Abstract
An approach for solving scattering problems, based on two quantum field theory methods, the heat-kernel method and the scattering spectral method, is constructed. This approach converts a method of calculating heat kernels into a method of solving scattering problems. This allows us to establish a method of scattering problems from a method of heat kernels. As an application, we construct an approach for solving scattering problems based on the covariant perturbation theory of heat-kernel expansions. In order to apply the heat-kernel method to scattering problems, we first calculate the off-diagonal heat-kernel expansion in the frame of covariant perturbation theory. Moreover, as an alternative application of the relation between heat kernels and partial-wave phase shifts presented in this paper, we give an example of how to calculate a global heat kernel from a known scattering phase shift.
Highlights
In this paper, based on two quantum field theory methods, heat-kernel method [1] and scattering spectral method [2], we present a new approach to solve scattering problems
Based on the heat-kernel expansion given by the covariant perturbation theory [10,11,12], by the relation between partial-wave phase shifts and heat kernels given by Eq (3), we establish an expansion for scattering phase shifts
First order The first-order contribution given by the covariant perturbation theory and the Born approximation can be directly obtained by substituting the potential (66) into Eqs. (20) and (63), respectively: δl(1)
Summary
In this paper, based on two quantum field theory methods, heat-kernel method [1] and scattering spectral method [2], we present a new approach to solve scattering problems. A method for calculating scattering phase shifts based on the covariant perturbation theory in the heat kernel theory is established as an example of our approach. 3, based on the relation between partial-wave phase shifts and heat kernels given, we establish an approach for the calculation of partial-wave phase shifts, based on an heatkernel expansion, the covariant perturbation theory. The main result of the present paper is the following theorem which reveals a relation between partial-wave scattering phase shifts and heat kernels. Page 3 of 14 294 obtain a partial-wave phase shift from a known heat kernel directly By this relation, what we can obtain is one method for scattering problems. In order to prove the theorem, we need to first find a relation between partial-wave heat kernels and heat kernels
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