Volume 38, Issue 2, February 1997
Index of content:

Operator ordering index method for multiple commutators and Suzuki’s quantum analysis
View Description Hide DescriptionGeneral formulas are presented for the multiple commutator involving a function of an operator A in terms of (multiple) commutators involving A by using “operator ordering indices.” It is pointed out that this analysis essentially reproduces the quantum analysis, proposed by Suzuki very recently, in a more transparent way. Extension to the case of several operators is also made in an elegant form.

Relativistic point interaction with Coulomb potential in one dimension
View Description Hide DescriptionThe Dirac Hamiltonian in one space dimension is investigated under the influence of a potential of the form −γ/x. The corresponding (fourparameter) family of all selfadjoint extensions is given and described via the boundary form. The resolvent is calculated and the spectrum is studied. Furthermore, we examine the zero mass case. In the nonrelativistic limit we obtain the fourparameter family of Schrödinger operators with the Coulomb potential.

Bound states in Galileaninvariant quantum field theory
View Description Hide DescriptionWe consider the nonrelativistic quantum mechanics of a model of two spinless fermions interacting via a twobody potential. We introduce quantum fields associated with the two particles as well as the expansion of these fields in asymptotic “in” and “out” fields, including such fields for bound states, in principle. We limit our explicit discussion to a twobody bound state. In this context we discuss the implications of the Galilean invariance of the model and, in particular, show how to include bound states in a strictly Galileaninvariant quantum field theory.

Octonionic representations of GL(8,R) and GL(4,C)
View Description Hide DescriptionOctonionic algebra being nonassociative is difficult to manipulate. We introduce left/right octonionic barred operators which enable us to reproduce the associative GL(8,R) group. Extracting the basis of GL(4,C), we establish an interesting connection between the structure of left/right octonionic barred operators and generic 4×4 complex matrices. As an application we give an octonionic representation of the fourdimensional Clifford algebra.

Canonical gauges in the path integral for parametrized systems
View Description Hide DescriptionIt is well known that—differing from ordinary gauge systems—canonical gauges are not admissible in the path integral for parametrized systems. This is the case for the relativistic particle and gravitation. However, a time dependent canonical transformation can turn a parametrized system into an ordinary gauge system. It is shown how to build a canonical transformation such that the fixation of the new coordinates is equivalent to the fixation of the original ones; this aim can be achieved only if the Hamiltonian constraint allows for an intrinsic global time. Thus the resulting action, describing an ordinary gauge system and allowing for canonical gauges, can be used in the path integral for the quantum propagator associated with the original variables.

Relativistic quantum mechanics on the SL(2,R) space–time
View Description Hide DescriptionThe Schrödingertype formalism of the Klein–Gordon quantum mechanics is adapted for the case of the SL(2,R) space–time. The free particle case is solved, the results of a recent work are reproduced while all the other, topologically nontrivial solutions and the antiparticle modes are also found, and a deeper insight into the physical content of the theory is given.

Massive spin2 propagators on de Sitter space
View Description Hide DescriptionWe compute the Pauli–Jordan, Hadamard, and Feynman propagators for the massive metrical perturbations on de Sitter space. They are expressed both in terms of mode sums and in invariant forms.

The quartic oscillator
View Description Hide DescriptionThe quantum quartic oscillator is treated on the basis of a singularityanalytic approach to the central twopoint connection problem of the triconfluent case of Heun’s differential equation. We split off the asymptotic factors by means of a specific linear transformation of the independent variable and represent the solution in terms of a Jaffé expansion. The result is a fourthorder linear difference equation of Poincaré–Perron type the asymptotic behavior of which is significant for the connection problem. This is investigated by means of the Birkhoffset of the difference equation and leads to the exact eigenvaluecondition of the problem.

Massless fields in plane wave geometry
View Description Hide DescriptionConformal isometry algebras of plane wave geometry are studied. Then, based on the requirement of conformal invariance, a definition of masslessness is introduced and gauge invariant equations of motion, subsidiary conditions, and corresponding gauge transformations for all plane wave geometry massless spin fields are constructed. Light cone representation for elements of conformal algebra acting as differential operators on wavefunctions of massless higher spin fields is also evaluated. Interrelation of plane wave geometry massless higher spin fields with ladder representation of u(2,2) algebra is investigated.

Fermion determinants and effective actions
View Description Hide DescriptionConfiguration space heatkernel methods are used to evaluate the determinant and hence the effective action for a SU(2) doublet of fermions in interaction with a covariantly constant SU(2) background field. Exact results are exhibited that are applicable to any Abelian background on which the only restriction is that (B^{2}−E^{2}) and E⋅B are constant. Such fields include the uniform field and the plane wave field. The fermion propagator is also given in terms of gauge covariant objects. An extension to include finite temperature effects is given and the probability for creation of fermions from the vacuum at finite temperature in the presence of an electric field is discussed.

Prepotentials of Yang–Mills theories coupled with massive matter multiplets
View Description Hide DescriptionWe discuss Yang–Mills gauge theories coupled with massive hypermultiplets in the weak coupling limit. We determine the exact massive prepotentials and the monodromy matrices around the weak coupling limit. We also study that the double scaling limit of these massive theories and find that the massive theory can be obtained from the massive theory. New formulae for the massive prepotentials and the monodromy matrices are proposed. In these formulae, dependences are clarified.

Quantumclassical correspondence using projection operators
View Description Hide DescriptionClassical properties, associated with a large phase space domain with a smooth boundary as compared to the Planck constant, are shown to be expressed quantummechanically by a family of quantum projection operators with the help of known theorems in microlocal analysis. Under the conditions of Egorov’s theorem, the conservation of this correspondence under classical/quantum dynamics is asserted.

Localization of relativistic particles
View Description Hide DescriptionIn order to discuss localization experiments and also to extend the consistent history interpretation of quantum mechanics to relativistic properties, the techniques introduced in a previous paper [J. Math. Phys. 38, 697 (1997)] are applied to the localization of a photon in a given region of space. An essential requirement is to exclude arbitrarily large wavelengths. The method is valid for a particle with any mass and spin. Though there is no proper position operator for a photon, one never needs one in practice. Causality is valid up to exponentially small corrections.

Aharonov–Bohm scattering: The role of the incident wave
View Description Hide DescriptionThe scattering problem under the influence of the Aharonov–Bohm (AB) potential is reconsidered. By solving the Lippmann–Schwinger (LS) equation we obtain the wave function of the scattering state in this system. In spite of working with a plane wave as an incident wave we obtain the same wave function as was given by Aharonov and Bohm. Another method to solve the scattering problem is given by making use of a modified version of Gordon’s idea, which was invented to consider the scattering by the Coulomb potential. These two methods give the same result, which guarantees the validity of taking an incident plane wave as usual to make an analysis of this scattering problem. The scattering problem by a solenoid of finite radius is also discussed, and we find that the vector potential of the solenoid affects the charged particles, even when the magnitude of the flux is an odd integer as well as a noninteger. It is shown that the unitarity of the S matrix holds provided that a plane wave is taken to be an incident one.

Natural renormalization
View Description Hide DescriptionA careful analysis of differential renormalization shows that a distinguished choice of renormalization constants allows for a mathematically more fundamental interpretation of the scheme. With this set of a priori fixed integration constants differential renormalization is most closely related to the theory of generalized functions. The special properties of this scheme are illustrated by application to the toy example of a free massive bosonic theory. Then we apply the scheme to the theory. The twopoint function is calculated up to five loops. The renormalization group is analyzed and the betafunction and the anomalous dimension are calculated up to fourth and fifth order, respectively.

Symmetries of decoherence functionals
View Description Hide DescriptionThe basic ingredients of the “consistent histories” approach to quantum theory are a space UP of “history propositions” and a space D of “decoherence functionals.” In this article we consider such history quantum theories in the case where UP is given by the set of projectors P(V) on some Hilbert spaceV. Using an analog of Wigner’s theorem in the context of history quantum theories proven earlier, we develop the notion of a “symmetry of a decoherence functional” and prove that all such symmetries form a group which we call “the symmetry group of a decoherence functional.” We calculate for the case of standard quantum mechanics—when looked at from the perspective of the history program—some of these symmetries explicitly and relate them to some discussions that have appeared previously.

Instantons and splitting
View Description Hide DescriptionIn this paper we compare various formulas for the leading term of the amplitude of the splitting of the eigenvalues of semiclassical Schrödinger operators with multiple wells.

Topological Casimir energy for a general class of Clifford–Klein space–times
View Description Hide DescriptionUsing zeta regularization we compute the vacuum energy for free massless scalar fields on ultrastatic space–times R×(Γ\X), where X is an arbitrary noncompact irreducible rank 1 symmetric space and Γ is a cocompact torsion free subgroup of isometries of X. The spaces X include hyperbolic manifolds on which previous authors have focused. Specifically, using a general trace formula, we extend the work of Bytsenko, Goncharov, Zerbini (and others), where X=SO_{1} (m,1)/SO(m), to the other classical rank 1 symmetric spaces X=SU(m,1)/U(m), SP(m,1)/(SP(m)×SP(1)), and the exceptional space X=F_{4(−20)} /Spin(9). We find in general that the trivial unitary character of Γ always induces a negative topological component of the energy.

Exact solutions of linearized Schwinger–Dyson equation of fermion selfenergy
View Description Hide DescriptionThe Schwinger–Dyson equation of fermion selfenergy in the linearization approximation is solved exactly in a theory with gauge and effective fourfermion interactions. Different expressions for the independent solutions, which, respectively, submit to irregular and regular ultraviolet boundary condition are derived and expounded.

Motion of strings in the plane: A solvable model. I
View Description Hide DescriptionBy taking appropriately the n→∞ limit of a (recently introduced) eightparameter family of solvable (classical, nonrelativistic) nbody problems in the plane, a solvable eightparameter family of models is introduced, each of which describes the motion in the plane of a string (possibly composed of several pieces). In this paper the equations of motion which characterize these models are displayed, the technique to “solve” them is outlined, and special solutions are exhibited (for certain models, quite explicitly). A more detailed analysis of the phenomenology of the string motions entailed by these models is postponed to future papers.