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OALib

The Significance of Particle’s Density in Quantum Field Theory

Eliahu Comay

Published Jan 1, 2021• 0 citations
Publisher

Abstract

The paper emphasizes the significance of density of an elementary massive quantum particle.In quantum field theory, a quantum function of an elementary particle takes the form of ( ) ,t ψ r .This kind of function is used for putting the inner product of the corresponding Hilbert space in the form of an appropriate integral, and the inner product of a function with itself depends on the particle's density.Density also affects the multi-particle Fock space, because this space relies on single-particle Hilbert space.This work shows a new reason where a coherent theoretical expression for the density of an elementary particle is required: A theoretical description of experiments that measure the transition of unstable states and the decay of an elementary quantum particle.This new aspect of density strengthens its meaning in quantum theories.The usefulness of this outcome is shown in its application to the decay of the muon and the electroweak's , W Z ± particles.It turns out that the Dirac theory provides a consistent description of the muon decay.In contrast, the electroweak theory fails to explain the decay of the , W Z ± particles.

DOI
10.4236/oalib.1107384

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The paper emphasizes the significance of density of an elementary massive quantum particle.In quantum field theory, a quantum function of an elementary particle takes the form of ( ) ,t ψ r .This kind of function is used for putting the inner product of the corresponding Hilbert space in the form of an appropriate integral, and the inner product of a function with itself depends on the particle's density.Density also affects the multi-particle Fock space, because this space relies on single-particle Hilbert space.This work shows a new reason where a coherent theoretical expression for the density of an elementary particle is required: A theoretical description of experiments that measure the transition of unstable states and the decay of an elementary quantum particle.This new aspect of density strengthens its meaning in quantum theories.The usefulness of this outcome is shown in its application to the decay of the muon and the electroweak's , W Z ± particles.It turns out that the Dirac theory provides a consistent description of the muon decay.In contrast, the electroweak theory fails to explain the decay of the , W Z ± particles.
Quantum mechanics of one particle In quantum mechanics the physical state of a particle is described in terms of a ket |Ψ〉. This ket belongs to a Hilbert space which is nothing but a vector space endowed with an inner product. The dimension of the Hilbert space is essentially fixed by our physical intuition; it is we who decide which kets are relevant for the description of the particle. For instance, if we want to describe how a laser works we can choose those energy eigenkets that get populated and depopulated and discard the rest. This selection of states leads to the well-known description of a laser in terms of a three-level system, four-level system, etc. A fundamental property following from the vector nature of the Hilbert space is that any linear superposition of kets is another ket in the Hilbert space. In other words we can make a linear superposition of physical states and the result is another physical state. In quantum mechanics, however, it is only the “direction” of the ket that matters, so |Ψ〉 and C |Ψ〉 represent the same physical state for all complex numbers C . This redundancy prompts us to work with normalized kets. What do we mean by that? We said before that there is an inner product in the Hilbert space.
The interpretation problems of quantum theory are considered. In the formalism of quantum theory the possible states of a system are described by a state vector. The state vector, which will be represented as |ψ> in Dirac notation, is the most general form of the quantum mechanical description. The central problem of the interpretation of quantum theory is to explain the physical significance of the |ψ>. In this paper we have shown that one of the best way to make of interpretation of wave function is to take the wave function as an operator.
The paper examines the Dirac-Pauli differences concerning the order of the primary differential equation of an elementary massive quantum particle.The analysis relies on several self-evident constraints that an acceptable quantum theory must satisfy, like conservation laws, compatibility with Maxwellian electrodynamics and the correspondence principle.The dimension of the quantum function that is used in the Lagrangian density of a given quantum theory together with the corresponding differential equations play an important role in the reasoning procedure.The paper proves that Dirac was right and that second-order quantum theories like the Klein-Gordon equation and the electroweak theory of the W ± bosons do not satisfy fundamental constraints.This outcome is inconsistent with the Standard Model of Particle Physics.