We study the quantum properties of a Galilean-invariant abelian gauge theory coupled to a Schrödinger scalar in 2+1 dimensions. At the classical level, the theory with minimal coupling is obtained from a null-reduction of relativistic Maxwell theory coupled to a complex scalar field in 3+1 dimensions and is closely related to the Galilean electromagnetism of Le-Bellac and Lévy-Leblond. Due to the presence of a dimensionless, gauge-invariant scalar field in the Galilean multiplet of the gauge-field, we find that at the quantum level an infinite number of couplings is generated. We explain how to handle the quantum corrections systematically using the background field method. Due to a non-renormalization theorem, the beta function of the gauge coupling is found to vanish to all orders in perturbation theory, leading to a continuous family of fixed points where the non-relativistic conformal symmetry is preserved.
In this paper, we construct a single Lagrangian for both limits of Galilean electrodynamics. The framework relies on a covariant formalism used in describing Newton-Cartan geometry. We write down the Galilean conformal algebra and its representation in this formalism. We also show that the Lagrangian is invariant under the Galilean conformal algebra in d = 4 and calculate the energy-momentum tensor.
Maxwell's Electrodynamics admits two distinct Galilean limits called the Electric and Magnetic limits. We show that the equations of motion in both these limits are invariant under the Galilean Conformal Algebra in D=4, thereby exhibiting non-relativistic conformal symmetries. Remarkably, the symmetries are infinite dimensional and thus Galilean Electrodynamics give us the first example of an infinitely extended Galilean Conformal Field Theory in D>2. We examine details of the theory by looking at purely non-relativistic conformal methods and also use input from the limit of the relativistic t…
We investigate the existence of action for both the electric and magnetic sectors of Galilean Electrodynamics using Helmholtz conditions. We prove the existence of unique action in magnetic limit with the addition of a scalar field in the system. The check also implies the non existence of action in the electric sector of Galilean electrodynamics. Dirac constraint analysis of the theory reveals that there are no local degrees of freedom in the system. Further, the theory enjoys a reduced but an infinite dimensional subalgebra of Galilean conformal symmetry algebra as global symmetries. The ful…
Revisiting the Symmetries of Galilean Electrodynamics
Trinity College Dublin（都柏林三一学院）；Hamilton Mathematics Institute, Trinity College Dublin（都柏林三一学院汉密尔顿数学研究所）；II. Institut für Theoretische Physik, Universität Hamburg（汉堡大学第二理论物理研究所）
In 2+1 dimensions, we propose a renormalizable non-linear sigma model action which describes the $\mathcal{N}=2$ supersymmetric generalization of Galilean Electrodynamics. We first start with the simplest model obtained by null reduction of the relativistic Abelian $\mathcal{N}=1$ supersymmetric QED in 3+1 dimensions and study its renormalization properties directly in non-relativistic superspace. Despite the existence of a non-renormalization theorem induced by the causal structure of the non-relativistic dynamics, we find that the theory is non-renormalizable. Infinite dimensionless, supersy…
We consider a Galilean N=2 supersymmetric theory in 2+1 dimensions with F-term couplings, obtained by null reduction of a relativistic Wess-Zumino model. We compute quantum corrections and we check that, as for the relativistic parent theory, the F-term does not receive quantum corrections. Even more, we find evidence that the causal structure of the non-relativistic dynamics together with particle number conservation constrain the theory to be one-loop exact.
The procedure of null reduction provides a concrete way of constructing field theories with Galilean invariance. We use this to examine Galilean gauge theories, viz. Galilean electrodynamics and Yang-Mills theories in spacetime dimensions 3 and 4. Different non-relativistic conformal symmetries arise in these contexts: Schr{ö}dinger symmetry in $d=3$ and Galilean conformal symmetry in $d=4$. A canonical analysis further reveals that the symmetries enhance to their infinite dimensional versions in phase space and pick up central extensions. In addition, for the Abelian theory, we discuss non-re…
We present the quantum field description of Galilean electrodynamics minimally coupled to massless Galilean fermion in (3 + 1) dimensions. At the classical level, the Lagrangian is obtained as a null reduction of a relativistic theory in one higher dimension. We use functional techniques to develop the quantum field description of the theory. Quantum corrections to the propagators and vertex are obtained upto first order and the theory is found to be renormalizable to this order. The beta function of the theory is found to grow linearly; the theory is not asymptotically free.
We construct a consistent model of Galileon scalar electrodynamics. The model satisfies three essential requirements: (1) The action contains higher-order derivative terms, and obey the Galilean symmetry, (2) Equations of motion also satisfy Galilean symmetry and contain only up to second-order derivative terms in the matter fields and, hence do not suffer from instability, and (3) local U(1) gauge invariance is preserved. We show that the non-minimal coupling terms in our model are different from that of the real scalar Galileon models; however, they match with the Galileon real scalar field …
We study renormalization group flow in a non-local version of quantum electrodynamics (QED). We determine the regime in which the theory flows to a local theory in the infrared and study a possible UV completion of four-dimensional QED. In addition, we find that there exist non-local conformal theories with a one-dimensional conformal manifold and non-local deformations of QED in three dimensions that are exactly marginal. Along the way we develop methods for coupling non-local derivatives to external sources and discuss unitarity and conformal vs. scale invariance of these theories.
We examine three versions of non-relativistic electrodynamics, known as the electric and magnetic limit theories of Maxwell's equations and Galilean electrodynamics (GED) which is the off-shell non-relativistic limit of Maxwell plus a free scalar field. For each of these three cases we study the couplings to non-relativistic dynamical charged matter (point particles and charged complex scalars). The GED theory contains besides the electric and magnetic potentials a so-called mass potential making the mass parameter a local function. The electric and magnetic limit theories can be coupled to tw…
We perform a detailed analysis of Galilean field theories, starting with free theories and then interacting theories. We consider non-relativistic versions of massless scalar and Dirac field theories before we go on to review our previous construction of Galilean Electrodynamics and Galilean Yang-Mills theory. We show that in all these cases, the field theories exhibit non-relativistic conformal structure (in appropriate dimensions). The surprising aspect of the analysis is that the non-relativistic conformal structure exhibited by these theories, unlike relativistic conformal invariance, beco…
The paper proposes an algorithm for regularization of the self-energy expressions for a Dirac particle that meets the relativistic and gauge invariance requirements.
We show that the general Lorentz- and CPT-violating extension of quantum electrodynamics is one-loop renormalizable. The one-loop Lorentz-violating beta functions are obtained, and the running of the coefficients for Lorentz and CPT violation is determined. Some implications for theory and experiment are discussed.
It has been conjectured in the literature that renormalizability of the $θ$-expanded noncommutative gauge theories improves when one takes into account full nonuniqueness of the Seiberg-Witten expansion, which relates noncommutative (`high-energy') with commutative (`low-energy') fields. In order to check this conjecture we analyze renormalizability of the noncommutative chiral electrodynamics: we quantize the action which contains all possible terms implied by the SW map. After renormalization we arrive at a different theory in which the relation between the coupling constants is changed. Thi…
It has been demonstrated that the effective potential V(ϕ) in a massless O(N) λϕ^4_4 model is determined completely by the renormalization group functions provided the renormalization condition \frac{d^4V}{d ϕ^4}|_{ϕ=μ}=λis used. This is shown to also hold in massless scalar electrodynamics. By employing a variant of the method of characteristics, the sums contributing to the leading-log, next-to-leading-log etc. contributions to V(ϕ) can be evaluated.
The effective equations of motion for a point charged particle taking account of radiation reaction are considered in various space-time dimensions. The divergencies steaming from the pointness of the particle are studied and the effective renormalization procedure is proposed encompassing uniformly the cases of all even dimensions. It is shown that in any dimension the classical electrodynamics is a renormalizable theory if not multiplicatively beyond d=4. For the cases of three and six dimensions the covariant analogs of the Lorentz-Dirac equation are explicitly derived.
Two-loop $β$-function and anomalous dimension are calculated for N=1 supersymmetric quantum electrodynamics, regularized by higher derivatives in the minimal subtraction scheme. The result for two-loop contribution to the $β$-function appears to be equal to 0, does not depend on the form of regularizing term and does not lead to anomaly puzzle. Two-loop anomalous dimension can be also made independent on parameters of higher derivative regularization by a special choice of subtraction scheme.
We formulate quantum electrodynamics on the basis of gauge (or BRST) covariant diffusion equations of fields. This is a particular example of the gradient flow exact renormalization group (GFERG). The resulting Wilson action fulfills a simple gauge Ward--Takahashi identity. We solve the GFERG equation around the Gaussian fixed point to the second order in gauge coupling and obtain the 1-loop beta function and anomalous dimensions. The anomalous dimension of the electron field coincides with that of the fermion field diffused by a gauge covariant flow equation of Lüscher.
In 1999, A. Connes and D. Kreimer have discovered the Hopf algebra structure on the Feynman graphs of scalar field theory. They have found that the renormalization can be interpreted as a solving of some Riemann -- Hilbert problem. In this work a generalization of their scheme to the case of quantum electrodynamics is proposed. The action of the gauge group on the Hopf algebra of diagrams are defined and the proof that this action is consistent with the Hopf algebra structure is given.
Starting from a formal Hamiltonian as found in the physics literature -- omitting photons -- we define a renormalized Hamiltonian through charge and mass renormalization. We show that the restriction to the one-electron subspace is well-defined. Our construction is non-perturbative and does not use a cut-off. The Hamiltonian is relevant for the description of the Lamb shift in muonic atoms.
The paper determines the anomalous magnetic moment and Lamb energy level shift in the second order of the perturbation theory using the algorithm of self-energy expression regularization in quantum electrodynamics that meets the relativistic and gauge invariance requirements; the comparison is made to the associated conventional quantum electrodynamics results. A limiting 4-impulse with an infinitely large temporal component and limited value of spatial components that depend quite weakly on the Dirac particle impulse variation is introduced within the algorithm. The answer in regard to the ag…
We study the renormalization flow of axion electrodynamics, concentrating on the non-perturbative running of the axion-photon coupling and the mass of the axion (like) particle. Due to a non-renormalization property of the axion-photon vertex, the renormalization flow is controlled by photon and axion anomalous dimensions. As a consequence, momentum-independent axion self-interactions are not induced by photon fluctuations. The non-perturbative flow towards the ultraviolet exhibits a Landau-pole-type behavior, implying that the system has a scale of maximum UV extension and that the renormaliz…
In this work, we employ algebraic renormalization technique to show the renormalizability to all orders in perturbation theory of the Lorentz and CPT violating QED. Essentially, we control the breaking terms by using a suitable set of external sources. Thus, with the symmetries restored, a perturbative treatment can be consistently employed. After showing the renormalizability, the external sources attain certain physical values, which allow the recovering of the starting physical action. The main result is that the original QED action presents the three usual independent renormalization param…
$ $In this paper we present a systematic treatment for fundamental renormalization of quantum electrodynamics in real space. Although the standard renormalization is an old school problem in this case, it has not yet been completely done in position space. The most important difference with well-known differential renormalization is that we do the whole procedure in coordinate space without need to transformation to momentum space. Specially, we directly derive the conterterms in real space. This problem becomes important when the translational symmetry of the system breaks somehow explicitly …
The BPHZL renormalization of a planar quantum electrodynamics up to 2-loops and beyond
Universidade Federal de Viçosa（维索萨联邦大学）；Ibitipoca Institute of Physics（伊比蒂波卡物理研究所）；Instituto Federal de Educação, Ciência e Tecnologia de Minas Gerais（米纳斯吉拉斯联邦教育、科学和技术学院）；Secretaria de Estado de Educação de Minas Gerais（米纳斯吉拉斯州教育厅）
Three-dimensional quantum electrodynamics exhibits a number of interesting properties, such as dynamical chiral symmetry breaking, weak confinement, and non-Fermi liquid behavior, and also has wide applications in condensed matter physics. We study the effects of random potentials, which exist in almost all realistic condensed-matter systems, on the low-energy behaviors of massless Dirac fermions by means of renormalization group method, and show that the role of random mass is significantly enhanced by the gauge interaction, whereas random scalar and vector potentials are insusceptible to the…
We consider the quantum loop effects in scalar electrodynamics on de Sitter space by making use of the functional renormalization group approach. We first integrate out the photon field, which can be done exactly to leading (zeroth) order in the gradients of the scalar field, thereby making this method suitable for investigating the dynamics of the infrared sector of the theory. Assuming that the scalar remains light we then apply the functional renormalization group methods to the resulting effective scalar theory and focus on investigating the effective potential, which is the leading order …
The possibility to construct inflationary models for the renormalization-group improved potentials corresponding to scalar electrodynamics and to $SU(2)$ and $SU(5)$ models is investigated. In all cases, the tree-level potential, which corresponds to the cosmological constant in the Einstein frame, is seen to be non-suitable for inflation. Rather than adding the Hilbert-Einstein term to the action, quantum corrections to the potential, coming from to the RG-equation, are included. The inflationary scenario is analyzed with unstable de Sitter solutions which correspond to positive values of the…
Burgers-Kardar-Parisi-Zhang (KPZ) scaling has recently (re-) surfaced in a variety of physical contexts, ranging from anharmonic chains to quantum systems such as open superfluids, in which a variety of random forces may be encountered and/or engineered. Motivated by these developments, we here provide a generalization of the KPZ universality class to situations with long-ranged temporal correlations in the noise, which purposefully break the Galilean invariance that is central to the conventional KPZ solution. We compute the phase diagram and critical exponents of the KPZ equation with $1/f$-…
We study three well known models of matter coupled to the ultraviolet cutoff, quantized radiation field and to the Coulomb potential of arbitrarily many nuclei. Two are nonrelativistic: the first uses the kinetic energy (p+eA(x))^2 and the second uses the Pauli-Fierz energy (p+eA(x))^2 +eσ\cdot B(x). The third, no-pair model, is relativistic and replaces the kinetic energy with the Dirac operator D(A), but restricted to its positive spectral subspace, which is the ``electron subspace''. In each case we are able to give an upper bound to the {\it binding} energy -- as distinct from the less dif…
Differential regularization is used to investigate the one-loop quantum corrections to Chern-Simons-Maxwell spinor and scalar electrodynamics. We illustrate the techniques to write the loop amplitudes in coordinate space. The short-distance expansion method is developed to perform the Fourier transformation of the amplitudes into momentum space and the possible renormalization ambiguity in Chern-Simons type gauge theories in terms of differential regularization is discussed. We also stress that the surface terms appearing in the differential regularization should be kept along for finite theor…
We consider a higher-derivative extension of QED modified by the addition of a gauge-invariant dimension-6 kinetic operator in the U(1) gauge sector. The Feynman diagrams at one-loop level are then computed. The modification in the spin-1 sector leads the electron self-energy and vertex corrections diagrams finite in the ultraviolet regime. Indeed, no regularization prescription is used to calculate these diagrams because the modified propagator always occurs coupled to conserved currents. Moreover, besides the usual massless pole in the spin-1 sector, there is the emergence of a massive one, …
The field theoretic renormalization study of reduced quantum electrodynamics (QED) is performed up to two loops. In the condensed matter context, reduced QED constitutes a very natural effective relativistic field theory describing (planar) Dirac liquids, e.g., graphene and graphene-like materials, the surface states of some topological insulators and possibly half-filled fractional quantum Hall systems. From the field theory point of view, the model involves an effective (reduced) gauge field propagating with a fractional power of the d'Alembertian in marked contrast with usual QEDs. The use …
In this work we use the framework of effective field theory to couple Einstein's gravity to scalar electrodynamics and determine the renormalization of the model through the study of physical processes below Planck scale, a realm where quantum mechanics and general relativity are perfectly compatible. We consider the effective field theory up to dimension six operators, corresponding to processes involving one graviton exchange. Studying the renormalization group functions we see that the beta function of the electric charge is positive and possesses no contribution coming from gravitational i…
Manifestly invariant renormalization scheme for supersymmetric gauge theories is proposed. This scheme is applied to supersymmetric quantum electrodynamics.
In this paper the nonperturbative analysis of the spectrum for one-particle excitations of the electron-positron field (EPF) is considered in the paper. A standard form of the quantum electrodynamics (QED) is used but the charge of the "bare" electron $e_0$ is supposed to be of a large value. It is shown that in this case the quasi-particle can be formed with a non-zero averaged value of the scalar component of the electromagnetic field (EMF). Self-consistent equations for the distribution of charge density in the "physical" electron (positron) are derived. A variational solution of these equa…
When using dimensional regularization, the bare couplings are expressed as a power series in (2 - n/2)^{-1} where n is the number of dimensions. It is shown how the renormalization group can be used to sum the leading pole, next to leading pole etc. contributions to these sums in scalar electrodynamics (or any theory with multiple couplings.)
Quantum parity conservation is verified at all orders in perturbation theory for a massless parity-even $U(1)\times U(1)$ planar quantum electrodynamics (QED$_3$) model. The presence of two massless fermions requires the Lowenstein-Zimmermann (LZ) subtraction scheme, in the framework of the Bogoliubov-Parasiuk-Hepp-Zimmermann-Lowenstein (BPHZL) renormalization method, in order to subtract the infrared divergences induced by the ultraviolet subtractions at 1- and 2-loops, however thanks to the superrenormalizability of the model the ultraviolet divergences are bounded up to 2-loops. Finally, it…
The Maxwell-Chern-Simons gauge theory with charged scalar fields is analyzed at two loop level. The effective potential for the scalar fields is derived in the closed form, and studied both analytically and numerically. It is shown that the U(1) symmetry is spontaneously broken in the massless scalar theory. Dimensional transmutation takes place in the Coleman-Weinberg limit in which the Maxwell term vanishes. We point out the subtlety in defining the pure Chern-Simons scalar electrodynamics and show that the Coleman-Weinberg limit must be taken after renormalization. Renormalization group ana…
We obtain analytical five-loop results for the renormalization group $β$-function of Quantum Electrodynamics with the single lepton in different renormalization schemes. The theoretical consequences of the results obtained are discussed.
In this paper is considered nonlinear electrodynamics (NE) which does not satisfy the linear superposition principle (LSP). Since the presentation of the special theory of relativity, it has been commonly accepted that a famous formula E = mc^2 = m_0c^2 /(1-v^2/c^2)^(1/2) expresses the energy of a free particle only. However, while studying the experiment for the annihilation of particle and antiparticle and the production of the photon in terms of the law of energy conservation of particle and field, we obtain a conclusion that E includes not only the energy of a free particle but also the en…
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.
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.
This paper presents divergent contributions of the radiative corrections for a Lorentz-violating extension of the scalar electrodynamics. We initially discuss some features of the model and extract the Feynman rules. Then we compute the one-loop radiative corrections using Feynman parametrization and dimensional regularization in order to evaluate the integrals. We also discuss Furry's theorem validity and renormalization in the present context.
We analyze quantization of noncommutative chiral electrodynamics in the enveloping algebra formalism in linear order in noncommutativity parameter $θ$. Calculations show that divergences exist and cannot be removed by ordinary renormalization, however they can be removed by the Seiberg-Witten redefinition of fields. Performing the redefinitions explicitly, we obtain renormalizable lagrangian and discuss the influence of noncommutativity on field propagation. Noncommutativity affects the propagation of chiral fermions only: half of the fermionic modes become massive and birefringent.
We consider the gauge invariance of the standard Yang-Mills model in the framework of the causal approach of Epstein-Glaser and Scharf and determine the generic form of the anomalies. The method used is based Epstein-Glaser approach to renormalization theory. In the case of quantum electrodynamics we obtain quite easily the absence of anomalies in all orders.
This article is an extension of the author's second master thesis [1]. It aims to introduce to the theory of perturbatively quantized General Relativity coupled to Spinor Electrodynamics, provide the results thereof and set the notation to serve as a starting point for further research in this direction. It includes the differential geometric and Hopf algebraic background, as well as the corresponding Lagrange density and some renormalization theory. Then, a particular problem in the renormalization of Quantum General Relativity coupled to Quantum Electrodynamics is addressed and solved by a g…
A classical field theory is proposed for the electric current and the electromagnetic field interpolating between microscopic and macroscopic domains. It represents a generalization of the density functional for the dynamics of the current and the electromagnetic field in the quantum side of the crossover and reproduces standard classical electrodynamics on the other side. The effective action derived in the closed time path formalism and the equations of motion follow from the variational principle. The polarization of the Dirac-see can be taken into account in the quadratic approximation of …
We investigate quantum longitudinal rescaling of electrodynamics, transforming coordinates as $x^{0,3}\toλx^{0,3}$ and $x^{1,2}\to x^{1,2}$, to one loop. We do this by an aspherical Wilsonian renormalization, which was applied earlier to pure Yang-Mills theory. We find the anomalous powers of $λ$ in the renormalized couplings. Our result is only valid for $λ\lesssim 1$, because perturbation theory breaks down for $λ\ll 1$.
We show that relativistic quantum electrodynamics in the Coulomb gauge satisfies the following bound, which establishes stability: let $H(Λ,V)$ denote the Hamiltonian of $QED_{1+3}$ on the three-dimensional torus of volume $V$ and with ultraviolet cutoff $Λ$. Then there exists a constant $0<μ(Λ,V)<\infty$ (the vacuum energy renormalization) such that the renormalized Hamiltonian is positive: $H_{ren}(Λ,V) \equiv H_{Λ,V}+μ_{Λ, V}\cdot \mathbb{1} \ge 0 $.
