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arXiv:1310.1295

What do kets represent?

Casey Blood

Published Oct 4, 2013• 0 citations
arXivHTMLPDF

Abstract

It is usually assumed that a ket represents the state of an actually existing particle. But one can show there is no evidence for particles. The particle-like properties of mass, spin and charge, as well as particle-like trajectories, the photoelectric effect, and localized effects from spread-out wave functions can be explained using quantum mechanics alone. It is therefore proposed instead that kets represent particle-like solutions to a pre-representational linear partial differential equation which has Poincaré and internal symmetries. This equation underlies the completely representational character, including mass, spin, charge, internal symmetries, and symmetric and antisymmetric statistics, of current quantum mechanics.

DOI
10.48550/arxiv.1310.1295

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It is usually assumed that a ket represents the state of an actually existing particle. But one can show there is no evidence for particles. The particle-like properties of mass, spin and charge, as well as particle-like trajectories, the photoelectric effect, and localized effects from spread-out wave functions can be explained using quantum mechanics alone. It is therefore proposed instead that kets represent particle-like solutions to a pre-representational linear partial differential equation which has Poincaré and internal symmetries. This equation underlies the completely representational character, including mass, spin, charge, internal symmetries, and symmetric and antisymmetric statistics, of current quantum mechanics.
The current form of quantum mechanics is very successful and is almost certainly correct. It is remarkable, however, that the entire structure-from the mass, spin and charge labels on particlelike states to antisymmetry to broken internal symmetries to gauge transformations to the equations of motion-is built upon concepts from group representation theory. That is, the theory is constructed exactly as if it were a representational form of an underlying theory. Our proposed form for the underlying theory is that it is based on a linear equation, OF(V)=0. F is a function of some set of independent, currently unknown variables V, with O being a linear, partial differential operator in those variables. The operator is assumed to be invariant under a group of transformations of the Vs, homomorphic to the direct product of the inhomogeneous Lorentz group and the internal symmetry group. In such a theory, a state vector, denoted by a ket with group-theoretic labels, would represent a function of the independent variables. In addition to explaining the group representational structure of quantum mechanics, an underlying theory offers insight into gauge theory.
Linearity allows several versions of reality to simultaneously exist in the state vector. But it implies that there is no interaction between versions, and that there will never be perception of more than one version. It also implies, in conjunction with group representation theory, that the particle-like properties of mass, energy, momentum, spin and charge are attributes of the state vectors. These results can be used to show there is no evidence for the objective existence of particles. The properties of the wave function are sufficient to explain all the particle-like properties of matter. Representation theory is also extensively employed in the Standard Model, with gauge fields transforming as representations of the internal symmetry group. And when applied to the permutation group, it is essential for understanding symmetric and antisymmetric states. In fact all of quantum mechanics is set up exactly as if it were the representation of an underlying pre-representational theory. A linear equation structure for the underlying theory is suggested, and it is shown in outline how quantum field theory emerges as a representational form of the pre-representational theory.
There are a number of experiments and observations that appear to argue for the existence of particles, including the photoelectric and Compton effects, exposure of only one film grain by a spread-out photon wave function, and particle-like trajectories in bubble chambers. It can be shown, however, that all the particle-like phenomena can be explained by using properties of the wave functions/state vectors alone. Thus there is no evidence for particles. Wave-particle duality arises because the wave functions alone have both wave-like and particle-like properties. Further the results of the Bell-Aspect experiment and other experiments on entangled systems, which seem to imply peculiar properties for particles if they exist, are easily and naturally understood if reality consists of the state vectors alone. The linear equation-Hilbert space structure for the state vectors, by itself, can explain every mystery in quantum mechanics except the origin of the probability law.
In this article, we discard the bra-ket notation and its correlative definitions, given by Paul Dirac. The quantum states are only described by the wave functions. The fundamental concepts and definitions in quantum mechanics is simplified. The operator, wave functions and square matrix are represented in the same expression which directly corresponds to the system of equations without additional introduction of the matrix representation of operator. It can make us to convert the operator relations into the matrix relations. According to the relations between the matrices, the matrix elements will be determined. Furthermore, the first order differential equations will be given to find the solution of equations. As a result, we unified the descriptions of the matrix mechanics and the wave mechanics.
In this article, we discard the bra-ket notation and its correlative definitions, given by Paul Dirac. The quantum states are only described by the wave functions. The fundamental concepts and definitions in quantum mechanics is simplified. The operator, wave functions and square matrix are represented in the same expression which directly corresponds to the system of equations without additional introduction of the matrix representation of operator. It can make us to convert the operator relations into the matrix relations. According to the relations between the matrices, the matrix elements will be determined. Furthermore, the first order differential equations will be given to find the solution of equations. As a result, we unified the descriptions of the matrix mechanics and the wave mechanics.
Abstract Chapter 1 opens with a brief review of some basic features of quantum mechanics, including the Schrödinger equation, linear and angular momentum and the theory of the hydrogenic atom: It also includes complete orthonormal sets of eigenfunctions, the translation operator, current, spin, equation of continuity, gauge transformation, determinant & permanent multiparticle energy eigenfunctions for noninteracting particles and the Pauli exclusion principle. Attention is then focused on Dirac bra-ket notation and complete sets of commuting observables. In this connection, representations and transformation among representations are discussed in detail for the Schrödinger system state vector and the eigenstates, as well as bra-ket matrix elements of operators. Finally, Schwinger’s interpretation of ket-bra matrix operator structures (Schwinger “Measurement Symbols”) in terms of annihilation and creation of systems in eigenstates is introduced.