# The heat kernel expansion for higher order minimal and non-minimal operators

A. O. Barvinsky<EMAIL_ADDRESS>Theory Department, Lebedev Physics
Institute, Leninsky Prospect 53, Moscow 119991, Russia  Institute for
Theoretical and Mathematical Physics, Lomonosov Moscow State University,
Leninskie Gory, GSP-1, Moscow, 119991, Russia  W. Wachowski
<EMAIL_ADDRESS>Theory Department, Lebedev Physics Institute, Leninsky
Prospect 53, Moscow 119991, Russia

###### Abstract

We build a systematic calculational method for the covariant expansion of the
two-point heat kernel $\hat{K}(\tau|x,x^{\prime})$ for generic minimal and
non-minimal differential operators of any order. This is the expansion in
powers of dimensional background field objects—the coefficients of the
operator and the corresponding spacetime and vector bundle curvatures,
suitable in renormalization and effective field theory applications. For
minimal operators whose principal symbol is given by an arbitrary power of the
covariant Laplacian $(-\Box)^{M}$, $M>1$, this result generalizes the well-
known Schwinger–DeWitt (or Seeley–Gilkey) expansion to the infinite series of
positive and negative fractional powers of the proper time $\tau^{1/M}$,
weighted by the generalized exponential functions of the dimensionless
argument $-\sigma(x,x^{\prime})/2\tau^{1/M}$ depending on the Synge world
function $\sigma(x,x^{\prime})$. The coefficients of this series are
determined by the chain of auxiliary differential operators acting on the two-
point parallel transport tensor, which in their turn follow from the solution
of special recursive equations. The derivation of these operators and their
recursive equations are based on the covariant Fourier transform in curved
spacetime. The series of negative fractional powers of $\tau$ vanishes in the
coincidence limit $x^{\prime}=x$, which makes the proposed method consistent
with the heat kernel theory of Seeley–Gilkey and generalizes it beyond the
heat kernel diagonal in the form of the asymptotic expansion in the domain
$\sigma(x,x^{\prime})\ll\tau^{1/M}$, $\tau\to 0$. Consistency of the method is
also checked by verification of known results for the minimal second-order
operators and their extension to the generic fourth-order operator. Possible
improvement of the suggested Fourier transform approach to the noncommutative
algebra of the covariant $\Box$ operator in the method of universal functional
traces is also briefly discussed.

##  1 Introduction

The purpose of this paper is to work out efficient method of calculating the
heat kernel of various differential operators in curved spacetime. It is
needless to say that the heat kernel is a powerful tool in mathematical
physics, which underlies numerous applications in quantum field theory, theory
of gauge fields and quantum gravity, theory of stochastic processes and
condensed matter systems. It is especially important in the renormalization
theory of quantum gravitational models where, when combined with the
background field method, it allows one to express in a closed form the Green
function of the theory and its one-loop effective action. Moreover, the
Schwinger–DeWitt [1, 2, 3, 4, 5] or Seeley–Gilkey [6, 7, 8, 9] heat kernel
method enables explicit calculation of local UV divergences not as their
expansion in powers of field perturbations on a simple symmetric background,
but as full nonlinear counterterms—local nonlinear functionals of the generic
background field. Starting with the pioneering application in quantum Einstein
theory [10], this method proved to be very efficient and now underlies
majority of results on renormalization of (super)gravitational models.

The basic tool of this method is the heat kernel expansion in powers of the
proper time (or Schwinger parameter). The coefficients of this expansion—the
so-called HaMiDeW (Hadamard–Minakshisundaram–DeWitt) [11] or Seeley–Gilkey
coefficients—represent the sequence of objects of growing dimensionality
directly related to the operators of field and gradient expansion in effective
field theory. Thus in the lowest orders corresponding to essential and
marginal operators they contain the information about UV renormalization
properties of the theory and also, within a needed accuracy below the cutoff,
incorporate all quantum corrections.

A well known difficulty with the extension of this method to a wider class of
field models is that it is directly applicable only to the theories with the
inverse propagator which is a second-order minimal differential operator

$\hat{F}(\nabla)=-\Box\,\hat{1}+\hat{P},$ (1.1)

whose derivatives form a covariant Laplacian $\Box=g^{ab}\nabla_{a}\nabla_{b}$
defined in a curved $d$-dimensional spacetime with the Riemannian metric
$g_{ab}(x)$.111In what follows we work in the Euclidean space version of the
theory related to the physical Lorentzian signature spacetime by Wick
rotation, so that $-\Box$ is positive-definite under appropriate boundary
conditions. Here the covariant derivatives (and the operator as a whole) are
acting on a generic set of fields $\varphi(x)=\varphi^{A}(x)$ with spin-tensor
labels $A$ of arbitrary nature, and the hat denotes matrices in the vector
space of $\varphi^{A}$, in particular $\hat{1}=\delta^{A}_{B}$ denoting the
corresponding unit matrix. Only in this particular case there exists the
Schwinger–DeWitt ansatz for its heat kernel
$\hat{K}(\tau|x,x^{\prime})=e^{-\tau\hat{F}(\nabla)}\delta(x,x^{\prime})$ of
the form

$\hat{K}(\tau|x,x^{\prime})=\frac{\Delta^{1/2}(x,x^{\prime})}{(4\pi\tau)^{d/2}}\,g^{1/2}(x^{\prime})\\\
\times\exp\left(-\frac{\sigma(x,x^{\prime})}{2\tau}\right)\sum\limits_{m=0}^{\infty}\tau^{m}\,\hat{a}_{m}(x,x^{\prime}).$
(1.2)

Here $\sigma(x,x^{\prime})$ is the Synge world function—one half of the square
of the geodetic distance between the points $x$ and $x^{\prime}$,
$\Delta(x,x^{\prime})$ is the dedensitized Pauli–Van Vleck–Morette determinant
(A.16), $g^{1/2}(x^{\prime})$ expresses the fact that the heat kernel as well
as the delta function are densities of weight one in the second argument and
$\hat{a}_{m}(x,x^{\prime})$ are well-known HaMiDeW or Schwinger–DeWitt
coefficients. These coefficients satisfy recurrent differential equations
which can be successively solved for $\hat{a}_{m}(x,x^{\prime})$ in the form
of covariant Taylor series in powers of
$\sigma^{a^{\prime}}=\nabla^{a^{\prime}}\\!\sigma$—the vector tangential at
the point $x^{\prime}$ to the geodetic connecting $x$ and $x^{\prime}$
(generalizing the notion of the flat space vector $x^{\prime}-x$ to curved
spacetime). The coefficients of this Taylor expansion are local functions of
spacetime metric, its curvature and background fields, and thus provide all
the goals of perturbative UV renormalization of local field models and their
effective field theory expansion.

Unfortunately, this powerful technique does not directly apply to higher order
operators or to non-minimal operators of any order, when their highest
derivative term is not a multiple of a unit matrix in the space of fields.
Examples of this happen already in the simplest case of the electromagnetic
field treated in a generic Lorenz covariant gauge with $\hat{F}(\nabla)\equiv
F^{a}_{b}(\nabla)=\Box\,\delta^{a}_{b}+\lambda\nabla^{a}\nabla_{b}+\ldots$,
for a wide class of such gauges in Einstein theory [3] and in higher-
derivative gravity theories [12, 13]. There exist indirect methods which
reduce the problems with higher-derivative and non-minimal operators to those
of the minimal second-order one — operator factorization into the product of
second-order ones and the method of the so-called universal functional traces
[14, 3]. The latter is based on the calculation of the sequence of coincidence
limits of the following two-point kernels

$\left.\nabla_{a_{1}}...\nabla_{a_{n}}\frac{1}{(-\Box)^{m}}\,\delta(x,x^{\prime})\,\right|_{x=x^{\prime}},$
(1.3)

which are calculable by the Schwinger–DeWitt technique of the above type,
because the $m$-th power of the inverse Laplacian is easily generated via the
proper time integral of the heat kernel of the minimal operator $\Box$.222The
essence of this method can be demonstrated on the example, say, of the higher
derivative theory with the inverse propagator $(-\Box)^{N}+P(\nabla)$. By
expanding the one-loop functional determinant,
$\operatorname{Tr}\ln\big{(}(-\Box)^{N}+P(\nabla)\big{)}=N\,\operatorname{Tr}\ln\Box+\operatorname{Tr}\ln\big{(}1+P(\nabla)/(-\Box)^{N}\big{)}$,
in powers of the nonlocal term $P(\nabla)/(-\Box)^{N}$ and commuting to the
left the powers of $P(\nabla)$ and to the right—the inverse powers of $\Box$,
one finds that the result will be given by the infinite series of terms (1.3)
multiplied by tensors of ever growing dimensionality. This method turned out
to be very efficient in the calculation of beta functions of the
$(3+1)$-dimensional Hořava gravity model [15]—the task which is apparently
impossible to accomplish within a conventional momentum space diagrammatic
technique on flat space background (because of the necessity to calculate the
contributions of hundreds of thousands of relevant Feynman graphs).

The efficiency of these indirect methods, the use of which is always
contingent on concrete peculiarities of the model, does not retract the
necessity of the general method applicable to a widest possible scope of
problems. Moreover, indirect methods usually allow one to calculate only the
heat kernel diagonal $\hat{K}(\tau|x,x)$ or the spacetime integral of this
coincidence limit, which is usually everything one needs in the one-loop
approximation of quantum field theory. However, one might need it beyond this
limit to obtain, for example, the two-point Green function and, what is even
more important, to apply it in multi-loop orders. In addition, even the
calculation of the expansion (1.2) requires to operate with $x^{\prime}\neq
x$, because the recurrent equations for $\hat{a}_{m}(x,x^{\prime})$ fully
involve their two-point functions rather than $\hat{a}_{m}(x,x)$. Extension
beyond the diagonal $x^{\prime}=x$ brings to life additional difficulty which
was briefly discussed in [16].

The series (1.2) can be considered as an expansion in powers of background
field dimensionality because in the coincidence limit every HaMiDeW-
coefficient has the structure of the sum of monomials in spacetime curvatures
of dimension two in units of inverse length $l$, $\mathcal{R}\sim 1/l^{2}$,
collectively denoted by $\mathcal{R}$, and their covariant derivatives

$\hat{a}_{m}(x,x)\propto\sum\nabla^{n}\mathcal{R}^{k},\qquad n+2k=2m.$ (1.4)

Denoting the dimension of such terms $\mathcal{O}\sim 1/l^{n}$ by
$\dim\mathcal{O}=n$, one has for the second order operator $\hat{F}(\nabla)$
and its proper time parameter $\tau$

$\dim\left(\nabla^{n}\mathcal{R}^{k}\right)=n+2k,\qquad\dim\tau=-2,$ (1.5)

so that every term in the expansion (1.2) is dimensionless, and the power of
$\tau$ can be regarded as grading the dimension of the corresponding
coefficient $\hat{a}_{m}$.333With two-point coefficients
$\hat{a}_{m}(x,x^{\prime})$ the situation is more complicated because they
also contain dimensional $\sigma^{a}(x,x^{\prime})$, $\dim\sigma^{a}=-1$, but
within the covariant Taylor expansion
$\hat{a}_{m}(x,x^{\prime})\sim\sum(-1)^{n}\sigma^{a_{1}}...\sigma^{a_{n}}(\nabla_{a_{1}}...\nabla_{a_{n}}a_{m}\,)_{x=x^{\prime}}/n!$,
and the dimension of $\sigma^{a}$’s is canceled by extra derivatives, so that
$\dim a_{m}=2m$.

For the coincidence limit $x^{\prime}=x$ the tensor structures
$\nabla^{n}\mathcal{R}^{k}$ and $\tau$ are everything that enters the
formalism, the dimensionality of $\tau$-powers compensating in each term of
the series the dimensionality of $\nabla^{n}\mathcal{R}^{k}$. Nothing else can
appear for dimensional reasons, and the expansion indeed runs in powers of the
background dimensionality. However, for separated points the additional
dimensionless structure

$\frac{\sigma^{a^{\prime}}(x^{\prime},x)}{\tau^{1/N}},$ (1.6)

with $N=2$—the order of the operator $\hat{F}$—enters the game. Its appearance
cannot be prohibited by any dimensional considerations, so that each tensor
structure $\nabla^{n}\mathcal{R}^{k}$ can enter the formalism with arbitrarily
high power of $\sigma^{a^{\prime}}/\tau^{1/2}$. So a priori one cannot look
for the heat kernel expansion ansatz in positive powers of the proper time.
Remarkably, for minimal second order operators infinite series of negative
powers of $\tau$ gets resummed into the Gaussian function of the argument
(1.6), which is regular at $x^{\prime}=x$ ($\sigma(x,x^{\prime})=0$) but has
essential singularity at $\tau=0$. The fact that it factorizes in (1.2) as an
overall multiplier is very reminiscent of the semiclassical approximation with
$\tau\to 0$ playing the role of the Planck constant $\hbar$ [16].

One could have expected that the same resummation in powers of (1.6) would be
possible for minimal higher order444 A typical example of higher-order minimal
operators are the so-called “conformally covariant differential operators”,
such as the 4th order Paneitz operator
$\Delta_{4}=\Box^{2}+2R^{ab}\nabla_{a}\nabla_{b}-\tfrac{2}{3}R\Box+\tfrac{1}{3}(\nabla^{a}R)\nabla_{a}$
and its higher-order analogs. There is a vast literature devoted to the study
of such operators, see, for example, [17, 18, 19] operators of generic order
$N=2M>2$

$\hat{F}(\nabla)=(-\Box)^{M}\hat{1}+\hat{P}(\nabla),$ (1.7)

where $\hat{P}(\nabla)$ is its lower derivative part (note that now
$\dim\tau=-N$). However, the attempt of such a resummation in terms of the
generalized exponential functions (GEF) fails [16], which is apparently
related to the failure of the relevant semiclassical approximation [20] in the
vicinity of $x^{\prime}=x$.555As discussed in [16], asymptotic expansion
obtained by the saddle-point approximation at $\tau\to 0$ works well as a
generalized function, satisfying in particular the required initial condition
$K(0|x,x^{\prime})=\delta(x,x^{\prime})$, but this expansion for $M>1$ is not
homogeneous for $x^{\prime}\to x$ and, therefore, gives singular results for
the physically important coincidence limit $K(\tau|x,x)$. This makes it
impossible to apply in this coincidence limit the semiclassical approximation
of [20]. This also invalidates the attempt, undertaken in [21] to build the
recurrent relations for the generalized HaMiDeW-coefficients for fourth-order
operators, because of the infinite series of negative powers in $\tau$.

Thus, the goal of this paper is to try circumventing these difficulties and
develop a workable calculational method for generic higher-derivative
operators. Rather than merely formulating general statements on the structure
of asymptotic expansion for the functional trace of the heat kernel, well
known from mathematics literature [6, 7, 8, 22, 23, 24, 25], we will derive
concrete algorithms for the expansion of the two-point heat kernel with
separate points. In view of the discussion above, for $x\neq x^{\prime}$ it
will not be an asymptotic expansion in $\tau\to 0$, because of the presence of
infinite series with negative powers of the proper time. Rather it will be an
expansion in the operators of positive dimension, the spacetime curvature,
background fields and their derivatives—what is just needed in UV
renormalization and effective field theory. Concrete algorithms for this
expansion will involve recurrent procedure for the calculation of what we call
generalized HaMiDeW coefficients—the coefficients of the series in $\tau$,
weighted by the sequence of GEFs that were introduced in [16]. For generic
higher-derivative operators these functions replace the overall usual
exponential function of the Schwinger–DeWitt series (1.2). We will show that
in the coincidence limit $x^{\prime}=x$ all generalized HaMiDeW-coefficients
of negative powers of $\tau$ vanish and, therefore, result in the conventional
asymptotic $\tau\to 0$ expansion for the heat kernel diagonal.

The method of derivation of all these results, is the generalized Fourier
transform in curved spacetime. Basically, it consists in the replacement of
the usual momentum space integrals in Cartesian coordinates by the integrals
over the momentum dual to the distinguished coordinate variable—the vector
$\sigma^{a^{\prime}}(x,x^{\prime})$ tangential to the geodetic interpolating
between the points $x$ and $x^{\prime}$. In fact, the Fourier integral method
underlies the original statements of spectral geometry of (pseudo)differential
operators of [6, 7]. Its covariant version was firstly introduced by H. Widom
[26, 27, 28] and then it was used in the works by Gusynin et al [29, 30, 31,
32, 33, 34] addressing various problems with second-order non-minimal and
minimal higher-order operators. Here we systematically use this method
starting with generic non-minimal operator of $N$-th order, then push forward
the resulting formalism to the case of minimal operators of arbitrary even
order $N=2M$. Then we show how the case of non-minimal operators, which
satisfy the so-called causality condition [3], can be fully reduced to the
minimal case.

The paper is organized as follows. Main results for two-point heat kernel are
listed in Sect. 2. Their derivation is given in Sects. 3 and 4, along with a
special emphasis on heat kernel diagonal elements. In Sect. 5 we recover known
results of the Schwinger-DeWitt technique for minimal second order operators.
In Sect. 6 these results are extended to generic minimal fourth-order
operators. Sect. 7 is devoted to relations between the heat kernels of
differential operators and their powers, including non-integer powers
corresponding to pseudo-differential case. Sect. 8 contains the discussion of
the suggested technique along with its computational limitations, which might
perhaps be circumvented within non-commutative algebra of the covariant $\Box$
operator in the method of universal functional traces of [3].

##  2 Summary of main results

We begin with the definition of the covariant matrix-valued differential
operator $\hat{F}(\nabla)=F^{A}_{B}(\nabla)$ of order $N$ acting on generic
set of fields $\varphi^{B}(x)$ in $d$-dimensional spacetime

$\hat{F}(\nabla)=\sum\limits_{k=0}^{N}\hat{F}_{k}^{a_{1}\ldots
a_{k}}(x)\nabla_{a_{1}}\ldots\nabla_{a_{k}},$ (2.1)

where $\hat{F}_{k}^{a_{1}\ldots a_{k}}(x)$ are its coefficients. Since
commutation of covariant derivatives leads to a decrease in the order, one can
consider these coefficients $\hat{F}_{k}(x)$ as completely symmetric tensors.
Hereinafter, we will systematically omit convolutions over repeated multiple
indices and use the following abbreviated notation

$\hat{F}_{k}*\nabla^{k}=\hat{F}_{k}^{a_{1}\ldots
a_{k}}\nabla_{a_{1}}\ldots\nabla_{a_{k}}.$ (2.2)

Thus,

$\hat{F}(\nabla)=\sum\limits_{k=0}^{N}\hat{F}_{k}(x)*\nabla^{k}.$ (2.3)

The main result for the heat kernel $\hat{K}(\tau|x,x^{\prime})$ of a generic
operator (2.1) can be formulated in terms of the following ingredients of the
geodetic interpolating between the points $x$ and $x^{\prime}$—the parallel
transport tensor $\hat{\mathcal{I}}(x,x^{\prime})$, the vectors
$\sigma^{a}(x,x^{\prime})$ and $\sigma^{a^{\prime}}(x,x^{\prime})$ tangent to
this geodetic respectively at $x$ and $x^{\prime}$ and the sequence of special
two-point tensor and matrix valued differential operators
$\hat{T}_{k,l}(\nabla)=\hat{T}_{k,l}^{a^{\prime}_{1}\ldots
a^{\prime}_{l}}(x,x^{\prime}|\nabla)$ which will be determined by the series
of recurrent equations in Sect.3. The resulting heat kernel expansion reads

$\displaystyle\hat{K}(\tau|x,x^{\prime})=\sum\limits_{k=0}^{\infty}\tau^{n}\\!\\!\\!\\!\sum\limits_{\;\;\;0\leq
l\leq
L_{n}}\\!\\!\\!\\!\hat{S}_{l}(\tau)*\hat{T}_{n,l}(\nabla)\,\hat{\mathcal{I}}(x,x^{\prime}),$
(2.4) $\displaystyle L_{n}(N)=\Big{(}N-\frac{1}{2}\,\Big{)}\,n,$ (2.5)

where $\hat{S}_{l}(\tau)=\hat{S}_{l,b^{\prime}_{1}\ldots
b^{\prime}_{l}}(\tau|x,x^{\prime})$ are the two-point matrix-valued tensors of
$l$-th rank at $x^{\prime}$ given by the following momentum integrals

$\hat{S}_{l,b^{\prime}_{1}\ldots
b^{\prime}_{l}}(\tau|x,x^{\prime})=\big{(}\det\bar{\sigma}^{a}_{b^{\prime}}\big{)}\,\bar{\sigma}^{a_{1}}_{b^{\prime}_{1}}\ldots\bar{\sigma}^{a_{l}}_{b^{\prime}_{l}}\\\
\times\int\frac{d^{d}p}{(2\pi)^{d}}(ip_{a_{1}})\ldots(ip_{a_{l}})\,\exp\left(-\tau\hat{F}_{N}(x,\bm{p})+ip_{a}\sigma^{a}\right)$
(2.6)

with $\hat{F}_{N}(x,\bm{p})$—the matrix-valued principal symbol of the
operator $\hat{F}(\nabla)$

$\hat{F}_{N}(x,\bm{p})=\hat{F}_{N}^{a_{1}\ldots
a_{N}}(x)\times(ip_{a_{1}})\ldots\times(ip_{a_{N}}).$ (2.7)

Here the tangent geodetic vectors $\sigma^{a}(x,x^{\prime})$,
$\sigma^{a^{\prime}}(x,x^{\prime})$ and the vector bundle parallel transport
tensor $\hat{\mathcal{I}}(x,x^{\prime})$ satisfy the equations

$\displaystyle\sigma^{a}\nabla_{a}\hat{\mathcal{I}}=0,\qquad\big{[}\,\hat{\mathcal{I}}\,\big{]}=\hat{1},$
(2.8)
$\displaystyle\sigma^{b}\nabla_{b}\sigma^{a^{\prime}}=\sigma^{a^{\prime}},\;\;\sigma^{b^{\prime}}\nabla_{b^{\prime}}\sigma^{a}=\sigma^{a},\;\;\big{[}\,\sigma^{a}\big{]}=\big{[}\,\sigma^{a^{\prime}}\big{]}=0,$
(2.9)

and in their turn determine the matrix
$\sigma^{b^{\prime}}_{a}=\sigma^{b^{\prime}}_{a}(x,x^{\prime})$ and its
inverse
$\bar{\sigma}^{a}_{b^{\prime}}=\bar{\sigma}^{a}_{b^{\prime}}(x,x^{\prime})$,

$\sigma^{b^{\prime}}_{a}=\nabla_{a}\sigma^{b^{\prime}},\quad\bar{\sigma}^{a}_{b^{\prime}}\sigma^{b^{\prime}}_{c}=\delta^{a}_{c},$
(2.10)

for any two-point function squared brackets are its coincidence limits

$[f(x,x^{\prime})]=f(x,x).$ (2.11)

It is important that, although the expansion (2.4) looks as the expansion in
$\tau\to 0$, in fact it is not a small time expansion of the heat kernel,
because the calculation of the integrals (2.6) can lead to arbitrarily high
negative powers of $\tau$. Rather, this is the expansion in powers of the
background dimensionality—what is exactly needed in renormalization and
effective field theory. This follows from the fact that the differential
operators $\hat{T}_{n,l}(\nabla)$ have the dimensionalities,

$\dim\,\hat{T}_{n,l}(\nabla)=nN-l\geq\frac{n}{2},$ (2.12)

monotonically growing with $n$ in view of the upper bound (2.5) on the
summation index $l$ in (2.4).

Note that a generic operator (2.1) does not assume the existence of the
spacetime metric and involves only the vector bundle connection in
$\nabla_{a}\varphi=(\partial_{a}+\hat{\varGamma}_{a})\,\varphi$ and the affine
connection which without loss of generality we assume to be symmetric.
Therefore, the tangent geodetic vectors at this stage are not the gradients of
the world function $\sigma(x,x^{\prime})$ which can be introduced only along
with the metric structure.

The metric structure emerges together with the definition of the minimal
differential operators. Their highest derivative term is defined as the power
of the covariant Laplacian $\Box=g^{ab}\nabla_{a}\nabla_{b}$ times the unit
matrix $\hat{1}$ in the vector space of the fields $\varphi$. Thus, a minimal
differential operator of general even order $N=2M$ is defined as

$\displaystyle\hat{F}(\nabla)=(-\Box)^{M}\,\hat{1}+\hat{P}(\nabla),$ (2.13)
$\displaystyle\hat{P}(\nabla)=\sum\limits_{k=0}^{2M-1}\hat{P}_{k}(x)*\nabla^{k},$
(2.14)

where $\hat{P}(\nabla)$ is the lower derivative part—an arbitrary differential
operator of the order $2M-1$.

As long as the spacetime metric $g_{ab}$ now enters the formalism, we will
assume that covariant derivatives conserve it, $\nabla_{a}g_{bc}=0$, and the
tangent geodetic vectors
$\sigma^{a^{\prime}}=\sigma^{a^{\prime}}(x^{\prime},x)$ and
$\sigma^{a}=\sigma^{a}(x,x^{\prime})$,

$\sigma^{a}=\nabla^{a}\sigma,\quad\sigma^{a^{\prime}}=\nabla^{a^{\prime}}\sigma,$
(2.15)

become the gradients of the world function $\sigma=\sigma(x,x^{\prime})$—the
squared geodetic distance between $x$ and $x^{\prime}$, which satisfies with
respect to its both arguments the equations

$\sigma=\frac{1}{2}\,\sigma^{a}\sigma_{a}=\frac{1}{2}\,\sigma^{a^{\prime}}\sigma_{a^{\prime}}.$
(2.16)

The principal symbol (2.7) of a minimal operator simplifies to

$\hat{F}_{N}(x,\bm{p})=\big{(}g^{ab}(x)\,p_{a}p_{b}\big{)}^{M}\,\hat{1}\equiv
p^{2M}\,\hat{1},$ (2.17)

and the set of integrals (2.6) expresses via a single basic integral in terms
of the generalized exponential function (GEF) $\mathcal{E}_{M,d/2}(z)$ of the
argument $z=-\sigma/2\tau^{1/M}=-\sigma^{a}\sigma_{a}/4\tau^{1/M}$,

$\int\frac{d^{d}p}{(2\pi)^{d}}\exp\left(-\tau
p^{2M}+ip_{a}\sigma^{a}\right)\\\
=\frac{g^{1/2}(x)}{\left(4\pi\tau^{1/M}\right)^{d/2}}\,\mathcal{E}_{M,d/2}\left(-\frac{\sigma}{2\tau^{1/M}}\right).$
(2.18)

This function was introduced in [16], and its main properties are presented in
Appendix B. Via the differentiation with respect to $\sigma^{b}$ it generates
a sequence of new fully symmetric tensor coefficients

$S_{p,l}=S_{p,l,\,a^{\prime}_{1}\ldots a^{\prime}_{l}}(x,x^{\prime}).$ (2.19)

These are the polynomials in powers of $\sigma_{a^{\prime}}$ and the factors
of the form
$\gamma_{a^{\prime}b^{\prime}}=\bar{\sigma}_{a^{\prime}}^{c}g_{cd}\bar{\sigma}^{d}_{b^{\prime}}$,
which arise according to the equation

$\Big{(}\prod\limits_{j=1}^{l}\bar{\sigma}^{b_{j}}_{a^{\prime}_{j}}\frac{\partial}{\partial\sigma^{b_{j}}}\Big{)}\,\mathcal{E}_{M,\frac{d}{2}}\left(-\frac{\sigma^{b}\sigma_{b}}{4\tau^{1/M}}\right)\\\
=\sum\limits_{p\geq\frac{l}{2}}^{l}\frac{S_{p,l,a^{\prime}_{1}\ldots
a^{\prime}_{l}}}{\left(-2\,\tau^{1/M}\right)^{p}}\,\mathcal{E}_{M,\frac{d}{2}+p}\left(-\frac{\sigma}{2\,\tau^{1/M}}\right),$
(2.20)

due to the main property (B.8) of $\mathcal{E}_{\nu,\mu}(z)$,
$\partial_{z}\mathcal{E}_{\nu,\mu}(z)=\mathcal{E}_{\nu,\mu+1}(z)$.

Main result for the heat kernel of a generic minimal operator of any order
$2M$ is then as follows. Its expansion in fractional powers of proper time
reads as

$\hat{K}(\tau|x,x^{\prime})=\frac{\Delta^{-1}(x,x^{\prime})}{(4\pi\tau^{1/M})^{d/2}}\,g^{1/2}(x^{\prime})\sum\limits_{m=-\infty}^{\infty}\tau^{m/M}\\\
\times\sum\limits_{\,n\geq
N_{m}}^{\infty}\mathcal{E}_{M,\frac{d}{2}+Mn-m}\left(-\frac{\sigma}{2\tau^{1/M}}\right)\,\hat{b}_{m,n}(x,x^{\prime}).$
(2.21)

where the lower bound on the summation index $n$ in each $m$-th order of
$\tau^{1/M}$ is

$\displaystyle N_{m}(M)=\begin{cases}\,\frac{m}{M},&m>0,\\\ &\\\
\,\frac{2|m|}{2M-1},&m<0,\end{cases}$ (2.22)

$\Delta(x,x^{\prime})$ is the Pauli–Van Vleck–Morette determinant (A.16) and
$\hat{b}_{m,n}(x,x^{\prime})$ are the generalized HaMiDeW coefficients given
by the following finite sums in terms of the above operators
$\hat{T}_{n,l}(\nabla)$ and tensor coefficient functions
$S_{p,l}(x,x^{\prime})$,

$\hat{b}_{m,n}(x,x^{\prime})=\frac{1}{(-2)^{Mn-m}}\\\
\times\sum\limits_{l=Mn-m}^{\lfloor
L_{m,n}\rfloor}S_{Mn-m,l}(x,x^{\prime})*\hat{T}_{n,l}(\nabla)\,\hat{\mathcal{I}}(x,x^{\prime}).$
(2.23)

Here the upper bound on the summation index $l$ is the integer part $\lfloor
L_{m,n}\rfloor$ of

$\displaystyle L_{m,n}$ $\displaystyle=$ $\displaystyle 2Mn-{\rm
max}\left\\{\,2m,\frac{n}{2}\,\right\\}.$ (2.24)

This is again effectively the expansion in background dimensionality, because
as it follows from (2.12) the dimensionalities of
$\hat{b}_{m,n}(x,x^{\prime})$ are always positive and grow with $m$ and $n$,

$\dim\,\hat{b}_{m,n}\geq\min_{\\{\,l\,\\}\vphantom{L^{L}}}\big{(}\dim\,\hat{T}_{n,l}\big{)}=\max\left\\{\,2m,\frac{n}{2}\,\right\\}.$
(2.25)

Below we go over to the derivation of the above results along with the
formulation of recurrent equations for the operators
$\hat{T}_{k,l}^{a^{\prime}_{1}\ldots a^{\prime}_{l}}(x,x^{\prime}|\nabla)$.
This will be followed by verification of the consistency of the obtained
formalism with known properties of the Schwinger–DeWitt technique.

##  3 General non-minimal operators

We define the heat kernel $\hat{K}(\tau|x,x^{\prime})$ of $\hat{F}$ as the
kernel of the operator $e^{-\tau\hat{F}}$, i.e. as the result of acting by
this operator on the two-point delta function
$\hat{\delta}(x,x^{\prime})=\delta^{A}_{B^{\prime}}(x,x^{\prime})$ in the
space of fields $\varphi=\varphi^{A}(x)$,

$\hat{K}(\tau|x,x^{\prime})=e^{-\tau\hat{F}}\,\hat{\delta}(x,x^{\prime})$
(3.1)

The delta function is itself defined by the convolution with $\varphi(x)$

$\varphi(x)=\int
d^{d}x^{\prime}\,\hat{\delta}(x,x^{\prime})\,\varphi(x^{\prime})=\int
d^{d}x^{\prime}\,\hat{\delta}^{A}_{B^{\prime}}(x,x^{\prime})\,\varphi^{B^{\prime}}(x^{\prime}),$
(3.2)

from which it follows that $\hat{\delta}(x,x^{\prime})$ is the density of unit
weight in $x^{\prime}$ and has zero weight in the first argument.666In view of
the point-like support of the delta-function, it can be written down as
$\hat{1}\,\delta(x,x^{\prime})=\delta^{A}_{B^{\prime}}\delta(x,x^{\prime})$,
but the distinction between the indices $A$ and $B^{\prime}$ associated
respectively with the first and the second spacetime points should be clearly
kept, especially when the delta function is acted upon by the covariant
derivatives in the vector bundle of $\varphi^{A}(x)$.

###  3.1 The generalized Fourier transform

The covariant Fourier transform method requires to consider the “plane-wave”
functions with the momentum vector $k_{b^{\prime}}$,

$\exp\left(ik_{b^{\prime}}\sigma^{b^{\prime}}(x,x^{\prime})\right),$ (3.3)

which are based on the coordinate vector
$\sigma^{b^{\prime}}(x,x^{\prime})$—the analogue of the vector $x^{\prime}-x$
in flat spacetime. This is the vector tangential at the point $x^{\prime}$ to
the geodetic curve interpolating between the points $x^{\prime}$ and $x$ and
satisfying the equation and initial conditions (2.9).777We remind that at this
point we do not need a metric $g_{ab}$ on the spacetime manifold, which is not
at all encoded in the generic differential operators of the form (2.1). The
metric structure and associated Synge world function $\sigma(x,x^{\prime})$
appears later for minimal differential operators, in which case the vector
$\sigma^{b^{\prime}}(x,x^{\prime})$ expresses via $\sigma(x,x^{\prime})$ as
$\sigma^{b^{\prime}}(x,x^{\prime})=\nabla^{b^{\prime}}\sigma(x,x^{\prime})$.
Note that with this definition the plane waves (3.3) are scalars at the point
$x$—bear in mind that $\sigma^{b^{\prime}}(x,x^{\prime})$ is here the scalar
as a function of $x$ and the vector cotangent to spacetime at the second point
$x^{\prime}$.

The generalized Fourier transform in curved spacetime is based on the
covariant integral representation for the delta-function. For the delta-
function acting on a scalar it follows from an obvious relation

$\delta(x,x^{\prime})=\delta\big{(}\sigma^{a^{\prime}}(x,x^{\prime})\big{)},$
(3.4)

justified by the fact that
$\big{[}\det\big{(}-\sigma^{a^{\prime}}_{b}\big{)}\big{]}=1$, and the
subsequent representation of $\delta\big{(}\sigma^{a^{\prime}}\big{)}$ in the
form of the Fourier integral over the momentum space of vectors
$k_{a^{\prime}}$ cotangent to the spacetime at the point $x^{\prime}$.

The generalization of this representation to generic vector bundle fields by a
simple multiplication with $\hat{1}$ will not be consistent for the following
reason. In what follows we will have to work out a covariant formalism for the
integrand of the corresponding momentum space integral, but the multiplication
of this integrand by $\hat{1}$ will not convert this object to a bi-tensor
quantity with correct transformation law of its both arguments $x$ and
$x^{\prime}$. Thus, the spacetime covariant definition of the delta-function
acting in the generic vector bundle of spin-tensor fields should read

$\hat{\delta}(x,x^{\prime})=\int\frac{d^{d}k}{(2\pi)^{d}}\exp\left(ik_{a^{\prime}}\sigma^{a^{\prime}}\right)\,\hat{\mathcal{I}}(x,x^{\prime}),$
(3.5)

where
$\hat{\mathcal{I}}(x,x^{\prime})=\mathcal{I}^{A}_{B^{\prime}}(x,x^{\prime})$
is some two-point spin-tensor quantity having correct transformation
properties with respect to the multindex $A$ associated with $x$ and the index
$B^{\prime}$ associated with $x^{\prime}$. The only restriction on the choice
of $\hat{\mathcal{I}}(x,x^{\prime})$ is the requirement that it should reduce
to the unit matrix at $x^{\prime}=x$. Otherwise it can be rather arbitrary,
and can generate different representations of one and the same quantity.

The simplest choice of $\hat{\mathcal{I}}(x,x^{\prime})$, one can figure out,
is apparently the parallel transport tensor satisfying the equations (2.8)
which provide the parallel translation of spin-tensor objects along the
geodetic connecting the points $x$ and $x^{\prime}$. This operator in fact
coincides with the zeroth HaMiDeW-coefficient in the Schwinger-DeWitt
expansion (1.2), $\hat{\mathcal{I}}(x,x^{\prime})=\hat{a}_{0}(x,x^{\prime})$
and in mathematics literature is usually associated with Widom calculus.

Note that within the definition (3.5) the exponential factor in the integrand
of the Fourier integral is the scalar of zero weight with respect to the
diffeomorphisms of $x$, and after the $k_{a^{\prime}}$-integration generates
the scalar density of weight one at $x^{\prime}$, which is in full accordance
with the transformation properties of the right hand side of (3.4).888Point-
like support of the delta-function fixes only its overall weight with respect
to diffeomorphisms of both arguments (which is one), but the way this weight
can be splitted between $x$ and $x^{\prime}$ in the integrand of the Fourier
integral representation depends on the choice of this representation—with
another choice, say $\delta\big{(}\sigma^{a}(x,x^{\prime})\big{)}$, the delta-
function would be a zero-weight scalar with respect to $x^{\prime}$, but
weight one with respect to $x$. In contrast to this, prior to this integration
the exponential function in question does not have good transformation
properties as a function of $x^{\prime}$ because of the contraction of
$\sigma^{a^{\prime}}$ with the diffeomorphism inert vector $k_{a^{\prime}}$,
but this drawback of the formalism can be disregarded, because we will develop
a fully covariant formalism (for the Fourier image of the heat kernel)
entirely at the point $x$.

Now we act by the operator $e^{-\tau\hat{F}(\nabla)}$ upon (3.5) and commute
the exponential plane-wave function with $\hat{F}(\nabla)$ according to the
following relation

$\exp\big{(}-ik_{b^{\prime}}\sigma^{b^{\prime}}\big{)}\hat{F}(\nabla_{a})\exp\big{(}ik_{b^{\prime}}\sigma^{b^{\prime}}\big{)}=\hat{F}\big{(}\nabla_{a}+ik_{b^{\prime}}\sigma^{b^{\prime}}_{a}\big{)},$
(3.6)

where $\sigma^{b^{\prime}}_{a}$ is the bi-tensor (2.10). Therefore, the heat
kernel takes the form

$\displaystyle\hat{K}(\tau|x,x^{\prime})=\int\frac{d^{d}k}{(2\pi)^{d}}\exp(ik_{a^{\prime}}\sigma^{a^{\prime}})\,\hat{\bm{K}}(\tau,\bm{k}|x,x^{\prime}),$
(3.7)
$\displaystyle\hat{\bm{K}}(\tau,\bm{k}|x,x^{\prime})=\exp\left(-\tau\,\hat{F}\big{(}\nabla_{a}+ik_{b^{\prime}}\sigma^{b^{\prime}}_{a}\big{)}\right)\hat{\mathcal{I}}(x,x^{\prime}),$
(3.8)

where its Fourier image $\hat{\bm{K}}(\tau,\bm{k}|x,x^{\prime})$, which we
will denote by bold letters, obviously satisfies the following initial Cauchy
problem

$\displaystyle\left(\partial_{\tau}+\hat{F}\big{(}\nabla_{a}+ik_{b^{\prime}}\sigma^{b^{\prime}}_{a}\big{)}\right)\hat{\bm{K}}(\tau,\bm{k}|x,x^{\prime})=0,$
(3.9)
$\displaystyle\hat{\bm{K}}(0,\bm{k}|x,x^{\prime})=\hat{\mathcal{I}}(x,x^{\prime}).$
(3.10)

Why would we need the solution of this problem if we already have it in a
closed form (3.8) and can explicitly expand it in powers of the exponential?
Point is that this expansion will not be a needed expansion in background
dimensionality $1/l$, because both (2.1) and (3.6) have terms of zeroth order
in $1/l$—see the leading term of (2.1) in derivatives with ${\rm
dim}\big{(}\hat{F}_{N}^{a_{1}\ldots a_{N}}(x))=0$. Thus, first we have to
disentangle from
$\hat{F}(\nabla_{a}+ik_{b^{\prime}}\sigma^{b^{\prime}}_{a}\big{)}$ this
$O(1/l^{0})$ term, explicitly find the zeroth order solution and then develop
the perturbation theory in $1/l$.

###  3.2 The solution for $\hat{\bm{K}}\left(\tau,\bm{k}|x,x^{\prime}\right)$

To do this, we first expand the operator
$\hat{F}(\nabla_{a}+ik_{b^{\prime}}\sigma^{b^{\prime}}_{a})$ in powers of
momenta $\bm{k}$,

$\hat{F}\left(\nabla_{a}+ik_{b^{\prime}}\sigma^{b^{\prime}}_{a}\right)=\sum\limits_{m=0}^{N}(i\bm{k})^{m}*\llbracket\hat{F}\rrbracket_{m},$
(3.11)

where we introduced the notations for monomials in momenta
$\bm{k}^{m}=k_{b^{\prime}_{1}}\ldots k_{b^{\prime}_{m}}$ and their coefficient
differential operators
$\llbracket\hat{F}\rrbracket_{m}=\llbracket\hat{F}\rrbracket_{m}^{b^{\prime}_{1}\ldots
b^{\prime}_{m}}(\nabla)$ and also use the same abbreviation of indices
contraction as in (2.2). The rule of constructing the operators
$\llbracket\hat{F}\rrbracket_{m}$ actually follows from the definition (3.11),

$\llbracket\hat{F}\rrbracket_{m}=\sum\limits_{k=m}^{N}\hat{F}_{k}(x)*\llbracket\nabla^{k}\rrbracket_{m},$
(3.12)

where the action of the operation $\llbracket\ldots\rrbracket_{m}$ on every
monomial of derivatives $\nabla^{k}=\nabla_{a_{1}}\ldots\nabla_{a_{k}}$
consists in replacing $m$ of them by the functions of the form
$\sigma^{b^{\prime}}_{a}$. This gives the sum of $\binom{k}{m}$ terms each
having extra $m$ contravariant primed indices
$b^{\prime}_{1},\ldots,b^{\prime}_{m}$. The original order of derivatives and
inserted factors $\sigma^{b^{\prime}}_{a}$ should be strictly retained, and
the additional obvious property holds: $\llbracket\nabla^{k}\rrbracket_{m}=0$
if $m>k$.

Thus, the operation $\llbracket\ldots\rrbracket_{m}$ reduces the order of
every differential operator by $m$ and adds to it extra $m$ primed indices
associated with the point $x^{\prime}$. In particular, for our $N$-th order
operator (2.2) its $\llbracket\hat{F}\rrbracket_{N}$ is not a differential
operator, but just a two-point function which is a scalar at $x$ and a tensor
at $x^{\prime}$ with $N$ contravariant indices

$\llbracket\hat{F}\rrbracket_{N}\equiv\llbracket\hat{F}\rrbracket_{N}^{b^{\prime}_{1}...b^{\prime}_{N}}=\hat{F}_{N}^{a_{1}\ldots
a_{N}}\sigma^{b^{\prime}_{1}}_{a_{1}}\ldots\sigma^{b^{\prime}_{N}}_{a_{N}}.$
(3.13)

When contracted with $(i\bm{k})^{N}$ it will play the role of the principal
symbol of $\hat{F}(\nabla)$ usually used in the Fourier analysis of
differential and pseudo-differential operators.

The dimensionality of these operators is equal to their order

$\dim\,\llbracket\hat{F}\rrbracket_{m}=N-m.$ (3.14)

Therefore, to develop an efficient expansion in powers of the background
dimensionality it would be sufficient to expand
$\hat{\bm{K}}(\tau,\bm{k}|x,x^{\prime})$ in powers of all
$\llbracket\hat{F}\rrbracket_{m}$ with $m\leq N-1$ while treating the leading
term $\llbracket\hat{F}\rrbracket_{N}$ exactly. This is of course possible,
because $\llbracket\hat{F}\rrbracket_{N}$ is not a differential operator but
just a matrix, and the lowest order solution of (3.9) is given by the matrix-
valued exponential function

$\hat{\bm{K}}_{0}(\tau,\bm{k})=\exp\left(-\tau(i\bm{k})^{N}*\llbracket\hat{F}\rrbracket_{N}\right)\hat{\mathcal{I}}.$
(3.15)

Therefore, we will look for the solution of Eqs.(3.9)–(3.10) in the form

$\displaystyle\hat{\bm{K}}(\tau,\bm{k})=\exp\Big{(}-\tau(i\bm{k})^{N}*\llbracket\hat{F}\rrbracket_{N}\Big{)}\,\hat{\bm{T}}(\nabla)\,\hat{\mathcal{I}},$
(3.16)

where
$\hat{\bm{T}}(\nabla)\equiv\hat{\bm{T}}(\nabla,\tau,\bm{k}|x,x^{\prime})$ is
some unknown operator to be found.

Substituting the ansatz (3.16) into the equation (3.9) we see that the
commutation of operators $\llbracket\hat{F}\rrbracket_{m}$ with the
exponential function
$\exp\big{(}-\tau(i\bm{k})^{N}*\llbracket\hat{F}\rrbracket_{N}\big{)}$ leads
again to elongation of derivatives

$\nabla_{a}\to\tilde{\nabla}_{a}=\nabla_{a}-\tau(i\bm{k})^{N}*\hat{D}_{a}(\tau,\bm{k}),$
(3.17)

where $\hat{D}_{a}(\tau,\bm{k})$ is given in quadratures as the following
generically nontrivial matrix-valued integral

$\displaystyle\bm{k}^{N}*\hat{D}_{a}(\tau,\bm{k})$
$\displaystyle=\frac{1}{\tau}\int_{0}^{\tau}d\tau^{\prime}\,e^{-\tau^{\prime}(i\bm{k})^{N}*\llbracket\hat{F}\rrbracket_{N}}$
$\displaystyle\times\big{(}\bm{k}^{N}*\nabla_{a}\llbracket\hat{F}\rrbracket_{N}\big{)}\,e^{\tau^{\prime}(i\bm{k})^{N}*\llbracket\hat{F}\rrbracket_{N}},$
(3.18)

which simplifies to $\nabla_{a}\llbracket\hat{F}\rrbracket_{N}$ for a
particular case of the vanishing commutator
$\big{[}\bm{k}^{N}*\llbracket\hat{F}\rrbracket_{N},\bm{k}^{N}*\nabla_{a}\llbracket\hat{F}\rrbracket_{N}\big{]}=0$.
The resulting operators can in their turn be expanded as,

$\llbracket\hat{F}\rrbracket_{m}(\tilde{\nabla})=\sum\limits_{n=0}^{N-m}\left(\tau(i\bm{k})^{N}\right)^{n}*\llbracket\hat{F}\rrbracket_{m,n}(\tau,\bm{k}),$
(3.19)

to give a new set of operators of the order $N-m-n$ with $m+Nn$ contravariant
indices $\llbracket\hat{F}\rrbracket_{m,n}^{b^{\prime}_{1}\ldots
b^{\prime}_{m}c^{\prime}_{1}\ldots c^{\prime}_{Nn}}$,

$\displaystyle\llbracket\hat{F}\rrbracket_{m,n}=\sum\limits_{k=m+n}^{N}\hat{F}_{k}(x)*\llbracket\nabla^{k}\rrbracket_{m,n},,$
(3.20) $\displaystyle\dim\,\llbracket\hat{F}\rrbracket_{m,n}=N-m,$ (3.21)

where $\llbracket\nabla^{k}\rrbracket_{m,n}$ are built by the same pattern as
$\llbracket\nabla^{k}\rrbracket_{m}$—they consist of $\binom{k}{m,n}$ terms,
in each of which $m$ covariant derivatives $\nabla_{a}$ are replaced by the
functions of the form $\sigma^{b^{\prime}}_{a}$, and $n$ covariant derivatives
$\nabla_{a}$ are replaced by the functions of the form
$-\hat{D}_{a}(\tau,\bm{k})$. It is easy to see that
$\llbracket\hat{F}\rrbracket_{m,0}=\llbracket\hat{F}\rrbracket_{m}$ and
$\llbracket\hat{F}\rrbracket_{m,N-m}$ are not operators but just tensor-valued
functions of $x$ and $x^{\prime}$.

After making the required commutation, one finds

$\displaystyle\left(\partial_{\tau}+\hat{\bm{F}}\right)\hat{\bm{T}}(\nabla,\tau,\bm{k})=0,$
(3.22) $\displaystyle\hat{\bm{T}}(\nabla,0,\bm{k})=\hat{1},$ (3.23)

where the new operator
$\hat{\bm{F}}(\nabla)\equiv\hat{\bm{F}}(\nabla,\tau,\bm{k}|x,x^{\prime})$
reads

$\hat{\bm{F}}(\nabla,\tau,\bm{k})=\sum\limits_{m=0}^{N-1}\sum\limits_{n=0}^{N-m}\tau^{n}(i\bm{k})^{m+Nn}*\llbracket\hat{F}\rrbracket_{m,n}(\tau,\bm{k})$
(3.24)

and has a positive dimensionality $\dim\,\hat{\bm{F}}(\nabla)\geq 1$. Now,
however, this “Hamiltonian” operator is time dependent and admits a
perturbative solution in the form of a $T$-ordered exponent

$\hat{\bm{T}}(\nabla,\tau,\bm{k})=T\exp\left(-\int_{0}^{\tau}dt\,\hat{\bm{F}}(\nabla,t,\bm{k})\right)\\\
=\sum\limits_{n=0}^{\infty}(-1)^{n}\int\limits_{0}^{\tau}dt_{1}\int\limits_{0}^{t_{1}}dt_{2}\cdots\\!\\!\int\limits_{0}^{t_{n-1}}dt_{n}\,\hat{\bm{F}}(t_{1})\cdots\hat{\bm{F}}(t_{n}),$
(3.25)

each term of which has at least of $n$-th order in background dimensionality.

In principle, every order of this expansion is explicitly calculable in terms
of the powers of operators $\llbracket\hat{F}\rrbracket_{m,n}$, introduced
above, and the powers of $\tau$ and $\bm{k}$. However, since the operator
(3.24) contains at least $N(N+1)/2$ terms, the number of terms in the $n$-th
order of this expansion will contain $[N(N+1)/2]^{n}$ monomials in powers of
$\llbracket\hat{F}\rrbracket_{m,n}$, not to mention that every
$\llbracket\hat{F}\rrbracket_{m,n}$ is a sufficiently long expression growing
with the operator order $N$.

Therefore, we will consider a simpler case of the vanishing commutator
$\big{[}\bm{k}^{N}*\llbracket\hat{F}\rrbracket_{N},\bm{k}^{N}*\nabla_{a}\llbracket\hat{F}\rrbracket_{N}\big{]}=0$,
when $\hat{D}_{a}=\nabla_{a}\llbracket\hat{F}\rrbracket_{N}$ and $\llbracket
F\rrbracket_{m,n}$ are both $(\tau,\bm{k})$-independent and, instead of
explicitly handling every order of the expansion (3.25), look for
$\hat{\bm{T}}(\tau,\bm{k})$ in the form of double series in powers of $\tau$
and $\bm{k}$. Then we try finding solvable recurrent relations for its
coefficients. This double series reads

$\hat{\bm{T}}(\nabla,\tau,\bm{k})=\sum\limits_{n=0}^{\infty}\\!\sum\limits_{\;\;\;0\leq
l\leq L_{n}}\\!\tau^{n}(i\bm{k})^{l}*\hat{T}_{n,l}(\nabla),$ (3.26)

where $\hat{T}_{n,l}(\nabla)=\hat{T}_{n,l}^{b^{\prime}_{1}\ldots
b^{\prime}_{l}}(\nabla|x,x^{\prime})$ are the unknown operator valued
coefficients which are $l$-th rank tensors at $x^{\prime}$ and $L_{n}$ is some
non-decreasing sequence of finite numbers.

The differentiation $\partial_{\tau}$ in (3.22) decreases the power in the
proper time $\tau$ by one. And the terms in the decomposition (3.24) of the
operator $\hat{\bm{F}}$, on the contrary, increase the power in $\tau$ by $n$
and at the same time the power in the momentum $i\bm{k}$ by $m+Nn$. Therefore,
if we substitute (3.24) and (3.26) into the equation (3.22), and then equate
to zero the terms with each power $\tau^{n}(i\bm{k})^{l}$, we obtain the
following system of recurrence relations for the operators
$\hat{T}_{n,l}(\nabla)$

$\displaystyle(n+1)\hat{T}_{n+1,l}=-\sum\limits_{p=0}^{N-1}\sum\limits_{q=0}^{N-p}\llbracket\hat{F}\rrbracket_{p,q}\,\hat{T}_{n-q,l-p-
Nq},$ (3.27) $\displaystyle\hat{T}_{0,0}=\hat{1},\qquad\hat{T}_{0,l}=0,\quad
l>0.$ (3.28)

This notation implies that $\hat{T}_{k,l}=0$ for $k<0$ or $l<0$ and, as it
should be, the composition of $(p+Nq)$-rank tensor
$\llbracket\hat{F}\rrbracket_{p,q}$ with the $(l-p-Nq)$-rank tensor
$\hat{T}_{n-q,l-p-Nq}$ on the right hand side of this equation gives the
$l$-rank tensor on the left. Initial values (3.28) follow from the initial
condition (3.23) and thus allow one to solve these recurrent relations for all
$\hat{T}_{n,l}$. An example of the calculation of these operators from those
of the lower orders is depicted in Fig. 1 for the case of
$\hat{T}_{6,5}(\nabla)$ and $N=2$.

Figure 1:  Operators $\hat{T}_{n,l}$ for the case $N=2$. The blue line shows
lowest value of $l$, $L_{n}(N)=(N-1/2)n$. Every dot at $(n,l)$-position on the
plane denotes the operator $\hat{T}_{n,l}$. Shaded domain of points shows the
pattern of their causal connection to the operator in the left hand side of
the recurrent equation (3.27) (in the example of $\hat{T}_{6,5}$).

Note that the ratio of the increment of the index $l$ to the increment of the
index $n$ in the transition from the right hand side of (3.27) to its left
hand side is maximal at $q=1$ and $p=N-1$

$\frac{\Delta l}{\Delta n}=\frac{p+Nq}{q+1},\qquad\max_{p,q}\frac{\Delta
l}{\Delta n}=N-\frac{1}{2}.$ (3.29)

This means that the recursive calculation of $\hat{T}_{n,l}$, starting from a
given value of $\hat{T}_{0,0}$, for a given $n$ involves only the values of
$l$ below the bound $L_{n}(N)$ given by Eq.(2.5). This specifies the finite
upper limit on the summation index $l$ that we introduced in (3.26).

Note that with the help of the relation

$\llbracket\hat{A}\hat{B}\rrbracket_{n}=\sum\limits_{k=0}^{n}\llbracket\hat{A}\rrbracket_{k}\llbracket\hat{B}\rrbracket_{n-k},$
(3.30)

which is valid for any two differential operators $\hat{A}(\nabla)$ and
$\hat{B}(\nabla)$, one can resolve the chain of recurrence relations (3.27)
for some of the coefficients $\hat{T}_{k,l}$, obtaining the following closed
expression

$\hat{T}_{k,l}=\frac{1}{k!}\left\llbracket(-\hat{F})^{k}\right\rrbracket_{l},\quad
l<N.$ (3.31)

In addition, the edge coefficients $\hat{T}_{2k,(2N-1)k}$ are also given by
the explicit formula

$\hat{T}_{2k,(2N-1)k}=\frac{1}{2^{k}k!}\left(-\llbracket\hat{F}\rrbracket_{N-1,1}\right)^{k}.$
(3.32)

However, an arbitrary coefficient $\hat{T}_{k,l}$ cannot be obtained by such
simple formulas. We can only say that it can be represented as a sum of terms
(with some coefficients) of the form

$\llbracket\hat{F}^{k_{1}}\rrbracket_{m_{1},n_{1}}\ldots\llbracket\hat{F}^{k_{p}}\rrbracket_{m_{p},n_{p}},$
(3.33)

where

$\sum\limits_{i=1}^{p}(k_{i}+n_{i})=k,\qquad\sum\limits_{i=1}^{p}(Nn_{i}+m_{i})=l,$
(3.34)

which is consistent with the T-exponent expansion (3.25).

###  3.3 Momentum space integration

After we have obtained the expansion for $\hat{\bm{K}}(\nabla,\tau,\bm{k})$,
it remains to integrate it over the momentum $\bm{k}$ to get the required
expansion. Substituting the expressions (3.16) and (3.26) into the integral
(3.7), we get the main result (2.4)-(2.5) for the heat kernel of a generic
positive differential operator of order $N$, where
$\hat{S}_{l}(\tau)=\hat{S}_{l,b^{\prime}_{1}\ldots
b^{\prime}_{l}}(\tau|x,x^{\prime})$ are the two-point matrix-valued tensors of
$l$-th rank at $x^{\prime}$ given by the following momentum integrals

$\hat{S}_{l}(\tau)=\int\frac{d^{d}k}{(2\pi)^{d}}(i\bm{k})^{l}\exp\left(-\tau(i\bm{k})^{N}\cdot\llbracket\hat{F}\rrbracket_{N}+ik_{a^{\prime}}\sigma^{a^{\prime}}\right).$
(3.35)

First of all, note that the replacement of the integration momentum
$k_{a^{\prime}}$ belonging to the cotangent space at the point $x^{\prime}$ of
spacetime to the the momentum $p_{a}$ cotangent at the point $x$,

$k_{a^{\prime}}\mapsto p_{a}=\sigma^{b^{\prime}}_{a}k_{b^{\prime}},$ (3.36)

leads to the expression (2.6) for $S_{l}(\tau|x,x^{\prime})$ with
$\bar{\sigma}^{a}_{b^{\prime}}$—the matrix inverse to
$\sigma^{b^{\prime}}_{a}$, (2.10), and with $\hat{F}_{N}(x,\bm{p})$—a usual
matrix-valued principal symbol (2.7) of $\hat{F}(\nabla)$. In the transition
to (2.6) we used the equation
$\bar{\sigma}^{a}_{b^{\prime}}\sigma^{b^{\prime}}=\sigma^{a}$—the corollary of
(2.9).

It is important that the expansion (2.4) is efficient from the viewpoint of
effective field theory, because with the growing $k$ the background
dimensionality of each term is also monotonically growing. Indeed, from the
recurrence relations (3.27) and Eq.(3.21), it follows that the operators
$\hat{T}_{n,l}(\nabla)$ have the following dimensions

$\dim\hat{T}_{n,l}(\nabla)=Nn-l,$ (3.37)

and in view of the upper limit (2.5) of the summation index $l$ the relation
(2.12) holds, so that (2.4) is really the expansion in powers of spacetime
curvature and dimensionful background fields.

It should be emphasized that in this counting the physical dimensionality of
$\hat{S}_{l}(\tau)$, $\dim\hat{S}_{l}(\tau)=l+d$, should not be accounted for,
because this factor only contributes powers of $\tau$ and the powers of the
dimensionless ratio (1.6) discussed in Introduction.999Powers of $\tau$ are
only grading the dimensionality of their dual operators in the expansion of
the quantity of some fixed overall dimensionality. On integration over $\tau$,
which generates a physical quantity like the effective action or the Green
function, this parameter gets replaced by the inverse of the cutoff, $\tau\to
1/\varLambda^{N}$, which suppresses the contributions of higher-dimensional
operators ${\cal O}_{n}$ of the dimension $n$, ${\cal
O}_{n}/{\cal\varLambda}^{n}$, in effective field theory or inessential
operators in UV renormalization. This is why the physical dimension of $\tau$
does not enter the background dimensionality. This follows from rescaling in
(3.35) the integration momentum by $\tau^{1/N}$ and expanding the result in
powers of $\sigma^{a^{\prime}}/\tau^{1/N}$,

$\hat{S}_{l}(\tau)=\tau^{-\frac{d+l}{N}}\sum\limits_{p=0}^{\infty}\tau^{-\frac{p}{N}}\,\hat{s}_{l,b^{\prime}_{1}\ldots
b^{\prime}_{p}}\,\sigma^{b^{\prime}_{1}}\ldots\sigma^{b^{\prime}_{p}}.$ (3.38)

With this expression for $\hat{S}_{l}(\tau)$ it becomes obvious that (2.4) is
not an expansion in $\tau$ at $\tau\to 0$, for its every $n$-th term of the
dimensionality $nN-l$ carries infinitely many negative powers of the proper
time. Obviously, they all vanish in the coincidence limit and do not break the
validity of the effective theory expansion.

##  4 Minimal operators

For the most general differential operator one cannot say much about the
analytical properties of the integrals (2.6) except their expansion (3.38). In
addition, the exact answer for the matrix integral (3.18),
$\hat{D}_{a}(\tau,\bm{k})$, with non-commutative
$\bm{k}*\llbracket\hat{F}\rrbracket_{N}$ and
$\bm{k}*\nabla_{a}\llbracket\hat{F}\rrbracket_{N}$ (which is needed beyond the
expansion in $\llbracket\hat{F}\rrbracket_{N}$ because ${\rm
dim}\llbracket\hat{F}\rrbracket_{N}=0$) is also hard to obtain in a closed
form. For special cases a partial resummation of this expansion is, however,
possible. First of all these are _minimal_ operators. Then we generalize this
simplest case to the more general non-minimal but _causal_ operators of
Lorentz invariant theories.

###  4.1 Minimal differential operator of general even order

For minimal operators of general even order $N=2M$, defined by
Eqs.(2.13)-(2.14) their principal symbol (2.7) simplifies to the expression
(2.17) and tangent geodetic vectors become gradients (2.15) of the Synge world
function which satisfies Eqs.(2.16). Moreover, since the principal symbol
matrix is trivial, $\llbracket\hat{F}\rrbracket_{N}\propto\hat{1}$, the
integral (3.18) yields
$\hat{D}_{a}(\tau,\bm{k})=\nabla_{a}\llbracket\hat{F}\rrbracket_{N}$ and leads
to relevant $(\tau,\bm{k})$-independent operators
$\llbracket\hat{F}\rrbracket_{m,n}$ and $\hat{T}_{n,l}(\nabla)$. Then the set
of integrals (2.6) expresses via the basic integral (2.18) of Sect.2 in terms
of the generalized exponential function (GEF) $\mathcal{E}_{M,d/2}(z)$ of the
argument

$z=-\frac{\sigma}{2\tau^{1/M}}=-\frac{\sigma^{a}\sigma_{a}}{4\tau^{1/M}}.$
(4.1)

This function, which was introduced in [16], partially performs resummation of
$1/\tau$ series in the heat kernel theory of higher-derivative operators, its
main properties being presented in Appendix B.

Bearing in mind that the powers of momenta $ip_{b}$ in (2.6) can be generated
by the differentiation of this basic integral with respect to $\sigma^{b}$ one
finds that the functions $\hat{S}_{l}(\tau)$ are given by the finite sum of
terms multiple of the unit matrix—tensor densities at $x^{\prime}$,

$\hat{S}_{l,a^{\prime}_{1}\ldots
a^{\prime}_{l}}=\frac{\Delta^{-1}(x,x^{\prime})\,g^{1/2}(x^{\prime})}{\left(4\pi\tau^{1/M}\right)^{d/2}}\\\
\times\sum\limits_{r\geq\frac{l}{2}}^{l}\frac{S_{r,l,a^{\prime}_{1}\ldots
a^{\prime}_{l}}}{\left(-2\,\tau^{1/M}\right)^{r}}\,\mathcal{E}_{M,\frac{d}{2}+r}\left(-{\sigma/2\,\tau^{1/M}}\right)\,\hat{1},$
(4.2)

where the new coefficients $\hat{S}_{r,l,a^{\prime}_{1}\ldots a^{\prime}_{l}}$
follow from Eq.(2.20) of Sect.2 on account of the property (B.8) of
$\mathcal{E}_{\nu,\mu}(z)$,
$\partial_{z}\mathcal{E}_{\nu,\mu}(z)=\mathcal{E}_{\nu,\mu+1}(z)$. Here
$\Delta(x,x^{\prime})$ is the dedensitized Van-Vleck determinant (A.16) and we
took into account that

$\det(-\bar{\sigma}^{a}_{b^{\prime}})=\frac{g^{1/2}(x^{\prime})}{g^{1/2}(x)}\,\Delta(x,x^{\prime}).$
(4.3)

The new tensor coefficients

$\displaystyle S_{r,l}=S_{r,l,\,a^{\prime}_{1}\ldots
a^{\prime}_{l}}(x,x^{\prime}),\quad r\geq 0,\quad l=r,\ldots,2r,$ (4.4)

are defined by the following simple rules. They are fully symmetric covariant
tensors of rank $l$, which consist of $2r-l$ factors $\sigma_{a^{\prime}}$ and
$l-r$ factors of the form
$\gamma_{a^{\prime}b^{\prime}}=\bar{\sigma}_{a^{\prime}}^{c}g_{cd}\bar{\sigma}^{d}_{b^{\prime}}$,
with the combinatorial coefficients equal to the number of different terms in
the symmetrization over $l$ indices. For example,

$\displaystyle S_{1,1}$ $\displaystyle=\sigma_{a^{\prime}},$ $\displaystyle
S_{1,2}$ $\displaystyle=\gamma_{a^{\prime}b^{\prime}},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S_{2,2}=\sigma_{a^{\prime}}\sigma_{b^{\prime}},$
(4.5) $\displaystyle S_{2,3}$
$\displaystyle=3\gamma_{(a^{\prime}b^{\prime}}\sigma_{c^{\prime})},$
$\displaystyle S_{2,4}$
$\displaystyle=3\gamma_{(a^{\prime}b^{\prime}}\gamma_{c^{\prime}d^{\prime})}$
$\displaystyle S_{3,3}$
$\displaystyle=\sigma_{a^{\prime}}\sigma_{b^{\prime}}\sigma_{c^{\prime}},\\!\\!\\!$
$\displaystyle S_{3,4}$
$\displaystyle=6\gamma_{(a^{\prime}b^{\prime}}\sigma_{c^{\prime}}\sigma_{d^{\prime})},\\!\\!\\!$

and so on. They have important property—in the coincidence limit,
$\sigma^{a^{\prime}}=0$, they are nonvanishing only for $l=2r$.

Thus, combining (2.4) with (2.20) we have

$\hat{K}(\tau|x,x^{\prime})=\frac{\Delta^{-1}(x,x^{\prime})\,g^{1/2}(x^{\prime})}{(4\pi\tau^{1/M})^{d/2}}\sum\limits_{n=0}^{\infty}\tau^{n}\sum\limits_{0\leq
l\leq L_{n}}\\\
\times\sum\limits_{r\geq\frac{l}{2}}^{l}\mathcal{E}_{M,\frac{d}{2}+r}\left(-\frac{\sigma}{2\tau^{1/M}}\right)\frac{S_{r,l}*\hat{T}_{k,l}}{\big{(}-2\tau^{1/M}\big{)}^{r}}\,\hat{\mathcal{I}}(x,x^{\prime})\\\
=\frac{\Delta^{-1}(x,x^{\prime})}{(4\pi\tau^{1/M})^{d/2}}\sum\limits_{n=0}^{\infty}\\!\\!\sum\limits_{\;\;\;0\leq
r\leq
L_{n}}\tau^{n-\frac{r}{M}}\mathcal{E}_{M,\frac{d}{2}+r}\left(-\frac{\sigma}{2\tau^{1/M}}\right)\\\
\times\sum\limits_{l=r}^{\lfloor
R_{r,n}\rfloor}\frac{S_{r,l}*\hat{T}_{n,l}}{(-2)^{r}}\,\hat{\mathcal{I}}(x,x^{\prime})\,g^{1/2}(x^{\prime}),$
(4.6)

where we interchanged the order of summation over $l$ and $r$ and introduced
the upper limit of summation over $l$—the integer part of

$R_{r,n}=\min\\{2r,L_{n}(2M)\\}.$ (4.7)

Then we replace the summation indices, $(n,r)\to(m,n)$, $m=Mn-r$, under which
their summation ranges change like

$\sum\limits_{n=0}^{\infty}\sum\limits_{\;\;0\leq r\leq
L_{n}}\\!\sum\limits_{l=r}^{\lfloor
R_{r,n}\rfloor}\to\sum\limits_{m=-\infty}^{\infty}\sum\limits_{\;\;n\geq
N_{m}}\\!\\!\sum\limits_{\;\;\;l=Mn-m}^{\lfloor L_{m,n}\rfloor},$ (4.8)

with $N_{m}(M)$ given by Eq.(2.22) and $L_{m,n}=R_{Mn-m,n}$. After making all
these replacements, we get the final expansion (2.21)-(2.24), where (2.24)
follows from (4.7) at $r=Mn-m$ in view of the expression (2.5) for
$L_{n}(2M)$.

This is the main result of the paper. For clarity we repeat the expression for
this expansion again

$\hat{K}(\tau|x,x^{\prime})=\frac{\Delta^{-1}(x,x^{\prime})}{(4\pi\tau^{1/M})^{d/2}}\,g^{1/2}(x^{\prime})\sum\limits_{m=-\infty}^{\infty}\tau^{m/M}\\\
\times\sum\limits_{\,n\geq
N_{m}}^{\infty}\mathcal{E}_{M,\frac{d}{2}+Mn-m}\left(-\frac{\sigma}{2\tau^{1/M}}\right)\,\hat{b}_{m,n}(x,x^{\prime}),$
(4.9)

which contains a double series with what we call _the generalized HaMiDeW
coefficients_ $\hat{b}_{m,n}(x,x^{\prime})$ defined by finite sums of Eqs.
(2.23)-(2.24). For a particular subset of indices $m=Mn$ these coefficients,
as it follows from (3.32), are universally expressed as

$\hat{b}_{Mn,n}(x,x^{\prime})=\hat{T}_{n,0}(\nabla)\hat{\mathcal{I}}=\frac{1}{n!}(-\hat{F})^{n}\hat{\mathcal{I}}.$
(4.10)

The main difference of the expansion (2.21) from the Schwinger-DeWitt
expansion (1.2) is the presence of infinitely many negative powers of $\tau$
and the absence of one overall exponential function multiplying the proper
time series. Instead, every single power $\tau^{m/M}$ and every single
generalized HaMiDeW-coefficient $\hat{b}_{m,n}$ are multiplied by their own
generalized exponential function explicitly depending on both indices $m$ and
$n$ of the double infinite series. In contrast to anticipations of [16], the
use of generalized exponential functions does not fully perform resummation of
all negative powers of $\tau$.

###  4.2 Heat kernel diagonal

Physically interesting and most commonly considered in mathematics is the
coincidence limit of the heat kernel. It involves a particular value of the
generalized exponential function $\mathcal{E}_{M,\frac{d}{2}+Mn-m}(0)$ (see
(B.3)) and the coincidence limits $[\,\hat{b}_{m,n}\,]$. Remarkably, for a
given $m$ the latter are vanishing outside of a limited range of $n$,

$[\,\hat{b}_{m,n}\,]=0,\quad n>4m,$ (4.11)

and, as a result, they are vanishing for all negative $m$, because the
summation index $n$ in (2.21) is always positive. This property follows from
the remark after the formula (4.5) that $[\,S_{l,p}\,]\neq 0$ only when
$l=2p$, so that the nonvanishing contribution in the sum (2.23) over $l$ is
possible only when $l=2(Mn-m)\leq L_{n}(2M)=(2M-1/2)n$.

Thus, the heat kernel coincidence limit has only a finite number of terms with
negative powers of $\tau$ and reads

$\hat{K}(\tau|x,x)=g^{1/2}(x)\sum\limits_{m=0}^{\infty}\tau^{\frac{m-d/2}{M}}\hat{E}_{2m}(x),$
(4.12)

where $\hat{E}_{2m}(x)$ represent the generalization of Seeley–Gilkey
coefficients [6, 7, 8] (usually given under the matrix trace over vector
bundle indices and integrated over the spacetime, which often leads to the
loss of total derivative terms),

$\displaystyle\hat{E}_{2m}(x)=\frac{1}{(4\pi)^{d/2}}\sum\limits_{n\geq\frac{m}{M}}^{4m}\frac{\Gamma\left(\frac{d/2-m}{M}+n\right)}{M\Gamma(\frac{d}{2}-m+Mn)}\,[\,\hat{b}_{m,n}\,],$
(4.13) $\displaystyle{\rm dim}\,\hat{E}_{2m}(x)=2m.$ (4.14)

As we see, the coincidence limit represents essential truncation of the
expansion (2.21), because the coefficient of every power of $\tau$ is given by
a finite number of terms.

It should be emphasized that (4.11) is not the only range where the
generalized HaMiDeW-coefficients or their coincidence limits are vanishing. As
will be shown below, some of the coefficients $\hat{b}_{m,n}(x,x^{\prime})$
are identically vanishing for all $x$ and $x^{\prime}$ due to the properties
of $\sigma(x,x^{\prime})$ and $\hat{\mathcal{I}}(x,x^{\prime})$. These
coefficients in the expansion (2.21) will be called _spurious_. Non-spurious
coefficients, in their turn, can be nonvanishing in the coincidence limit
$x=x^{\prime}$—in this case we will call them _essential_. Other non-spurious
coefficients vanishing at $x^{\prime}=x$, $[\hat{a}_{m,n}]=0$, we will call
_marginal_. Thus, in the range of indices (4.11) all the coefficients are
obviously marginal. Essential and marginal coefficients for the case $M=2$ are
shown in Fig. 2.

Figure 2:  Coefficients $\hat{b}_{m,n}$ for the case $M=2$. The blue lines at
the left and right are given by the expressions (2.22) for $N_{m}(2)$. Black
dots show essential coefficients, and gray dots show marginal ones, they are
distinguished by the blue line, determined by the condition (4.11)

.

The coefficients with $m<0$ at separate points $x$ and $x^{\prime}$ are not
automatically excluded in the expansion (2.21) even in the case of the Laplace
type operator $M=1$, which seems to contradict the standard Schwinger-DeWitt
expansion (1.2). This means that apparently for the $M=1$ case all the
coefficients are spurious. This will be explicitly checked below for several
low order coefficients and proven for a wide class of minimal higher-
derivative operators given by an arbitrary power of the minimal second-order
one. Otherwise, the expansion (2.21) perfectly passes the consistency test
with the Schwinger-DeWitt expansion (1.2) because all of its various
generalized exponential kernels for $M=1$ degenerate to a single function
$\exp(-\sigma/2\tau)$ in view of the property (B.4) of
$\mathcal{E}_{\nu,\mu}(z)$, $\mathcal{E}_{1,\mu}(z)=e^{z}$.

###  4.3 Causal differential operators

For the class of causal operators the coefficient matrix of its highest-
derivative term is built of the combination of spacetime metric and Kronecker
delta symbols.101010Causality connotation of these operators is associated
with the fact that matrix determinant of their symbol reads as a power of the
momentum squared, ${\rm
det}\,\hat{F}_{N}(p)\propto(g^{ab}(x)\,p_{a}p_{b})^{N\operatorname{tr}\hat{1}/2}$,
so that the characteristic surface of the hyperbolic wave equation in
spacetime with the Lorentzian signature, which is defined by the equation
$\det\hat{F}_{N}(p)=0$, corresponds to the light cone in the metric $g_{ab}$
[3]. Their principal symbol matrix can be diagonalized in terms of projectors
$\hat{\varPi}_{J}=\hat{\varPi}_{J}(n)$ on different field polarizations,
constructed in terms of the unit vector $n_{a}=p_{a}/p$,
$p=\sqrt{g^{ab}(x)\,p_{a}p_{b}}$, and satisfying the orthogonality relations
$\hat{\varPi}_{J}\,\hat{\varPi}_{K}=\delta_{JK}\hat{\varPi}_{J}$,

$\hat{F}_{N}(p)=p^{N}\sum\limits_{J}f_{J}\hat{\varPi}_{J}.$ (4.15)

Therefore the matrix valued exponential function in (2.6) and (3.18) reduces
to the sum of c-number ones with the corresponding purely numerical
(independent of $p$) eigenvalues $f_{J}$,

$\exp\left(-\tau\hat{F}_{N}(p)\right)=\sum\limits_{J}\hat{\varPi}_{J}\,\exp\left(-\tau
f_{J}p^{N}\right),$ (4.16)

and, in particular, allows one to perform integration in (3.18) in a closed
form. At least for even $N=2M$ this diagonalization procedure reduces the
calculations to the case of a minimal differential operators of arbitrary even
order, for which the expansion formalism can be further developed without
restrictions.

##  5 Comparison with Schwinger-DeWitt technique

In this section, we demonstrate our calculations of the coefficients
$\hat{b}_{m,n}(x,x^{\prime})$ and their coincidence limits $[\hat{b}_{m,n}]$
for a minimal operator of the second order,

$\hat{F}(\nabla)=-\Box\,\hat{1}+\hat{P}_{1}^{a}\nabla_{a}+\hat{P}_{0}.$ (5.1)

Redefining the connection,
$\nabla_{a}\mapsto\nabla_{a}+\frac{1}{2}\hat{P}_{1a}$, allows one to nullify
the first-order term, and adding for brevity of the formalism the term
$\hat{1}R/6$ finally brings $\hat{F}(\nabla)$ to the form adopted in
[3]111111This notation differs from [3] by an overall sign, providing its
positivity under Dirichlet boundary conditions at infinity.

$\hat{F}(\nabla)=-\Box\,\hat{1}+\hat{P}+\frac{\hat{1}}{6}\,R.$ (5.2)

In this case all generalized exponential functions
$\mathcal{E}_{M,d/2-n}\left(-\sigma/2\tau^{1/M}\right)$ degenerate into the
usual exponential function $\exp\left(-\sigma/2\tau\right)$ which can be
factored out to give the following answer

$\hat{K}(\tau|x,x^{\prime})=\frac{\Delta^{-1}(x,x^{\prime})\,g^{1/2}(x^{\prime})}{(4\pi\tau)^{d/2}}\,\exp\left(-\frac{\sigma}{2\tau}\right)\\\
\times\sum\limits_{m=-\infty}^{\infty}\tau^{m}\hat{b}_{m}(x,x^{\prime}),$
(5.3)

where each of the coefficients $\hat{b}_{m}(F|x,x^{\prime})$ is given by the
sum of an infinite number of coefficients $\hat{b}_{m,n}(F|x,x^{\prime})$

$\hat{b}_{m}(x,x^{\prime})=\sum\limits_{n\geq
N_{m}}^{\infty}\hat{b}_{m,n}(x,x^{\prime}),$ (5.4)

where

$N_{m}=\begin{cases}m,&m>0,\\\ -2m,&m<0.\end{cases}$ (5.5)

First of all, using the formulas (3.11) and (3.19), we find
$\llbracket\hat{F}\rrbracket_{m,n}$:

$\displaystyle\llbracket\hat{F}\rrbracket_{1}^{a^{\prime}}=-2\sigma^{a^{\prime}b}\nabla_{b}-\sigma^{a^{\prime}b}{}_{b},$
(5.6)
$\displaystyle\llbracket\hat{F}\rrbracket_{2}^{a^{\prime}_{1}a^{\prime}_{2}}=\sigma^{a^{\prime}_{1}}{}_{b}\sigma^{a^{\prime}_{2}b},$
(5.7)
$\displaystyle\llbracket\hat{F}\rrbracket_{1,1}^{a^{\prime}_{1}a^{\prime}_{2}a^{\prime}_{3}}=2i\sigma^{a^{\prime}_{1}b_{1}}\nabla_{b_{1}}(\sigma^{a^{\prime}_{2}}_{b_{2}}\sigma^{a^{\prime}_{3}b_{2}}),$
(5.8)
$\displaystyle\llbracket\hat{F}\rrbracket_{0,2}^{a^{\prime}_{1}a^{\prime}_{2}a^{\prime}_{3}a^{\prime}_{4}}=-\nabla_{b_{3}}(\sigma^{a^{\prime}_{1}}_{b_{1}}\sigma^{a^{\prime}_{2}b_{1}})\nabla^{b_{3}}(\sigma^{a^{\prime}_{3}}_{b_{2}}\sigma^{a^{\prime}_{4}b_{2}}),$
(5.9)
$\displaystyle\llbracket\hat{F}\rrbracket_{0,1}^{a^{\prime}_{1}a^{\prime}_{2}}=2\nabla^{b_{2}}(\sigma^{a^{\prime}_{1}}_{b_{1}}\sigma^{a^{\prime}_{2}b_{1}})\nabla_{b_{2}}+\nabla_{b_{2}}\nabla^{b_{2}}(\sigma^{a^{\prime}_{1}}_{b_{1}}\sigma^{a^{\prime}_{2}b_{1}}).$
(5.10)

Let us find the expressions we need for $\hat{T}_{k,l}$ by the formula (3.27)
(the coefficients with $l=0,1$ are given by the formula (3.30), and the
coefficients of the form $\hat{T}_{2k,3k}$ are given by the formula (3.32))

$\displaystyle\hat{T}_{2,2}=\frac{1}{2}\left(\llbracket\hat{F}\rrbracket_{1}^{2}-\llbracket\hat{F}\rrbracket_{0,1}\right),$
(5.11)
$\displaystyle\hat{T}_{3,2}=-\frac{1}{3!}\left(\hat{F}\llbracket\hat{F}\rrbracket_{1}^{2}+\llbracket\hat{F}\rrbracket_{1}\llbracket\hat{F}^{2}\rrbracket_{1}-\hat{F}\llbracket\hat{F}\rrbracket_{0,1}\right.$
$\displaystyle\qquad\qquad\left.-2\llbracket\hat{F}\rrbracket_{0,1}\hat{F}\right),$
(5.12)
$\displaystyle\hat{T}_{3,3}=-\frac{1}{3!}\left(\llbracket\hat{F}\rrbracket_{1}^{3}-\llbracket\hat{F}\rrbracket_{1}\llbracket\hat{F}\rrbracket_{0,1}-2\llbracket\hat{F}\rrbracket_{0,1}\llbracket\hat{F}\rrbracket_{1}\right.$
$\displaystyle\qquad\qquad\left.-\hat{F}\llbracket\hat{F}\rrbracket_{1,1}-2\llbracket\hat{F}\rrbracket_{1,1}\hat{F}\right),$
(5.13)
$\displaystyle\hat{T}_{3,4}=\frac{1}{3!}\left(\llbracket\hat{F}\rrbracket_{1}\llbracket\hat{F}\rrbracket_{1,1}+2\llbracket\hat{F}\rrbracket_{1,1}\llbracket\hat{F}\rrbracket_{1}-2\llbracket\hat{F}\rrbracket_{0,2}\right),$
(5.14)

#### Negative powers.

This expansion differs from the standard DeWitt expansion by the presence of
negative powers of $\tau$ ($m<0$). This visible discrepancy is removed by the
observation that the coefficients $\hat{b}_{m,n}$ for $m<0$ are in fact
spurious, i.e. they vanish identically not only in the coincidence limit, but
also with the separated points $x\neq x^{\prime}$. Although we could not prove
this property in general, we verified it by direct calculations for the first
four coefficients of this form (i.e. for $\hat{b}_{-1,2}$, $\hat{b}_{-1,3}$,
$\hat{b}_{-1,4}$ and $\hat{b}_{-2,4}$). It turns out that by virtue of the
relation $\sigma_{a^{\prime}bc}\sigma^{b}\sigma^{c}\equiv 0$ (A.11) and other
more complicated relations of the same type, these four first coefficients
really vanish identically. Here are the explicit expressions

$\displaystyle\hat{b}_{-1,2}=-\frac{1}{4}\sigma_{a^{\prime}bc}\sigma^{a^{\prime}}\sigma^{b}\sigma^{c}\hat{\mathcal{I}}\equiv
0,$ (5.15)
$\displaystyle\hat{b}_{-1,3}=\frac{1}{12}\sigma^{a^{\prime}_{1}}\sigma^{a^{\prime}_{2}}\sigma^{b_{1}}\sigma^{b_{2}}(\sigma_{a^{\prime}_{1}}{}^{c}{}_{b_{1}}\sigma_{a^{\prime}_{2}cb_{2}}$
$\displaystyle\qquad\quad+\sigma_{a^{\prime}_{1}b_{1}}{}^{c}\sigma_{a^{\prime}_{2}cb_{2}}+\sigma_{a^{\prime}_{1}b_{1}}{}^{c}\sigma_{a^{\prime}_{2}b_{2}c})\,\hat{\mathcal{I}}\equiv
0,$ (5.16)
$\displaystyle\hat{b}_{-2,4}=\frac{1}{32}\left(\sigma_{a^{\prime}bc}\sigma^{a^{\prime}}\sigma^{b}\sigma^{c}\right)^{2}\hat{\mathcal{I}}\equiv
0.$ (5.17)

We will not give an expression for the coefficient $\hat{b}_{-1,4}$ due to its
complexity, but only indicate that it contains convolutions of the form

$\sigma^{a^{\prime}_{1}}\sigma^{a^{\prime}_{2}}\sigma^{a^{\prime}_{3}}\sigma^{b_{1}}\sigma^{b_{2}}\sigma_{a^{\prime}_{1}b_{1}c_{1}}\sigma_{a^{\prime}_{2}b_{2}c_{2}}\sigma_{a^{\prime}_{3}}{}^{c_{1}c_{2}}.$
(5.18)

All terms of this kind are also identically zero.

#### Positive powers.

We now turn to the discussion of positive powers. Another discrepancy between
our expansion (5.4) and the standard expansion (1.2) is that instead of the
standard prefactor $\Delta^{1/2}(x,x^{\prime})$ we have the inverse
determinant $\Delta^{-1}(x,x^{\prime})$. Therefore, the following relation
should hold between the original Schwinger-DeWitt coefficients and the
generalized ones

$\hat{a}_{m}(x,x^{\prime})=\Delta^{-3/2}(x,x^{\prime})\,\hat{b}_{m}(x,x^{\prime})\\\
=\Delta^{-3/2}(x,x^{\prime})\\!\sum\limits_{n\geq
N_{m}}^{\infty}\hat{b}_{m,n}(x,x^{\prime}),\quad m\geq 0.$ (5.19)

Note that if we are only interested in the coincidence limits then, as it
follows from (4.13), only the first $3m+1$ terms (up to $k=4m$) from the whole
infinite series (5.4) will contribute to $[\hat{b}_{m}]$. (Actually, as we
will see later, even less due to symmetry reasons.)

We compare now the results obtained by our method with the well-known results
for the coincidence limits of the first HaMiDeW coefficients and their
derivatives (they can be found in [2] or in [3]),

$\displaystyle[\hat{a}_{0}]=\hat{1},\quad[\nabla_{a}\hat{a}_{0}]=0,\quad[\nabla_{a}\nabla_{b}\hat{a}_{0}]=\frac{1}{2}\hat{\mathcal{R}}_{ab},$
(5.20)
$\displaystyle[\hat{a}_{1}]=-\hat{P},\quad[\nabla_{a}\hat{a}_{1}]=-\frac{1}{2}\nabla_{a}\hat{P}-\frac{1}{6}\nabla^{b}\hat{\mathcal{R}}_{ba},$
(5.21)
$\displaystyle[\hat{a}_{2}]=\frac{1}{180}\left(R_{abcd}R^{abcd}-R_{ab}R^{ab}+\Box
R\right)\hat{1}$
$\displaystyle\qquad\quad+\frac{1}{2}\hat{P}^{2}+\frac{1}{12}\hat{\mathcal{R}}_{ab}\hat{\mathcal{R}}^{ab}-\frac{1}{6}\Box\hat{P}.$
(5.22)

In order to compare the coincidence limits for their derivatives, one would
need the coincidence limits for the Van Vleck prefactor $\Delta^{-3/2}$ in
(5.19),

$[\Delta]=1,\quad[\nabla_{a}\Delta]=0,\quad[\nabla_{a}\nabla_{b}\Delta]=\frac{1}{3}R_{ab}.$
(5.23)

Let us start with the coefficients with $m=0$. Two-point expressions for
several lowest $\hat{b}_{0,n}(x,x^{\prime})$ are

$\displaystyle\hat{b}_{0,0}=\hat{\mathcal{I}},\quad\hat{b}_{0,1}=-\frac{1}{2}\sigma_{a^{\prime}}\sigma^{a^{\prime}b}{}_{b}\,\hat{\mathcal{I}},$
(5.24)
$\displaystyle\hat{b}_{0,2}=\frac{1}{2}\sigma^{b^{\prime}}\sigma^{c}\left(\sigma_{b^{\prime}ac}+\sigma_{b^{\prime}ca}\right)\hat{\mathcal{I}}_{a}+\frac{1}{8}\left(\sigma^{a^{\prime}}\big{(}4\sigma_{a^{\prime}b}{}^{b}\right.$
$\displaystyle\qquad+\sigma^{b^{\prime}}(2\sigma_{a^{\prime}}{}^{cd}\sigma_{b^{\prime}cd}+\sigma_{a^{\prime}c}{}^{c}\sigma_{b^{\prime}d}{}^{d})+2\sigma^{b}(\sigma_{a^{\prime}bc}{}^{c}+\sigma_{a^{\prime}c}{}^{c}{}_{b})\big{)}$
$\displaystyle\qquad\left.+4\sigma^{a}\bar{\sigma}_{cb^{\prime}}\big{(}\sigma^{b^{\prime}}{}_{a}{}^{c}+\sigma^{b^{\prime}c}{}_{a}\big{)}\right)\hat{\mathcal{I}}.$
(5.25)

Their coincidence limits and coincidence limits of their lowest order
derivatives immediately follow as

$\displaystyle[\hat{b}_{0,0}]=\hat{1},\;\;[\nabla_{a}\hat{b}_{0,0}]=0,\;\;[\nabla_{a}\nabla_{b}\hat{b}_{0,0}]=\frac{1}{2}\hat{\mathcal{R}}_{ab},$
(5.26)
$\displaystyle[\hat{b}_{0,1}]=0,\;\;[\nabla_{a}\hat{b}_{0,1}]=0,\;\;[\nabla_{a}\nabla_{b}\hat{b}_{0,1}]=\frac{2}{3}R_{ab}\hat{1},$
(5.27)
$\displaystyle[\hat{b}_{0,2}]=0,\;\;[\nabla_{a}\hat{b}_{0,2}]=0,\;\;[\nabla_{a}\nabla_{b}\hat{b}_{0,2}]=-\frac{\hat{1}}{6}\,R_{ab}.$
(5.28)

We do not give explicit expressions for the coefficients $\hat{b}_{0,3}$ and
$\hat{b}_{0,4}$ due to their complexity (for example, the expression for the
latter contains about 750 terms), however, for their coincidence limits, the
calculation gives:

$\displaystyle[\hat{b}_{0,3}]=[\hat{b}_{0,4}]=0,$ (5.29)
$\displaystyle[\nabla_{a}\hat{b}_{0,3}]=[\nabla_{a}\hat{b}_{0,4}]=0,$ (5.30)
$\displaystyle[\nabla_{a}\nabla_{b}\hat{b}_{0,3}]=[\nabla_{a}\nabla_{b}\hat{b}_{0,4}]=0.$
(5.31)

Collecting these results we obtain for $m=0$

$[\hat{b}_{0}]=\hat{1},\quad[\nabla_{a}\hat{b}_{0}]=0,\quad[\nabla_{a}\nabla_{b}\hat{b}_{0}]=\frac{1}{2}\big{(}\hat{\mathcal{R}}_{ab}+\hat{1}R_{ab}\big{)}.$
(5.32)

Using these relations in Eq.(5.19) for $\hat{a}_{m}(x,x^{\prime})$ in terms of
$\hat{b}_{m,n}(x,x^{\prime})$ we recover on account of (5.23) the expressions
(5.20) well known from the solution of DeWitt recursion relations.

Similarly, the results for the first two $\hat{b}_{1,n}(x,x^{\prime})$

$\displaystyle\hat{b}_{1,1}=-\hat{F}(\nabla)\hat{\mathcal{I}},$ (5.33)
$\displaystyle\hat{b}_{1,2}=-\frac{1}{2}\sigma_{a}\big{(}\hat{\mathcal{I}}^{ab}{}_{b}+\hat{\mathcal{I}}_{b}{}^{ba}\big{)}-\frac{1}{2}\big{(}2+\sigma^{a^{\prime}}\sigma_{a^{\prime}b}{}^{b}\big{)}\,\hat{\mathcal{I}}_{c}^{c}$
$\displaystyle-\sigma^{a^{\prime}}\sigma_{a^{\prime}bc}\,\hat{\mathcal{I}}^{bc}-\frac{1}{2}\Big{(}\sigma^{a^{\prime}}(\sigma_{a^{\prime}bc}{}^{c}+\sigma_{a^{\prime}c}{}^{c}{}_{b})+2\sigma^{a^{\prime}c}{}_{c}\bar{\sigma}_{ba^{\prime}}$
$\displaystyle+2(\sigma_{a^{\prime}bc}+\sigma_{a^{\prime}cb})\bar{\sigma}^{ca^{\prime}}\Big{)}\,\hat{\mathcal{I}}^{b}-\frac{1}{4}\Big{(}2(\sigma^{a^{\prime}bc}{}_{c}+\sigma^{a^{\prime}c}{}_{c}{}^{b})\bar{\sigma}_{ba^{\prime}}$
$\displaystyle+\sigma^{a^{\prime}}\sigma_{a^{\prime}b}{}^{bc}{}_{c}+(\sigma^{a^{\prime}c}{}_{c}\sigma_{b^{\prime}d}{}^{d}+2\sigma^{a^{\prime}cd}\sigma_{b^{\prime}cd})\bar{\sigma}_{ea^{\prime}}\bar{\sigma}^{eb^{\prime}}\Big{)}\,\hat{\mathcal{I}}$
$\displaystyle+\frac{1}{2}\Big{(}\sigma^{a^{\prime}}\sigma_{a^{\prime}b}{}^{b}\big{(}\hat{P}+\frac{\hat{1}}{6}R\big{)}+\sigma^{a}\big{(}\nabla_{a}\hat{P}+\frac{\hat{1}}{6}\nabla_{a}R\big{)}\Big{)}\,\hat{\mathcal{I}},$
(5.34)

lead to the coincidence limits

$\displaystyle[\hat{b}_{1,1}]=-\hat{P}-\frac{\hat{1}}{6}R,$ (5.35)
$\displaystyle[\nabla_{a}\hat{b}_{1,1}]=-\nabla_{a}\hat{P}-\frac{\hat{1}}{6}\nabla_{a}R-\frac{1}{3}\nabla^{b}\hat{\mathcal{R}}_{ab},$
(5.36) $\displaystyle[\hat{b}_{1,2}]=\frac{\hat{1}}{6}\,R,$ (5.37)
$\displaystyle[\nabla_{a}\hat{b}_{1,2}]=\frac{1}{2}\nabla_{a}\hat{P}+\frac{\hat{1}}{3}\nabla_{a}R+\frac{1}{6}\nabla^{b}\hat{\mathcal{R}}_{ab}.$
(5.38)

We do not present the expressions for $\hat{b}_{1,3}$ and $\hat{b}_{1,4}$ (for
example, the expression for the latter contains about 3500 terms), but their
coincidence limits are very concise,

$\displaystyle[\hat{b}_{1,3}]=[\hat{b}_{1,4}]=0,$ (5.39)
$\displaystyle[\nabla_{a}\hat{b}_{1,3}]=-\frac{\hat{1}}{6}\,\nabla_{a}R,\quad[\nabla_{a}\hat{b}_{1,4}]=0.$
(5.40)

Collecting these results in (5.4), we obtain for $m=1$

$[\hat{b}_{1}]=-\hat{P},\quad[\nabla_{a}\hat{b}_{1}]=-\frac{1}{2}\nabla_{a}\hat{P}+\frac{1}{6}\nabla^{b}\hat{\mathcal{R}}_{ab},$
(5.41)

which again confirm the relations (5.21) for the HaMiDeW coefficient
$\hat{a}_{1}(x,x^{\prime})$.

Finally, consider the results for the coincidence limits of the first
coefficients $\hat{b}_{2,n}(x,x^{\prime})$ needed for the verification of the
known result for $\hat{a}_{2}(x,x)$,

$\displaystyle[\hat{b}_{2,2}]=\frac{1}{2}\Big{(}\hat{P}+\frac{\hat{1}}{6}R\Big{)}^{2}=\frac{1}{2}\Box\Big{(}\hat{P}+\frac{\hat{1}}{6}R\Big{)}$
$\displaystyle\qquad\quad+\frac{1}{4}\hat{\mathcal{R}}_{ab}\hat{\mathcal{R}}^{ab},$
(5.42)
$\displaystyle[\hat{b}_{2,3}]=\frac{1}{540}\,(12R_{abcd}R^{abcd}-12R_{ab}R^{ab}-15R^{2}$
$\displaystyle\quad\quad-102\Box
R)\hat{1}-\frac{1}{6}\,(\hat{\mathcal{R}}_{ab}\hat{\mathcal{R}}^{ab}+R\hat{P}+2\Box\hat{P}),$
(5.43)
$\displaystyle[\hat{b}_{2,4}]=\frac{1}{360}\,(-6R_{abcd}R^{abcd}+6R_{ab}R^{ab}+5R^{2}$
$\displaystyle\qquad\quad+36\Box R),$ (5.44)
$\displaystyle[\hat{b}_{2,5}]=[\hat{b}_{2,6}]=[\hat{b}_{2,7}]=[\hat{b}_{2,8}]=0.$
(5.45)

The sum of these expressions exactly generates the well-known expression
(5.22) which plays a very important role because it exhausts one-loop
divergences of a generic 4-dimensional field theory.

Our checks give sufficient evidence that the application of the generalized
Fourier method, in all parts, yields the results coinciding with those of the
standard DeWitt technique. There are, however, differences between these
methods. In the Schwinger-DeWitt technique to calculate the coincidence limit
$[\hat{a}_{m}]$ one needs to know the coincidence limits of all lower $m$
coefficients and a number their derivatives. On the contrary, in our method,
the recurrent procedure is carried out not for the coefficients
$\hat{b}_{m,n}$ themselves, but for the operators $\hat{T}_{n,l}(\nabla)$, and
the coefficients are obtained independently of each other in the form of
contractions of these operators with the tensors $S_{r,l}$ by Eq. (2.23).
Moreover, this gives closed expressions for the coefficients
$\hat{b}_{m,n}(x,x^{\prime})$ at separate points $x\neq x^{\prime}$ and via
Eq.(5.19) yields entirely new representation for HaMiDeW coefficients.

##  6 Fourth order minimal operator

Here we consider a generic minimal fourth order operator,

$\hat{F}(\nabla)=\Box^{2}+\hat{\varOmega}^{abc}\nabla_{a}\nabla_{b}\nabla_{c}+\hat{D}^{ab}\nabla_{a}\nabla_{b}+H^{a}\nabla_{a}+\hat{P},$
(6.1)

where $\hat{\varOmega}^{abc}=\hat{\varOmega}^{(abc)}$ and
$\hat{D}^{ab}=\hat{D}^{(ab)}$. Relevant computations and especially the
representation of the final results become excessively involved in this case
if the coefficient $\hat{\varOmega}^{abc}$, ${\rm
dim}\,\hat{\varOmega}^{abc}=1$, is nonzero. Therefore we present here the
results for the case of $\hat{\varOmega}^{abc}=0$ relegating to Appendix D the
generalization to nonzero value of this coefficient. Again, the chain of
differential operators $\hat{T}_{n,l}(\nabla)$ and generalized Schwinger-
DeWitt coefficients $\hat{b}_{m,n}(x,x^{\prime})$ are so complicated, that we
will skip initial calculational steps and go directly to the results for
several lowest order coincidence limits. For the coefficients of
dimensionality two they are rather simple ($\hat{D}=\hat{D}_{a}^{a}$),

$\displaystyle[\hat{b}_{1,1}]$
$\displaystyle=\frac{1}{2}\hat{D}-\frac{\hat{1}}{3}R,$ (6.2)
$\displaystyle[\hat{b}_{1,2}]$ $\displaystyle=\frac{\hat{1}}{6}(d+2)(d+4)R,$
(6.3) $\displaystyle[\hat{b}_{1,3}]$ $\displaystyle=[\hat{b}_{1,4}]=0,$ (6.4)

whereas for dimensionality four they involve all operator coefficients and
read

$\displaystyle[\hat{b}_{2,1}]=$
$\displaystyle-\hat{P}-\frac{1}{2}\hat{\mathcal{R}}_{ab}\hat{\mathcal{R}}^{ab},$
$\displaystyle[\hat{b}_{2,2}]=$
$\displaystyle\frac{\hat{1}}{15}(2d+7)\Big{(}R_{ab}^{2}-R_{abcd}^{2}-6\Box
R\Big{)}+\frac{\hat{1}}{6}R^{2}$
$\displaystyle+\frac{1}{4}(d+2)(d+4)\hat{\mathcal{R}}_{ab}^{2}+\frac{1}{8}\hat{D}^{2}+\frac{1}{4}\hat{D}_{ab}\hat{D}^{ab}$
$\displaystyle-\frac{1}{3}(d+3)\hat{D}^{ab}R_{ab}-\frac{1}{6}\hat{D}R+\frac{1}{2}(d+2)\nabla_{a}\hat{H}^{a}$
$\displaystyle+\nabla_{a}\nabla_{b}\hat{D}^{ab}+\frac{1}{4}(d+4)\Box\hat{D},$
(6.5) $\displaystyle[\hat{b}_{2,3}]=$
$\displaystyle-(d+4)(d+6)\Big{(}\frac{d+11}{45}\big{(}R_{ab}^{2}-R_{abcd}^{2}$
$\displaystyle-6\Box
R\big{)}\,\hat{1}+\frac{\hat{1}}{6}R^{2}+\frac{1}{6}(d+2)\hat{\mathcal{R}}_{ab}^{2}-\frac{1}{6}\hat{D}^{ab}R_{ab}$
$\displaystyle-\frac{1}{12}\hat{D}R+\frac{1}{3}\nabla_{a}\nabla_{b}\hat{D}^{ab}+\frac{1}{6}\Box\hat{D}\Big{)},$
(6.6) $\displaystyle[\hat{b}_{2,4}]=$ $\displaystyle(d+4)(d+6)(d+8)(d+10)$
$\displaystyle\times\Big{(}\frac{\hat{1}}{60}\big{(}R_{ab}^{2}-R_{abcd}^{2}-6\Box
R\big{)}+\frac{\hat{1}}{72}R^{2}\Big{)},$ (6.7)
$[\hat{b}_{2,5}]=[\hat{b}_{2,6}]=[\hat{b}_{2,7}]=[\hat{b}_{2,8}].$ (6.8)

These results show that the coincidence limits $[\hat{b}_{m,n}]$ for a given
$m$ start vanishing essentially earlier than at $n>4m$. This happens due to
the properties of symmetrized covariant derivatives of Synge world function
and, apparently, starts at $n>2m$, though we do not yet have the proof of this
property in higher orders.

Thus we have the Seeley–Gilkey coefficients (4.13) of the heat kernel diagonal
of the minimal fourth order operator. The dimensionality two coefficient is
contributed by nonvanishing $[\hat{b}_{1,1}]$ and $[\hat{b}_{1,2}]$ and reads

$\displaystyle\hat{E}_{2}(x)=\frac{1}{(4\pi)^{d/2}}\frac{\Gamma\left(\frac{d/2-1}{2}\right)}{2\Gamma(\frac{d}{2}-1)}\bigg{\\{}\frac{1}{2d}\hat{D}+\frac{\hat{1}}{6}R\bigg{\\}}.$
(6.9)

The dimensionality four coefficient includes nonvanishing contributions of the
above $[\hat{b}_{2,1}]$,…,$[\hat{b}_{2,4}]$ coincidence limits and equals

$\displaystyle\hat{E}_{4}(x)$
$\displaystyle=\frac{1}{(4\pi)^{d/2}}\frac{\Gamma\left(\frac{d}{4}\right)}{4\Gamma\left(\frac{d}{2}\right)}\Bigg{\\{}(d-2)\Big{(}\frac{\hat{1}}{90}R^{2}_{abcd}-\frac{\hat{1}}{90}R^{2}_{ab}$
$\displaystyle+\frac{\hat{1}}{36}R^{2}+\frac{1}{6}\hat{\mathcal{R}}^{2}_{ab}+\frac{\hat{1}}{15}\,\Box
R\,\Big{)}-\frac{1}{3}\hat{D}^{ab}R_{ab}+\frac{1}{6}\hat{D}R$
$\displaystyle+\frac{1}{d+2}\Big{(}\frac{1}{2}\hat{D}_{ab}\hat{D}^{ab}+\frac{1}{4}\hat{D}^{2}-\frac{2}{3}(d+1)\nabla_{a}\nabla_{b}\hat{D}^{ab}$
$\displaystyle+\frac{1}{6}(d+4)\Box\hat{D}\Big{)}-2\hat{P}+\nabla_{a}\hat{H}^{a}\Bigg{\\}}.$
(6.10)

Both expressions fully correspond to the results of [3, 30]. The paper [3]
contains also the contribution of nonzero $\varOmega^{abc}$, but without the
total derivative terms. In view of very lengthy terms with $\varOmega^{abc}$
their complete contribution is given in Appendix D.

The off-diagonal heat kernel of (6.1) is too complicated to be presented here
even in the lowest orders of its expansion. Therefore we give here only its
part which should be indicative of such new features of higher derivative
operators as the origin of negative powers of the proper time. This is the
marginal (that is vanishing on the heat kernel diagonal, but non-zero outside
of it) coefficient $\hat{b}_{-1,1}(x,x^{\prime})$ of the negative power of
$\tau$.

Using (2.23) and (2.24) we have the integer part of $\lfloor
L_{-1,1}\rfloor=3$, so that

$\hat{b}_{-1,1}(x,x^{\prime})=-\frac{1}{8}\,S_{3,3}*T_{1,3}(\nabla)\,\mathcal{I}(x,x^{\prime}),$
(6.11)

where the tensor $S_{3,3}$ is defined in Eq.(4.5) and the differential
operator $T_{1,3}(\nabla)=-\llbracket F(\nabla)\rrbracket_{3,0}=-\llbracket
F(\nabla)\rrbracket_{3}$, as it follows from (3.27)–(3.28). So it turns out to
be

$\displaystyle T_{1,3}^{a^{\prime}b^{\prime}c^{\prime}}(\nabla)$
$\displaystyle=-4\sigma^{a^{\prime}b}\sigma^{b^{\prime}}_{b}\sigma^{c^{\prime}c}\nabla_{c}-2\sigma^{a^{\prime}b}\sigma^{b^{\prime}}_{b}\sigma^{c^{\prime}d}{}_{d}$
$\displaystyle-4\sigma^{a^{\prime}bc}\sigma^{b^{\prime}}_{b}\sigma^{c^{\prime}}_{c}-\varOmega^{abc}\sigma^{a^{\prime}}_{a}\sigma^{b^{\prime}}_{b}\sigma^{c^{\prime}}_{c}.$
(6.12)

Therefore, in view of the parallel transport equation for
$\mathcal{I}(x,x^{\prime})$ and the relation
$\sigma^{ab^{\prime}}\sigma_{b^{\prime}}=\sigma^{a}$ we have

$\hat{b}_{-1,1}(x,x^{\prime})=\frac{1}{8}\,(4\sigma\sigma_{a^{\prime}}\Box\sigma^{a^{\prime}}+\hat{\varOmega}^{abc}\sigma_{a}\sigma_{b}\sigma_{c})\,\hat{\mathcal{I}}(x,x^{\prime}),$
(6.13)

which is explicitly non-vanishing for $x\neq x^{\prime}$. This proves that
negative powers of the proper time indeed arise in the expansion (2.21) for
$M>1$.

Calculations of this and the previous sections show that their complexity is
much higher than in the recursive procedure of the Schwinger-DeWitt technique.
Without using computer symbolic manipulations, that were performed by means of
Wolfram Mathematica and tensor computer algebra packages xAct and xTras, a
completion of the above checks would be impossible. Obviously the level of
complexity grows if one goes to higher derivative operators with $M>1$. In
connection with this we would like to point out to an unsolved and somewhat
mysterious feature of our formalism, which followed from the above computer
simulations throughout various consistency checks of the above type.

Interesting point is that there exist two versions of the generalized Fourier
transform. One of them is covariant and corresponds to the delta-function
representation (3.5), whereas the non-covariant one proceeds with
$k_{a^{\prime}}\sigma^{a^{\prime}}(x^{\prime},x)$ replaced with
$k_{a}\sigma^{a}(x,x^{\prime})$. This means that the auxiliary momentum space
tangent to spacetime manifold is introduced at the point $x$ rather than at
$x^{\prime}$. The non-covariant version has the calculational advantage of
having in the game only one unprimed type of indices, but it violates the
basic commutation equation (3.6) by extra non-covariant terms proportional to
the covariant derivatives of the diffeomorphisms inert momentum vector,
$(\nabla_{b}k_{a})\sigma^{a}(x,x^{\prime})=-k_{c}\varGamma_{ba}^{c}\sigma^{a}(x,x^{\prime})$.
Paradoxically, naive omission of these terms has led in the computer
simulation of the above type to exactly the same coincidence limits, and this
happens despite the fact that $\sigma^{a}(x,x^{\prime})$ in the expression
above gets numerously differentiated and thus is nonzero in this limit. Thus
far we do not have explanation for this digitally generated observation, which
however might be useful in symbolic computations.

##  7 Powers of a minimal operator

In this section we will study the direct and inverse Mellin transform in heat
kernel theory which allows one to relate the curvature expansions for the
minimal operator $\hat{F}(\nabla)$ and its integer or non-integer power
$\hat{F}^{s}(\nabla)$. Since we will have the formalism relating heat kernels,
the Green function and $\hat{b}_{m,n}$ coefficients for different operators,
we will mark them by additional subscript or superscript.

The Green function of the operator $\hat{F}^{s}(\nabla)$ or the $s$-th power
of the Green function of $\hat{F}(\nabla)$ is given by the Mellin
transform—the proper time integral of the form

$\displaystyle\hat{G}_{F^{s}}(x,x^{\prime})$
$\displaystyle\equiv\frac{\hat{1}}{F^{s}(\nabla)}\delta(x,x^{\prime})$
$\displaystyle=\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty}d\tau\,\tau^{s-1}\hat{K}_{F}(\tau|x,x^{\prime}).$
(7.1)

The inverse Mellin transform allows one to recover the heat kernel of the
operator $\hat{F}(\nabla)$ from $\hat{G}_{F^{s}}(x,x^{\prime})$

$\hat{K}_{F}(\tau|x,x^{\prime})=\frac{1}{2\pi
i}\int\limits_{w-i\infty}^{w+i\infty}ds\,\tau^{-s}\,\Gamma(s)\,\hat{G}_{F^{s}}(x,x^{\prime})\,,$
(7.2)

where the integration runs along the contour parallel to imaginary axis in the
complex plane of $s$ for sufficiently big positive $w$. We do not discuss the
range of $s$ where both transforms are well defined and only assume positive-
definiteness of $\hat{F}$ and the possibility to make needed analytic
continuation. Brief derivation of these transforms is given in Appendix C.

The idea of relating heat kernel curvature expansions for
$\hat{F}^{s}(\nabla)$ and $\hat{F}(\nabla)$ is based on the sequence of
transformations

(7.7)

where ${\mathcal{M}_{\tau t}}$ denotes the Mellin transform from the function
of the proper time $\tau$ to its image as a function of $t$ and
${\mathcal{M}^{-1}_{t\tau}}$ is the respective inverse Mellin transform.

###  7.1 The Green function of the operator $\hat{F}^{t}$.

Using (7) and integrating term by term the expansion (2.21) we get

$\hat{G}_{F^{t}}(x,x^{\prime})=\Delta^{-1}(x,x^{\prime})\,g^{1/2}(x^{\prime})\\\
\times\sum\limits_{m=-\infty}^{\infty}\sum\limits_{\,n\geq
N_{m}}^{\infty}\mathbb{G}^{m,n}(t,\sigma(x,x^{\prime}))\,\hat{b}^{F}_{m,n}(x,x^{\prime}),$
(7.8)

where $\hat{b}^{F}_{m,n}(x,x^{\prime})$ are the generalized HaMiDeW
coefficients for the operator $\hat{F}$ and their coefficients, which one
might call _the basis Green functions_ , equal

$\mathbb{G}^{m,n}(t,\sigma)=\frac{1}{(4\pi)^{d/2}}\left(\frac{\sigma}{2}\right)^{-s}\frac{M}{\Gamma(t)}\\\
\times\int\limits_{0}^{\infty}dz\,z^{s-1}\mathcal{E}_{M,d/2+Mn-m}(-z)\\\
=\frac{1}{(4\pi)^{d/2}}\frac{\Gamma\left(\frac{d}{2}-m-Mt\right)\Gamma(n+t)}{\Gamma(t)\,\Gamma\big{(}M(n+t)\big{)}}\left(\frac{\sigma}{2}\right)^{Mt+m-\frac{d}{2}}.$
(7.9)

In the derivation of this expression we changed the integration variable,
$\tau\to z=\sigma/2\tau^{1/M}$ and used the Mellin transform of the
generalized exponential function — Eq.(B.6) with $s=d/2-m-Mt$.

###  7.2 The heat kernel of the operator $\hat{F}^{s}$.

The inverse Mellin transform (7.2) now gives the result of the last step in
the diagram (7.7)—the asymptotic heat kernel expansion for the operator
$\hat{F}^{s}$,

$\hat{K}_{F^{s}}(\tau|x,x^{\prime})=\frac{1}{2\pi
i}\int\limits_{w-i\infty}^{w+i\infty}dt\,\tau^{-t}\,\Gamma(t)\,\hat{G}_{F^{st}}(x,x^{\prime}).$
(7.10)

Term by term integration of the expansion (7.8) with $t\to st$ gives new
coefficient functions of $\hat{b}_{m,n}$

$\frac{1}{2\pi
i}\int\limits_{w-i\infty}^{w+i\infty}dt\,\tau^{-t}\,\Gamma(t)\,\mathbb{G}^{m,n}(st,\sigma)\,\\\
=\frac{\tau^{m/sM}}{(4\pi\tau^{1/sM})^{d/2}}\,\mathcal{E}_{M,d/2+Mn-m}^{s,d/2-m}\left(-\frac{\sigma}{2\tau^{1/sM}}\right).$
(7.11)

where $\mathcal{E}_{\beta,b}^{\alpha,a}(-z)$ are the generalized exponential
functions of the second order (B.10)— some new hyper-geometric type functions.
Their properties and the derivation of this equation are briefly presented in
Appendices B and C. As a result

$\hat{K}_{F^{s}}(\tau|x,x^{\prime})=\frac{\Delta^{-1}(x,x^{\prime})}{(4\pi\tau^{1/sM})^{d/2}}\,g^{1/2}(x^{\prime})\sum\limits_{m=-\infty}^{\infty}\tau^{m/sM}\\\
\times\sum\limits_{\,n\geq
N_{m}}^{\infty}\mathcal{E}_{M,d/2+Mn-m}^{s,d/2-m}\left(-\frac{\sigma}{2\tau^{1/sM}}\right)\,\hat{b}^{F}_{m,n}(x,x^{\prime}).$
(7.12)

This relation for off-diagonal elements of heat kernels is also very
interesting for the coincidence limit $x^{\prime}=x$ case. In view of the
expression (B.13) for $\mathcal{E}_{\beta,b}^{\alpha,a}(0)$ the second order
generalized exponential functions here reduce to the gamma-function factors in
Eq.(4.13) modulo independent of $n$ coefficient
$\Gamma(\frac{d/2-m}{sM})/s\Gamma(\frac{d/2-m}{M})$, so that the Seeley–Gilkey
coefficients of the operator $\hat{F}^{s}$,

$\hat{K}_{F^{s}}(\tau|x,x)=g^{1/2}(x)\sum\limits_{m=0}^{\infty}\tau^{\frac{m-d/2}{sM}}\hat{E}^{F^{s}}_{m}(x),$
(7.13)

express via those of $\hat{F}$,

$\hat{E}^{F^{s}}_{m}(x)=\frac{\Gamma\left(\frac{d/2-m}{sM}\right)}{s\,\Gamma\left(\frac{d/2-m}{M}\right)}\,\hat{E}^{F}_{m}(x).$
(7.14)

This is an extension of a well known Fegan–Gilkey relation between functional
traces of these two operators [24] to the case of their diagonal
elements—functions of a spacetime point $x$.

This formula can be, in particular, used to check the consistency of
Eqs.(6.9)-(6.10) for Seeley–Gilkey coefficients of the fourth order operator.
A squared version $\hat{F}^{2}(\nabla)$ of a minimal second order operator
$\hat{F}(\nabla)=-\Box+\hat{P}$ is the fourth order operator (6.1) with

$\displaystyle\hat{\varOmega}^{abc}_{F^{2}}=0,\quad\hat{D}^{ab}_{F^{2}}=-2g^{ab}\hat{P},$
(7.15)
$\displaystyle\hat{H}^{a}_{F^{2}}=-2\nabla_{a}\hat{P},\quad\hat{P}_{F^{2}}=\hat{P}^{2}-(\Box\hat{P}).$
(7.16)

Substituting these expressions into Eqs.(6.9)-(6.10) one easily confirms
Fegan-Gilkey relations (7.14) between $E^{F^{2}}_{2,4}(x)$ and
$E^{F}_{2,4}(x)$.

The above relations hold also for non-integer powers $s$, for which case we do
not see the presence of $\log\tau$-terms advocated to be possible in
mathematical literature [35, 36]. The presence of such terms occurring due to
merging poles might be an artifact of the method which was carried out only in
the coincidence limit $x=x^{\prime}$ and required using the $\zeta$-functional
regularization. In contrast, our method operates with objects at separated
points and does not require additional regularization. The difference between
the two methods can be clearly visualized by the lack of closure of the
following diagram, where $\hat{\zeta}_{F}(s,x)=\hat{G}_{F^{s}}(x,x)$ denotes
the regularized zeta-function in the coincidence limit $x^{\prime}=x$,

$\textstyle{\hat{K}_{F}(\tau|x,x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{M}_{\tau
t}}$$\scriptstyle{x\to
x^{\prime}}$$\textstyle{\hat{G}_{F^{s}}(x,x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F\to
F^{s}}$$\textstyle{\hat{G}_{F^{st}}(x,x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{M}^{-1}_{t\tau}}$$\textstyle{\hat{K}_{F^{s}}(\tau|x,x^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\to
x^{\prime}}$$\textstyle{\hat{K}_{F}(\tau|x,x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{M}_{\tau
t}}$$\textstyle{\hat{\zeta}_{F}(t,x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F\to
F^{s}}$$\textstyle{\hat{\zeta}_{F^{s}}(t,x)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{M}^{-1}}$$\textstyle{\hat{K}_{F^{s}}(\tau|x,x).}$
(7.17)

###  7.3 Integer powers of a Laplace type operator

Thus far we considered generic fractional and even complex powers $s$ of the
operator. Now consider the integer power $s=M$ of the minimal second order
operator $\hat{F}(\nabla)$ and note that this time $\hat{F}(\nabla)$ and
$\hat{F}^{M}(\nabla)$ are both minimal differential operators. Then for the
operator $\hat{F}^{M}(\nabla)$ we have two different expansions: one obtained
using the general Fourier method (2.21), and another obtained from the
expansion for the operator $\hat{F}(\nabla)$ using the Mellin transform
(7.12). It is natural to compare these expansions. For the sake of simplicity,
we will consider the case when $\hat{F}^{M}(\nabla)$ is a Laplace type
operator (as we showed above in Sect. 5, the expansion which in this case
generates the Fourier method is equivalent to the standard DeWitt expansion).

The Fourier method expansion (2.21) gives

$\hat{K}_{F^{M}}(\tau|x,x^{\prime})=\frac{\Delta^{-1}(x,x^{\prime})}{(4\pi\tau^{1/M})^{d/2}}\,g^{1/2}(x^{\prime})\sum\limits_{m=-\infty}^{\infty}\tau^{m/M}\\\
\times\\!\\!\\!\\!\sum\limits_{\,n\geq
N_{m}(M)}^{\infty}\mathcal{E}_{M,\frac{d}{2}+Mn-m}\left(-\frac{\sigma}{2\tau^{1/M}}\right)\,\hat{b}^{F^{M}}_{m,n}(x,x^{\prime}),$
(7.18)

while exponentiation using the Mellin transform leads to the formula (which is
a special case of (7.12))

$\hat{K}_{F^{M}}(\tau|x,x^{\prime})=\frac{\Delta^{-1}(x,x^{\prime})}{(4\pi\tau^{1/M})^{d/2}}\,g^{1/2}(x^{\prime})\\\
\times\sum\limits_{m=0}^{\infty}\tau^{m/M}\,\mathcal{E}_{M,d/2-m}\left(-\frac{\sigma}{2\tau^{1/M}}\right)\,\hat{b}^{F}_{m}(x,x^{\prime}).$
(7.19)

It is clearly seen that the two expansions are substantially different and
cannot be directly compared term by term. Comparison of (7.18) and (7.19)
would only be possible if we additionally expand both expressions in powers of
the argument of generalized exponential functions $-\sigma/2\tau^{1/M}$. This
in its turn would correspond to double covariant Taylor series in two
independent variables $\tau$ and
$\sigma^{a^{\prime}}(x,x^{\prime})/\tau^{1/2M}$. The coefficients of these
series are, obviously, the coincidence limits of multiple covariant
derivatives
$\nabla_{a_{1}}\cdots\nabla_{a_{n}}\hat{K}(\tau|x,x^{\prime})\,|_{\,x^{\prime}=x}$—exactly
the ones that were considered in Sect.5 when comparing our results with
conventional Schwinger–DeWitt technique.

This, in particular, explains why the result (7.19) for integer power of the
minimal second order operator (with its absence of $m<0$ terms) does not
contradict the nonzero value (6.13) of $\hat{b}_{-1,1}(x,x^{\prime})$ for the
fourth order operator (6.1). Even if one adjusts its coefficients
$\hat{\varOmega}^{abc}$, $\hat{D}^{ab}$, $H^{a}$ and $\hat{P}$ so that the
operator becomes the square of the second order minimal one, its
$\hat{b}_{-1,1}(x,x^{\prime})$ does not identically vanish because of the
first term of (6.13)—this is a simple demonstration that term by term
comparison of (7.18) and (7.19) does not work.

##  8 Discussion and conclusions

We have suggested a systematic calculational method for the covariant
expansion of the two-point heat kernel for generic minimal and non-minimal
differential operators of any order. This expansion for off-diagonal heat
kernel elements is not an expansion in positive powers of the proper time,
like it happens for the minimal second order operators within a conventional
Schwinger–DeWitt technique. Rather this is an expansion in background
dimensionality of relevant background field objects describing the
dimensionful coefficients of the operator and the corresponding spacetime and
vector bundle curvatures. Any order of this expansion in powers of these
curvatures and their covariant derivatives can be reached in a finite number
of calculational steps, which is what one usually needs in local gradient
expansion in the energy domain below the effective field theory cutoff. For
the coincidence limit of the heat kernel and its arbitrarily high covariant
derivatives this expansion becomes the series in positive fractional powers of
the proper time. This makes our results fully consistent with the structure of
asymptotic expansion of the functional trace of the heat kernel well known
from mathematics literature on differential and pseudo-differential operators
on curved manifolds [6, 7, 8].

Main results of the paper for generic non-minimal operators of order $N$ are
given by Eqs.(2.4)-(2.6), where the set of auxiliary differential operators
$\hat{T}_{n,l}(\nabla)$ is defined by the recurrence relations (3.27). The
efficiency of this result depends on the users skill of calculating the
momentum space integral (2.6) for a given principal symbol of the operator
(2.7). For general minimal operators, whose principal term represents $M$-th
power of Laplacian, this integral can be universally calculated in terms of
generalized exponential functions (2.18) and symmetric tensors (4.2). This
leads to the expansion (2.21)-(2.22) with the generalized HaMiDeW coefficients
$\hat{b}_{m,n}(x,x^{\prime})$ defined by Eqs.(2.23)-(2.24).

There are two basic differences of this expansion from the $M=1$ case of
Schwinger-DeWitt technique. Firstly, one overall exponential function of the
argument $-\sigma(x,x^{\prime})/2\tau^{1/M}$ gets replaced by various
generalized exponential functions of the same argument, multiplying individual
fractional powers of the proper time. Secondly, fractional powers of the
proper time series extend to minus infinity. In the coincidence limit
$x^{\prime}=x$, however, the coefficients of the negative powers vanish in
full accordance with the known properties of the heat kernel trace [6, 7, 8].

Note that the main recursive ingredient of the heat kernel technique is the
solution of the chain of relations (3.27). Even though the anticipation of the
“bottomless” chain of HaMiDeW coefficients for higher-derivatives operators,
expressed in [21], is correct, the recursive procedure for them turns out to
be possible. The structure of these recurrent equations is more complicated as
compared to the Schwinger-DeWitt expansion, but they can be successively
solved. Their formulation crucially depends on the decomposition (3.11) of the
original operator into the auxiliary operators
$\llbracket\hat{F}\rrbracket_{m}$ and $\llbracket\hat{F}\rrbracket_{m,n}$
introduced in Sect.3.

Complexity of the minimal higher derivative operators manifests itself in the
double series nature of the expansion (2.21). As shown in Sect. 7, for minimal
second order operators double series reduces with the aid of Eqs.(5.19) and
(5.4) to the original expansion with the DeWitt coefficients
$\hat{a}_{m}(x,x^{\prime})$ composed from their generalized double indexed
version $\hat{b}_{m,n}(x,x^{\prime})$. In fact, Eq.(5.19) constitutes an
alternative representation of the off-diagonal Schwinger-DeWitt coefficients
$\hat{a}_{m}(x,x^{\prime})$. This representation in terms of the infinite
series (5.4) of the $\hat{b}_{m,n}$ coefficients is not a priori equivalent to
the covariant Taylor expansion of $\hat{a}_{m}(x,x^{\prime})$. Thus, it might
perhaps open the prospects of their explicit (not recursive) calculation and
as a byproduct partial resummation of Schwinger-DeWitt series, discussed from
a somewhat other viewpoint in [37, 38]. This resummation might be facilitated
by the fact that certain subsets of the operators $\hat{T}_{n,l}(\nabla)$,
participating in the construction of $\hat{b}_{m,n}(x,x^{\prime})$
coefficients, can be obtained beyond recursive procedure, see Eqs.(3.31),
(3.32) and (4.10).

It should be emphasized that our results for two-point heat kernels are
essentially of local nature. In particular, for a general curved spacetime the
transition to the locally geodetic coordinate system of
$\sigma^{a^{\prime}}(x^{\prime},x)$ centered at the point $x^{\prime}$ is
restricted by the vicinity of this point where $\Delta(x,x^{\prime})$ is
nonsingular. This domain is bounded by the caustic points $x$ conjugated to
$x^{\prime}$ with $\Delta(x,x^{\prime})=\infty$, at which the relation (5.19)
becomes singular. For practical purposes of UV renormalization this
restriction is not very important, because the coincidence limit of the heat
kernel and all its derivatives is basically everything what one needs in
effective field theory. In this respect, however, it is quite interesting that
the generalized Fourier method does not recover in the heat kernel expansion
(2.21) a one-half power of $\Delta(x,x^{\prime})$ typical for the Schwinger-
DeWitt expansion (1.2). Instead it gives the inverse of
$\Delta(x,x^{\prime})$. Thus, naively this would mean that approaching the
caustic gives a decreasing amplitude of the physical signal rather than its
enhancement which is characteristic of semiclassical approximation.

The explanation of this paradox apparently consists in the observation that
for higher derivative operators with $M>1$ our Fourier method is critically
different from the semiclassical expansion. If one calculates the momentum
space integrals (2.18) by the saddle point method for $\tau\to 0$ then one
would get the expansion in powers of $\tau^{1/(2M-1)}$ rather than
$\tau^{1/M}$ and, moreover, the terms of this expansion will be singular at
$\sigma(x,x^{\prime})\to 0$ [16]. For $M>1$ the semiclassical asymptotic
expansion at $\tau\to 0$ is not homogeneous for $x^{\prime}\to x$ but conveys
a right behavior for large $\sigma(x,x^{\prime})$ even when approaching the
caustic. This, in particular, does not allow us to use the semiclassical
approximation method of Maslov and Fedoriuk [20] in this limit. Conversely,
our background dimensionality expansion is regular at $x\to x^{\prime}$, but
fails for large distances. Interestingly, for minimal operators with $M=1$
these two expansions look equivalent in the vicinity of the heat kernel
diagonal. Here $\Delta(x,x^{\prime})$ is close to one, and in the relation
(5.19) only affects nontrivial match between multiple covariant derivatives of
$\hat{a}_{m}(x,x^{\prime})$, $\Delta(x,x^{\prime})$ and
$\hat{b}_{m}(x,x^{\prime})$ what was explicitly checked in Sect.5.

Local nature of our method can be scrutinized also from another viewpoint.
Obviously, the expansion up to infinitely positive and negative powers of
$\tau$ cannot be regarded as asymptotic for $\tau\to 0$. But the infinite tail
of negative powers of $\tau$ should be regarded as an expansion in positive
powers of the ratio $\sigma^{a^{\prime}}(x,x^{\prime})/\tau^{1/2M}$, so that
the total result is the sum of two expansions which are both asymptotic at
$\tau\to 0$ in the domain $\sigma^{a^{\prime}}(x,x^{\prime})\ll\tau^{1/2M}$.
The coefficients of both expansions are local curvature invariants of growing
dimensionality, which explains the efficiency of this method in effective
field theory and in renormalization theory.

One might notice that on top of very high generality of the suggested
expansions and their detailed analysis no new results were obtained in
concrete applications of this technique. This can be explained by the fact
that any step beyond second order minimal operators is practically impossible
without using powerful symbolic manipulating computer programs. Even the
simplest examples of Sect.5, focused on the comparison of the new method with
well known calculations of the lowest orders of the Schwinger–DeWitt
expansion, would be impossible without using a computer code. While the
solution of DeWitt’s recurrent equations for $\hat{a}_{m}(x,x)$ with small
$m=0,1,2,3$ can be easily performed manually, the same calculation by means of
generalized DeWitt coefficients $\hat{b}_{m,n}(x,x)$ turns out to be much more
complicated, and it was accomplished above only by using the xAct package of
Mathematica. Extension beyond the case of $M=1$ minimal operators is even more
complicated and, in fact, is impossible without the algorithms obtained above.
This is undoubtedly true modulo exceptional cases and special tools, like the
method of universal functional traces [14, 3], the method of squaring the low-
derivative operator, dimensional reduction [15], etc. So the ultimate goal of
this work is to develop on the basis of above algorithms an efficient code for
calculating the heat kernel expansion and using it in high energy
applications.

One of the motivations for this project is the problem of UV renormalization
in Hořava gravity theory, which is computationally extremely complicated
because of Lorentz invariance violation. Extension of the heat kernel method
to Lorentz violating models is also possible [39, 40, 41] and includes a
recent application of the Fourier method [42] along the lines of the Gusynin
method for the operator resolvent [29, 30]. In four dimensions this problem
involves non-minimal operators of the sixth order [15] and requires the whole
set of special methods involving the above mentioned universal functional
traces [3], 3-dimensional reduction on a static background, square root
extraction for the sixth-order operator having hundreds of terms, etc. At
present these are the only available tools of the background field method
needed to avoid calculation of humongous number of Feynman graphs, that was,
for example, undertaken in the proof of asymptotic freedom in
(2+1)-dimensional Hořava gravity [40]. However, these auxiliary tools are not
always possible and often lengthen the needed computations like it happens
with the method of [15], which relies on Sylvester procedure for extracting
the square root of the sixth-order operator in covariant derivatives. The
presented technique can and should circumvent these difficulties, and the
comparison of its efficiency with the approach of [15] is one of our main
future goals.

We accomplish the paper with a couple of brief remarks on possible future
refinement of the suggested formalism. The failure to perform a complete
resummation of negative powers of the proper time suggests that such a
resummation is perhaps, in principle, impossible. This might be explained by
the fact that negative powers of $\tau$ entail positive powers of the vector
$\sigma^{a}(x,x^{\prime})$ rather than the scalar function
$\sigma(x,x^{\prime})$. This conclusion seems to be corroborated by
expressions like (6.13). Therefore the coefficients of this expansion are
tensors of growing rank, and this invalidates universality of the needed
resummation, leaving however the option of summing up the series of particular
tensor structures.

Another remark is that our Fourier method consists of multiple
stages—construction of auxiliary operators $\llbracket F\rrbracket_{m,n}$ in
the decompositions (3.12) and (3.19), recursive solution of equations for the
operators $\hat{T}_{n,l}(\nabla)$, necessity of finding coincidence limits of
multiple derivatives of $\sigma$, presence of primed and unprimed indices,
etc. All this essentially slows down symbolic computations. This suggests the
necessity of an alternative approach combining the advantages of the
commutator technique of the universal functional traces of [3, 15] and the
above Fourier method. Such approach indeed can be worked out, and it is
anticipated that in the new method all generalized heat kernel coefficients
$\hat{b}_{m,n}(x,x^{\prime})$ could be expressed in terms of the conventional
DeWitt coefficients $\hat{a}_{m}^{(-\Box)}(x,x^{\prime})$ of the simplest
minimal operator $-\Box$. This can be achieved via acting upon them by the set
of differential operators which are formed of the products of multiple
commutators of $(-\Box)^{M}$ with the lower derivative part $\hat{P}(\nabla)$
of the full minimal operator (2.13),

$[(-\Box)^{M},[(-\Box)^{M},...[(-\Box)^{M},\hat{P}(\nabla)],...]],$ (8.1)

This is the idea of the method of universal functional traces [3, 15],
exploiting a simple fact that every commutation with $\Box$ increases the
dimensionality of any object at least by one. This algorithm, which is
currently under study [43], is expected to be especially suitable for the
needs of effective field theory, because it clearly shows how a needed
precision of curvature expansion truncates the orders of multiple commutators.

## Acknowledgments

The authors are deeply grateful to Alexander Kurov for his assistance in
computer symbolic calculations. We are also indebted to Darren Grasso for the
correction of matrix algebra in the case of generic non-minimal operators. The
work was partially supported by the Russian Foundation for Basic Research
grant 20-02-00297 and by the Foundation for Theoretical Physics Development
“Basis”.

##  Appendix A The world function and the parallel transport tensor

First of all agree on basic definitions and notation. Let us be given some
$d$-dimensional manifold $\mathcal{M}$, which is the base of the vector bundle
$\pi\colon\mathcal{E}\to\mathcal{M}$. We will use lowercase Latin letters for
spacetime indices, and we will usually omit indices in the bundle, denoting
endomorphisms in fibers by hats. Further, let the connection $\nabla_{a}$ be
given on the sections of this bundle. We everywhere assume that this
connection is symmetric, so that there is no torsion $T^{c}{}_{ab}=0$ and
covariant derivatives commute on scalars. Then the Riemann tensor and
curvature in the bundle are defined as

$\displaystyle[\nabla_{a},\nabla_{b}]v^{c}=R^{c}{}_{dab}v^{d},$ (A.1)
$\displaystyle[\nabla_{a},\nabla_{b}]\varphi=\hat{\mathcal{R}}_{ab}\varphi.$
(A.2)

For the action of $[\nabla_{a},\nabla_{b}]$ on a matrix-valued function, we
obtain

$[\nabla_{a},\nabla_{b}]\hat{P}=[\hat{\mathcal{R}}_{ab},\hat{P}].$ (A.3)

The convolution of the Riemann tensor with respect to a pair of indices
determine the Ricci tensor

$R_{ab}=R^{c}{}_{acb}.$ (A.4)

On a manifold with connection, we can introduce geodesics, as lines, the
tangent vector to which goes into itself by a parallel transport along them.
This leads to the following definition of tangent vectors
$\sigma^{a}(x,x^{\prime})$

$\sigma^{b}\nabla_{b}\sigma^{a}=\sigma^{a},\qquad\big{[}\,\sigma^{a}\big{]}=0.$
(A.5)

Similarly, we can transport along geodesics a tensor of arbitrary nature with
an index in the bundle. The condition on the parallel transport tensor
$\hat{\mathcal{I}}(x,x^{\prime})$ will have the form

$\sigma^{a}\nabla_{a}\hat{\mathcal{I}}=0,\qquad\big{[}\,\hat{\mathcal{I}}\,\big{]}=\hat{1}.$
(A.6)

For the derivatives of $\sigma^{a}(x,x^{\prime})$ and
$\hat{\mathcal{I}}(x,x^{\prime})$, we introduce the following abbreviations

$\displaystyle\sigma^{a}{}_{b_{1}\ldots b_{n}}$
$\displaystyle=\nabla_{b_{n}}\cdots\nabla_{b_{1}}\sigma^{a},$ (A.7)
$\displaystyle\hat{\mathcal{I}}_{b_{1}\ldots b_{n}}$
$\displaystyle=\nabla_{b_{n}}\ldots\nabla_{b_{1}}\hat{\mathcal{I}}.$ (A.8)

Then the definition (A.5) takes the form

$\sigma^{a}{}_{b}\sigma^{b}=\sigma^{a}.$ (A.9)

Differentiating (A.9), we get
$\sigma_{a}{}^{b}\sigma_{b}{}^{c}+\sigma_{a}{}^{bc}\sigma_{b}=\sigma_{a}{}^{c}$,
which in the coincidence limit leads to

$[\sigma^{a}{}_{b}]=\delta_{b}^{a}.$ (A.10)

Sequentially acting on the relations (A.9) and (A.6) by the operator
$\sigma^{c}\nabla_{c}$ and simplifying the expressions using (A.9), we obtain
the important relations

$\displaystyle\sigma^{a}{}_{b_{1}\ldots
b_{n}}\sigma^{b_{1}}\ldots\sigma^{b_{n}}=0,\quad n\geq 2,$ (A.11)
$\displaystyle\sigma^{a_{1}}\ldots\sigma^{a_{k}}\hat{\mathcal{I}}_{a_{1}\ldots
a_{k}}=0.$ (A.12)

We emphasize once again that the introduction of geodesics and related objects
$\sigma^{a}(x,x^{\prime})$ and $\hat{\mathcal{I}}(x,x^{\prime})$ and their
properties does not require the existence of the metric $g_{ab}(x)$ (although,
of course, in this case there is no natural parameter along the geodesic).

If $\mathcal{M}$ is a Riemannian manifold with metric $g_{ab}(x)$, then the
conditions of covariant constancy of the metric $\nabla_{a}g_{bc}=0$ determine
the Christoffel connection

$\displaystyle\nabla_{a}v^{c}=\partial_{a}v^{c}+\Gamma^{c}{}_{ab}v^{b},$
(A.13)
$\displaystyle\Gamma^{c}{}_{ab}=\frac{1}{2}g^{cd}\left(\partial_{a}g_{bd}+\partial_{b}g_{ad}-\partial_{d}g_{ab}\right).$
(A.14)

The convolution of the Ricci tensor determines the scalar curvature

$R=R^{a}{}_{a}.$ (A.15)

The geodesics divergence rate $\sigma_{a}{}^{a}$ is closely related with the
Pauli–Van Vleck–Morette determinant

$\displaystyle\Delta(x,x^{\prime})=\frac{\det\left(-\nabla_{a}\nabla_{b^{\prime}}\sigma\right)}{g^{1/2}(x)\,g^{1/2}(x^{\prime})},$
(A.16) $\displaystyle\sigma^{a}\nabla_{a}\Delta=\Delta(d-\sigma_{a}{}^{a}).$
(A.17)

If the point $x$ lies on the caustic of the point $x^{\prime}$ (that is, if
the geodesics outgoing from $x^{\prime}$ intersect at $x$), the determinant
$D(x,x^{\prime})$ becomes infinite. However, we are everywhere interested only
in the local (ultraviolet) properties of the quantities under consideration,
i.e. the case when $x$ and $x^{\prime}$ are close enough to each other.

Finally, we can calculate the coincidence limits $[\sigma_{a_{1}\cdots
a_{n}}]$ and $[\hat{\mathcal{I}}_{a_{1}\cdots a_{n}}]$. To do this, one needs
to differentiate (A.9) and (A.6) the necessary number of times, go to the
coincidence limits in the expressions obtained, and order the derivatives in
them using (A.1)–(A.2). As a result, we get

$\displaystyle[\sigma_{abc}]$ $\displaystyle=0,$ (A.18)
$\displaystyle[\sigma_{a_{1}a_{2}a_{3}a_{4}}]$
$\displaystyle=\frac{2}{3}R_{a_{1}(a_{3}a_{4})a_{2}},$ (A.19)
$\displaystyle[\sigma_{a_{1}a_{2}a_{3}a_{4}a_{5}}]$
$\displaystyle=\frac{3}{2}\nabla_{(a_{3}}R_{|a_{1}|a_{4}a_{5})a_{2}},$ (A.20)
$\displaystyle[\hat{\mathcal{I}}_{a}]$ $\displaystyle=0,$ (A.21)
$\displaystyle[\hat{\mathcal{I}}_{ab}]$
$\displaystyle=\frac{1}{2}\hat{\mathcal{R}}_{ab},$ (A.22)
$\displaystyle[\hat{\mathcal{I}}_{abc}]$
$\displaystyle=\frac{2}{3}\nabla_{(a}\hat{\mathcal{R}}_{b)c}$ (A.23)

and so on.

##  Appendix B The generalized exponential functions (GEF)

In the articles [44, 16] we showed that the heat kernel of a power of the
Laplacian $(-\Box)^{\nu}$ in a flat $d$-dimensional space with the Euclidean
metric has the form

$\mathbb{K}_{\nu,d}(\tau;x)=\exp\left(-\tau(-\Box)^{\nu}\right)\delta(x)\\\
=\frac{1}{\big{(}4\pi\tau^{1/\nu}\big{)}^{d/2}}\,\mathcal{E}_{\nu,d/2}\left(-\frac{|x|^{2}}{4\tau^{1/\nu}}\right),$
(B.1)

where $\mathcal{E}_{\nu,\alpha}(z)$ is a two-parameter family of some new
special functions defined by the Taylor series

$\mathcal{E}_{\nu,\alpha}(z)=\frac{1}{\nu}\sum\limits_{m=0}^{\infty}\frac{\Gamma\left(\frac{\alpha+m}{\nu}\right)}{\Gamma(\alpha+m)}\frac{z^{m}}{m!}.$
(B.2)

In particular

$\mathcal{E}_{\nu,\alpha}(0)=\frac{\Gamma(\alpha/\nu)}{\nu\Gamma(\alpha)}.$
(B.3)

The expression (B.1) is a direct generalization of the standard heat kernel
for the Laplacian $-\Box$ for $\nu=1$

$\displaystyle\mathcal{E}_{1,\alpha}(z)=\exp(z),$ (B.4)
$\displaystyle\mathbb{K}_{1,d}(\tau;x)=\frac{1}{(4\pi\tau)^{d/2}}\exp\left(-\frac{|x|^{2}}{4\tau}\right),$
(B.5)

in which the functions $\mathcal{E}_{\nu,d/2}(z)$ are in place of the usual
exponential functions. So we call these new functions _the generalized
exponential functions (GEF)_. We investigated their properties in detail in
[44, 16]. The most important of them are the direct and inverse Mellin
transform and the differentiation rule

$\displaystyle\varepsilon_{\alpha,a}(s)\equiv\int\limits_{0}^{\infty}z^{s-1}\mathcal{E}_{\alpha,a}(-z)dz=\frac{\Gamma(s)\Gamma\left(\frac{a-s}{\alpha}\right)}{\alpha\Gamma(a-s)},$
(B.6) $\displaystyle\mathcal{E}_{\alpha,a}(-z)=\frac{1}{2\pi
i}\int\limits_{C}\varepsilon_{\alpha,a}(s)z^{-s}ds,$ (B.7)
$\displaystyle\frac{d^{b}}{dz^{b}}\mathcal{E}_{\alpha,a}(z)=\mathcal{E}_{\nu,a+b}(z).$
(B.8)

Let us also introduce new functions which we will call _the GEF of the second
order_

$\displaystyle\varepsilon_{\nu,b}^{\alpha,a}(s)=\frac{\Gamma(s)\Gamma\left(\frac{a-s}{\alpha\nu}\right)\Gamma\left(\frac{b-s}{\nu}\right)}{\alpha\nu\Gamma\left(\frac{a-s}{\nu}\right)\Gamma(b-s)},$
(B.9) $\displaystyle\mathcal{E}_{\nu,b}^{\alpha,a}(-z)=\frac{1}{2\pi
i}\int\limits_{C}\varepsilon_{\nu,b}^{\alpha,a}(s)\,z^{-s}\,ds.$ (B.10)

Their properties are:

$\displaystyle\mathcal{E}_{1,b}^{\alpha,a}(z)=\mathcal{E}_{\alpha,a}(z),\qquad\mathcal{E}_{\nu,b}^{1,a}(z)=\mathcal{E}_{\nu,b}(z),$
(B.11)
$\displaystyle\mathcal{E}_{\nu,b}^{\alpha,a}(z)=\sum\limits_{n=0}^{\infty}\frac{\Gamma\left(\frac{a+n}{\alpha\nu}\right)\Gamma\left(\frac{b+n}{\nu}\right)}{\alpha\nu\Gamma\left(\frac{a+n}{\nu}\right)\Gamma(b+n)}\frac{z^{n}}{n!}.$
(B.12)

In particular

$\mathcal{E}_{\nu,b}^{\alpha,a}(0)=\frac{\Gamma(a/\alpha\nu)\Gamma(b/\nu)}{\alpha\nu\Gamma(a/\nu)\Gamma(b)}.$
(B.13)

On account of these properties of GEF and second order GEF, the derivation of
Eq.(7.11) consists in the change of integration variable $t\to\mu=d/2-m-Mst$,

$\frac{1}{2\pi
i}\int\limits_{w-i\infty}^{w+i\infty}dt\,\tau^{-t}\,\Gamma(t)\,\mathbb{G}^{m,n}(st,\sigma)\,\\\
=\frac{\tau^{m/sM}}{(4\pi\tau^{1/sM})^{d/2}}\int\limits_{C}\frac{d\mu}{2\pi
i}\,z^{-\mu}\,\varepsilon_{M,d/2+Mn-m}^{s,d/2-m}(\mu),$ (B.14)

and the use of Eqs.(B.9)-(B.10) with the argument $z=\sigma/2\tau^{1/\alpha
M}$.

##  Appendix C Complex powers

The complex power $F^{-s}$ of the operator $F$ for $s\neq 0,-1,-2,\ldots$ can
be formally defined as the following integral of the operator $e^{-\tau F}$

$F^{-s}=\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty}d\tau\,\tau^{s-1}e^{-\tau\hat{F}}.$
(C.1)

For $s\in\mathbb{N}$ the relation (C.1) can be verified by alternately acting
on it by the operator $F$ and integrating by parts $s$ times. Note that from
the definition (C.1), using the properties of gamma functions, it is not
difficult to obtain properties that are naturally expected from complex
powers, for example, $F^{a}\,F^{b}=F^{a+b}$.

The inverse transform is given by the formula

$e^{-\tau F}=\frac{1}{2\pi
i}\int\limits_{w-i\infty}^{w+i\infty}\frac{\tau^{-s}\Gamma(s)}{F^{s}}ds,$
(C.2)

The last expression is easy to understand as follows: the function $\Gamma(s)$
has simple poles at the points $s_{n}=-n$ with residues $(-1)^{n}/n!$. Then
the integral over $s$ reduces to the sum of the residues at these poles, which
exactly gives the standard Taylor series expansion for the operator exponent
$e^{-\tau F}$.

Note that mathematics of spectral geometry and related topics is usually
limited to compact manifolds, which leads to significant simplifications
associated with the discrete spectrum of $F$. Indeed, in this case, for a
strictly positive elliptic differential operator $F=\hat{F}(\nabla)$, there
exists a discrete orthonormal basis of eigenfunctions

$|n\rangle=\phi_{n}(x),\qquad
F|n\rangle=\lambda_{n}|n\rangle,\qquad\lambda_{n}>0.$ (C.3)

Then the operator itself and functions of it can be expressed in a simple way
in terms of eigenfunctions and eigenvalues

$\displaystyle F=\sum_{n}\lambda_{n}|n\rangle\langle n|,$ (C.4) $\displaystyle
e^{-\tau F}=\sum_{n}e^{-\tau\lambda_{n}}|n\rangle\langle n|,$ (C.5)
$\displaystyle\frac{1}{F+\lambda}=\sum_{n}\frac{|n\rangle\langle
n|}{\lambda_{n}+\lambda},$ (C.6) $\displaystyle
F^{-s}=\sum_{n}\lambda_{n}^{-s}|n\rangle\langle n|.$ (C.7)

Using the functional trace allows to determinate important global quantities
such as the heat kernel trace and the standard operator zeta function

$\displaystyle\operatorname{Tr}e^{-\tau F}=\sum_{n}e^{-\tau\lambda_{n}},$
(C.8)
$\displaystyle\zeta_{F}(s)=\operatorname{Tr}F^{-s}=\sum_{n}\lambda_{n}^{-s}.$
(C.9)

On compact manifolds these global quantities can be obtained by the
integration of local ones

$\displaystyle\operatorname{Tr}e^{-\tau\hat{F}}=\int
d^{d}x\,g^{1/2}(x)\,\operatorname{tr}\hat{K}_{F}(\tau|x,x).$ (C.10)

For non-compact manifolds and manifolds with boundaries, which are important
for physical applications, the definition of global quantities requires
additional care because of boundary conditions. The difficulties associated
with this can be circumvented by working directly with local (i.e., point-
dependent) quantities which asymptotically in small $\tau$ and
$\sigma^{a}/\tau^{1/2M}$ are not sensitive to the presence of boundaries. This
is the case we consider in this paper.

##  Appendix D $\varOmega$-terms

Additional contributions $\hat{E}_{2}^{\varOmega}$ and
$\hat{E}_{4}^{\varOmega}$ of nonzero $\hat{\varOmega}^{abc}$ to the Gilkey-
Seeley coefficients $\hat{E}_{2}$ and $\hat{E}_{4}$ of the minimal fourth-
order operator (6.1) can be relatively concisely written down with the aid of
the totally symmetric tensor

$g_{a_{1}\cdots a_{2n}}=[S_{n,2n}]=\frac{(2n)!}{2^{n}n!}g_{(a_{1}a_{2}}\cdots
g_{a_{2n-1}a_{2n})},$ (D.1)

where symmetrization is taken over all $2n$ indices (with the coefficient
$1/(2n)!$).

The expression for $\hat{E}_{2}^{\varOmega}$ is rather simple

$\hat{E}_{2}^{\varOmega}=-\frac{1}{(4\pi)^{d/2}}\frac{3\,\Gamma\left(\frac{d/2-1}{2}\right)}{8\,d\,\Gamma(\frac{d}{2}-1)}\\\
\times\Big{\\{}\frac{1}{4(d+4)}\Big{(}2\hat{\varOmega}_{abc}\hat{\varOmega}^{abc}+3\hat{\varOmega}_{a}\hat{\varOmega}^{a}\Big{)}+\nabla_{a}\hat{\varOmega}^{a}\Big{\\}},$
(D.2)

whereas for $\hat{E}_{4}^{\varOmega}$ it is a sum of the contribution of
$\hat{\varOmega}^{abc}$ alone and the cross terms of $\hat{\varOmega}^{abc}$
with Ricci curvature, fibre bundle curvature and the coefficients of the
operator $\hat{D}^{ab}$ and $\hat{H}^{a}$,

$\hat{E}_{4}^{\varOmega}=\frac{1}{(4\pi)^{d/2}}\frac{\Gamma\left(\frac{d}{4}\right)}{2\Gamma\left(\frac{d}{2}\right)}\Big{\\{}\hat{B}_{\varOmega}+\hat{B}_{\varOmega
R}+\hat{B}_{\varOmega\mathcal{R}}+\hat{B}_{\varOmega DH}\Big{\\}}.$ (D.3)

Separately these contributions look as follows. The most complicated is the
first term $\hat{B}_{\varOmega}$ which is represented as a quartic polynomial
in $\hat{\varOmega}^{abc}$ and its derivatives,

$\begin{split}&\hat{B}_{\varOmega}=-\frac{1}{8}(\nabla_{a}\Box+\Box\nabla_{a})\hat{\varOmega}^{a}+\frac{d}{8(d+2)}(\nabla_{b}\nabla_{a}\nabla^{b}\hat{\varOmega}^{a}+2\nabla_{a}\nabla_{b}\nabla_{c}\hat{\varOmega}^{abc})\\\
&\qquad+\frac{1}{32(d+2)(d+6)}\bigg{\\{}3(d+4)\Big{(}2\hat{\varOmega}^{abc}\nabla_{a}\nabla_{b}\hat{\varOmega}_{c}+2(\nabla_{a}\nabla_{b}\hat{\varOmega}_{c})\hat{\varOmega}^{abc}+2(\nabla_{c}\nabla^{d}\hat{\varOmega}_{abd})\hat{\varOmega}^{abc}+2(\nabla^{d}\nabla_{c}\hat{\varOmega}_{abd})\hat{\varOmega}^{abc}\\\
&\qquad+2(\nabla_{a}\nabla_{b}\hat{\varOmega}^{abc})\hat{\varOmega}_{c}+(\nabla_{b}\nabla_{a}\hat{\varOmega}^{b})\hat{\varOmega}^{a}-(\nabla_{a}\nabla_{b}\hat{\varOmega}^{b})\hat{\varOmega}^{a}\Big{)}+(d+8)\Big{(}-3\hat{\varOmega}^{a}\nabla_{a}\nabla_{b}\hat{\varOmega}^{b}-6(\nabla_{a}\hat{\varOmega}^{abc})\nabla_{b}\hat{\varOmega}_{c}\\\
&\qquad+6\hat{\varOmega}^{abc}\nabla_{c}\nabla^{d}\hat{\varOmega}_{abd}+3(\nabla_{a}\hat{\varOmega}_{b})\nabla^{b}\hat{\varOmega}^{a}+3(\nabla_{a}\hat{\varOmega}_{b})\nabla^{a}\hat{\varOmega}^{b}-3\hat{\varOmega}^{a}\nabla_{b}\nabla_{a}\hat{\varOmega}^{b}-6\hat{\varOmega}^{abc}\nabla^{d}\nabla_{c}\hat{\varOmega}_{abd}-2\hat{\varOmega}_{abc}\Box\hat{\varOmega}^{abc}\\\
&\qquad-3\hat{\varOmega}_{a}\Box\hat{\varOmega}^{a}-2(\Box\hat{\varOmega}_{abc})\hat{\varOmega}^{abc}-3(\Box\hat{\varOmega}_{abc})\hat{\varOmega}^{abc}-6(\nabla_{d}\hat{\varOmega}_{abc})\nabla^{c}\hat{\varOmega}^{abd}+2(\nabla_{d}\hat{\varOmega}_{abc})\nabla^{d}\hat{\varOmega}^{abc}\Big{)}\\\
&\qquad+3(3d+16)\Big{(}2(\nabla_{a}\hat{\varOmega}^{acd})\nabla^{b}\hat{\varOmega}_{bcd}+2(\nabla_{a}\hat{\varOmega}_{b})\nabla_{c}\hat{\varOmega}^{abc}+(\nabla_{a}\hat{\varOmega}^{a})^{2}+2\hat{\varOmega}_{a}\nabla_{b}\nabla_{c}\hat{\varOmega}^{abc}\Big{)}\bigg{\\}}\\\
&\qquad-\frac{g_{abcdefgh}}{32(d+2)(d+6)}\Big{(}\hat{\varOmega}^{iab}\hat{\varOmega}^{cde}\nabla_{i}\hat{\varOmega}^{fgh}+\hat{\varOmega}^{abc}\hat{\varOmega}^{ide}\nabla_{i}\hat{\varOmega}^{fgh}+\hat{\varOmega}^{iab}(\nabla_{i}\hat{\varOmega}^{cde})\hat{\varOmega}^{fgh}\Big{)}\\\
&\qquad+\frac{g_{abcdefghij}}{384(d+2)(d+6)}\Big{(}(\nabla^{a}\hat{\varOmega}^{bcd})\hat{\varOmega}^{efg}\hat{\varOmega}^{hij}+2\hat{\varOmega}^{abc}(\nabla^{d}\hat{\varOmega}^{efg})\hat{\varOmega}^{hij}+\hat{\varOmega}^{abc}\hat{\varOmega}^{def}\nabla^{g}\hat{\varOmega}^{hij}\Big{)}\\\
&\qquad+\frac{g_{abcdefghijkl}}{1536(d+2)(d+6)(d+10)}\hat{\varOmega}^{abc}\hat{\varOmega}^{def}\hat{\varOmega}^{ghi}\hat{\varOmega}^{jkl}.\end{split}$
(D.4)

The cross terms correspondingly read

$\displaystyle\hat{B}_{\varOmega
R}=-\frac{1}{8}R\nabla_{a}\hat{\varOmega}^{a}+\frac{1}{4}R_{ab}\nabla_{c}\hat{\varOmega}^{abc}+\frac{1}{32(d+2)}\Big{(}6R_{ab}\hat{\varOmega}^{a}\hat{\varOmega}^{b}+6R_{ab}\hat{\varOmega}_{c}\hat{\varOmega}^{abc}-3R\hat{\varOmega}_{a}\hat{\varOmega}^{a}-2R\hat{\varOmega}_{abc}\hat{\varOmega}^{abc}$
$\displaystyle\qquad+4\hat{\varOmega}^{abc}\nabla_{a}R_{bc}+2\hat{\varOmega}^{a}\nabla_{a}R+8R_{ab}\nabla_{c}\hat{\varOmega}^{abc}\Big{)},$
(D.5)
$\displaystyle\hat{B}_{\varOmega\mathcal{R}}=\frac{1}{16(d+2)}\Big{(}4(d+4)(\nabla_{b}\hat{\varOmega}_{a})\hat{\mathcal{R}}^{ab}-8\hat{\varOmega}_{b}\nabla_{a}\hat{\mathcal{R}}^{ab}-6\hat{\varOmega}_{acd}\hat{\varOmega}_{b}{}^{cd}\hat{\mathcal{R}}^{ab}-3\hat{\varOmega}_{a}\hat{\varOmega}_{b}\hat{\mathcal{R}}^{ab}\Big{)},$
(D.6) $\displaystyle\hat{B}_{\varOmega