar
S_{\rm jacobian}[b]
& =& -\frac{ i \hbar}{2} {\rm Tr ln} g_{mn}[b] \delta (t -t^\prime )
\nonumber \\
& =& i \hbar \delta(0)
\left[ \sum_{k \in Z} 1 \right]
\sum_{j=1}^{\infty} \frac{(-\alpha N)^j}{j}
\int dt B_{j(N-1)} ,
\label{eq:m1j}
\eeqar
where we have dropped an infinite constant
$- i\hbar \delta(0) {\rm ln }[2 \pi R] [\sum_{k \in Z} 1 ] /2 .$
Finally, the extra term arising from the stochastic
nature of the path integral may be evaluated (see appendix B) to give
a contribution to the action of
\beqar
S_{\rm extra}[b] &=& -\frac{\hbar^2}{8}
\frac{(N-1)^2}{2 \pi R} \left[ \sum_{k \in Z} 1 \right]^2
\sum_{X=0}^\infty (-\alpha N)^{X+2}
\left( \begin{array}{c} X+3 \\ 3 \end{array} \right) \nonumber \\
& & \times \int dt \: \: B_{X(N-1)+2(N-2)}[b] .
\label{eq:m1e}
\eeqar
Note the presence of the infinite sum over
modes in equation~\ref{eq:m1j} and the same sum squared in
equation~\ref{eq:m1e}.
\subsection{Method 2}
Alternately, we may choose to perform the field redefinition in the original
quantum field theory, and then afterwards expand in the Kaluza Klein modes.
Upon direct substitution, the field redefinition of equation~\ref{eq:Ntran}
introduces derivative interaction terms into the action
\beq
S_0[\varphi] = -\half \int d^2 x \; \eta^{\mu \nu}
(1+ 2 N \alpha \varphi^{N-1} + N^2 \alpha^2 \varphi^{2(N-1)} )
\partial_\mu \varphi (x) \partial_\nu \varphi (x) .
\eeq
The exponentiated jacobian determinant gives a contribution
to the action
\beq
S_{\rm jacobian}[\varphi] = - i \hbar \delta^{(2)}(0) \int d^2 x \:
{\rm ln}[1 + N \alpha \varphi^{N-1}] .
\eeq
Assuming the conventional arguments hold would imply that
the total action is given by $S_0[\varphi] + S_{\rm jacobian}[\varphi].$
We now convert this action to an effectively quantum mechanical one by
integrating out the angular coordinate.
The terms in the action for the $b$ modes obtained from method
2 will be written with tildes to distinguish them from those of
method 1.
Upon integrating out the $\theta$ dependence, $S_0[\varphi]$ becomes
\beq
\tilde S_0[b] = \int dt \sum_{m \in Z} \sum_{p \in Z}
\left[ \half g_{mp}[b] \dot{b}^{(m)} \dot{b}^{(p)}
+ \frac{m p}{2 R^2} g_{mp}[b] b^{(m)} b^{(p)} \right]
\label{eq:m20}
\eeq
where $g^{(b)}_{mp}$ is exactly the metric appearing in the
previous section.
Similarly, expanding the logarithm and using equation~\ref{eq:defB},
one obtains the expression for the jacobian
\beq
\tilde S_{\rm jacobian}[b] = 2 \pi R i \hbar \delta^{(2)}(0)
\sum_{j=1}^\infty \frac{(-\alpha N )^j}{j} \int dt B_{j(N-1)} .
\label{eq:m2j}
\eeq
\subsection{Comparison of Methods}
If the standard lore holds true, one would expect to find that
the two actions for the effective quantum mechanics theory
are equivalent, i.e.
\beq
S_0[b] + S_{\rm jacobian}[b] + S_{\rm extra} [b] = \tilde S_0[b] +
\tilde S_{\rm jacobian}[b] .
\eeq
Firstly, we note that using identity~\ref{eq:p2ident} from
appendix A allows one to prove the equality of the kinetic terms, i.e.
$S_0[b] = \tilde S_0[b]$ (see equations~\ref{eq:m10} and~\ref{eq:m20} ) .
Secondly, comparison of the jacobian factors
(equations~\ref{eq:m1j} and~\ref{eq:m2j}) indicates that the following
correspondence must hold
\beq
\sum_{k \in Z} 1 = 2 \pi R \: \: \delta (0)
\label{eq:sumtodelta}
\eeq
where the delta function at zero argument on the right hand side of this
equation is a delta fuction in the coordinate $x$, i.e.
$\delta(0) = \5{x \rightarrow 0}{\rm Lim } \delta(x).$ If we express
the delta function in terms of the angular coordinate $\theta = x/R$,
the $R$ dependence in the above equation drops out.
With this identification, we find $ S_{\rm jacobian}[b]
= \tilde S_{\rm jacobian}[b]$ up to the infinite constant that we dropped.
Finally, we turn our attention to the most interesting term, namely
$S_{\rm extra}[\varphi].$
In the second method where we performed the nonlinear field
redefinition in the quantum field theory using the standard arguments,
we did not pick up any such term.
Taking the pragmatic point of view that this term can not
be rationalized to vanish, we ask the following question
``could this term arise as an extra term generated upon nonlinear
field redefinition in the original quantum field theory?''
Remarkably, it is possible (see appendix B) to perform the resummation
of the infinite series in equation~\ref{eq:m1e}. The resulting term is
\beq
S_{\rm extra}[\varphi] = -\frac{\hbar^2}{8}
\frac{\left[ \sum_{k \in Z} 1 \right]^2}{(2 \pi R)^2} \alpha^2 N^2 (N-1)^2
\int d^2 x \frac{\varphi^{2N-4}}{(1+\alpha N\varphi^{N-1})^4} .
\label{eq:VE}
\eeq
We still must contend with the peculiar infinite summations to
relate this potential to the two dimensional field theory.
Using the result of the comparison of the jacobian terms, i.e.
equation~\ref{eq:sumtodelta}, we conclude that the extra term
becomes
\beq
S_{\rm extra}[\varphi] = -\frac{\hbar^2}{8} (\delta(0))^2
\alpha^2 N^2 (N-1)^2
\int d^2 x \frac{\varphi^{2N-4}}{(1+\alpha N\varphi^{N-1})^4} .
\label{eq:kkextra}
\eeq
Interestingly, note that this term diverges as the
square of the $\delta (0) $ in the spatial coordinate,
and {\it not} like the two-dimensional delta function at zero argument.
This feature (which persists in $d$-dimensions quantum field theory where
an extra term containing $(\delta^{(d-1)} (0))^2$ appears)
occurs for an important reason which becomes evident when one
evaluates Feynman diagrams.
We could have elected to present the material in the intermediate step
of Method 2 of this exercise in a different, more complete manner.
That is, after the field redefinition, we should add to the action an
infinite series of counterterms with unspecified coefficients
$c_\ell,$ i.e.
\beqarn
S[\varphi] &=& -\half \int d^2 x \; \eta^{\mu \nu}
(1+ 2 N \alpha \varphi^{N-1} + N^2 \alpha^2 \varphi^{2(N-1)} )
\partial_\mu \varphi (x) \partial_\nu \varphi (x) \\
& & - i \hbar \delta^{(2)}(0) \int d^2 x \:
{\rm ln}[1 + N \alpha \varphi^{N-1}]
+ \int d^2x \: \sum_{\ell=1}^\infty c_\ell \varphi^\ell (\varphi) .
\eeqarn
With this action, one would have to determine the infinite series of
unknown coefficients, the $c_\ell$'s, by calculating physically significant
quantities and matching with the original free theory.
On the other hand, our exercise has enabled us to pin down the precise
counterterm without having to resort to evaluating Feynman diagrams, i.e.
the series in $c_\ell$ is replaced by the simple expression in
equation~\ref{eq:kkextra}.
To conclude this section, we see evidence for a single extra term
being generated in the $1+1$ dimensional quantum field theory
on a Minkowskian cylinder upon making a nonlinear field redefinition.
Note that the field theory involved in this example is one with nontrivial
spacetime topology. In the following section, we will back up and
address instead the issue of extra terms in $d$ dimensional quantum field
theories with {\it trivial} spacetime topology more
directly by considering a discretized version of the path integral.
\section{Discretization of the Path Integral}
The derivation of the extra term in the
path integral of quantum mechanics involves discretizing
the time coordinate, so one is inclined to consider a
discretization of the spacetime coordinates in the quantum field theory case.
\subsection{Quantum Mechanics Preliminary}
As a warmup to the quantum field theory case, let us review
the salient features of the quantum mechanics case.
For the most part,
we will follow the derivation of the well-known extra term in quantum
mechanics by Gervais and Jevicki~\cite{gervais}.
We will, however, treat the jacobian term in a different, but
completely equivalent way.
Consider the simple case of an $n$-dimensional quantum mechanical particle
\beq
Z = \int \prod_{a=1}^n D q^a \: \: e^{\frac{i}{\hbar} \int dt \left[ \half
\sum_{a=1}^n ({\dot q}^a)^2 - V_0(q) \right]} .
\eeq
In particular, we will assume that there are no
time derivatives in the potential.
Discretizing the time interval $t_f - t_0$
into $N$ segments of length $\epsilon $ such that
$t_k = t_0 + \epsilon k $ and $t_f \equiv t_N,$
and using the abbreviated notation
$q^a(k) = q^a(t_k) $, one arrives at
\beq
Z = \int \prod_{a=1}^n \prod_{k=1}^{N-1} D q^a (k) \prod_{k=1}^{N-1}
e^{\frac{i}{\hbar} \left[ \frac{1}{2 \epsilon}
\sum_{a=1}^n ( q^a(k+1) - q^a(k) )^2 - \epsilon V_0( q(k)) \right] }.
\eeq
The limit $N \rightarrow \infty$ and $\epsilon \rightarrow 0$
while $N \epsilon = t_f - t_0$ is implied.
A key observation to make is that the one dimensional time
differential $dt \propto \epsilon$ carries a single power of
$\epsilon , $ so any terms of ${\cal O}(\epsilon^2)$
in the exponential will vanish in the small $\epsilon$ limit.
After a field transformation,
the only ``extra'' terms that need be retained in the path
integral action are either ${\cal O}(\epsilon) $ or possibly divergent.
Under a coordinate redefinition $q^a(t) = F^a(Q(t)) $
from the $q^a$ to the $Q^i$ ($i= 1,\ldots n$),
the path integral becomes
%\beqar
%Z & &= \left[ \det g_{ij}(Q_f) \det g_{ij}(Q_0) \right]^{- \frac{1}{4}}
% \int \prod_{i=1}^n \prod_{k=1}^{N-1} D Q^i (k) \nonumber \\
%& \times & \prod_{k=0}^{N-1} J(Q(k+1),Q(k))
%e^{\frac{i}{\hbar} \left[ \frac{1}{2 \epsilon}
%( F^a(Q(k+1)) - F^a(Q(k)) )^2 - \epsilon V_0( F(Q(k))) \right]}
%\eeqar
%where the jacobian factors are
%$$
%J(Q(k+1),Q(k))=\left[ \det g_{ij}(Q(k+1))
% \det g_{ij}(Q(k)) \right]^{\frac{1}{4}}
%$$
\beqar
Z &= & \int \prod_{i=1}^n \prod_{k=1}^{N-1} D Q^i (k)
\prod_{k=1}^{N-1} \left[ \det{g_{ij}(Q(k))} \right]^\half \\
& \times & \prod_{k=1}^{N-1} e^{\frac{i}{\hbar} \left[ \frac{1}{2 \epsilon}
( F^a(Q(k+1)) - F^a(Q(k)) )^2 - \epsilon V_0( F(Q(k))) \right]}
\eeqar
where the metric in the new coordinates is
$$
g_{ij}(Q) = \sum_{a=1}^n \frac{\partial F^a(Q)}{\partial Q^i}
\frac{\partial F^a(Q)}{\partial Q^j}.
$$
Choosing the midpoint prescription
\beqar
\bar{Q}^i(k) &\equiv& \half \left( Q^i(k+1) + Q^i(k) \right) \nonumber \\
\Delta Q^i(k) &\equiv& Q^i(k+1) - Q^i(k)
\eeqar
we Taylor expand the jacobian determinant
and the kinetic term about the midpoint,
and integrate out terms in $\Delta Q$ to put the action back in its
original ``canonical'' form.
By rescaling $Q^i$ by $\epsilon^{\half}$, it is
clear that the normalized integral of $\Delta Q^i \Delta Q^j$
is proportional to $\epsilon$, therefore one need only keep
terms in the jacobian to quadratic order in $\Delta Q$ and
terms in the kinetic term to quartic order in $\Delta Q$.
Note that since integrating out the fluctuations about
the midpoints only
brings in positive powers of $\epsilon$, the extra terms in the action
will always be less singular than the $\epsilon^{-1}$ kinetic term.
Explicitly, writing the path integral normalization as
%\beqarn
%\langle 1 \rangle &=& \left[ \det g_{ij}(Q_f) \det g_{ij}(Q_0)
%\right]^{- \frac{1}{4}} \prod_{k=0}^{N-1} (\det g_{ij}(\bar Q(k)))^\half \\
%& & \times
%\int \prod_{\ell=1}^n \prod_{k=1}^{N-1} {\cal D} \Delta Q^\ell (k) \:
%e^{\frac{i}{\hbar} \frac{1}{2 \epsilon} g_{ij}(\bar Q (k) )
%\Delta Q^i (k) \Delta Q^j (k) }
% \eeqarn
$$
\langle 1 \rangle = \prod_{k=1}^{N-1} \left[
\det g_{ij}(\bar Q(k)) \right]^\half \times
\int \prod_{\ell=1}^n \prod_{k=1}^{N-1} {\cal D} \Delta Q^\ell (k) \:
e^{\frac{i}{\hbar} \frac{1}{2 \epsilon} g_{ij}(\bar Q (k) )
\Delta Q^i (k) \Delta Q^j (k) }
$$
we may use the relations
\beqar
\langle \Delta Q^\ell \Delta Q^m \rangle &=& \langle 1 \rangle i \hbar
\epsilon g^{\ell m} \\
\langle \Delta Q^i \Delta Q^j \Delta Q^k \Delta Q^\ell \rangle
& = & \langle 1 \rangle (i \hbar \epsilon)^2 ( g^{ij} g^{k \ell}
+ g^{i k} g^{j \ell} + g^{i \ell} g^{jk} ) \label{eq:qmint} \\
\langle \Delta Q^{i_1} \Delta Q^{i_2} \ldots \Delta Q^{i_{2n+1}} \rangle
& = & 0
\eeqar
to rewrite the following extra terms which appear in the lagrangian
\beqarn
\frac{i}{\hbar} {\cal L}_{\rm extra}(\bar Q) & = &
\frac{1}{16} \left[ g(\bar Q)^{ij}_{,\ell} g(\bar Q)_{ij,m}
+ g(\bar Q)^{ij} g(\bar Q)_{ij,\ell m} \right] \Delta Q^\ell \Delta Q^m \\
& & + \frac{i}{24 \hbar \epsilon} F^a(\bar Q)_{,i} F^a(\bar Q)_{,jk\ell}
\Delta Q^i \Delta Q^j \Delta Q^k \Delta Q^\ell .
\eeqarn
The result is the extra potential introduced in the previous section
$$
S_{\rm extra}(\bar Q) = - \hbar^2 \int dt \sum_{k,\ell, p, n \in Z}
\frac{1}{8}
g^{\ell n}(\bar Q) \Gamma^k_{\ell p} (\bar Q) \Gamma^p_{nk} (\bar Q) .
$$
\subsection{Quantum Field Theory}
How does this change when we consider quantum field theory?
Let us focus on the simple case of a massless, noninteracting
scalar field theory existing in a spacetime of trivial topology.
In order to maintain Poincare invariance
in quantum field theory, one expects to discretize all spacetime
coordinates with the same parameter $\epsilon.$
Consider a $d$-dimensional Euclidean spacetime and
discretize~\footnote{If spacetime is finite
volume, then as in the quantum mechanical case we have
$N \rightarrow \infty$ and $\epsilon \rightarrow 0$
as the product $N \epsilon^d$ goes to a finite value.
If, on the other hand, spacetime is infinite,
$N$ must diverge more rapidly than $\epsilon^d.$
One may imagine even that there exists
a lower length scale so that $\epsilon$
can not be taken all the way to zero.}
the spacetime into $N$ hypercubic volumes of size
$\epsilon^d.$ The points of the latticized spacetime are labeled
by a $d$-dimensional vector $k$. The action for a noninteracting
massless scalar field $\phi(x)$ in a flat $d$-dimensional spacetime is
\beq
Z = \int \prod_{k} {\cal D} \phi_k
\: \prod_k e^{- \frac{1}{\hbar} \epsilon^{d-2}
\frac{1}{2} \sum_{i=1}^{2d} \half (\phi_{k+n_i} - \phi_k )^2 }
\eeq
where the $n_i$ are the unit vectors connecting the vector $k$ to
its $2d$ nearest neighbors. In $d=2$, for example, one may write
$n_1 = \hat x$, $n_2 = \hat y$, $n_3 = - \hat x$ and $n_4 = -\hat y.$
The factor of $1/2$ has been inserted to
counter the fact the kinetic term sums over forward and backward
directions.
Clearly, an equivalent way of writing this action is to drop the
factor of $\half$ and allow the
sum to run over only the $d$ independent directions.
If spacetime has a boundary where the field values are fixed,
the product over lattice sites $k$ excludes these points.
Let us now implement a change of variables $\phi = F[\varphi ]$
on the lattice. The integrand above simply becomes
\beq
\prod_{k} \: e^{- \frac{1}{ \hbar} \epsilon^{d-2}
\sum_{i=1}^{d} \half (F(\varphi_{k+n_i})-F(\varphi_k))^2 } .
\eeq
The issue of the measure is slightly more complicated, and
before addressing it, it will be advantageous
to look ahead to the prescription
that we will use to calculate any extra terms.
We will choose an analog to the midpoint prescription.
Since the kinetic term is the sum of differences in $d$ directions,
the analog is rather a ``midpoints'' prescription.
We define $d$ midpoint variables and $d$ difference variables
\beqar
\bar{\varphi}_{k,n_i} &\equiv& \half \left( \varphi_{k+n_i} +
\varphi_k \right) \\
\Delta \varphi_{k,n_i} &\equiv& \varphi_{k+n_i} - \varphi_k .
\eeqar
Inverting these equations, one may write the field values at the
neighboring lattice sites to $k$ uniquely as
\beq
\varphi_{k+n_i} = \bar \varphi_{k,n_i} + \half \Delta \varphi_{k,n_i} .
\eeq
On the other hand, one may write $\varphi_k$
asymmetrically by using a single $n_i$
\beq
\varphi_{k} = \bar \varphi_{k,n_i} - \half \Delta \varphi_{k,n_i}
\quad \quad {\rm for } \: {\rm any } \:\: n_i ,
\label{eq:asym}
\eeq
or symmetrically by averaging over all directions
\beq
\varphi_k = \frac{1}{d} \sum_{i=1}^{d} \left( \bar \varphi_{k,n_i} -
\half \Delta \varphi_{k,n_i} \right) ,
\eeq
or in a multitude of other ways since we are using redundant variables.
The issue of how to handle a function of $\varphi_k$ in the action,
i.e. whether one writes
$$
{\cal G}(\varphi_k) = \frac{1}{d} \sum_{i=1}^{d}
{\cal G}( \bar \varphi_{k,n_i} - \half \Delta \varphi_{k,n_i})
$$
versus, for example,
$$
{\cal G}(\varphi_k) = {\cal G}( \frac{1}{d} \sum_{i=1}^{d}
(\bar \varphi_{k,n_i} - \half \Delta \varphi_{k,n_i}))
$$
could be resolved by choosing the most technically convenient expression,
as all forms are formally equivalent. The constraints among the new
difference variables will be
reflected in the measure, which will ensure the equivalence of these
expressions. In writing the measure,
the essential question is how to go from the $d \phi_k$ differential
(or equivalently the $d \Delta \phi_k$ differential)
to the $(d)$ $d \Delta \varphi_{k,n_i}$ differentials.
Let us label the $d$ different yet equivalent expressions for
$\varphi_k$ by $\varphi_k^{A_i}$, i.e.
\beq
\varphi_{k} = \bar \varphi_{k,n_i} - \half \Delta \varphi_{k,n_i}
\equiv \varphi_k^{A_i} .
\eeq
After making the change of variables from $\phi_k$ to $\varphi_k$ in the
measure, we insert a factor of one using
delta functions constraints in order to
introduce the $d$ equivalent representations of $\varphi_k$
\beqar
d \phi_k &=& d \varphi_k \: \frac{\partial F}{\partial \varphi} (\varphi_k)\\
&=& d \varphi_k \: \frac{\partial F}{\partial \varphi} (\varphi_k)
\int \prod_{i=1}^d d \varphi_k^{A_i} \delta(\varphi_k - \varphi_k^{A_i}) .
\nonumber
\eeqar
Notice this form is completely symmetric and there is no preferred direction.
In practice these integrals will be done by integrating first with respect to
$\varphi_k$ and employing any one of the $d$ delta functions.
For example one could use the first delta function as follows
\beq
\int d \phi_k {\cal H}\left[\phi_k\right]
= \int \prod_{i=1}^d d \varphi_k^{A_i}
\times \prod_{j=2}^d \delta(\varphi_k^{A_1} - \varphi_k^{A_j})
\times \frac{\partial F}{\partial \varphi} (\varphi_k^{A_1})
{\cal H}\left[F(\varphi_k^{A_1})\right]
\eeq
\noindent
Let us return to the integral that we must perform
\beqar
Z &=& \int \prod_{k} d \phi_k \: \prod_k e^{- \frac{1}{\hbar} \epsilon^{d-2}
\frac{1}{2} \sum_{i=1}^{2d} \half (\phi_{k+n_i} - \phi_k )^2 }
\nonumber \\
&=& \int d \varphi_k \prod_{i=1}^d d \varphi_k^{A_i}
\times \prod_{j=1}^d \delta(\varphi_k - \varphi_k^{A_j}) \nonumber \\
& & \times \prod_{k} \: e^{- \frac{1}{ \hbar} \left[ \epsilon^{d-2}
\sum_{i=1}^{d} \half (F(\varphi_{k+n_i})-F(\varphi_k))^2
- \hbar {\rm ln} \frac{\partial F}{\partial \varphi} (\varphi_k) \right] } .
\eeqar
We will evaluate this integral by expanding in the $d$ fluctuation variables
about the $d$ midpoint variables for each $k$
and using perturbation theory about the
canonical Gaussian action.
Specifically, we must expand the jacobian and the kinetic terms
and integrate out terms in
$\Delta \varphi_{k,n_i}$ to put the action back in its original
``canonical'' form.
The first few terms in the expansion of the kinetic term,
using a convenient choice for $\varphi_k$ in each term, are
\beqarn
& & \sum_{i=1}^{d}
\half \left( F[\varphi_{k+n_i}] - F[\varphi_{k}] \right)^2 \\
&=& \half \sum_{i=1}^{d} \left[ \sum_{p=0}^\infty \frac{2}{(2p+1)!}
\left( \frac{\Delta \varphi_{k,n_i}}{2} \right)^{2p+1}
\frac{\partial^{2p+1} F}{\partial \varphi^{2p+1}}
|_{\bar \varphi_{k,n_i}} \right]^2 \\
&=& \sum_{i=1}^{d} \left[ \half g(\bar \varphi_{k,n_i}) (\Delta
\varphi_{k,n_i})^2 + \frac{1}{24} F^\prime (\bar \varphi_{k,n_i})
F^{\prime \prime \prime} (\bar \varphi_{k,n_i}) (\Delta
\varphi_{k,n_i})^4 \right. \\
& & \left.
+ \frac{1}{1152} \left( (F^{\prime \prime \prime} (\bar \varphi_{k,n_i}) )^2
+\frac{3}{5} F^\prime (\bar \varphi_{k,n_i}) F^{(v)} (\bar
\varphi_{k,n_i}) \right)
(\Delta \varphi_{k,n_i})^6
+ {\cal O} ((\Delta \varphi_{k,n_i})^8 ) \right] ,
\eeqarn
where the first term is the usual canonical kinetic term in the new
coordinates. All terms but the first will be integrated out.
The first terms in the expansion of the logarithm of the jacobian are
\beqarn
& & {\rm ln } F'( \varphi_k )
= {\rm ln } F'(\bar \varphi_{k,n_i} )
- \half \frac{ F''(\bar \varphi_{k,n_i} ) }{F'(\bar \varphi_{k,n_i} )}
\Delta \varphi_{k,n_i} \\
& &
+ \left( - \frac{1}{8} \frac{F''(\bar \varphi_{k,n_i} )^2
}{(F'(\bar \varphi_{k,n_i} ))^2 } + \frac{1}{8} \frac{F'''(\bar \varphi_{k,n_i}
)}{ F'(\bar \varphi_{k,n_i} ) } \right) (\Delta \varphi_{k,n_i})^2 \\
& &
+ \left( -\frac{1}{24} \frac{ (F''(\bar \varphi_{k,n_i} ))^3}{
(F'(\bar \varphi_{k,n_i} ) )^3 } + \frac{1}{16} \frac{ F''(\bar
\varphi_{k,n_i} ) F'''(\bar \varphi_{k,n_i} ) }{ (F'(\bar
\varphi_{k,n_i} ) )^2} - \frac{1}{48} \frac{F^{(iv)}(\bar
\varphi_{k,n_i} ) }{ F'(\bar \varphi_{k,n_i} ) } \right)
(\Delta \varphi_{k,n_i})^3 \\
& &
+ \left( - \frac{1}{64} \frac{ (F''(\bar \varphi_{k,n_i} ))^4}{
(F'(\bar \varphi_{k,n_i} ) )^4} + \frac{1}{32} \frac{(F''(\bar
\varphi_{k,n_i} ) )^2 F'''(\bar \varphi_{k,n_i} ) }{ (F'(\bar
\varphi_{k,n_i} ) )^3 } - \frac{1}{128} \frac{ (F'''(\bar \varphi_{k,n_i} )
)^2 }{ (F'(\bar \varphi_{k,n_i} ) )^2 } \right. \\
& & \left.
- \frac{1}{96} \frac{F''(\bar \varphi_{k,n_i} ) F^{(iv)}(\bar
\varphi_{k,n_i} ) }{ (F'(\bar \varphi_{k,n_i} ) )^2 }
+ \frac{1}{384} \frac{ F^{(v)}(\bar \varphi_{k,n_i} ) }{F'(\bar
\varphi_{k,n_i} ) } \right)
(\Delta \varphi_{k,n_i})^4 + {\cal O}( (\Delta \varphi_{k,n_i})^5) .
\eeqarn
As in the quantum mechanics case, all but the first expressions in the
expansion of the kinetic and jacobian terms
will be integrated out to put the path integral into the canonical form.
The path integral normalization, denoted with bra-ket notation, is as follows
\beqar
\langle 1 \rangle &=& \int \prod_k d \varphi_k \prod_{i=1}^d d \varphi_k^{A_i}
\prod_{j=1}^d \delta( \varphi_k - \varphi_k^{A_i} ) \\
& & \times \: \prod_k \: e^{-\frac{1}{ \hbar} \epsilon^{d-2}
\sum_{i=1}^{d} \half g(\bar \varphi_{k,n_i} ) \left(\Delta
\varphi_{k,n_i} \right)^2 } \:
e^{{\rm ln } \frac{\partial F}{\partial \varphi}(\bar \varphi_k )}
\nonumber \\
&=& \int \prod_{i=1}^d d \varphi_k^{A_i}
\prod_{j=2}^d \delta( \varphi_k^{A_1} - \varphi_k^{A_i} ) \\
& & \times \: \prod_k \: e^{-\frac{1}{ \hbar} \epsilon^{d-2}
\sum_{i=1}^{d} \half g(\bar \varphi_{k,n_i} ) \left(\Delta
\varphi_{k,n_i} \right)^2 } \:
e^{{\rm ln } \frac{\partial F}{\partial \varphi}(\bar \varphi_k^{A_1} )} ,
\nonumber
\eeqar
where the bar in the argument of the logarithm indicates that
there are no fluctuations in that term. In other words,
the leading jacobian term is a function of a midpoint variable and
is thus inert under the integration.
Making a linear change to the proper integration variables for
integrating out the fluctuations, i.e.
\beq
d \varphi^{A_i}_k = - \half d \Delta \varphi_{k,n_i} ,
\eeq
the integral becomes
\beqarn
\langle 1 \rangle &=&
(-\half)^d \int \prod_{i=1}^d d \Delta \varphi_{k,n_i}
\prod_{j=2}^d \delta( \bar \varphi_{k,n_1} - \bar \varphi_{k,n_i}
- \half \Delta \varphi_{k,n_1} + \half \Delta \varphi_{k,n_i} ) \\
& & \times \: \prod_k \: e^{-\frac{1}{ \hbar} \epsilon^{d-2}
\sum_{i=1}^{d} \half g(\bar \varphi_{k,n_i} ) \left(\Delta
\varphi_{k,n_i} \right)^2 } \:
e^{{\rm ln } \frac{\partial F}{\partial \varphi}(\bar \varphi_{k,n_1})} .
\nonumber
\eeqarn
Temporarily using the abbreviation
$$
a_i = \frac{1}{2 \hbar} \epsilon^{d-2} g(\bar \varphi_{k,n_i} )
$$
one finds upon evaluation
\beq
\langle 1 \rangle = \sqrt{\frac{\pi}{\sum_{i=1}^d a_i }} \:\times
e^{{\rm ln } \frac{\partial F}{\partial \varphi}(\bar \varphi_{k,n_1})}
\times {\rm exp}\left[ 4 \left(
\frac{(\sum_{i=1}^d a_i \bar \varphi_{k,n_i} )^2}{\sum_{i=1}^d a_i } -
\sum_{i=1}^d a_i \bar \varphi_{k,n_i}^2 \right) \right] .
\eeq
In the limit of vanishing epsilon, all barred variables reach the
same value, which we will denote $\bar \varphi_k.$
This has the consequence that the quantity in the square
brackets above goes to zero.
Integrals over fluctuations may be obtained in the usual way by
differentiation with respect to the $a_i.$
One can easily show that the quantity in square brackets above
is of no consequence to these calculations, and hence we will drop
it here. From the expression for $\langle 1 \rangle,$ one deduces
\beq
\langle \Delta \varphi_{k,n_i}^{2m} \rangle = \langle 1 \rangle
\frac{ (2 m -1)!! \:\:\: \hbar^m \epsilon^{(2-d) m } }{
\left( \sum_{i=1}^d
(\frac{\partial F}{\partial \varphi } (\bar \varphi_{k,n_i}))^2 \right)^m} ,
\eeq
\beq
\langle \Delta \varphi_{k,n_i}^{2m} \: \Delta \varphi_{k,n_j}^{2p}
\rangle = \langle 1 \rangle
\frac{ (2 (m+p) -1)!! \:\:\: \hbar^{m+p} \epsilon^{(2-d) (m+p) } }{
\left( \sum_{i=1}^d
(\frac{\partial F}{\partial \varphi } (\bar \varphi_{k,n_i}))^2 \right)^m
\left( \sum_{j=1}^d
(\frac{\partial F}{\partial \varphi } (\bar\varphi_{k,n_j}))^2 \right)^p}
\eeq
and so on, while integrals over odd powers of $\Delta \varphi_{k,n_i}$ vanish.
Symbolically, we have the result
\beq
\langle \Delta \varphi^{2m} \rangle = \langle 1 \rangle
(2 m -1)!! \left( \frac{ \hbar \epsilon^{2-d } }{
d F'[\bar \varphi ]^2 } \right)^m .
\eeq
Note that unlike the case of quantum mechanics, in quantum field
theory an infinite number of terms potentially contribute.
Taking the ${\cal O}(\Delta \varphi_{k,n_i}^2)$ terms coming
from the jacobian and the
${\cal O}(\Delta \varphi_{k,n_i}^4)$ terms coming
from the kinetic term, one finds in the discretized action a term
$$
S_{\rm extra}^{(1)} = \epsilon^d \frac{\hbar^2}{8}
\frac{1}{d^2} \sum_{i=1}^{d} \frac{ (F^{\prime
\prime}[\varphi_{k,n_i}])^2}{(F^\prime[\varphi_{k,n_i}])^4 }
\epsilon^{2-2d}.
$$
Thus the first extra term (corresponding to the
extra ${\cal O} (\hbar^2) $ term in quantum mechanics)
that arises in the quantum field theory upon a change of variables is
\beq
S_{\rm extra}^{(1)} [\varphi] =
\int d^dx \frac{\hbar^2}{8d} \frac{ F^{\prime
\prime}[\varphi]^2}{F^\prime[\varphi]^4 } \epsilon^{2-2d}.
\eeq
Note that in contrast to the situation in quantum mechanics,
the extra term $S_{\rm extra}^{(1)} [\varphi] $
in $d$ dimensional field theory contains a {\it divergent} quantity
$\epsilon^{2-2d}$ or equivalently $(\delta^{(d-1)} (0))^2.$
In four dimensions, this term is the $( \delta^{(3)} (0) )^2$ term
previously mentioned by Dowker and Mayes~\cite{dowker},
Charap~\cite{charap} and Suzuki and Hattori~\cite{suzuki}, especially
in the context of chiral dynamics.
The second extra term comes from the
${\cal O}((\Delta\varphi_{k,n_i})^4)$ term in the jacobian expansion
together with the
${\cal O}((\Delta \varphi_{k,n_i})^6)$ term in the kinetic term expansion
\beqar
S_{\rm extra}^{(2)} [\varphi]
&=& \frac{\hbar^3 }{192 d^2} \int d^d x \:\: \left[
\frac{F^{(5)}[\varphi]}{(F^\prime[\varphi])^5}
+ 6 \frac{ F^{\prime \prime}[\varphi] F^{(4)}[\varphi]}{
(F^\prime[\varphi])^6}
+ \frac{26 }{3}
\frac{(F^{\prime \prime \prime}[\varphi])^2}{(F^\prime[\varphi])^6}
\right. \nonumber \\
& & \left. -18 \frac{(F^{\prime \prime}[\varphi])^2
F^{\prime \prime\prime}[\varphi] }{(F^\prime[\varphi])^7}
+ 9 \frac{(F^{\prime \prime}[\varphi])^4}{(F^\prime[\varphi])^8} \right]
\epsilon^{4-3d} .
\eeqar
More generally, the $j$-th extra term in the action will be
${\cal O}(\hbar^{j+1}),$ have $d$ dependence like $d^{-j}$
and diverge as $\epsilon^{2j - d (j+1) }.$
\bigskip
Although the analysis in this section applies to a theory on a trivial
spacetime, it is interesting to compare these results
to those for the scalar field living on a cylindrical spacetime
that we considered in the previous section.
In the case of a field redefinition of the form
$\phi = F[\varphi] \equiv \varphi + \alpha \varphi^N,$ the first
extra term is
\beq
S_{\rm extra}^{(1)} [\varphi]=\int d^dx \frac{\hbar^2}{8 d} \epsilon^{2-2d}
\frac{\alpha^2 (N -1)^2 N^2 \varphi^{2N-4}}{(1 + \alpha N
\varphi^{N-1})^4} .
\eeq
Comparing this to the extra term generated in the Kaluza Klein example
(see equation~\ref{eq:VE}) , we see that the first term above gives precisely
that found in the Kaluza Klein example, provided we identify
$\epsilon^{-2}$ with $\delta(0)^2,$ take into account the overall minus sign
in going from Euclidean to Minkowskian signature, and finally
we set the $d$ in the denominator equal to 1 {\em not} 2. The justification
for this last proviso will become clear shortly. But first one may ask,
``what about the string of extra terms that we have found in this section?"
In the Kaluza Klein example, there was only evidence of
a single extra term, not a string of extra terms. What is going on?
We argue that this is an indication that the correct treatment of
theories on nontrivial spacetime topologies involves a different
discretization than the one performed here.
For example, if a dimension is compactified, Poincare invariance of
the total space is lost and one should use different $\epsilon$
factors as necessary in discretizing the spacetime.
In the example of the scalar field on the Minkowskian cylinder,
one would use $\epsilon_1$ and $\epsilon_2$ to discretize the space.
If one is allowed to take the limit of the epsilons going to zero
separately, then the problem is effectively reduced to the
quantum mechanical case (hence $d=1$), and only one term will be generated
since $\theta$ is integrated out.
\section{Feynman Diagram Analysis}
To illustrate the important role of the divergent extra terms that
arise in a quantum field theory after nonlinear field redefinitions are made,
we turn to explicit Feynman diagram calculations.
In the process on exploring this issue, we will gain insight into
what is really happening when one employs the unphysical procedure
of dimensional regularization.
We will work with the real, massless, noninteracting $d$-dimensional scalar
field theory considered in the previous section
\beq
Z = \int {\cal D} \phi \: \: e^{- \frac{1}{\hbar}
\int d^d x \half \partial^\mu \phi \partial_\mu \phi }.
\eeq
and will take the concrete example of a field redefinition
involving a cubic term
\beq
\phi = F[\varphi] \equiv \varphi + \alpha \varphi^3 .
\eeq
To properly analyze this situation using Feynman diagrams, we will
evaluate the Feynman diagrams using the lattice action.
This is necessary since the continuum limit must be taken in a
consistent manner.
In particular, we must use the correct form for the propagator and
for the derivative interactions.
Extracting lattice Feynman rules for the non-derivative interactions
is trivial.
Under the above nonlinear transformation, the previous section implies
that the non-derivative part of the action consists of the
``usual" jacobian along with an infinite series of extra terms.
The usual lattice jacobian term is
$$
S_{ \cal J} [ \bar\varphi]
= \sum_x \epsilon^{d}
\left[ -\hbar \epsilon^{-d}
{\rm ln} \left(1+3 \alpha \bar\varphi^2 \right) \right] ,
$$
and the first couple of extra terms are
$$
S_{\rm extra}^{(1)} [\bar\varphi]
= \sum_x \epsilon^{d} \left[
\frac{ 9 \hbar^2 \alpha^2}{2 d}\epsilon^{2-2d}
\frac{\bar\varphi^2}{\left( 1 + 3 \alpha \bar\varphi^2 \right)^4} \right]
$$
and
$$
S_{\rm extra}^{(2)} [\bar\varphi] =
\sum_x \epsilon^{d} \left[
\frac{\hbar^3 \alpha^2}{4 d^2} \epsilon^{4-3d} \left(
\frac{27}{\left( 1 + 3 \alpha \bar\varphi^2 \right)^8}
- \frac{27}{\left( 1 + 3 \alpha \bar\varphi^2 \right)^7}
+ \frac{13}{2 \left( 1 + 3 \alpha \bar\varphi^2 \right)^6} \right) \right].
$$
Extracting the lattice Feynman rules for the derivative terms requires
a little more work. (The apparent continuum expression is
$$ \left.
S_0[ F[\varphi]] = \int d^d x \half \partial_\mu \varphi
\partial^\mu \varphi \left( 1 + 3 \alpha \varphi^2 \right)^2 .\right)
$$
For example, the derivative part of the discretized action contains the term,
with slight notational changes from the previous section
(to avoid confusion due to the fact that we perform a Fourier transform),
$$
S[ \varphi] = \epsilon^{d-2} \sum_{x \in \epsilon Z^d}
\frac{1}{2} \sum_{\mu =1}^d
\left( \varphi_{x + \epsilon \hat n_\mu} - \varphi_x \right)^2
$$
where $n_\mu$ is a unit vector in the $\mu$ direction.
Inserting the Fourier transformed field
$$
\varphi_x = \int d^dp \tilde \varphi (p) e^{i p \cdot x}
$$
into the action, and taking care to note that
the momentum integrations cover only the first Brillioun zones
$$
-\frac{\pi}{\epsilon} \leq p_\mu \leq \frac{\pi}{\epsilon}
\quad \quad \quad \mu = 1 \ldots d,
$$
we find
\begin{eqnarray*}
& & \frac{1}{\epsilon^2} \sum_{x \in \epsilon Z^d}
\frac{1}{2} \sum_{\mu =1}^d
\left( \varphi_{x + \epsilon \hat n_\mu} - \varphi_x \right)^2 \\
&=& \sum_{x \in \epsilon Z^d} \frac{1}{2 \epsilon^2} \int d^dp
\int d^d p^\prime
\tilde \varphi(p) \tilde \varphi(p^\prime ) e^{i (p+ p^\prime ) \cdot x}
\left( e^{i p \cdot n_\mu \epsilon} -1 \right)
\left( e^{i p^\prime \cdot n_\mu \epsilon} -1 \right) \\
&=& \frac{(2 \pi)^d}{\epsilon^2}
\frac{1}{2} \int d^dp \tilde \varphi (p) \tilde \varphi (-p)
\left| e^{i p \cdot n_\mu \epsilon} -1 \right|^2
\end{eqnarray*}
This is the only term both quadratic in $\varphi$ and order ${\cal
O}(\alpha^0)$ in the action, and our unrenormalized
lattice propagator for $\varphi$ is thus
\begin{equation}
\Delta_\epsilon (p) = \left[ \sum_{\mu=1}^d \frac{1}{\epsilon^2}
\left| e^{i p \cdot n_\mu \epsilon} -1 \right|^2 \right]^{-1}
= \frac{1}{p^2} + \frac{\epsilon^2}{12} \frac{ \sum_{\mu=1}^d
p_\mu^4 }{p^4} + \cdots .
\end{equation}
We only expand the propagator here to make contact with the
usual form of the continuum propagator. Note that taking the usual
form for the propagator assumes that the
components of the momentum are very small in magnitude compared to the natural
cutoff dictated by the lattice spacing, i.e. taking the usual
continuum propagator means that for each component you stay away from the
edges of the Brillioun zone. We will not truncate the propagator
in this way as there is not a meaningful expansion in $\epsilon$ for
all values of $p.$ We stress that the small $\epsilon$ limit does not
imply the usual $1/p^2$ form for the propagator from the lattice
point of view, unless one is willing to introduce an ad hoc cutoff
$\Lambda$ which is much smaller than the natural cutoff
$\frac{\pi}{\epsilon}$.
In a similar manner, one may derive the Feynman vertices for the
non-quadratic
derivative interactions. For the four-field interaction one finds
\begin{equation}
\lambda^{(4)}_\epsilon (p,k) = \frac{12 i \alpha}{\epsilon^2}
\sum_{\mu =1}^d \left( e^{i p \cdot n_\mu \epsilon} -1 \right)
\left( e^{i k \cdot n_\mu \epsilon} -1 \right) ,
\end{equation}
while for the six-field interaction one finds
\begin{equation}
\lambda^{(6)}_\epsilon (p,k) = \frac{216 i \alpha^2}{\epsilon^2}
\sum_{\mu =1}^d \left( e^{i p \cdot n_\mu \epsilon} -1 \right)
\left( e^{i k \cdot n_\mu \epsilon} -1 \right) .
\end{equation}
In evaluating Feynman diagrams, it proves expedient to make use of the
fact that
$$
\lambda^{(4)}_\epsilon (p,-p) = 12 i \alpha \left[ \Delta_\epsilon (p)
\right]^{-1}
$$
and
$$
\lambda^{(6)}_\epsilon (p,-p) = 216 i \alpha^2 \left[ \Delta_\epsilon (p)
\right]^{-1} .
$$
The lattice Feynman rules extracted from the
$\varphi$ theory are illustrated in Figure 1.
These Feynman rules taken by themselves would appear to describe an
interacting scalar field theory, but knowing their origin, we
are aware that this theory is simply a disguised free theory.
This complicated action for $\varphi$ should yield results
physically equivalent to the trivial free action for $\phi .$
In particular,
one would expect that the pole of the renormalized $\varphi$
propagator should be equal to the original physical pole.
The parameter $\alpha$ is not necessarily a small parameter and
we have presented the exact action to ${\cal O}(\hbar^2)$, but
now we will treat $\alpha$ as small and
compute the order ${\cal O}(\alpha)$ and
${\cal O}(\alpha^2)$ corrections to the lattice propagator.
One may verify that for sufficiently small $\alpha$, that perturbation
theory about $\varphi =0$ makes sense.
At ${\cal O} (\alpha)$, there are three graphs (see figure 2)
contributing to the propagator with external momentum $k$.
The divergent contribution coming from the
derivative interaction loop is precisely canceled by an
equal but opposite contribution from the mass graph arising from the
jacobian term, i.e.
\beqar
& & \half \int \frac{d^dp}{(2 \pi)^d} \int \frac{d^dp^\prime}{(2 \pi)^d}
\delta^{(d)}(p+p^\prime) i \hbar \Delta_\epsilon(p)
\lambda^{(4)}_\epsilon(p,p^\prime)
+ 6 \alpha \hbar \epsilon^{-d} \nonumber \\
&=& - \half \int \frac{d^dp}{(2 \pi)^d} i \hbar \Delta_\epsilon(0)
12 i \alpha (\Delta_\epsilon(p))^{-1}
+ 6 \alpha \hbar \epsilon^{-d} \nonumber \\
&=& - 6 \alpha \hbar \int \frac{d^dp}{(2 \pi)^d}
\: + \: 6 \alpha \hbar \epsilon^{-d} \: = \: 0 .
\eeqar
The remaining less divergent diagram from the derivative-interaction
loop, where the derivatives are not acting on the loop but on the
external legs with momentum $k,$ gives
\beq
\half \lambda^{(4)}_\epsilon(k,-k)
\int \frac{d^dp}{(2 \pi)^d} i \hbar \Delta_\epsilon(p)
= - 6 \alpha \hbar (\Delta_\epsilon(k))^{-1}
\int \frac{d^dp}{(2 \pi)^d} \Delta_\epsilon(p)
\eeq
which is harmless in that it is an overall wavefunction renormalization.
This cancellation illustrates that the role of the usual jacobian term
with its diverging factor, i.e. $\epsilon^{-d}$ on the lattice or
$\delta^{(d)} (0)$ in the continuum, is to cancel off the divergent
contribution to the propagator which would have shifted the value
of the mass. There is no need to think of $\delta^{(d)} (0)$ or
$\epsilon^{-d}$ being set zero, or as being absorbed into some redefinition.
The situation is more interesting at order ${\cal O}(\alpha^2),$
where the so-called extra terms begin to arise.
Including the vertex arising from the first ``extra'' term we found
in the previous section,
there are seven graphs (see figure 3) which potentially contribute to
a mass shift. Figure 3 does not include graphs which would
affect only wave function renormalization.
Graphs (d) and (e) cancel each other
as one would expect from the cancellation at ${\cal O} (\alpha)$.
Graphs (a), (b) and (c) may be shown to add up to
\beq
-i 72 \alpha^2 \hbar^2 \int \frac{d^dk_1}{(2 \pi)^d} \frac{d^dk_2}{(2 \pi)^d}
\Delta_\epsilon(k_1) .
\label{eq:athruc}
\eeq
Note that a change of integration variables from the $k_i$ to
$\ell_i \equiv \epsilon k_i$
reveals the degree of divergence of this integral as $\epsilon \rightarrow 0$
to be $\epsilon^{2-2d}$,
since all $\epsilon$ dependence leaves both the integrand and the limits
of integration and goes into a prefactor to the integral.
The lattice ``setting sun'' diagram (f) is much more complicated to
evaluate
\beqarn
& & -i \hbar^2 \half \int \frac{d^dk_1}{(2 \pi)^d} \frac{d^dk_2}{(2 \pi)^d}
\frac{d^dk_3}{(2 \pi)^d} \delta^{(d)}(k_1+k_2+k_3)
\Delta_\epsilon(k_1) \Delta_\epsilon(k_2) \Delta_\epsilon(k_3) \\
& & \times \frac{1}{3} \left[
\lambda^{(4)}_\epsilon(k_1,k_2) \lambda^{(4)}_\epsilon(k_1,k_2) +
%\lambda^{(4)}_\epsilon(k_2,k_3) \lambda^{(4)}_\epsilon(k_2,k_3) +
%\lambda^{(4)}_\epsilon(k_3,k_1) \lambda^{(4)}_\epsilon(k_3,k_1)
{\rm cyclic}\: {\rm in} \: (k_1,k_2,k_3) \right] .
\eeqarn
More explicitly, the integral is
\beqarn
&=& i 24 \alpha^2 \hbar^2 \epsilon^2 \int \frac{d^dk_1}{(2 \pi)^d}
\frac{d^dk_2}{(2 \pi)^d} \frac{d^dk_3}{(2 \pi)^d} \delta^{(d)}(k_1+k_2+k_3=0)
\\ & & \times
\frac{\left[ \left(
\sum_{\mu =1}^d \left( e^{i k_1 \cdot n_\mu \epsilon} -1 \right)
\left( e^{i k_2 \cdot n_\mu \epsilon} -1 \right) \right)^2
+ {\rm cyclic}\: {\rm in} \: (k_1,k_2,k_3) \right]}{
\sum_{\nu =1}^d \left| e^{i k_1 \cdot n_\nu \epsilon} -1 \right|^2
\sum_{\rho =1}^d \left| e^{i k_2 \cdot n_\rho \epsilon} -1 \right|^2
\sum_{\sigma =1}^d \left| e^{i k_3 \cdot n_\sigma \epsilon} -1 \right|^2 },
\eeqarn
where the integration domains cover the first Brillioun zones,
i.e. $-\frac{\pi}{\epsilon} \leq k_\mu \leq +\frac{\pi}{\epsilon}$.
The first thing to observe about this integral is that a change of
integration variables from the $k_i$ to $\ell_i \equiv \epsilon k_i$
reveals that it will diverge like $\epsilon^{2-2d}$
in the limit $\epsilon \rightarrow 0.$
The exact evaluation of this integral is prohibitive, since there is
no meaningful expansion in $\epsilon.$
Although we have not been able to exactly evaluate this integral, all
indications support that this integral does not exactly cancel
the contributions from the graphs (a) through (e), and that the
extra term graph (g) is necessary to ensure that all contributions add up to
zero at ${\cal O}(\alpha^2)$ to keep the propagator's pole intact.
Note that the extra (counter)term
$$
-i \frac{9}{d} \alpha^2 \hbar^2 \epsilon^{2-2d}
$$
has the correct divergence structure in $\epsilon$
to cancel off the ${\cal O}(\alpha^2)$
term that otherwise would shift the physical mass.
To lend credibility to the assertion that graph (g) is necessary to
maintain the pole of the renormalized propagator,
let us obviate the technical difficulties mentioned above
by evaluating the diagrams using the ``standard'' limiting forms for
the propagator and the four-field vertex.
The integral above for the ``setting sun'' diagram becomes
\beqarn
&=& i 24 \alpha^2 \hbar^2 \int \frac{d^dk_1}{(2 \pi)^d}
\frac{d^dk_2}{(2 \pi)^d} \frac{d^dk_3}{(2 \pi)^d} \delta^{(d)}(k_1+k_2+k_3)
\\ & & \times
\frac{\left[ (k_1 \cdot k_2)^2 + (k_2 \cdot k_3)^2 + (k_3 \cdot k_1)^2
\right]}{ k_1^2 \: k_2^2 \: k_3^2 }
\eeqarn
where the limits of integration are unchanged.
A careful evaluation of this integral yields
\beq
= i 36 \alpha^2 \hbar^2
\left[ \int \frac{d^dk_1}{(2 \pi)^d} \frac{d^dk_2}{(2 \pi)^d}
\frac{1}{k_1^2} \:\: + \5{|(k_1+k_2)_\mu | \leq \frac{\pi}{\epsilon} }{\int}
\frac{d^dk_1}{(2 \pi)^d} \frac{d^dk_2}{(2 \pi)^d} \frac{1}{k_1^2} \right]
\eeq
where the integration region in the second term in restricted from the
previous integrals.
Furthermore, given that we are integrating
an even function in each component of both $k_1$ and $k_2,$
the restricted integration region may be rewritten to give the
following result
\beq
i 72 \alpha^2 \hbar^2 \prod_{i=1}^d
\left[ \int_{-\frac{\pi}{\epsilon}}^{\frac{\pi}{\epsilon}} dk_i
\int_{-\frac{\pi}{\epsilon}}^{\frac{\pi}{\epsilon}} dp_i
- 2\int_0^{\frac{\pi}{\epsilon}} dp_i \left( \frac{\pi}{\epsilon} -p_i \right)
\right] \frac{1}{p^2} .
\eeq
One of the $2^d$ resulting integrals, namely the one where each
component of each momenta is integrated over the entire region,
precisely cancels the correspondingly approximated integral of
equation~\ref{eq:athruc}, i.e.
$$
i 72 \alpha^2 \hbar^2 \int d^dk \int d^dp \frac{1}{p^2} .
$$
% Due to the fact that we have been working with a hypercubic lattice,
% the region of integration for the integrals
%above is a $d$-dimensional hypercube.
%Writing for each component of the momentum,
%$p_{\rm max} = + \frac{\lambda \: \pi}{\epsilon}$ and
% $p_{\rm min } = -\frac{\lambda \: \pi}{\epsilon},$ one of the two integrals
%we must evaluate is trivial, i.e.
%$$
%\int \frac{d^dp}{(2 \pi)^d} = \int_{-\frac{\lambda \:\pi}{\epsilon}}^{
% +\frac{\lambda \: \pi}{\epsilon}} \frac{d p_1}{2 \pi} \ldots
%\int_{-\frac{\lambda \: \pi}{\epsilon}}^{+\frac{\lambda \: \pi}{\epsilon}}
%\frac{d p_d}{2 \pi} = \lambda^d \:\epsilon^{-d} .
%$$
%The other integral is complicated by the hypercubic integration domain
%\beqarn
%\int \frac{d^dp}{(2 \pi)^d} \frac{1}{p^2} &=&
%\int \frac{d^dp}{(2 \pi)^d} \int_0^\infty d \beta e^{-\beta p^2}
%= \int_0^\infty d \beta \prod_{i=1}^d
%\int_{-\frac{\pi}{\epsilon}}^{\frac{\pi}{\epsilon}} \frac{d p_i}{2 \pi}
%e^{-\beta p_i^2} \\
%&=& \int_0^\infty d \beta \frac{\pi^{\frac{d}{2} }}{\beta^{\frac{d}{2}}}
%\:\:({\rm erf}(\sqrt{\beta} \frac{\pi}{\epsilon}))^d
%= 2 \pi^\frac{d}{2} \left( \frac{ \lambda \: \pi}{\epsilon} \right)^{d-2}
%\int_0^\infty d y y \left( \frac{ {\rm erf}(y) }{y} \right)^d .
%\eeqarn
The remaining $2^d -1$ integrals give a nonzero contribution that must
be canceled by the extra term, however we will not carry out the many
integrals necessary to show this precise cancellation.
We have indicated the role of the extra term as a mechanism for canceling off
the potential mass shift, using the $1/p^2$ limit of the propagator and
$p\cdot q$ limit of the four-field vertex; however, theoretically it
should be possible to show this with the strict lattice forms of the
propagator and the vertex.
Since physically significant quantities must be invariant under
field redefinition, this process must continue, with successive
extra terms properly canceling off any would-be shifts in the
physical mass. It is encouraging that a cursory look at the divergences
at the next order in $\alpha$ go like $\epsilon^{4-3d}$ which is
precisely the divergence with which the extra term at ${\cal O}(\alpha^3)$
appears.
We conjecture that proper inclusion of the extra terms will
ensure that the pole of the propagator remains at its physical value,
and more generally that all physically significant quantities will
remain unchanged via the same cancellation mechanism.
\section{Conclusion}
Using a discretized version of the path integral for a
free massless scalar quantum field in $d$ dimensions,
we have shown that beyond the usual jacobian term,
an infinite series of divergent ``extra'' terms
arises in the action whenever a nonlinear field redefinition is made.
The explicit forms for the first couple of these terms were given.
We have argued that in the strictest sense
these terms should not be neglected, since for example one role of
these terms is to
cancel off contributions which would change the value of physical
quantities such as the pole of the renormalized propagator.
Using a different method than discretization of the path integral,
i.e. dimensional reduction down to a
quantum mechanical model where known results could be exploited,
we analyzed a $1+1$ dimensional free massless scalar theory on
a Minkowskian cylinder, and saw that only a single term appeared (as one
finds in quantum mechanics). The fact that a single term
arose in this case is a consequence of the nontrivial spacetime
topology which would affect a discretization and
reduce the problem to two essentially one dimensional cases.
The role that spacetime topology plays in the discretization
of the path integral will be explored further in future articles~\cite{us}.
Let us clarify the effect of these divergent extra terms
which arise upon making nonlinear field redefinitions.
Consider a quantum field theory with some number of bare parameters.
Physically significant renormalized quantities in this original theory
are functions of the bare parameters and the cutoff in the original action.
At least in principal, the bare parameters may be fixed by knowledge
of experimental data.
Suppose one now performs a nonlinear field redefinition, and
computes the renormalized quantities as a function of the same
bare parameters, the cutoff and any field redefinition parameters.
If the divergent extra terms are neglected, the computed values of
some physical quantities will differ from those previously computed.
We argue that the extra terms generated in the field transformation are
precisely what is necessary to cancel the divergences
that one normally sweeps into the ad hoc addition of counterterms and/or
the redefinition of the bare parameters. From this point of view,
what we have presented in this paper are the exact counterterms for the
quantum field theory.
Thus, it is important to include these terms if one is making
nonlinear field redefinitions in a theory where values of some coefficients
are known.
Feynman diagrammatic calculations have enabled us to see explicitly
that the extra terms are essential to understanding why
the unphysical practice of dimensional regularization works at all,
at least in this context. From the pragmatist's point
of view where $\delta^{(d)}(0) \sim
a^{-d}$ blows up as the lattice spacing goes to zero, this work
is especially pleasing since we have shown that it is not necessary to
make even a formal assertion that $\delta^{(d)}(0) = 0$.
We have indicated how the divergent extra counterterms
cancel out unwanted divergent contributions to physical quantities so that
the result is {\em consistent} with simply setting these divergences equal
to zero in evaluations of Feynman diagrams.
Note that this is completely a different matter than setting factors
such as $\delta^{(d)}(0)$ equal to zero in the action!
This cancellation mechanism by the extra terms is of course
valid whether or not dimensional regularization is applicable.
In instances where dimensional regularization is not valid, care
should be taken that extra terms be included properly in calculations
involving nonlinear field redefinitions.
A further reason for interest in these so-called extra terms is
the exciting possibility that these extra terms that we have found could
in fact allow for the presence of anomalies of higher orders in $\hbar$
in quantum field theories with nonlinear symmetries.
This issue is currently under investigation, and could
potentially be of extreme importance to, for example,
gravity theories and currently popular theories possessing extended
conformal ${\cal W}$ algebra symmetries.
Work on these issues is in progress~\cite{us}.
\section*{Acknowledgements}
\hspace{\parindent}%
We would like to thank Cliff Burgess and Joe Polchinski for
reading a preliminary version of this paper and for insightful
comments.
This research was supported in part by Robert A.~Welch Foundation,
NSF Grant PHY 9009850 (UT Austin) and
Department of Education grant DE-FG05-87ER40367 (Vanderbilt).
\bigskip
\appendix
\section{Useful identities involving modes}
In this appendix, we prove some identities used in obtaining
results in this paper.
Consider the expansion in modes of a function $\varphi (\theta , t)$
with a periodic coordinate $\theta$
\beq
\varphi(\theta,t) = \sum_{p \in Z} b^{(p)}(t) e^{i p \theta} .
\eeq
We call $p$ the mode number.
Define the following object which is multilinear in $N$ modes
and where the sum of the mode numbers is $\ell$
\beq
E^{(\ell)}_N [b] \equiv \sum_{p_1 \in Z} \ldots \sum_{p_N \in Z}
b^{(p_1)} \ldots b^{(p_N)} \delta^{\ell - \sum_{i=1}^N p_i =0} .
\eeq
Note the special case when the sum of the mode numbers is zero
\beq
E^{(0)}_N [ b ] = \int \frac{d\theta}{2 \pi} \varphi(\theta,t)^N
\equiv B_N[b] .
\label{eq:Bmode}
\eeq
One may straightforwardly relate the sum over a product of two $E$'s
to a single $E$ in the following way
\beqar
& & \sum_{p \in Z} E^{(\ell + p)}_M [b] E^{(-p+m)}_N [b] \nonumber \\
&=& \sum_{p \in Z}
\sum_{\begin{array}{c} {p_1,\ldots p_M \in Z} \\
\ell_1,\ldots \ell_N \in Z \end{array}}
b^{(p_1)} \ldots b^{(p_N)} b^{(\ell_1)} \ldots b^{(\ell_N)}
\delta^{p+ \ell - \sum_{i=1}^M p_i =0} \delta^{-p +m - \sum_{j=1}^N
\ell_j = 0} \nonumber \\
&=& \sum_{p \in Z} \sum_{p_1 \in Z} \ldots \sum_{p_{M+N} \in Z}
b^{(p_1)} \ldots b^{(p_{M+N})}
\delta^{p+ \ell - \sum_{i=1}^M p_i =0} \delta^{-p +m - \sum_{j=M+1}^{M+N}
p_j = 0} \nonumber \\
&=& \sum_{p_1 \in Z} \ldots \sum_{p_{M+N} \in Z}
b^{(p_1)} \ldots b^{(p_{M+N})} \delta^{\ell + m - \sum_{j=1}^{M+N} p_j = 0}
\nonumber \\
&=& E^{(\ell + m)}_{M+N} [b] .
\eeqar
Consider three $E$'s where the number of modes may be different
and where the sum of all mode numbers is zero
\beqar
& & \sum_{\ell \in Z} \sum_{n \in Z} \sum_{p \in Z}
E^{(\ell + n)}_L [b] \: \: \: E^{(-\ell+p)}_M [b] E^{(-p-n)}_N [b]
\nonumber \\
&=& \sum_{n \in Z} \sum_{p \in Z} E^{(n+p)}_{L+M} [b] \:\:\: E^{(-p-n)}_N [b]
\nonumber \\
&=& \left[ \sum_{n \in Z} 1\right]
\sum_{p^\prime \in Z} E^{(p^\prime)}_{L+M} [b] \:\:\: E^{(-p^\prime)}_N [b]
\nonumber \\
&=& \left[ \sum_{n \in Z} 1 \right] B_{L+M+N} [b] .
\eeqar
Finally consider the following trace of $j$ $E_M$'s where
the sum of all mode numbers is zero (summation convention assumed)
\beqar
{\rm tr } (E_M^j) &=& E^{(-n_j + n_1)}_M [b] E^{(-n_1 + n_2)}_M [b] \ldots
E^{(-n_{j-1}+n_j)}_M [b] \nonumber \\
&=& E^{(-n_j + n_2)}_{2M} [b] E^{(-n_2 + n_3)}_M [b] \ldots
E^{(-n_{j-1}+n_j)}_M [b] \nonumber \\
&=& \left[ \sum_{n \in Z} 1 \right] B_{j(N-1)} [b] .
\eeqar
This identity is useful when evaluating the jacobian determinant in
section 2 of this paper.
A final identity which is useful in comparing the kinetic terms in
the example of section 2 is
\beqar
& & \sum_{p \in Z} p^2 E_N^{(p)} E_1^{(-p)} \nonumber \\
&= & \sum_{p \in Z} \sum_{\ell_1 \in Z} \cdots \sum_{\ell_N \in Z}
p^2 b^{(\ell_1)} \cdots b^{(\ell_N)} b^{(-p)}
\delta^{p-\sum_{i=1}^N \ell_i =0} \nonumber \\
&= & \sum_{\ell_1 \in Z} \cdots \sum_{\ell_N \in Z}
\left( \sum_{i=1}^N \ell_i \right)^2
b^{(\ell_1)} \cdots b^{(\ell_N)} b^{(-\sum_{i=1}^N \ell_i)}
\nonumber \\
&=& \sum_{\ell_1 \in Z} \cdots \sum_{\ell_N \in Z}
\left( \sum_{i=1}^N (\ell_i)^2 + \sum_{i \neq j} \ell_i \ell_j \right)
b^{(\ell_1)} \cdots b^{(\ell_N)} b^{(-\sum_{i=1}^N \ell_i)}
\nonumber \\
&=& \sum_{\ell_1 \in Z} \cdots \sum_{\ell_N \in Z}
\left( N (\ell_N)^2 + N \ell_N \sum_{i=1}^{N-1} \ell_i \right)
b^{(\ell_1)} \cdots b^{(\ell_N)} b^{(-\sum_{i=1}^N \ell_i)}
\nonumber \\
&=& N \sum_{\ell_1 \in Z} \cdots \sum_{\ell_N \in Z}
\ell_N \left( \sum_{i=1}^{N-1} \ell_i \right)
b^{(\ell_1)} \cdots b^{(\ell_N)} b^{(-\sum_{i=1}^N \ell_i)}
\nonumber \\
&=& -N \sum_{p \in Z} \sum_{\ell_1 \in Z} \cdots \sum_{\ell_N \in Z}
\ell_N p b^{(\ell_1)} \cdots b^{(\ell_N)} b^{(p)}
\delta^{(-p-\sum_{i=1}^N \ell_i =0 )}
\nonumber \\
&=& -N \sum_{p \in Z} \sum_{\ell_1 \in Z} \cdots \sum_{\ell_{N-1} \in Z}
\sum_{\ell \in Z} \ell p b^{(\ell_1)} \cdots b^{(\ell_{N-1})} b^{(\ell)}
b^{(p)} \delta^{(-p-\ell -\sum_{i=1}^{N-1} \ell_i =0 )} \nonumber \\
&=& -N \sum_{p \in Z} \sum_{\ell \in Z} \ell p
E_1^{(p)} E_1^{(\ell)} E_{N-1}^{(-\ell-p)}
\label{eq:p2ident} .
\eeqar
\section{Details of Kaluza Klein calculation}
The metric in terms of the $b^{(m)}$ modes in the Kaluza Klein calculation
is
\beqarn
g_{mn}[b] &=& 2 \pi R
(\delta_{m+n=0} + 2 \alpha N E^{(-m-n)}_{N-1}[b] + N^2 \alpha^2
E^{(-m-n)}_{2N-2}[b] ) \\
&=& 2 \pi R (1 + \alpha N E_{N-1}[b] )^2_{m+n} ,
\eeqarn
where $1_{m+n} $ is shorthand for $\delta_{m+n=0},$
$(E_M)_{m+n}$ is shorthand for $E_M^{(-m-n)}$, and
$(E_M^2)_{m+n}$ is shorthand for $E_M^{(-m-p)} E_M^{(p-n)}$.
Subtracting out an infinite constant due to the $2 \pi R,$
we compute the jacobian contribution
\beqar
{\rm Tr ln } [\frac{g_{mn}[b]}{2 \pi R}] \delta(t-t^\prime)
&=& 2 {\rm Tr ln } (1 + \alpha N E_{N-1}[b] )_{m+n} \delta(t-t^\prime)
\nonumber \\
&=& -2 {\rm Tr }\sum_{j=1}^\infty\frac{(-\alpha N)^j}{j} (E^j_{N-1}[b])_{m+n}
\delta(t-t^\prime) \nonumber \\
&=& -2 \delta(0) \sum_{j=1}^\infty \frac{(-\alpha N)^j}{j} \int dt
\left[ E^{(-m+n_1)}_{N-1}[b] \cdots E^{(n_{j-1}+m)}_{N-1}[b] \right]
\nonumber \\
&=& -2 \delta(0) \sum_{j=1}^\infty \frac{(-\alpha N)^j}{j}
\left[ \sum_{k \in Z} 1 \right] \int dt \: B_{j(N-1)}[b] .
\eeqar
We now illustrate how the inverse metric is obtained to all orders in
$\alpha$ via a recursion equation. Write
\beq
g^{km}[b] = \frac{1}{2 \pi R}
\sum_{Q=0}^\infty f_Q (\alpha) E^{(k+m)}_{Q}[b].
\eeq
and compute $g^{km}[b] g_{mn}[b] = \delta^{k-n=0}$ as follows
\beqarn
\delta^{k-n=0} &=& \sum_{Q=0}^\infty f_Q (\alpha) E^{(k-n)}_{Q}[b]
+ 2 \alpha N \sum_{Q=0}^\infty f_Q (\alpha) E^{(k+m)}_{Q}[b]
E^{(-m-n)}_{N-1}[b] \\
& & + \alpha^2 N^2 \sum_{Q=0}^\infty
f_Q (\alpha) E^{(k+m)}_{Q}[b] E^{(-m-n)}_{N-2}[b] \\
&=& \sum_{Q=0}^\infty f_Q (\alpha) E^{(k-n)}_{Q}[b]
+ 2 \alpha N \sum_{Q=0}^\infty f_Q (\alpha) E^{(k-n)}_{Q+N-1}[b] \\
& & + \alpha^2 N^2 \sum_{Q=0}^\infty f_Q (\alpha) E^{(k-n)}_{Q+2N-2}[b] \\
&=& \sum_{Q=0}^{N-2} f_Q (\alpha) E^{(k-n)}_{Q}[b]
+ \sum_{Q=0}^{N-2} \left( f_{Q+N-1} (\alpha)
+ 2 \alpha N f_{Q} (\alpha) \right) E^{(k-n)}_{Q+N-1}[b] \\
& & + \sum_{Q=0}^\infty \left( f_{Q+2N-2} (\alpha) +
2 \alpha N f_{Q+N-1} (\alpha) +
\alpha^2 N^2 f_{Q} (\alpha) \right) E^{(k-n)}_{Q+2N-2}[b].
\eeqarn
The first sum on the righthand side after the last equal side
must be a Kroeneker delta function, and the other two sums
must vanish. From the first sum, one finds $f_0(\alpha) =1$ and
$f_1(\alpha) = \cdots = f_{N-2}(\alpha) =0.$ From the second sum,
one finds $f_{N-1}(\alpha) =-2 \alpha N$ and
$f_{N}(\alpha) = \cdots = f_{2N-3}(\alpha) =0.$ From the third sum,
it is clear that $f_{Q}(\alpha) =0$
if $Q$ is not divisible by $N-1.$
Furthermore, making the ansatz
$$
f_{(N-1)q}(\alpha) = (-\alpha N)^q C_q ,
$$
we may determine the unknown function $C_q$ by plugging into
the third sum.
Explicitly, the condition that the third sum vanishes is
$$
f_{(N-1)(2+q)}(\alpha) = -2 \alpha N f_{(N-1)(1+q)}(\alpha)
- \alpha^2 N^2 f_{(N-1)q}(\alpha),
$$
and taking into account the boundary condition $C_0=1,$
this leads to the solution
$$
f_{(N-1)q}(\alpha) = (-\alpha N)^q (q+1) .
$$
Thus, we obtain the inverse metric to all orders in the parameter $\alpha$
\beq
g^{km}[b] = \frac{1}{2 \pi R}
\sum_{j=0}^\infty (-\alpha N)^j (j+1) E^{(k+m)}_{j(N-1)}[b].
\eeq
Using this expression, the connection coefficients are computed to be
\beq
\Gamma^k_{\ell p}[b] = - \sum_{j=0}^\infty (-\alpha N)^{j+1} (N-1)
E^{(k-p-\ell)}_{j(N-1)+N-2}[b].
\eeq
Note that $\Gamma^k_{\ell p}[b]$ is dependent only on the total mode
number $k-p-\ell$ so it will prove useful to write
$\Gamma^k_{\ell p}[b] = \Gamma^{k-p-\ell}[b]$.
The metric and inverse metric also depend only on the sum of their indices.
Using these expressions, the extra term may be found by computing
\beqar
& & \sum_{k,\ell, p, n \in Z}
\frac{1}{8} g^{\ell n}[b] \Gamma^k_{\ell p}[b] \Gamma^p_{nk}[b]
\nonumber \\
&=& \sum_{k,\ell, p, n \in Z} \frac{1}{8} g^{\ell n}[b] \Gamma^{k -\ell- p}[b]
\Gamma^{p -n -k}[b] \nonumber \\
&=& \sum_{k \in Z} 1 \sum_{\ell, p, n \in Z} \frac{1}{8} g^{\ell n}[b]
\Gamma^{-\ell-p}[b] \Gamma^{p-n}[b] \nonumber \\
%&=& [ \sum_{k \in Z} 1 ]
%\frac{(N-1)^2}{8} \sum_{L,M,J=0}^\infty (-\alpha N)^{L+M+J+2} (L+1)
%\sum_{\ell, p, n \in Z} E^{(\ell+n)}_{L(N-1)}[b]
%E^{(-p-\ell)}_{M(N-1)+N-2}[b] E^{(p-n)}_{J(N-1)+N-2}[b]
%\nonumber \\
&=& \left[ \sum_{k \in Z} 1 \right]^2
\frac{(N-1)^2 }{8\cdot 2 \pi R} \sum_{L,M,J=0}^\infty
(-\alpha N)^{L+M+J+2} (L+1) E^{(0)}_{(M+J+L)(N-1) +2(N-2) }[b]
\nonumber \\
&=& \left[\sum_{k \in Z} 1 \right]^2 \frac{(N-1)^2 }{8\cdot 2 \pi R}
\sum_{L,M,J=0}^\infty (-\alpha N)^{L+M+J+2}
\left(\frac{L+M+J}{3} +1 \right) \nonumber \\
& &\:\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
\quad \quad
\times B_{(M+J+L)(N-1) +2(N-2) }[b] \nonumber \\
&=& \left[ \sum_{k \in Z} 1 \right]^2 \frac{(N-1)^2}{8\cdot 2 \pi R}
\sum_{X=0}^\infty {\cal P}_3(X) (-\alpha N)^{X+2} \left(\frac{X}{3}+1 \right)
B_{X(N-1) +2(N-2) }[b] \nonumber \\
&=& \frac{(N-1)^2}{8\cdot 2 \pi R} \left[ \sum_{k \in Z} 1 \right]^2
\sum_{X=0}^\infty (-\alpha N)^{X+2}
\left( \begin{array}{c} X+3 \\ 3 \end{array} \right) B_{X(N-1) +2(N-2) }[b] .
\eeqar
We have used the fact that ${\cal P}_3(X) = \half (X+1)(X+2) $
is the number of ways to write $X$ as the sum of 3 non-negative integers.
To rewrite this expression in terms of $\varphi$, we use
equation~\ref{eq:Bmode}
\beqar
&=& \frac{(N-1)^2}{48 \cdot 2 \pi R}
\left[ \sum_{k \in Z} 1 \right]^2
\sum_{X=0}^\infty (-\alpha N)^{X+2} (X+1) (X+2) (X+3)
\nonumber \\
& & \times \int \frac{d\theta}{2 \pi} \varphi^{X(N-1) +2(N-2) }[b] \nonumber \\
&=& \frac{(N-1)^2 }{48 \cdot 2 \pi R} \left[ \sum_{k \in Z} 1 \right]^2
\int \frac{d\theta}{2 \pi}
\frac{1}{(-\alpha N)} \varphi^{2N-4} \frac{d^3 \:\:\:}{d(\varphi^{N-1})^3}
\sum_{X=0}^\infty (- \alpha N \varphi^{N-1} )^{X+3} \nonumber \\
&=& \frac{(N-1)^2}{48\cdot 2 \pi R} \left[ \sum_{k \in Z} 1 \right]^2
\int \frac{d\theta}{2 \pi} \:\: \frac{1}{(-\alpha N)}
\varphi^{2N-4} \frac{d^3 \:\:\:}{d(\varphi^{N-1})^3}
\left[ \frac{ (-\alpha N \varphi^{N-1})^3}{1+ \alpha N \varphi^{N-1}}
\right] \nonumber \\
&=& \frac{1}{8} \frac{\alpha^2 N^2 (N-1)^2}{2 \pi R}
\left[ \sum_{k \in Z} 1 \right]^2
\int \frac{d\theta}{2 \pi} \:\:
\frac{(\varphi^{N-2} )^2}{(1 + \alpha N \varphi^{N-1})^4} .
\eeqar
\newpage
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\newpage
% DRAW THE FEYNMAN VERTICES
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\newpage
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% Order \alpha diagrams
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\put(10,-10){\rm Figure 2: ${\cal O}(\alpha)$ Corrections to the Propagator}
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%% 1st row
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%% 2nd row
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\end{document}