Renormalization character and quantum S-matrix for a classically integrable theory

The quantum theory of a N-component generalization of the sine-Gordon model is investigated. We find at the one-loop order that the model is renormalizable only when the corresponding classical theory is completely integrable: N = 1 (sine-Gordon model) and N = 2 (reduced O(4) σ-model). Moreover the coupling constant does nor renormalize in these two cases. Although the S-matrix for N = 2 is factorizable at the tree level, an anomaly appears at the one-loop order. Its effect is like a local quartic coupling.     

Time-dependent mean field S-matrix theory

By using the path integral approach to many-body systems, we formulate a time-dependent mean field S-matrix theory of nuclear reactions. Many-body channel eigenstates are constructed by using projection techniques. In this way the S-matrix between the channel eigenstates is expressed as a superposition of S-matrix elements between wave-packet-like states localized in space and time. A field operator representation of the interaction picture S-matrix is derived which enables one to apply the path integral approach. Applying the stationary phase approximation to the path integral representation of the interaction picture S-matrix between the localized states an asymptotically constant time-dependent mean field approximation to this S-matrix is obtained. Finally, the S-matrix between the projected channel eigenstates is obtained by evaluating the integral, arising from the projections, over the space-time positions of the localized states in the stationary phase approximation. The stationary phase conditions select those localized states from the projected channel states for which the mean field values of energy and momentum coincide with their corresponding channel eigenvalues.     

Topics in the S-matrix theory of massless particles

We discuss how massless particle reactions may be incorporated into standard S-matrix theory. The crucial element for doing so is a low-energy zero. Examples of reactions where such zeros occur are weak interaction processes involving neutrinos, chirally symmetric massless pion scattering, and two-photon exchange between neutral systems. These zeros make two-body unitarity a good approximation for sufficiently low energy despite the coalescence of multiparticle thresholds. Through two-body unitarity, these zeros produce lines of zeros in the absorptive parts and double spectral functions. These lines of zeros are the S-matrix analog of the requirement of an infrared finite field theory. Not only do they produce finite total cross sections at finite energies, but they also allow both upper and lower bounds to be derived for these cross sections at high energies. This upper bound is our main result. If a plausible smoothness assumption is made, we find σ_tot <s^? (where ? is arbitrarily small). In particular, the experimentally observed linear rise of the neutrino proton cross section cannot continue indefinitely.     

The time-dependent mean-field S-matrix theory in the adiabatic limit

To elucidate the spirit of the recently developed time-dependent mean-field S-matrix theory we apply it to low-energy elastic collisions and solve the corresponding temporally non-local mean-field equation under the assumption that the relative motion proceeds adiabatically. Neglecting the exchange processes this assumption leads to a pure potential scattering picture. The phase shift extracted from the corresponding mean-field S-matrix coincides with the WKB result of potential scattering.     

Yang-Mills instantons and the S-matrix

We present a scheme for calculating gauge-invariant S-matrix elements in the presence of instantons. We exploit the conformal invariance of the zero-mass field equations. The asymptotic in and out states are defined by their values on null infinity $\text{J}$. We use this method to calculate to lowest-order S-matrix elements for scalar particles and fermions in a dilute gas of SU(2) instantons and anti-instantons. The scalar particles acquire an effective mass and an effective interaction of the form $\text{exp}\text{(?(?}{}^2\text{/16π) ?}\text{?}\text{)}$, where ? is the scale of the instanton, plus other interactions which cannot be presented by a local effective lagrangian. The fermions acquire the effective lagrangian obtained by 't Hooft. In the case of a single flavour of fermions, this corresponds to a mass term.     

S-matrix and unitarity bounds for three-channel systems,with application to low-energy photoproduction of pions from nucleons

A parametrization of the S-matrix for three-channel systems is given, where we state explicitly the unitarity bounds on the free parameters. The free parameters are directly related to measurable quantities. The results are applied to low-energy photoproduction of pions from nucleons, and lower bounds for the electric dipole amplitudes E₀₊ for π⁰ production are obtained. These lower bounds follow without using the Fermi-Watson final-state theorem.     

Unitarity and the energy averaged S-matrix

A unitarity relation between mean width $\text{γ}$, level distance D and the absorptive part of the energy averaged S-matrix is proved and discussed. Consequences for the determination of $\text{γ}\text{D}$ are pointed out.     

Gauge invariance and gauge independence of the S-matrix in nonrelativistic quantum mechanics and relativistic quantum field theories

The gauge independence of transition rates as opposed to the gauge invariance of the equations of motion and gauge dependence of operators and state vectors is critically examined and explicitly demonstrated, both in nonrelativistic quantum mechanics and quantum field theory. Time independent as well as time dependent gauge transformations are explicitly analyzed using several techniques in order to clarify the physical content and significance of gauge independence and the conditions for its applicability.     

S-matrix,Feynman zigzag and Einstein correlation

An inherent binding between Einstein correlations and the S-matrix formalism entails full relativistic covariance, complete time symmetry, and spacelike connexions via Feynman zigzags. The relay is in the past for predictive correlations between future measurements, and in the future for retrodictive correlations between past preparations (Pflegor and Mandel).An analogy and a partial binding exist between intrinsic symmetry together with factlike asymmetry of (1) “blind statistical” prediction and retrodiction (retarded and advanced waves, information as cognizance and as will) and (2) positive and negative frequencies (particles and antiparticles). As advanced waves are required for completeness of expansions, “antiphysics” obeying blind statistical retrodiction should show up in appropriate contexts, “parapsychology” being submitted as one of them.     

The background field method and the S-matrix

We prove that the S-matrix can be correctly obtained from the gauge-invariant effective action in the background field approach to gauge theories. In addition, we present a computation of the two-loop fermionic contributions to the Yang-Mills β-function.     

Complete S-matrix of the massive thirring model

On the basis of Zamolodchikov's S-matrix for two sine-Gordon solitons we derive the S-matrix for the scattering of an arbitrary number of particles including bound states.     

All possible generators of supersymmetries of the S-matrix

A generator of a symmetry or supersymmetry of the S-matrix has to have three simple properties (see sect. 2). Starting from these properties one can give a complete analysis of the possible structure of the pseudo Lie algebra of these generators. In a theory with non-vanishing masses one finds that the only extension of previously known relations is the possible appearance of “central charges” as anticommutators of Fermi charges. In the massless case (disregarding infrared problems and symmetry breaking) the Fermi charges may generate the conformal group together with a unitary internal symmetry group.     

Some properties of the S-matrix in a model coupled-channel calculation

In this paper we give the explicit solution to a coupled-channel problem consisting of two open channels coupled by a separable interaction. It is shown that the cross sections exhibit one rather narrow anomaly which displaces to higher energies as the coupling is increased and at the same time its width decreases and increases alternatively. This behavior is analyzed in terms of the eigenphaseshifts and also in terms of the motion of the poles of the S-matrix as a function of the coupling strength.     

Threshold phenomena in R-matrix theory

An R-matrix treatment of threshold phenomena in nuclear reactions is developed. It results, in addition to cusp phenomena, in a class of threshold anomalies induced by a resonance located at the threshold of the opening channel. The influence of compound nucleus and single-particle resonances and of direct interaction on threshold phenomena is discussed for s- and p- waves.     

Strong absorption and the distribution of zeros of the S-matrix

The only distributions normally included in a discussion of the statistical theory of nuclear resonance reactions are the distributions of the widths (Γ) and spacings (D) of the levels of the compound nucleus. However, as the usual Hauser-Feshbach theory makes clear, $\text{Γ}$ and $\text{D}$ alone are not sufficient to determine the ratio $\text{σ}{}_{\mathrm{cc\prime }}{}^{\mathrm{<mtext>dir</mtext>}}\text{σ}{}_{\mathrm{cc\prime }}{}^{\mathrm{<mtext>fl</mtext>}}$. In an attempt to determine what further statistical information is sufficient to determine this ratio, in the special limit that it tends to zero for all cc′, c ≠ c′ (the “strong-absorption” limit), we study several “picket fence” S-matrix models, as well as a random-residue model exhibiting Ericson fluctuations. These models indicate that the strong-absorption limit $\text{(}\text{S}{}_{\mathrm{cc\prime }}\text{= 0,}\text{all}\text{cc′)}$ is directly related to the distribution of the zeros of S_cc′(E) in the upper half of the complex E-plane, and that strong absorption is reached only if these zeros are distributed with a high density in the region E → + i ∞. As a by-product, we obtain a generalization of the theorem of Moldauer and Simonius $\text{[|}\text{det}\text{S}\text{| =}\text{exp}\text{(?π}\text{Γ}\text{/}\text{D}\text{)]}$. Our generalization applies to individual optical S-matrix elements (and so to direct-reaction cross sections) rather than just to their determinant.     

S-matrix regularities of two-dimensional σ-models of stiefel manifolds

The S-matrices of the two-dimensional nonlinear O(n+m)/O(n) and O(n+m)/O(n) XO(m) σ-models corresponding to Stiefel and Grassmann manifolds, respectively, are compared in leading order in $\text{1}\text{n}$. It is shown, that after averaging over O(m) labels of the incoming and outgoing particles, the S-matrices of both models become identical. This result explains why commonly expected regularities of the Grassmann models, in particular absence of particle production, are found, modulo an O(m) average, also in Stiefel models.     

Exact factorized S-matrix of the chiral field in two dimensions

The exact factorized S-matrix and the spectrum for the two-dimensional SU(N)-principal chiral field are presented.     

Matching conditions in the Einstein-Cartan theory of gravitation

In this paper matching and junction conditions in the Einstein-Cartan theory are presented. It is shown that a Weyssenhoff fluid sphere collapses to a singularity.Supported in part by the National Science Foundation Grant GF-36217.Part of this work was done during the stay of the author in Department of Mechanics, Paris University VI and in Collège de France.