F. S. Gama, J. R. Nascimento, A. Yu. Petrov, P. J. Porfirio
We investigate the spontaneous symmetry breaking in a nonlocal version of the scalar QED. When the mass parameter $m^2$ satisfies the requirement $m^2>0$, we find that all fields, including the Nambu-Goldstone field, acquire a non-zero mass dependent on the nonlocal scales. On the other hand, when $m^2=0$, we find that the nonlocal corrections to the masses are very small and can be neglected.
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The existence of the invariant measure in nonlocal regularized actions is discussed. It is shown that the measure for nonlocally regularized QED, as presented in\cite{Moff-Wood}, exists to all orders, and is precisely what is required to maintain gauge invariance at one loop and guarantees perturbative unitarity. We also demonstrate how the given procedure breaks down in anomalous theories, and discuss its generalization to other actions.
We show that the anomaly of nonlocal symmetry can be canceled by the well-known Wess-Zumino acton which cancels the gauge anomaly in the two-dimensional chiral electrodynamics.
The Letter reconsiders a result obtained by Chrétien and Peierls in 1954 within nonlocal QED in 4D [Proc. Roy. Soc. London A 223, 468]. Starting from secondly quantized fermions subject to a nonlocal action with the kernel $[ i\not\partial_x a(x) - m b(x)]$ and gauge covariantly coupled to an external U(1) gauge field they found that for $a = b$ the induced gauge field action cannot be made finite irrespectively of the choice of the nonlocality $a$ $(= b)$. But, the general case $a \neq b$ naturally to be studied admits a finitely induced gauge field action, as the present Letter demonstrates.
In this paper we use the Batalin-Fradkin-Vilkovisky formalism to study a recently proposed nonlocal symmetry of QED. In the BFV extended phase space we show that this symmetry stems from a canonical transformation in the ghost sector.
Although axial QED suffers from a gauge anomaly, gauge invariance may be maintained by the addition of a nonlocal counterterm. Such nonlocal conterterms, however, are expected to ruin unitarity of the theory. We explicitly investigate some relevant Feynman diagrams and show that, indeed, unitarity is violated, contrary to recent claims.
The well-known formula $det(A\cdot B)=\det A \cdot \det B$ can be easily proved for finite dimensional matrices but it may be incorrect for the functional determinants of differential operators, including the ones which are relevant for Quantum Field Theory applications. Considerable work has been done to prove that this equality can be violated, but in all previously known cases the difference could be reduced to renormalization ambiguity. We present the first example, where the difference between the two functional determinants is a nonlocal expression and therefore can not be explained by t…
Carroll-Field-Jackiw term in nonlocal Lorentz-violating QED
研究非局域洛伦兹违反量子电动力学中CFJ项的辐射产生，通过全非局域形状因子引入非局域性，利用导数展开推导诱导贡献，分析紫外行为，两类形状因子分别自然收敛或经阿贝尔正则化使CFJ系数有限，为控制紫外贡献提供自然框架。
The spontaneous magnetization in Chern-Simons QED_3 is discussed in a finite temperature system. The thermodynamical potential is analyzed within the weak field approximation and in the fermion massless limit. We find that there is a linear term with respect to the magnetic field with a negative coefficient at any finite temperature. This implies that the spontaneous magnetic field does not vanish even at high temperature. In addition, we examine the photon spectrum in the system. We find that the bare Chern-Simons coefficient is cancelled by the radiative effects. The photons then become topo…
We demonstrate how nonlocal regularization is applied to gauge invariant models with spontaneous symmetry breaking. Motivated by the ability to find a nonlocal BRST invariance that leads to the decoupling of longitudinal gauge bosons from physical amplitudes, we show that the original formulation of the method leads to a nontrivial relationship between the nonlocal form factors that can appear in the model.
Scalar QED is studied with higher order derivatives for the scalar field kinetic energy. A local potential is generated for the gauge field due to the covariant derivatives and the vacuum with non-vanishing expectation value for the scalar field and the vector potential is constructed in the leading order saddle point expansion. This vacuum breaks the global gauge and Lorentz symmetry spontaneously. The unitarity of time evolution is assured in the physical, positive norm subspace and the linearized equations of motion are calculated. Goldstone theorem always keeps the radiation field massless…
Spontaneous magnetization in the Maxwell QED$_{2+1}$ at finite fermion density is studied. It is shown that at low fermion densities the one-loop free energy has its minimum at some nonvanishing value of the magnetic field. The magnetization is due to the asymmetry of the fermion spectrum of the massive QED$_{2+1}$ in an external magnetic field.
We investigate the effects of thermal fluctuations on the spontaneous magnetic condensate in three dimensional QED coupled with P-odd Dirac fermions. Our results show that the phenomenon of the spontaneous generation of the constant background magnetic field survives to the thermal corrections even at infinite temperature. We also study the thermal corrections to the fermionic condensate in presence of the magnetic field.
Confinement, Nonlocality and Haag Duality Violation in the Algebraic Structure of 1+1D QED
本文在AQFT框架下直接构造1+1维QED的局域可观测量代数网，证明禁闭表现为DHR超选择扇区缺失，并揭示Haag对偶性破坏与纠缠楔重构失效的联系。
It is known that in the 2+1 dimensional quantum electrodynamics with Chern-Simons term, spontaneous magnetic field induces Lorentz symmetry breaking. In this paper, thermodynamical characters, especially the phase structure of this model are discussed. To see the behavior of the spontaneous magnetic field at finite temperature, the effective potential in the finite temperature system is calculated within the weak field approximation and in the fermion massless limit. We found that the spontaneous magnetic field never vanishes at any finite temperature. This result doesn't change even when the …
We study two-dimensional QED with unequal charges at finite temperature, and show that there is a phase with a spontaneously broken $Z_n$ symmetry. In spite of this, we were not able to establish the presence of domain walls. The relevance for QCD in higher dimensions is discussed.
We construct a relativistically covariant symmetry of QED. Previous local and nonlocal symmetries are special cases. This generalized symmetry need not be nilpotent, but nilpotency can be arranged with an auxiliary field and a certain condition. The Noether charge generating the symmetry transformation is obtained, and it imposes a constraint on the physical states.
A recent experiment with squeezed light has shown that two-photon absorption by an atom can occur with a linear intensity dependence. We point out that this result verifies a prediction made by us more than a decade ago from an analysis of a nonlocal model of QED. This model had earlier been proposed by us in an ad hoc manner to interpret certain features of multiphoton double ionization and above-threshold ionization in an atom placed in a strong laser field ; in this paper we show that the model can be obtained field- theoretically by demanding covariance of the field Lagrangian under a nonl…
We reconsider an alternative theory of the QED with the photon as a massless vector Nambu-Goldstone boson and show that the underlying spontaneous Lorentz violation caused by the vector field vacuum expectation value, while being superficial in gauge invariant theory, becomes physically significant in the QED with a tiny gauge non-invariance. This leads, through special dispersion relations appearing for charged fermions, to a new class of phenomena which could be of distinctive observational interest in particle physics and astrophysics. They include a significant change in the GZK cutoff for…
The vacuum structure of N = 2 linear supersymmetry (LSUSY) invariant QED, which is equivalent to N = 2 nonlinear supersymmetry (NLSUSY) model, is studied explicitly in two dimensional space-time (d = 2). Two different isometries SO(1,3) and SO(3,1) appear for the vacuum field configuration corresponding to the various parameter regions. Two different field configurations of SO(3,1) isometry describe the two different physical vacua, i.e. one breaks spontaneously both U(1) and SUSY and the other breaks spontaneously SUSY alone.
The electromagnetic field inside an isotropic optically active medium is quantized on the base of Fedorov's model for optical activity. The modified photon propagator is derived. Using this result it is shown that the QED correction to the electron mass contains chiral terms which nevertheless are parity invariant. The spontaneous emission rate of a two-level atom is shown to increase in the medium. The spontaneously emitted radiation is partially polarized.
It is by now clear that infrared sector of QED has an intriguingly complex structure. Based on earlier pioneering works on this subject, two of us recently proposed a simple modification of QED by constructing a generalization of the $U(1)$ charge group of QED to the "Sky" group incorporating the known spontaneous Lorentz violation due to infrared photons, but still compatible in particular with locality. There it was shown that the "Sky" group is generated by the algebra of angle dependent charges and a study of its superselection sectors has revealed a manifest description of spontaneous bre…
Nonlocal regularization of QED is shown to possess an axial anomaly of the same form as other regularization schemes. The Noether current is explicitly constructed and the symmetries are shown to be violated, whereas the identities constructed when one properly considers the contribution from the path integral measure are respected. We also discuss the barrier to quantizing the fully gauged chiral invariant theory, and consequences.
We study the connection of the chiral dynamics in QED and QCD in a strong magnetic field with noncommutative field theories (NCFT). It is shown that these dynamics determine complicated nonlocal NCFT. In particular, although the interaction vertices for electrically neutral composites in these gauge models can be represented in the space with noncommutative spatial coordinates, there is no field transformation that could put the vertices in the conventional form considered in the literature. It is unlike the Nambu-Jona-Lasinio (NJL) model in a magnetic field where such a field transformation c…
In this paper, we present a parity-preserving QED3 with spontaneous breaking of a local U(1)-symmetry. The breaking is accomplished by a potential of the \vf^6-type. It is shown that a net attractive interaction appears in the Møller scattering (s and p-wave scattering between two electrons) as mediated by the gauge field and a Higgs scalar. This might favour a pair-condensation mechanism.
We argue that a generic trigger for photon and other gauge fields to emerge as massless Nambu-Goldstone modes could be spontaneously broken supersymmetry rather than physically manifested Lorentz violation. We consider supersymmetric QED model extended by an arbitrary polynomial potential of vector superfield that induces the spontaneous SUSY violation in the visible sector. As a consequence, massless photon appears as a companion of massless photino being Goldstone fermion state in tree approximation. Remarkably, the photon masslessness appearing at tree level is further protected against rad…
It has been shown that QED in (1+4)-dimensional space-time, with the fifth dimension compactified on a circle, leads to CP violation (CPV). Depending on fermionic boundary conditions, CPV may be either explicit (through the Scherk--Schwarz mechanism), or spontaneous (via the Hosotani mechanism). The fifth component of the gauge field acquires (at the one-loop level) a non-zero vacuum expectation value. In the presence of two fermionic fields, this leads to spontaneous CPV in the case of CP-symmetric boundary conditions. Phenomenological consequences are illustrated by a calculation of the elec…
The phase diagram of massless quantum electrodynamics in three space-time dimensions as a function of fermion flavor number $N$ exhibits two well-known phases: at large $N > N_c^{conf}$ the system is in a conformal gapless state, while for small $N < N_c^{χSB}$ the fermions are expected to develop a dynamical mass due to spontaneous chiral symmetry breaking. Using $ε$ expansion near the lower critical dimension of 2, as well as the recent results on the generalization of the $F$ theorem to continuous dimension, we show that $N_c^{conf} > N_c^{χSB}$. There is therefore an intermediate range of …
The Bound State in QED is described in systematic way by means of nonlocal irreducible representations of the nonhomogeneous Poincare group and Dirac's method of quantization. As an example of application of this method we calculate triangle diagram $Para-Positronium \to γγ$. We show that the Hamiltonian approach to Bound State in QED leads to anomaly-type contribution to creation of pair of parapositronium by two photon.
We present a kind of model of quantum electrodynamics with nonlocal interaction, all the action and the equations of motion of charged particle and electromagnetic field are given. The main characteristics of the theory are: the model obeys the action principle; free charged particle and free electromagnetic field obey the Dirac equation and the Maxwell equation of free fields, respectively; for the case of interaction, both the equations of motion of charged particle and electromagnetic field lead to the normal current conservation naturally; the theory is Lorentz invariant and gauge invarian…
We calculate the effective action for Quantum Electrodynamics (QED) in D=2,3 dimensions at the quadratic approximation in the gauge fields. We analyse the analytic structure of the corresponding nonlocal boson propagators nonperturbatively in k/m. In two dimensions for any nonzero fermion mass, we end up with one massless pole for the gauge boson . We also calculate in D=2 the effective potential between two static charges separated by a distance L and find it to be a linearly increasing function of L in agreement with the bosonized theory (massive Sine-Gordon model). In three dimensions we fi…
We investigate (2+1)-dimensional QED coupled with Dirac fermions both at zero and finite temperature. We discuss in details two-components (P-odd) and four-components (P-even) fermion fields. We focus on P-odd and P-even Dirac fermions in presence of an external constant magnetic field. In the spontaneous generation of the magnetic condensate survives even at infinite temperature. We also discuss the spontaneous generation of fermion mass in presence of an external magnetic field.
Introducing a chemical potential in the functional method, we construct the effective action of QED$_3$ with a Chern-Simons term. We examine a possibility that charge condensation $\langleψ^\daggerψ\rangle$ remains nonzero at the limit of the zero chemical potential. If it happens, spontaneous magnetization occurs due to the Gauss' law constraint which connects the charge condensation to the background magnetic field. It is found that the stable vacuum with nonzero charge condensation is realized only when fermion masses are sent to zero, keeping it lower than the chemical potential. This resu…
The physics of spontaneous chiral symmetry breaking in the case of the simultaneous presence of a magnetic field and a fermionic quartic interaction is discussed for both local and nonlocal kernels in 2+1 and 3+1 dimensions. The approach is based on the use of Valatin-Bogoliubov canonical transformations, which allow, in the absence of fermionic quartic terms, to completely diagonalize the Hamiltonian and construct the vacuum state.
We investigate two-loop quantum corrections to non-minimally coupled Maxwell-Chern-Simons theory. The non-minimal gauge interaction represents the magnetic moment interaction between the charged scalar and the electromagnetic field. We show that the one-loop renormalizability of the theory found in previous work does not survive to the two-loop level. However, with an appropriate choice of the non-minimal coupling constant, it is possible to renormalize the two-loop effective potential and hence render it potentially useful for a detailed analysis of spontaneous symmetry breaking induced by ra…
Frohlich, Morchio and Strocchi long ago proved that Lorentz invariance is spontaneously broken in QED because of infrared effects. We develop a simple model where consequences of this breakdown can be explicitly and easily calculated. For this purpose, the superselected U(1) charge group of QED is extended to a superselected "Sky" group containing direction-dependent gauge transformations at infinity. It is the analog of the Spi group of gravity. As Lorentz transformations do not commute with Sky, they are spontaneously broken. These abelian considerations and model are extended to non-Abelian…
We investigate a lattice version of QED by numerical simulations. For the renormalized charge and mass we find results which are consistent with the renormalized charge vanishing in the continuum limit. A detailed study of the relation between bare and renormalized quantities reveals that the Landau pole lies in a region of parameter space which is made inaccessible by spontaneous chiral symmetry breaking.
Quantum electrodynamics (QED) of electrons confined in a plane and that yet can undergo interactions mediated by an unconstrained photon has been described by the so-called {\it pseudo-QED} (PQED), the (2+1)-dimensional version of the equivalent dimensionally reduced original QED. In this work, we show that PQED with a nonlocal Chern-Simons term is dual to the Chern-Simons Higgs model at the quantum level. We apply the path-integral formalism in the dualization of the Chern-Simons Higgs model to first describe the interaction between quantum vortex particle excitations in the dual model. This …
It might be expected that only global symmetries are fundamental symmetries of Nature, whereas local symmetries and associated massless gauge fields could solely emerge due to spontaneous breaking of underlying spacetime symmetries involved, such as relativistic invariance and supersymmetry. This breaking, taken in the form of the nonlinear sigma-model type pattern for vector fields or superfields, puts essential restrictions on geometrical degrees of freedom of a physical field system that makes it to adjust itself in such a way that its global internal symmetry G turns into the local symmetr…
We study the out-of-equilibrium properties of $1+1$ dimensional quantum electrodynamics (QED), discretized via the staggered-fermion Schwinger model with an Abelian $\mathbb{Z}_{n}$ gauge group. We look at two relevant phenomena: first, we analyze the stability of the Dirac vacuum with respect to particle/antiparticle pair production, both spontaneous and induced by an external electric field; then, we examine the string breaking mechanism. We observe a strong effect of confinement, which acts by suppressing both spontaneous pair production and string breaking into quark/antiquark pairs, indic…
It is shown that the chiral symmetry is spontaneously broken by a constant magnetic field in QED. The dynamical mass of fermions (energy gap in the fermion spectrum) is $m_{dyn}\simeq C\sqrt{eB}\exp\left[-\left(π/α\right) ^{1/2}\right]$, where $B$ is the magnetic field, the constant $C$ is of order one and $α=e^2/4π$ is the renormalized coupling constant. Possible applications of this effect are discussed.
After imposing current conservation together with the Gauss law as initial conditions on the Abelian Nambu model, we prove that the resulting theory is equivalent to standard QED in the non-linear gauge $\left(A_{μ}A^{μ}-n^{2}M^{2}\right)=0$, to all orders in perturbation theory. We show this by writing both models in terms of the same variables, which produce identical Feynman rules for the interactions and propagators. A crucial point is to verify that the Faddeev-Popov ghosts arising from the gauge fixing procedure in the QED sector decouple to all orders. We verify this decoupling by follo…
The spontaneous symmetry breaking (and Higgs) mechanism in the theory quantized on the light-front ({\it l.f.}), in the {\it discretized formulation}, is discussed. The infinite volume limit is taken to obtain the {\it continuum version}. The hamiltonian formulation is shown to contain a new ingredient in the form of nonlocal {\it constraint eqs.} which lead to a {\it nonlocal l.f. Hamiltonian}. The description of the broken symmetry here has the same physical content as in the conventional formulation though arrived at through a different mechanism.
The coupled system of renormalized Dyson-Schwinger equations for the electron self-energy and the photon propagator are supplied with the tree level vertex as Ansatz for the renormalized three point function. The system is investigated numerically. In the case of a massive electron, the theory is ``weakly renormalizable'', i.e.\ cutoff independent for values of the cutoff below an upper limit. In this regime of cutoff independence, the quenched approximation yields good results for the electron self-energy. In the chiral limit, a logarithmic cutoff dependence of the electron self-energy is fou…
We introduce a 3D compact U(1) lattice gauge theory having nonlocal interactions in the temporal direction, and study its phase structure. The model is relevant for the compact QED$_3$ and strongly correlated electron systems like the t-J model of cuprates. For a power-law decaying long-range interaction, which simulates the effect of gapless matter fields, a second-order phase transition takes place separating the confinement and deconfinement phases. For an exponentially decaying interaction simulating matter fields with gaps, the system exhibits no signals of a second-order transition.
Dynamical symmetry breaking in three dimensional QED with $N$ flavors, which has been mostly analyzed by solving the Schwinger-Dyson equations, is investigated by means of the approximated Wilson, or non-perturbative, renormalization group (RG). We study the RG flows of the gauge coupling and the general four-fermi couplings allowed by the symmetry with concentrating our interest on study of the phase structure. The RG equations have no gauge parameter dependence in our approximation scheme. It is found that there exist chirally broken and unbroken phases for $N > N_{\rm cr}$ ($3 < N_{\rm cr} …
We investigate textbook QED in the framework of the exact renormalization group. In the strong-coupling region, we study the influence of fluctuation-induced photonic and fermionic self-interactions on the nonperturbative running of the gauge coupling. Our findings confirm the triviality hypothesis of complete charge screening if the ultraviolet cutoff is sent to infinity. Though the Landau pole does not belong to the physical coupling domain owing to spontaneous chiral symmetry breaking (chiSB), the theory predicts a scale of maximal UV extension of the same order as the Landau pole scale. In…
We study and analyse the questions regarding breakdown of global symmetry on noncommutative sphere. We demonstrate this by considering a complex scalar field on a fuzzy sphere and isolating Goldstone modes. We discuss the role of nonlocal interactions present in these through geometrical considerations.
We consider violation of Lorentz invariance in QED induced by a very high frequency background wave. An effective theory is obtained by averaging observables over the rapid field oscillations. This preserves Ward identities and restores translation invariance below the high frequency scale, but only partial Lorentz invariance: we show that the effective theory is C-invariant SIM(2)-QED in Very Special Relativity. Averaging generates the nonlocal terms familiar from SIM(2) theories, while the short-distance behaviour of the background field fermion propagator generates the infinite number of hi…
Comments on QED$_3$ in a Magnetic Field
本文研究2+1维磁场中无质量狄拉克费米子与光子相互作用的低能动力学，发现真空自发破缺全局对称性，并确定激发谱和低能有效作用量，方法借鉴量子霍尔铁磁体技术。
Staggered fermions are constructed for the transverse lattice regularization scheme. The weak perturbation theory of transverse lattice non-compact QED is developed in light-cone gauge, and we argue that for fixed lattice spacing this theory is ultraviolet finite, order by order in perturbation theory. However, by calculating the anomalous scaling dimension of the link fields, we find that the interaction Hamiltonian becomes non-renormalizable for $g^2(a) > 4π$, where $g(a)$ is the bare (lattice) QED coupling constant. We conjecture that this is the critical point of the chiral symmetry breaki…
Dynamical symmetry breaking in three-dimensional QED with N fermion flavours is considered at finite temperature, in the large $N$ approximation. Using an approximate treatment of the Schwinger-Dyson equation for the fermion self-energy, we find that chiral symmetry is restored above a certain critical temperature which depends itself on $N$. We find that the ratio of the zero-momentum zero-temperature fermion mass to the critical temperature has a large value compared with four-fermion theories, as had been suggested in a previous work with a momentum-independent self-energy. Evidence of a te…
We show that QED in the Coulomb gauge can be considered as a low energy linear approximation of a non-linear $σ$-type model where the photon emerges as a vector Goldstone boson related to the spontaneous breakdown of Lorentz symmetry down to its spatial rotation subgroup at some high scale $M$. Starting with a general massive vector field theory one naturally arrives at this model if the pure spin-1 value for the vector field $A_μ(x)$ provided by the Lorentz condition $\partial_μA_μ(x)=0$ is required. The model coincides with conventional QED in the Coulomb gauge in the limit of M going to inf…
It is shown that an alternative to the standard scalar QED is possible. In this new version there is only global gauge invariance as far as the charged scalar fields are concerned although local gauge invariance is kept for the electromagnetic field. The electromagnetic coupling has the form $j_μ(A^μ +\partial^μB)$ where $B$ is an auxiliary field and the current $j_μ$ is $A_μ$ independent so that no "sea gull terms" are introduced. In a model of this kind spontaneous breaking of symmetry does not lead to photon mass generation, instead the Goldstone boson becomes a massless source for the elec…
The problem of gauge invariance in an ultraviolet complete quantum field theory (QFT) with nonlocal interactions is investigated. For local fields that couple through a nonlocal interaction, it is demonstrated that the quantum electrodynamic (QED) sector of the electroweak (EW) model without a Higgs particle is gauge invariant. The non-Abelian QFT with massless gluons and nonlocal interactions is shown to be gauge invariant, and a perturbative S-matrix formalism in the interaction representation is investigated. The finite self-energy of a fermion and vacuum polarization are derived to first o…
We discuss one-loop radiatively-induced magnetic moment in four-dimensional quantum electrodynamics (QED) with anisotropic coupling, and examine various cases which may be of interest in effective gauge theories of antiferromagnets, whose planar limit coresponds to highly anisotropic QED couplings. We find a different scaling with the magnetic field intensity in case there are extra statistical gauge interactions in the model with spontaneous symmetry breaking. Such a case is encountered in the CP-1 sigma-model sector of effective spin-charge separated gauge theories of antiferromagnetic syste…
We study dynamical symmetry breaking in three-dimensional QED with a Chern-Simons (CS) term, considering the screening effect of $N$ flavor fermions. We find a new phase of the vacuum, in which both the fermion mass and a magnetic field are dynamically generated, when the coefficient of the CS term $κ$ equals $N e^2/4 π$. The resultant vacuum becomes the finite-density state half-filled by fermions. For $κ=N e^2/2 π$, we find the fermion remains massless and only the magnetic field is induced. For $κ=0$, spontaneous magnetization does not occur and should be regarded as an external field.
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