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Physical Review

Generalization of Quantum Mechanics

T. E. Phipps

Published Jun 15, 1960• 16 citations

Abstract

The possibility of generalizing quantum mechanics in such a way as to retain its predictive results, while comprehending additional solutions, is examined. It is found that this can be done through a perfected formal correspondence with Hamilton-Jacobi mechanics, by which one is led to consider generalizations of the Heisenberg postulate of the form ${p}_{k}{q}_{j}\ensuremath{-}{q}_{j}{p}_{k}=S{\ensuremath{\delta}}_{\mathrm{jk}}$, where $S$ is a quantum analog of Hamilton's principal function. The formalism is shown to be equivalent to a simple change in Hamiltonian, with transformed momentum operators satisfying conventional commutation relations, and with an additional relationship involving formal analogs of the classical "initial constants" adjoined. A particular choice of $S(=\frac{\ensuremath{\hbar}}{i})$ leads to a theory identical with wave mechanics a part from a constant (unobservable) phase factor on the wave function. The fact that $S$ may possess other, nonconstant values, demonstrated by a specific example, suggests the ability of the mechanical equations to describe a broader class of physical states than has hitherto been investigated.

DOI: 10.1103/physrev.118.1653

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The possibility of generalizing quantum mechanics in such a way as to retain its predictive results, while comprehending additional solutions, is examined. It is found that this can be done through a perfected formal correspondence with Hamilton-Jacobi mechanics, by which one is led to consider generalizations of the Heisenberg postulate of the form

{p}_{k}{q}_{j}\ensuremath{-}{q}_{j}{p}_{k}=S{\ensuremath{\delta}}_{\mathrm{jk}}

, where

S

is a quantum analog of Hamilton's principal function. The formalism is shown to be equivalent to a simple change in Hamiltonian, with transformed momentum operators satisfying conventional commutation relations, and with an additional relationship involving formal analogs of the classical "initial constants" adjoined. A particular choice of

S(=\frac{\ensuremath{\hbar}}{i})

leads to a theory identical with wave mechanics a part from a constant (unobservable) phase factor on the wave function. The fact that

S

may possess other, nonconstant values, demonstrated by a specific example, suggests the ability of the mechanical equations to describe a broader class of physical states than has hitherto been investigated.

We here consider a generalization of the Klein-Gordon scalar wave equation which involves a single arbitrary function. The quantization may be viewed as allowing

\hbar

to be a function of the momentum or wave vector rather than a constant. The generalized theory is most easily viewed in the wave vector space analog of the Lagrangian. We need no reference to spacetime. In the generalized theory the de Broglie relation between wave vector and momentum is generalized, as are the canonical commutation relations and the uncertainty principle. The generalized uncertainty principle obtained is the same as has been derived from string theory, or by a general consideration of gravitational effects during the quantum measurement process. The propagator of the scalar field is also generalized, and an illustrative example is given in which it factors into the usual propagator times a "propagator form factor."

Hamilton-Jacobi theory is a fundamental subject of classical mechanics and has also an important role in the development of quantum mechanics. Its conceptual framework results from the advantages of transformation theory and, for this reason, relates to the features of the generating function of canonical transformations connecting a specific Hamiltonian H to a new Hamiltonian K chosen to simplify Hamilton's equations of motion. Usually, the choice is between K=0 and a cyclic K depending on the new conjugate momenta only. Here, we investigate more general representations of the new Hamiltonian. Furthermore, it is pointed out that the two common alternatives of K=0 and a cyclic K should be distinguished in more detail. An attempt is made to clearly discern the two regimes for the harmonic oscillator and, not surprisingly, some correspondences to the quantum harmonic oscillator appear. Thanks to this preparatory work, generalized coordinates and momenta associated with the Hamilton's principal function S (i.e., K=0) are used to determine dependences in the Hamilton's characteristic function W (i.e., cyclic K). The procedure leads to the Schrodinger's equation where the Hamilton's characteristic function plays the role of the Schrodinger's wave function, whereas the first integral invariant of Poincare' takes the place of the reduced Planck constant. This finding advances the pedagogical value of the Hamilton-Jacobi theory in view of an introduction to Schrodinger's wave mechanics without the tools of quantum mechanics.

It is pointed out that the classical analog to a wave function is not a single orbit but a set of paths as defined by the solution of the Hamilton-Jacobi equation. A review is given of Hamilton-Jacobi theory in language appropriate to these path concepts. It is shown that there is a classical analog to a wave packet. Quantum-mechanically, constants of motion are not arbitrary but must be characteristic values of commuting operators. It is shown by examples that the possible classical states are similarly restricted. The conditions are discussed under which a classical set of paths is derivable from the solution of a Hamilton-Jacobi equation.

We propose a new way to perform path integrals in quantum mechanics by using a quantum version of Hamilton-Jacobi (HJ) theory. In classical mechanics, Hamilton-Jacobi theory is a powerful formalism, however, its utility is not explored in quantum theory beyond approximation schemes. The canonical transformation enables one to set the new Hamiltonian to constant or zero, but keeps the information about solution in Hamilton’s characteristic function. To benefit from this in quantum theory, one must work with a formulation in which classical Hamiltonian is used. This uniquely points to phase space path integral. However, the main variable in HJ formalism is energy, not time. Thus, we are led to consider the Fourier transform of the path integral, the spectral path integral Z ˜ ( E ) . The evaluation of path integrals reduces to determining the quantum Hamilton characteristic functions (which can be achieved via an asymptotic analysis) and a discrete sum over the quantum period lattice, generalizing Gutzwiller’s sum.