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Register a new userMathematical surprises and Diracs formalism in quantum mechanics
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Authors
F. Gieres
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By a series of simple examples, we illustrate how the lack of mathematical concern can readily lead to surprising mathematical contradictions in wave mechanics. The basic mathematical notions allowing for a precise formulation of the theory are then summarized and it is shown how they lead to an elucidation and deeper understanding of the aforementioned problems. After stressing the equivalence between wave mechanics and the other formulations of quantum mechanics, i.e. matrix mechanics and Diracs abstract Hilbert space formulation, we devote the second part of our paper to the latter approach: we discuss the problems and shortcomings of this formalism as well as those of the bra and ket notation introduced by Dirac in this context. In conclusion, we indicate how all of these problems can be solved or at least avoided.
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We propose a history state formalism for a Dirac particle. By introducing a reference quantum clock system it is first shown that Diracs equation can be derived by enforcing a timeless Wheeler-DeWitt-like equation for a global state. The Hilbert space of the whole system constitutes a unitary representation of the Lorentz group with respect to a properly defined invariant product, and the proper normalization of global states directly ensures standard Diracs norm. Moreover, by introducing a second quantum clock, the previous invariant product emerges naturally from a generalized continuity equation. The invariant parameter $tau$ associated with this second clock labels history states for different particles, yielding an observable evolution in the case of an hypothetical superposition of different masses. Analytical expressions for both space-time density and electron-time entanglement are provided for two particular families of electrons states, the former including Pryce localized particles.
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A neo-classical relativistic mechanics model is presented where the spin of an electron is a natural part of its space-time path. The fourth-order equation of motion corresponds to the same Lagrangian function in proper time as in special relativity except for an additional spin energy term. The total motion can be decomposed into a sum of a local circular motion giving the spin and a global motion of the spin center, each being governed by a second-order differential equation. The local spin motion corresponds to Schrodingers zitterbewegung and is a perpetual motion; it produces magnetic and electric dipoles through the Lorentz force on the electrons point charge. The global motion is sub-luminal and described by Newtons second law in proper time, the time for a clock fixed at the spin center, where the inertia to acceleration resides. The total motion occurs at the speed of light c, consistent with the eigenvalues of Diracs velocity operators having magnitude c. A spin tensor is introduced that is the angular momentum of the electrons total motion about its spin center. The fundamental equations of motion expressed using this tensor are identical to those of the Barut-Zanghi theory; they can be expressed in an equivalent form using the same operators as in Diracs theory for the electron but applied to a state function of proper time satisfying a Dirac-Schrodinger spinor equation. This state function produces a neo-classical wave function that satisfies Diracs relativistic wave equation for the free electron when the Lorentz transformation is used to express proper time in terms of an observers space-time coordinates. In summary, the theory provides a hidden-variable model for spin that leads to Diracs relativistic wave equation and explains the electrons moment coupling to an electro-magnetic field, albeit with a magnetic moment that is one half of that in Diracs theory.
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