Statistical origin of pseudo-Hermitian supersymmetry and pseudo-Hermitian fermions

Abstract

We show that the metric operator for a pseudo-supersymmetric Hamiltonian that has at least one negative real eigenvalue is necessarily indefinite. We introduce pseudoHermitian fermion (phermion) and abnormal phermion algebras and provide a pair of basic realizations of the algebra of N = 2 pseudo-supersymmetric quantum mechanics in which pseudo-supersymmetry is identified with either a boson-phermion or a bosonabnormal-phermion exchange symmetry. We further establish the physical equivalence (non-equivalence) of phermions (abnormal phermions) with ordinary fermions, describe the underlying Lie algebras, and study multi-particle systems of abnormal phermions.The latter provides a certain bosonization of multi-fermion systems.

Two- and four-dimensional representations of the PT - and CPT -symmetric fermionic algebras

“…Let us examine the anticommutator of η with η PT . According to [2], we obtain a fermionic algebra with a minus sign:…”

Pseudo-Hermitian Systems, Involutive Symmetries and Pseudofermions

“…. , N, corresponding to eigenvalue E (1) , while b annihilates all states |ψ N +k , corresponding to eigenvalue E (2) . These two operators are ladder operators for the 2N-level Hamiltonian,…”

Nonlinearn-pseudo fermions

“…At n = 1 and b(1) = a # (1) the phermion relation aa # + a # a = 1 is recovered [11,12]. In [3] a suggestion is made that if a pseudo-fermion operator a(1) ≡ a admits a vacuum (a|0 = 0) then b † also admits a vacuum and b is η-pseudo adjoint to a, i.e.…”

“…Physical example of phermions is given in [12], where (phermions were called pseudofermions and) the first bi-overcomplete family of PF coherent states was established. The standard fermions and the pseudo-fermions so far considered [11,12] are defined through linear in terms of the corresponding creation and annihilation operators anticommutation relations.…”