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Progress of Theoretical Physics

On the Quantization of Physical Space-time Operators

Ichie Watanabe

Published Sep 1, 1960• 1 citation
Publisher

Abstract

In order to find a clue to extending the conventional field theory in a consistent way, a physical interpretation of Dirac-Bergmann's field theory is proposed from the point of view of the uncertainty principle. The physical space-time operators introduced by them are connected with the measurement of the position. Further a new formulation of the auxiliary condition, which is the direct consequence of the quantization of the physical space-time operators, is constructed, and then the consistency and the detailed structure of the formalism are made clear.

DOI
10.1143/ptp.24.465

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In order to find a clue to extending the conventional field theory in a consistent way, a physical interpretation of Dirac-Bergmann's field theory is proposed from the point of view of the uncertainty principle. The physical space-time operators introduced by them are connected with the measurement of the position. Further a new formulation of the auxiliary condition, which is the direct consequence of the quantization of the physical space-time operators, is constructed, and then the consistency and the detailed structure of the formalism are made clear.
When we quantize a system consisting of a single particle, the proper time τττ and the rest mass mmm are usually dealt with as parameters. In the present article, however, we introduce a new quantization rule by which these quantities are regarded as operators in addition to the position and the momentum. Applying this new rule to a scalar particle and to a particle with spin $ 1/2 $, we analyze the time evolution of the operator τττ. In the former case, the evolution of the proper time perfectly matches several well-established classical formulae. In the case of the particle with spin 1/2, our new rule implies that an oscillation appears in the time evolution of the operator τττ. This oscillation is similar to Zitterbewegung which is well-known in the ordinary Dirac theory. We formulate one physical effect of this oscillation by considering the interaction with a gravitational field, and estimate how small it is.
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.
A method for quantizing a classical system, based upon the notion of localization on phase space, is developed. This method, to which is given the name prime quantization, is based upon the mathematical theory of positive-operator-valued measures and their relationship to reproducing kernel Hilbert spaces, and to a notion of generalized kinematical variables related to phase space. For a system with openRn{\mathrm{openR}}^{n}openRn as its configuration space, it has the advantage of being able to connect the well-known problem of ordering in quantum mechanics to the choice of a measuring apparatus in a joint measurement of position and momentum (within the limits of the uncertainty principle). Moreover, for such a system, a choice of ordering in the present context also turns out to be a choice of polarization in the method of geometric quantization.
By taking into account the quantum aspects intrinsic to any operational definition of spatio-temporal relationships, a stochastic concept of spacetime emerges. In relation to its classical counterpart is realized as a stochastic mean around which quantum fluctuations become negligible only in the limit of macroscopic spacetime intervals. The test-particle propagators used in the proposed quantum concept of spacetime are derived by solving in a consistent manner the localizability problem for relativistic particles. This is achieved in the framework of the stochastic phase space formulation of quantum mechanics, which in the nonrelativistic context is shown to result from systems of imprimitivity related to phase space conserved probability currents derivable from bona fide convariant probability densities in stochastic phase spaces of one particle systems, which can be interpreted as due to measurements performed with extended rather than pointlike test particles. The associated particle propagators can be therefore consistently related to coordinate probability densities measurable by the exchange of photons in between test particles from a chosen standard. Quantum spacetime is defined as the family of propagators corresponding to all conceivable coherent flows of test particles. This family of free-fall propagators has to satisfy certain self-consistency conditions as well as consistent laws ofmore » motion which inplicitly determine the stochastic geometro-dynamics of quantum space-time. Field theory on quantum spacetime retains many of the formal features of conventional quantum field theory. On a fundamental epistemological level stochastic geometries emerge as essential prerequisites in the construction of spacetime models that would be operationally based and yet consistent with the relativity principle as well as with the uncertinty principle.« less
The question of time in quantum mechanics has been a topic of intense discussion since its inception.Dating back to the works of the founders of the subject, the notion of time and how it fit in the theory was already a point with a lack of a formal explanation, especially in aspects such as the uncertainty principle and canonical commutation relations.A theorem by Pauli set in stone that any attempt at trying to understand time with the formalism in use at the time for quantum mechanics was impossible.Through studies on the logical basis of the subject and establishing different sets of axioms, more modern works were able to relax sufficiently the impositions on the concept of observables so that time could then be understood as a 'generalized' observable.Not only that, but it was shown that the correct way to interpret eventual time measurements was through the formalism of Positive Operator Valued Measures.Latest works then were focused in the direction of constructing either direct measures and time operators, or, in the case that was followed throughout this work, to establish a procedure to construct measures.This work was then an in-depth study of the work done by Brunetti and Fredenhagen, with the intent to do the first steps in generalizing their ideas by establishing their method to be covariant with respect to Poincaré transformations.
It is well known that nonrelativistic quantum mechanics presents a clear asymmetry between space and time. Much of this asymmetry is attributed to the lack of Lorentz invariance of the theory. Nonetheless, a recent work [Phys. Rev. A \textbf{95}, 032133 (2017)] showed that even though this is partially true, there is a broader physical scenario in which space and time can be handled in nonrelativistic quantum theory in a more symmetric way. In this space-time-symmetric formalism, an additional Hilbert space is defined so that time is raised to the status of operator and position becomes a parameter. As a consequence, the Hilbert space now requires a space-conditional quantum state governed by a new quantum dynamics. In this manuscript, we reveal some physical and mathematical properties of the space-time-symmetric formalism such as: symmetries between the Hamilton-Jacobi and the space-conditional equation; the general solution for a time-independent potential; and a new Lagrangian for a spinless particle in one dimensional. Finally, we present the space-conditional equation for a particle under the effect of an electromagnetic field, and the gauge invariance of this equation is proved.