This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103900982

Zeta function regularization of path integrals in curved spacetime

Lectures on Cauchy’s Problem in Linear Partial Differential Equations. By J. Hadamard. Pp. viii+316. 15s.net. 1923. (Per Oxford University Press.)

The Generalized Schwinger-Dewitt Technique in Gauge Theories and Quantum Gravity

Observations on Conformal Anomalies

In 1973 two Salam proteges (Derek Capper and the author) discovered that the conformal invariance under Weyl rescalings of the metric tensor displayed by classical massless-field systems in interaction with gravity no longer survives in the quantum theory. Since then these Weyl anomalies have found a variety of applications in black-hole physics, cosmology, string theory and statistical mechanics. We give a nostalgic review.

https://iopscience.iop.org/article/10.1088/0264-9381/11/6/004/pdf

Twenty years of the Weyl anomaly

Regularization of the stress-energy tensor for vector and scalar particles propagating in a general background metric

We calculate the renormalization constants describing nucleon and pion matrix elements of scalar, pseudoscalar, and tensor ($S, P, T$) current densities. For certain of the constants, expressions can be obtained using standard S${\mathrm{U}}_{3}$ and chiral S${\mathrm{U}}^{3}$\ensuremath{\bigotimes}S${\mathrm{U}}^{3}$ methods. To get the remaining constants, we employ the quark model with spherically symmetric quark wave functions to relate the $S$, $P$, $T$ renormalization constants to known parameters of the usual vector and axial-vector ($V,A$) currents. We also evaluate the renormalization constants using the MIT "bag" model quark wave functions. We summarize our results in tabular form, compare the results of the various calculational methods used, and attempt to estimate the accuracy of our predictions.

Renormalization constants for scalar, pseudoscalar, and tensor currents

A set of ratios characterizing the production of the isospin-1/2 resonances D$sub 13$(1514) and S$sub 11$(1505), as well as the isospin-3/2 resonance P$sub 33$(1232), by electromagnetic and weak (charged and neutral) currents are defined. Assuming the most general weak neutral current with V, A spatial structure, the matrix elements needed to extract the production ratios are calculated in three dynamical models for resonance production: the Born- approximation model, the nonrelativistic quark model, and the quark-bag model. For the special case of the Weinberg-Salam theory neutral current, the three models suggest that for 0 < or = sin$sup 2$theta/subW/ < or = 0.5 the ratios D$sub 13$/P$sub 33$ and S$sub 11$/P$sub 33$ observed in neutral-current $pi$$sup 0$ or eta production on a free nucleon target should be substantially smaller (by a factor of 2 to 4) than the corresponding ratios observed in charged-current $pi$$sup 0$ or eta production. An attempt is made to predict the ratios D$sub 13$/P$sub 33$ and S$sub 11$/P$sub 33$ to be expected in charged-current production from the corresponding ratios observed in electroproduction, but here the three models do not agree, and so it is possible only to state a wide possible range in which the values of the ratios may lie. The charge-exchange corrections needed for comparison with experiments on complex nuclear targets are briefly discussed; it is found that for an aluminum target the reduction factors quoted above are halved, so that in neutral-current-induced reactions the ratios D$sub 13$/P$sub 33$ and S$sub 11$/P$sub 33$ are smaller by a factor of 1 to 2 than the corresponding ratios observed in charged-current--induced reactions. (AIP)

/pdf/isospin-1-2-nucleon-resonance-production-by-a-v-a-weak-1kgkz91hvw.pdf

Isospin- 1/2 nucleon-resonance production by a V, A weak neutral current

We develop the cross-section formulas needed to correlate information on neutral currents obtained from deep-inelastic neutrino scattering, neutrino-proton elastic scattering, and neutral-current-induced soft-pion production, in the case of neutral currents with $S$, $P$, $T$ spatial structure. (The necessary $S$, $P$, $T$ current renormalization constants were estimated by us in a previous paper.) The pion-emission amplitude is obtained by current-algebra soft-pion techniques, with the effects of (3,3)-resonance excitation taken into account to leading nonvanishing order in the static approximation. We analyze recently reported Brookhaven National Laboratory results for neutral-current-induced soft-pion production under the simplifying assumption of a purely isoscalar $S$, $P$, $T$ neutral current, while simultaneously imposing existing bounds on neutrino-proton elastic scattering and fitting existing data on neutral-current-induced deep-inelastic scattering. If all $S$, $P$, $T$ renormalization constants are given their central quark-model values, the elastic-scattering and deep-inelastic restrictions constrain the pion-production cross section to be too low compared with experiment; if apparently reasonable deviations of the parameters from the quark-model values are permitted, satisfactory fits to all data are obtained with $S$, $T$, with $P$, $T$ or with $S$, $P$, $T$ mixtures. An isovector tensor or pseudoscalar neutral current is found to lead to a strong (3,3) peak in $\ensuremath{\pi}N$ invariant-mass plots, but an isovector scalar neutral current can be present without producing a visible (3,3) peak, even when ratios of the various $\ensuremath{\pi}N$ charge states produced by the neutral current are appreciably changed from the values which they have in the isoscalar-current case. Two other interesting qualitative features of $\mathrm{CP}$-conserving $S$, $P$, $T$ structures are the following: (1) constructive $T$ interference with $S$ (or $S$ and $P$) in $\ensuremath{\nu}+N\ensuremath{\rightarrow}\ensuremath{\nu}+N+\ensuremath{\pi}$ can accompany destructive interference in $\ensuremath{\nu}+p\ensuremath{\rightarrow}\ensuremath{\nu}+p$, and vice versa, and (ii) observation of unequal neutrino- and antineutrino-induced neutral-current cross sections would not be accompanied by neutral-current-induced parity-violating effects in the $\mathrm{pp}$, $\mathrm{ep}$, and $\ensuremath{\mu}p$ interactions.

Application of Current Algebra Techniques to Soft Pion Production by the Weak Neutral Current: S,P,T Case

We show that general phase-space considerations permit the scattering amplitude for two interacting massless particles to develop a weak-coupling singularity in the ladder approximation. In the case of photon-photon scattering induced by electron vacuum polarization, spin factors prevent such singularities from actually occurring in unaccelerated flat spacetime. However, in the case of conformally flat Riemannian spacetimes, which can be studied using Feynman rules similar to those in the Minkowski-space case, a reevaluation of the photon-photon scattering ladder sum shows that weak-coupling singularities do occur. We conjecture that such instabilities are a general feature of the non-Minkowski case and may provide a microscopic basis for gravitation, with the gravitational fields identified with photon pairing amplitudes of a superconductive type. According to this conjecture, the ''graviton'' would not be described by a conventional local quantum field. (AIP)

Judy Lieberman

Papers

Regularization of the stress-energy tensor for vector and scalar particles propagating in a general background metric

Renormalization constants for scalar, pseudoscalar, and tensor currents

Isospin- 1/2 nucleon-resonance production by a V, A weak neutral current

Application of Current Algebra Techniques to Soft Pion Production by the Weak Neutral Current: S,P,T Case

Photon pairing instabilities: A microscopic origin for gravitation?