Equal-time Commutators Research Articles

Quantum theory of electromagnetic field for a dielectric–dielectric interface in three dimensions

The three dimensional scheme of the Green function approach of the electromagnetic field quantization is extended to the background medium made up of two adjacent semi-infinite dielectrics which show both loss and dispersion. It is shown that this system separates into a singlet system providing the normal polarization state of the electric field and a doublet system giving both the parallel state of polarization and the longitudinal part of the field. The longitudinal field expression is also derived by using the usual approach of vector decomposition. The overall results enable us to present the form of the field operators for each state of polarization as well as the longitudinal field contribution. The equal-time canonical commutation relations between the vector potential operator and its conjugate generalized momentum are verified in each case.

THE MASTER WARD IDENTITY

In the framework of perturbative quantum field theory (QFT) we propose a new, universal (re)normalization condition (called 'master Ward identity') which expresses the symmetries of the underlying classical theory. It implies for example the field equations, energy-momentum, charge- and ghost-number conservation, renormalized equal-time commutation relations and BRST-symmetry. It seems that the master Ward identity can nearly always be satisfied, the only exceptions we know are the usual anomalies. We prove the compatibility of the master Ward identity with the other (re)normalization conditions of causal perturbation theory, and for pure massive theories we show that the 'central solution' of Epstein and Glaser fulfills the master Ward identity, if the UV-scaling behavior of its individual terms is not relatively lowered. Application of the master Ward identity to the BRST-current of non-Abelian gauge theories generates an identity (called 'master BRST-identity') which contains the information which is needed for a local construction of the algebra of observables, i.e. the elimination of the unphysical fields and the construction of physical states in the presence of an adiabatically switched off interaction.

GAUGE-INVARIANT FIELDS IN THE TEMPORAL GAUGE, COULOMB-GAUGE FIELDS, AND THE GRIBOV AMBIGUITY

We examine the relation between Coulomb-gauge fields and the gauge-invariant fields constructed in the temporal gauge for two-color QCD by comparing a variety of properties, including their equal-time commutation rules and those of their conjugate chromoelectric fields. We also express the temporal-gauge Hamiltonian in terms of gauge-invariant fields and show that it can be interpreted as a sum of the Coulomb-gauge Hamiltonian and another part that is important for determining the equations of motion of temporal-gauge fields, but that can never affect the time evolution of "physical" state vectors. We also discuss multiplicities of gauge-invariant temporal-gauge fields that belong to different topological sectors and that, in a previous work, were shown to be based on the same underlying gauge-dependent temporal-gauge fields. We argue that these multiplicities of gauge-invariant fields are manifestations of the Gribov ambiguity. We show that the differential equation that bases the multiplicities of gauge-invariant fields on their underlying gauge-dependent temporal-gauge fields has nonlinearities identical to those of the "Gribov" equation, which demonstrates the nonuniqueness of Coulomb-gauge fields. These multiplicities of gauge-invariant fields — and, hence, Gribov copies — appear in the temporal gauge, but only with the imposition of Gauss' law and the implementation of gauge invariance; they do not arise when the theory is represented in terms of gauge-dependent fields and Gauss' law is left unimplemented.

Two-Dimensional Gravitational Anomalies, Schwinger Terms, and Dispersion Relations

We are dealing with two-dimensional gravitational anomalies, specifically with the Einstein anomaly and the Weyl anomaly, and we show that they are fully determined by dispersion relations independent of any renormalization procedure (or ultraviolet regularization). The origin of the anomalies is the existence of a superconvergence sum rule for the imaginary part of the relevant formfactor. In the zero mass limit the imaginary part of the formfactor approaches a δ-function singularity at zero momentum squared, exhibiting in this way the infrared feature of the gravitational anomalies. We find an equivalence between the dispersive approach and the dimensional regularization procedure. The Schwinger terms appearing in the equal time commutators of the energy momentum tensors can be calculated by the same dispersive method. Although all computations are performed in two dimensions the method is expected to work in higher dimensions too.

Mode expansion and photon operators in dispersive and absorbing dielectrics

We present a one-dimensional quantization scheme for the electromagnetic field in arbitrary planar absorbing and dispersive dielectrics, taking into account explicitly the finite extent of media. The complete form of the electric field operator includes a part that corresponds to the free fields incident from the vacuum towards the medium and a particular solution which can be expressed by using the classical Green-function integral representation of the electromagnetic field. By expressing the classical Green function in terms of the light modes of the dielectric structure, we obtain a natural generalization to absorbing media of the familiar method of mode expansion which strictly applies to non-absorbing media. As an application of our mode expansion, we obtain general quantum-optical input-output relations relating the output photon operators in vacuum to the input photon operators and to the reservoir noise operators. The input-output relations obtained are uniquely determined by the knowledge of the classical light modes of the planar system. We finally show that our quantization scheme for arbitrary planar dielectrics satisfies the prescribed equal-time commutation relations of QED.

Second-quantized models for optical solitons in nonlinear fibers: Equal-time versus equal-space commutation relations

The dynamics of a quantum soliton propagating in nonlinear optical fibers is analyzed in the second-quantized picture in a fully canonical way. The quantum nonlinear Schr\"odinger equation with equal-time commutation relations for the field operators is solved using the formalism developed by Haus and Lai [J. Opt. Soc. Am. B $7,$ 386 (1990)]. The canonical approach is compared with the treatment of Haus and Lai based on phenomenological equal-space commutation relations. The two theories are shown to provide similar outcomes of a homodyne measurement. This result justifies the use of equal-space commutation relations in quantum studies of soliton propagation.

Improvements to the method of dispersion relations forBnonleptonic decays

We bring some clarifications and improvements to the method of dispersion relations in the external masses variables that we proposed recently for investigating the final state interactions in the B nonleptonic decays. We first present arguments for the existence of an additional term in the dispersion representation, which arises from an equal-time commutator in the Lehmann-Symanzik-Zimmerman formalism and can be approximated by the conventional factorized amplitude. The reality properties of the spectral function and the Goldberger-Treiman procedure to perform the hadronic unitarity sum are analyzed in more detail. We also improve the treatment of the strong interaction part by including the contributions of both t- and u-channel trajectories in the Regge amplitudes. Applications to the ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{K}^{+}$ decays are presented.

Schwinger terms in the fully quantized 1 + 1 dimensional model – exact solution

We calculate the equal-time commutator of two fermionic currents within the framework of the 1+1 dimensional fully quantized theory, describing the interaction of fermions with a vector boson. It is shown that the interaction does not change the result obtained within the theory of free fermions.

Classical interventions in quantum systems. II. Relativistic invariance

If several interventions performed on a quantum system are localized in mutually space-like regions, they will be recorded as a sequence of ``quantum jumps'' in one Lorentz frame, and as a different sequence of jumps in another Lorentz frame. Conditions are specified that must be obeyed by the various operators involved in the calculations so that these two different sequences lead to the same observable results. These conditions are similar to the equal-time commutation relations in quantum field theory. They are sufficient to prevent superluminal signaling. (The derivation of these results does not require most of the contents of the preceding article. What is needed is briefly summarized here, so that the present article is essentially self-contained.)

Three-dimensional electromagnetic field quantization in absorbing and dispersive bounded dielectrics

We present a quantization scheme for the electromagnetic field in dispersive and lossy dielectrics with planar interfaces, incorporating propagation in all the spatial directions, and including both TE and TM polarized fields. Field quantization is carried out starting from a microscopic model for the polarization quanta interacting with the electromagnetic field. Dissipation is described by considering the coupling of the polarization quanta of the system with the reservoir oscillators in the usual Langevin approach. The method is applied to a semi-infinite medium and expressions for the electromagnetic field operators are obtained in the two different spatial regions, i.e., in the vacuum and in the dielectric. As a check of our quantization scheme, we have verified that the obtained vector potential operator and its conjugate momentum satisfy equal-time canonical commutation rules, giving rise to the prescribed transverse Dirac $\ensuremath{\delta}$ function.

Nonfactorizable contributions in hadronic weak decays of charm mesons

Two body decays of charm mesons are studied by describing their amplitude in terms of a sum of factorizable and non-factorizable ones. The former is estimated by using a naive factorization while the latter is calculated by using a hard pseudo-scalar-meson approximation. The hard pseudo-scalar-meson amplitude is given by a sum of the so-called equal-time commutator term and surface term which contains all possible pole contributions of various mesons, not only the ordinary $\{q\bar q\}$ but also four-quark $\{qq\bar q\bar q\}$, hybrid $\{q\bar qg\}$ and glue-balls. Naively factorized amplitudes for the spectator decays which lead to too big rates can interfere destructively with exotic meson pole amplitudes and the total amplitudes can reproduce their observed rates. The non-factorizable contributions can supply sufficiently large contributions to the color suppressed decays which are strongly suppressed in the naive factorization. A possible solution to the long standing puzzle that the ratio of decay rates for $D^0\to K^+K^-$ to $D^0\to \pi^+\pi^-$ is around 2.5 is given by different contributions of exotic meson poles.

QED commutation relations for inhomogeneous Kramers-Kronig dielectrics

Recently a quantization scheme for the phenomenological Maxwell theory of the full electromagnetic field in an inhomogeneous three-dimensional, dispersive, and absorbing dielectric medium has been developed and applied to a system consisting of two infinite half-spaces with a common planar interface (H.T. Dung, L. Knoell, and D.-G. Welsch, Phys. Rev. A 57, in press). Here we show that the scheme, which is based on the classical Green-tensor integral representation of the electromagnetic field, applies to any inhomogeneous medium. For this purpose we prove that the fundamental equal-time commutation relations of QED are preserved for an arbitrarily space-dependent, Kramers--Kronig consistent permittivity. Further, an extension of the quantization scheme to linear media with bounded regions of amplification is given, and the problem of anisotropic media is briefly addressed.

Three-dimensional quantization of the electromagnetic field in dispersive and absorbing inhomogeneous dielectrics

A quantization scheme for the phenomenological Maxwell theory of the full electromagnetic field in an inhomogeneous three-dimensional, dispersive, and absorbing dielectric medium is developed. The classical Maxwell equations with spatially varying and Kramers-Kronig consistent permittivity are regarded as operator-valued field equations, introducing additional current- and charge-density operator fields in order to take into account the noise associated with the dissipation in the medium. It is shown that the equal-time commutation relations between the fundamental electromagnetic fields $\mathbf{E}^$ and $\mathbf{B}^$ and the potentials $\mathbf{A}^$ and $\stackrel{^}{\ensuremath{\varphi}}$ in the Coulomb gauge can be expressed in terms of the Green tensor of the classical problem. From the Green tensors for bulk material and an inhomogeneous medium consisting of two bulk dielectrics with a common planar interface it is explicitly proven that the well-known equal-time commutation relations of QED are preserved.

Schwinger model in the light-cone gauge

The Schwinger model, defined in the space interval $\ensuremath{-}L<~x<~L$, with (anti)periodic boundary conditions, is canonically quantized in the light-cone gauge ${A}_{\ensuremath{-}}=0$ by means of equal-time (anti)commutation relations. The transformation diagonalizing the complete Hamiltonian is explicitly constructed, thereby giving spectrum, chiral anomaly, and condensate. The structures of Hilbert spaces related both to free and to interacting Hamiltonians are completely exhibited. Besides the usual massive field, two chiral massless fields are present, which can be consistently expunged from the physical space by means of a subsidiary condition of a Gupta-Bleuler type. The chiral condensate does provide the correct nonvanishing value in the decompactification limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$.

Schwinger terms, gerbes, and operator residues

These notes explain recent developments concerning chiral anomalies and hamiltonian quantization, their relation to the theory of gerbes, and extensions of generalized loop algebras using the residue calculus of pseudodifferential operators. The renormalization of the Dirac field, leading to Schwinger terms in equal time commutation relations, is treated in a mathematically rigorous manner. The same renormalization can be used to prove the existence of S-operator in background field problems.

Current Algebra and Bosonization in Three Dimensions

We consider the fermion-boson mapping in three dimensional space-time, in the Abelian case, from the current algebra point of view. We show that in a path-integral framework one can derive a general bosonization recipe leading, in the bosonic language, to the correct equal-time current commutators of the original free fermionic theory.

Quantum optics of plasmon polaritons and velocity sum rules.

We demonstrate that the reciprocal-space Hamiltonian method for quantizing the electromagnetic fields in a dispersive dielectric medium leads to inconsistencies when applied to the cases of plasmon polaritons and coupled phonon-plasmon polaritons. In particular, we show that the use of the standard expressions for the dielectric functions in these two cases leads to the violation of the Huttner-Barnett velocity-ratio sum rule and, hence, to the breakdown of the equal-time canonical field commutation relations. We show that correct behavior is recoverable after the dielectric functions are appropriately regularized. Regularization effectively leads in each case to the introduction of an additional transverse low-frequency polariton branch. This is sufficient to rectify the theory and there is no need to invoke the presence of the longitudinal modes. The theory is illustrated with reference to the coupled phonon-plasmon polaritons of Si-doped GaAs. \textcopyright{} 1996 The American Physical Society.

Microscopic theory of quantization of radiation in molecular dielectrics: Normal-mode representation of operators for local and averaged (macroscopic) fields.

Quantization of radiation has been performed from first principles in a realistic molecular medium, represented by an arbitrary number of energy levels (electronic, vibrational, rotational, etc.) for each constituent molecule. Adopting a polariton model, the field operators have been expanded in terms of normal Bose operators for polariton creation and annihilation. The expansion coefficients have been derived explicitly for the normal modes characterized by wavelengths exceeding considerably the characteristic distance of separation between the molecules. Accordingly, the formalism applies to the long-wavelength region of the spectrum for which description in terms of the macroscopic refractive index is relevant; furthermore, consideration is restricted to the nonabsorbing areas of the spectrum. The theory has been formulated in a manner that made possible a parallel and comparative consideration of operators for both the averaged (macroscopic) and local fields. Consequently, the mode expansions derived cover both the local displacement-field operator and also the averaged (macroscopic) operators for the electric, displacement, magnetic, and polarization fields, and the vectorial potential. The expansions, involving summation over an arbitrary number of branches of polariton dispersion, manifestly embody the refractive influences as well. To this end, the local-field effects intrinsically emerge within the present formalism that treats systematically the photon umklapp processes. Relations have been established between the expansion components of the local and averaged field operators. The relationships support some previous attempts to link the amplitudes of local and macroscopic field operators phenomenologically, and are also consistent with the familiar results of classical electrodynamics. Equal-time commutation relations have been demonstrated to be preserved, expressing the operators for the averaged fields in terms of the normal Bose operators. On the other hand, the commutation relations between the macroscopic fields are of the same form as those for their microscopic counterparts, subject to the coarse-graining procedure. Finally, the present study dealing with the macroscopic and local operators provides a tool for combined investigation of both propagation of the quantized fields in molecular dielectrics and also interaction of the fields with the embedded molecules or atoms. \textcopyright{} 1996 The American Physical Society.

Green-function approach to the radiation-field quantization for homogeneous and inhomogeneous Kramers-Kronig dielectrics.

A quantization scheme for the radiation field in dispersive and absorptive linear dielectrics is developed, which applies to both bulk material and multilayer dielectric structures. Starting from the phenomenological Maxwell equations, where the properties of the dielectric are described by a permittivity consistent with the Kramers-Kronig relations, an expansion of the field operators is performed that is based on the Green function of the classical Maxwell equations and preserves the equal-time canonical field commutation relations. In particular, in frequency intervals with approximately vanishing absorption the concept of quantization through mode expansion for dispersive dielectrics is recognized. The theory further reveals that weak absorption gives rise to space-dependent mode operators that spatially evolve according to quantum Langevin equations in the space domain. To illustrate the applicability of the theory to inhomogeneous structures, the quantization of the radiation field in a dispersive and absorptive one-interface dielectric is performed. \textcopyright{} 1996 The American Physical Society.

Current algebra results for theB - D systems

Using the equal time commutation relations for the components of the vector and axialvector currents and keeping single particle states we obtain relations for the weak form factors for theB - D systems. In the heavy quark effective theory (HQET) limit these relations determine the Isgur-Wise function.