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Covariant perturbation expansion of off-diagonal heat kernel

Covariant perturbation expansion is an important method in quantum field theory. In this paper an expansion up to arbitrary order for off-diagonal heat kernels in flat space based on the covariant perturbation expansion is given. In literature, only diagonal heat kernels are calculated based on the covariant perturbation expansion.


Introduction
The heat-kernel method is widely used in physics [1,2] and mathematics [1,3]. The heatkernel method can extract the information embedded in an operator; in physics the operator is often the Hamiltonian [1,2]. In quantum field theory, for example, one can calculate oneloop effective actions, vacuum energies, and other spectral functions by the heat-kernel method [1,4,5]. In statistical mechanics, the heat-kernel method is very useful since the global heat kernel, the trace of the heat kernel, is indeed the partition function [6][7][8]. In scattering theory, the heat-kernel method can be used to calculate scattering phase shifts [9,10].
Developing various asymptotic expansions is a core issue in the study of the heatkernel method. There is much research on the perturbation calculation of heat kernels. Nevertheless, most literature dealing with only diagonal heat kernel [1,[11][12][13], while there are only a very few literature calculates off-diagonal heat kernel [14][15][16][17][18]. Nevertheless, rather than the diagonal one, the off-diagonal heat kernel contains all information of an operator, and many problems need off-diagonal, e.g., the heat-kernel method for scattering [9,10].
The covariant perturbation expansion, among various heat-kernel algorithms, is a very effective method [19][20][21][22][23][24][25][26][27][28][29]. It can account for non-local effects [30]. Particularly, the covariant perturbation expansion is uniformly convergent [20,27]. The special advantage of the covariant perturbation, as an example, is shown in Ref. [9]. It is provided in Ref. [9] a heat-kernel method to calculate scattering phase shifts. By this method, each method of calculating heat kernels can be converted into a method of solving a scattering problem. It is shown that the perturbation expansion of phase shift obtained by the covariant perturbation expansion is more accurate than the Born approximation.
In a series of papers, only the diagonal heat kernel is considered [5]. In this paper, we expand the off-diagonal heat kernel to arbitrary order for flat space cases by the covariant perturbation theory.
In section 2, we provide an expansion of heat kernels based on the covariant perturbation expansion. In section 3, we give an expansion of off-diagonal heat kernels. In section 4, we give an alternative expression of off-diagonal heat kernels. In section 5, the first three orders of diagonal and global heat kernels are given. The conclusion is given in section 6.

The heat kernel
For an operator D, we can define the heat kernel operator [1], (2.1) The heat kernel operator is determined by the heat equation with the initial condition K (0) = 1.
The heat kernel is the matrix element of the heat kernel operator [5] K (τ ; and the global heat kernel is the trace of the heat kernel where φ λ (x) and λ are the eigenfunction and eigenvalue of the operator D, determined by the eigenequation Dφ λ = λφ λ .

The covariant perturbation expansion
In Refs. [5,20,25,26], the covariant perturbation expansion is applied to calculate diagonal heat kernels to first three orders. In Ref. [9], for calculating the scattering phase shift, we use the covariant perturbation expansion to calculate the non-diagonal heat kernel. In this paper, we give an expansion of non-diagonal heat kernels to n-th order by the covariant perturbation method. The expansion of a heat kernel operator in flat space can be expressed as [5] K D (τ ) = K 0 (τ ) + K 1 (τ ) + K 2 (τ ) + · · · , (2.5) where K n (τ ) is the n-th order heat kernel operator. In our cases, In order to achieve the non-diagonal heat kernel, from Eqs. (2.2) and (2.5), we achieve the following recurrence relations: (2.7) The solution of Eq. (2.7) can be expressed formally as Note that C n (τ ) here is an operator. Derivating Eq. (2.8) with respect to τ gives From Eqs. (2.7) and (2.10), we obtain By integrating both sides of Eq. (2.11) with respect to τ , we obtain where C n (0) = 0. By Eq. (2.8), we obtain (2.14) By Eqs. (2.12), (2.13), and (2.14), we arrive at Repeatedly using Eq. (2.15), we obtain

The off-diagonal heat kernel
Before we calculate the n-th order matrix elements of heat kernels, we first need to calculate the matrix elements of the leading order heat kernel. The matrix element of a leading order heat kernel in the coordinate representation reads [5] x |K 0 (τ )| y = 1 where ν denotes the dimension of space. Now we calculate the n-th order matrix elements of a heat kernel x |K n | y based on Eq. (2.16).
Inserting the identity 1 = d ν z |z z| into x |K n | y , we have By Eq. (3.1), we arrive at where The integral in I 1 can be performed by using the Gaussian integral [5], Then we arrive at (3.9) Introducing and using Eq. (3.5), we obtain 4 An alternative expression of the n-th order heat kernels In this section, we suggest an alternative expression of the n-th order matrix element of heat kernel x |K n | y , in which the integral with variable upper limit given in the above section is replaced by definite integrals.
The integral in Eq. (3.11) is a variable upper limit integral. In the following we rewrite Eq. (3.11) by a definite integral.

The first three orders of diagonal and global heat kernels
In this section, based on the above results, we give two equivalent represents of the first three orders of diagonal and global heat kernels: one is represented byṼ (k n ) and the other is represented by V (R n ).

The heat kernel with respect toṼ (k n )
First-order contribution. The first-order heat kernel x |K 1 | y in the coordinate representation, by Eq. (3.9), is The first-order diagonal heat kernel is then The first-order global heat kernel then reads Second-order contribution. The second-order heat kernel x |K 2 | y in the coordinate representation, by Eq. (3.9), is The second-order diagonal heat kernel is The second-order global heat kernel then reads where erfi (z) gives the imaginary error function erf (z) /i. Third-order contribution. The third-order heat kernel x |K 3 | y in the coordinate representation, by Eq. (3.9), is The third-order diagonal heat kernel x |K 3 | x is The third-order global heat kernel then reads (5.9)

The heat kernel with respect to V (R n )
In this section we give an alternative expression of first three orders of diagonal and global heat kernels, which is represented V (R n ), based on Eq. (3.11).
By Eq. (3.11), the first-order heat kernel x |K 1 | y is 10) The first-order diagonal heat kernel is The first-order global heat kernel is The second-order heat kernel is The second-order diagonal heat kernel is 14) The second-order global heat kernel is The third-order heat kernel x |K 3 | y is The third-order diagonal heat kernel is The third-order global heat kernel is

The heat kernel with respect to u n
In this section, we give an equivalent expression of the first three orders of diagonal and global heat kernels, which is represented u n , based on Eq. (4.2). By Eq. (4.2), the first-order heat kernel x |K 1 | y is The first-order diagonal heat kernel is The first-order global heat kernel is The second-order heat kernel x |K 2 | y is The second-order diagonal heat kernel is The second-order global heat kernel is The third-order heat kernel x |K 3 | y is The third-order diagonal heat kernel is The third-order global heat kernel is

Conclusion
The heat-kernel method is widely used in physics and mathematics. In application, there are few heat kernel which can be solved exactly. Thus in most cases, we need to solve the heat kernel approximately. As a kind of approximation methods, there are many asymptotic method for solving heat kernels, such as the Seeley-DeWitt type expansion [1] and the covariant perturbation expansion [19][20][21][22][23][24][25][26][27][28][29].
In literature, the covariant perturbation expansion is only used to calculate diagonal heat kernels. In this paper, we use the covariant perturbation expansion to calculate nondiagonal heat kernel.
The non-diagonal heat kernel contains all information of an operator. Therefore, starting from a non-diagonal heat kernel, we can calculate many physical quantities, such as effective actions, vacuum energies [4], spectral counting functions [8], thermodynamic quantities, and scattering phase shifts [9,10]. In the further work, we will calculate such quantities by the result given by the present paper.