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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Space-Time Position Operators

Arthur A. Broyles

Published Feb 15, 1970• 22 citations
Publisher

Abstract

A formalism is presented giving the development of $\ensuremath{\psi}(\overline{x})$ in time with the interpretation that $\ensuremath{\psi}(\overline{x})$ is the probability amplitude for observing an event at a space-time point $\overline{x}$. No properties other than the four space-time coordinates are associated with an event. A Hilbert space is defined in which $\ensuremath{\psi}(\overline{x})$ is the result of a scalar product. The space-time position operators defined in this Hilbert space have no association with particle properties, such as mass. These operators cannot be defined in the Hilbert space spanned by solutions of a Schr\"odinger equation, since the operators lead out of the Hilbert subspace belonging to a given mass. It is shown that state vectors in Hilbert space that are eigenvectors of ${P}_{\ensuremath{\mu}}{P}^{\ensuremath{\mu}}$ produce position amplitudes satisfying the Klein-Gordon equation. The relation between this Hilbert space and the one introduced by Dirac is discussed. Spin is not considered.

DOI
10.1103/physrevd.1.979

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A formalism is presented giving the development of \ensuremathψ(x‾)\ensuremath{\psi}(\overline{x})\ensuremathψ(x) in time with the interpretation that \ensuremathψ(x‾)\ensuremath{\psi}(\overline{x})\ensuremathψ(x) is the probability amplitude for observing an event at a space-time point x‾\overline{x}x. No properties other than the four space-time coordinates are associated with an event. A Hilbert space is defined in which \ensuremathψ(x‾)\ensuremath{\psi}(\overline{x})\ensuremathψ(x) is the result of a scalar product. The space-time position operators defined in this Hilbert space have no association with particle properties, such as mass. These operators cannot be defined in the Hilbert space spanned by solutions of a Schr"odinger equation, since the operators lead out of the Hilbert subspace belonging to a given mass. It is shown that state vectors in Hilbert space that are eigenvectors of P\ensuremathμP\ensuremathμ{P}_{\ensuremath{\mu}}{P}^{\ensuremath{\mu}}P\ensuremathμ​P\ensuremathμ produce position amplitudes satisfying the Klein-Gordon equation. The relation between this Hilbert space and the one introduced by Dirac is discussed. Spin is not considered.
We review the postulates of quantum mechanics with respect to the representation of physical states and measurable quantities, their time evolution, and the interpretation of measurements. We first formulate the postulates in terms of wave functions and differential operators, and then reformulate them in the abstract Hilbert space of state vectors, using Dirac’s notations. Improper states subject to Dirac’s delta normalization are introduced, and the space of physical states is extended to include them. The postulates are rationalized by associating each Hermitian linear operator with a complete orthonormal system of its eigenvectors, where measurement probabilities depend on the projections of these eigenvectors on the system’s state vector. Particularly, wave functions are identified as projections of state vectors on the position operator eigenstates. State vectors representing multidimensional systems are formulated as tensor products of vectors in their subspaces. Finally, we address the general uncertainty relations in simultaneous measurements of different observables.
Abstract Relativistic quantum‐field theory provides the machinery for calculating wave functions or probability amplitudes depending upon space‐time coordinates. The currently accepted theory, however, fails to provide position operators and a means of measuring particle coordinates that are consistent with Dirac's properties of physical observables. This is because it calls for a space position probability distribution at a specified time. This paper shows, however, that space‐time event coordinate operators, together with a corresponding measurement procedure, can be found that are consistent with Dirac's requirements. This is done through a reinterpretation of the amplitudes computed by field theory and does not involve any change in that mathematical formalism. The measurement of the space‐time coordinates of an event is accomplished by detecting the absorption of a photon by a particle from each of two light pulses designed to overlap at a given point at a given time. If a final emitted photon has an energy whose sum with the final particle energy approximately equals the sum of the mean energies of the pulses, then the absorption of the two pulse photons must certainly have taken place within a distance the order of a Compton wavelength of the small space‐time region of overlapping pulses. This is clear from the fact that the high energy required to confine the pulses to very small volumes must throw a particle absorbing them far off the mass shell. Thus the absorption of the two photons throws the particle into a narrowly confined spatial wave function that must decay extremely rapidly—to within a Compton wavelength, a delta function in space‐time. This delta function is the eigenfunction of space‐time coordinate operators Xμ and is the scalar product of vectors in a Hilbert space spanned by spin–space‐time kets large enough to contain the operators of the Poincaré group. These event operators transform properly under the action of Poincaré operators but do not commute with the mass. If the Compton wavelength is not negligible compared to the accuracy desired in the coordinate measurements, individual coordinate measurements are no longer possible. Nevertheless, a large number of repeated coordinate measurements can be carried out to produce a coordinate probability distribution. This distribution can be unfolded to find a true coordinate probability distribution if the charge form factor is known from basic theory. An analysis of laboratory particle detection techniques shows that they actually determine space coordinates and energy rather than spatial coordinates at a given time. When this fact is included, the Klein–Nishina formula can be derived using the electromagnetic four‐vector potential as the photon probability amplitude wave. To clarify the meaning of the observables, a mass‐momentum measurement is described.
A field-theoretical space-time position operator can be properly defined for the Dirac field, the Klein-Gordon field and the photon field, where its temporal component can also represent an observable according to the formalism of POVMs. To achieve this, the spinor representation of the photon field and the Feshbach-Villars representation of the Klein-Gordon field are used. Such a space-time position operator plays the role of a generalized Noether charge associated with a local U(1) symmetry, and its second-quantized form shows that quantum fields possess zero-point times, which implies that a zero-point fluctuation of energy must be accompanied with a zero-point fluctuation of time, and then in agreement with the time-energy uncertainty principle. In practice, some possible physical effects associated with the presence of zero-point times, remain to be found.