Department ofScience Educationand Research Institute forBasicSciences, EwhaWomansUniversity, Seoul 120-750 Korea(Dated: August 15, 2011)We construct a relativistic quantum mechanics for a boson which sustains the positive definiteparticle probability for the relativistic boson. In this theory, we settle down the well-known 85 yearsold unsolved problem on the particle probability of the Klein-Gordon quantum mechanics. Thesolutions for a boson obtained from our theory possess a negative solution as well as a positive one,similar to those of the Dirac quantum mechanics for a fermion. We next study Lorentz transforma-tion, and a vacuum state with all fermion and boson negative energy levels filled and all fermion andboson positive energy levels empty, so that we can have unification of the bosonic sea and the Diracfermionic one. In this unified sea we investigate the bosonic and fermionic antiparticles associatedwith the unified vacuum on an equal footing, and we obtain a charge conjugation operator to studydecay modes of the particles and antiparticles.I. INTRODUTION
As currently understood since its discovery, the bare Klein-Gordon theory consists of negative quantum probabilities which are considered to be physically meaningless if not outright obsolete. Despite this annoying setback, these negative probabilities are what led the great Paul Dirac in 1928 to the esoteric discovery of the Dirac Equation. The Dirac Equation led to one of the greatest advances in our understanding of the physical world. In this reading, we ask the seemingly senseless question, “Do negative probabilities exist in quantum mechanics?” In an effort to answer this question, we arrive at the conclusion that depending on the choice one makes of the quantum probability current, one will obtain negative probabilities. We thus propose a new quantum probability current of the Klein-Gordon theory. This quantum probability current leads directly to positive definite quantum probabilities. Because these negative probabilities are in the bare Klein-Gordon theory, intrinsically a result of negative energies, the fact that we here arrive at a theory with positive probabilities, means that negative energy particles are not to be considered problematic as is the case in the bare Klein-Gordon theory. From an abstract—objective stand-point; in comparison with positive energy particles, the corollary is that negative energy particles should have equal chances to exist. As to why these negative energy particles do not exist, this is analogous to asking why is it that Dirac’s antimatter does not exist in equal proportions with matter. This problem of why negative energy particles do not exist in equal proportions with positive energy particles is a problem that needs to be solved by a future theory.
We show that the concept of particles and antiparticles is related to the Weyl-Heisenberg group
WH3. Its generators are the position operators
x^a, the momentum operators
p^a, and the central element
Qv, which defines a subgroup U(1)
v of
WH3. We show that the wave functions have a charge relative to the group U(1)
v equal to either
q+=1/ℏ (particle
ψ+) or
q−=−1/ℏ (antiparticle
ψ−). We call these eigenvalues of the operator
Qv quantum charges:
Qvψ±=q±ψ±. The wave functions
ψ+ and
ψ− belong to the Hilbert spaces
H+ and
H−, on which unitarily inequivalent representations
πℏ and
π−ℏ of the Weyl-Heisenberg group
WH3 are defined. These representations are related to each other by the antilinear charge conjugation operator
C^, which maps
H+,q+ to
H−,q−. Charges
q± are intrinsic characteristics of the wave functions
ψ±, similar to spin and electric charge. The wave function possesses one of these charges,
q+ or
q−, prior to the specification of any equations. We show that negative probabilities/energies and the Klein paradox in Klein-Gordon and Dirac theories are consequences of the violation of U(1)
v-invariance in these theories. Restoring this invariance separates particle and antiparticles and eliminates all paradoxes at the level of first-quantized relativistic theories.
As currently understood since its discovery, the bare Klein-Gordon theory consists of negative quantum probabilities which are considered to be physically meaningless if not outright obsolete. Despite this annoying setback, these negative probabilities are what led the great Paul Dirac in 1928 to the esoteric discovery of the Dirac equation. The Dirac equation led to one of the greatest advances in our understanding of the physical World. In this reading, we ask the seemingly senseless question, �Do negative probabilities exist in quantum mechanics?��. In an effort to answer this question, we arrive at the conclusion that depending on the choice one makes of the quantum probability current, one will obtain negative probabilities. We thus propose a new quantum probability current of the Klein-Gordon theory. This quantum probability current leads directly to positive definite quantum probabilities. Because these negative probabilities are in the bare Klein-Gordon theory, intrinsically a result of negative energies, the fact that we-here arrive at a theory with positive probabilities, it means that negative energy particles are not to be considered problematic as is the case in the bare Klein-Gordon theory. From an abstract-objective stand-point; in comparison with positive energy particles, the corollary is that negative energy particles should have equal chances to exist. As to why these negative energy particles do not exist, this is redolent to asking why is it that Dirac�s antimatter does not exist in equal proportions with matter. This problem of why negative energy particles not exist in equal proportions with positive energy particles is a problem that needs to be solved by a future theory.
It is generally acknowledged that neither the Klein-Gordon equation nor the Dirac Hamiltonian can produce sound solitary-particle relativistic quantum mechanics due to the ill effects of their negative-energy solutions; instead their field-quantized wavefunctions are reinterpreted as dealing with particle and antiparticle simultaneously--despite the clear physical distinguishability of antiparticle from particle and the empirically known slight breaking of the underlying CP invariance. The natural square-root Hamiltonian of the free relativistic solitary particle is iterated to obtain the Klein-Gordon equation and linearized to obtain the Dirac Hamiltonian, steps that have calculational but not physical motivation, and which generate the above-mentioned problematic negative-energy solutions as extraneous artifacts. Since the natural square root Hamiltonian for the free relativistic solitary particle contrariwise produces physically unexceptionable quantum mechanics, this article focuses on extending that Hamiltonian to describe a solitary particle (of either spin 0 or spin one-half) in relativistic interaction with an external electromagnetic field. That is achieved by use of Lorentz-covariant solitary-particle four momentum techniques together with the assumption that well-known nonrelativistic dynamics applies in the particle's rest frame. Lorentz-invariant solitary particle actions, whose formal Hamiltonization is an equivalent alternative approach, are as well explicitly displayed. It is proposed that two separate solitary-particle wavefunctions, one for a particle and the other for its antiparticle, be independently quantized in lieu of "reinterpreting" negative energy solutions--which indeed don't even afflict proper solitary particles.