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Physical Science & Biophysics Journal

Proof Formalism General Quantum Density Commutator Matrix Physics

Rajan N. Iyer, Manuel Malaver

Published Aug 2, 2021• 8 citations
Publisher

Abstract

Formalism proofing general derivation, applying matrix properties operations, showing fundamental relationships with inner product to outer product has been advanced here. This general proof formalism has direct application with physics to quantify quantum density at micro scale level to time commutator at macro scale level. System of operator algebraic equations has been rigorously derived to obtain analytic solutions which are physically acceptable. Extended physics application will include metricizing towards unitarization to achieve gaging Hamiltonian mechanics to electromagnetic gravitational strong theory, towards grand unifying physics atomistic to astrophysics or vice versa via quantum relativistic general physics thereby patching to classical physics fields energy.

DOI
10.23880/psbj-16000185

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Formalism proofing general derivation, applying matrix properties operations, showing fundamental relationships with inner product to outer product has been advanced here. This general proof formalism has direct application with physics to quantify quantum density at micro scale level to time commutator at macro scale level. System of operator algebraic equations has been rigorously derived to obtain analytic solutions which are physically acceptable. Extended physics application will include metricizing towards unitarization to achieve gaging Hamiltonian mechanics to electromagnetic gravitational strong theory, towards grand unifying physics atomistic to astrophysics or vice versa via quantum relativistic general physics thereby patching to classical physics fields energy.
One of us has shown that the statistical properties of the measurements of a quantum mechanical system assume their simplest form when expressed in terms of a certain hypercomplex algebra which is commutative but not associative.1 This algebra differs from the non-commutative but associative matrix algebra usually considered in that one is concerned with the commutative expression ½(A × B + B × A) instead of the associative product A × B of two matrices. It was conjectured that the laws of this commutative algebra would form a suitable starting point for a generalization of the present quantum mechanical theory. The need of such a generalization arises from the (probably) fundamental difficulties resulting when one attempts to apply quantum mechanics to questions in relativistic and nuclear phenomena.
The objective of this series of three papers is to axiomatically derive quantum mechanics from classical mechanics and two other basic axioms. In this first paper, Schreodinger's equation for the density matrix is fist obtained and from it Schroedinger's equation for the wave functions is derived. The momentum and position operators acting upon the density matrix are defined and it is then demonstrated that they commute. Pauli's equation for the density matrix is also obtained. A statistical potential formally identical to the quantum potential of Bohm's hidden variable theory is introduced, and this quantum potential is reinterpreted through the formalism here proposed. It is shown that, for dispersion free {\it ensembles% }, Schroedinger's equation for the density matrix is equivalent to Newton's equations. A general non-ambiguous procedure for the construction of operators which act upon the density matrix is presented. It is also shown how these operators can be reduced to those which act upon the wave functions.
A bstract In this paper, we investigate the general formalism and properties of the spacetime density matrix, which captures correlations among different Cauchy surfaces and can be regarded as a natural generalization of the standard density matrix defined on a single Cauchy surface. We present the construction of the spacetime density matrix in general quantum systems and its representation via the Schwinger-Keldysh path integral. We further introduce a super-operator framework, within which the spacetime density matrix appears as a special case, and discuss possible generalizations from this perspective. We also show that the spacetime density matrix satisfies a Liouville-von Neumann type equation of motion. When considering subsystems, a reduced spacetime density matrix can be defined by tracing over complementary degrees of freedom. We study the general properties of its moments and, in particular, derive universal short-time behavior of the second moment. We find that coupling between subsystems plays a crucial role in obtaining nontrivial results. Assuming weak coupling, we develop a perturbative method to compute the moments systematically.
A new pre-quantum level physical theory underlying the quantum mechanics has been developed and its epistemological features are mainly derived from the fluctuating classical electromagnetic zeropoint field which induces random oscillations on charged particle. In complex vector space, the average random oscillations of the charged particle are treated as complex rotations so that a particle has extended structure. It is found that the quantum nature of particles arises from the superposition of internal oscillations on the center of mass motion. Using Lagrangian mechanics in complex vector space, a generalized Newton’s equation is derived and which is shown to be acceptable for both micro and macro mechanics. The generalized Newton’s equation contains additional relativistic terms at the micro level. Further, a complex Hamilton-Jacobi equation is derived and which is used to solve quantum mechanical problems in classical approach. The presence of zeropoint field also produces modifications in the central potential and the additional terms are responsible for the effects like advance of planets perihelion and deflection of light near massive objects at macro level. Further, in the case of atoms such terms lead to the estimation of Lamb shift using stochastic electrodynamics. The consideration of extended particle structure in stochastic electrodynamics allows deriving mass correction and charge correction and consequently the estimation of anomalous magnetic moment. The fundamental deeper level theory developed here, may further enhance the efforts for research connecting micro and macro aspects of matter. Keywords: Foundations of Quantum Mechanics, Geometric Algebra, Stochastic Electrodynamics, Zeropoint Energy, Gravitation. DOI : 10.7176/APTA/88-04 Publication date : June28th 2024
Unitarity is a cornerstone of quantum theory, ensuring the conservation of probability and information. Although non-Hermitian Hamiltonians are typically associated with open or dissipative systems, pseudo-Hermitian quantum mechanics shows that real spectra and unitary evolution can still emerge through a suitably defined inner product. Motivated by this insight, we extend the pseudo-Hermitian framework to relativistic quantum field theory and construct a consistent formulation of scattering processes. A novel structural feature of this theory is the presence of distinct metric operators for the in and out sectors, connected through a nontrivial metric projector that guarantees global probability conservation under pseudo-unitary time evolution. We further develop a general symmetry formalism, showing that each symmetry generally corresponds to two pseudo-unitary operators associated with the in and out metrics, respectively. Within this framework, the scattering matrix admits a perturbative expansion through the Dyson series and remains Lorentz invariant and unitary, remarkably in complete agreement with the conventional Hermitian case. The fundamental CPT theorem is also shown to hold. Our results provide a rigorous foundation for interacting pseudo-Hermitian quantum field theories and open new directions for exploring their possible physical implications beyond the standard Hermitian paradigm.