aainstitutetext: Instituto de Física La Plata, CONICET and Universidad
Nacional de La Plata,
CC 67 (1900) La Plata, Argentina.

# Worldline approach for spinor fields in manifolds with boundaries

Lucas Manzo<EMAIL_ADDRESS>
###### Abstract

The worldline formalism is a useful scheme in Quantum Field Theory which has
also become a powerful tool for numerical computations. It is based on the
first quantization of an auxiliary point-particle whose transition amplitudes
correspond to the heat-kernel of the operator of quantum fluctuations of the
field theory. However, to study a quantum field theory in a bounded manifold
one needs to restrict the path integration domain of the auxiliary point-
particle to a specific subset of worldlines enclosed by those boundaries. In
the present article it is shown how to implement this restriction for the case
of a spinor field in a two-dimensional curved half-plane under MIT bag
boundary conditions, and compute the first few heat-kernel coefficients as a
verification of the proposed construction. This construction admits several
generalisations.

##  1 Introduction

The worldline formalism is a method to compute different quantities in Quantum
Field Theory, such as effective actions, amplitudes, anomalies and partition
functions. Unlike the traditional procedure involving Feynman diagrams, this
formalism is characterised by the introduction of an auxiliary point-particle
whose dynamics is described using a first quantization scheme. The transition
amplitudes of this particle are used to compute the aforementioned quantities
in the quantum field theory, which for unbounded manifolds can be computed
using traditional quantum mechanical path integrals. Hence, first quantization
path integration can be formally used in this framework to extract information
in quantum field theories.

For manifolds without boundaries, this worldline formalism is a well
established and computationally efficient tool. Since the foundational works
of Z. Bern and D. W. Kosower BernKosower , and M. J. Strassler Strassler , it
has been used for computing scattering amplitudes and effective actions for a
variety of quantum field theories Schubert2001 . Scalar, fermionic and vector
fields have been established by Strassler in Strassler  for flat manifolds,
and were later analysed in BastianelliZirotti ; BastianelliCorradiniZirotti ;
BastianelliBenincasaGiombi  for an arbitrary gravitational background. Higher-
spin fields in conformally flat manifolds were addressed in
BastianelliCorradiniLatini ; Corradini2010 ; BastianelliBonezziCorradiniLatini
.

In spite of the many models studied and applications considered, field
theories in manifolds with boundaries have been more elusive. In order to
construct a worldline formulation in these manifolds, one needs to restrict
the path integration domain of the auxiliary point-particle within its
boundaries. Furthermore, this restriction also needs to take into account the
specific boundary condition imposed on the field, which translates into
imposing boundary conditions on the transition amplitudes of the auxiliary
point-particle. Therefore, the problem of formulating a worldline prescription
for field theories in bounded manifolds reduces to the problem of constructing
a path integral representation for the transition amplitudes in quantum
mechanics.

Scalar fields in manifolds with boundaries have been extensively studied. For
instance, Dirichlet boundary conditions on a $(D-1)$-dimensional surface
$\Sigma$ can be modelled on the whole $\mathbb{R}^{D}$ through the coupling
$\lambda\delta_{\Sigma}(x)$ to a delta-function with support on $\Sigma$: in
the limit of infinite coupling $\lambda\rightarrow\infty$ one reproduces
Dirichlet boundary conditions. This approach was introduced in the worldline
context in GiesLangfeldMoyaerts  (for a similar mechanism for Neumann boundary
conditions, see FoscoLombardoMazzitelli ). However, this procedure is
generally not suitable for usual perturbative calculations: indeed, such
procedures would lead to an expansion in positive powers of $\lambda$ and the
limit $\lambda\rightarrow\infty$ usually appears to be ill-defined (for a
strong coupling approach involving Padé approximants, see
EdwardsGonzalezDominguezHuetTrejo ). An exception occurs for the free scalar
field in flat space, where a resummation that leads to the correct heat-trace
expansion for the Dirichlet propagator in the limit of infinite coupling is
possible FranchinoMazzitelli  (for a similar resummation involving the Neumann
propagator, see AhmadiniazFranchinoManzoMazzitelli ). If instead of the free
scalar field one considers a potential $V(x)$, then the aforementioned
resummation for the free Dirichlet propagator can be used to compute the
contribution to the heat-trace at different powers of $V(x)$, even for strong
coupling Franchino . When the coupling constant is left finite, the delta-
function coupling to the scalar field reproduces a semitransparent mirror,
which was studied in the context of the worldline formalism in FranchinoPisani
.

A rather different approach to the problem of a bounded field in the worldline
formalism involves the use of image charges. This method was used in
BastianelliCorradiniPisani2007 ; BastianelliCorradiniPisaniSchubert  for a
scalar field with either Dirichlet or Neumann boundary conditions on the
$D$-dimensional half-space $M=\mathbb{R}^{D-1}\times\mathbb{R}^{+}$ limited by
an infinite flat hyperplane. A generalization of the Neumann case to include
Robin boundary conditions was later introduced in
BastianelliCorradiniPisani2008  for the same manifold. This technique strongly
relies on the fact that the boundary is flat, so for other types of geometries
an adaptation is necessary. In this context, the case of a scalar field
confined to a $D$-dimensional ball $B^{D}$ (that is, the interior of the
hypersphere $S^{D-1}$) was analysed for Dirichlet and Neumann boundary
conditions in CorradiniEdwardsHuetManzoPisani , where a conformal
transformation was performed to map the ball into the half-space. This same
analysis was extended to include Robin boundary conditions in TrabajoDiploma .

In the present work, a similar procedure in terms of image charges is
performed for the case of a spinor field minimally coupled to an Abelian gauge
field, confined inside certain smooth $2$-dimensional curved manifolds $M$
under MIT bag boundary conditions.111In some contexts, these are also known as
Berry-Mondragón boundary conditions, or infinite-mass boundary conditions. The
procedure, which singles out in the path integral the contributions of
worldlines which reach the boundary from those which lie entirely in the bulk,
allows one to determine both diagonal and off-diagonal heat-kernel elements.
The first ones correspond to closed worldlines within $M$ that allow for the
computation of the heat-trace asymptotics —and, thus, functional
determinants—, while the second ones correspond to open worldlines that are
useful for the computation of tree-level $n$-point functions within the
context of the worldline formalism. Besides constructing a representation for
the heat-kernel useful for either closed or open paths, the former scenario is
considered in this article for the computation of the first heat-trace
asymptotic coefficients (Seeley-DeWitt coefficients).

The reason for considering MIT bag boundary conditions is twofold. On the
practical side, these local conditions imply that half of the spinor
components satisfy a Dirichlet boundary condition while the other half satisfy
a specific Robin boundary condition, and both types of conditions where
previously analysed in the context of the worldline formalism for scalar
fields. Other local conditions for fermions do not lead to a Robin boundary
condition but instead to a more complicated first-order condition, such as the
chiral bag boundary condition for which half of the spinor components satisfy
a condition involving tangential derivatives, which have not so far been
applied within a worldline framework. On the other hand, MIT bag boundary
conditions are used in a vast number of physical situations: since they where
introduced in ChodosJaffeJohnsonThornWeisskopf  as part of an extended model
for hadrons, they have also been applied to other contexts such as gauged
supergravity BreitenlohnerFreedman , fermionic billiards BerryMondragon ,
superstrings SheikhJabbari , spinning membranes LuckockMoss  and graphene
devices BeneventanoSantangelo . It is expected that an analysis of MIT bag
boundary conditions in terms of the worldline formalism will allow the
implementation of worldline tools in these areas.

The procedure carried in the present article begins by representing the
manifold $M$ in coordinates that span the entire half-plane
$\mathbb{R}\times\mathbb{R}^{+}$. In this coordinate representation, the
manifold presents a certain metric $g_{\mu\nu}$, and the boundary $\partial M$
is represented via the straight line $\mathbb{R}$. The next step is to
duplicate $M$ to build up another region $\widetilde{M}\approx\mathbb{R}^{2}$
by reflecting the half-plane through its boundary and endowing the resulting
full plane with the symmetric extension of the original metric. The curvature
of the region $\widetilde{M}$ is different from the curvature of $M$ because
the symmetric extension introduces a Heaviside-function on the metric, which
is thus non-smooth at the interface $\mathbb{R}$. Besides, path integration of
a point-like particle on curved space corresponds to a $0+1$ sigma model which
requires certain counterterms —specific to each regularization— that are
necessary to maintain general coordinate invariance GreenBook . In particular,
the counterterm required by time-slicing renormalization contains a term
proportional to the curvature of $\widetilde{M}$, which is given by a delta-
function with support at the interface. As a consequence, the computation of
the heat-trace in these coordinates amounts to obtaining the point-particle
expectation values of combinations of delta- and Heaviside-functions. Besides
these curvature considerations, a symmetric extension of the tangential
component of the gauge field in $\widetilde{M}$ is performed, as well as an
antisymmetric extension of its normal component. The latter component is made
continuous through a gauge fixing condition.

Image charges are used to separate “direct” and “indirect” contributions to
the transition amplitude, according to whether the end-point of the trajectory
lies in the physical region $M\subset\widetilde{M}$ or not. Finally, to
illustrate the whole procedure, the leading direct and indirect contributions
to the heat-trace are computed —which correspond to the volumes of $M$ and its
boundary $\partial M$— as well as the next-to-leading contribution to obtain
the Seeley-DeWitt coefficient $a_{2}$ which gives the trace anomaly. A
worldline formulation in phase space is employed for this application.

The organisation of the article is as follows. In Section 2 the relation
between the heat-trace and the effective action for a quantum
$\frac{1}{2}$-spin field coupled to an electromagnetic background is
presented. The manifold $M$ and the construction of the doubled manifold
$\widetilde{M}$, along with some geometrical properties, is described in
Section 3. After this, geometrical and electromagnetic quantities in $M$ must
be “reflected” appropriately to $\widetilde{M}$, which is done in Section 4.
This completes the setup in terms of image quantities, from which path
integration can be constructed in $\widetilde{M}$. Since the field is
fermionic, this path integration must take the Dirac gamma matrices into
account. For this purpose, coherent states are chosen in the present article,
so they are reviewed in Section 5. Next, the transition amplitudes for a
fermionic point-particle in curved manifolds are introduced in Section 6,
using path integrals in a phase space representation. This completes the setup
for using first-quantization path integrals to compute quantities in a
fermionic Quantum Field Theory in manifolds with boundaries. This whole
structure is then used considering a specific boundary condition: the MIT bag
ones. For completeness, they are described briefly in Section 7. Then Section
8 contains the main result of this article, where an ansatz for a fermionic
heat-kernel in $M$ under MIT bag boundary conditions is presented. As an
application of this construction, the first three Seeley-DeWitt coefficients
of the heat-trace are computed in Section 9, showing perfect consistency with
previous results. Finally, some considerations on the applications of this
construction are drawn in Section 10, with particular emphasis on extensions
to other boundary conditions for fermions and other kinds of fields, and also
the implementations of the procedure to worldline numerical computations.

##  2 Effective action

Consider a spin$-\frac{1}{2}$ field $\Psi$ of mass $m$ confined to a
$D-$dimensional Euclidean manifold $x\in M$, minimally coupled to gravity and
to an Abelian gauge field. The action reads

$S[A_{\mu}]=\int_{M}d^{D}x\,\sqrt{g}\,\Psi^{\dagger}\big{(}-i\not{D}-im\big{)}\Psi\;,$
(1)

where $g$ is the determinant of the metric in $M$ and222The convention
established for this article is that Greek letters from the middle of the
alphabet (such as $\mu$, $\nu$, $\rho$) denote curved indices in the bulk,
while Latin letters from the middle of the alphabet (such as $i$, $j$, $k$)
denote flat indices in the bulk. Also, Greek letters from the beginning of the
alphabet (such as $\alpha$, $\beta$, $\gamma$) denote curved indices in the
boundary, while Latin letters from the beginning of the alphabet (such as $a$,
$b$, $c$) denote flat indices in the boundary. For a $D-$dimensional manifold
with boundaries, bulk indices range from $1$ to $D$ and boundary indices range
from $1$ to $D-1$..

$-i\not{D}=-i\,e^{\mu}_{k}\,\gamma^{k}\,D_{\mu}\;,\qquad
D_{\mu}=\partial_{\mu}+\frac{1}{4}\omega_{\mu
ij}\gamma^{i}\,\gamma^{j}+iA_{\mu}\;,$ (2)

is the (massless) Dirac operator. It contains an Abelian gauge connection
$A_{\mu}$ as well as a spin connection

$\omega_{\mu
ij}=-\delta_{ik}e^{\nu}_{j}\big{[}(\partial_{\mu}e^{k}_{\nu})-\Gamma^{\rho}_{\mu\nu}e^{k}_{\rho}\big{]}\;,$
(3)

where $e^{\mu}_{i}$ and $e^{i}_{\mu}$ are vielbeins such that
$g_{\mu\nu}=e^{i}_{\mu}e^{j}_{\nu}\delta_{ij}$,
$\delta^{ij}=e^{i}_{\mu}e^{j}_{\nu}g^{\mu\nu}$ and
$e^{\mu}_{i}=g^{\mu\nu}\delta_{ij}e_{\nu}^{j}$. The (flat) Dirac matrices
$\gamma^{i}$ are constant and obey the Clifford algebra
$\\{\gamma^{i},\gamma^{j}\\}=2\delta^{ij}$. In the present work an even-
dimensional Euclidean manifold is considered, so Dirac matrices satisfy
$(\gamma^{i})^{\dagger}=\gamma^{i}$ and the chiral Dirac matrix
$\gamma^{\text{ch}}=(-1)^{D/2}\,i\gamma^{1}\dots\gamma^{D}$ is well-defined.

The effective action up to one-loop order is

$\displaystyle\Gamma[A_{\mu}]$
$\displaystyle=S[A_{\mu}]+\hbar\log\text{Det}\big{(}-i\not{D}-im\big{)}$ (4)
$\displaystyle=S[A_{\mu}]+\frac{\hbar}{2}\log\text{Det}\big{(}-\not{D}^{2}+m^{2}\big{)}\;.$
(5)

Using Schwinger’s proper time regularization Schwinger , the one-loop
correction becomes

$\frac{1}{2}\log\text{Det}\big{(}-\not{D}^{2}+m^{2}\big{)}=-\frac{1}{2}\int_{0}^{\infty}\frac{dT}{T}\,e^{-Tm^{2}}\,\text{Tr}\,e^{-T(-\not{D}^{2})}\;,$
(6)

which represents the divergent functional determinant in terms of the heat-
trace of $-\not{D}^{2}$.

By means of the Schrödinger-Lichnerowicz formula Schrodinger ; Lichnerowicz ,
the Dirac operator squared can be written as a coordinate operator in the form

$-\not{D}^{2}=-\frac{1}{\sqrt{g}}D_{\mu}\big{(}\sqrt{g}g^{\mu\nu}D_{\nu}\big{)}-\frac{i}{4}[\gamma^{i},\gamma^{j}]e^{\mu}_{i}e^{\nu}_{j}F_{\mu\nu}+\frac{1}{4}R\;,$
(7)

where $R$ is the scalar curvature of the manifold and
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor. In this form it is clear that $-\not{D}^{2}$ is
a Laplace-type operator with the fully covariant333Here, “fully covariant”
means that the Laplacian is written in terms of covariant derivatives with two
types of connections: a curvature connection (the spin connection in this
case) and also a gauge connection (which for the present article only includes
the Abelian gauge potential). Laplacian

$\Delta=-\frac{1}{\sqrt{g}}D_{\mu}\big{(}\sqrt{g}g^{\mu\nu}D_{\nu}\big{)}\;.$
(8)

Under quite general conditions, the heat-trace of $-\not{D}^{2}$ admits the
following asymptotic expansion:

$\text{Tr}\,e^{-T(-\not{D}^{2})}\sim\frac{1}{(4\pi
T)^{D/2}}\sum_{n=0}^{\infty}a_{n}(-\not{D}^{2},M)\,T^{n/2}\;,$ (9)

where the Seeley-DeWitt coefficients $a_{n}(-\not{D}^{2},M)$ can be computed
in terms of invariants that only contain the gauge field $A_{\mu}$, geometric
elements of $M$ and its boundary $\partial M$, and parameters appearing in the
boundary conditions. Expressions (6) and (9) show that the coefficients
$a_{n}(-\not{D}^{2},M)$ with $0\leq n\leq D$ give the one-loop divergences of
the effective action $\Gamma[\Psi,\Psi^{\dagger}]$, while the coefficients
with $n>D$ contribute to the finite part of $\Gamma[\Psi,\Psi^{\dagger}]$. In
particular, the coefficient $a_{0}(-\not{D}^{2},M)\sim\text{Vol}(M)$ is given
only by the volume of the manifold $M$ and does not depend on any other
geometric property of space. Similarly,
$a_{1}(-\not{D}^{2},M)\sim\text{Vol}(\partial M)$ depends only on the volume
of the boundary $\partial M$ as well as the boundary condition
HeatKernelManual .

This is just one example of the applications of heat-kernel techniques to the
perturbative study of quantum field theories. In this article it is shown how
to compute the coefficients $a_{n}(-\not{D}^{2},M)$ for certain smooth
2-dimensional manifolds using worldline techniques.

##  3 Geometry of $M$ and the ‘doubled’ manifold $\widetilde{M}$

In Section 6 it is shown that the heat-trace asymptotics (9) for the Laplace-
type operator $-\not{D}^{2}$ in $M$ is determined by the path integral over
closed trajectories of a non-relativistic particle. In order to study the
dynamics of this particle, it is convenient to identify $M$ with a
$2$-dimensional half-plane which is then embedded into a whole plane
$\mathbb{R}^{2}$, denoted $\widetilde{M}$, that represents two copies of the
original manifold $M$ glued together along the interface $\partial M$ as in
Figure 1.

(a)

(b)

Figure 1: Worldlines in two dimensions. Picture 1(a) shows a worldline in the
manifold $M$ from initial point $\bullet$ (black) to final point $\bullet$
(green), hitting the boundary once in the intermediate point $\bullet$ (blue).
Picture 1(b) represents $\widetilde{M}$, where the boundary turns into an
interface. Point $\bullet$ (pink) in the lower half-plane is the reflection of
$\bullet$ (green) along this interface. A typical curve from $\bullet$ (black)
to $\bullet$ (green) that hits the boundary once at $\bullet$ (blue) has a
corresponding curve from $\bullet$ (black) to $\bullet$ (pink): the segment
from $\bullet$ (black) to $\bullet$ (blue) remains the same, while the segment
from $\bullet$ (blue) to $\bullet$ (green) gets symmetrically reflected from
$\bullet$ (blue) to $\bullet$ (pink). The contribution of the original curve
is called ‘direct’, and the contribution of the second one is called
‘indirect’. Both curves measure equally.

Hence, consider a smooth two-dimensional manifold representable in half-plane
coordinates $(x,y)$ such that $-\infty<x<\infty$ and $0\leq y<\infty$. In
particular, $y=0$ represents the boundary $\partial M$. Let $g_{\mu\nu}$ be
the metric in this coordinate representation, which will be considered as non-
singular and diagonal, so in its matrix form one can write

$g_{\mu\nu}=\begin{pmatrix}h&0\\\ 0&g/h\end{pmatrix}$ (10)

where both $h\equiv h(x,y)$ and $g\equiv g(x,y)$ are non-singular functions
without zeros. Both of them have a straightforward geometrical interpretation:
on one hand, when evaluated at $y=0$, the function $h(x,0)$ is the metric of
the boundary $\partial M$, so $h(x,y)$ is an extension of $h(x,0)$ to the bulk
of $M$. On the other hand, $g=\det\,g_{\mu\nu}$.

Next, extend this metric to the whole plane $\mathbb{R}^{2}$ by making a
symmetric reflection with respect to the line $y=0$. This “doubled manifold”,
which is denoted as $\widetilde{M}$, has metric

$\widetilde{g}_{\mu\nu}(x,y)=g_{\mu\nu}(x,|y|)\;.$ (11)

note that the doubled metric may no longer be analytic for it may have a
discontinuous normal derivative at the fixed points $y=0$. The corresponding
integration measure is

$\sqrt{\widetilde{g}(x,y)}=\sqrt{g(x,|y|)}\;.$ (12)

Denoting objects computed in $\widetilde{M}$ with a tilde “ $\,\widetilde{}\,$
”, one can express the Christoffel symbols as

$\widetilde{\Gamma}^{\rho}_{\mu\nu}=\Gamma^{\rho}_{\mu\nu}-\theta(-y)\,g^{\rho\sigma}\Big{[}\delta^{2}_{\mu}\,\partial_{2}g_{\nu\sigma}+\delta^{2}_{\nu}\,\partial_{2}g_{\mu\sigma}-\delta^{2}_{\sigma}\,\partial_{2}g_{\mu\nu}\Big{]}$
(13)

where $\theta(y)$ is the Heaviside step function. In this and all subsequent
geometrical expressions –unless something different is explicitly stated–,
every object computed in $\widetilde{M}$ is evaluated at $(x,y)$ and objects
computed in $M$ are evaluated at $(x,|y|)$. The scalar curvature and the full
contraction of two Christoffel symbols are

$\displaystyle\widetilde{R}$
$\displaystyle=R-\frac{2}{g}\delta(y)(\partial_{2}h)\;,$ (14)
$\displaystyle\widetilde{\Gamma}^{2}$
$\displaystyle\equiv\widetilde{g}^{\mu\nu}\widetilde{\Gamma}^{\sigma}_{\mu\rho}\widetilde{\Gamma}^{\rho}_{\nu\sigma}=g^{\mu\nu}\Gamma^{\sigma}_{\mu\rho}\Gamma^{\rho}_{\nu\sigma}\equiv\Gamma^{2}\;.$
(15)

note that the difference between the scalar curvatures of $M$ and
$\widetilde{M}$ has support at the boundary.

Path integration in curved manifolds requires the introduction of an
additional counterterm potential to ensure coordinate invariance. In the
present manuscript, phase space path integration will be used, which in curved
spaces without boundaries is suitably described in terms of a Time Slicing
formulation which involves the counterterm GreenBook

$\Delta H_{TS}=-\frac{1}{4}\widetilde{R}+\frac{1}{4}\widetilde{\Gamma}^{2}\;.$
(16)

Time Slicing construction for the computation of transition amplitudes is
suitably defined in the manifold $\widetilde{M}$, which has no boundaries.
Therefore, the idea of the present article is to use path integrals in
$\widetilde{M}$ and relate them to the transition amplitudes in $M$ under MIT
bag boundary conditions.

Going back to $M$, one needs to consider the inward-pointing normal unit
vector $n^{\mu}$ at the boundary and extend it in some way to the interior of
the manifold. From the metric (10) it follows

$n_{\mu}\big{|}_{\partial M}=\sqrt{g/h}\Big{|}_{\partial
M}\,\delta^{2}_{\mu}\;.$ (17)

As for the bulk of the manifold, a parallel transport extension along
geodesics normal to the boundary will be considered. Therefore, $n_{\mu}$
obeys in $M$ the parallel transport equation

$n^{\nu}\nabla_{\nu}n^{\mu}=0\;.$ (18)

The second fundamental form in $\partial M$ has a single component $L_{11}$
and is given by

$L_{11}\equiv-\nabla_{1}n_{1}\big{|}_{\partial
M}=-\sqrt{g/h}\,\Gamma^{2}_{11}\Big{|}_{\partial
M}=-\frac{1}{2}\sqrt{h/g}\,(\partial_{2}h)\Big{|}_{\partial M}\;.$

Then the extrinsic curvature $L$, which is defined as the trace of the second
fundamental form and is an invariant in $\partial M$, is

$L\equiv
g^{11}L_{11}=-\frac{1}{2}\frac{1}{\sqrt{h\,g}}\,(\partial_{2}h)\Big{|}_{\partial
M}\;.$ (19)

Then (14) becomes

$\widetilde{R}=R+4L\sqrt{\frac{h}{g}}\,\delta(y)\;,$ (20)

which is an expression relating scalar invariants in $M$, $\partial M$ and
$\widetilde{M}$.

Going back to $M$, one needs to analyse the vielbeins $e^{i}_{\mu}$ that are
present in the Dirac operator squared, both explicitly an implicitly through
the spin connection $\omega_{\mu ij}$. In the present article, the following
choice will be used to set the vielbeins: at any point in $M$, let
$e^{i=2}_{\mu}$ be equal to the normal unit vector $n_{\mu}$. It follows that
$e^{i=1}_{\mu}$ must be equal to a unit vector perpendicular to $n_{\mu}$.
This leads to a sign (directional) ambiguity that is solved by picking the
direction that obeys the following 2-dimensional identity:

$\text{det }e^{i}_{\mu}=\sqrt{g}\;.$ (21)

At this point it must be remarked that with the aforementioned vielbein
choice, every geometrical element in (7) must be non-singular. This ensures
that the transition amplitudes to be constructed are well-defined in the
entire manifold $M$.

Finally, to simplify expressions in two dimensions one could use the
antisymmetry of the spin connection with respect to the flat indices to define
$\omega_{\mu}\equiv\omega_{\mu 12}$ (‘$1$’ and ‘$2$’ are flat indices in this
definition), which from (3) can be written as
$\omega_{\mu}=-e^{\nu}_{i=2}\nabla_{\mu}e^{i=1}_{\nu}=e^{i=1}_{\nu}\nabla_{\mu}e^{\nu}_{i=2}$.
This definition implies $n^{\mu}\omega_{\mu}=0$, as one can easily see using
(18) and the vielbein choice discussed in the previous paragraph. In
particular, this means that $\omega_{2}|_{y=0}=0$.

##  4 Symmetry of functions and operators reflected to the lower half-plane

In the spirit of the method of images, one expects paths in the upper half-
plane to have the same measure as reflected paths in the lower half-plane. To
ensure this, several considerations about how to extend functions and
operators must be taken into account. This task is carried out in the present
Section. Firstly, the extension of geometrical quantities in $M$ such as the
metric $g_{\mu\nu}$, the normal vector $n^{\mu}$, the Riemann covariant
derivative $\nabla_{\mu}$ and the spin connection $\omega_{\mu ij}$ is
performed. Secondly, electromagnetic quantities such as the background gauge
field $A_{\mu}$ and the field tensor $F_{\mu\nu}$ are taken in consideration
by analysing the geometry of gauge invariance. The results obtained lead to a
natural extension of operators that depend on this geometrical and
electromagnetic factors.

###  4.1 Reflection of geometrical quantities

If paths in the upper half-plane and their reflected paths in the lower half-
plane must have the same value, then the line element
$ds=g_{\mu\nu}x^{\mu}x^{\nu}$ must be the same in both half-planes. For this
purpose, the metric $g_{\mu\nu}$ was extended symmetrically. In mathematical
form and using (10), this reads

$\displaystyle g(x,y)$ $\displaystyle=g(x,-y)\;,$ (22) $\displaystyle h(x,y)$
$\displaystyle=h(x,-y)\;,$

where this extension means both functions have domain in
$\widetilde{M}$.444Unless otherwise stated, the domain of every function $f$
in the present Section is $\widetilde{M}$, and is an extension of the
corresponding function $f$ defined in $M$. They should not be confused with
the function $\widetilde{f}$, whose domain is also $\widetilde{M}$. For
example, the scalar curvature $R$ with domain in $\widetilde{M}$ is obtained
by first computing $R$ in $M$ (that is, computing the corresponding
derivatives of $g_{\mu\nu}$ in $M$) and then extending the result
symmetrically to $\widetilde{M}$, whereas $\widetilde{R}$ is obtained by first
extending the metric $g_{\mu\nu}$ symmetrically to $\widetilde{M}$ and then
computing the corresponding derivatives. To ensure that the lower half-plane
is a physically equivalent reflection of the upper half-plane, every spinor in
the theory is chosen to be reflected symmetrically as well:

$\displaystyle\Psi(x,y)$ $\displaystyle=\Psi(x,-y)\;,$ (23)
$\displaystyle\Psi^{\dagger}(x,y)$ $\displaystyle=\Psi^{\dagger}(x,-y)\;.$

Next, consider the partial derivative operator $\partial_{\mu}$ acting on
spinors. Since the spinors are reflected symmetrically, their partial
derivative in the $y$-direction should be antisymmetrical:

$\displaystyle\partial_{1}|_{y>0}$ $\displaystyle=\partial_{1}|_{y<0}\;,$ (24)
$\displaystyle\partial_{2}|_{y>0}$ $\displaystyle=-\partial_{2}|_{y<0}\;.$

From metric (10) one can see that if an even (odd) number of indices in the
Christoffel symbol are equal to 2, then the Christoffel symbol is proportional
to a derivative in the $x$-direction ($y$-direction) of a metric component.
Then, since the metric is reflected symmetrically, one gets

$\displaystyle\Gamma^{\rho}_{\mu\nu}(x,y)=\Gamma^{\rho}_{\mu\nu}(x,-y)$
$\displaystyle\qquad\text{if an even number of indices are equal to 2}\;,$
(25) $\displaystyle\Gamma^{\rho}_{\mu\nu}(x,y)=-\Gamma^{\rho}_{\mu\nu}(x,-y)$
$\displaystyle\qquad\text{if an odd number of indices are equal to 2}\;,$

which leads to

$\displaystyle\nabla_{1}|_{y>0}$ $\displaystyle=\nabla_{1}|_{y<0}\;,$ (26)
$\displaystyle\nabla_{2}|_{y>0}$ $\displaystyle=-\nabla_{2}|_{y<0}\;.$

Reflections (25) also imply that the scalar curvature $R$ and $\Gamma^{2}$ are
reflected symmetrically.

Moving on to the normal unit vector $n^{\mu}$, the chosen extension consist in
symmetrizing the component $n^{2}$ and antisymmetrizing the component $n^{1}$,
which has the advantage that the normal vector is not discontinuous at the
interface $y=0$. In mathematical form,

$\displaystyle n^{1}(x,y)$ $\displaystyle=-n^{1}(x,-y)\;,$ (27) $\displaystyle
n^{2}(x,y)$ $\displaystyle=n^{2}(x,-y)\;.$

Since $n_{\mu}=e^{i=2}_{\mu}$ in the upper half-plane, the same vielbein
choice can be set for the lower half-plane, and then

$\displaystyle e^{i=2}_{\mu=1}(x,y)$ $\displaystyle=-e^{i=2}_{\mu=1}(x,-y)\;,$
(28) $\displaystyle e^{i=2}_{\mu=2}(x,y)$
$\displaystyle=e^{i=2}_{\mu=2}(x,-y)\;.$

Retaining (21) in the lower half-plane leads to the reflection of
$e^{i=1}_{\mu}$ according to

$\displaystyle e^{i=1}_{\mu=1}(x,y)$ $\displaystyle=e^{i=1}_{\mu=1}(x,-y)\;,$
(29) $\displaystyle e^{i=1}_{\mu=2}(x,y)$
$\displaystyle=-e^{i=1}_{\mu=2}(x,-y)\;.$

Recalling the expression
$\omega_{\mu}=-e^{\nu}_{i=2}\nabla_{\mu}e^{i=1}_{\nu}$ and using (26), (28)
and (29) it follows

$\displaystyle\omega_{1}(x,y)$ $\displaystyle=-\omega_{1}(x,-y)\;,$ (30)
$\displaystyle\omega_{2}(x,y)$ $\displaystyle=\omega_{2}(x,-y)\;.$

###  4.2 Reflection of electromagnetic quantities

Consider first the local Abelian gauge transformation of the spinor field

$\Psi(x,y)\rightarrow e^{i\alpha(x,y)}\Psi(x,y)\;.$ (31)

If the symmetry property (23) is meant to be satisfied for an arbitrary gauge,
it follows that

$\alpha(x,y)=\alpha(x,-y)\;.$ (32)

Now let $v^{\mu}$ be an arbitrary unit vector. Since (31) is a local
transformation, $\Psi(x,y)$ and $\Psi(x+\varepsilon v^{1},y+\varepsilon
v^{2})$ transform differently for $\varepsilon\neq 0$. Therefore, a partial
derivative is not a very sensible way to compare both quantities because it
depends on the difference $\Psi(x+\varepsilon v^{1},y+\varepsilon
v^{2})-\Psi(x,y)$. To construct a gauge covariant derivative
$\mathcal{D}_{\mu}$, one then introduces a comparator $U(x+\varepsilon
v^{1},y+\varepsilon v^{2};x,y)$ that transforms under

$U(x+\varepsilon v^{1},y+\varepsilon v^{2};x,y)\rightarrow
e^{i\alpha(x+\varepsilon v^{1},y+\varepsilon v^{2})}U(x+\varepsilon
v^{1},y+\varepsilon v^{2};x,y)e^{-i\alpha(x,y)}\;,$ (33)

so both $\Psi(x+\varepsilon v^{1},y+\varepsilon v^{2})$ and $U(x+\varepsilon
v^{1},y+\varepsilon v^{2};x,y)\Psi(x,y)$ obey the same transformation law. At
zero separation $\varepsilon=0$, one sets $U(x,y;x,y)=1$. Then one defines the
gauge covariant derivative in the direction of $v^{\mu}$ as

$v^{\mu}\mathcal{D}_{\mu}\Psi(x,y)=\lim_{\varepsilon\rightarrow
0}\frac{1}{\varepsilon}\Big{[}\Psi(x+\varepsilon v^{1},y+\varepsilon
v^{2})-U(x+\varepsilon v^{1},y+\varepsilon v^{2};x,y)\Psi(x,y)\Big{]}\;.$ (34)

Considering $U(x+\varepsilon v^{1},y+\varepsilon v^{2};x,y)\Psi(x,y)$ as a
spinor, it must satisfy the reflection condition (23). Then

$U(x+\varepsilon v^{1},y+\varepsilon v^{2};x,y)=U(x+\varepsilon
v^{1},-y-\varepsilon v^{2};x,-y)\;.$ (35)

Therefore

$v^{\mu}\mathcal{D}_{\mu}\Psi(x,-y)=\tilde{v}^{\mu}\mathcal{D}_{\mu}\Psi(x,y)$
(36)

where $(\tilde{v}^{1},\tilde{v}^{2})=(v^{1},-v^{2})$. In particular, setting
$v^{\mu}\propto\delta^{\mu}_{1}$ or $v^{\mu}\propto\delta^{\mu}_{2}$ one gets

$\displaystyle\mathcal{D}_{1}|_{y>0}$
$\displaystyle=\mathcal{D}_{1}|_{y<0}\;,$ (37)
$\displaystyle\mathcal{D}_{2}|_{y>0}$
$\displaystyle=-\mathcal{D}_{2}|_{y<0}\;.$

One can now expand

$U(x+\varepsilon v^{1},y+\varepsilon v^{2};x,y)=1-i\varepsilon
v^{\mu}A_{\mu}(x,y)+\mathcal{O}(\varepsilon^{2})\;,$ (38)

where $A_{\mu}$ is the electromagnetic gauge field, to obtain

$\mathcal{D}_{\mu}=\partial_{\mu}+iA_{\mu}\;.$ (39)

From (35) and (38), or from (39), (37) and (24), one finds

$\displaystyle A_{1}(x,y)$ $\displaystyle=A_{1}(x,-y)\;,$ (40) $\displaystyle
A_{2}(x,y)$ $\displaystyle=-A_{2}(x,-y)\;.$

In a theory where the partial derivative $\partial_{2}A_{2}$ is found, such as
in the Laplacian (8), it is convenient to chose a gauge that ensures the
continuity of $A_{2}$ at $y=0$ to avoid the unnecessary appearance of some
delta functions $\delta(y)$. Therefore, the gauge ‘semi-fixing’555The term
‘semi-fixing’ comes from the fact that (41) is not a condition imposed in the
entire manifold $\widetilde{M}$ but only at the interface $y=0$. condition

$A_{2}(x,0)=0$ (41)

will be imposed throughout this article.

Finally, the electromagnetic tensor field is defined as

$F_{\mu\nu}\Psi(x,y)=-i[\mathcal{D}_{\mu},\mathcal{D}_{\nu}]\Psi(x,y)\;.$ (42)

Since it is antisymmetric in its indices, the only independent component in
two dimensions is $F\equiv F_{12}$. Using (37) it follows

$F(x,y)=-F(x,-y)\;.$ (43)

###  4.3 Reflection of operators

Since the goal of the present manuscript is to express transition amplitudes
in the manifold $M$ (with boundary) in terms of transition amplitudes in the
manifold $\widetilde{M}$ (without boundaries), then every operator
$\hat{O}:\mathcal{H}(M)\rightarrow\mathcal{H}(M)$ defined in the Hilbert space
$\mathcal{H}(M)$ must be somehow extended to a new operator
$\hat{O}:\mathcal{H}(\widetilde{M})\rightarrow\mathcal{H}(\widetilde{M})$
defined in the Hilbert space $\mathcal{H}(\widetilde{M})$.666To simplify the
notation in the rest of the article, the name of the operator $\hat{O}$ has
not been changed after being extended from $\mathcal{H}(M)$ to
$\mathcal{H}(\widetilde{M})$. This extension is written as

$\hat{O}=\begin{cases}\hat{O}^{>}&\text{if }y>0\\\ \hat{O}^{<}&\text{if
}y<0\end{cases}\;,$ (44)

where $\hat{O}^{>}$ is the same operator as
$\hat{O}:\mathcal{H}(M)\rightarrow\mathcal{H}(M)$, and $\hat{O}^{<}$ is
computed by copying $\hat{O}^{>}$ to the lower half-plane and introducing the
reflection properties of Sections 4.1 and 4.2. This will ensure that any
transition amplitude contained within the upper half-plane measure the same as
the reflected transition amplitude contained within the lower half-plane.

To set ideas, consider the Dirac operator squared (7). Using (10) one can
write, in two dimensions,

$\displaystyle-\not{D}^{2}$
$\displaystyle=-\frac{1}{\sqrt{g}}D_{\mu}\big{(}\sqrt{g}g^{\mu\nu}D_{\nu}\big{)}+g^{-1/2}F\,\gamma^{\text{ch}}+\frac{1}{4}R\;,$
(45) $\displaystyle D_{\mu}$
$\displaystyle=\partial_{\mu}+\frac{i}{2}\omega_{\mu}\gamma^{\text{ch}}+iA_{\mu}\;.$
(46)

Of course, this is defined in $\mathcal{H}(M)$. The extension to
$\mathcal{H}(\widetilde{M})$ reads

$\displaystyle-\not{D}^{2}$
$\displaystyle=\begin{cases}(-\not{D}^{2})^{>}&\text{if }y>0\\\
(-\not{D}^{2})^{<}&\text{if }y<0\end{cases}\;,$ (47)
$\displaystyle(-\not{D}^{2})^{>}$
$\displaystyle=-\frac{1}{\sqrt{g}}D_{\mu}\big{(}\sqrt{g}g^{\mu\nu}D_{\nu}\big{)}+g^{-1/2}F\,\gamma^{\text{ch}}+\frac{1}{4}R\;,$
(48) $\displaystyle D_{\mu}$
$\displaystyle=\partial_{\mu}+\frac{i}{2}\omega_{\mu}\gamma^{\text{ch}}+iA_{\mu}\;,$
(49) $\displaystyle(-\not{D}^{2})^{<}$
$\displaystyle=-\frac{1}{\sqrt{g}}\overline{D}_{\mu}\big{(}\sqrt{g}g^{\mu\nu}\overline{D}_{\nu}\big{)}-g^{-1/2}F\,\gamma^{\text{ch}}+\frac{1}{4}R\;,$
(50) $\displaystyle\overline{D}_{\mu}$
$\displaystyle=\partial_{\mu}-\frac{i}{2}\omega_{\mu}\gamma^{\text{ch}}+iA_{\mu}\;.$
(51)

That is, $(-\not{D}^{2})^{<}$ is operationally similar to
$(-\not{D}^{2})^{>}$, which corresponds to the Dirac operator squared in the
upper half-plane, except for the change of sign in the electromagnetic field
tensor $F$ and the spin connection $\omega_{\mu}$.

##  5 Fermionic coherent states

If one aims to consider transition amplitudes of a non-relativistic particle
on the manifold $M$, it is necessary to integrate not only on configuration
space $|x\rangle$ or phase space $|x\rangle\otimes|p\rangle$, but also on the
internal degrees of freedom. For fermions, these degrees of freedom are given
by the different spinor components. Two approaches exist for this purpose,
namely the “Weyl symbol” method BerezinMarinov ; FradkinGitman ;
HenneauxTeitelboim ; AhmadiniazBandaGuzmanBastianelliCorradiniEdwardsSchubert
and the “coherent state” method OhnukiKashiwa ; BordiCasalbuoni ;
HentyHoweTownsend ; vanHolten ; DHokerGagne1 ; DHokerGagne2 ; Bhattacharya .
The latter will be used in this manuscript and goes as follows.777The present
description of the coherent state method follows GreenBook .

Firstly, recall the extension (47) of the Dirac operator squared, written
entirely in terms of the fermionic operators $1$ and $\gamma^{\text{ch}}$.
Instead of representing operators acting on a spinor $\Psi$ through these
matrices, define the operators

$\psi\equiv\frac{1}{2}(\gamma^{1}+i\gamma^{2})\;,\qquad\psi^{\dagger}\equiv\frac{1}{2}(\gamma^{1}-i\gamma^{2})\;,$
(52)

which obey the anticommutation relations $\\{\psi,\psi^{\dagger}\\}=1$ and
$\\{\psi,\psi\\}=\\{\psi^{\dagger},\psi^{\dagger}\\}=0$. Then, instead of
describing the fermionic parts of $\Psi$ through the spinor components
$\Psi_{1}$ and $\Psi_{2}$, one can construct the “vacuum state” $|0\rangle$
and the “excited state” $|1\rangle=\psi^{\dagger}\,|0\rangle$ such that the
vacuum obeys $\psi\,|0\rangle=0$ and $\langle 0|0\rangle=1$. One can now
define

$\displaystyle|\eta\rangle$ $\displaystyle\equiv
e^{-\eta\psi^{\dagger}}|0\rangle\equiv(1-\eta\psi^{\dagger})|0\rangle\equiv|0\rangle-\eta|1\rangle\;,$
(53) $\displaystyle\langle\bar{\eta}|$ $\displaystyle\equiv\langle
0|\,e^{-\psi\bar{\eta}}\equiv\langle 0|(1-\psi\bar{\eta})\equiv\langle
0|-\langle 1|\bar{\eta}\;.$

Both $\eta$ and $\bar{\eta}$ are Grassmann numbers defined in such a way as to
commute with the vacuum state $|0\rangle$ and to anticommute with both
operators $\psi$ and $\psi^{\dagger}$. These states are coherent with respect
to the operators $\psi$ and $\psi^{\dagger}$:

$\displaystyle\psi|\eta\rangle$
$\displaystyle=\eta|\eta\rangle=|\eta\rangle\eta\;,$ (54)
$\displaystyle\langle\bar{\eta}|\psi^{\dagger}$
$\displaystyle=\langle\bar{\eta}|\bar{\eta}=\bar{\eta}\langle\bar{\eta}|\;.$

From these definitions, it can be proved that the trace of any zero-degree
operator $\hat{O}\equiv O(\psi,\psi^{\dagger})$ is given by

$\text{Tr}(\hat{O})=\int
d\eta\,d\bar{\eta}\,e^{\bar{\eta}\eta}\langle\bar{\eta}|\hat{O}|\eta\rangle=\langle
0|\hat{O}|0\rangle+\langle 1|\hat{O}|1\rangle\;,$ (55)

where Grassmann integration obeys

$\displaystyle\int d\eta\,1=\int d\bar{\eta}\,1=0\;,\qquad\int
d\eta\,\eta=\int d\bar{\eta}\,\bar{\eta}=1\;.$ (56)

Finally, the chiral Dirac matrix is identified with the operator

$\gamma^{\text{ch}}=-i\gamma^{1}\gamma^{2}\equiv\psi\psi^{\dagger}-\psi^{\dagger}\psi\equiv
2(\psi\psi^{\dagger})_{A}\;,$ (57)

where the subscript $A$ stands for “antisymmetrized”.

##  6 Transition amplitudes of a fermionic point-particle

The transition amplitudes of a fermionic point-particle in a two-dimensional
manifold are now at hand. Let $H$ be the Hamiltonian of the particle. Now
symmetrize it in terms of the bosonic operators $x$ and $p$ and antisymmetrize
it in terms of the fermionic operators $\psi$ and $\psi^{\dagger}$. This
version of $H$, which is written as
$H_{W}=[g^{\mu\nu}p_{\mu}p_{\nu}]_{S}+V_{W}$, is the “Weyl ordered”
Hamiltonian of the particle, where $V_{W}$ is the corresponding effective
potential, also Weyl ordered.888An arbitrary potential will be considered in
the present Section. The subscript $S$ stands for “symmetrized”.

Consider the evolution of this particle at time $T$ from state
$|x,y,\eta\rangle$, with fermionic coherent number $\eta$ and located at
position $(x,y)$, to state $\langle x^{\prime},y^{\prime},\bar{\eta}|$. For
convenience, the corresponding trajectories will be described as
$x(t)=x_{0}(t)+q^{1}(t)$ and $y(t)=y_{0}(t)+q^{2}(t)$, where $x_{0}(t)$ and
$y_{0}(t)$ consist on the straight line

$x_{0}(t)=x+\frac{t}{T}(x^{\prime}-x)\quad\text{and}\quad
y_{0}(t)=y+\frac{t}{T}(y^{\prime}-y)$ (58)

that connects the point $\big{(}x_{0}(0)=x\,,\,y_{0}(0)=y\big{)}$ with
$\big{(}x_{0}(T)=x^{\prime}\,,\,y_{0}(T)=y^{\prime}\big{)}$, and
$\big{(}q^{1}(t),q^{2}(t)\big{)}$ representing quantum fluctuations under
homogeneous Dirichlet conditions $q^{1}(0)=q^{2}(0)=q^{1}(T)=q^{2}(T)=0$.
Similarly, paths in the coherent space will be described as
$\eta(t)=\eta+\psi(t)$ and $\bar{\eta}(t)=\bar{\eta}+\psi^{\dagger}(t)$, where
the quantum fluctuations obey $\psi(0)=\psi^{\dagger}(T)=0$. Then the
transition amplitudes can be represented in terms of a bosonic path integral
in phase space and a fermionic path integral in coherent-space GreenBook :

$\langle
x^{\prime},y^{\prime},\bar{\eta}\,|\,e^{-TH}\,|x,y,\eta\rangle=\Big{(}\sqrt{g(x,y)}\sqrt{g(x^{\prime},y^{\prime})}\Big{)}^{-1/2}e^{\bar{\eta}\eta}\times\\\
\int\mathcal{D}p\mathcal{D}q\mathcal{D}\psi^{\dagger}\mathcal{D}\psi\;e^{-\int_{0}^{T}dt\big{\\{}g^{\mu\nu}p_{\mu}p_{\nu}-ip_{1}\big{(}\frac{x^{\prime}-x}{T}+\dot{q}^{1}\big{)}-ip_{2}\big{(}\frac{y^{\prime}-y}{T}+\dot{q}^{2}\big{)}+\psi^{\dagger}\dot{\psi}+V_{W}\big{\\}}}\;.$
(59)

In the integrand, both $g^{\mu\nu}$ and $V_{W}$ are evaluated at
$x_{0}(t)+q^{1}(t)$ and $y_{0}(t)+q^{2}(t)$. The effective potential $V_{W}$
is also evaluated at $\eta+\psi(t)$ and $\bar{\eta}+\psi^{\dagger}(t)$.
Evaluating the inverse metric at a fixed point $(x,y)$, which in the present
manuscript will be taken as the initial point, path integrals turn out to be
normalised according to

$\displaystyle\int\mathcal{D}p\mathcal{D}q\,e^{-\int_{0}^{T}dt\big{\\{}g^{\mu\nu}(x,y)\,p_{\mu}p_{\nu}-ip_{\mu}\dot{q}^{\mu}\big{\\}}}$
$\displaystyle=\frac{\sqrt{g(x,y)}}{4\pi T}\;,$ (60)
$\displaystyle\int\mathcal{D}\psi^{\dagger}\mathcal{D}\psi\,e^{-\int_{0}^{T}dt\,\psi^{\dagger}\dot{\psi}}$
$\displaystyle=1\;.$ (61)

In order to keep track of the different powers of the (small) variable $T$, it
is useful to turn to dimensionless quantities: $t\rightarrow Tt$,
$p_{\mu}(t)\rightarrow p_{\mu}(t)/\sqrt{T}$ and
$q^{\mu}(t)\rightarrow\sqrt{T}q^{\mu}(t)$. Next, it is convenient to expand
$g^{\mu\nu}$ in the kinetic factor around the initial point $(x,y)$ as

$\displaystyle g^{\mu\nu}$
$\displaystyle\Big{(}x_{0}(t)+\sqrt{T}q^{1}(t)\,,\,y_{0}(t)+\sqrt{T}q^{2}(t)\Big{)}=g^{\mu\nu}(x,y)+\Delta
g^{\mu\nu}\;,$ (62) $\displaystyle\Delta g^{\mu\nu}$ $\displaystyle\equiv
g^{\mu\nu}\Big{(}x+t(x^{\prime}-x)+\sqrt{T}q^{1}(t)\,,\,y+t(y^{\prime}-y)+\sqrt{T}q^{2}(t)\Big{)}-g^{\mu\nu}(x,y)\;,$

where $\Delta g^{\mu\nu}$ is meant to be expanded in a power series around
$(x,y)$. It is also convenient to shift the momentum variables in the exponent
and complete squares to get rid of the factors $p_{1}\,(x^{\prime}-x)$ and
$p_{2}\,(y-y^{\prime})$. Defining

$\Delta
x^{\mu}=(x^{\prime}-x,y^{\prime}-y)\;,\quad\xi^{2}=g_{\mu\nu}(x,y)\,\Delta
x^{\mu}\,\Delta
x^{\nu}\;\quad\text{and}\quad\pi_{\mu}=p_{\mu}+\frac{i}{2}g_{\mu\nu}(x,y)\,\Delta
x^{\nu}$

one gets

$\langle
x^{\prime},y^{\prime},\bar{\eta}\,|\,e^{-TH}\,|x,y,\eta\rangle=\Big{(}\sqrt{g(x,y)}\sqrt{g(x^{\prime},y^{\prime})}\Big{)}^{-1/2}\,e^{-\xi^{2}/4T}\,e^{\bar{\eta}\eta}\times\\\
\int\mathcal{D}p\mathcal{D}q\mathcal{D}\psi^{\dagger}\mathcal{D}\psi\;e^{-\int_{0}^{1}dt\big{\\{}g^{\mu\nu}(x,y)\,p_{\mu}p_{\nu}+\Delta
g^{\mu\nu}\pi_{\mu}\pi_{\nu}-ip_{\mu}\dot{q}^{\mu}+\psi^{\dagger}\dot{\psi}+T\,V_{W}\big{\\}}}\;,$
(63)

where $\Delta g^{\mu\nu}$ is evaluated according to (62) and $V_{W}$ is
evaluated at $\pi_{\mu}$, $x+t(x^{\prime}-x)+\sqrt{T}q^{1}(t)$,
$y+t(y^{\prime}-y)+\sqrt{T}q^{2}(t)$, $\eta+\psi(t)$ and
$\bar{\eta}+\psi^{\dagger}(t)$.

In order to turn (63) into a computational efficient expression, it is useful
to define the expectation value of any arbitrary functional of the fields as

$\Big{\langle}F[p,q,\psi,\psi^{\dagger}]\Big{\rangle}\\\ =\frac{4\pi
T}{\sqrt{g(x,y)}}\int\mathcal{D}p\mathcal{D}q\mathcal{D}\psi^{\dagger}\mathcal{D}\psi\,e^{-\int_{0}^{1}dt\big{\\{}g^{\mu\nu}(x,y)\,p_{\mu}p_{\nu}-ip_{\mu}\dot{q}^{\mu}+\psi^{\dagger}\dot{\psi}\big{\\}}}F[p,q,\psi,\psi^{\dagger}]\;.$
(64)

The prefactor is chosen to fix the normalization according to $\langle
1\rangle=1$. Then (63) turns into

$\langle
x^{\prime},y^{\prime},\bar{\eta}\,|\,e^{-TH}\,|x,y,\eta\rangle=\frac{e^{\bar{\eta}\eta}}{4\pi
T}\bigg{(}\frac{g(x,y)}{g(x^{\prime},y^{\prime})}\bigg{)}^{1/4}\,e^{-\xi^{2}/4T}\bigg{\langle}e^{-\int_{0}^{1}dt\big{\\{}\Delta
g^{\mu\nu}p_{\mu}p_{\nu}+T\,V_{W}\big{\\}}}\bigg{\rangle}\;.$ (65)

From the bosonic side, the expectation value in (65) represents the phase
space integration over trajectories which satisfy homogeneous Dirichlet
conditions $q^{\mu}(0)=q^{\mu}(1)=0$ in configuration space and no restriction
at all in momentum space. This makes the relevant quadratic operator in the
Gaussian measure of (65) invertible. From the fermionic side, this expectation
value represents the coherent-space integration over trajectories in terms of
the “coherent fields” $\psi(t)$ and $\psi^{\dagger}(t)$ which satisfy
$\psi(0)=\psi^{\dagger}(1)=0$.

Finally, in order to compute the expectation values it is convenient to define
the generating functional as

$Z[j,k,\bar{\kappa},\kappa]\equiv\bigg{\langle}e^{\int_{0}^{1}dt\big{\\{}i\,j_{\mu}q^{\mu}+i\,k^{\mu}p_{\mu}+\bar{\kappa}\psi-\psi^{\dagger}\kappa\big{\\}}}\bigg{\rangle}\;,$
(66)

where $j_{\mu}$, $k^{\mu}$, $\bar{\kappa}$ and $\kappa$ are “six” source
fields —of which $\bar{\kappa}$ and $\kappa$ are Grassmann fields. Completing
squares one finds

$Z[j,k,\bar{\kappa},\kappa]=\exp\bigg{\\{}\\!-\\!\int_{0}^{1}\\!dt\\!\int_{0}^{1}\\!dt^{\prime}\,\bigg{(}\frac{1}{4}g_{\mu\nu}(x,y)\,k^{\mu}(t)k^{\nu}(t^{\prime})+i\,^{\bullet}G(t,t^{\prime})\,k^{\mu}(t)j_{\mu}(t^{\prime})\\\
+g^{\mu\nu}(x,y)\,G(t,t^{\prime})\,j_{\mu}(t)j_{\nu}(t^{\prime})+K(t,t^{\prime})\,\bar{\kappa}(t)\kappa(t^{\prime})\bigg{)}\bigg{\\}}\;,$
(67)

with the Green functions given by

$\displaystyle G(t,t^{\prime})$
$\displaystyle\equiv-\frac{1}{2}|t-t^{\prime}|+\frac{1}{2}(t+t^{\prime})-tt^{\prime}\;,$
(68) ${}^{\bullet}G(t,t^{\prime})$
$\displaystyle\equiv-\frac{1}{2}\epsilon(t-t^{\prime})+\frac{1}{2}-t^{\prime}\;,$
(69) $\displaystyle K(t,t^{\prime})$
$\displaystyle\equiv\frac{1}{2}\epsilon(t-t^{\prime})+\frac{1}{2}\;,$ (70)

where $\epsilon(t-t^{\prime})$ is the sign of $t-t^{\prime}$ with the
convention $\epsilon(0)=0$. Any expectation value of a functional that is
polynomial in the integrated fields can be computed as functional derivatives
of the generating functional in terms of the source fields, with the
successive evaluation at zero source. Other sources give the expectation value
for other specific functionals (for example, see
CorradiniEdwardsHuetManzoPisani  for the computation of the expectation value
of an exponential). In perturbative calculations, the relevant expectation
values can usually be computed using Wick Theorem, for which only the two-
point functions are needed. In the present case:

$\displaystyle\big{\langle}p_{\mu}(t)\,p_{\nu}(t^{\prime})\big{\rangle}$
$\displaystyle=\frac{1}{2}g_{\mu\nu}(x,y)\;,$ (71)
$\displaystyle\big{\langle}p_{\mu}(t)\,q^{\nu}(t^{\prime})\big{\rangle}$
$\displaystyle=i\,^{\bullet}G(t,t^{\prime})\delta^{\nu}_{\mu}\;,$ (72)
$\displaystyle\big{\langle}q^{\mu}(t)\,q^{\nu}(t^{\prime})\big{\rangle}$
$\displaystyle=2\,G(t,t^{\prime})\,g^{\mu\nu}(x,y)\;,$ (73)
$\displaystyle\big{\langle}\psi(t)\,\psi^{\dagger}(t^{\prime})\big{\rangle}$
$\displaystyle=K(t,t^{\prime})\;,$ (74)

and every other two-point function is just zero.

##  7 MIT bag boundary conditions

In this section, a brief introduction to (Euclidean) MIT bag boundary
conditions in a two-dimensional manifold $M$ is provided in terms of Dirac
matrices as well as in terms of coherent states. Following Luckock Luckock1991
, consider the projector

$\Pi_{-}=\frac{1}{2}\big{(}1-i\not{n}\gamma^{\text{ch}}\big{)}$ (75)

whose rank is equal to one-half of the total number of spinor components. For
$D=2$, the rank is then equal to one. The MIT bag boundary conditions are
given by

$\Pi_{-}\Psi|_{\partial M}=0\;.$ (76)

These local conditions are sufficient to ensure the self-adjointness of the
Dirac operator $-i\not{D}$, which is a first-order differential operator.
Equation (76) imposes Dirichlet boundary conditions on one-half of the spinor
components.

When working with the second-order differential operator $-\not{D}^{2}$, an
additional consideration is made as usual: the relevant functional space is
spanned by the eigenfunctions of the Dirac operator $-i\not{D}$. Therefore,
not only does $\Psi$ obey the MIT bag boundary condition but so does
$\not{D}\Psi$, implying $\Pi_{-}\not{D}\Psi|_{\partial M}=0$. Commuting
$\not{D}$ with $\Pi_{-}$ one finds

$\bigg{(}n^{\mu}D_{\mu}-\frac{1}{2}L\bigg{)}\Pi_{+}\Psi|_{\partial M}=0\;,$
(77)

where $L$ is the extrinsic curvature and

$\Pi_{+}=1-\Pi_{-}=\frac{1}{2}\big{(}1+i\not{n}\gamma^{\text{ch}}\big{)}$ (78)

is the projector complementary to $\Pi_{-}$. Hence, while the $\Pi_{-}$
projection of a spinor satisfies the “Dirichlet-like” boundary condition (76),
the complementary $\Pi_{+}$ projection satisfies the “Robin-like” boundary
condition (77). When working with the Laplace-type operator $-\not{D}^{2}$,
both (76) and (77) are used, leading to a set of mixed boundary conditions.
The total number of these boundary conditions now becomes exactly equal to the
total number of spinor components.

The same approach can be repeated in terms of operators acting on coherent
states rather than using Dirac matrices. Focusing on the manifold $M$
represented as a half-plane (whose boundary is the straight line $y=0$) and
recalling the vielbein convention established in Section 3, one finds
$\not{n}=g_{\mu\nu}\,e^{\mu}_{i}n^{\nu}\,\gamma^{i}=\gamma^{2}$. Thus,
defining the difference between the complementary projectors as $\chi$, it
follows

$\chi\equiv\Pi_{+}-\Pi_{-}=i\not{n}\gamma^{\text{ch}}=-\gamma^{1}\equiv-(\psi+\psi^{\dagger})\;.$
(79)

Then the mixed MIT bag boundary conditions (76) and (77) become

$\displaystyle\big{(}1+\psi+\psi^{\dagger}\big{)}|x,y,\eta\rangle\,\Big{|}_{y=0}=0\;,$
(80)
$\displaystyle\bigg{(}n^{\mu}D_{\mu}-\frac{1}{2}L\bigg{)}\big{(}1-\psi-\psi^{\dagger}\big{)}|x,y,\eta\rangle\,\Big{|}_{y=0}=0\;.$
(81)

##  8 Heat-kernel in the manifold $M$

This section summarises the main result of this article: a worldline
representation for the heat-kernel of the Laplace-type operator $-\not{D}^{2}$
on the manifold $M$ expressed as a curved half-plane under MIT bag boundary
conditions. Transition amplitudes (65) in the manifold $\widetilde{M}$ without
boundaries are used for this purpose.

Let

$K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)=\langle
x^{\prime},y^{\prime},\bar{\eta}\,|\,e^{-T(-\not{D})^{2}}\,|x,y,\eta\rangle_{M}$
(82)

be the heat-kernel on $M$, (i.e. the transition amplitudes of a fermionic
point-particle evolving with Hamiltonian $-\not{D}^{2}$). The ansatz proposed
in the present article is

$K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)=\langle
x^{\prime},y^{\prime},\bar{\eta}\,|\,e^{-T\widetilde{H}}\,|x,y,\eta\rangle+\langle
x^{\prime},-y^{\prime},\bar{\eta}\,|\,\chi\,e^{-T\widetilde{H}}\,|x,y,\eta\rangle\;,$
(83)

where the terms in the RHS represent transition amplitudes of a point-particle
in $\widetilde{M}$ evolving with Hamiltonian

$\widetilde{H}(x^{\prime},y^{\prime})=\big{(}\\!-\not{D}^{2}\big{)}^{\\!>}(x^{\prime},|y^{\prime}|)+\Pi_{+}\sqrt{\frac{h}{g}}\,L\,\delta(y^{\prime})\;.$
(84)

In this Hamiltonian, $\big{(}\\!-\not{D}^{2}\big{)}^{\\!>}$ is given by (48)
and represents the Dirac operator squared $-\not{D}^{2}$ as in
$\mathcal{H}(M)$, but symmetrized by being evaluated at
$(x^{\prime},|y^{\prime}|)$. Note that since the RHS of (83) is defined in the
entire double manifold $\widetilde{M}$, then the LHS is defined in
$\widetilde{M}$ as well. Therefore, the heat-kernel in the LHS of (83) is,
formally, an extension of the heat-kernel in the LHS of (82) from $M$ to
$\widetilde{M}$. The first (second) term in the RHS of (83) is referred as a
‘direct’ (‘indirect’) contribution to the heat-kernel.

To make sense of the ansatz (83), first is necessary to specify what is meant
when, for an arbitrary operator $\hat{O}$ defined as in (44), one writes
$\hat{O}K_{M}$. It specifically means that the operator is acting on the final
point of the heat-kernel:

$\hat{O}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)=\langle
x^{\prime},y^{\prime},\bar{\eta}\,|\hat{O}^{>}\,e^{-T\widetilde{H}}\,|x,y,\eta\rangle+\langle
x^{\prime},-y^{\prime},\bar{\eta}\,|\hat{O}^{<}\,\chi\,e^{-T\widetilde{H}}\,|x,y,\eta\rangle\;.$
(85)

For instance, if one considers the projectors $\Pi_{\pm}$ (which satisfy
$\Pi_{\pm}^{>}=\Pi_{\pm}^{<}=\Pi_{\pm}$) then

$\displaystyle\Pi_{\pm}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)$
$\displaystyle=\pm\Pi_{\pm}K_{M}(x,y,\eta;x^{\prime},-y^{\prime},\bar{\eta};T)\;,$
(86) $\displaystyle\Pi_{\pm}\frac{\partial}{\partial
y^{\prime}}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)$
$\displaystyle=\mp\Pi_{\pm}\frac{\partial}{\partial
y^{\prime}}K_{M}(x,y,\eta;x^{\prime},-y^{\prime},\bar{\eta};T)\;.$ (87)

This results are a direct consequence of the factor $\chi$ in the last term of
(83). Hence, the $\Pi_{-}$ ($\Pi_{+}$) projection of the heat-kernel is
antisymmetric (symmetric) in terms of a reflection with respect to the
boundary. This is a desired property of $K_{M}$, because previous research for
scalar fields involving Dirichlet, Neumann and Robin boundary conditions
BastianelliCorradiniPisani2007 ; BastianelliCorradiniPisaniSchubert ;
BastianelliCorradiniPisani2008 ; CorradiniEdwardsHuetManzoPisani ;
TrabajoDiploma  has shown that if Dirichlet (Robin) boundary conditions are
satisfied, then the heat-kernel is antisymmetric (symmetric). Equation (84) is
also inspired by these results for scalar fields: in the literature, the
original Hamiltonian in $\mathcal{H}(M)$ has always been reflected
symmetrically to the lower half-plane and, if Robin boundary conditions where
obeyed, an additional delta term was introduced. The $\Pi_{+}$ projector
accompanying the delta term in (84) ensures that this delta acts only on the
projection that obeys a Robin boundary condition.

To verify if ansatz (83) is indeed equal to the heat-kernel (82) for
$y^{\prime}\geq 0$, it is necessary to check if (83) obeys the following
conditions:

  * •

the heat equation in the bulk (that is, for $y^{\prime}\neq 0$)

$\bigg{(}-\not{D}^{2}+\frac{d}{dT}\bigg{)}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)=0\;,$
(88)

  * •

the Dirichlet boundary condition for the $\Pi_{-}$ projection

$\Pi_{-}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)\big{|}_{y^{\prime}=0}=0\;,$
(89)

  * •

the Robin boundary condition for the $\Pi_{+}$ projection

$\bigg{(}n^{\mu}D_{\mu}-\frac{1}{2}L\bigg{)}\Pi_{+}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)\bigg{|}_{y^{\prime}=0}=0\;,$
(90)

  * •

and the initial condition in the bulk

$K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T=0)=\langle
x^{\prime}|x\rangle\langle
y^{\prime}|y\rangle\langle\bar{\eta}|\eta\rangle=e^{\bar{\eta}\eta}\,\delta(x-x^{\prime})\delta(y-y^{\prime})\;.$
(91)

In the following, all the aforementioned requirements will be proved valid for
(83).

The Dirichlet condition (89) follows trivially from (86). The initial
condition (91) is straightforward: from (83) it is trivial that
$K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T=0)=\langle
x^{\prime}|x\rangle\langle
y^{\prime}|y\rangle\langle\bar{\eta}|\eta\rangle+\langle
x^{\prime}|x\rangle\langle-y^{\prime}|y\rangle\langle\bar{\eta}|\chi|\eta\rangle$,
and since $y^{\prime}>0$ in the bulk, one also finds
$\langle-y^{\prime}|y\rangle=\delta(y+y^{\prime})=0$ for every $y\geq 0$.

To prove the heat equation (88), expression (85) is used with
$\hat{O}=-\not{D}^{2}$, which is given by (47). Then, considering the relation
$(-\not{D}^{2})^{<}\;\chi=\chi\;(-\not{D}^{2})^{>}$ which follows from
properties of the Dirac matrices, one gets

$-\not{D}^{2}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)=\\\ \langle
x^{\prime},y^{\prime},\bar{\eta}\,|\,(-\not{D}^{2})^{>}\;e^{-T\widetilde{H}}\,|x,y,\eta\rangle+\langle
x^{\prime},-y^{\prime},\bar{\eta}\,|\,\chi\;(-\not{D}^{2})^{>}\;e^{-T\widetilde{H}}\,|x,y,\eta\rangle\;.$
(92)

On the other hand,

$-\frac{d}{dT}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)=\langle
x^{\prime},y^{\prime},\bar{\eta}\,|\widetilde{H}\,e^{-T\widetilde{H}}\,|x,y,\eta\rangle+\langle
x^{\prime},-y^{\prime},\bar{\eta}\,|\,\chi\,\widetilde{H}\,e^{-T\widetilde{H}}\,|x,y,\eta\rangle\;.$
(93)

From (92) and (93) it follows

$\bigg{(}-\not{D}^{2}+\frac{d}{dT}\bigg{)}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)=-\sqrt{\frac{h}{g}}\,L\,\delta(y^{\prime})\,\Pi_{+}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)\;.$
(94)

Although (94) is valid in the entire manifold $\widetilde{M}$, note that its
RHS is exactly zero in the bulk (where $y^{\prime}>0$), from which (88) is
obtained.

The final requirement to prove is the Robin condition (90). Firstly, consider
the integral of (94), multiplied by $\sqrt{g}$, with respect to $y^{\prime}$
from $0^{-}$ to $0^{+}$:

$\int_{0^{-}}^{0^{+}}\\!\\!dy^{\prime}\sqrt{g}\bigg{(}-\not{D}^{2}+\frac{d}{dT}\bigg{)}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)+\sqrt{h}\,L\,\Pi_{+}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)\bigg{|}_{y^{\prime}=0}=0\;.$
(95)

Next, note from equation (86) that $\Pi_{-}K_{M}$ is not necessarily
continuous at the interface $y^{\prime}=0$, but as long as the discontinuity
is finite, the integral of $K_{M}$ is zero. The same conclusion is drawn for
any derivative of $K_{M}$ with respect to any variable other than
$y^{\prime}$. Those observations hold even if $K_{M}$ or its derivatives are
multiplied by any function with finite discontinuity. Therefore, (95) becomes

$\int_{0^{-}}^{0^{+}}\\!\\!dy^{\prime}D_{2}\bigg{(}\sqrt{g}\frac{h}{g}D_{2}\bigg{)}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)-\sqrt{h}\,L\,\Pi_{+}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)\bigg{|}_{y^{\prime}=0}=0\;.$
(96)

This integral is solved easily if one considers the fact that
$\omega_{2}|_{y^{\prime}=0}=0$ (see Section 3), the gauge semi-fixing
condition (41) and the symmetry property (87):

$2\frac{h}{\sqrt{g}}\frac{\partial}{\partial
y^{\prime}}\Pi_{+}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)\bigg{|}_{y^{\prime}=0}-\sqrt{h}\,L\,\Pi_{+}K_{M}(x,y,\eta;x^{\prime},y^{\prime},\bar{\eta};T)\bigg{|}_{y^{\prime}=0}=0\;.$
(97)

The Robin condition (90) follows trivially from (97) and (17).

##  9 Application: Seeley-DeWitt coefficients

As an application of the previous construction, the first three Seeley-DeWitt
coefficients $a_{0}$, $a_{1}$ and $a_{2}$ for the asymptotic heat-trace (9)
are computed in the present Section. In two dimensions the coefficient $a_{2}$
represents the (integrated) conformal anomaly of the theory.

The computation will be performed in phase space to avoid the introduction of
ghost fields. Therefore, a phase space representation of the point-particle
Hamiltonian (84) is needed. From the usual identification
$\partial_{\mu}=ig^{1/4}\hat{p}_{\mu}\,g^{-1/4}$, introducing the counterterms
(16), and taking Section 5 into consideration to work in terms of coherent
states instead of Dirac matrices, one finds the Weyl ordered expression

$\displaystyle\widetilde{H}=$
$\displaystyle\Big{[}g^{\mu\nu}(\hat{p}_{\mu}+A_{\mu})(\hat{p}_{\nu}+A_{\nu})\Big{]}_{S}+\frac{1}{4}\omega^{\mu}\omega_{\mu}+\frac{1}{4}\Gamma^{2}-\frac{1}{2}\Big{(}1+\psi+\psi^{\dagger}\Big{)}\,L\sqrt{\frac{h}{g}}\,\delta(y)$
(98)
$\displaystyle+2\,\Big{\\{}\Big{[}\omega^{\mu}(\hat{p}_{\mu}+A_{\mu})\Big{]}_{S}+g^{-1/2}\,F\Big{\\}}\,(\psi\psi^{\dagger})_{A}\;.$

where the subscript $S$ stands for “symmetrized” with respect to the bosonic
operators $\hat{p}_{\mu}$ and $\hat{x}^{\mu}$. The heat-trace then admits the
following expression:

$\text{Tr }K_{M}=\text{Tr }K_{M}^{\text{dir}}+\text{Tr }K_{M}^{\text{ind}}\;,$
(99)

with the ‘direct trace’ $\text{Tr }K_{M}^{\text{dir}}$ and the ‘indirect
trace’ $\text{Tr }K_{M}^{\text{ind}}$ given by

$\displaystyle\text{Tr }K_{M}^{\text{dir}}$
$\displaystyle=\int_{M}dx\,dy\sqrt{g}\int
d\eta\,d\bar{\eta}\,e^{\bar{\eta}\eta}\langle
x,y,\bar{\eta}\,|\,e^{-T\widetilde{H}}\,|x,y,\eta\rangle\;,$ (100)
$\displaystyle\text{Tr }K_{M}^{\text{ind}}$
$\displaystyle=\int_{M}dx\,dy\sqrt{g}\int
d\eta\,d\bar{\eta}\,e^{\bar{\eta}\eta}\langle
x,-y,\bar{\eta}\,|\,\chi\,e^{-T\widetilde{H}}\,|x,y,\eta\rangle\;.$ (101)

The computation for each contribution will be carried out in two steps: first,
a general non-perturbative expression will be obtained by solving the
Grassmann trace. Then, a perturbative calculation up to order $T^{0}$ will
provide the SDW coefficients $a_{0}$, $a_{1}$ and $a_{2}$.

###  9.1 Direct trace

From Sections 6 and 8 it is now known that one can write (100) as

$\text{Tr }K_{M}^{\text{dir}}=\frac{1}{4\pi T}\int_{M}dx\,dy\sqrt{g}\int
d\eta\,d\bar{\eta}\,e^{2\bar{\eta}\eta}\Big{\langle}e^{-\int_{0}^{1}dt\,\widetilde{H}_{\text{dir}}}\Big{\rangle}$
(102)

where the exponent in the mean value is given by

$\displaystyle\widetilde{H}_{\text{dir}}$
$\displaystyle=\Big{[}g^{\mu\nu}\big{(}x+\sqrt{T}q^{1},\big{|}y+\sqrt{T}q^{2}\big{|}\big{)}-g^{\mu\nu}(x,y)\Big{]}p_{\mu}\,p_{\nu}+2\sqrt{T}\,g^{\mu\nu}\,p_{\mu}\,A_{\nu}+T\,g^{\mu\nu}A_{\mu}A_{\nu}$
(103)
$\displaystyle\quad+\frac{T}{4}g^{\mu\nu}\omega_{\mu}\omega_{\nu}+\frac{T}{4}\Gamma^{2}-\frac{T}{2}\Big{(}1+\psi+\psi^{\dagger}+\eta+\bar{\eta}\Big{)}L\sqrt{\frac{h}{g}}\,\delta\big{(}y+\sqrt{T}q^{2}\big{)}$
$\displaystyle\quad+2\sqrt{T}\Big{\\{}g^{\mu\nu}\omega_{\mu}p_{\nu}+\sqrt{T}g^{\mu\nu}\omega_{\mu}A_{\nu}+\sqrt{T}g^{-1/2}F\Big{\\}}\,(\psi\psi^{\dagger}+\psi\bar{\eta}+\eta\psi^{\dagger}+\eta\bar{\eta})\;.$

Every function of the coordinates in (103) is evaluated at
$\big{(}x+\sqrt{T}q^{1},\big{|}y+\sqrt{T}q^{2}\big{|}\big{)}$ unless
explicitly stated. Note how $\widetilde{H}_{\text{dir}}$ is of order
$\sqrt{T}$ at least, which eases the perturbative expansion in powers of $T$.
Expanding $e^{-\int_{0}^{1}dt\,\widetilde{H}_{\text{dir}}}$ in powers of
$\eta$ and $\bar{\eta}$ and then solving the Grassmannian trace yields

$\text{Tr }K_{M}^{\text{dir}}=\frac{1}{2\pi
T}\\!\int_{M}\\!\\!dx\,dy\sqrt{g}\Big{\langle}e^{-\int_{0}^{1}dt\,\mathcal{H}_{\text{dir}}}\\\
\times\Big{(}1+\\!\sqrt{T}\,b_{\text{dir}}+2T\,\theta_{\text{dir}}\bar{\theta}_{\text{dir}}-\frac{T^{3/2}}{2}(\theta_{\text{dir}}+\bar{\theta}_{\text{dir}})\Delta_{\text{dir}}-\frac{T^{2}}{8}(\Delta_{\text{dir}})^{2}\Big{)}\Big{\rangle}$
(104)

with

$\displaystyle\mathcal{H}_{\text{dir}}$
$\displaystyle=\Big{[}g^{\mu\nu}\big{(}x+\sqrt{T}q^{1},\big{|}y+\sqrt{T}q^{2}\big{|}\big{)}-g^{\mu\nu}(x,y)\Big{]}p_{\mu}\,p_{\nu}+2\sqrt{T}\,g^{\mu\nu}\,p_{\mu}\,A_{\nu}+T\,g^{\mu\nu}A_{\mu}A_{\nu}$
$\displaystyle\quad+\frac{T}{4}g^{\mu\nu}\omega_{\mu}\omega_{\nu}+\frac{T}{4}\Gamma^{2}-\frac{T}{2}\Big{(}1+\psi+\psi^{\dagger}\Big{)}L\sqrt{\frac{h}{g}}\,\delta\big{(}y+\sqrt{T}q^{2}\big{)}$
$\displaystyle\quad+2\sqrt{T}\Big{\\{}g^{\mu\nu}\omega_{\mu}p_{\nu}+\sqrt{T}g^{\mu\nu}\omega_{\mu}A_{\nu}+\sqrt{T}g^{-1/2}F\Big{\\}}\,\psi\psi^{\dagger}\;,$
(105) $\displaystyle b_{\text{dir}}$
$\displaystyle=\int_{0}^{1}dt\,\Big{\\{}g^{\mu\nu}\omega_{\mu}p_{\nu}+\sqrt{T}\,g^{\mu\nu}\omega_{\mu}A_{\nu}+\sqrt{T}\,g^{-1/2}F\Big{\\}}\;,$
(106) $\displaystyle\theta_{\text{dir}}$
$\displaystyle=\int_{0}^{1}dt\,\Big{\\{}g^{\mu\nu}\omega_{\mu}p_{\nu}+\sqrt{T}\,g^{\mu\nu}\omega_{\mu}A_{\nu}+\sqrt{T}\,g^{-1/2}F\Big{\\}}\psi\;,$
(107) $\displaystyle\bar{\theta}_{\text{dir}}$
$\displaystyle=\int_{0}^{1}dt\,\Big{\\{}g^{\mu\nu}\omega_{\mu}p_{\nu}+\sqrt{T}\,g^{\mu\nu}\omega_{\mu}A_{\nu}+\sqrt{T}\,g^{-1/2}F\Big{\\}}\psi^{\dagger}\;,$
(108) $\displaystyle\Delta_{\text{dir}}$
$\displaystyle=\int_{0}^{1}dt\,L\sqrt{\frac{h}{g}}\,\delta\big{(}y+\sqrt{T}q^{2}\big{)}\;.$
(109)

In expressions (105)-(109) every function of the coordinates is again
evaluated at $\big{(}x+\sqrt{T}q^{1},\big{|}y+\sqrt{T}q^{2}\big{|}\big{)}$
unless the evaluation point is explicitly stated. Equation (104) gives a non-
perturbative result for the direct contribution of the heat-trace.

To compute the heat-trace up to any desired Seeley-DeWitt coefficient $a_{k}$,
now one expands (104) in terms of $T$ up to order $T^{-1+k/2}$. Noticing that
$\mathcal{H}_{\text{dir}}$ is of order $\sqrt{T}$ at least, this means that
one needs to expand the exponential containing this factor up to order $k$ in
the power series, thus “lowering” $\mathcal{H}_{\text{dir}}$ up to $k$ times
in total. Then, defining

$\text{Tr }K_{M}^{\text{dir}(k)}=\frac{(-1)^{k}}{2\pi
T}\\!\int_{M}\\!\\!dx\,dy\sqrt{g}\,\Big{\langle}\int_{0}^{1}dt_{1}\dots
dt_{k}\,\mathcal{H}_{\text{dir}}(t_{1})\dots\mathcal{H}_{\text{dir}}(t_{k})\\\
\times\Big{(}1+\\!\sqrt{T}\,b_{\text{dir}}+2T\,\theta_{\text{dir}}\bar{\theta}_{\text{dir}}-\frac{T^{3/2}}{2}(\theta_{\text{dir}}+\bar{\theta}_{\text{dir}})\Delta_{\text{dir}}-\frac{T^{2}}{8}(\Delta_{\text{dir}})^{2}\Big{)}\Big{\rangle}$
(110)

one can see that

$\text{Tr }K_{M}^{\text{dir}}=\sum_{k=0}^{\infty}\frac{1}{k!}\,\text{Tr
}K_{M}^{\text{dir}(k)}\;.$

In particular, computing up to the coefficient $a_{2}$ means expanding up to
order $T^{0}$, thus “lowering” $\mathcal{H}_{\text{dir}}$ up to $2$ times in
total. Hence, one can readily write

$\text{Tr }K_{M}^{\text{dir}}=\text{Tr }K_{M}^{\text{dir}(0)}+\text{Tr
}K_{M}^{\text{dir}(1)}+\frac{1}{2}\,\text{Tr
}K_{M}^{\text{dir}(2)}+\mathcal{O}(T^{1/2})\;.$ (111)

To compute $\text{Tr }K_{M}^{\text{dir}(0)}$, one first expands up to order
$T^{0}$ obtaining

$\text{Tr }K_{M}^{\text{dir}(0)}=\frac{1}{2\pi
T}\\!\int_{M}\\!\\!dx\,dy\sqrt{g}\,\Big{\langle}1+\\!\sqrt{T}\,b_{\text{dir}}+2T\,\theta_{\text{dir}}\bar{\theta}_{\text{dir}}\Big{\rangle}+\mathcal{O}(T^{1/2})\;,$
(112)

where $b_{\text{dir}}$ is expanded up to order $T^{1/2}$, and in the product
$\theta_{\text{dir}}\bar{\theta}_{\text{dir}}$ one only retains the lowest
order which is $T^{0}$. In this scenario, using the mean values obtained at
the end of Section 6, one gets

$\displaystyle\langle b_{\text{dir}}\rangle$
$\displaystyle=\sqrt{T}\big{(}g^{\mu\nu}\omega_{\mu}A_{\nu}+g^{-1/2}F\big{)}+\mathcal{O}(T)\;,$
$\displaystyle\langle\theta_{\text{dir}}\bar{\theta}_{\text{dir}}\rangle$
$\displaystyle=\frac{1}{2}g^{\mu\nu}\omega_{\mu}\omega_{\nu}\int_{0}^{1}dtdt^{\prime}K(t,t^{\prime})+\mathcal{O}(T^{1/2})\;,$

with the RHS of both equations evaluated at $(x,|y|)$. The integral of the
Green function $K(t,t^{\prime})$ yields $1/2$, so replacing into (112) one
gets

$\text{Tr }K_{M}^{\text{dir}(0)}=\frac{1}{2\pi
T}\text{Vol}(M)+\frac{1}{4\pi}\int_{M}\\!\\!dx\,dy\sqrt{g}\bigg{(}g^{\mu\nu}\omega_{\mu}\omega_{\nu}+2g^{\mu\nu}\omega_{\mu}A_{\nu}+2g^{-1/2}F\bigg{)}+\mathcal{O}(T^{1/2})\;.$
(113)

Moving on to $\text{Tr }K_{M}^{\text{dir}(1)}$ and expanding up to order
$T^{0}$ one obtains

$\text{Tr }K_{M}^{\text{dir}(1)}=\frac{-1}{2\pi
T}\\!\int_{M}\\!\\!dx\,dy\sqrt{g}\int_{0}^{1}dt_{1}\,\Big{\langle}\mathcal{H}_{\text{dir}}(t_{1})\Big{(}1+\\!\sqrt{T}\,b_{\text{dir}}\Big{)}\Big{\rangle}+\mathcal{O}(T^{1/2})\;.$
(114)

Focusing first on the factor $1$ in the parenthesis, the mean value
$\langle\mathcal{H}_{\text{dir}}(t_{1})\rangle$ needs to be computed up to
order $T$. To start this calculation, consider the “$\delta$-term” first:

$\int_{M}\\!\\!dx\,dy\sqrt{g}\int_{0}^{1}\\!\\!dt_{1}\bigg{[}-\frac{T}{2}L\sqrt{\frac{h}{g}}\bigg{\langle}\Big{(}1+\psi+\psi^{\dagger}\Big{)}\,\delta\Big{(}y+\sqrt{T}q^{2}(t_{1})\Big{)}\bigg{\rangle}\bigg{]}\\\
=-\frac{T}{2}\int_{M}\\!\\!dx\,dy\sqrt{h}\,L\int_{0}^{1}\\!\\!dt_{1}\Big{\langle}\delta\Big{(}y+\sqrt{T}q^{2}(t_{1})\Big{)}\Big{\rangle}\;.$
(115)

This expectation value can be computed by expressing the delta distribution in
its Fourier representation:

$\Big{\langle}\delta\Big{(}y+\sqrt{T}q^{2}(t_{1})\Big{)}\Big{\rangle}=\int_{-\infty}^{\infty}\frac{dk}{2\pi}e^{-iky}\Big{\langle}e^{-ik\sqrt{T}\,q^{2}(t_{1})}\Big{\rangle}\;.$

Then the problem of computing the mean value of a delta turns into the problem
of computing the mean value of an exponential, which can be achieved by using
equations (66) and (67) with the source
$j_{\mu}(t)=-\omega\,\delta_{\mu}^{2}\,\delta(t-t_{1})$ (see
CorradiniEdwardsHuetManzoPisani  for a list with several mean values
containing an exponential). This yelds $\langle
e^{-ik\sqrt{T}\,q^{2}(t_{1})}\rangle=e^{-Tk^{2}g^{22}G(t_{1},t_{1})}$. Then

$\Big{\langle}\delta\Big{(}y+\sqrt{T}q^{2}(t_{1})\Big{)}\Big{\rangle}=\bigg{(}\frac{g}{4\pi
T\,h\,G(t_{1},t_{1})}\bigg{)}^{\\!\\!1/2}\,e^{-y^{2}g/4T\,h\,G(t_{1},t_{1})}\;.$

Inserting this into the integral on the manifold $M$ and rescaling
$y\rightarrow\sqrt{4T}y$ yields

$\int_{M}\\!\\!dx\,dy\sqrt{h}\,L\\!\int_{0}^{1}dt_{1}\,\Big{\langle}\delta\Big{(}y+\sqrt{T}q^{2}(t_{1})\Big{)}\Big{\rangle}=\frac{1}{\sqrt{\pi}}\\!\int_{M}\\!\\!dx\,dy\sqrt{g}\,L\\!\int_{0}^{1}\\!\frac{dt_{1}}{\sqrt{G(t_{1},t_{1})}}e^{-y^{2}g/h\,G(t_{1},t_{1})}\;,$

with the functions $g$ and $h$ on the RHS evaluated at $(x,\sqrt{4T}y)$.
Expanding them in powers of $T$ leaves them evaluated at $(x,0)$ plus terms of
order $T^{1/2}$. The integral in $y$ is then trivial at the lowest order:

$\int_{M}\\!\\!dx\,dy\sqrt{h}\,L\\!\int_{0}^{1}dt_{1}\,\Big{\langle}\delta\Big{(}y+\sqrt{T}q^{2}(t_{1})\Big{)}\Big{\rangle}=\frac{1}{2}\int_{\partial
M}\\!\\!dx\,\sqrt{h}\,L+\mathcal{O}(T^{1/2})\;.$

Now that the $\delta$-term in $\langle\mathcal{H}_{\text{dir}}(t_{1})\rangle$
has been computed, one needs to solve the mean values of the other terms, each
of them being proportional to some functions evaluated at
$\big{(}x+\sqrt{T}q^{1},\big{|}y+\sqrt{T}q^{2}\big{|}\big{)}$. The general
ideas about how to compute these mean values is left for Appendix A. The final
result is

$\int_{M}\\!\\!dx\,dy\sqrt{g}\int_{0}^{1}\\!dt_{1}\,\big{\langle}\mathcal{H}_{\text{dir}}(t_{1})\big{\rangle}=\;\frac{T}{12}\int_{M}\\!\\!dx\,dy\sqrt{g}\bigg{\\{}g_{\mu\nu}g^{\rho\sigma}\partial_{\rho}\partial_{\sigma}g^{\mu\nu}-\partial_{\mu}\partial_{\nu}g^{\mu\nu}+3\Gamma^{2}\\\
\begin{aligned}
&+12\,g^{\mu\nu}A_{\mu}A_{\nu}+3\,g^{\mu\nu}\omega_{\mu}\omega_{\nu}+12\,g^{\mu\nu}\omega_{\mu}A_{\nu}+12g^{-1/2}F\bigg{\\}}\\\
&-\frac{T}{12}\int_{\partial
M}\\!\\!dx\,\sqrt{h}\,L+\mathcal{O}(T^{3/2})\;.\end{aligned}$ (116)

Going back to (114) and focusing now on the term with the factor
$\sqrt{T}\,b_{\text{dir}}$ in the parenthesis, the mean value
$\langle\mathcal{H}_{\text{dir}}(t_{1})\,b_{\text{dir}}\rangle$ needs to be
expanded up to order $T^{1/2}$. Since that is exactly the lowest order of
$\mathcal{H}_{\text{dir}}$, then one only needs to retain the lowest order of
both $\mathcal{H}_{\text{dir}}$ and $b_{\text{dir}}$. Through similar and less
involved calculations as before, this results in

$\int_{M}\\!\\!dx\,dy\sqrt{g}\\!\int_{0}^{1}\\!\\!dt_{1}\big{\langle}\mathcal{H}_{\text{dir}}(t_{1})\,b_{\text{dir}}\big{\rangle}=\;\frac{\sqrt{T}}{2}\int_{M}\\!\\!dx\,dy\sqrt{g}\bigg{\\{}2\,g^{\mu\nu}A_{\mu}\omega_{\nu}+\,g^{\mu\nu}\omega_{\mu}\omega_{\nu}\bigg{\\}}+\mathcal{O}(T)\;.$
(117)

Inserting (116) and (117) into (114) gives

$\text{Tr
}K_{M}^{\text{dir}(1)}=\;\frac{1}{24\pi}\int_{M}\\!\\!dx\,dy\sqrt{g}\bigg{\\{}-g_{\mu\nu}g^{\rho\sigma}\partial_{\rho}\partial_{\sigma}g^{\mu\nu}+\partial_{\mu}\partial_{\nu}g^{\mu\nu}-3\Gamma^{2}\\\
\begin{aligned}
&-12\,g^{\mu\nu}A_{\mu}A_{\nu}-9\,g^{\mu\nu}\omega_{\mu}\omega_{\nu}-24\,g^{\mu\nu}\omega_{\mu}A_{\nu}-12g^{-1/2}F\bigg{\\}}\\\
&+\frac{1}{24\pi}\int_{\partial
M}\\!\\!dx\,\sqrt{h}\,L+\mathcal{O}(T^{1/2})\;.\end{aligned}$ (118)

Only the term $\text{Tr }K_{M}^{\text{dir}(2)}$ remains to be computed. Its
expansion up to order $T^{0}$ is

$\text{Tr }K_{M}^{\text{dir}(2)}=\frac{1}{2\pi
T}\\!\int_{M}\\!\\!dx\,dy\sqrt{g}\,\Big{\langle}\int_{0}^{1}dt_{1}dt_{2}\,\mathcal{H}_{\text{dir}}(t_{1})\mathcal{H}_{\text{dir}}(t_{2})\Big{\rangle}+\mathcal{O}(T^{1/2})\;.$

Next, one needs to expand the product
$\mathcal{H}_{\text{dir}}(t_{1})\mathcal{H}_{\text{dir}}(t_{2})$ up to order
$T$, which is simply the lowest order. Through similar calculations as before:

$\frac{1}{2}\text{Tr
}K_{M}^{\text{dir}(2)}=\frac{1}{24\pi}\\!\int_{M}\\!\\!dx\,dy\sqrt{g}\bigg{\\{}\frac{1}{4}g^{\mu\nu}(\partial_{\mu}\log
g)(\partial_{\nu}\log g)+(\partial_{\mu}g^{\mu\nu})(\partial_{\nu}\log g)\\\
\begin{aligned}
&-\frac{1}{2}g^{\mu\nu}(\partial_{\mu}g_{\rho\sigma})(\partial_{\nu}g^{\rho\sigma})-g^{\mu\nu}(\partial_{\rho}g_{\mu\sigma})(\partial_{\nu}g^{\rho\sigma})\\\
&+12\,g^{\mu\nu}A_{\mu}A_{\nu}+3\,g^{\mu\nu}\omega_{\mu}\omega_{\nu}+12\,g^{\mu\nu}\omega_{\mu}A_{\nu}\bigg{\\}}+\mathcal{O}(T^{1/2})\;.\end{aligned}$
(119)

Adding equations (113), (118) and (119), and after some geometrical
identities, yields

$\text{Tr }K_{M}^{\text{dir}}=\frac{1}{2\pi
T}\text{Vol}(M)-\frac{1}{24\pi}\int_{M}\\!\\!dx\,dy\sqrt{g}\,R+\frac{1}{24\pi}\int_{\partial
M}\\!\\!dx\,\sqrt{h}\,L+\mathcal{O}(T^{1/2})\;.$ (120)

###  9.2 Indirect trace

The indirect trace is a bit more involved that the direct one for several
reasons. First, note that the transition amplitude in (101) includes the term
$\langle x,-y,\bar{\eta}\,|\,\chi$. Since the present model uses
$\chi=-\psi-\psi^{\dagger}$, one can write

$\langle
x,-y,\bar{\eta}\,|\,\chi=-\bigg{(}\bar{\eta}+\frac{d}{d\bar{\eta}}\bigg{)}\langle
x,-y,\bar{\eta}\,|\;.$

This removes the operator $\chi$ in the transition amplitude at the expense of
introducing the factor in parenthesis in the previous equation. Without
$\chi$, the transition amplitude to be integrated is $\langle
x,-y,\bar{\eta}\,|\,e^{-T\widetilde{H}}\,|x,y,\eta\rangle$, which can be
expressed in terms of (65). The next complication is that now there is a
multiplicative factor which is equal to $e^{-y^{2}g/hT}$. This term, when
expanded as a power series, provides negative powers of $T$. To get rid of
them, one can perform the rescaling $y\rightarrow\sqrt{T}y$. With all these
ingredients, equation (101) turns into

$\text{Tr
}K_{M}^{\text{ind}}=\frac{-1}{4\pi\sqrt{T}}\int_{M}dx\,dy\sqrt{g(x,\sqrt{T}y)}\,e^{-y^{2}g(x,\sqrt{T}y)/h(x,\sqrt{T}y)}\\\
\times\int
d\eta\,d\bar{\eta}\,e^{\bar{\eta}\eta}\bigg{(}\bar{\eta}+\frac{d}{d\bar{\eta}}\bigg{)}e^{\bar{\eta}\eta}\Big{\langle}e^{-\int_{0}^{1}dt\,\widetilde{H}_{\text{ind}}}\Big{\rangle}$

where the exponent in the mean value is given by

$\displaystyle\widetilde{H}_{\text{ind}}$
$\displaystyle=\Big{[}g^{\mu\nu}\big{(}x+\sqrt{T}q^{1},\sqrt{T}\big{|}(1-2t)y+q^{2}\big{|}\big{)}-g^{\mu\nu}(x,\sqrt{T}y)\Big{]}\pi_{\mu}\,\pi_{\nu}$
(121)
$\displaystyle\quad+2\sqrt{T}\,g^{\mu\nu}\,\pi_{\mu}\,A_{\nu}+T\,g^{\mu\nu}A_{\mu}A_{\nu}$
$\displaystyle\quad+\frac{T}{4}g^{\mu\nu}\omega_{\mu}\omega_{\nu}+\frac{T}{4}\Gamma^{2}-\frac{\sqrt{T}}{2}\Big{(}1+\psi+\psi^{\dagger}+\eta+\bar{\eta}\Big{)}L\sqrt{\frac{h}{g}}\,\delta\big{(}(1-2t)y+q^{2}\big{)}$
$\displaystyle\quad+2\sqrt{T}\Big{\\{}g^{\mu\nu}\omega_{\mu}\pi_{\nu}+\sqrt{T}g^{\mu\nu}\omega_{\mu}A_{\nu}+\sqrt{T}g^{-1/2}F\Big{\\}}\,(\psi\psi^{\dagger}+\psi\bar{\eta}+\eta\psi^{\dagger}+\eta\bar{\eta})\;,$

where $\pi_{1}=p_{1}$ and $\pi_{2}=p_{2}-i\,y\,g(x,\sqrt{T}y)/h(x,\sqrt{T}y)$.
Every function of the coordinates in (121) is evaluated at
$\big{(}x+\sqrt{T}q^{1},\sqrt{T}\big{|}(1-2t)y+q^{2}\big{|}\big{)}$ unless
explicitly stated. $\widetilde{H}_{\text{ind}}$ is of order $\sqrt{T}$ at
least, just like $\widetilde{H}_{\text{dir}}$. Expanding
$e^{-\int_{0}^{1}dt\,\widetilde{H}_{\text{ind}}}$ in powers of $\eta$ and
$\bar{\eta}$ and then solving the Grassmannian trace yields

$\text{Tr
}K_{M}^{\text{ind}}=\frac{1}{4\pi}\\!\int_{M}\\!dx\,dy\sqrt{g(x,\sqrt{T}y)}\,e^{-y^{2}g(x,\sqrt{T}y)/h(x,\sqrt{T}y)}\Big{\langle}e^{-\int_{0}^{1}dt\,\mathcal{H}_{\text{ind}}}\Big{(}2(\bar{\theta}_{\text{ind}}-\theta_{\text{ind}})-\Delta_{\text{ind}}\Big{)}\\!\Big{\rangle}$
(122)

with

$\displaystyle\mathcal{H}_{\text{ind}}$
$\displaystyle=\Big{[}g^{\mu\nu}\big{(}x+\sqrt{T}q^{1},\sqrt{T}\big{|}(1-2t)y+q^{2}\big{|}\big{)}-g^{\mu\nu}(x,\sqrt{T}y)\Big{]}\pi_{\mu}\,\pi_{\nu}$
$\displaystyle\quad+2\sqrt{T}\,g^{\mu\nu}\,\pi_{\mu}\,A_{\nu}+T\,g^{\mu\nu}A_{\mu}A_{\nu}$
$\displaystyle\quad+\frac{T}{4}g^{\mu\nu}\omega_{\mu}\omega_{\nu}+\frac{T}{4}\Gamma^{2}-\frac{\sqrt{T}}{2}\Big{(}1+\psi+\psi^{\dagger}\Big{)}L\sqrt{\frac{h}{g}}\,\delta\big{(}(1-2t)y+q^{2}\big{)}$
$\displaystyle\quad+2\sqrt{T}\Big{\\{}g^{\mu\nu}\omega_{\mu}\pi_{\nu}+\sqrt{T}g^{\mu\nu}\omega_{\mu}A_{\nu}+\sqrt{T}g^{-1/2}F\Big{\\}}\,\psi\psi^{\dagger}\;,$
(123) $\displaystyle\theta_{\text{ind}}$
$\displaystyle=\int_{0}^{1}dt\,\Big{\\{}g^{\mu\nu}\omega_{\mu}\pi_{\nu}+\sqrt{T}\,g^{\mu\nu}\omega_{\mu}A_{\nu}+\sqrt{T}\,g^{-1/2}F\Big{\\}}\psi\;,$
(124) $\displaystyle\bar{\theta}_{\text{ind}}$
$\displaystyle=\int_{0}^{1}dt\,\Big{\\{}g^{\mu\nu}\omega_{\mu}\pi_{\nu}+\sqrt{T}\,g^{\mu\nu}\omega_{\mu}A_{\nu}+\sqrt{T}\,g^{-1/2}F\Big{\\}}\psi^{\dagger}\;,$
(125) $\displaystyle\Delta_{\text{ind}}$
$\displaystyle=\int_{0}^{1}dt\,L\sqrt{\frac{h}{g}}\,\delta\big{(}(1-2t)y+q^{2}\big{)}\;.$
(126)

Once again, in expressions (123)-(126) every function of the coordinates is
evaluated at
$\big{(}x+\sqrt{T}q^{1},\sqrt{T}\big{|}(1-2t)y+q^{2}\big{|}\big{)}$ unless the
evaluation point is explicitly stated. Equation (122) gives a non-perturbative
result for the indirect contribution of the heat-trace.

One interesting result that follows from equation (122) is the fact that for
boundaries $\partial M$ obeying $L=0$ one finds $\text{Tr
}K_{M}^{\text{ind}}=0$. Indeed, if $L=0$ then $\Delta_{\text{ind}}=0$ and the
mean value in the integrand becomes
$\big{\langle}e^{-\int_{0}^{1}dt\,\mathcal{H}_{\text{ind}}}(\bar{\theta}_{\text{ind}}-\theta_{\text{ind}})\big{\rangle}$.
Since $\bar{\theta}_{\text{ind}}-\theta_{\text{ind}}$ is linear in the fields
$\psi$ and $\psi^{\dagger}$, it turns out that only those terms of
$e^{-\int_{0}^{1}dt\,\mathcal{H}_{\text{ind}}}$ that are also linear in the
fields $\psi$ and $\psi^{\dagger}$ could give a non-zero mean value. But such
terms do not exist if $L=0$, proving that $\text{Tr }K_{M}^{\text{ind}}=0$ if
the boundary has zero extrinsic curvature.

Another interesting fact lies in the aforementioned rescaling
$y\rightarrow\sqrt{T}y$. Due to it, the evaluation point $(x,y)$ of any
function is replaced by $(x,\sqrt{T}y)$. Hence, the coefficients in the
subsequent expansion in powers of $T$ correspond to functions evaluated at
$(x,0)$, that is, at the boundary. Therefore, indirect terms contribute only
to boundary terms in the heat-trace expansion.

Note now that the RHS of equation (122) is of order $T^{0}$ at least, which
implies that the indirect terms do not contribute to the Seeley-DeWitt
coefficients $a_{0}$ and $a_{1}$. For the coefficients $a_{k}$ with $k\geq 2$,
one expands (122) in terms of $T$ up to order $T^{-1+k/2}$. Note that
$\mathcal{H}_{\text{ind}}$ is of order $\sqrt{T}$ at least, which means that
one needs to expand the exponential containing this factor up to order $k-2$
in its power series, thus “lowering” $\mathcal{H}_{\text{ind}}$ up to $k-2$
times in total. In particular, computing up to the coefficient $a_{2}$ means
that one can completely ignore the exponential containing the factor
$\mathcal{H}_{\text{ind}}$. Therefore,

$\displaystyle\text{Tr }K_{M}^{\text{ind}}$
$\displaystyle=\frac{-1}{4\pi}\\!\int_{M}\\!dx\,dy\sqrt{g(x,0)}\,e^{-y^{2}g(x,0)/h(x,0)}\Big{\langle}\Delta_{\text{ind}}\Big{\rangle}+\mathcal{O}(T^{1/2})$
$\displaystyle=\frac{-1}{4\pi}\\!\int_{M}\\!dx\,dy\sqrt{h(x,0)}\,L\,e^{-y^{2}g(x,0)/h(x,0)}\\!\\!\int_{0}^{1}\\!dt\Big{\langle}\delta\Big{(}(1-2t)y+q^{2}(t)\Big{)}\Big{\rangle}+\mathcal{O}(T^{1/2})\;.$

The mean value of the delta is solved analogously as in the direct case. Thus
one gets

$\text{Tr }K_{M}^{\text{ind}}=-\frac{1}{8\pi}\int_{\partial
M}\\!dx\sqrt{h}\,L+\mathcal{O}(T^{1/2})\;.$ (127)

###  9.3 The trace anomaly

Adding equations (120) and (127) leads to the heat-trace on the manifold $M$:

$\text{Tr }K_{M}=\frac{1}{2\pi
T}\text{Vol}(M)-\frac{1}{24\pi}\int_{M}\\!\\!dx\,dy\sqrt{g}\,R-\frac{1}{12\pi}\int_{\partial
M}\\!\\!dx\,\sqrt{h}\,L+\mathcal{O}(T^{1/2})\;.$

By comparison with (9), the first three Seeley-DeWitt coefficients are
obtained:

$a_{0}=2\,\text{Vol}(M)\;,\qquad\quad a_{1}=0\;,\qquad\quad
a_{2}=-\frac{1}{6}\int_{M}\\!\\!dx\,dy\sqrt{g}\,R-\frac{1}{3}\int_{\partial
M}\\!\\!dx\,\sqrt{h}\,L\;.$

These results are in perfect agreement with HeatKernelManual ;
BransonGilkeyKirstenVassilevich . In particular, the coefficient $a_{2}$ leads
to the integrated trace anomaly:

$\int_{M}\\!\\!dx\,dy\,\sqrt{g}\,\langle
T^{\mu}_{\;\;\mu}\rangle=\frac{a_{2}}{4\pi}=-\frac{1}{24\pi}\int_{M}\\!\\!dx\,dy\sqrt{g}\,R-\frac{1}{12\pi}\int_{\partial
M}\\!\\!dx\,\sqrt{h}\,L\;.$ (128)

##  10 Conclusions and future work

In this article, an approach to consistently implement the worldline formalism
(WLF) for spinor particles in manifolds with boundaries was introduced. The
boundary condition under consideration was the MIT bag boundary condition. To
better fix ideas, the construction was carried out in an arbitrary two-
dimensional manifold $M$ representable in half-plane coordinates $(x,y)$, such
that the bulk of the manifold consists on $y>0$ and the boundary $\partial M$
is the straight line $y=0$. The construction relies on the method of images:
the manifold was mirrored symmetrically with respect to the boundary,
generating a new two-dimensional manifold $\widetilde{M}$ without boundaries.
The old boundary became an interface were a singular curvature developed. To
analyse the applicability of the method, the first three Seeley-DeWitt
coefficients of the heat-trace were computed. In this approach, all curvature
terms must be non-singular in the $(x,y)$ coordinate representation of the
manifold.

The procedure that has been set up admits generalisations and concrete
applications. Up to now, worldline formulations on quantum fields with
boundaries have been established only for scalar fields: firstly, Dirichlet
and Neumann boundary conditions were analysed in flat space using the method
of images BastianelliCorradiniPisani2007 , and Robin boundary conditions were
considered later using the same method BastianelliCorradiniPisaniSchubert .
Some years later, the method was extended to incorporate curved manifolds and
curved boundaries, including compact manifolds CorradiniEdwardsHuetManzoPisani
; TrabajoDiploma . A different approach involving delta potentials, also for
scalar fields, was analysed as well: firstly, a free scalar field with
semitransparent Dirichlet boundary conditions was studied FranchinoPisani ;
FranchinoMazzitelli , and semitransparent Neumann boundary conditions were
analysed more recently AhmadiniazFranchinoManzoMazzitelli . Shortly after, the
case of semitransparent Dirichlet boundary conditions for scalar fields with a
potential was studied using the same method Franchino . None of the
aforementioned worldline approaches considered, until now, spinor fields (but
see Grosche1995 ; Grosche1999  for previous path integral approaches for Dirac
particles in the presence of delta-like potentials). The procedure described
in the present article is expected to open the path to include new scenarios,
including new fields and new boundary conditions, in the context of the WLF.

For the case of new fields, the approach followed in the present article, as
described above, could be regarded as splitting the projections of the field
components according to whether they obey Dirichlet or Robin boundary
conditions. The same procedure should be valid for any field obeying the same
mixed boundary conditions regardless of the Lorentzian nature of the field.
For instance, a multiplet of several scalar fields or a gauge (vector) field
are scenarios where the present approach is expected to hold. For the latter
case, physically meaningful mixed boundary conditions are present in the
literature BransonGilkey ; MossPoletti ; Vassilevich1995 ; Vassilevich1998
(see also Section 3.4 in HeatKernelManual  for a brief summary). The
implementation of the procedure described in the present article to gauge
fields with these conditions is currently under consideration.

For the case of other boundary conditions, it is well known that spinors could
satisfy a large variety of them. In the area of local boundary conditions, MIT
bag conditions are just one of many possibilities. Another closely related,
physically meaningful possibility are the chiral bag boundary conditions,
first introduced in ChodosThorn  to construct a chirally symmetric theory in
Quantum Chromodynamics. Since then, they were applied in other areas such as
fermionic monopoles Yamagishi , fermionic billiards BerryMondragon , graphene
devices ZhangDietz  and Weyl semimetals IvanovKurkovVassilevich . These
conditions can be written as $\Pi_{-}\Psi|_{\partial M}=0$ using the non-
Hermitian projector
$\Pi_{-}=\frac{1}{2}\big{(}1-i\not{n}\gamma^{\text{ch}}\,e^{r(x^{\alpha})\gamma^{\text{ch}}}\big{)}$,
where $r(x^{\alpha})$ is a real-valued function of the tangential coordinates.
The main difficulty with these conditions is that, for $r(x^{\alpha})\neq 0$,
the first-order induced boundary condition contains partial derivatives
EspositoKirsten . Furthermore, first-order boundary conditions with tangential
derivatives are completely unexplored within the worldline frame. It would be
interesting to see if one could implement a worldline procedure, similar to
the one of the present article, to include these boundary conditions. Besides,
they are reduced to MIT bag boundary conditions for $r(x^{\alpha})=0$, so the
present article could be use to check the construction of any possible
procedure.

In the area of global boundary conditions, the Atiyah-Patodi-Singer (APS)
conditions, sometimes referred as spectral boundary conditions, are well known
in the literature AtiyahPatodiSinger ; Abrikosov . They are connected to index
theorems Witten ; DabholkarJainRudra ; FukayaFurutaMatsuoOnogiYamaguchi ;
KobayashiYonekura  and raised interest in the study of topological devices
AsoreyBalachandranPerezPardo . The Seeley-DeWitt coefficients of the heat-
kernel asymptotics for these boundary conditions have been studied in the past
DowkerGilkeyKirsten ; GilkeyKirsten . They can be written in terms of
projectors $\Pi_{\pm}$, but they are usually expressed in terms of eigenvector
components which are not perfectly suited for worldline computations. It would
be interesting to see if APS boundary conditions can be implemented within a
worldline perspective, where the familiar calculational efficiency of
worldline techniques is expected to hold.

Besides the aforementioned generalisations, concrete applications are also at
hand. On the one hand, fermions in manifolds with boundaries have been used to
study quantities in Quantum Field Theories such as anomalies FoscoSilva  and
the Casimir effect SundbergJaffe . These approaches are based on delta-like
potentials instead of the method of images. On the other hand, worldline
numerical computations have been employed on the same areas
GiesLangfeldMoyaerts ; EdwardsGonzalezDominguezHuetTrejo ; GiesKlingmuller ;
EdwardsGerberSchubertTrejoWeber ; EdwardsGerberSchubertTrejoTsiftsiWeber ,
mostly considering rigid boundaries and scalar fields. The approach provided
in the present article could be used to study these phenomena analytically for
spinor fields, as well to test numerical computations in manifolds with rigid
boundaries.

###### Acknowledgements.

The author thanks Santiago Christiansen Murguizur, Horacio Falomir, Joaquín
Liniado and Pablo Pisani for valuable discussions during the research. This
work was supported by CONICET (under the program “Becas Internas Doctorales”,
RESOL-2020-129-APN-DIR#CONICET) and UNLP (under project 11/X909).

##  Appendix A Mean value of symmetrically reflected functions

Consider an arbitrary function $f(x,y)$ that is finite at $y=0$. If one
replaces $y\rightarrow|y|$ it is possible to write
$f(x,|y|)=f(x,y)-2\theta(-y)f_{A}(x,y)$ with
$f_{A}(x,y)\equiv\big{\\{}f(x,y)-f(x,-y)\big{\\}}/2$. Consider then the
problem of computing the following mean value:

$\Big{\langle}f\big{(}x+\sqrt{T}q^{1},|y+\sqrt{T}q^{2}|\big{)}\mathcal{M}\Big{\rangle}=\Big{\langle}f\big{(}x+\sqrt{T}q^{1},y+\sqrt{T}q^{2}\big{)}\mathcal{M}\Big{\rangle}\\\
-2\,\Big{\langle}\theta\big{(}-y-\sqrt{T}q^{2}\big{)}\,f_{A}\big{(}x+\sqrt{T}q^{1},y+\sqrt{T}q^{2}\big{)}\mathcal{M}\Big{\rangle}\;,$
(129)

which appears frequently in the calculation of the direct trace. $\mathcal{M}$
represents any monomial of the fields $q^{\mu}$ and $p_{\mu}$. In the
following it will be shown how to compute (129) based on the ideas set in
CorradiniEdwardsHuetManzoPisani .

To solve the mean value with the step function, perform the Fourier
transformation

$\theta(-y)=\\!\\!\int_{-\infty}^{\infty}\frac{dk}{2\pi
i}\frac{e^{-iky}}{k-i0}=\\!\\!\int_{-\infty}^{\infty}dk\,\frac{e^{-iky}}{2\pi
i}\bigg{[}\text{PV}\bigg{(}\frac{1}{k}\bigg{)}+i\pi\delta(k)\bigg{]}\;.$

Then (129) becomes

$\Big{\langle}f\big{(}x+\sqrt{T}q^{1},|y+\sqrt{T}q^{2}|\big{)}\mathcal{M}\Big{\rangle}=\Big{\langle}f\big{(}x+\sqrt{T}q^{1},y+\sqrt{T}q^{2}\big{)}\mathcal{M}\Big{\rangle}\\\
-\int_{-\infty}^{\infty}\\!\\!dk\,\frac{e^{-iky/\sqrt{T}}}{i\pi}\bigg{[}\text{PV}\bigg{(}\frac{1}{k}\bigg{)}+i\pi\delta(k)\bigg{]}\Big{\langle}e^{-ikq^{2}}\,f_{A}\big{(}x+\sqrt{T}q^{1},y+\sqrt{T}q^{2}\big{)}\mathcal{M}\Big{\rangle}\;.$
(130)

The computation of the mean values in the RHS of (130) can now be carried out
by expanding $f$ and $f_{A}$ as power series with respect to the fields
$q^{\mu}$. The mean value with the exponential is computed employing the
generating functional given by (66) and (67) with
$j_{\mu}=-k\,\delta^{2}_{\mu}\,\delta(t-t^{\prime})$, where $t^{\prime}$ is
the time at which the field $q^{2}(t^{\prime})$ in the exponent is evaluated.
An explicit list of several mean values involving this exponential is given in
CorradiniEdwardsHuetManzoPisani .

In calculations of the heat-trace, equation (130) appears integrated in the
manifold $M$ with measure $\sqrt{g}$. In this scenario, introducing the
rescaling $y\rightarrow\sqrt{T}y$ yields

$\int_{M}\\!\\!dx\,dy\sqrt{g}\Big{\langle}f\big{(}x+\sqrt{T}q^{1},|y+\sqrt{T}q^{2}|\big{)}\mathcal{M}\Big{\rangle}=\int_{M}\\!\\!dx\,dy\sqrt{g}\,\Big{\langle}f\big{(}x+\sqrt{T}q^{1},y+\sqrt{T}q^{2}\big{)}\mathcal{M}\Big{\rangle}\\\
-\sqrt{T}\\!\\!\int_{M}\\!\\!dx\,dy\sqrt{g(x,\sqrt{T}y)}\\!\\!\int_{-\infty}^{\infty}\\!\\!\\!dk\,\frac{e^{-iky}}{i\pi}\bigg{[}\text{PV}\bigg{(}\frac{1}{k}\bigg{)}\\!\\!+\\!i\pi\delta(k)\bigg{]}\\\
\times\Big{\langle}e^{-ikq^{2}}f_{A}\big{(}x\\!+\\!\sqrt{T}q^{1},\sqrt{T}(y\\!+\\!q^{2})\big{)}\mathcal{M}\Big{\rangle}_{y\rightarrow\sqrt{T}y},$
(131)

where the result of the mean value in the third line should be evaluated at
$(x,\sqrt{T}y)$ due to the rescaling. Note that the last term (second and
third line) in the RHS is at least of order $T$: indeed, the lowest explicit
order is $\sqrt{T}$, but its coefficient is proportional to $f_{A}(x,0)=0$.

Several particular applications of equation (131) are of interest for the
calculations of Section 9. As an example, consider the monomial
$\mathcal{M}=p_{\mu}p_{\nu}$ and $f=g^{\mu\nu}$. Up to order $T$, equation
(131) yields

$\int_{M}\\!\\!dx\,dy\sqrt{g}\,\Big{\langle}\Big{[}g^{\mu\nu}\Big{(}x\\!+\\!\sqrt{T}q^{1},|y\\!+\\!\sqrt{T}q^{2}|\Big{)}-g^{\mu\nu}(x,y)\Big{]}p_{\mu}p_{\nu}\Big{\rangle}\\\
\begin{aligned}
&=\frac{T}{2}\\!\int_{M}\\!\\!\\!dx\,dy\sqrt{g}\Big{\\{}g_{\mu\nu}g^{\rho\sigma}\partial_{\rho}\partial_{\sigma}g^{\mu\nu}G(t,t)-2\partial_{\mu}\partial_{\nu}g^{\mu\nu}\,{}^{\bullet}G(t,t)^{2}\Big{\\}}\\\
&\quad+\frac{T}{2}\int_{-\infty}^{\infty}dx\sqrt{g}\,\partial_{2}g^{\mu\nu}\Big{\\{}g^{22}g_{\mu\nu}\,G(t,t)-2\delta_{\mu}^{2}\delta_{\nu}^{2}\,{}^{\bullet}G(t,t)^{2}\Big{\\}}\Big{|}_{y=0}+\mathcal{O}(T^{3/2})\;,\end{aligned}$
(132)

where each curvature element in the integral of the last line is being
evaluated at $(x,0)$.

As a final remark, the present procedure to compute mean values can be
extended to the case of products of several functions $\langle f_{1}f_{2}\dots
f_{l}\rangle$, where each function $f_{i}$ is being evaluated at
$\big{(}x+\sqrt{T}q^{1}(t_{i}),\big{|}y+\sqrt{T}q^{2}(t_{i})\big{|}\big{)}$.
For each of them it is possible to write, as before, a “bulk term” plus a
“$\theta$-term”, with its own Fourier transformation for each step function.

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