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Paul Federbush

Publications and source records attributed to Paul Federbush.

54 records · Page 3Linked to original sources
arXiv2003

A Polymer Expansion for the Random Walk on the Permutation Group Associated to the Quantum Heisenberg Ferromagnet

For a long time one has associated to the Quantum Heisenberg Ferromagnet on a lattice, a random walk on the permutation group of the lattice vertices. We here present a polymer expansion for the solution of the heat equation coupled to the random walk. We work on a finite lattice, there is no question of convergence. We leave to future work bounding terms in the expansion necessary to extend the result to an infinite lattice.

math-ph↗
arXiv2002

For the Quantum Heisenberg Ferromagnet, Tao to the Proof of a Phase Transition

We present the outline of a proof for the 3-d phase transition which we hope to carry forth. At the same time this paper provides some physical understanding of the phase transition, in the flavor of relatively simple arguments from an undergraduate course. A number of directions for mathematical research, interesting in their own right, will be suggested by aspects of the development. We hope and believe that readers will be enticed by the naturalness and beauty of the path; some perhaps even, big game veterans, sniffing the quarry, will be ready to join the hunt. The central construct views the trace, Tr(exp(-beta*H)), as a lattice gas of polymers, each representing a cycle in the permutation group, with hard core interactions. The activities of the polymers have expressions as arising from the main conjecture of the paper. The estimates lead to a phase transition in 3-d, but not 2-d. This occurs via the same argument that a random walk in 2-d has certain return to the origin, but not so for a random walk in 3-d.

math-ph↗
arXiv2001

For the Quantum Heisenberg Ferromagnet, a Polymer Expansion and its High T Convergence

We let Psi_0 be a wave function for the Quantum Heisenberg ferromagnet sharp i sigma_zi and Psi_mu = exp(-mu*H)Psi_0. We study expectations similar to the form / for which we present a formal polymer expansion, whose convergence we prove for sufficiently small mu. The approach of the paper is to relate the wavefunction Psi_mu to an approximation to it that is a product function. In the jth spot of the product approximation the upper component is phi_mu(j), and the lower component is (1-phi_mu(j)), where phi satisfies the lattice heat equation. This is shown via a cluster or polymer expansion. The present work began in a previous paper, primarily a numerical study, and provides a proof of results related to Conjecture 3 of this previous paper.

math-ph↗
arXiv2001

For the Quantum Heisenberg Ferromagnet, Some Conjectured Approximations

We present some conjectured approximations for spin expectations in a Quantum Heisenberg system. The conjectures are based on numerical experimentation, some theoretical insights and underpinning, and aesthetic value. We hope theoretical developments will follow from these ideas, even leading to a proof of the phase transition (in three dimensions).

math-ph↗
arXiv2000

Linear Inflation in Curvature-Quadratic Gravity

We continue our study of gravity described by the action density (-g)^(1/2)(R_ik^2+bR^2); and look for cosmological solutions of gravity coupled to dust, for the closed isotropic model. There is a solution that at t approaches zero has for the radius, a(t)=t/3^(1/2); in the absence of dust this solution holds for all time.

hep-th↗
arXiv2000

A Note on Singular Instantons

We point out the existence of some singular, radial, spin-0 instantons for curvature-quadratic gravity theories. They are complex.

hep-th↗
arXiv1999

Some Generalized BRS Transformations. II A Quantum Gravity Model

Generalized BRS transformations such as introduced in Part I (hep-th/9906245) are applied to a model of quantum gravity. This development is technically complex; but at the least should illustrate how much less rigid and more general of application are the new BRS transformations.

hep-th↗
arXiv1999

Some Generalized BRS Transformations.I The Pure Yang-Mills Case

Some generalized BRS transformations are developed for the pure Yang-Mills theory, and a form of quantum gravity. Unlike the usual BRS transformations: these are nonlocal; may be infinite formal power series in the gauge fields; and do not leave the action invariant, but only the product exp(-S) with the Jacobian. Similar constructions should exist for many other field theory situations.

hep-th↗
arXiv1999

A Speculative Approach to Quantum Gravity

The bare bones of a theory of quantum gravity are exposed. It may have the potential to solve the cosmological constant problem. Less certain is its behavior in the Newtonian limit.

hep-th↗
arXiv1999

e to the A, in a New Way

Apparently new expressions are given for the exponential of a hermitian matrix,A, in the 2x2,3x3,and 4x4 cases. Replacing A by iA these are explicit formulas for the Fourier transform of exp(iA).

math-ph↗
arXiv1997

A Diffeomorphism-Invariant Cut-Off Regularization of the Determinant of a Scalar Particle in a Euclidean Gravitational Field

Continuing the thrust of our recent work, but with an important new idea, we find a cut-off regularization of the determinant of a scalar particle in a classical Euclidean gravitational field. The field is assumed asymptotically flat, and the regularization is diffeomorphism-invariant under coordinate changes that are the identity at infinity. The scalar field is expanded in term of variables that depend on the gravitational field, with wavelet localization of each variable. A renormalization group structure is thus automatically present. A similar construction is carried out for the determinant of a scalar field in a background Yang-Mills field.

hep-th↗
arXiv1996

The Axial Anomaly Revisited

We consider theories with gauged chiral fermions in which there are abelian anomalies, and no nonabelian anomalies (but there may be nonabelian gauge fields present). We construct an associated theory that is gauge-invariant, renormalizable, and with the same particle content, by adding a finite number of terms to the action. Alternatively one can view the new theory as arising from the original theory by using another regularization, one that is gauge-invariant. The situation is reminiscent of the mechanism of adding Fadeev-Popov ghosts to an unsatisfactory gauge theory, to arrive at the usual quantization procedure. The models developed herein are much like the abelian Wess-Zumino model (an abelian effective theory with a Wess-Zumino counterterm), but unlike the W-Z model are renormalizable! Details of the approach are worked out explicitly for the special case of a single massless Dirac fermion, for which we couple one abelian gauge field to the vector current and another abelian gauge field to the axial current.

hep-th↗
arXiv1996

Chiral Anomalies? A Voice for Choice

We consider theories with gauged chiral fermions in which there are abelian anomalies, and no nonabelian anomalies (but there may be nonabelian gauge fields present). We construct an associated theory that is gauge invariant, renormalizable, and with the same particle content, by adding a finite number of terms to the action. Alternatively one can view the new theory as arising from the original theory by using another regularization, one that is gauge invariant. The situation is reminiscent of the mechanism of adding Fadeev-Popov ghosts to an unsatisfactory gauge theory, to arrive at the usual quantization procedure. The models developed herein are much like the abelian Wess-Zumino model (an abelian effective theory with a Wess-Zumino counter term), but unlike the W-Z model are renormalizable!

hep-th↗
arXiv1995

A Gauge-Invariant Regularization of the Weyl Determinant Using Wavelets

In line with a previous paper, a gauge-invariant regularization is developed for the Weyl determinant of a Euclidean gauged chiral fermion. We restrict ourselves to gauge configurations with the $A$ field going to zero at infinity in Euclidean space; and thus restrict gauge transformations to those with $U$ the identity at infinity. For each finite cutoff one gets a strictly gauge-invariant expression for the Weyl determinant. Full Euclidean invariance is only to be sought in the limit of removing the cutoff. We expect the limit to be Euclidean invariant, but this has not yet been proved. One need not enforce the no-anomaly condition on the representation of the gauge group! We leave to future research relating the present results to conventional physics wisdom.

hep-th↗
arXiv1995

A New Formulation and Regularization of Gauge Theories Using a Non-Linear Wavelet Expansion

The Euclidean version of the Yang-Mills theory is studied in four dimensions. The field is expressed non-linearly in terms of the basic variables. The field is developed inductively, adding one excitation at a time. A given excitation is added into the ``background field'' of the excitations already added, the background field expressed in a radially axial gauge about the point where the excitation is centered. The linearization of the resultant expression for the field is an expansion $$ A_μ(x) \ \cong \ \sum_α\; c_α\; ψ_μ^α(x) $$ where $ψ^α_μ(x)$ is a divergence-free wavelet and $c_α$ is the associated basic variable, a Lie Algebra element of the gauge group. One is working in a particular gauge, regularization is simply cutoff regularization realized by omitting wavelet excitations below a certain length scale. We will prove in a later paper that only the usual gauge-invariant counterterms are required to renormalize perturbation theory. Using related ideas, but essentially independent of the rest of paper, we find an expression for the determinant of a gauged boson or fermion field in a fixed ``small'' external gauge field. This determinant is expressed in terms of explicitly gauge invariant quantities, and again may be regularized by a cutoff regularization. We leave to later work relating these regularizations to the usual dimensional regularization.

hep-ph↗