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C. R. Hagen

Publications and source records attributed to C. R. Hagen.

At least 19 recordsLinked to original sources
arXiv2017

Group theory, coherent states, and the N-dimensional oscillator

The isotropic harmonic oscillator in N dimensions is shown to have an underlying symmetry group O(2,1)X O(N)which implies a unique result for the energy spectrum of the system. Raising and lowering operators analogous to those of the one-dimensional oscillator are given for each value of the angular momentum parameter. This allows the construction of an infinite number of coherent states to be carried out. In the N=1 case there is a twofold family of coherent states, a particular linear combination of which coincides with the single set already well known for that case. Wave functions are readily derived which require only the solution of a first order differential equation, an attribute generally characteristic of group theoretical approaches.

quant-ph↗
arXiv2014

Arbitrary Spin Galilean Oscillator

The so-called Dirac oscillator was proposed as a modification of the free Dirac equation which reproduces many of the properties of the simple harmonic oscillator but accompanied by a strong spin-orbit coupling term. It has yet to be extended successfully to the arbitrary spin S case primarily because of the unwieldiness of general spin Lorentz invariant wave equations. It is shown here using the formalism of totally symmetric multispinors that the Dirac oscillator can, however, be made to accommodate spin by incorporating it into the framework of Galilean relativity. This is done explicitly for spin zero and spin one as special cases of the arbitrary spin result. For the general case it is shown that the coefficient of the spin-orbit term has a 1/S behavior by techniques which are virtually identical to those employed in the derivation of the g-factor carried out over four decades ago.

quant-ph↗
arXiv2014

Where Have All the Goldstone Bosons Gone?

According to a commonly held view of spontaneously broken symmetry in gauge theories, troublesome Nambu-Goldstone bosons are effectively eliminated by turning into longitudinal modes of a massive vector meson. This note shows that this is not in fact a consistent view of the role of Nambu-Goldstone bosons in such theories. These particles necessarily appear as gauge excitations whenever they are formulated in a manifestly covariant gauge. The radiation gauge provides therefore the dual advantage of circumventing the Goldstone theorem and making evident the disappearance of these particles from the physical spectrum.

hep-th↗
arXiv2012

Group-Theoretical Derivation of Aharonov-Bohm Phase Shifts

The phase shifts of the Aharonov-Bohm effect are generally determined by means of the partial wave decomposition of the underlying Schrodinger equation. It is shown here that they readily emerge from an o(2,1) calculation of the energy levels employing an added harmonic oscillator potential which discretizes the spectrum.

math-ph↗
arXiv2004

Spin and Statistics in Galilean Covariant Field Theory

The existence of a possible connection between spin and statistics is explored within the framework of Galilean covariant field theory. To this end fields of arbitrary spin are constructed and admissible interaction terms introduced. By explicitly solving such a model in the two particle sector it is shown that no spin and statistics connection can be established.

quant-ph↗
arXiv2002

Casimir effect for the sphere revisited

In a recent work Brevik \emph{et al.} have offered formal proofs of two results which figure prominently in calculations of the Casimir pressure on a sphere. It is shown by means of simple counterexamples that each of those proofs is necessarily incorrect.

quant-ph↗
arXiv2002

Second central extension in Galilean covariant field theory

The second central extension of the planar Galilei group has been alleged to have its origin in the spin variable. This idea is explored here by considering local Galilean covariant field theory for free fields of arbitrary spin. It is shown that such systems generally display only a trivial realization of the second central extension. While it is possible to realize any desired value of the extension parameter by suitable redefinition of the boost operator, such an approach has no necessary connection to the spin of the basic underlying field.

quant-ph↗
arXiv2001

Cutoff dependence and Lorentz invariance of the Casimir effect

The Casimir force between two conducting planes is considered in both the electromagnetic and scalar field cases. This is done by the usual summation over energy eigenmodes of the system as well as by a calculation of the stress tensor in the region between the planes. The latter case requires that careful attention be given to singular operator products, an issue which is accommodated here by invoking the point separation method in conjunction with a scalar cutoff. This is shown to yield cutoff dependent and divergent contributions to the Casimir pressure which are dependent on the separation parameters, but entirely consistent with Lorentz covariance. Averaging over the point splitting parameters allows finite results to be obtained, but fails to yield a unique Casimir force.

quant-ph↗
arXiv2000

Cutoff dependence of the Casimir effect

The problem of calculating the Casimir force on two conducting planes by means of the stress tensor is examined. The evaluation of this quantity is carried out using an explicit regularization procedure which has its origin in the underlying (2+1) dimensional Poincaré invariance of the system. The force between the planes is found to depend on the ratio of two independent cutoff parameters, thereby rendering any prediction for the Casimir effect an explicit function of the particular calculational scheme employed. Similar results are shown to obtain in the case of the conducting sphere.

quant-ph↗
arXiv1999

Soluble field theory with a massless gauge invariant limit

It is shown that there exists a soluble four parameter model in (1+1) dimensions all of whose propagators can be determined in terms of the corresponding known propagators of the vector coupling theory. Unlike the latter case, however, the limit of zero bare mass is nonsingular and yields a nontrivial theory with a rigorously unbroken gauge invariance.

hep-th↗
arXiv1999

Galilean Lee Model of the Delta Function Potential

The scattering cross section associated with a two dimensional delta function has recently been the object of considerable study. It is shown here that this problem can be put into a field theoretical framework by the construction of an appropriate Galilean covariant theory. The Lee model with a standard Yukawa interaction is shown to provide such a realization. The usual results for delta function scattering are then obtained in the case that a stable particle exists in the scattering channel provided that a certain limit is taken in the relevant parameter space. In the more general case in which no such limit is taken finite corrections to the cross section are obtained which (unlike the pure delta function case) depend on the coupling constant of the model.

quant-ph↗
arXiv1999

Fractional spin and the Pauli term

It has recently been claimed that the inclusion of a Pauli term in (2+1) dimensions gives rise to a new type of anomalous spin term. The form of that term is shown to contradict the structure relations for the inhomogeneous Lorentz group.

hep-th↗
arXiv1999

Casimir Energy for Spherical boundaries

Calculations of the Casimir energy for spherical geometries which are based on integrations of the stress tensor are critically examined. It is shown that despite their apparent agreement with numerical results obtained from mode summation methods, they contain a number of serious errors. Specifically, these include (1) an improper application of the stress tensor to spherical boundaries, (2) the neglect of pole terms in contour integrations, and (3) the imposition of inappropriate boundary conditions upon the relevant propagators. A calculation which is based on the stress tensor and which avoids such problems is shown to be possible. It is, however, equivalent to the mode summation method and does not therefore constitute an independent calculation of the Casimir energy.

hep-th↗
arXiv1998

Casimir Energy of a Spherical Shell

The Casimir energy for a conducting spherical shell of radius $a$ is computed using a direct mode summation approach. An essential ingredient is the implementation of a recently proposed method based on Cauchy's theorem for an evaluation of the eigenfrequencies of the system. It is shown, however, that this earlier calculation uses an improper set of modes to describe the waves exterior to the sphere. Upon making the necessary corrections and taking care to ensure that no mathematically ill-defined expressions occur, the technique is shown to leave numerical results unaltered while avoiding a longstanding criticism raised against earlier calculations of the Casimir energy.

hep-th↗