Inverse Propagator Research Articles

An overview of the inverse problem for sound reconstruction in interior spaces

Calculation of the sound field between noise sources and a closed two-dimensional measurement surface is an inverse propagation problem. A powerful and popular method for the reconstruction of sound fields in this space is based on boundary element methods applied to the Helmholtz integral equation. Unfortunately, due to the numerical nature of these solutions, the physics associated with backtracking (backpropagating) the field from the measurement surface to the source surfaces is hidden. Analytic models are used to view this backpropagation as a deconvolution operation on the measured field. The nature of this deconvolving function (called the inverse propagator) is examined; it is a singular function, it is spatially local, and it unsoothes the measured pressure field. An understanding of the nature of the inverse propagator is critical to successful solution of the inverse problem by BEM methods (especially the regularization process), as well as the sucessful development of three-dimensional sound projection methods for sound synthesis. Furthermore, this understanding leads to new insights into the solution of the inverse problem. [Work supported by ONR and NASA.]

Finite-size effects and operator product expansions in a CFT for d>2

The large momentum expansion for the inverse propagator of the auxiliary field λ( x) in the conformally invariant O( N) vector model is calculated to leading order in 1/ N, in a strip-like geometry with one finite dimension of length L for 2< d<4. Its leading terms are identified as contributions from λ( x) itself and the energy momentum tensor, in agreement with a previous calculation based on conformal operator product expansions. It is found that a non-trivial cancellation takes place by virtue of the gap equation. The leading coefficient of the energy momentum tensor contribution is shown to be related to the free energy density.

On Ward-Takahashi identities for the Parisi spin glass

The introduction of ``small permutations'' allows us to derive Ward-Takahashi identities for the spin-glass, in the Parisi limit of an infinite number of steps of replica symmetry breaking. The first identities express the emergence of a band of Goldstone modes. The next identities relate components of (the Replica Fourier Transformed) 3-point function to overlap derivatives of the 2-point function (inverse propagator). A jump in this last function is exhibited, when its two overlaps are crossing each other, in the special simpler case where one of the cross-overlaps is maximal.

Corrections to finite-size scaling in two-dimensional O( N) σ-models

We have considered the corrections to the finite-size-scaling functions for a general class of O(N) σ-models with two-spin interactions in two dimensions for N = ∞. We have computed the leading corrections finding that they generically behave as ( f( z) log L + g( z))/ L 2 where z = m( L) L and m( L) is a mass scale; f( z) vanishes for Symanzik improved actions for which the inverse propagator behaves as q 2 + O( q 6), for small q, but not for on-shell improved ones. We also discuss a model with four-spin interactions which shows a much more complicated behaviour.

One-way representations of seismic data

A general one-way representation of seismic data can be obtained by substituting a Green's one-way wavefield matrix into a reciprocity theorem of the convolution type for one-way wavefields. From this general one-way representation, several special cases can be derived. By introducing a Green's one-way wavefield matrix for primaries, a generalized Bremmer series representation is obtained. Terminating this series after the first-order term yields a primary representation of seismic reflection data. According to this representation, primary seismic reflection data are proportional to a reflection operator, ‘modified’ by primary propagators for downgoing and upgoing waves. For seismic imaging, these propagators need to be inverted. Stable inverse primary propagators can easily be obtained from a one-way reciprocity theorem of the correlation type. By introducing a Green's one-way wavefield matrix for generalized primaries, an alternative representation is obtained in which multiple scattering is organized quite differently (in comparison with the generalized Bremmer series representation). According to the generalized primary representation, full seismic reflection data are proportional to a reflection operator, ‘modified’ by generalized primary propagators for downgoing and upgoing waves. Internal multiple scattering is fully included in the generalized primary propagators {either via a series expansion or in a parametrized way). Stable inverse generalized primary propagators can be obtained from the one-way reciprocity theorem of the correlation type. These inverse propagators are the nucleus for seismic imaging techniques that take the angle-dependent dispersion effects due to fine-layering into account.

Self-consistent couplings of pions and Δ-hole states in nuclear matter

Dyson equations for the propagation of pions and Δ's in nuclear matter are solved self-consistently using extended hadronic vertex functions to regularize the ultraviolet divergences of the loop integrals. The real part of the pion inverse propagator vanishes at only one energy for each momentum, unlike the conventional picture of level mixing and level repulsion for the pionic and Δ-hole states, because of the width of the Δ-hole excitations. This self-consistent Δ-hole model is compared to data on charge-exchange reactions and used to study the absorption of photons by nuclei in the Δ-resonance region. The comparison of the baryonic vertex form factors obtained for pionic and electromagnetic probes agrees with their interpretation as effective hadronic structure functions.

The nonabelian screening potential beyond the leading order

The nonabelian screening potential is calculated in the temporal axial gauge. The Slavnov-Taylor identity is used to construct the three-gluon vertex function from the inverse gluon propagator. After solving the Schwinger-Dyson equation beyond leading order we find that the obtained momentum dependence of the gluon self-energy at high temperature does not correspond to an attractive QCD-Debye potential, but instead it is repulsive and power behaved ( ⋍ 1/r 6 ) at large distance.

Gauge invariance and the electromagnetic current of composite pions.

The Global Color-symmetry Model of QCD is extended to deal with a background electromagnetic field and the associated conserved current is identified for the finite size q\ifmmode\bar\else\textasciimacron\fi{}q pion modes at tree level. We work within a well-defined truncation that factorizes the bilocal pion field into a local field variable and a hadronic form factor having a ladder Bethe-Salpeter content. The associated pion charge form factor is formulated. These developments are used to provide an illustration of how an effective hadronic action containing form factors may be electrmagnetically coupled in a gauge invariant way that is accountable to its field substructure. In particular, the Ward-Takahashi identity for the photon vertex appropriate to the localized pion fields is seen to contain the hadronic form factors. In this context, gauge invariance of the effective hadronic action also requires recognition of the fact that the free inverse propagator for the localized pion field gauge transforms due to the substructure field content that has been absorbed into it.

Nonrenormalizability of the topological charge as a result of the index theorem for the inverse gluon propagator

It is shown that in Yang-Mills theory the vanishing of the radiative corrections to the matrix element of the topological charge is a manifestation of the index theorem for the operator, which enters the bilinear part of the Yang-Mills action in the background field gauge.

Mass-gap equations within the QED framework: II. Closed sets of mass equations

The QED Dyson-Schwinger equation for the inverse fermion propagator is examined for composite fermions. The photon couples to the constituent scalar of the fermions constituting the radial excitations of a hypercolour composite. A set of nonlinear mass-gap equations for the individual states is found. After having estimated the contribution of the intermediate states lying above the compositeness scale Lambda , this set is solved for a number of states below Lambda ranging from 10 to 14. Though the most elementary estimate is chosen, the results are seen to confirm those of part I: the mass spectra exhibit both the observed large mass ratios and the electron mass scale if the running coupling vanishes at hypercolour energies. In the case of a sharp cutoff, the precise values of the fine-structure constant and the electron-muon mass ratio can be reproduced; fourteen states below Lambda are predicted for Lambda approximately=1015 GeV. The muon-tau mass ratio comes out to be too large and, therefore, nonsharp cutoffs are studied. The complete solution of the mass equations is presented. This reveals a pronounced decrease of the muon-tau mass ratio. The effect is independent of the kind of non-sharp cutoff chosen and persists when extending the mass loop integration to larger values of q2 or when varying the number of states below Lambda while keeping alpha at 1/137 and m1/m0 at 207. Extending the loop integration to still larger values of q2, distinctive effects of 'spectral inversion' are found, in particular, the effect mL/m tau >tau /mmu (mL denotes the fourth charged lepton mass). These results suggest the existence of 11-12 charged lepton states below Lambda approximately=1015 GeV and 40 GeV<mL<50 GeV. Sequential Dirac neutrinos should exist above MZ/2.

Minimal transverse operators in string field theory

Gives a new method for deriving transverse inverse propagators in string field theory from the knowledge of null secondary vectors in (super) conformal field theory in two dimensions. The author proves that the locality of such operators is equivalent to the vanishing of the singular subset of the null secondary vectors at particular eigenvalues of the Hamiltonian H=L0. Furthermore, the author gives a new method of deriving the 'minimal' inverse propagators introduced originally by Friedan (1985).

Quantum Regge calculus in the Lorentzian domain and its Hamiltonian formulation

The author sets up a formalism for quantum Regge calculus in the Lorentzian domain, calculating the inverse propagator in the free field case. The author relates the variables in the Arnowitt-Deser-Misner 3+1 formulation of general relativity to the Regge calculus variables.

Dynamics of the Gribov horizon: A one-mode treatment

Properties of the Gribov horizon in Coulomb-gauge nonabelian Yang-Mills theory are studied in an approximation which retains global color symmetry but truncates the spin-space degrees of freedom of the transverse gauge field to a single mode. The facilities an explicit evaluation of the contribution of the Faddeev-Popov zeroes to the effective quantum potential, which is accomplished with a sturmian expansion in the zero modes of the inverse ghost propagator. In particular, the singularities of the one-mode hamiltonian at the horizon are isolated and analyzed for their effect on the spectrum. Quantizing on the compact manifold S 3 and utilizing the hyperspherical techniques developed by Cutkosky, the formalism is applied to pure gluonic systems of long-wave-length modes. It is found that in the limit a cutoff on integration over ghost momenta is removed to infinity, the hamiltonian reduces to a trivial (quadratic) theory with a mass gap which is a simple multiplicative modification of the free-field result.

Continuous regularizations in strong coupling expansions from a generalized inverse propagator method

A strong coupling expansion is derived in terms of a generalized inverse propagator and is used to construct a regularization scheme which, for some regulators, differs from the canonical one. In particular, the Gaussian cut-off is shown to induce unbiased results, in contrast to a previous analysis using this regulator in the standard way. Several continuous and discrete regulations are then compared through a numerical example which suggests their equivalence.

Continuum strong-coupling expansion for quantum electrodynamics

We derive from the path integral a continuum strong-coupling expansion for QED in $d$-dimensional Euclidean space-time. It is a double expansion in the fermion and boson kinetic energy (inverse free propagators), which leads to a double power series for the Green's functions of the cutoff theory in terms of $\frac{1}{{e}^{2}}$ and $\frac{{\ensuremath{\Lambda}}^{2}}{{M}^{2}}$. $\ensuremath{\Lambda}$ is a smooth cutoff in Euclidean momentum space, and $M$ is an infrared regulator mass for the photons needed to define the local part of the path integral. We demonstrate how dimensional continuation is necessary to control the broken gauge invariance of the cutoff theory. Restricting to $d=2$ (the Schwinger model) we show how to remove the cutoff using Pad\'e approximants. We find some evidence that as $\frac{{\ensuremath{\Lambda}}^{2}}{{M}^{2}}\ensuremath{\rightarrow}\ensuremath{\infty}$ gauge invariance is restored and we calculate the vector-meson mass, keeping the first three terms in the expansion in powers of the bare photon inverse propagator.

Continuum regulation of the strong-coupling expansion for quantum field theory

We study two continuum methods of regulating the formal strong-coupling expansion of the Green's functions, obtained by expanding the path integral in powers of the kinetic energy (inverse free propagator). Our continuum regulations amount to introducing either a hard ($\ensuremath{\theta}$ function) or soft (Gaussian) cutoff $\ensuremath{\Lambda}$ in momentum space. The cutoff takes the place of the usual spatial cutoff, the lattice spacing, which arises when the path integral is defined as the continuum limit of ordinary integrals on a Euclidean space-time lattice. We find, by investigating free field theory and $g{\ensuremath{\varphi}}^{4}$ field theory in one dimension, that the $\ensuremath{\theta}$-function regulation is more accurate than the Gaussian and, unlike the Gaussian, preserves certain continuum Green's-function identities. The extension to field theories with fermions is trivial and we give the strong-coupling graphical rules for an arbitrary field theory with fermions and bosons in $d$ dimensions.

Three-particle structure ofφ4interactions and the scaling limit

We show that the inverse propagator $\ensuremath{\Gamma}(x)$ and the proper self-energy part $\ensuremath{\Pi}(x)$ are positive for all $x\ensuremath{\ne}0$. These results combined with some postulates on the sign or the decay of the vertex functions, in particular, on the decay rate for $\ensuremath{\Gamma}(x)$, yield the absence of three-particle bound states and the existence of the scaling (Gell-Mann, Low, Wilson) limit for the ${\ensuremath{\varphi}}^{4}$ interaction.

Direct and inverse time-evolution propagators. General properties and application to plasma turbulence

Abstract. — The solution of a general evolution equation [formule] — describing the time evolution of a function ƒ in terms of a general time-dependent differential operator C(0 — is given in terms of time evolution operators. The explicit solution for the direct and inverse propagators are shown to satisfy separately a semi-group property. A simple prescription concerning the time-ordering allows us to define a general operator, describing both direct and inverse time propagations, satisfying the group property. An interaction representation is given to derive useful expressions involving new operators which are dressed either by an external force, or by a free propagation. As an application, useful compact expressions are obtained for the free propagator in a constant magnetic field and for the propagator of Vlasov plasma turbulence. On the other hand, two examples are given to show how this interaction representation can be used to define two new approximations which could be useful to go beyond the usual weak coupling approximation of plasma turbulence.

Spontaneous Breaking of Chiral Symmetry in a Vector-Gluon Model

Solutions of the self-consistent equation for fermion propagator in a vector-gluon model are fully examined. The equation is characterized by a set of parameters, i.e., the coupling constant g, the bare mass of the fermion m0 and the cutoff Λ. It is proved that with a suitable gauge chosen, the equation without cutoff has solutions only in the case m0 = 0. It is then shown that, if g2/4π< (16/33)2π, the number of the solution is infinity of continuum. This situation does not come from the freedom of fixing the mass scale since the gluon mass is chosen to be non-vanishing. When the cutoff is introduced, the equation has a unique solution for g2/4π< π/4. In this case, however, β turns out to be identically zero if we put m0 = 0, which means that any “superconducting” solutions do not exist for such a value of g irrespective of cutoff momentum; that is, there are no Nambu-Goldstone bosons. When g2 is large enough, such a “superconducting” solution does exist in the model in which the vector part of the inverse fermion propagator is identically set equal to unity. Furthermore the existence of many “superconducting” solutions is inferred in this model. It is also found that in the region g2/4π > 8π, the “normal-state” solution for the equation without cutoff, even if it existed, should necessarily have an unphysical singularity. This fact implies that the “normal-state” solution becomes unstable for a sufficiently large value of g2.

Study of the Hypothesis of Threshold Dominance in a Renormalized LinearσModel

We examine Li and Pagels's hypothesis of threshold dominance (h.t.d.) in a linear SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) $\ensuremath{\sigma}$ model renormalized up to one loop. The exact inverse two-point functions ($\ensuremath{\sigma}$ and $\ensuremath{\pi}$ inverse propagators) at zero momentum transfer are calculated. These values are compared with that obtained by following the h.t.d. procedure. The threshold-dominance values show a sensitive dependence on the "cutoff," and the error ranges from 8% to 43%.