Quaternionic Quantum Mechanics and Quantum Fields

Abstract

In this talk I shall first make some brief remarks on quaternionic quantum mechanics, and then describe recent work with A.C. Millard in which we show that standard complex quantum field theory can arise as the statistical mechanics of an underlying noncommutative dynamics.

Dyson indices and Hilbert–Schmidt separability functions and probabilities

“…Here we allow the single pair of non-null off-diagonal entries to be quaternionic in nature [41,42] [43, sec. IV].…”

Alternative Algebraic Structures From Bi-Hamiltonian Quantum Systems

“…Therefore the quaternionic group of bi-unitary transformations is contained in the commutant G ′ of the operator G. In the case that G admits discrete spectrum, its spectral decomposition reads [10] Then, the quaternionic group of bi-unitary transformations is given by…”

“…However, if we consider standard Poisson structures and Euclidean structures in the (q, p) variables, the quadratic functions associated with Pauli matrices will define the expected product. and the dynamics of our quantum system is then described by the Schroedinger equation [10] d dt…”

“…Moreover, a systematic formulation of quantum mechanics on quaternionic Hilbert spaces exists [10] and, in the last few years, there have been some suggestions that quaternionic quantum mechanics (QQM) may be useful to classify positive maps in standard quantum mechanics. [11] Thus, we introduce the notion of compatible quaternionic quantum bi-Hamiltonian dynamical systems and discuss their Schroedinger and Heisenberg representations.…”

On anti-centro-symmetric solutions of quaternion matrix equation AXA*=B

“…If A and B are both square matrices, define A • B = (a ij b ij ) to be Hadamard product of them. Quaternion matrix equations play important roles in both theoretical studies and numerical computations of quaternion application disciplines [2,3,4,5], and have been studied by many experts [3,4,5]. For quaternion matrix equation…”