Volume 35, Issue 6, June 1994
Index of content:

Local automorphism invariance: Gauge boson mass without a Higgs particle
View Description Hide DescriptionThe consequences of the assumption of invariance of a spinor theory under local automorphism transformations of the Clifford algebra basis elements are explored. This invariance is equivalent to allowing the orthonormal basis spinors of the spinor space to be chosen arbitrarily at each point in space–time and is analogous to the situation in general relativity where the orthonormal basis vectors of the tangent space are allowed to be chosen arbitrarily at each point in space–time. This invariance then dictates that the Clifford algebra generators be functions of space–time and is implemented by introducing new fields, the drehbeins (‘‘spin legs’’), which are somewhat akin to the vielbeins introduced in general relativity to invoke the concept of local Lorentz invariance. However, in contrast to general relativity, the covariant derivatives of the Clifford algebra generators do not vanish. The dynamical variables of the theory are then the spinors, the gauge fields of the automorphism group, and the drehbeins. The invariant Lagrangian density and the concomitant field equations for this theory are discussed. Interestingly, the ‘‘kinetic’’ Lagrangian density term for the drehbein fields induces a gauge invariant mass term for the gauge fields. This constitutes a new mass generation mechanism, of different character and complementary to the familiar Higgs mechanism. Although the idea of local automorphism invariance is a natural generalization of the principle of equivalence, herein attention is restricted to the case of nondynamic flat space–time.

Levinson’s theorem, zero‐energy resonances, and time delay in one‐dimensional scattering systems
View Description Hide DescriptionThe one‐dimensional Levinson’s theorem is derived and used to study zero‐energy resonances in a double‐potential system. The low energy behavior of time delay is also investigated. In particular, it is shown that the quantum mechanical time delay admits a classical lower bound, in the low energy limit, if the potential has no bound‐state solutions.

Schwinger–Dyson Becchi–Rouet–Stora–Tyutin symmetry and the Batalin–Vilkovisky Lagrangian quantization of gauge theories with open or reducible gauge algebras
View Description Hide DescriptionIn this short paper we extend the results of Alfaro and Damgaard on the origin of antifields to theories with a gauge algebra that is open or reducible.

A soluble model for scattering and decay in quaternionic quantum mechanics. I. Decay
View Description Hide DescriptionThe Lee–Friedrichs model has been very useful in the study of decay‐scattering systems in the framework of complex quantum mechanics. Since it is exactly soluble, the analytic structure of the amplitudes can be explicitly studied. It is shown in this article that a similar model, which is also exactly soluble, can be constructed in quaternionic quantum mechanics. The problem of the decay of an unstable system is treated here. The use of the Laplace transform, involving quaternion‐valued analytic functions of a variable with values in a complex subalgebra of the quaternion algebra, makes the analytic properties of the solution apparent; some analysis is given of the dominating structure in the analytic continuation to the lower half plane. A study of the corresponding scattering system will be given in a succeeding article.

A soluble model for scattering and decay in quaternionic quantum mechanics. II. Scattering
View Description Hide DescriptionIn a previous article, it was shown that a soluble model can be constructed for the the description of a decaying system in analogy to the Lee–Friedrichs model of complex quantum theory. It is shown here that this model also provides a solublescattering theory, and therefore constitutes a model for a decay‐scattering system. Generalized second resolvent equations are obtained for quaternionic scattering theory. It is shown explicitly for this model, in accordance with a general theorem of Adler, that the scattering matrix is complex subalgebra valued. It is also shown that the method of Adler, using an effective optical potential in the complex sector to describe the effect of the quaternionic interactions, is equivalent to the general method of Green’s functions described here.

(2+1)‐dimensional models of relativistic particles with curvature and torsion
View Description Hide Description(2+1)‐dimensional models of relativistic particles with the action depending arbitrarily on the world‐trajectory curvature and torsion are investigated. A special class of models, described by the action with a maximal symmetry allowing only spin internal degrees of freedom, is singled out. A classical analysis of the systems from this special class is carried out, and the quantization procedure for them is described. Within this class, tachyon‐free models are pointed out and the systems having classical and quantum spectra with only massive spin states (whose number in the quantum case may be both finite and infinite) are considered as concrete examples of such models.

Parastatistics as Lie‐supertriple systems
View Description Hide DescriptionParastatistics are reformulated herein in terms of Lie‐supertriple systems. In this way, various new kinds of parastatistics discovered recently by Palev in addition to the standard one are reproduced. Also, bosonic and fermionic operators may not necessarily commute with each other.

q‐deformed relativistic wave equations
View Description Hide DescriptionBased on the representation theory of the q‐deformed Lorentz and Poincaré symmetries q‐deformed relativistic wave equation are constructed. The most important cases of the Dirac, Proca, Rarita–Schwinger, and Maxwell equations are treated explicitly. The q‐deformed wave operators look structurally like the undeformed ones but they consist of the generators of a noncommutative Minkowski space. The existence of the q‐deformed wave equations together with previous results on the representation theory of the q‐deformed Poincaré symmetry solve the q‐deformed relativistic one particle problem.

Topological aspects of SU(2) Weyl fermion and global anomaly
View Description Hide DescriptionIt is argued here that the global anomaly of SU(2) Weyl fermions is related to the residual topological properties of massive Dirac fermions leading to the topological index corresponding to the fermion number. As in the case of the chiral anomaly with Dirac fermions, the SU(2) anomaly with a Weyl fermion vanishes when the effect of this topological property is taken into account in the Lagrangian formulation.

Representations of commutation relations in BRST quantization
View Description Hide DescriptionIt is shown that the asymptotic state space in the BRST quantization is a Krein space. The equivalence of the representations, which are realized by the asymptotic fields in such spaces, is proved.

Study on the one dimensional boson–fermion model
View Description Hide DescriptionThe one dimensional boson–fermion model is introduced and the general eigenstates of the model Hamiltonian are constructed. The Cooper‐pairlike bound state is given. It is concluded that the present model is exactly soluble via the Bethe ansatz.

Symmetry scattering for SU(2,2) with applications
View Description Hide DescriptionIn the framework of symmetry scattering, SU(2,2)‐invariant differential equations on the homogeneous space X=SU(2,2)/S(U(2)⊗U(2)) are studied. The radial Schrödinger equation for a family of one or two dimensional potentials or for two particles arise. From the asymptotic behavior of the solutions exact partial wave scattering amplitudes are derived.

Hamiltonian formalism for nonlocal Lagrangians
View Description Hide DescriptionA Hamiltonian formalism is set up for nonlocal Lagrangian systems. The method is based on obtaining an equivalent singular first order Lagrangian, which is processed according to the standard Legendre transformation and then, the resulting Hamiltonian formalism is pulled back onto the phase space defined by the corresponding constraints. Finally, the standard results for local Lagrangians of any order are obtained as a particular case.

Jump conditions for a radiating relativistic gas
View Description Hide DescriptionAnalytical solutions of the Rankine–Hugoniot conditions for a radiating gas are presented both for shocks and subshocks. The case of Synge gas is studied in detail. In order to study subshocks, radiation is described by the moment equations closed with a variable Eddington factor.

Two‐dimensional integrable systems and self‐dual Yang–Mills equations
View Description Hide DescriptionThe relation between two‐dimensional integrable systems and four‐dimensional self‐dual Yang–Mills equations is considered. Within the twistor description and the zero‐curvature representation a method is given to associate self‐dual Yang–Mills connections with integrable systems of the Korteweg–de Vries and nonlinear Schrödinger type or principal chiral models. Examples of self‐dual connections are constructed that as points in the moduli do not have two independent conformal symmetries.

Two classes of dynamical systems all of whose bounded trajectories are closed
View Description Hide DescriptionAccording to the Bertrand theorem, the Kepler problem and the harmonic oscillator are the only central force dynamical systems that have closed orbits for all bounded motions. In this article, other dynamical systems having such a closed orbit property are found on T*(R ^{3}−{0}). Consider a natural dynamical system on T*(R ^{4}−{0}) whose Hamiltonian function is composed of kinetic and potential energies, and invariant under a SO(2) action. Then one can reduce the system to a Hamiltonian system on T*(R ^{3}−{0}) by the use of the Kustaanheimo–Stiefel transformation. If the original potential on R ^{4}−{0} is a central one, Bertrand’s method is applicable to the reduced system for determining the potential so that any bounded motions may be periodic. As a result, two types of potential functions will be found; one is linear in the radial variable and the other proportional to the inverse square root of that. The dynamical systems obtained are capable of physical interpretation. In particular, the dynamical system with the inverse square root potential may be called the twofold Kepler system, whose bounded trajectories have a self‐intersection point.

Caustics in 1+1 integrable systems
View Description Hide DescriptionCaustics arising in the asymptotic description of rapidly decaying solutions of the equations integrable via the inverse scattering transform are defined. Different asymptotic approaches to the description are considered. The appearance of members (Ρ^{ n } _{2}) of the hierarchy of the second Painlevé equation as special functions of wave catastrophies is discussed. The ‘‘adjoining’’ problem Ρ^{ n } _{2} → Ρ^{ k } _{2}(n≳k) for the simplest example Ρ^{2} _{2} → Ρ_{2} is considered in detail.

Asymptotics of the multipositon–soliton τ function of the Korteweg–de Vries equation and the supertransparency
View Description Hide DescriptionThe asymptotics of the τ function generating the N‐positon–M‐soliton solution of the Korteweg–de Vries equation is calculated. This allows to prove that solitons do not experience any phase shift in a collision with positons. The positons themselves survive mutual collisions unchanged. This phenomenon is called the supertransparency or super‐reflectionless property of the multipositon solutions. The linear aspects of this phenomenon are also discussed. It is demonstrated that positons acquire two additional but always finite phase shifts in collision with solitons. This result admits a natural extension to any number of solitons and positons in interaction.

A hierarchy of nonlinear evolution equations and finite‐dimensional involutive systems
View Description Hide DescriptionA spectral problem and an associated hierarchy of nonlinear evolution equations are presented in this article. In particular, the reductions of the two representative equations in this hierarchy are given: one is the nonlinear evolution equation r _{ t }=−αr _{ x }−2iαβ‖r ^{2}‖r which looks like the nonlinear Schrödinger equation, the other is the generalized derivative nonlinear Schrödinger equation r _{ t }= 1/2iαr _{ xx }−iα‖r‖^{2} r−αβ(‖r‖^{2} r)_{ x } −αβ‖r‖^{2} r _{ x }−2iαβ^{2}‖r‖^{4} r which is just a combination of the nonlinear Schrödinger equation and two different derivative nonlinear Schrödinger equations [D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 789 (1978); M. J. Ablowitz, A. Ramani, and H. Segur, J. Math. Phys. 21, 1006 (1980)]. The spectral problem is nonlinearized as a finite‐dimensional completely integrable Hamiltonian system under a constraint between the potentials and the spectral functions. At the end of this article, the involutive solutions of the hierarchy of nonlinear evolution equations are obtained. Particularly, the involutive solutions of the reductions of the two representative equations are developed.

An involutive system and integrable C. Neumann system associated with the modified Korteweg–de Vries hierarchy
View Description Hide DescriptionIn this article, a system of finite‐dimensional involutive functions is presented and proven to be integrable in the Liouville sense. By using the nonlinearization method, the C. Neumann system associated with the modified Korteweg–de Vries (mKdV) hierarchy is obtained. Thus, the C. Neumann system is shown to be completely integrable via a gauge transformation between it and an integrable Hamiltonian system. Finally, the solution of a stationary mKdV equation and the involutive solutions of the mKdV hierarchy are secured. As two examples, the involutive solutions are given for the mKdV equation: v _{ t }+1/4v _{ xxx }−3/2v ^{2} v _{ x }=0 and the 5th mKdV equation v _{ t }−1/16v _{ xxxxx }+5/8v ^{2} v _{ xxx } +5/2vv _{ x } v _{ xx } +5/8v ^{3} _{ x }−3/40v ^{4} v _{ x }=0.