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arXiv:q-alg/9703017

Traces of creation-annihilation operators and Fredholm's formulas

Alexander V Chervov

Published Mar 9, 1997• 0 citations
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Abstract

We prove the formula for the traces of certain class of operators in bosonic and fermionic Fock spaces. Vertex operators belong to this class. Traces of vertex operators can be used for calculation of correlation functions and formfactors of integrable models (XXZ, Sine-Gordon, etc.), that is why we are interested in this problem. Also we show that Fredholm's minor and determinant can be expressed by such traces. We obtain a short proof of the Fredholm's formula for the solution of an integral equation. Analogue of Fredholm's formula in terms of permanents is obtained.

DOI
10.48550/arxiv.q-alg/9703017

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Chapter 3

Perturbation Theory

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We prove the formula for the traces of certain class of operators in bosonic and fermionic Fock spaces. Vertex operators belong to this class. Traces of vertex operators can be used for calculation of correlation functions and formfactors of integrable models (XXZ, Sine-Gordon, etc.), that is why we are interested in this problem. Also we show that Fredholm's minor and determinant can be expressed by such traces. We obtain a short proof of the Fredholm's formula for the solution of an integral equation. Analogue of Fredholm's formula in terms of permanents is obtained.
We prove the formula for the traces of certain class of operators in bosonic and fermionic Fock spaces. Vertex operators belong to this class. Traces of vertex operators can be used for calculation of correlation functions and formfactors of integrable models (XXZ, Sine-Gordon, etc.), that is why we are interested in this problem. Also we show that Fredholm's minor and determinant can be expressed by such traces. We obtain a short proof of the Fredholm's formula for the solution of an integral equation.
In this chapter we derive the perturbation series for the partition function. Since it is given by a trace of an exponential of an operator H0 +λV , we first expand the exponential with respect to λ. The result is a power series with operator valued coefficients. Then we compute the trace of the n’th order coefficient. This can be done with Wick’s theorem which lies at the bottom of any diagrammatic expansion. It states that the expectation value, with respect to exp[−βH0], of a product of 2n annihilation and creation operators is given by a sum of terms where each term is a product of expectation values of only two annihilation and creation operators. Depending on whether the model is fermionic or bosonic or complex or scalar, this sum is given by a determinant, a pfaffian, a permanent or simply the sum over all pairings which is the bosonic analog of a pfaffian. For the many-electron system, one obtains an n × n determinant. The final result is summarized in Theorem 3.2.4 below.
Trace formulas for the heat semi-groups of second quantization operators and their perturbations in the abstract Boson Fock space are given in terms of path (functional) integral representations.As applications, an inequality of Golden-Thompson type and a classical limit are derived for the trace of the heat semi-group of a perturbed second quantization operator.The abstract results are applied to a model of P(∅)P(\emptyset)P(∅) -type in quantum field theory.