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.     

Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models

The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed. The factorization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different properties for the cases N = 2 and N 3. For N = 2 the general solution depends on one parameter (of coupling constant type), whereas the solution for N 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliton S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N 3 there are two “minimum” solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear σ-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.     

Modification of Hauser-Feshbach calculations by direct-reaction channel coupling

If open channels are strongly coupled by direct reactions, the traditional Hauser-Feshbach method of calculating fluctuation cross sections is invalid, because of non-statistical correlations which the direct channel-coupling induces between resonance partial widths in different channels. The fluctuation cross sections can still be computed from the optical S-matrix elements, however, and the formulas necessary for doing so are obtained here with the aid of an “optical background” representation of the full S-matrix. The resulting compound-elastic cross section is increased over the Hauser-Feshbach expression by a factor of 2(Γ ? D) or 3(Γ ? D) in the large-N limit, and compound-reaction cross sections are increased by roughly a factor of $\text{(N + 1)}\text{N}$, where N is the number of directly-coupled open channels.     

Statistical theory of nuclear reactions and the Gaussian orthogonal ensemble

Using methods developed in field theory and statistical mechanics, especially in the context of the Anderson model as generalised by Wegner, a novel approach to the statistical theory of nuclear reactions is developed. A finite set of N bound states, coupled to each other by an ensemble of Gaussian orthogonal matrices, is considered and coupled to a set of channels via fixed coupling matrix elements. The ensemble average and the variance of the elements of the nuclear scattering matrix are evaluated, using the method of a generating function combined with the replica trick, followed by the Hubbard-Stratonovitch transformation and a modified loop expansion. In the limit N → ∞, it is shown quite generally that, aside from a trivial dependence on average S-matrix elements, the variance depends only on the transmission coefficients, and that the correlation width of a pair of S-matrix elements is given by a universal function of the transmission coefficients. A modified loop expansion yields an asymptotic series valid for strong absorption. The terms in this series are partly novel, and partly coincide with results obtained earlier in the framework of a model which did not take account of the GOE eigenvalue fluctuations. This suggests that average cross sections are mainly sensitive to the stiffness of the GOE spectrum. Fluctuation properties are also derived, and the link to Ericson fluctuation theory is established.     

Relativistic effects in the 3N continuum and the A y puzzle

The three-nucleon (3N) Faddeev equation is solved in a Poincaré-invariant model of the three-nucleon system. Two-body interactions are generated so that when they are added to the two-nucleon invariant mass operator (rest energy) the two-nucleon S-matrix is identical to the non-relativistic S-matrix with a CD Bonn interaction. Cluster properties of the three-nucleon S-matrix determine how these two-nucleon interactions are embedded in the three-nucleon mass operator. Differences in the predictions of the relativistic and corresponding non-relativistic models for elastic and breakup processes are investigated. Of special interest are the lowering of the A _y maximum in elastic nucleon-deuteron (Nd) scattering below ≈25?MeV caused by the Wigner spin rotations and the significant changes of the breakup cross sections in certain regions of the phase space.     

Scattering theory of three-body systems

The Faddeev amplitude is expressed in the N/D form in terms of the real reciprocal matrix K. The S-matrix is written in the unitary form (1 + iπK)S = 1 ?iπK. The Breit-Wigner formula for the three-body system including the break-up channel is derived. In the present method, the three-body problem is reduced to solve the eigenvalue problem for the real symmetric kernel.     

A dynamical theory for the off-shell continuation of the πNt-matrix

A dynamical theory, based on analyticity and dispersion theory, for the half-off-shell continuation of the on-shell πNt-matrix is proposed and developed. The resulting half-shell t-matrix is covariant, unitary, crossing symmetric, and based on a field-theoretic foundation. The dynamical information required to continue half off shell is obtained from field theory and consists of the off-shell amplitudes corresponding to the exchanges of the nucleon in the s- and u-channels and the ? and σ mesons in the t-channel. A coupled system of integral equations is derived for the partial wave half-shell t-matrix, which is truncated at the S- and P-waves and solved numerically. The results are compared with those obtained from various separable models of the πNt-matrix. The half-shell t-matrix is examined for separability and is found to be approximately separable in the P₃₃ and P₃₁ states. The dynamical content of the half-shell t-matrix is further illustrated by modeling the dynamical equation.     

Factorizable Z(N) models

The factorizable S-matrix with Z(N) symmetry is constructed. It is speculated that the field theory belonging to this S-matrix matrix is related to the scaling limit of Z(N) generalizations of the Ising model.     

Collision theory for waves in two dimensions and a characterization of models with trivialS-matrix

Within the framework of local relativistic quantum theory in two space-time dimensions, we develop a collision theory for waves (the set of vectors corresponding to the eigenvalue zero of the mass operator). Since among these vectors there need not be one-particle states, the asymptotic Hilbert spaces do not in general have Fock structure. However, the definition and “physical interpretation” of anS-matrix is still possible. We show that thisS-matrix is trivial if the correlations between localized operators vanish at large timelike distances.     

Weak LSZ limits and reduction formula for the non-relativistic coulombic interaction

The asymptotic condition is stated in the case of a non-relativistic Coulombic potential (long-range interaction), and the corresponding LSZ reduction formula is established for the S-matrix elements of the theory. Since the Coulombic Green function is known in closed (non-perturbative) form, we are able to rigorously prove that the resulting S-matrix is free of infinite Coulombic phases and coincides with Dollard's S-matrix. A brief discussion of the forward-scattering amplitude is also given.     

P-Matrix analysis of the dibaryon1 D 2 resonance

We study the relation between intermediate energy nucleon-nucleon scattering and the eigenstates of the bag model using theP-matrix formalism. Data of existing phase-shift analyses are employed to calculate theP-matrix for the coupled¹ D ₂(pp) and⁵ S ₂(NΔ) channels in the energy region above the Δ-isobar production threshold. TheP-matrix calculated for the equivalent hadronic bag radiib=1.4?1.5 fm is shown to have a pole in the mass range 2.31–2.34 GeV in agreement with the MIT bag model prediction of theI=1,J ^P =2⁺ 6-quark state with the mass 2.34 GeV. The hadronic shift of this state is shown to be ≈200 MeV; the dibaryon pole of theS-matrix is located at the energy 2.15–2.17 GeV with the width ≈100–200 MeV.     

Comparison between γN→π and πN→ϱN in the resonance region up to s=1.75 GeV via the vector-dominance-model

We display the helicity amplitudes for the N_? system for each J^P state of the N_π system in the reaction Nπ→N?. These amplitudes are obtained from a K-matrix fit to amplitudes of the reaction Nπ→Nπ and Nπ→Nππ (via the isobar model). We then take either these amplitudes or the resonance couplings estimated from these amplitudes and compare them to the photoproduction amplitudes via the V.D.M. Two possible kinematics are considered for the N_? system below its threshold. These four methods give a range in which the V.D.M. can be accomodated even at this low energy.     

Eigenvalue spectrum with chebyshev polynomial approximation of the transport equation in slab geometry

Using certain well-known properties of chebyshev polynomials, a simple and highly efficient approach to evaluate eigenvalue in radiation transport is presented. The spectrum of eigenvalues has been studied for slabs with isotropic scattering of different magnitudes of the cross section parameter c (i.e., the mean number of neutrons emitted per collision). It is shown that in the presence of the chebyshev polynomial approximation (T_N) there are both discrete and continuum of eigenvalues. It is found that the T_N method gives very good agreement with conventional spherical harmonics approximation (P_N).     

More about the S-matrix of the chiral SU(N) Thirring model

We fix the bound state poles of the S-matrix of the chiral SU(N) Thirring model by general arguments. Avoiding an infrared problem by using a modified $\text{1}\text{N}$ expansion, the result is confirmed in leading order.     

Standard forms of the eigenvalue problems associated with gyroscopic systems

The eigenvalue problem arising in the free vibration and stability analysis of gyroscopic systems is associated with a λ-matrix in which λ as well as its square appears. The characteristic polynomial of a non-dissipative gyroscopic system however, is a function of λ², and an equivalent standard eigenvalue formulation involving only λ² as an eigenvalue, therefore, seems to be a more appropriate representation of the physical system. Such forms and the related transformations are discussed herein. Similarly, when the loading parameter also appears explicitly in the λ-matrix, an equivalent double-eigenvalue problem involving λ² and the loading parameter as eigenvalues is generated. The extremum properties of the Rayleigh Quotient leads to a convenient proof of an upper bound theorem on λ². The use of left and right eigenvectors and the new double-eigenvalue formulation allows for the establishment of a flutter condition similar to one obtained for circulatory systems earlier. An example illustrating some of the concepts is presented.     

Relation between Bethe-Salpeter and Schrödinger wave functions for many-particel systems

The connection is made between a many-time approach to S-matrix elements and energy eigenvalues, which naturally arises from a field theoretical point of view, and the single time Schrödinger- and Breit-like formalism often used in detailed calculations for many-particle systems, such as many-electron atoms. Specifically, the many-particle Bethe-Salpeter equation is expressed in terms of the corresponding Schrödinger equation for the non-relativistic case in which the Bethe-Salpeter kernel consists of only two-particle local static interactions. Also, the one-photon transition matrix element for this case in the Bethe-Salpeter formalism is shown to be equivalent to the corresponding well-known Schrödinger result. The treatment developed is well suited to systematic relativistic generalization.     

Antiparticles as bound states of particles in the factorized S-matrix framework

We show that in a factorized two-dimensional S-matrix having SU(N) (N > 2) symmetry the antiparticles (transforming according to $\left\{\overline{N}\right\}$) are bound states of particles and vice versa and argue that this S-matrix is the one of the chiral Gross-Neveu model with screened U(1) charge and pseudocharge.     

Yang-Baxter algebras of monodromy matrices in integrable quantum field theories

We consider a large class of two-dimensional integrable quantum field theories with non-abelian internal symmetry and classical scale invariance. We present a general procedure to determine explicitly the conserved quantum monodromy operator generating infinitely many non-local charges. The main features of our method are a factorization principle and the use of $\text{P}$, $\text{T}$, and internal symmetries. The monodromy operator is shown to satisfy a Yang-Baxter algebra, the structure constants (i.e. the quantum R-matrix) of which are determined by two-particle S-matrix of the theory. We apply the method to the chiral SU(N) and the O(2N) Gross-Neveu models.     

On Eigenvalues of the Sum of Two Random Projections

We study the behavior of eigenvalues of matrix P _N +Q _N where P _N and Q _N are two N-by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P _N +Q _N is not universal in the usual sense.