Free Quantum Fields on the Poincare' Group

A class of free quantum fields defined on the Poincare' group, is described by means of their two-point vacuum expectation values. They are not equivalent to fields defined on the Minkowski spacetime and they are "elementary" in the sense that they describe particles that transform according to irreducible unitary representations of the symmetry group, given by the product of the Poincare' group and of the group SL(2, C) considered as an internal symmetry group. Some of these fields describe particles with positive mass and arbitrary spin and particles with zero mass and arbitrary helicity or with an infinite helicity spectrum. In each case the allowed SL(2, C) internal quantum numbers are specified. The properties of local commutativity and the limit in which one recovers the usual field theories in Minkowski spacetime are discussed. By means of a superposition of elementary fields, one obtains an example of a field that present a broken symmetry with respect to the group Sp(4, R), that survives in the short-distance limit. Finally, the interaction with an accelerated external source is studied and and it is shown that, in some theories, the average number of particles emitted per unit of proper time diverges when the acceleration exceeds a finite critical value.Comment: 49 pages, plain tex with vanilla.st

Field on Poincare group and quantum description of orientable objects

We propose an approach to the quantum-mechanical description of relativistic orientable objects. It generalizes Wigner's ideas concerning the treatment of nonrelativistic orientable objects (in particular, a nonrelativistic rotator) with the help of two reference frames (space-fixed and body-fixed). A technical realization of this generalization (for instance, in 3+1 dimensions) amounts to introducing wave functions that depend on elements of the Poincare group $G$. A complete set of transformations that test the symmetries of an orientable object and of the embedding space belongs to the group $\Pi =G\times G$. All such transformations can be studied by considering a generalized regular representation of $G$ in the space of scalar functions on the group, $f(x,z)$, that depend on the Minkowski space points $x\in G/Spin(3,1)$ as well as on the orientation variables given by the elements $z$ of a matrix $Z\in Spin(3,1)$. In particular, the field $f(x,z)$ is a generating function of usual spin-tensor multicomponent fields. In the theory under consideration, there are four different types of spinors, and an orientable object is characterized by ten quantum numbers. We study the corresponding relativistic wave equations and their symmetry properties.Comment: 46 page

Difficulty with a kinematic concept of unstable particles: the SZ.-Nagy extension and the Matthews-Salam-Zwanziger representation

We discuss the possibility of describing unstable systems, or dissipative systems in general, by vectors in a Hilbert space, evolving in time according to some non-unitary group or semigroup of translations. If the states of the unstable or dissipative system are embedded in a larger Hilbert space containing “decay products” as well, so that the time evolution of the system as a whole becomes unitary, we show that the infinitesimal generator necessarily has all energies from minus to plus infinity in its spectrum. This result supplements and extends the well-known fact that a positive energy spectrum is incompatible with a decay law bounded by a decreasing exponential. As an example of both facts, we discuss Zwanziger's irreducible, nonunitary representation of the Poincaré group; and we find its minimal, unitary extension (the Sz.-Nagy construction). The answer provides a mathematically canonical approach to the Matthews-Salam theory of wave functions for unstable, elementary particles, where the spectrum difficulty was already recognized. We speculate on the possibility that the Matthews-Salam-Zwanziger representation might be a strong coupling approximation in the relativistic version of the Wigner-Weisskopf theory, but we have not shown the existence of a physically acceptable model where that is so.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46496/1/220_2005_Article_BF01645753.pd

“Biological Geometry Perception”: Visual Discrimination of Eccentricity Is Related to Individual Motor Preferences

In the continuum between a stroke and a circle including all possible ellipses, some eccentricities seem more “biologically preferred” than others by the motor system, probably because they imply less demanding coordination patterns. Based on the idea that biological motion perception relies on knowledge of the laws that govern the motor system, we investigated whether motorically preferential and non-preferential eccentricities are visually discriminated differently. In contrast with previous studies that were interested in the effect of kinematic/time features of movements on their visual perception, we focused on geometric/spatial features, and therefore used a static visual display.In a dual-task paradigm, participants visually discriminated 13 static ellipses of various eccentricities while performing a finger-thumb opposition sequence with either the dominant or the non-dominant hand. Our assumption was that because the movements used to trace ellipses are strongly lateralized, a motor task performed with the dominant hand should affect the simultaneous visual discrimination more strongly. We found that visual discrimination was not affected when the motor task was performed by the non-dominant hand. Conversely, it was impaired when the motor task was performed with the dominant hand, but only for the ellipses that we defined as preferred by the motor system, based on an assessment of individual preferences during an independent graphomotor task.Visual discrimination of ellipses depends on the state of the motor neural networks controlling the dominant hand, but only when their eccentricity is “biologically preferred”. Importantly, this effect emerges on the basis of a static display, suggesting that what we call “biological geometry”, i.e., geometric features resulting from preferential movements is relevant information for the visual processing of bidimensional shapes