Heat semigroups on Weyl algebra

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Abstract

We study the algebra of semigroups of Laplacians on the Weyl algebra. We consider first-order partial differential operators i± forming the Lie algebra [j±,k±]=iRjk± and [j+,k]=i12(Rjk++Rjk) with some anti-symmetric matrices Rij± and define the corresponding Laplacians Δ±=g±iji±j± with some positive matrices g±ij. We show that the heat semigroups exp(tΔ±) can be represented as a Gaussian average of the operators expξ,± and use these representations to compute the product of the semigroups, exp(tΔ+)exp(sΔ) and the corresponding heat kernel.

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 L, Θr(β)=Trexp(βω),where ω=L and β is a positive parameter (that plays the role of the temperature), as well as the quantum heat traces, Θb,f(β,μ)=TrEb,fβωμ.where μ is another real parameter (not necessarily positive, that plays the role of the chemical potential) and Eb,f are functions defined by Eb,f(x)=1ex1.These functions come from the quantum statistical physics with the indices b and f 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 L±, of the form Ψ(t,s)=Trexp(tL+)exp(tL)exp(sL+)exp(sL).In our recent paper [7] we introduced another invariant, called the Bogolyubov invariant, Bb(β)=Tr{Ef(βω+)Ef(βω)}{Eb(βω+)Eb(βω)}, 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 X(t,s)=Trexp(tL+)exp(tL).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 L± 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 Rn 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 L+ and L, U(t,s)=exp(tL+)exp(sL),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 L± can be represented in the form L±=G±ABA±B±,where GAB is a positive symmetric matrix and A are some first-order partial differential operators. Suppose further that the operators A+,B form a closed Lie algebra, say, [A+,B+]=C+CABC++AB+,[A,B]=CCABC+AB,[A+,B]=E+CABC++ECABC+RAB, where C±CAB, AB±, E±CAB and RAB are some constants that we collectively call curvatures (since they determine the commutators). Here we assume that there are N operators A+ and N operators A, so the indices range over 1 to N. Then, by using purely algebraic methods one can try to represent the heat semigroups in the form exp(tL±)=RNdξΦ±(t,ξ)expξ,±,where ξ,±=ξAA± and Φ±(t,ξ) 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 U(t,s)=R2NdηdξΦ+(t,ξ)Φ(s,η)expη,+expξ,.And, finally, one can use the Campbell–Hausdorff type formulas to compute the convolution of the operators expη,+expξ,=expF+(η,ξ),++F(η,ξ),,where F±(η,ξ) 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 Rn, 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 Rn in the form that will be convenient for us later. In particular, we introduced a Gaussian average of functions on Rn 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 Rij 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, (1,,n,x1,,xn,i), where j=j12iRjkxk,Given a positive matrix g we introduce an operator called the Laplacian by Δg=gijij,where gij is the inverse matrix. We consider all these operators acting in the space C0(Rn) of smooth functions with compact support. Since this space is dense in the Hilbert space L2(Rn), the extension of these operators to the whole Hilbert space L2(Rn) is well defined. Moreover, the operators i 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 R=(Rij) be an anti-symmetric matrix and g=(gij) be a positive symmetric matrix. Let D(t)=(Dij) be a symmetric matrix defined by D(t)=iRcothtg1iR, T(t) be a matrix defined by T(t)=D(t)+iR,and Ω(t) be a function defined by Ω(t)=detT(t)12=detg1sinh(tg1iR)g1iR12. Let i:C0(Rn)C0(Rn) be anti-self-adjoint first-order partial differential operators of the form k=k12iRkjxj acting on the space of smooth functions in Rn forming the Lie algebra [j,k]=iRjk,and Δg be the operator defined by Δg=gijij.Then exp(tΔg)=(4π)n2Ω(t)Rndξexp14ξ,D(t)ξexpξ,.

In Section 8 we consider two sets of such anti-self-adjoint first-order partial differential operators i+ and j and prove the following theorem.

Theorem 2

Let R±=(Rij±) be two anti-symmetric matrices, g±=(gij±) be two positive symmetric matrices and let D±(t) be two symmetric matrices defined by D±(t)=iR±coth(tg±1iR±).Let ̃=(̃AB) be a 2n×2n anti-symmetric matrix defined by ̃=R+RRR, where R=12R++R,and Q̃(t,s)=(Q̃AB) and G̃1(t,s)=(G̃AB) be the 2n×2n symmetric matrices defined by Q̃(t,s)=D+(t)iRiRD(s),G̃1(t,s)=tanh1Q̃1(t,s)ĩi1. Let i±:C0(Rn)C0(Rn) be anti-self-adjoint first-order partial differential operators of the form k±=k12iRkj±xj acting on the space of smooth functions over Rn forming the Lie algebra [i+,j+]=iRij+,[i,j]=iRij,[i+,j]=iRij, and Δ± be two operators defined by Δ±=g±iji±j±.Let (D̃1,,D̃n,D̃n+1,,D̃2n)=(1+,,n+,1,,n) and (t,s) be the operator defined by (t,s)=D̃,G̃1(t,s)D̃.Then exp(tΔ+)exp(sΔ)=exp(t,s).

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 R±, g±, D±(t), and the operators ± be defined as in Theorem 2.

Theorem 3

Let T±(t), D(t,s) and Z(t,s) be the matrices defined by T±(t)=D±(t)+iR,D(t,s)=D+(t)+D(s),Z(t,s)=D+(t)D(s)2iR, and Ω(t,s) be a function defined by Ω(t,s)=detT+(t)12detT(s)12detD12(t,s).Let H(t,s) be a symmetric matrix defined by H(t,s)=14D(t,s)ZT(t,s)D1(t,s)Z(t,s).Let i and Xj be the operators defined by i=12(i++i),Xi=i+i. Then exp(tΔ+)exp(sΔ)=(4π)n2Ω(t,s)expX,D1(t,s)X×Rndαexp14α,H(t,s)α12α,ZT(t,s)D1(t,s)Xexpα,.

In Section 9 we compute the convolution of the heat kernels and prove the following theorem. Let the matrices R±, g±, D±(t), T±(t) and the operators ± be defined as in Theorem 3.

Theorem 4

Let A±(t,s) and B(t,s) be matrices defined by A+(t,s)=D+(t)T+T(t)D1(t,s)T+(t),A(t,s)=D(s)TT(s)D1(t,s)T(s),B(t,s)=T+(t)D1(t,s)T(s), and S be a function defined by S(t,s;x,x)=14x,A+(t,s)x+14x,A(t,s)x12x,B(t,s)x. Then the kernel of the product of the semigroups is U(t,s;x,x)=exp(tΔ+)exp(sΔ)δ(xx),=detSxx(t,s)2π12expS(t,s;x,x).

Section snippets

Gaussian integrals

We will make extensive use of Gaussian integrals. We denote by , the standard pairing in Rn. Let γ be a symmetric n×n matrix with positive definite real part. Then for any vector A there holds (see, e.g. [17]) Rndξexp14ξ,γξ+A,ξ=(4π)n2detγ12expA,γ1A.Let S:Rn be a quadratic polynomial with positive definite real quadratic part. Such a polynomial can always be written in the form S(ξ)=S0+12Sξ,Sξξ1Sξ,where S0 is a constant, Sξ and Sξξ are the vector of first partial derivatives and the

Gaussian kernels

Let S be a quadratic polynomial on Rn×Rn of the form S(x,y)=14x,Ax12x,Cy+14y,By12v,x12w,y+r,where A and B are real symmetric positive matrices, C is a complex matrix, v,w are some vectors and r is a complex number. We introduce the following notation for the derivatives of the function S S1(x,y)=Sx(x,y)=12AxCyvS2(x,y)=Sy(x,y)=12CTx+BywS11(x,y)=Sxx(x,y)=12A,S12(x,y)=Sxy(x,y)=12C,S22(x,y)=Syy(x,y)=12B.

Let A be the 2n×2n matrix A=ACCTB.We assume that the matrix ReA is non-negative, ReA0

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 g be a Lie algebra. For any operator Xg we define the operator adX:gg by adXY=[X,Y].Let P and X be some operators in some Lie algebra g. Our goal is to compute the product (expP)(expX). We proceed rather formally.

We consider a smooth path, Qt, t[0,1] in a Lie algebra. We prove a useful lemma.

Lemma 1

The derivative of the exponential expQt is given by texpQt=exp(Qt)1exp(adQ

Curvature

Let Mn(R) be the algebra n×n matrices with the standard matrix product and R=(Rij)Mn(R) be a fixed real antisymmetric matrix that we call the curvature. We define a bilinear binary operation (that we will call a R-bracket) {,}:Mn(R)×Mn(R)Mn(R)as follows: for any two matrices A=(Aij) and B=(Bij) {A,B}=ARBBRA.Obviously, this bracket is anti-symmetric, {A,B}={B,A} and satisfies the Jacobi identity {A,{B,C}}+{B,{C,A}}+{C,{A,B}}=0Let SnMn(R) and LnMn(R) be the subspaces of symmetric and

Weyl algebra

The Lie algebra hn of the Heisenberg group H2n+1 is generated by the (2n+1) operators (p1,,pn,x1,,xn,i) satisfying the commutation relations [pk,xj]=iδkj,[pk,pj]=[xk,xj]=[pk,i]=[xk,i]=0. We will just call it the Heisenberg algebra. We will use another basis of the Heisenberg algebra (P1,,Pn,x1,,xn,i) defined by Pk=pk+12Rkjxj;satisfying the commutation relations [Pk,xj]=iδkj,[Pk,Pj]=iRkj,[xk,xj]=[Pk,i]=[xk,i]=0. Obviously, the operators (Pj,i) form a subalgebra of the Heisenberg algebra.

Its

Noncommutative Gaussian integrals

Let f(ξ) be a function depending on the operators i and f(ξ)t be the Gaussian average defined by (2.7). Suppose that we can find operators L(t) such that this average satisfies the differential equation tf(ξ)t=L(t)f(ξ)t.Recall that f(ξ)0=f(0), which serves as the initial condition. Then, if the operators L(t) and L(s) commute for any t and s, one can solve this equation to obtain f(ξ)t=exp0tdτL(τ)f(0).

We use this idea to study the Gaussian average of the operator expξ,, expξ,t=(4πt)n2(det

Product of semigroups

We consider the set of anti-self-adjoint operators (1+,,n+,1,,n,i) acting on the space C0(Rn) of smooth functions of compact support forming the Lie algebra [i+,j+]=iRij+,[i,j]=iRij,[i+,j]=iRij, where Rij=12Rij++Rij.This algebra can be written in a more compact form by introducing the operators (D̃A), A=1,,2n, by D̃1=1+,,D̃n=n+,D̃n+1=1,,D̃2n=n. We will use the convention that the capital Latin indices run over 1,,2n. The operators D̃A form the algebra [D̃A,D̃B]=ĩA

Convolution of heat kernels

We work with operators i±=i12iRij±xj,acting on smooth functions in Rn. These operators form the Lie algebra (8.3). The operators i and Xi defined in (8.9), (8.10) then take the form i=i12iRijxj,Xi=iFijxj, with the matrices Rij and Fij defined in (8.4), (8.15). The corresponding Laplacians are Δ±=g±iji±j±.

Recall the definition of the matrices D±(t) and T±(t) by (8.25), (7.39). Also, notice that the functions Ω±(t) defined by (7.31) are determined by the determinant of the matrices T± Ω

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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