A dynamical system is called localizable if its wave functions can be expressed in terms of variables, each referring to physical conditions at only one point in space-time. These variables may be at points on any three-dimensional space-like surface in space-time.A general investigation is made of how the wave function varies when the surface is varied in any way. The variation of the wave function is given by equations of the Schr\"odinger type involving certain operators
Hn(u) which play the role of Hamiltonians. The commutation relations for these operators are obtained (Eqs. (50), (51)). The theory works entirely with relativistic concepts and it provides the general pattern which any relativistic quantum theory must conform to, provided the dynamical system is localizable.
The conventional correspondence basis for quantum dynamics is here replaced by a self-contained quantum dynamical principle from which the equations of motion and the commutation relations can be deduced. The theory is developed in terms of the model supplied by localizable fields. A short review is first presented of the general quantum-mechanical scheme of operators and eigenvectors, in which emphasis is placed on the differential characterization of representatives and transformation functions by means of infinitesimal unitary transformations. The fundamental dynamical principle is stated as a variational equation for the transformation function connecting eigenvectors associated with different spacelike surfaces, which describes the temporal development of the system. The generator of the infinitesimal transformation is the variation of the action integral operator, the spacetime volume integral of the invariant lagrange function operator. The invariance of the lagrange function preserves the form of the dynamical principle under coordinate transformations, with the exception of those transformations which include a reversal in the positive sense of time, where a separate discussion is necessary. It will be shown in Sec. III that the requirement of invariance under time reflection imposes a restriction upon the operator properties of fields, which is simply the connection between the spin and statistics of particles. For a given dynamical system, changes in the transformation function arise only from alterations of the eigenvectors associated with the two surfaces, as generated by operators constructed from field variables attached to those surfaces. This yields the operator principle of stationary action, from which the equations of motion are obtained. Commutation relations are derived from the generating operator associated with a given surface. In particular, canonical commutation relations are obtained for those field components that are not restricted by equations of constraint. The surface generating operator also leads to generalized Schr\"odinger equations for the representative of an arbitrary state. Action integral variations which correspond to changing the dynamical system are discussed briefly. A method for constructing the transformation function is described, in a form appropriate to an integral spin field, which involves solving Hamilton-Jacobi equations for ordered operators. In Sec. III, the exceptional nature of time reflection is indicated by the remark that the charge and the energy-momentum vector behave as a pseudoscalar and pseudovector, respectively, for time reflection transformations. This shows, incidentally, that positive and negative charge must occur symmetrically in a completely covariant theory. The contrast between the pseudo energy-momentum vector and the proper displacement vector then indicates that time reflection cannot be described within the unitary transformation framework. This appears most fundamentally in the basic dynamical principle. It is important to recognize here that the contributions to the lagrange function of half-integral spin fields behave like pseudoscalars with respect to time reflection. The non-unitary transformation required to represent time reflection is found to be the replacement of a state vector by its dual, or complex conjugate vector, together with the transposition of all operators. The fundamental dynamical principle is then invariant under time reflection if inverting the order of all operators in the lagrange function leaves an integral spin contribution unaltered, and reverses the sign of a half-integral spin contribution. This implies the essential commutativity, or anti-commutativity, of integral and half-integral field components, respectively, which is the connection between spin and statistics.
Abstract The quantization of the dynamics of point systems is closely connected with the Hamilton-Jacobi theory of the calculus of variations for simple integrals, the latter being a suitable mathematical formalism for describing the classical laws of point dynamics. This connexion is evident from the fact that, in order to quantize a dynamical system, one has first to know what the pairs of canonically conjugate variables are, and also from the fact that the quantum laws, if expressed in terms of commutation brackets, have exactly the same form as the classical laws, if expressed in terms of Poisson brackets. In dealing with the quantization of the dynamics of continuous media, e.g. the electromagnetic and other fields, which has been developed along different lines, following Heisenberg and Pauli (1929), it seems tempting to try to base the method of quantization on an extended Hamilton-Jacobi theory of the calculus of variations for multiple integrals, the latter being the appropriate formalism for describing most relevant systems of continuum physics. In a previous paper (Weiss 1936), referred to as I, such an attem pt has been made, by extending the notion of pairs of conjugate variables. That attem pt led to quantum relations on an arbitrary space-like section in space-time, provided that one postulated that, if the space-like section becomes especially a space section, the quantum relations should go over into those of Heisenberg and Pauli.
Abstract The purpose of this article is to construct an explicit relation between the field operators in Quantum Field Theory and the relevant operators in Quantum Mechanics for a system of N identical particles, which are the symmetrised functions of the canonical operators of position and momentum, thus providing a clear relation between Quantum Field Theory and Quantum Mechanics. This is achieved in the context of the non-interacting Klein–Gordon field. Though this procedure may not be extendible to interacting field theories, since it relies crucially on particle number conservation, we find it nevertheless important that such an explicit relation can be found at least for free fields. It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations. The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton–Wigner position operator. We make some clarifications on the interpretation of Newton–Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations. From a complex Klein–Gordon field, position and momentum operators can be constructed for both particles and antiparticles.
I: Stochastic Quantum Mechanics.- 1 / Nonrelativistic Stochastic Quantum Mechanics.- 1.1 Stochastic Values and Stochastic Phase Spaces.- 1.2. Sharp Localizability and Systems of Imprimitivity.- 1.3. Stochastic Localizability and Systems of Co variance.- 1.4. Phase-Space Representations of the Galilei Group.- 1.5. Localizability in Nonrelativistic Stochastic Phase Spaces.- 1.6. Stochastic Phase-Space Systems of Covariance.- 1.7. Resolution Generators and Gauge Freedom in Stochastic Phase Spaces.- 1.8. Stochastic Probability Currents and Their Sharp-Point Limits.- 1.9. Propagators in Nonrelativistic Stochastic Phase Spaces.- 1.10. Quantum Stochastic Processes and Phase-Space Path Integrals.- 1.11. Semiclassical Propagators in Stochastic Phase Spaces.- 1.12. The Physical Nature and the Measurement of Proper Wave Functions and of Stochastic Position and Momentum.- Notes.- 2 / Relativistic Stochastic Quantum Mechanics.- 2.1. The Nonexistence of Relativistic Probability Currents at Sharp Space time Points.- 2.2. The Unfeasibility of Sharp Localization in Relativistic Quantum Mechanics.- 2.3. Phase-Space Representations of the Poincare Group.- 2.4. Resolution Generators and Relativistic Phase-Space Representations.- 2.5. Localizability in Relativistic Stochastic Phase Spaces.- 2.6. Systems of Covariance in Relativistic Stochastic Phase Spaces.- 2.7. Relativistic Canonical Commutation Relations and Gauge Freedom.- 2.8. Relativistically Covariant and Conserved Probability and Charge Currents.- 2.9. Propagators in Relativistic Stochastic Phase Spaces.- 2.10. Relativistic Extended Quantum Particles in External Electromagnetic Fields.- Notes.- 3 / Statistical Mechanics on Stochastic Phase Spaces.- 3.1. Phase-Space Representations of Quantum Statistical Mechanics.- 3.2. Informational Equivalence and Completeness of POV Measures.- 3.3. Generalized Stochastic Phase-Space Systems of Covariance.- 3.4. Classical and Quantum Mechanics in L2 (?N).- 3.5. Classical and Quantum Statistical Mechanics in Master Liouville Space ?(?N).- 3.6. Quantum Observables as Density Distributions on Stochastic Phase Spaces.- 3.7. Unified Formulations of Classical and Quantum Boltzmann Equations and Scattering Cross-Sections.- Notes.- II: Quantum Spacetime.- 4 / Reciprocity Theory and the Geometrization of Stochastic Quantum Mechanics.- 4.1. The Operational Foundations of Special and General Relativity.- 4.2. Coordinate Fluctuation Amplitudes for Coherent Flows of Quantum Test Particles.- 4.3. Quantum Lorentz Frames and Minkowski Quantum Spacetime.- 4.4. General Quantum Spacetimes and Stochastic Geometries.- 4.5. Reciprocity Theory and Born's Quantum Metric Operator.- 4.6. Unitary Internal Symmetries and Generalized Representations of Relativistic Canonical Commutation Relations.- Notes.- 5 / Field Theory on Quantum Spacetime.- 5.1. Free Scalar Fields on Minkowski Quantum Spacetime.- 5.2. LSZ Formalism and Causality on Quantum Spacetime.- 5.3. Lagrangian Field Theory on Quantum Spacetime.- 5.4. Dirac Bispinors and Free Fields on Quantum Spacetime.- 5.5. Photon Localizability and the Electromagnetic Field on Quantum Spacetime.- 5.6. Reciprocally Invariant Quantum Electrodynamics for Extended Fermions.- Notes.- Appendix A / Elements of Measure Theory.- Appendix B / Elements of Operator Theory on Hilbert Spaces.- Notes.- References.
The study of the change of the Schroedinger wave functionals on space-like surfaces as the surface changes is formulated simply. In this formulation, we let
S be any space-like surface with the curvilinear coordinates
u, introduce
〈q(u)S∣〉 as the Schroedinger functional on it, with the nature of the functions
q(u) not specified for the moment, and write the Schroedinger equation as $\ensuremath{\hbar}i{〈q(u),{S}^{\ensuremath{'}}|〉\ensuremath{-}〈q(u),S|〉}=J〈q(u),S|〉$ where ${S}^{\ensuremath{'}}$ is a surface slightly different from
S, and
J is an operator operating on
q(u) and depending on
S, ${S}^{\ensuremath{'}}$, and the choice of their coordinates
u. Obviously, the only necessary conditions for
J is that
J is Hermitian and that (1) is integrable. To get a theory resembling the existing ones, we require further that if we construct the expectation value of a certain field quantity at a point
P from
〈q(u),S∣〉,
S being a surface passing through
P, the expectation value is independent of choice of
S and its coordinates
u, and satisfies certain differential equations.When these differential equations follow from a Langrangian principle, an expression for
J was effectively given by Weiss. His work is given here in a complete form and the proof of the satisfaction of all requirements completed.Finally, by a simple transformation on the wave functions and the observables, we deduce from the Weiss's formalism the Tomonaga's formalism. As a consequence, it is pointed out that the Tomonaga's formalism will encounter essentially the same difficulties as the orthodox Heisenberg-Pauli formalism.