We examine a covariant quantization of electromagnetic fields by using an operator derived from a constant scalar that can be called extended Lorentz gauge. The quantization can avoid an inconsistency between Lorentz gauge and a commutation relation, which can eliminate the need for introduction of physical state defined by a subsidiary condition and auxiliary field in Lagrangian density in Lorentz gauge. By using this quantization and indefinite metric straightforwardly, all quantum phenomena can be provided without enigmatic and paradoxical "probability interpretation".
We examine a covariant quantization of electromagnetic fields by using an operator derived from a constant scalar that can be called extended Lorentz gauge. The quantization can avoid an inconsistency between Lorentz gauge and a commutation relation, which can eliminate the need for introduction of physical state defined by a subsidiary condition and auxiliary field in Lagrangian density in Lorentz gauge. By using this quantization and indefinite metric straightforwardly, all quantum phenomena can be provided without enigmatic and paradoxical "probability interpretation".
Quantization of electromagnetic fields is investigated in the framework of stochastic variational method (SVM). Differently from the canonical quantization, this method does not require canonical form and quantization can be performed directly from the gauge invariant Lagrangian. The gauge condition is used to choose dynamically independent variables. We verify that, in the Coulomb gauge condition, SVM result is completely equivalent to the traditional result. On the other hand, in the Lorentz gauge condition, SVM quantization can be performed without introducing the indefinite metric. The temporal and longitudinal components of the gauge filed, then, behave as c-number functionals affected by quantum fluctuation through the interaction with charged matter fields. To see further the relation between SVM and the canonical quantization, we quantize the usual gauge Lagrangian with the Fermi term and argue a stochastic process with a negative second order correlation is introduced to reproduce the indefinite metric.
The quantum theory of electromagnetic field, developed by Fermi and Dirac, revealed some difficulties in quantizing this field. Although these difficulties have been considered in numerous publications, the situation was far from clear. Currently, this problem is even more urgent, because quantum properties of electromagnetic field are proposed to use in a number of devices. In this paper we consider the quantization problems anew. Some features of scalar potential quantization are revealed. A basis system of stationary states, which is conventionally normalized, is constructed. This construction made it possible to reject an unnecessary design: the so-called indefinite metric. The requirements to the field state with respect to the Lorentz condition are formulated. A state satisfying the Lorentz condition and describing the state of the Coulomb field is constructed. It is shown that the component of the four-potential of electromagnetic field (the difference component), which describes the Coulomb field, has zero quantum-mechanical uncertainty. The other (sum) component of the four-potential of electromagnetic field has an infinite quantum dispersion or, in other words, infinite quantum-mechanical uncertainty. Apparently, specifically this circumstance served a basis for numerous statements about unobservability of these potentials. However, it is only the aforementioned sum component of the four-potential of electromagnetic field that is in fact unobservable.
The covariant quantization of the electromagnetic field in the Lorentz gauge gives rise to longitudinal and scalar photons in addition to the usual transverse photons. It is shown here that the exchange of longitudinal and scalar photons can produce entanglement between two distant atoms or harmonic oscillators. The form of the entangled states produced in this way is very different from that obtained in the Coulomb gauge, where the longitudinal and scalar photons do not exist. A generalized gauge transformation is used to show that all physically observable effects are the same in the two gauges, despite the differences in the form of the entangled states. An approach of this kind may be useful for a covariant description of the dynamics of quantum information processing.
We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive-definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive-energy photon. With invariant plane-wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The photon probability amplitude is the real part of the projection of the photon's state vector onto a basis of position eigenvectors and its square implements the Born rule. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Lorenz gauge.
In this paper, we propose an alternate interpretation of the quantum theory using objective physical reality that does not depend on the conventional probability interpretation. As typical physical phenomena for the probability interpretation, we consider the single-photon interference, single-electron interference, and EPR correlation experiments using photon polarization. For the calculation using the alternate interpretation, the minus sign derived from the covariant quantization of Maxwell's equations, which is associated with the scalar potential of time axis component of four-vector, is taken as it is as an inevitable request from the theory. We show that both conventional and alternate interpretation derive the identical calculation results for these single photon, single electron interference, and EPR correlation. These alternate calculation processes describe that there is the scalar potential in whole space-time and when there is some geometrical arrangement in the space, the scalar potential forms the oscillatory field of the potential according to the arrangement. It reveals the objective physical reality that the single-photon, single-electron interference, and EPR correlation are generated by the movement of the photons and electrons in the oscillatory field with interference. In addition, we show that the oscillatory field formation of the scalar potential depending on the geometrical arrangement causes energy fluctuation in the space, which enables removal of infinite zero-point energy and causes spontaneous symmetry breaking and Casimir effect. By recognizing the electromagnetic field as an unitary U(1) gauge field and generalizing it to a special unitary SU(2), we also show the uncharged particle, e.g. neutron, interferences are generated by the geometrical arrangement of the SU(2) gauge field. Furthermore, we discuss the origin of the scalar potential by distinguishing the space where the substance exists and the vacuum. Finally, by introducing the extended Lorentz gauge, we propose the alternate solution without physical state and subsidiary condition for the contradiction between Lorentz gauge as an operator equation and the commutation relation in the covariant canonical quantization of Maxwell's equations with the conventional Lorentz gauge. This paper contains the compilation of published author's papers \cite{Morimoto1,Morimoto2} in addition to featured discussions such as the physical reality, uncharged particle interference and alternate proposal for the contradiction between Lorentz gauge as an operator equation and commutation relation.
Classically, electromagnetic pulses are described by real fields that couple to charged matter and propagate causally. We will show here that real fields of the form used in standard classical electromagnetic theory have a quantum mechanical interpretation in which the probability density for a photon to be at x is positive definite and operators representing all of the standard physical observables exist. A covariant alternative to the ${\ensuremath{\omega}}_{k}^{1/2}$ dependence that appears in most (but not all) presentations of quantum optics and quantum field theory is presented and real Lorentz scalar one-photon advanced and retarded potentials are derived.