Symmetry information beneath wave mechanics is re-examined. Homogeneity of space is the symmetry, fundamental to the quantum free particle. The unitary information of the Canonical Commutation Relation is shown not to be implied by that symmetry. Keywords:quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space unitary, non-unitary.
Symmetry information beneath wave mechanics is re-examined. Homogeneity of space is the symmetry, fundamental to the quantum free particle. The unitary information of the Canonical Commutation Relation is shown not to be implied by that symmetry. Keywords:quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space unitary, non-unitary.
Abstract
Abstract Textbook theory says that the Canonical Commutation Relation derives from the homogeneity of space. This paper shows that the Canonical Commutation Relation does not derive from homogeneity of space or the homogeneity symmetry itself, but derives from a duality viewpoint of homogeneity, seen both from the viewpoint of position space, and from the viewpoint of momentum space, combined. Additionally, a specific particular fixed scale factor, relating position space with momentum space is necessary. It is this additional scaling information which enables complementarity between the system variables and makes the system unitary. Without this particular scaling, the Canonical Commutation Relation is left non-unitary and broken. Indeed, unitarity is separate information, unconnected and logically independent of the quantum system's underlying symmetry. This single counter-example contradicts the current consensus that foundational symmetries, underlying quantum systems, are ontologically, intrinsically and unavoidably unitary. And thus removes ‘unitary ontology’, as reason, for axiomatically imposing unitarity (or self-adjointness) — by Postulate — on quantum mechanical systems.
Keywords
foundations of quantum theory, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary.
The homogeneity symmetry is re-examined and shown to be non-unitary. This is motivated by the prospect that logical independence in elementary algebra, entering quantum mathematics, will constitute the basis for a theory explaining quantum randomness. Keywords: foundations of quantum theory, quantum physics, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary, non-unitary, unitarity, mathematical logic, formal system, elementary algebra, information, axioms, mathematical propositions, logical independence, quantum indeterminacy, quantum randomness.
Abstract
The homogeneity symmetry is re-examined and shown to be non-unitary, with no requirement for the imaginary unit. This removes symmetry, as reason, for imposing unitarity (or self-adjointness) -- by Postulate. The work here is part of a project researching logical independence in quantum mathematics, for the purpose of advancing a full and complete theory of quantum randomness.
Keywords
foundations of quantum theory, quantum physics, quantum mechanics, wave mechanics, Canonical Commutation Relation, symmetry, homogeneity of space, unitary, non-unitary, unitarity, mathematical logic, formal system, elementary algebra, information, axioms, mathematical propositions, logical independence, quantum indeterminacy, quantum randomness.
We propose six principles as the fundamental principles of quantum mechanics: principle of space and time, Galilean principle of relativity, Hamilton's principle, wave principle, probability principle, and principle of indestructibility and increatiblity of particles. We deductively develop the formalism of quantum mechanics on the basis of them: we determine the form of the Lagrangian that satisfies requirements of these principles, and obtain the Schroedinger equation from the Lagrangian. We also derive the canonical commutation relations. Then we adopt the following four guide lines. First, we do not premise the relations between dynamical variables in classical mechanics. Second, we define energy, momentum, and angular momentum as the constants of motion that are derived from homogeneity and isotropy in space and time on the basis of principle of space and time. Since energy and momentum are quantitatively defined in classical mechanics, we define them in quantum mechanics so that the corresponding conservation laws are satisfied in a coupling system of a quantum particle and a classical particle. Third, we define Planck's constant and the mass of a particle as proportionality constants between energy and frequency due to one of Einstein-de Broglie formulas and between momentum and velocity, respectively. We shall obtain the canonical commutation relations and the Schroedinger equation for a particle in an external field in the definitive form. We shall also prove that relations between dynamical variables in quantum mechanics have the same forms for those in classical mechanics.
The free Schrodinger equation is shown to be a consequence of space-time homogeneity in the non-relativistic domain. First, we show that space-time translation invariance of natural laws, which is a conseqiuence of space-time homogeneity, entails that the state of a free particle with definite momentum and energy assumes the plane wave form when the time evolution of the state is linear. Next, we show that the conservation of energy and momentum, which is a consequence of space-time translation invariance, may further determine the energy-momentum relation In the non-relativistic domain. These results then lead to the free Schrodinger equation naturally. The new integrated analysis may help understand the origin of the wave equations in quantum theory.
The physical world is quantum. However, our description of the quantum physics still relies much on concepts in classical physics and in some cases with 'quantized' interpretations. The most important case example is that of spacetime. We examine the picture of the physical space as described by simple, so-called non-relativisitic, quantum mechanics instead of assuming the Newtonian model. The key perspective is that of (relativity) symmetry representation, and the idea that the physical space is to be identified as the configuration space for a free particle. Parallel to the case of the phase space, we have a model of the quantum physical space which reduces to the Newtonian classical model under the classical limit. The latter is to be obtained as a contraction limit of the relativity symmetry.