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 ${H}^{n}(u)$ 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.
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.
Expressions for the operators in Dirac's general theory of quantum mechanics of localizable dynamical systems are explicitly constructed and their commutation laws directly worked out (\textsection{}2). Conditions for the expectation values of dynamical variables or their densities to be independent of the parametrization of the surface are given and it is shown that of the two conjugate field variables in Weiss' theory, one is a parametrization-independent variable and the other is a parametrization-independent density (\textsection{}3).Finally it is shown that Pn in Dirac's theory does not have any simple expression other than that given in Weiss' theory. This, together with the results described in the above paragraph, shows that for all practical purposes the two theories are equivalent (\textsection{}5).
Abstract The new mechanics of the atom introduced by Heisenberg may be based on the assumption that the variables that describe a dynamical system do not obey the commutative law of multiplication, but satisfy instead certain quantum conditions. One can build up a theory without knowing anything about the dynamical variables except the algebraic laws that they are subject to, and can show that they may be represented by matrices whenever a set of uniformising variables for the dynamical system exists. It may be shown, however (see 3), that there is no set of uniformising variables for a system containing more than one electron, so that the theory cannot progress very far on these lines. A new development of the theory has recently been given by Schrödinger. Starting from the idea that an atomic system cannot be represented by a trajectory, i. e., by a point moving through the co-ordinate space, but must be represented by a wave in this space, Schrödinger obtains from a variation principle a differential equation which the wave function ψ must satisty. This differential equation turns out to be very closely connected with the Hamiltonian equation which specifies the system, namely, if H (qr, Pr - W = 0 is the Hamiltonian equation of the system, where the qr, Pr are canonical variables, then the wave equation for ψ is {H(qr, ih ∂/∂q) - W} ψ = 0.
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.
In non-relativistic quantum mechanics of N particles in three spatial dimensions, the wave function ψ ( q 1 , ..., q N , t ) is a function of 3 N position coordinates and one time coordinate. It is an obvious idea that in a relativistic setting, such functions should be replaced by ϕ (( t 1 , q 1 ), ..., ( t N , q N )), a function of N space-time points called a multi-time wave function because it involves N time variables. Its evolution is determined by N Schrödinger equations, one for each time variable; to ensure that simultaneous solutions to these N equations exist, the N Hamiltonians need to satisfy a consistency condition. This condition is automatically satisfied for non-interacting particles, but it is not obvious how to set up consistent multi-time equations with interaction. For example, interaction potentials (such as the Coulomb potential) make the equations inconsistent, except in very special cases. However, there have been recent successes in setting up consistent multi-time equations involving interaction, in two ways: either involving zero-range ( δ potential) interaction or involving particle creation and annihilation. The latter equations provide a multi-time formulation of a quantum field theory. The wave function in these equations is a multi-time Fock function, i.e., a family of functions consisting of, for every n = 0, 1, 2, ..., an n -particle wave function with n time variables. These wave functions are related to the Tomonaga–Schwinger approach and to quantum field operators, but, as we point out, they have several advantages.
Although the concept of localization in non-relativistic quantum mechanics is mathematically well-defined there is no obvious and unambiguous way to generalize it to relativistic quantum theory. After a brief review of localization in quantum mechanics the Newton-Wigner localization scheme is introduced and it is shown by the example of a massive spinless system how it leads to a position operator with seemingly unphysical properties. This motivated the development of several theorems which claim to rule out the existence of localizable particles. A particularly important one of them is presented and its validity discussed. In the last section however it is shown that if localization is considered with respect to spacelike hyperplanes, the properties of the Newton-Wigner position operators are not necessarily unphysical and localizable particles can exist.
For a nonrelativistic quantum system of N particles, the wave function is a function of 3N spatial coordinates and one temporal coordinate. The relativistic generalization of this wave function is a function of N time variables known as the multitime wave function, and its evolution is described by N Schrödinger equations, one for each time variable. To guarantee the existence of a nontrivial common solution of these N equations, the N Hamiltonians must satisfy a compatibility condition known as the integrability condition. In this work, the integrability condition is expressed in terms of Lagrangians. The time evolution of a wave function with N time variables is derived in Feynman’s picture of quantum mechanics. However, these evolutions are compatible if and only if the N Lagrangians satisfy a certain relation called the consistency condition, which can be expressed in terms of Wilson line. As a consequence of this consistency condition, the evolution of the wave function gives rise to a key feature called the “path-independence” property on the space of time variables. This suggests that one must consider all possible paths not only on the space of dependent variables (spatial variables) but also on the space of independent variables (temporal variables). Geometrically, this consistency condition can be regarded as a zero-curvature condition and the multitime evolutions can be treated as compatible parallel transport processes on the flat space of time variables.