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Field-theoretical space-time position operator

Zhiyong Wang, Cai-Dong Xiong

Published Dec 12, 2006• 0 citations
OpenAlex

Abstract

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.

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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 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 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 $\ensuremath{\psi}(\overline{x})$ 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.
Generalized parity [Formula: see text], time-reversal [Formula: see text], and charge-conjugation [Formula: see text] operators were initially defined in the study of the pseudo-Hermitian Hamiltonians. We construct a concrete realization of these operators for Klein–Gordon fields and show that in this realization [Formula: see text] and [Formula: see text] operators respectively correspond to the ordinary time-reversal and charge-grading operations. Furthermore, we present a complete description of the quantum mechanics of Klein–Gordon fields that is based on the construction of a Hilbert space with a relativistically invariant, positive-definite, and conserved inner product. In particular we offer a natural construction of a position operator and the corresponding localized and coherent states. The restriction of this position operator to the positive-frequency fields coincides with the Newton–Wigner operator. Our approach does not rely on the conventional restriction to positive-frequency fields. Yet it provides a consistent quantum mechanical description of Klein–Gordon fields with a genuine probabilistic interpretation.
Time-symmetric relativity theory (TSR) based on Zisman-Stueckelberg-Feynman (ZSF) interpretation includes reference frames going backward in time, maximally expanding their symmetry group to general Lorentz group O(1,3) with 4-inversion (Article 1). Similar time-symmetric (TS) extensions of relativistic quantum mechanics (TS RQM) (Article 2) and QFT (TS QFT) (this article) avoid problems of former RQM and QFT. Former QFT introduced unfounded hypotheses about physical quantum fields in entire space for each type of particle, and these hypotheses led to insoluble problems. TS QFT is limited to known external interaction fields, and describes particles using theory of many-particle systems. In second quantization method, single-particle wave functions and operators, which increase or decrease number of particles in system, form operator amplitudes of transition probabilities. These amplitudes are complex, act in their state spaces for given energy sign, and lead to positive state norm. As in quantum mechanics, they, together with conjugate amplitude, form canonical pair. Canonical formalism in relativity theory is consistent at using Hamilton-Jacobi (HJ) formalism, where 3-momentum and Hamiltonian enter into HJ equation in the same degree. Differential form of relativistic HJ equation is Klein-Gordon (KG) equation, expressing the relationship between energy and momentum, and is general equation for physical components of all operator amplitudes. For bosons, KG equation suffices; for fermions, linearization of HJ equation is required, leading to special first-order equations, such as Dirac equation. Operators of observables are normally ordered, and there are no zero-point fluctuations in ground states. Probability current is determined by 4-momentum and has the same sign; interaction currents are also proportional to probability current and change sign with 4-momentum, which corrects the currents in fermion theory. Introducing interactions into HJ equation leads to new contact terms. Time-symmetric orderings of operator amplitudes lead to causal propagators and usual diagram technique. Taking into account external gravitational field of quanta at Planck length, gravitational radius of the system, and dilation of their proper times in this field lead to gravitational self-regularization of loop diagrams. This makes perturbation theory series in Standard Model (SM) and quantum gravity finite and convergent. Contributions of chiral anomalies are finite and cancel out in SM.
A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.
We construct a concrete realization of the generalized parity (P), time-reversal (T), and charge-conjugation (C) operators, that were initially defined in the study of the PT-symmetric and pseudo-Hermitian Hamiltonians, for Klein-Gordon fields. We show that PT and C operators, that signify certain symmetries of the system, correspond to the ordinary time-reversal and charge-conjugation transformations, respectively. Furthermore, we construct a positive-definite and relativistically invariant inner product on the space of Klein-Gordon fields and show that the generator h of time-translations viewed as acting in the Hilbert space calH{cal H}calH defined by this inner product is Hermitian. The quantum system having calH{cal H}calH as its Hilbert space and h as its Hamiltonian is unitarily equivalent to the one defined by a Hermitian Hamiltonian acting in the Hilbert space L2(R3)oplusL2(R3)L^2(R^3)oplus L^2(R^3)L2(R3)oplusL2(R3). We give the explicit form of the corresponding unitary transformation and use it to construct position operators, localized and coherent states, and a set of wave functions for the Klein-Gordon fields. Our approach provides a consistent quantum mechanical description of the Klein-Gordon fields that admits a genuine probabilistic interpretation.
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.