# On the spectra of fermionic second quantization operators

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On the spectra of fermionic second quantization operators Shinichiro Futakuchi∗and Kouta Usui† Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan. June 13, 2013 Abstract We derive several formulae for the spectra of the second quantization operators in abstract fermionic Fock spaces. 1 Introduction Abstract theory of Fock spaces [1, 2, 3, 4] provides powerful mathematical tools when one analyzes models of quantum field theory, the most promising physical theory which is expected to describe the fundamental interactions of elementary particles. This results from the fact that quantum filed theory deals with a quantum system with infinitely many degrees of freedom, including particles which may be created or annihilated, and that Fock spaces are furnished with suitable structure to describe particle creation or annihilation. In mathematical physics, two different types of Fock spaces, bosonic (or symmetric) Fock space and fermionic (or antisymmetric) Fock spaces, are considered, reflecting the fact that there are two different sorts of elementary particles in Nature — bosons and fermions — . In mathematical analyses of quantum theories, one of the most important problems includes to determine the spectra of various self-adjoint operators representing physical observables, especially, that of a Hamilto- nian, which represents the total energy of the system under consideration. To each self-adjoint operator A acting in an underlying one particle Hilbert space H, bosonic or fermionic second quantization is defined as an operator which naturally “lifts” A up to the bosonic or fermionic Fock space over H, respectively. In a bosonic Fock space, the spectra of second quantization operators were well investigated and useful formulae for them have been available. However, as far as we know, the corresponding useful formulae in a fermionic Fock space are still missing. The main motivation of the present work is to derive such formulae in fermionic Fock spaces to fill the gap.

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