Heat semigroups on Weyl algebra
Introduction
Elliptic partial differential operators on manifolds play a crucial role in global analysis, spectral geometry and mathematical physics [1], [3], [5], [10], [13]. The spectrum of elliptic operators does, of course, depend on the geometry of the manifold. Therefore, one can ask the question: “Does the spectrum of an elliptic operator describe the geometry?” It is well known now that the answer to this question is negative, that is, there are non-isometric manifolds that have the same spectrum (see, e.g. [10], [14], [18]). Another area where elliptic operators are of great importance is the quantum field theory and quantum gravity (see, for example, [12], [15], [16], [19]). In this setting the resolvent and the heat kernel of an elliptic operator enable one to study the Green functions of quantum fields and the corresponding effective action. This is applied to study, in particular, the creation of particles in strong time-dependent gravitational and electromagnetic fields [7]. An important tool to study the spectra of elliptic operators is the heat semigroup and the associated heat kernel and its trace. However, the heat trace describes only the eigenvalues of the operators but not their eigensections. That is why, it makes sense to study more general invariants of partial differential operators, or, even, a collection of operators, that might contain more information about the geometry of the manifold. Such invariants are not necessarily spectral invariants that only depend on the eigenvalues of the operators; they depend, rather, on both the eigenvalues and the eigensections.
In our paper [9] we (with B. J. Buckman) initiated the study of a new invariant of second-order elliptic partial differential operators that we called heat determinant. In our paper [6] we studied so called relativistic heat trace of a Laplace type operator , where and is a positive parameter (that plays the role of the temperature), as well as the quantum heat traces, where is another real parameter (not necessarily positive, that plays the role of the chemical potential) and are functions defined by These functions come from the quantum statistical physics with the indices and standing for the bosonic and fermionic cases. It was shown that such traces can be reduced by some integral transform to the usual heat traces.
In the paper [8] we studied so called relative spectral invariants of two operators , of the form In our recent paper [7] we introduced another invariant, called the Bogolyubov invariant, and applied it to the study of particle creation in quantum field theory and quantum gravity. It was shown that these invariants can be reduced to the study of the combined traces of the form The long term goal of this project is to develop a comprehensive methodology for such invariants in the same way as the theory of the standard heat trace invariants. The primary motivation for this study is spectral geometry and quantum field theory.
It is impossible to compute the combined traces (1.6) exactly for the general Laplace type operators on manifolds. One can make some progress towards calculation of such traces for operators and manifolds with some symmetries. This has been initiated for scalar Laplacians with constant magnetic fields in our paper [2], [3] in and for scalar Laplacians on symmetric spaces. Finally, these ideas enabled us to compute the heat trace for Laplacians on homogeneous bundles over symmetric spaces in [4].
In the present paper we study the product of the semigroups of two operators and , and the corresponding kernels by using purely algebraic tools.
The main idea of this approach can be described as follows. Suppose that two Laplace type operators can be represented in the form where is a positive symmetric matrix and are some first-order partial differential operators. Suppose further that the operators form a closed Lie algebra, say, where , , and are some constants that we collectively call curvatures (since they determine the commutators). Here we assume that there are operators and operators , so the indices range over to . Then, by using purely algebraic methods one can try to represent the heat semigroups in the form where and are some functions that depend on the curvatures. Such representations were found in our papers [2], [3], [4] in some special cases.
Then one can use this representation to compute the convolution And, finally, one can use the Campbell–Hausdorff type formulas to compute the convolution of the operators where are some functions. The main point of this idea is that it is much easier to compute the convolution of the exponentials of the first-order differential operators that form some Lie algebra than to compute the convolution of the exponential of the second-order differential operators.
For this idea to work we consider in the present paper a rather simple very special non-geometric setup of operators on , such that they form a nilpotent algebra described below. This problem has a direct application to the creation of particles in time dependent magnetic fields [7].
In Section 2 we describe the standard theory of Gaussian integrals in in the form that will be convenient for us later. In particular, we introduced a Gaussian average of functions on and study its properties. In Section 3 we study so called Gaussian kernels. We show that the set of Gaussian kernel is a semigroup with respect to the convolution and study some of its sub-semigroups. In Section 4 we explore some of the well known formulas related to the Campbell–Hausdorff series and prove a couple of useful lemmas. In Section 5 we introduce a real antisymmetric matrix that we call curvature study some canonical functions of this matrix.
In Section 6 we consider the Heisenberg algebra and its universal enveloping algebra. We introduce a particular representation of the Heisenberg algebra related to the Weyl algebra of differential operators with polynomial coefficients, , where Given a positive matrix we introduce an operator called the Laplacian by where is the inverse matrix. We consider all these operators acting in the space of smooth functions with compact support. Since this space is dense in the Hilbert space , the extension of these operators to the whole Hilbert space is well defined. Moreover, the operators are anti-self-adjoint so that the Laplacian is self-adjoint.
In Section 7 we compute integrals of functions of the operators . We prove the following theorem.
Theorem 1 Let be an anti-symmetric matrix and be a positive symmetric matrix. Let be a symmetric matrix defined by
be a matrix defined by and be a function defined by Let be anti-self-adjoint first-order partial differential operators of the form acting on the space of smooth functions in forming the Lie algebra and be the operator defined by Then
In Section 8 we consider two sets of such anti-self-adjoint first-order partial differential operators and and prove the following theorem.
Theorem 2 Let be two anti-symmetric matrices, be two positive symmetric matrices and let be two symmetric matrices defined by Let be a anti-symmetric matrix defined by where and and be the symmetric matrices defined by Let be anti-self-adjoint first-order partial differential operators of the form acting on the space of smooth functions over forming the Lie algebra and be two operators defined by Let and be the operator defined by Then
We should remark here that Eq. (1.27) should be understood in terms of a power series; it is well defined even if the matrix is not invertible.
We also prove the following theorem. Let the matrices , , , and the operators be defined as in Theorem 2.
Theorem 3 Let , and be the matrices defined by and be a function defined by Let be a symmetric matrix defined by Let and be the operators defined by Then
In Section 9 we compute the convolution of the heat kernels and prove the following theorem. Let the matrices , , , and the operators be defined as in Theorem 3.
Theorem 4 Let and be matrices defined by and be a function defined by Then the kernel of the product of the semigroups is
Section snippets
Gaussian integrals
We will make extensive use of Gaussian integrals. We denote by the standard pairing in . Let be a symmetric matrix with positive definite real part. Then for any vector there holds (see, e.g. [17]) Let be a quadratic polynomial with positive definite real quadratic part. Such a polynomial can always be written in the form where is a constant, and are the vector of first partial derivatives and the
Gaussian kernels
Let be a quadratic polynomial on of the form where and are real symmetric positive matrices, is a complex matrix, are some vectors and is a complex number. We introduce the following notation for the derivatives of the function
Let be the matrix We assume that the matrix is non-negative,
Campbell–Hausdorff formula
We describe some of the well-known facts about the Campbell–Hausdorff formula (for a detailed exposition see, e.g. [11]). Let be a Lie algebra. For any operator we define the operator by Let and be some operators in some Lie algebra . Our goal is to compute the product . We proceed rather formally.
We consider a smooth path, , in a Lie algebra. We prove a useful lemma.
Lemma 1 The derivative of the exponential is given by
Curvature
Let be the algebra matrices with the standard matrix product and be a fixed real antisymmetric matrix that we call the curvature. We define a bilinear binary operation (that we will call a -bracket) as follows: for any two matrices and Obviously, this bracket is anti-symmetric, and satisfies the Jacobi identity Let and be the subspaces of symmetric and
Weyl algebra
The Lie algebra of the Heisenberg group is generated by the operators satisfying the commutation relations We will just call it the Heisenberg algebra. We will use another basis of the Heisenberg algebra defined by satisfying the commutation relations Obviously, the operators form a subalgebra of the Heisenberg algebra.
Its
Noncommutative Gaussian integrals
Let be a function depending on the operators and be the Gaussian average defined by (2.7). Suppose that we can find operators such that this average satisfies the differential equation Recall that , which serves as the initial condition. Then, if the operators and commute for any and , one can solve this equation to obtain
We use this idea to study the Gaussian average of the operator ,
Product of semigroups
We consider the set of anti-self-adjoint operators acting on the space of smooth functions of compact support forming the Lie algebra where This algebra can be written in a more compact form by introducing the operators , , by We will use the convention that the capital Latin indices run over . The operators form the algebra
Convolution of heat kernels
We work with operators acting on smooth functions in . These operators form the Lie algebra (8.3). The operators and defined in (8.9), (8.10) then take the form with the matrices and defined in (8.4), (8.15). The corresponding Laplacians are .
Recall the definition of the matrices and by (8.25), (7.39). Also, notice that the functions defined by (7.31) are determined by the determinant of the matrices
Acknowledgment
I am very grateful to an anonymous referee for pointing out some missing details in the original version of the proof of Lemma 7, which led to the significant improvement of the paper.
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