This work aims to explore the algebraic and spectral properties of a novel parametric Hermitian non-Pauli spin basis. The primary motivation for this work is to introduce an alternative to the Pauli spin basis for investigating systems where conventional quantum statistics no longer apply. This study explores the symmetries, commutation relations, Lie algebra structures, and ladder operators of the proposed spin matrices. Additionally, it provides discussions on key findings, potential applications, and concludes with some final remarks. An example for modeling the spectrum of a quantum system is provided.
In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here. In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators. Bosonic and fermionic matrices and the cyclic trace identities We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.
After a brief review of various mappings of fermion pairs to bosons, we rigorously derive a general approach. Following the methods of Marumori and Otsuka, Arima, and Iachello, our approach begins with mapping states and constructs boson representations that preserve fermion matrix elements. In several cases these representations factor into finite, Hermitian boson images times a projection or norm operator that embodies the Pauli principle. We pay particular attention to truncated boson spaces, and describe general methods for constructing Hermitian and approximately finite boson image Hamiltonians, including effective operator theory to account for excluded states. This method is akin to that of Otsuka, Arima, and Iachello introduced in connection with the Interacting Boson Model, but is more rigorous, general, and systematic.
Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that ${\mathcal{T}}^{2}=\ensuremath{-}1$ for fermionic systems. In
PT-symmetric quantum mechanics an operator has both
PT and
CPT adjoints. Fermionic operators $\ensuremath{\eta}$, which are quadratically nilpotent (${\ensuremath{\eta}}^{2}=0$), and algebras with
PT and
CPT adjoints can be constructed. These algebras obey different anticommutation relations: $\ensuremath{\eta}{\ensuremath{\eta}}^{\mathcal{PT}}+{\ensuremath{\eta}}^{\mathcal{PT}}\ensuremath{\eta}=\ensuremath{-}\mathbb{1}$, where ${\ensuremath{\eta}}^{\mathcal{PT}}$ is the
PT adjoint of $\ensuremath{\eta}$, and $\ensuremath{\eta}{\ensuremath{\eta}}^{\mathcal{CPT}}+{\ensuremath{\eta}}^{\mathcal{CPT}}\ensuremath{\eta}=\mathbb{1}$, where ${\ensuremath{\eta}}^{\mathcal{CPT}}$ is the
CPT adjoint of $\ensuremath{\eta}$. This paper presents matrix representations for the operator $\ensuremath{\eta}$ and its
PT and
CPT adjoints in two and four dimensions. A
PT-symmetric second-quantized Hamiltonian modeled on quantum electrodynamics that describes a system of interacting fermions and bosons is constructed within this framework and is solved exactly.