In this work, we consider the generalized quantum electrodynamics proposed by Podolsky in Heisenberg picture via Kallen methodology. We investigate the effects of higher-order derivatives to understand the qualitative and quantitative aspects of vacuum polarization. In addition, the most general structure of induced current and polarization tensor that emerge naturally by a perturbative scheme "a la" Kallen is also obtained. Afterward, we discuss the physical implication of charge renormalization in the perspective of unitary and stable Podolsky theory.
We examine the generalized quantum electrodynamics as a natural extension of the Maxwell electrodynamics to cure the one-loop divergence. We establish a precise scenario to discuss the underlying features between photon and fermion where the perturbative Maxwell electrodynamics fails. Our quantum model combines stability, unitarity, and gauge invariance as the central properties. To interpret the quantum fluctuations without suffering from the physical conflicts proved by Haag's theorem, we construct the covariant quantization in the Heisenberg picture instead of the Interaction one. Furthermo…
Scalar Quantum Electrodynamics is investigated in the Heisenberg picture via the Duffin-Kemmer-Petiau gauge theory. On this framework, a perturbative method is used to compute the vacuum polarization tensor and its corresponding induced current for the case of a charged scalar field in the presence of an external electromagnetic field. Charge renormalization is brought into discussion for the interpretation of the results for the vacuum polarization.
High order harmonic generation by extremely intense, interacting, electromagnetic waves in the quantum vacuum is investigated within the framework of the Heisenberg-Euler formalism. Two intersecting plane waves of finite duration are considered in the case of general polarizations. Detailed finite expressions are obtained for the case where only the first Poincaré invariant does not vanish. Yields of high harmonics in this case are most effective.
We calculate the lowest-order corrections to the static potential for both the generalized Born-Infeld Electrodynamics and an Euler-Heisenberg-like model, in the presence of a constant external magnetic field. Our analysis is carried out within the framework of the gauge-invariant but path-dependent variables formalism. The calculation reveals a long-range correction ($ {\raise0.7ex\hbox{$1$} \mathord{\left/ {\vphantom {1 {r^5}}}\right.\kern-\nulldelimiterspace} \lower0.7ex\hbox{${r^5}$}}$-type) to the Coulomb potential for the generalized Born-Infeld Electrodynamics. Interestingly enough, in …
We consider Heisenberg-Euler-type model of nonlinear electrodynamics with two parameters. Heisenberg-Euler electrodynamics is a particular case of this model. Corrections to Coulomb's law at $r\rightarrow\infty$ are obtained and energy conditions are studied. The total electrostatic energy of charged particles is finite. The charged black hole solution in the framework of nonlinear electrodynamics is investigated. We find the asymptotic of the metric and mass functions at $r\rightarrow\infty$. Corrections to the Reissner-Nordström solution are obtained.
In this paper, we investigate from the framework of generalized electrodynamics the differential cross section of the electron-electron scattering process $e^- e^- \rightarrow e^- e^-$, i.e., Møller scattering, in $(2+1)$ dimensions in the Heisenberg picture. To this goal, one starts within the stable and unitary framework of planar generalized electrodynamics, instead of Maxwell one. We argue the Haag's theorem strongly suggests the study of the differential cross section in the Heisenberg representation. Afterward, we explore the influence of Podolsky mass cutoff and calculate the differenti…
We consider a particular 4-dimensional generalization of the transition from the Heisenberg to the Schrödinger picture. The space-time independent expansion with respect to the unitary irreducible representations of the Lorentz group is applied, as Fourier transformation in the Heisenberg picture, to the states of a massive relativistic particle. A new Hamiltonian operator has been found for such a particle with spin one.
Multiple scale techniques are well-known in classical mechanics to give perturbation series free from resonant terms. When applied to the quantum anharmonic oscillator, these techniques lead to interesting features concerning the solution of the Heisenberg equations of motion and the Hamiltonian spectrum.
We consider the (2+1)-dimensional gauged Thirring model in the Heisenberg picture. In this context we evaluate the vacuum polarization tensor as well as the corrected gauge boson propagator and address the issues of generation of mass and dynamics for the gauge boson (in the limits of QED$_3$ and Thirring model as a gauge theory, respectively) due to the radiative corrections.
We consider both massive Euler-Heisenberg-like and Euler-Heisenberg-like Electrodynamics in the approximation of the strong-field limit. Our analysis shows that massive Euler-Heisenberg-type Electrodynamics displays the vacuum birefringence phenomenon. Afterwards, we calculate the lowest-order modifications to the interaction energy for both classes of Electrodynamics, within the framework of the gauge-invariant path-dependent variables formalism. Interestingly enough, for massive Euler-Heisenbeg-like electrodynamics (Whichmann-Kroll) we obtain a new long-range ($1/r^3$- type) correction, apar…
The new method based on the operator formalism proposed by Abe and Nakanishi is applied to the quantum nonlinear abelian gauge theory in two dimension. The soluble models in this method are extended to wider class of quantum field theories. We obtain the exact solution in the canonical-quantization operator formalism in the Heisenberg picture. So this analysis might shed some light on the analysis of gravitational theory and non-polynomial field theories.
The Euler-Heisenberg action for a $U(1)\times U(1)$ dyon quantum electrodynamics
Instituto de Física, Facultad de Ingeniería, Universidad de la República（乌拉圭共和国大学工程学院物理研究所）；Departamento de Física, Universidade Federal de Viçosa（维索萨联邦大学物理系）；Ibitipoca Institute of Physics (IbitiPhys)（伊比蒂波卡物理研究所）
Within the framework of the gauge-invariant, but path-dependent, variables formalism, we study the manifestations of vacuum electromagnetic nonlinearities in $D=3$ models. For this we consider both generalized Born-Infeld and Pagels-Tomboulis-like electrodynamics, as well as, an Euler-Heisenberg-like electrodynamics. We explicitly show that generalized Born-Infeld and Pagels-Tomboulis-like electrodynamics are equivalent, where the static potential profile contains a long-range (${\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle {r^2}$}}$-type) correctio…
We investigate the quantization in the Heisenberg representation of a model which represents a simplification of the Hopfield model for dielectric media, where the electromagnetic field is replaced by a scalar field $ϕ$ and the role of the polarization field is played by a further scalar field $ψ$. The model, which is quadratic in the fields, is still characterized by a nontrivial physical content, as the physical particles correspond to the polaritons of the standard Hopfield model of condensed matter physics. Causality is also taken into account and a discussion of the standard interaction r…
We discuss the Heisenberg-Wigner phase-space formalism in quantum electrodynamics as well as scalar quantum electrodynamics with respect to transverse fields. In regard to the special characteristics of such field types we derive modified transport equations such that particle momenta perpendicular to the propagation direction of the waves show up as external parameters only. In case of spatially oscillating fields we further demonstrate how to transform momentum derivative operators of infinite order into simple coupling terms.
We advocate the study of external-field quantum electrodynamics with $N$ charged particle flavors. Our main focus is on the Heisenberg-Euler effective action for this theory in the large $N$ limit which receives contributions from all loop orders. The contributions beyond one loop stem from one-particle reducible diagrams. We show that specifically in constant electromagnetic fields the latter are generated by the one-loop Heisenberg-Euler effective Lagrangian. Hence, in this case the large $N$ Heisenberg-Euler effective action can be determined explicitly at any desired loop order. We demonst…
Contrary to the conventional view point of quantization that breaks the gauge symmetry, a gauge invariant formulation of quantum electrodynamics is proposed. Instead of fixing the gauge, some frame is chosen to yield the locally invariant fields. We show that all the formulations, such as the Coulomb, the axial, and the Lorentz gauges, can be constructed and that the explicit LSZ mapping connecting Heisenberg operators to those of the asymptotic fields is possible. We also make some comments on gauge transformations in quantized field theory.
We discuss the following recent applications of the ``string-inspired'' worldline technique to calculations in quantum electrodynamics: i) Photon splitting in a constant magnetic field, ii) The two-loop Euler-Heisenberg Lagrangian, iii) A progress report on a recalculation of the three-loop QED beta -- function.
We study the geodesic deviation equation for a quantum particle in a linearized quantum gravitational field. Particle's Heisenberg equations of motion are treated as stochastic equations with a quantum noise. We explore the stochastic equation beyond its local approximation as a differential. We discuss the squeezed states resulting from an inflationary evolution. We calculate the noise in the thermal and squeezed states.
We present the Heisenberg-picture approach to the quantum evolution of the scalar fields in an expanding FRW universe which incorporates relatively simply the initial quantum conditions such as the vacuum state, the thermal equilibrium state, and the coherent state. We calculate the Wightman function, two-point function, and correlation function of a massive scalar field. We find the quantum evolution of fluctuations of a self-interacting field perturbatively and discuss the renormalization of field equations.
In this paper, we give an update on divergent problems concerning the radiative corrections of quantum electrodynamics in $(3+1)$ dimensions. In doing so, we introduce a geometric adaptation for the covariant photon propagator by including a higher derivative field. This derivation, so-called generalized quantum electrodynamics, is motivated by the stability and unitarity features. This theory provides a natural and self-consistent extension of the quantum electrodynamics by enlarging the space parameter of spinor-gauge interactions. In particular, Haag$'$s theorem undermines the perturbative …
Quantum regression theorem is a very useful result in open quantum system and extensively used for computing multi-point correlation functions. Traditionally it is derived for two-time correlators in the Markovian limit employing the Schrödinger picture. In this paper we make use of the Heisenberg picture to derive quantum regression theorems for multi-time correlation functions which in the special limit reduce to the well known two-time regression theorem. For multi-time correlation function we find that the regression theorem takes the same form as it takes for two-time correlation function…
We demonstrate the existence of quantum hair in electrodynamics and gravity using effective action techniques. In the case of electrodynamics we use the Euler-Heisenberg effective action while in the case of quantum gravity we use the unique effective action. We give a general formulation of these effects which applies to both theories and discuss analogies and differences between them. Furthermore, we present a QED analog to black hole evaporation. Spontaneous pair production in the external field of a ball of charge is analogous to Hawking radiation from black holes. Assuming spherical symme…
On this 75th anniversary of the publication of the Heisenberg-Euler paper on the full non-perturbative one-loop effective action for quantum electrodynamics I review their paper and discuss some of the impact it has had on quantum field theory.
The effects of the entanglement of the inflaton field which initially entangled with those of another field on observables like power spectrum are known in the context of the Schrödinger field theory. To clarify this effect in Heisenberg picture, there were some attempts to construct the initial entangled state by making use of an entangled transformation (like Bogoliubov transformation) between the Bunch-Davies vacuums and squeezed states. We study the role of the time-dependent entangled transformation in the Schrödinger field theory. We derive the relation between two vacuum states which th…
We study the effective interactions of external electromagnetic fields induced by fluctuations of virtual particles in the vacuum of quantum electrodynamics. Our main focus is on these interactions at two-loop order. We discuss in detail the emergence of the renowned Heisenberg-Euler effective action from the underlying microscopic theory of quantum electrodynamics, emphasizing its distinction from a standard one-particle irreducible effective action. In our explicit calculations we limit ourselves to constant and slowly varying external fields, allowing us to adopt a locally constant field ap…
It is well known that the unboundedness of operators in Hilbert space entails domain troubles. It is also well known that most domain troubles can be surmounted by extending the Hilbert space to a rigged Hilbert space. In this note, we point out another of such troubles, namely the correspondence between the Schrodinger and the Heisenberg pictures for unbounded operators, and sketch the solution of this problem within the rigged Hilbert space.
We consider the process of photon decay in quantum electrodynamics with a CPT-violating Chern-Simons-like term added to the action. For a simplified model with only the quadratic Maxwell and Chern-Simons-like terms and the quartic Euler-Heisenberg term, we obtain a nonvanishing probability for the decay of a particular photon state into three others.
The Heisenberg picture for non-Hermitian but $η$-pseudo-Hermitian Hamiltonian systems is suggested. If a non-Hermitian but $η$-pseudo-Hermitian Hamiltonian leads to real second order equations of motion, though their first order Heisenberg equations of motion are complex, we can construct a Hermitian counterpart that gives the same second order equations of motion. In terms of a similarity transformation we verify the iso-spectral property of the Hermitian and non-Hermitian Hamiltonians and obtain the related eigenfunctions. This feature can be used to determine real eigenvalues for such non-H…
The annihilation-creation operators of the harmonic oscillator, the basic and most important tools in quantum physics, are generalised to most solvable quantum mechanical systems of single degree of freedom including the so-called `discrete' quantum mechanics. They admit exact Heisenberg operator solution. We present unified definition of the annihilation-creation operators (a^{(\pm)}) as the positive/negative frequency parts of the exact Heisenberg operator solution.
The quantization scheme is suggested for a spatially inhomogeneous 1+1 Bianchi I model. The scheme consists in quantization of the equations of motion and gives the operator (so-called quasi-Heisenberg) equations describing an explicit evolution of a system. Some particular gauge suitable for quantization is proposed. The Wheeler-DeWitt equation is considered in the vicinity of zero scale factor and it is used to construct a space, where the quasi-Heisenberg operators act. Spatial discretization as a UV regularization procedure is suggested for the equations of motion.
Relativistic accretion process onto rotating black holes in Einstein-Euler-Heisenberg nonlinear electrodynamic gravity
American University of the Middle East（中东美国大学）；Zhejiang Normal University（浙江师范大学）；Khazar University（阿塞拜疆哈萨尔大学）；Virtual University of Pakistan（巴基斯坦虚拟大学）；Echahid Cheikh Larbi Tebessi University（特贝西大学）；University of Management and Technology（管理与技术大学）；Urgench State University（Urgench国立大学）
Thermal aspects and particle dynamics of Euler-Heisenberg AdS black hole in 4D Einstein Gauss-Bonnet gravity
Faculté des Sciences Exactes, Université Constantine 1（康斯坦丁大学精确科学学院）；Department of Physics, Zhejiang Normal University（浙江师范大学物理系）；Research Center of Astrophysics and Cosmology, Khazar University（哈萨尔大学天体物理学与宇宙学研究中心）
Some of the basic notions of nonlinear optics are summarized and then applied to the case of the Dirac vacuum, as described by the Heisenberg-Euler effective one-loop Lagrangian. The theoretical and experimental basis for the appearance of nonlinear optical phenomena, such as the Kerr effect, Cotton-Mouton effect, and four-wave mixing are discussed. Further effects due to more exotic assumptions on the structure of spacetime, such as gravitational curvature and the topology of the Casimir vacuum are also presented.
Modified Euler-Heisenberg effective action and Proper-Time Method in Lorentz-Violating Scalar QED
Universidade Federal de Lavras（拉夫拉斯联邦大学）；Universidade Federal de São João del Rei（圣若昂德尔雷伊联邦大学）；Universidade Federal do Pará（帕拉联邦大学）；Universidade Federal da Paraíba（帕拉伊巴联邦大学）
We provide a theoretical foundation for the notion of the quantum coherent state of the electrostatic field of a static external charge distribution introduced in a 1998 paper and rederive formulae there for the inner products of a pair of such states. Contrary to what one might expect, these inner products are non-zero whenever the total charges of the two charge distributions are equal, even if the charge distributions themselves differ. We actually display two different frameworks for these same coherent states, in the second of which Gauss's law only holds in expectation value. We propose …
Leon Rosenfeld published in 1930 the first systematic Hamiltonian approach to Lagrangian models that possess a local gauge symmetry. The application of this formalism to theories with local internal symmetries, such as electromagnetism in interaction with charged matter fields, is valid and complete, and predates by two decades the work by Dirac and Bergmann. Although he provided a group-theoretical justification for gauge fixing procedures that had just been implemented in the first expositions of quantum electrodynamics by Heisenberg and Pauli, and also by Fermi, his contribution went largel…
We consider the possibility that photons of noncommutative QED can make bound states. Using the potential model, developed based on the constituent gluon picture of QCD glue-balls, arguments are presented in favor of existence of these bound states. The basic ingredient of potential model is that the self-interacting massless gauge particles may get mass by inclusion non-perturbative effects.
Spontaneous chiral symmetry breaking in three dimensional ($d=3$) quantum electrodynamics is understood as annihilation of an infrared-stable fixed point that describes the large-N conformal phase by another unstable fixed point at a critical number of fermions $N=N_c$. We discuss the root of universality of $N_c$ in this picture, together with some features of the phase boundary in the $(d,N)$ plane. In particular, it is shown that as $d\rightarrow 4$, $N_c\rightarrow 0$ with a constant slope, our best estimate of which suggests that $N_c = 2.89$ in $d=3$.
Quantum electrodynamics predicts the vacuum to behave as a non-linear medium, including effects such as birefringence. However, for experimentally available field strengths, this vacuum polarizability is extremely small and thus very hard to measure. In analogy to the Heisenberg limit in quantum metrology, we study the minimum requirements for such a detection in a given strong field (the pump field). Using a laser pulse as the probe field, we find that its energy must exceed a certain threshold depending on the interaction time. However, a detection at that threshold, i.e., the Heisenberg lim…
