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Theory of Longitudinal Photons in Quantum Electrodynamics
Abstract
The radiation field is quantized by introducing four types of photons - two transverse, one longitudinal, and one scalar. The scalar photons are treated by using an indefinite metric, and it is found necessary to modify the usual supplementary condition slightly. The present theory offers a justification for the symmetrical treatment of the four components of the electromagnetic potential, recently applied by a number of authors, and proves to be very convenient in applications. The results of physical interest, however, are the same as obtained from the ordinary formulation.
... This guarantees the expectation that one-particle Hilbert space of the quantum theory and the classical field configurations are in one-to-one relation. This is the standard procedure introduced by Julian Schwinger in 1948 [11,12] and developed further in 1950 by Gupta [13] and Bleuler [14]. See [9] for a review of the Gupta-Bleuler method which is a well-known but old and somewhat outdated quantization method. ...
... In our case, φ i − φ 0 i and C i (or the positive frequency parts of them) do not vanish over H t but are zero over H p . To be precise, the Schwinger-Gupta-Bleuler quantization scheme [11,12,13,14], or the "right-action quantization scheme" requires 3 ...
... We close this part by the remark that it was already noted in the old literature [13,14] (see also [16,17]) that (7) are physically what one needs for quantization. However, in the Gupta-Bleuler quantization only the gauge-fixing condition was required to hold as the sandwich condition (the left equation in (7)) and in practice it was imposed and solved by the right-action requirement; the EoM for the gauge d.o.f C i = 0 was not considered. ...
Quantization of field theories with gauge symmetry is an extensively discussed and well-established topic. In this short note, we revisit this old problem. The gauge degrees of freedom have vanishing momenta, and hence their equations of motion appear as constraints on the system. We argue that to ensure consistency of quantization one can impose these constraints as "sandwich conditions": The physical Hilbert space of the theory consists of all states for which the constraints sandwiched between any two physical states vanish. We solve the sandwich constraints and show they have solutions not discussed in the gauge field theory literature. We briefly discuss the physical meaning of these solutions and implications of the "sandwich quantization scheme".
... The presence of the q µ q ν /M W 2 term in the conventional Feynman propagator is due to the fact that only 3 of the 4 components of the field of a massive vector boson are quantized in the unitary gauge of the Standard Model [20]. An alternative approach that quantizes all 4 components of the field in a covariant way using the indefinite metric [21][22][23][24][25][26] will be considered. This eliminates the divergent tensor product in the propagator and ensures that the charge-changing weak interaction is negligible for right-handed fermions. ...
... The alternative approach is based on an analogy with the covariant quantization of the electromagnetic field A µ using the indefinite metric. All four polarization components are included: two transverse, one longitudinal, and one time-like, as first suggested by Gupta [21][22][23][24][25][26]. The indefinite metric ensures that the longitudinal and time-like photons are not observable in a freely-propagating (radiative) field. ...
... The new time-like polarization mode is nonphysical. As in quantum optics [25,46] and quantum electrodynamics [21][22][23][24], this requires that the commutation relations for the creation and annihilation operators b † (0, s) and b(0, s) in the rest frame have the form ...
Experiments have shown that the charge-changing weak interaction is purely left-handed, which is taken into account in the Standard Model by the inclusion of a left-handed projection operator in the Lagrangian. Nevertheless, it will be shown here that the Standard Model predicts charge-changing weak interactions for right-handed fermions that can be larger than those for left-handed fermions if the mass is sufficiently large, as is the case for the top quark. Here we are using the conventional terminology in which a massive fermion with its spin parallel to its momentum is referred to as being right-handed in the relativistic limit, where it is in an approximate eigenstate of the chirality operator. These effects are due to the way in which the field of the W boson is quantized, which gives a divergent tensor product in the Feynman propagator in the unitary gauge. It will be shown that the off-diagonal terms in the propagator can convert a left-handed projection operator into a right-handed projection operator, which allows an interaction with right-handed fermions even though the Lagrangian is left-handed. Experiments to date have only demonstrated charge-changing weak interactions for left-handed particles, and an alternative quantization approach that eliminates the divergent off-diagonal terms in the W boson propagator and avoids these difficulties will be considered. The alternative approach appears to be in agreement with existing experiments, but additional high-energy experiments may be required in order to distinguish its predictions from those of the Standard Model.
... The Wightman framework of local, relativistic quantum field theory (QFT) turned out to be too narrow for theoretical physicists, who were interested in handling situations involving in particular gauge fields (like in quantum electrodynamics). For several reasons which are intimately connected with the needs of the standard procedure of the perturbative calculation of the scattering matrix (for a detailed discussion, see [32]), the concept of QFT with indefinite metric was introduced, where a probability interpretation is possible only on Hilbert subspaces singled out by a gauge condition in the sense of Gupta [18] and Bleuler [9]. On the other hand, "ghosts", which are quantum fields with the "wrong" connection of spin and statistics, entered the physical scene in connection with the Fadeev-Popov determinant in perturbation theory [15]. ...
... In Section 4 we construct a class of QFTs with indefinite metric and nontrivial scattering behaviour fitting into the frame of Section 3. The main ingredient of this section is a sequence of local, relativistic truncated Wightman functions called the 'structure functions', which have been introduced and studied in [1][2][3][4][5]8,17,18,25]. The non trivial scattering behaviour of the structure functions has been observed in [3,16,24]. ...
... The functional F G ∈ S ext ′ is defined by the following formulae for the Fourier transform of the componentsF G(a1,...,an) n , a l =in/loc/out, l = 1, . . . , n: (18) and F G(a1,...,an) ...
In this work, we discuss the scattering theory of local, relativistic quantum fields with indefinite metric. Since the results of Haag--Ruelle theory do not carry over to the case of indefinite metric, we propose an axiomatic framework for the construction of in- and out- states, such that the LSZ asymptotic condition can be derived from the assumptions. The central mathematical object for this construction is the collection of mixed vacuum expectation values of local, in- and out- fields, called the ``form factor functional'', which is required to fulfill a Hilbert space structure condition. Given a scattering matrix with polynomial transfer functions, we then construct interpolating, local, relativistic quantum fields with indefinite metric, which fit into the given scattering framework.
... In this case, the scalar and longitudinal modes are presented in the theory. However one can remove such modes in a Lorentzcovariant way using the Gupta-Blueler approach [33,34]. For our propose, the manifest Lorentz covariance can be dropped out choosing the Coulomb gauge. ...
... The principal result of this Section is given by the conditions given by Eq. (27) and Eqs. (31)(32)(33). These expression give us all the possible values of the surface mode frequencies for the electromagnetic field inside the dielectric cavity. ...
... where L z is the dimension of the large non-compactified dimension of the waveguide and the sum r refers to roots or zeros of the boundary condition equations given by Eq. (27) and Eqs. (31)(32)(33). For example, in order to consider a X-mode frequency it must be ensure that this frequency is a root of the functions that defines the boundary conditions, i.e., F X (n, k, ω X r ) = 0, where we must consider the function indicated in Eq. (27) or Eq. ...
We analyze the Casimir-Lifshitz effect associated with the electromagnetic field in the presence of a rectangular waveguide consisting of two distinct dielectric materials in a (3 + 1)-dimensional spacetime. We employ the surface mode technique to derive a generalized Lifshitz formula for this specific geometry. Our formulation accounts for the unique dielectric properties of the materials composing the waveguide, leading to a precise calculation of the Casimir-Lifshitz energy. In the asymptotic limit, our results recover the classical expressions for perfect reflecting boundaries. The behavior of the force on the rectangular cavity walls admit us to identify the system as a Casimir-Lifshitz anharmonic oscillator, that may vibrate due only the to vacuum stress if we allow two parallel non-fixed walls.
... The solutions are chosen to be consistent with the Lorentz invariance [59,60] for H 0 → 0 in both gauges, and the de Sitter invariance [46] for ǫ → 0 in the deceleration gauge, ...
... This gauge should not be confused with the higher derivative Eastwood-Singer gauge[54][55][56][57][58].3 This approach can be seen as a reformulation of the Gupta-Bleuler quantization procedure[59,60] in the canonical language, adapted for covariant and noncovariant gauges alike, and divorced from symmetry requirements. For mathematically oriented Gupta-Bleuler quantization in covariant gauges on globally hyperbolic spaces see e.g.[61,62]. ...
Photon propagators for power-law inflation are constructed in two one-parameter families of noncovariant gauges, in an arbitrary number of spacetime dimensions. In both gauges photon propagators take relatively simple forms expressed in terms of scalar propagators and their derivatives. These are considerably simpler compared to their general covariant gauge counterpart. This makes feasible performing dimensionally regulated loop computations involving massless vector fields in inflation.
... (B.23) 12 It is worth pointing out that, since the Lorentz gauge-fixing term has been added to equation (B.14), it is necessary to introduce the so-called Gupta-Bleuler supplementary condition [74,75] ...
... Such a condition ensures that there are no negative norm states and that the Hamiltonian for the free Maxwell field is bounded from below. While we do not examine such a condition in detail here, we refer the reader to [68,74,75] for details. ...
In the context of ghost-free infinite derivative gravity, we consider the single graviton exchange between two spinless particles, a spinless particle and a photon, or between a spinless particle and a spin-1/2 particle. To this end, we compute the gravitational potential for the three aforementioned cases and derive the O(Gℏ2) correction that arises at the linearised level. In the local theory, it is well-known that such a correction appears in the form of a Dirac delta function. Here, we show that this correction is smeared out for the nonlocal theory and, in contrast to the local theory, takes on non-zero values for a non-zero separation between the two particles. In the case of the single graviton exchange between a spinless particle and a spin-1/2 particle, we also compute the O(Gℏ) correction that arises in the non-static case within the non-relativistic approximation and show that it is finite in the nonlocal theory.
... The longitudinal photons are forbidden by the transversality conditions (11) and (12), although they can appear as virtual photons in QED [31]. Furthermore, since the spacetime fields E and B are real, the momentum space fields must satisfy E −1 ðkÞ ¼ E Ã 1 ð−kÞ and B −1 ðkÞ ¼ B Ã 1 ð−kÞ, and thus the transverse antiphotons are not independent of the transverse photons. ...
... (47) holds when v 1;AE and v 2;AE are expressed as column vectors in the ðθ;φÞ basis. From Eqs. (28)- (31), all θ derivatives of the column vectors v 1AE and v 2AE vanish, so the lhs vanishes for α ≥ 1. Since also g ðαÞ ðπ=2Þ ¼ 0 for α ≥ 1, all θ derivatives ofFðθ; ϕÞ vanish as θ → π=2 þ , so the rhs is also 0 for α ≥ 1. ...
The topology of photons in vacuum is interesting because there are no photons with k ¼ 0, creating a hole in momentum space. We show that while the set of all photons forms a trivial vector bundle γ over this momentum space, the R and L photons form topologically nontrivial subbundles γ AE with first Chern numbers ∓2. In contrast, γ has no linearly polarized subbundles, and there is no Chern number associated with linear polarizations. It is a known difficulty that the standard version of Wigner's little group method produces singular representations of the Poincaré group for massless particles. By considering representations of the Poincaré group on vector bundles we obtain a version of Wigner's little group method for massless particles which avoids these singularities. We show that any massless bundle representation of the Poincaré group can be canonically decomposed into irreducible bundle representations labeled by helicity, which in turn can be associated to smooth irreducible Hilbert space representations. This proves that the R and L photons are globally well defined as particles and that the photon wave function can be uniquely split into R and L components. This formalism offers a method of quantizing the electromagnetic field without invoking discontinuous polarization vectors as in the traditional scheme. We also demonstrate that the spin-Chern number of photons is not a purely topological quantity. Lastly, there has been an extended debate on whether photon angular momentum can be split into spin and orbital parts. Our work explains the precise issues that prevent this splitting. Photons do not admit a spin operator; instead, the angular momentum associated with photons' internal degree of freedom is described by a helicity-induced subalgebra corresponding to the translational symmetry of γ.
... The next step is to present a single qubit as two qubits in the real representation in order to exhibit the role played by entanglement. A comparison is then made with the Gupta-Bleuler method of quantizing the electromagnetic field [7,8]. Finally, we discuss the implications of our results and speculate on possible extensions of quantum physics. ...
... where A + µ is the positive frequency part of the vector potential (the one containing the annihilation field operators). This condition, known as the Gupta-Bleuer constraint [7,8], must hold at all space points and at all times. It ensures that the expectation value of ∂A µ /∂x µ , ⟨Ψ|∂A µ /∂x µ |Ψ⟩, vanishes at all points and times. ...
We show how imaginary numbers in quantum physics can be eliminated by enlarging the Hilbert space followed by an imposition of—what effectively amounts to—a superselection rule. We illustrate this procedure with a qubit and apply it to the Mach–Zehnder interferometer. The procedure is somewhat reminiscent of the constrained quantization of the electromagnetic field, where, in order to manifestly comply with relativity, one enlarges the Hilbert Space by quantizing the longitudinal and scalar modes, only to subsequently introduce a constraint to make sure that they are actually not directly observable.Quanta 2023; 12: 164–170.
... The longitudinal photons are forbidden by the transversality conditions (11) and (12), although they can appear as virtual photons in QED [31]. Furthermore, since the spacetime fields E and B are real, the momentum space fields must satisfy E −1 ðkÞ ¼ E Ã 1 ð−kÞ and B −1 ðkÞ ¼ B Ã 1 ð−kÞ, and thus the transverse antiphotons are not independent of the transverse photons. ...
... (47) holds when v 1;AE and v 2;AE are expressed as column vectors in the ðθ;φÞ basis. From Eqs. (28)- (31), all θ derivatives of the column vectors v 1AE and v 2AE vanish, so the lhs vanishes for α ≥ 1. Since also g ðαÞ ðπ=2Þ ¼ 0 for α ≥ 1, all θ derivatives ofFðθ; ϕÞ vanish as θ → π=2 þ , so the rhs is also 0 for α ≥ 1. ...
The topology of photons in vacuum is interesting because there are no photons with k = 0, creating a hole in momentum space. We show that while the set of all photons forms a trivial vector bundle γ over this momentum space, the R-and L-photons form topologically nontrivial subbundles γ ± with first Chern numbers ±2. In contrast, γ has no linearly polarized subbundles, and there is no Chern number associated with linear polarizations. It is a known difficulty that the standard version of Wigner's little group method produces singular representations of the Poincaré group for massless particles. By considering representations of the Poincaré group on vector bundles we obtain a version of Wigner's little group method for massless particles which avoids these singularities. We show that any massless bundle representation of the Poincaré group can be canonically decomposed into irreducible bundle representations labeled by helicity. This proves that the R-and L-photons are globally well-defined as particles and that the photon wave function can be uniquely split into R-and L-components. This formalism offers a method of quantizing the EM field without invoking discontinuous polarization vectors as in the traditional scheme. We also demonstrate that the spin-Chern number of photons is not a purely topological quantity. Lastly, there has been an extended debate on whether photon angular momentum can be split into spin and orbital parts. Our work explains the precise issues that prevent this splitting. Photons, as massless irreducible bundle representations of the Poincaré group, do not admit a spin operator. Instead, the angular momentum associated with photons' internal degree of freedom is described by a helicity-induced subalgebra, which is 3D and commuting, corresponding to the translational symmetry of γ.
... where I use the positive time signature for the Minkowski metric, i. e. g = diag(1, −1, −1, −1). Actually, in their original papers [7,8], the authors consider the particular case ξ = 1, nowadays known as Feynman gauge. This case is the most thoroughly studied in the literature, see for instance [9] for a rigorous treatment. ...
... At this point one usually takes ξ = 1 (Feynman gauge) that, as we said, was the original proposal of Gupta and Bleuler [7,8]. Of course, the reason for that choice is that in the Feynman gauge the equations have a basis of plane wave solutions of the form ...
We carry out the canonical quantization of the electromagnetic field in arbitrary ξ -gauge and compute its propagator. In this way we fill a gap in the literature and clarify some existing confusion about Feynman i ϵ prescription for the propagator of the electromagnetic field. We also discuss the BRST quantization and investigate the apparent singularities present in the theory when the gauge parameter ξ takes the value −1. We find that this is a mere artifact due to the choice of basic modes and show that in the appropriate basis the commutation relations and the BRST transformation are, in fact, independent of the gauge parameter. The latter only appears as the coefficient of a BRST exact term in the Hamiltonian, which constitutes an extremely simple proof of the independence of any physical process on the gauge parameter ξ .
... In order to find more methods to reconstruct signals, various generalizations of dual frame have been proposed such as oblique dual frame [4][5][6][7][8][9][10][11], approximate dual frame [12][13][14], approximate oblique dual frame [15,16], generalized dual frame [17,18], and biframe [19][20][21]. Krein space, as a generalization of classical Hilbert space, is the setting for establishing the Markov hypothesis and Heisenberg elementary particle theory and that some quantum field theories such as quantum electrodynamics, vector meson field, and the Tsung-Dao Lee model can be quantized in a self-consistent form only by introducing Krein space [22][23][24][25][26][27]. We refer to [26][27][28] for basic results in Krein spaces. ...
Due to its significance in mathematics and engineering, the operator theory of Krein spaces and Krein space approaches have been being attracted attention of many mathematicians. Recently, the concept of frame has been introduced in Krein spaces. This paper addresses the approximate oblique dual frames for Krein spaces. We present some parametric expressions and constructions of approximate oblique dual frames of a given frame sequence and prove that there exists a bijection between the approximate oblique dual frames of a given frame sequence and its portrait under a bounded invertible operator on ℓ2 and between the approximate oblique dual frames of original and perturbed frame sequences. Also, we estimate the deviation of the canonical approximate oblique dual frames of original and perturbed frame sequences and give an explicit characterization for the best approximation of the approximate oblique dual frame of original frame sequence using that of perturbed frame sequence. Finally, applying our results to shift‐invariant systems, we derive some new results in shift‐invariant spaces.
... Therefore, the noise constraint has to be recovered in a different way when working in covariant gauges. It turns out that one can make it explicit by using a quantisation condition similar to Gupta-Bleuler condition [35,36] ⟨Ψ| ∂ µÂ µ |Ψ⟩ = 0, (4.133) ...
A bstract
In many scenarios of interest, a quantum system interacts with an unknown environment, necessitating the use of open quantum system methods to capture dissipative effects and environmental noise. With the long-term goal of developing a perturbative theory for open quantum gravity, we take an important step by studying Abelian gauge theories within the Schwinger-Keldysh formalism. We begin with a pedagogical review of general results for open free theories, setting the stage for our primary focus: constructing the most general open effective field theory for electromagnetism in a medium. We assume locality in time and space, but allow for an arbitrary finite number of derivatives. Crucially, we demonstrate that the two copies of the gauge group associated with the two branches of the Schwinger-Keldysh contour are not broken but are instead deformed by dissipative effects. We provide a thorough discussion of gauge fixing, define covariant gauges, and calculate the photon propagators, proving that they yield gauge-invariant results. A notable result is the discovery that gauge invariance is accompanied by non-trivial constraints on noise fluctuations. We derive these constraints through three independent methods, highlighting their fundamental significance for the consistent formulation of open quantum gauge theories.
... Dirac first formalized and quantized the electromagnetic (EM) fields in 1927 [2,3]. In quantum electrodynamics, Gupta [4] studied the canonical quantization of EM fields under the Lorentz gauge and indefinite metric condition [5,6]. The results show that the contributions of time-like photons and longitudinal photons are opposite, cancel each other out to zero, and only two transverse photons have observable effects. ...
This paper has proposed a single-photon structure model based on both the Maxwell wave equation and canonical quantization results of free electromagnetic fields under gauge conditions. Model assumption: There are only right-spin single photon and left-spin single photon, and their spin angular frequency (ω=2πν) is their spin angular velocity. By studying the bunching properties of two single photons, it was found that homo-frequency right-spin photons and left-spin photons could be superimposed into linearly polarized (LP) 2-photon monomers. Two homo-frequency and parallel LP 2-photon monomers with a phase difference of 90° can be superimposed onto an LP4-photon monomer. This monomer has been proven to be a normal unit of light.
... Therefore, the noise constraint has to be recovered in a different way when working in covariant gauges. It turns out that one can make it explicit by using a quantisation condition similar to Gupta-Bleuler condition [28,29] ⟨Ψ| ∂ µÂ µ |Ψ⟩ = 0, (4.133) where |Ψ⟩ is a physical state. ...
In many scenarios of interest, a quantum system interacts with an unknown environment, necessitating the use of open quantum system methods to capture dissipative effects and environmental noise. With the long-term goal of developing a perturbative theory for open quantum gravity, we take an important step by studying Abelian gauge theories within the Schwinger-Keldysh formalism. We begin with a pedagogical review of general results for open free theories, setting the stage for our primary focus: constructing the most general open effective field theory for electromagnetism in a medium. We assume locality in time and space, but allow for an arbitrary finite number of derivatives. Crucially, we demonstrate that the two copies of the gauge group associated with the two branches of the Schwinger-Keldysh contour are not broken but are instead deformed by dissipative effects. We provide a thorough discussion of gauge fixing, define covariant gauges, and calculate the photon propagators, proving that they yield gauge-invariant results. A notable result is the discovery that gauge invariance is accompanied by non-trivial constraints on noise fluctuations. We derive these constraints through three independent methods, highlighting their fundamental significance for the consistent formulation of open quantum gauge theories.
... One of them is the excess number of degrees of freedom in the description of the electromagnetic fields. In this context the approach of Gupta [1] and of Bleuler [2] is well-known. An alternative is found in the work of Creutz [3]. ...
A simple transformation of field variables eliminates Coulomb forces from the theory of quantum electrodynamics. This suggests that Coulomb forces may be an emergent phenomenon rather than being fundamental. This possibility is investigated in the context of reducible quantum electrodynamics. It is shown that states exist which bind free photon and free electron fields. The binding energy peaks in the long-wavelength limit. This makes it plausible that Coulomb forces result from the interaction of the electron/positron field with long-wavelength transversely polarized photons.
... The definition of all basic free fields A i (x) and the Fock spaces on which they act is standard [Sch16,Wei95,Der14]. In the case of the vector field we use the Gupta-Bleuler approach [Gup50,Ble50] which is nicely summarized in the monographs [Sch16,Str13]. Let us only mention that in this approach the vector field is defined on the Krein space with two inner products. ...
We construct the Wightman and Green functions in a large class of models of perturbative QFT in the four-dimensional Minkowski space in the Epstein-Glaser framework. To this end we prove the existence of the weak adiabatic limit, generalizing the results due to Blanchard and Seneor. Our proof is valid under the assumption that the time-ordered products satisfy certain normalization condition. We show that this normalization condition may be imposed in all models with interaction vertices of canonical dimension 4 as well as in all models with interaction vertices of canonical dimension 3 provided each of them contains at least one massive field. Moreover, we prove that it is compatible with all the standard normalization conditions which are usually imposed on the time-ordered products. The result applies, for example, to quantum electrodynamics and non-abelian Yang-Mills theories.
... Alternatively, following Ref. [31,32] the hermitean K-matrix can be considered as an approximation to the scattering amplitude, which can be obtained order by order in perturbation theory using Eq. (2.6). ...
We propose a new approach to the LHC dark matter search analysis within the effective field theory (EFT) framework by utilising the K-matrix unitarisation formalism. This approach provides a reasonable estimate of the dark matter production cross section at high energies, and hence allows reliable bounds to be placed on the cut-off scale of relevant operators without running into the problem of perturbative unitarity violation. We exemplify this procedure for the effective operator D5 in monojet dark matter searches in the collinear approximation. We compare our bounds to those obtained using the truncation method and identify a parameter region where the unitarisation prescription leads to more stringent bounds.
... A similar symmetry reduction scheme exists for scalar field models [9], where the non-Hermitian operators analogous toË[f, g] are annihilation operators on Fock space and the conditions analogous toË[f, g]|ψ = 0 assert that all non-symmetric field modes are unexcited. (The Gupta-Bleuler quantization [43,44] of the electromagnetic field rests on essentially the same idea.) Importantly, this quantum symmetry reduction of scalar fields commutes with quantization in a precise sense [9]. ...
In this paper we work out in detail a new proposal to define rigorously a sector of loop quantum gravity at the diffeomorphism invariant level corresponding to homogeneous and isotropic cosmologies, and propose how to compare in detail the physics of this sector with that of loop quantum cosmology. The key technical steps we have completed are (a) to formulate conditions for homogeneity and isotropy in a diffeomorphism covariant way on the classical phase space of general relativity, and (b) to translate these conditions consistently using well-understood techniques to loop quantum gravity. To impose the symmetry at the quantum level, on both the connection and its conjugate momentum, the method used necessarily has similiarities to the Gupta-Bleuler method of quantizing the electromagnetic field. Lastly, a strategy for embedding states of loop quantum cosmology into this new homogeneous isotropic sector, and using this embedding to compare the physics, is presented.
... However in this paper we challenge the implications of the Gauss's law with respect to the existence of longitudinal electric waves by considering the electric dipole radiation. A theoretical method for quantizing the radiation field using transverse, longitudinal and scalar photons was introduced in [8]. An attempt to consider the existence of longitudinal electric waves using general framework of classical electrodynamics was tried in [26]. ...
In this work by using the assumptions that wavelength is much smaller than charge separation distance of an electric dipole, which in turn is much smaller than a distance up to the point of observation, the new results for radiation of an electric dipole were obtained. These results generalize and extend the standard classical solution, and they indicate that under the above assumptions the electric dipole emits both long-range longitudinal electric and transverse electromagnetic waves. For a specific values of the dipole system parameters the longitudinal and transverse electric fields are displayed. Total power emitted by electric and electromagnetic waves are calculated and compared. It was shown that under the standard assumption of charge separation distance being much smaller than wavelength: a) classical solution correctly describes the transverse electromagnetic waves only; b) longitudinal electric waves are non-negligible; c) total radiated power is proportional to the fourth degree of frequency and to the second degree of the charge separation distance; d) transverse component of our solution reduces to classical solution. In case wavelength is much smaller than charge separation distance: a) the classical solution is not valid and it overestimates the total radiated power; b) longitudinal electric waves are dominant and transverse electromagnetic waves are negligible; c) total radiated power is proportional to the third degree of frequency and to the charge separation distance; d) most of the power is emitted in a narrow beam along the dipole axis, thus emission of waves is focused as with lasers.
... This implies that the number of degrees of freedom of the electromagnetic field is 2 rather than 3 or 4. There is no need for the construction of Gupta [15] and Bleuler [16], which intends to remove the nonphysical degrees of freedom. This is an important simplification, which however raises a number of questions. ...
This paper discusses an attempt to develop a mathematically rigorous theory of Quantum Electrodynamics (QED). It deviates from the standard version of QED mainly in two aspects: it is assumed that the Coulomb forces are carried by transversely polarized photons, and a reducible representation of the canonical commutation and anti-commutation relations is used. Both interventions together should suffice to eliminate the mathematical inconsistencies of standard QED.
... Following the original idea in [60,61], we only admitted vector states |Ψ for which the expectation value of the gauge condition (18) is satisfied ...
Canonical quantization is reviewed here for the Abelian Lee-Wick model by using the Dirac constraints method, a Gupta-Bleuler-like prescription is implemented and the BRST charge operator for this model is built. New degrees of freedom associated to the (higher derivatives) Lee-Wick ghost are excluded from the (physical) space of observables by the imposition of a Piguet-Nielsen-type BRST-extended symmetry for the Lee-Wick mass term, removing the Lee-Wick mass parameter of the theory, which implies remove all the higher-derivative sector. In addition, another prescription based on a new subsidiary condition of invariance under a purely higher-derivative BRST-type new symmetry is proposed.
... (2) λ ) such that:â (2) λ |n, λ = − √ n |n − 1 and a †(2) λ |n, λ = √ n + 1 |n + 1 , and also redefine the inner product as n|O|n so that the inner product comes out positive for all states |n . Here, O is an operator that has the following properties: This way of avoiding the difficulty of negative norm was first carried out by Gupta [10] and independently by Bleuler in 1950. ...
Feynman's modification to electrodynamics and its application to the calculation of self-energy of a free spin- particle, appearing in his 1948 Physical Review paper, is shown to be applicable for the self-energy calculation of a free spin-0 particle as well. Feynman's modification to electrodynamics is shown to be equivalent to a Hamiltonian approach developed by Podolsky.
... Such a representation cannot be unitary. Nevertheless, the famous Gupta-Bleuler formalism in quantum electrodynamics was developed in such a setting [4,10]. Indecomposable representations with invariant indefinite inner products have seen considerable applications within this formalism (e.g., [7,8,9]). ...
We generalize a result of Araki (1985) on indecomposable group representations with invariant (necessarily indefinite) inner product and irreducible subrepresentation to Hopf -algebras. Moreover, we characterize invariant inner products on the projective indecomposable representations of small quantum groups at odd roots of unity and on the indecomposable representations of generalized Taft algebras .
... In this case, the scalar and longitudinal modes are presented in the theory. However, one can remove such modes in a Lorentzcovariant way using the Gupta-Blueler approach [36,37]. For our proposal, the manifest Lorentz covariance can be dropped out, of choosing the Coulomb gauge. ...
... Because of the noncovariant gauge fixing (1), and because of the reduced symmetry of cosmological spaces compared to maximally symmetric ones, we have to understand how to compute photon two-point functions without relying on a large number of symmetries that usually simplify problems. To this end we consider the canonical quantization of the photon field similar to the Gupta-Bleuler quantization [46,47]. The approach taken here mainly differs by not relying on the on symmetries of the system. ...
Canonical quantization of the photon—a free massless vector field—is considered in cosmological spacetimes in a two-parameter family of linear gauges that treat all the vector potential components on equal footing. The goal is setting up a framework for computing photon two-point functions appropriate for loop computations in realistic inflationary spacetimes. The quantization is implemented without relying on spacetime symmetries, but rather it is based on the classical canonical structure. Special attention is paid to the quantization of the canonical first-class constraint structure that is implemented as the condition on the physical states. This condition gives rise to subsidiary conditions that the photon two-point functions must satisfy. Some of the de Sitter space photon propagators from the literature are found not to satisfy these subsidiary conditions, bringing into question their consistency.
Published by the American Physical Society 2024
... While we have sacrificed manifest Lorentz covariance by our choice of Coulomb gauge, this was simply due to our interest in the cavity QED situation of the quantization in a resonator. If wanted, retaining Lorentz covariance and determining relativistically invariant analogs of the expressions, (48), for the wave functionals is possible by resorting to the Gupta-Bleuler [44,45] method or the more general approach of BRST quantization [46,47]. For a modern discussion contrasting these approaches as applied to electromagnetism in -gauge, a generalization of Lorenz gauge, we refer to Ref. [47]. ...
We show that despite the fundamentally different situations, the wave functional of the vacuum in a resonator is identical to that of free space. The infinite product of Gaussian ground state wave functions defining the wave functional of the vacuum translates into an exponential of a sum rather than an integral over the squares of mode amplitudes weighted by the mode volume and a power of the mode wave number. We express this sum by an integral of a bilinear form of the field containing a kernel given by a function of the square root of the negative Laplacian acting on a transverse delta function. For transverse fields it suffices to employ the familiar delta function which allows us to obtain explicit expressions for the kernels of the vector potential, the electric field and the magnetic induction. We show for the example of the vector potential that different mode expansions lead to different kernels. Lastly, we show that the kernels have a close relationship with the Wightman correlation functions of the fields.
... While we have sacrificed manifest Lorentz covariance by our choice of Coulomb gauge, this was simply due to our interest in the cavity QED situation of the quantization in a resonator. If wanted, retaining Lorentz covariance and determining relativistically invariant analogs of the expressions, (48), for the wave functionals is possible by resorting to the Gupta-Bleuler [44,45] method or the more general approach of BRST quantization [46,47]. For a modern discussion contrasting these approaches as applied to electromagnetism in -gauge, a generalization of Lorenz gauge, we refer to Ref. [47]. ...
We show that despite fundamentally different situations, the wave functional of the vacuum in a resonator is identical to that of free space. The infinite product of the Gaussian ground state wave functions defining the wave functional of the vacuum translates into an exponential of a sum rather than an integral over the squares of mode amplitudes weighted by the mode volume and power of the mode wave number. We express this sum by an integral of a bilinear form of the field containing a kernel given by a function of the square root of the negative Laplacian acting on a transverse delta function. For transverse fields, it suffices to employ the familiar delta function, which allows us to obtain explicit expressions for the kernels of the vector potential, the electric field, and the magnetic induction. We show for the example of the vector potential that different mode expansions lead to different kernels. Lastly, we show that the kernels have a close relationship with the Wightman correlation functions of the fields.
... which is the set of states satisfying the conventional Gupta-Bleuler condition [41,42] for free Maxwell fields. Then a class of solutions of Eq. (34) is given by 11 ...
We study the S-matrix and inclusive cross-section for general dressed states in quantum electrodynamics. We obtain an infrared factorization formula of the S-matrix elements for general dressed states. This enables us to study which dressed states lead to infrared-safe S-matrix elements. The condition for dressed states can be interpreted as the memory effect, which is nothing but the conservation law of the asymptotic symmetry. We derive the generalized soft photon theorem for general dressed states. We also compute an inclusive cross-section using general dressed states. It is necessary to use appropriate initial and final dressed states to evaluate interference effects, which cannot be computed correctly by using Fock states due to the infrared divergence.
... The perception bundle description of photon as presented in this section is similar to the Gupta-Bleuler quantization ( [3,14]) of electromagnetic fields since both include the process of quotienting out the longitudinal polarization D e . ...
Recently, a bundle theoretic description of massive single-particle state spaces, which is better suited for Relativistic Quantum Information Theory than the ordinary Hilbert space description, has been suggested. However, the mathematical framework presented in that work does not apply to massless particles. It is because, unlike massive particles, massless particles cannot assume the zero momentum state and hence the mass shell associated with massless particles has non-trivial cohomology. To overcome this difficulty, this paper suggests a new framework that can be applied to massless particles. Applications to the cases of massless particles with spin-1 and 2, namely photon and graviton, will reveal that the field equations, the gauge conditions, and the gauge freedoms of Electromagnetism and General Relativity naturally arise as manifestations of an inertial observer's perception of the internal quantum states of a photon and a graviton, respectively. Finally, we show that gauge freedom is exhibited by all massless particles, except those with spin-0 and 1/2.
... At first sight this situation appears rather academic or mind-boggling, since it shows by inspection of Eq. (129) that a physical system can contain two "spurious" fields whose contributions to scattering processes seem to cancel exactly. Nonetheless could it be very fruitfully used to explain the also mind-boggling contradiction that the Lorentz covariant vector field A µ of the photon possessing four-components A 0 , A 1 , A 2 and A 3 appears in Nature only with two transverse components A 1 and A 2 while the components A 0 and A 3 seem to be -in the spirit of S. N. Gupta [44] and K. Bleuler [45] -invisible or "spurious" in the above sense. ...
A system of two independent Bosonic Harmonic Oscillators is converted into the respective fourth-order derivative Pais-Uhlenbeck oscillator model. The conversion procedure displays transparently how the quantization of the fourth-order derivative Pais-Uhlenbeck oscillator has to be performed in order not to suffer from the divergence problems of the vacuum state and path integrals as conjectured most recently by P. D. Mannheim in his article ``Determining the normalization of the quantum field theory vacuum, with implications for quantum gravity" [arXiv:2301.13029 [hep-th]]. In order to make the case we present the construction of the path integral, generating functionals and vacuum persistence amplitudes for PT-symmetry completed systems in Quantum Mechanics and Quantum Field Theory and discuss some implications to Quantum Field Theory and Particle Physics.
... Because of the non-covariant gauge-fixing (1.1), and because of the reduced symmetry of cosmological spaces compared to maximally symmetric ones, we have to be able to compute photon two-point functions without relying on a large number of symmetries that usually simplify problems. To this end we consider the canonical quantization of the photon field similar to the Gupta-Bleuler quantization [34,35]. The main difference is that the approach utilized here does not rely on symmetries of the system, but rather on the classical canonical structure in the multiplier gauge defined by (1.1). ...
Canonical quantization of the photon -- a free massless vector field -- is considered in cosmological spacetimes in a two-parameter family of linear non-covariant gauges that treat all the vector potential components on equal footing. The goal is to set up a framework for computing convenient photon two-point functions appropriate for loop computations in realistic inflationary spacetimes. The quantization is implemented without relying on spacetime symmetries, but rather it is based on the classical canonical structure. Special attention is paid to the quantization of the canonical first-class constraint structure, that is implemented via the subsidiary condition on the physical states. This subsidiary condition gives rise to subsidiary conditions on the photon two-point functions that serve as convenient consistency conditions. Some of the de Sitter space photon propagators from the literature are found not to satisfy these subsidiary conditions, bringing into question their consistency.
... The QED Lagrangian is defined to contain a nonzero photon mass μ which regulates infrared divergences [39,40], as well as a gauge-fixing term that makes the quantized theory well defined. I use the Gupta-Bleuler formalism [41,42] in particular for the gauge-fixing terms (although the Nakanishi-Lautrup formalism [43][44][45][46] exists as an alternative). ...
Noether’s first and second theorems both imply conserved currents that can be identified as an energy-momentum tensor (EMT). The first theorem identifies the EMT as the conserved current associated with global spacetime translations, while the second theorem identifies it as a conserved current associated with local spacetime translations. This work obtains an EMT for quantum electrodynamics and quantum chromodynamics through the second theorem, which is automatically symmetric in its indices and invariant under the expected symmetries [e.g., Becchi-Rouet-Stora-Tyutin (BRST) invariance] without the need for introducing an ad hoc improvement procedure.
Periodically driven systems can host many interesting and intriguing phenomena. The irradiated two-dimensional Dirac systems, driven by circularly polarized light, are the most attractive thanks to intuitive physical view of the absorption and emission of photon near Dirac cones. Here, we assume that the light is incident in the two-dimensional plane, and choose to treat the light-driven Dirac systems by making a unitary transformation to capture the photon-mediated electronic correlation effects, instead of using usual Floquet theory. In this approach, the electron-photon interaction terms can be cancelled out and the resultant effective electron-electron interactions can produce important effects. These effective interactions will produce a topological band structure in the case of 2D Fermion system with one Dirac cone, and can lift the energy degeneracy of the Dirac cones for graphene. This method can be applicable to similar light-driven Dirac systems to investigate photon-mediated electronic effects in them.
This paper addresses the dilation problem on (dual) frames for Krein spaces. We characterize Riesz bases for Krein spaces and equivalence ([Formula: see text]-unitary equivalence) between frames for Krein spaces; prove that every frame (dual frame pair) for a Krein space can be dilated to a Riesz basis (dual Riesz basis pair) for a larger Krein space, and that the corresponding [Formula: see text]-orthogonal complementary frame ([Formula: see text]-joint complementary frame) is unique up to equivalence ([Formula: see text]-joint equivalence). Also we illustrate that two equivalent Parseval frames for Krein spaces need not be [Formula: see text]-unitarily equivalent and that not every Parseval frame can be dilated to a [Formula: see text]-orthonormal basis for a larger Krein space, and derive a result on matrices of finite size as application.
We study the Maxwell field with a general gauge fixing (GF) term in the radiation-dominant (RD) and matter-dominant (MD) stages of expanding Universe, as a continuation to the previous work in the de Sitter space. We derive the exact solutions, perform the covariant canonical quantization and obtain the stress tensor in the Gupta–Bleuler (GB) physical states, which is independent of the GF constant and is also invariant under the quantum residual gauge transformation. The transverse stress tensor is similar in all flat Robertson–Walker spacetimes, and its vacuum part is ∝k4 and becomes zero after the 0th-order adiabatic regularization. The longitudinal-temporal stress tensor, in both the RD and MD stages, is zero due to a cancelation between the longitudinal and temporal parts in the GB states, and so is the particle part of the GF stress tensor. The vacuum GF stress tensor, in the RD stage, contains k4,k2 divergences and becomes zero by the 2nd-order regularization, however, in the MD stage, contains k4,k2,k0 divergences and becomes zero by the 4th-order regularization. So, the order of adequate regularization depends not only upon the type of fields, but also upon the background spacetimes. In summary, in both the RD and MD stages, as in the de Sitter space, the total regularized vacuum stress tensor is zero, independent of the GF constant, only the transverse photon part remains, there is no trace anomaly, and the vanishing GF stress tensor cannot be a candidate for the dark energy.
A bstract
Photon propagator for power-law inflation is considered in the general covariant gauges within the canonical quantization formalism. Photon mode functions in covariant gauges are considerably more complicated than their scalar counterparts, except for the special choice of the gauge-fixing parameter we call the simple covariant gauge. We explicitly construct the position space photon propagator in the simple covariant gauge, and find the result considerably more complicated than its scalar counterpart. This is because of the need for explicitly inverting the Laplace operator acting on the scalar propagator, which results in Appell’s fourth function. Our propagator correctly reproduces the de Sitter and flat space limits. We use this propagator to compute two simple observables: the off-coincident field strength-field strength correlator and the energy-momentum tensor, both of which yield consistent results. As a spinoff of our computation we also give the exact expression for the Coulomb gauge propagator in power-law inflation in arbitrary dimensions.
Autoregularization, a new divergence-free framework for calculating scattering amplitudes, uses a Lorentz-invariant scale harvested from the kinematics of a scattering process to regularize the amplitude of the process [N. Prabhu, .]. Preliminary validation studies show that autoregularization’s predictions are in good agreement with experimental data—across several scattering processes and a wide range of energy scales. Further, tree-level calculation of the vacuum energy density of the free fields in the Standard Model, using autoregularization, is shown to yield a value that is smaller than the current estimate of the cosmic critical density. In this paper, we prove that the scattering amplitudes in QED, calculated using autoregularization, are gauge invariant. Our proof, which is valid both for autoregularization and current theory, is stronger in that it shows the amplitude of every Feynman diagram is gauge invariant in contrast to previous proofs, which establish gauge invariance only for sum of amplitudes of Feynman diagrams of a process. Next, we show that—unlike in the standard quantization framework, which requires modification of both the quantization framework itself as well as the Lagrangian in order to quantize gauge fields in covariant gauge—in autoregularization the gauge field in QED can be quantized, in covariant gauge, without modifying the standard quantization procedure or the Lagrangian and without introducing the ghost field. Finally, we illustrate renormalization based on autoregularization up to 1-loop in φ 4 theory. Since perturbative corrections are finite in autoregularization, the counterterms are not designed to remove divergences but to implement renormalization prescriptions at every order of perturbation. We also derive the renormalization group equation (RGE). Unlike in some regularization schemes (such as dimensional regularization), in which the physical meaning of the fictitious scale introduced by regularization is unclear, in autoregularization the scale in RGE has a transparent physical meaning—it is the Lorentz-invariant kinematic scale of the scattering process of interest. The increasing simplifications resulting within autoregularization and the agreement between its predictions and experimental data, together with the underlying thermodynamic argument, which shows that the framework is essential for a complete description of quantum fields, all converge to suggest that autoregularization provides the proper framework for the description of quantum fields.
Published by the American Physical Society 2024
Atomic high-precision measurements have become a competitive and essential technique for tests of fundamental physics, the Standard Model, and our theory of gravity. It is therefore self-evident that such measurements call for a consistent relativistic description of atoms that eventually originates from quantum field theories like quantum electrodynamics. Most quantum metrological approaches even postulate effective field-theoretical treatments to describe a precision enhancement through techniques like squeezing. However, a consistent derivation of interacting atomic quantum gases from an elementary quantum field theory that includes both the internal structure as well as the center of mass of atoms, has not yet been addressed. We present such a subspace effective field theory for interacting, spin carrying, and possibly charged ensembles of atoms composed of nucleus and electron that form composite bosons called cobosons, where the interaction with light is included in a multipolar description. Relativistic corrections to the energy of a single coboson, light-matter interaction, and the scattering potential between cobosons arise in a consistent and natural manner. In particular, we obtain a relativistic coupling between the coboson’s center-of-mass motion and internal structure encoded by the mass defect. We use these results to derive modified bound-state energies, including the motion of ions, modified scattering potentials, a relativistic extension of the Gross-Pitaevskii equation, and the mass defect applicable to atomic clocks or quantum clock interferometry.
Published by the American Physical Society 2024
Quantum electrodynamics (QED) is the most accurate of all experimentally verified physical theories. How QED and other theories of fundamental interactions couple to gravity through special unitary symmetries, on which the standard model of particle physics is based, is, however, still unknown. Here we develop a coupling between the electromagnetic field, Dirac electron-positron field, and the gravitational field based on an eight-component spinorial representation of the electromagnetic field. Our spinorial representation is analogous to the well-known representation of particles in the Dirac theory but it is given in terms of 8 × 8 bosonic gamma matrices. In distinction from earlier works on the spinorial representations of the electromagnetic field, we reformulate QED using eight-component spinors. This enables us to introduce the generating Lagrangian density of gravity based on the special unitary symmetry of the eight-dimensional spinor space. The generating Lagrangian density of gravity plays, in the definition of the gauge theory of gravity and its symmetric stress-energy-momentum tensor source term, a similar role as the conventional Lagrangian density of the free Dirac field plays in the definition of the gauge theory of QED and its electric four-current density source term. The fundamental consequence, the Yang-Mills gauge theory of unified gravity, is studied in a separate work [], where the theory is also extended to cover the other fundamental interactions of the standard model. We devote ample space for details of the eight-spinor QED to provide solid mathematical basis for the present work and the related work on the Yang-Mills gauge theory of unified gravity.
Published by the American Physical Society 2024
We present the modern covariant quantization of perturbative string theory á la BRST. We determine the physical spectrum of the several string theories, write down the vertices of physical states, and prove unitarity. We discuss the role of the picture charge in the superstring, and deduce several general properties of perturbative (super)string theory. In the three preliminary sections we discuss subtle points of bosonization, light-cone quantization, and the old covariant quantization. In the last section, the Chan–Paton degrees of freedom are introduced and studied.
Atomic high-precision measurements have become a competitive and essential technique for tests of fundamental physics, the Standard Model, and our theory of gravity. It is therefore self-evident that such measurements call for a consistent relativistic description of atoms that eventually originates from quantum field theories like quantum electrodynamics. Most quantum-metrological approaches even postulate effective field-theoretical treatments to describe a precision enhancement through techniques like squeezing. However, a consistent derivation of interacting atomic quantum gases from an elementary quantum field theory that includes both the internal structure as well as the center of mass of atoms, has not yet been addressed. We present such an effective quantum field theory for interacting, spin-carrying, and possibly charged ensembles of atoms composed of nucleus and electron that form composite bosons called cobosons, where the interaction with light is included in a multipolar description. Relativistic corrections to the energy of a single coboson, light-matter interaction, and the scattering potential between cobosons arise in a consistent and natural manner. In particular, we obtain a relativistic coupling between the coboson's center-of-mass motion and internal structure encoded by the mass defect, together with an ion spin-orbit coupling. We use these results to derive modified bound-state energies including the motion of ions, modified scattering potentials, a relativistic extension of the Gross-Pitaevskii equation, and the mass defect applicable to atomic clocks or quantum-clock interferometry. Our theory does not only combine and generalize aspects of effective field theories, quantum optics, scattering theory, and ultracold quantum gases, but it also bridges the gap between quantum electrodynamics and effective field theories for ultracold quantum gases.
We examine the quantum gravitational entanglement of two test masses in the context of linearized general relativity with specific nonlocal interaction with matter. To accomplish this, we consider an energy-momentum tensor describing two test particles of equal mass with each possessing some nonzero momentum. After discussing the quantization of the linearized theory, we compute the gravitational energy shift, which is operator valued in this case. As compared to the local gravitational interaction, we find that the change in the gravitational energy due to the self-interaction terms is finite. We then move on to study the quantum-gravity-induced entanglement of masses for two different scenarios. The first scenario involves treating the two test masses as harmonic oscillators with an interaction Hamiltonian given by the aforesaid gravitational energy shift. In the second scenario, each of the test masses is placed in a quantum spatial superposition of two locations, based on their respective spin states, and their entanglement being induced by the gravitational interaction and the shift in the vacuum energy. For these two scenarios, we compute both the concurrence and the von Neumann entropy, showing that an increase in the nonlocality of the gravitational interaction results in a decrease in both of these quantities.
Dilatons [ϕ(x)] are a class of bosonic scalar particles associated with scaling symmetry and its compensation (under the violations of the same). They are capable of interacting gravitationally with other massive bodies. As they have coupling to two photons (γ), they are (also) capable of decaying to the two photons. However, the decay time is long and that makes them a good candidate for dark matter. Furthermore due to two photon coupling, they can produce optical signatures in a magnetic field. In a vacuum or plain matter they couple to one of the transversely polarized states of the photon. But in magnetized matter, they couple to both the transversely polarized state of photons (due to the emergence of a parity violating part of the photon self-energy contribution from magnetized matter). Being spin zero scalar, they could mix with spin zero longitudinal part of photons but they do not. A part of this work is directed towards understanding this issue of mixing the scalar with various polarization states of photons in a medium (magnetized or unmagnetized) due to the constraints from different discrete symmetries, e.g., charge conjugation (C), parity (P) and time reversal (T) associated with the interaction. Based on these symmetry aided arguments, the structure of the mixing matrix is found to be 3×3, as in the case of neutrino flavor mixing matrix. Thus there exists nonzero finite probabilities of oscillation between different polarization states of photon to dilaton. Our analytical and numerical analysis show no existence of periodic oscillation length either in temporal or spatial direction for the most general values of the parameters in the theory. Possible astrophysical consequences of these results can be detected through the discussed observations.
The theory of gauge-fixed Maxwell equations in linear isotropic dielectrics is developed using a generalization of the standard gauge-fixing term. In static space times, the theory can be quantized using the Gupta-Bleuler method, which is worked out explicitly for optical fibers either in flat space time or at a constant gravitational potential. This yields a consistent first-principles description of gravitational fiber-optic interferometry at the single-photon level within the framework of quantum field theory in curved space times.
The covariant formulation of quantum electrodynamics, developed in a previous paper, is here applied to two elementary problems—the polarization of the vacuum and the self-energies of the electron and photon. In the first section the vacuum of the non-interacting electromagnetic and matter fields is covariantly defined as that state for which the eigenvalue of an arbitrary time-like component of the energy-momentum four-vector is an absolute minimum. It is remarked that this definition must be compatible with the requirement that the vacuum expectation values of a physical quantity in various coordinate systems should be, not only covariantly related, but identical, since the vacuum has a significance that is independent of the coordinate system. In order to construct a suitable characterization of the vacuum state vector, a covariant decomposition of the field operators into positive and negative frequency components is introduced, and the properties of these associated fields developed. It is shown that the state vector for the electromagnetic vacuum is annihilated by the positive frequency part of the transverse four-vector potential, while that for the matter vacuum is annihilated by the positive frequency part of the Dirac spinor and of its charge conjugate. These defining properties of the vacuum state vector are employed in the calculation of the vacuum expectation values of quadratic field quantities, specifically the energy-momentum tensors of the independent electromagnetic and matter fields, and the current four-vector. It is inferred that the electromagnetic energy-momentum tensor, and the current vector must vanish in the vacuum, while the matter field energy-momentum tensor vanishes in the vacuum only by the addition of a suitable multiple of the unit tensor. The second section treats the induction of a current in the vacuum by an external electromagnetic field. It is supposed that the latter does not produce actual electron-positron pairs; that is, we consider only the phenomenon of virtual pair creation. This restriction is introduced by requiring that the establishment and subsequent removal of the external field produce no net change in state for the matter field. It is demonstrated, in a general manner, that the induced current at a given space-time point involves the external current in the vicinity of that point, and not the electromagnetic potentials. This gauge invariant result shows that a light wave, propagating at remote distances from its source, induces no current in the vacuum and is therefore undisturbed in its passage through space. The absence of a light quantum self-energy effect is thus indicated. The current induced at a point consists, more precisely, of two parts: a logarithmically divergent multiple of the external current at that point, which produces an unobservable renormalization of charge, and a more involved finite contribution, which is the physically significant induced current. The latter agrees with the results of previous investigations. The modification of the matter field properties arising from interaction with the vacuum fluctuations of the electromagnetic field is considered in the third section. The analysis is carried out with two alternative formulations, one employing the complete electromagnetic potential together with a supplementary condition, the other using the transverse potential, with the variables of the supplementary condition eliminated. It is noted that no real processes are produced by the first order coupling between the fields. Accordingly, alternative equations of motion for the state vector are constructed, from which the first order interaction term has been eliminated and replaced by the second order coupling which it generates. The latter includes the self action of individual particles and light quanta, the interaction of different particles, and a coupling between particles and light quanta which produces such effects as Compton scattering and two quantum pair annihilation. It is concluded from a comparison of the alternative procedures that, for the treatment of virtual light quantum processes, the separate consideration of longitudinal and transverse fields is an inadvisable complication. The light quantum self-energy term is shown to vanish, while that for a particle has the anticipated form for a change in proper mass, although the latter is logarithmically divergent, in agreement with previous calculations. To confirm the identification of the self-energy effect with a change in proper mass, it is shown that the result of removing this term from the state vector equation of motion is to alter the matter field equations of motion in the expected manner. It is verified, finally, that the energy and momentum modifications produced by self-interaction effects are entirely accounted for by the addition of the electromagnetic proper mass to the mechanical proper mass—an unobservable mass renormalization. An appendix is devoted to the construction of several invariant functions associated with the electromagnetic and matter fields.
DOI:https://doi.org/10.1103/PhysRev.75.535
A simple method of obtaining the induced charge-density four vector on the basis of the subtraction formalism of the positron theory is given. Further, in the general case of time-dependent fields the result is calculated directly without use of the Lorentz invariance of the theory.
The charge distribution, the electromagnetic field and the self-energy of an electron are investigated. It is found that, as a result of Dirac's positron theory, the charge and the magnetic dipole of the electron are extended over a finite region; the contributions of the spin and of the fluctuations of the radiation field to the self-energy are analyzed, and the reasons that the self-energy is only logarithmically infinite in positron theory are given. It is proved that the latter result holds to every approximation in an expansion of the self-energy in powers of e2hc. The self-energy of charged particles obeying Bose statistics is found to be quadratically divergent. Some evidence is given that the "critical length" of positron theory is as small as h(mc).exp(-hce2).
A relativistic cut-off of high frequency quanta, similar to that suggested by Bopp, is shown to produce a finite invariant self-energy for a free electron. The electromagnetic line shift for a bound electron comes out as given by Bethe and Weissk opf's wave packet prescription. The scattering of an electron in a potential, without radiation, is discussed. The cross section remains finite. The problem of polarization of the vacuum is not solved. Otherwise, the results will in general agree essentially with those calculated by the prescription of Schwinger. An alternative cut-off procedure analogous to one proposed by Wataghin, which eliminates high frequency intermediate states, is shown to do the same things but to offer to solve vacuum polarization problems as well.
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