Physical Propagator Research Articles

Physical and unphysical regimes of self-consistent many-body perturbation theory

In the standard framework of self-consistent many-body perturbation theory, the skeleton series for the self-energy is truncated at a finite order N and plugged into the Dyson equation, which is then solved for the propagator G_NGN. We consider two examples of fermionic models, the Hubbard atom at half filling and its zero space-time dimensional simplified version. First, we show that G_NGN converges when N\to∞N→∞ to a limit G_∞\,G∞, which coincides with the exact physical propagator G_{exact}Gexact at small enough coupling, while G_∞ ≠ G_{exact}G∞≠Gexact at strong coupling. This follows from the findings of [Phys. Rev. Lett. 114, 156402 (2015)] and an additional subtle mathematical mechanism elucidated here. Second, we demonstrate that it is possible to discriminate between the G_∞=G_{exact}G∞=Gexact and G_∞≠G_{exact}G∞≠Gexact regimes thanks to a criterion which does not require the knowledge of G_{exact}Gexact, as proposed in [Phys. Rev. B 93, 161102 (2016)].

Mean-field description of aging linear response in athermal amorphous solids

We study the linear response to strain in a mean-field elastoplastic model for athermal amorphous solids, incorporating the power-law mechanical noise spectrum arising from plastic events. In the ``jammed'' regime of the model, where the plastic activity exhibits a nontrivial slow relaxation referred to as aging, we find that the stress relaxes incompletely to an age-dependent plateau, on a timescale which grows with material age. We determine the scaling behavior of this aging linear response analytically, finding that key scaling exponents are universal and independent of the noise exponent $\ensuremath{\mu}$. For $\ensuremath{\mu}>1$, we find simple aging, where the stress relaxation timescale scales linearly with the age ${t}_{\mathrm{w}}$ of the material. At $\ensuremath{\mu}=1$, which corresponds to interactions mediated by the physical elastic propagator, we find instead a ${t}_{\mathrm{w}}^{1/2}$ scaling arising from the stretched exponential decay of the plastic activity. We compare these predictions with measurements of the linear response in computer simulations of a model jammed system of repulsive soft athermal particles, during its slow dissipative relaxation towards mechanical equilibrium, and find good agreement with the theory.

On direct integration for mirror curves of genus two and an almost meromorphic Siegel modular form

This work considers aspects of almost holomorphic and meromorphic Siegel modular forms from the perspective of physics and mathematics. The first part is concerned with (refined) topological string theory and the direct integration of the holomorphic anomaly equations. Here, a central object to compute higher genus amplitudes, which serve as the generating functions of various enumerative invariants, is provided by the so-called propagator. We derive a universal expression for the propagator for geometries that have mirror curves of genus two which is given by the derivative of the logarithm of Igusa's cusp form of weight 10. In addition, we illustrate our findings by solving the refined topological string on the resolutions of the three toric orbifolds of order three, five and six. In the second part, we give explicit expressions for lowering and raising operators on Siegel modular forms, and define almost holomorphic Siegel modular forms based on them. Extending the theory of Fourier-Jacobi expansions to almost holomorphic Siegel modular forms and building up on recent work by Pitale, Saha, and Schmidt, we can show that there is no analogue of the almost holomorphic elliptic second Eisenstein series. In the case of genus 2, we provide an almost meromorphic substitute for it. This, in particular, leads us to a generalization of Ramanujan's differential equation for the second Eisenstein series. The two parts are intertwined by the observation that the meromorphic analogue of the almost holomorphic second Eisenstein series coincides with the physical propagator. In addition, the generalized Ramanujan identities match precisely the physical consistency conditions that need to be imposed on the propagator.

Scattering theory for two-body quantum systems with singular potentials in a time-periodic electric field

Recently, the first author [Adachi, “Asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field,” Commun. Math. Phys. 275, 443 (2007)] proved the asymptotic completeness for N-body quantum systems with long-range interactions in a time-periodic electric field whose mean in time is nonzero by obtaining propagation estimates for the physical propagator. However, in his work, it is needed that potentials under consideration are sufficiently smooth. In this paper, when N=2, we prove the asymptotic completeness of (modified) wave operators under the assumption that the potential has the local singularity of type |x|−1+ϵ when d≥3 and 0<ϵ<1, where d is the space dimension. We also discuss the modifiers in the position representation, which are used in the definition of the modified wave operators in the long-range case.

Equation of state of gluon plasma and renormalization of local action

We consider a local, renormalizable, BRST-invariant action for QCD in Coulomb gauge that contains auxiliary bose and fermi ghost fields. It possess a non-perturbative vacuum that spontaneously breaks BRST-invariance. The vacuum condition leads to a gap equation that introduces a mass scale. Calculations are done to one-loop order in a perturbative expansion about this vacuum. They are free of the finite-T infrared divergences found by Linde and which occur in the order g6 corrections to the Stefan-Boltzmann equation of state. We obtain a finite result for these corrections. Renormalization and renormalization-group flow are described. We calculate the ghost propagator and color-Coulomb potential to one-loop and find that they are long range, whereas the 3-dimensionally transverse would-be physical gluon propagator is suppressed like k2 at small |k|. These one-loop results accord with the Gribov scenario in Coulomb gauge and with recent numerical determinations of these quantities. When the auxiliary fields are integrated out, one obtains the standard Coulomb gauge action with a cut-off at the Gribov horizon.

Gauge-invariant dressed fermion propagator in massless QED3

The infrared behaviour of the gauge-invariant dressed fermion propagator in massless QED3 is discussed for three choices of dressing. It is found that only the propagator with the isotropic (in three Euclidean dimensions) choice of dressing is acceptable as the physical fermion propagator. It is explained that the negative anomalous dimension of this physical fermion does not contradict any field-theoretical requirement.

Confinement and gluon propagator in Coulomb gauge QCD

We consider the role of the Faddeev-Popov determinant in the Coulomb gauge on the confinement properties of the QCD vacuum. We show that the determinant is needed to regularize the otherwise divergent functional integrals near the Gribov horizon but still enables large field configurations to generate IR enhanced running coupling. The physical gluon propagator is found to be strongly suppressed in the IR consistent with expectations from lattice gauge calculations.

Anomalous dimensions of gauge-invariant amplitudes in massless effective gauge theories of strongly correlated systems

We use the radial gauge to calculate the recently proposed ansatz for the physical electron propagator in such effective models of strongly correlated electron systems as the $QED_3$ theory of the pseudogap phase of the cuprates. The results of our analysis help to settle the recent dispute about the sign and the magnitude of the anomalous dimension which characterizes the gauge invariant amplitude in question and set the stage for computing other, more physically relevant, ones.

Elusive physical electron propagator in QED-like effective theories

We study the previously conjectured form of the physical electron propagator and its allegedly Luttinger type of behavior in the theory of the pseudogap phase of high-temperature copper-oxide superconductors and other effective QED-like models. We demonstrate that, among a whole family of seemingly gauge-invariant functions, the conjectured “stringy ansatz” for the electron propagator is the only one that is truly invariant. However, contrary to the results of the earlier works, it appears to have a negative anomalous dimension, which makes it a rather poor candidate to the role of the physical electron propagator. Instead, we argue that the latter may, in fact, feature a “super-Luttinger” behavior characterized by a faster than any power-law decay G phys( x)∝exp(−const·ln 2| x|).

Gauge-invariant Green functions of Dirac fermions coupled to gauge fields

We develop a unified approach to both infrared and ultraviolet asymptotics of the fermion Green functions in the condensed matter systems that allow for an effective description in the framework of the Quantum Electrodynamics. By applying a path integral representation to the previously suggested form of the physical electron propagator we demonstrate that in the massless case this gauge invariant function features a "stronger-than-a-pole" branch-cut singularity instead of the conjectured Luttinger-like behavior. The obtained results alert one to the possibility that construction of physically relevant amplitudes in the effective gauge theories might prove more complex than previously thought.

Confinement made simple in the Coulomb gauge

In Gribov's scenario in Coulomb gauge, confinement of color charge is due to a long-range instantaneous color-Coulomb potential V( R). This may be determined numerically from the instantaneous part of the gluon propagator D 44,inst = V( R)δ( t). Confinement of gluons is reflected in the vanishing at k = 0 of the equal-time three-dimensionally transverse would-be physical gluon propagator D tr( k). We present exact analytic results on D 44 and D tr (which have also been investigated numerically, A. Cucchieri, T. Mendes, and D. Zwanziger, this conference), in particular the vanishing of D tr( k) at k = 0, and the determination of the running coupling constant from x 0 g 2( k) = k 2 D 44,inst, where x 0 = 12 N/(11 N − 2 N f ).

Elusive physical electron propagator in QED-like effective theories

Elusive physical electron propagator in QED-like effective theories

Fit to gluon propagator and Gribov formula

We report a numerical study of SU(2) lattice gauge theory in the minimal Coulomb gauge at beta = 2.2 and 9 different volumes. After extrapolation to infinite volume, our fit agrees with a lattice discretization of Gribov's formula for the transverse equal-time would-be physical gluon propagator, that vanishes like |k| at k = 0, whereas the free equal-time propagator (2|k|)^{-1} diverges. Our fit lends reality to a confinement scenario in which the would-be physical gluons leave the physical spectrum while the long-range Coulomb force confines color.

Geometro-Differential Conception of Extended Particles and the Semigroup of Trajectories in Minkowski Space-Time

The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of extended particles with internal Poincare symmetry and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup.

The physical propagator of a slowly moving charge

We consider an electron which is electromagnetically dressed in such a way that it is both gauge invariant and that it has the associated electric and magnetic fields expected of a moving charge. We study the propagator of this dressed electron and, for small velocities, show explicitly at one loop that at the natural (on-shell) renormalisation point, p 0 = m, p = mv, one can renormalise the propagator multiplicatively. Furthermore the renormalisation constants are infra-red finite. This shows that the dressing we use corresponds to a slowly moving, physical asymptotic field.

On the physical propagators of QED

The true variables in QED are the transverse photon components and Dirac's physical electron, constructed out of the fermionic field and the longitudinal components of the photon. We calculate the propagators in terms of these variables to one loop and demonstrate their gauge invariance. The physical electron propagator is shown not to suffer from infrared divergences in any gauge. In general, all physical Green functions are gauge invariant and infrared-finite.

Effect of the gluon mass on dynamical chiral-symmetry breaking in QCD.

We incorporate a physical gluon propagator containing a dynamical gluon mass into the Schwinger-Dyson equation for the effective quark propagator in QCD, from which we find solutions to the dynamical quark mass $B(p)$. Numerical results showing the dependence of $B(0)$ on the gluon mass $m$ indicate that the chiral-symmetry phase transition takes place for $m<\mathcal{M}$ (where $\mathcal{M}$ is the QCD renormalization-group mass).

Many-body propagators of a self-coupled spinor field in Edwards-Lieb’s approximation

The continuous integral representation of the complete, many-body propagators of a self-coupled spinor field is deduced and investigated. The continuous integrals are calculated by means ofEdwards—Lieb’s non-perturbative approximation method. In first approximation, the physical one-body propagator agrees with the propagator of a free, dressed particle and the physical two-body propagator corresponds to the one obtained in the “closed-loop chain” approximation ofAbrikosov et al.