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arXiv:physics/0509131

Kinematical formalism of elementary spinning particles

M. Rivas

Published Sep 15, 2005• 5 citations
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Abstract

The concept of elementary particle rests on the idea that it is a physical system with no excited states, so that all possible kinematical states of the particle are just kinematical modifications of any one of them. The way of describing the particle attributes is equivalent to describe the collection of consecutive inertial observers who describe the particle in the same kinematical state. The kinematical state space of an elementary particle is a homogeneous space of the kinematical group. By considering the largest homogeneous spaces of both, Galilei and Poincaré groups, it is shown how the spin structure is related to the different degrees of freedom. The formalism is quantized by means of Feynman's path integral approach and special attention is paid to the classical model which satisfies Dirac's equation. Dirac's algebra is related to the classical observables, in particular to the orientation variables. Several spin effects are also analyzed.

DOI
10.48550/arxiv.physics/0509131

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The concept of elementary particle rests on the idea that it is a physical system with no excited states, so that all possible kinematical states of the particle are just kinematical modifications of any one of them. The way of describing the particle attributes is equivalent to describe the collection of consecutive inertial observers who describe the particle in the same kinematical state. The kinematical state space of an elementary particle is a homogeneous space of the kinematical group. By considering the largest homogeneous spaces of both, Galilei and Poincaré groups, it is shown how the spin structure is related to the different degrees of freedom. The formalism is quantized by means of Feynman's path integral approach and special attention is paid to the classical model which satisfies Dirac's equation. Dirac's algebra is related to the classical observables, in particular to the orientation variables. Several spin effects are also analyzed.
The Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.
Spin of elementary particles is the only kinematic degree of freedom not having classical corre- spondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of relativistic quantum field theory acting on multiple-component quantum fields non-unitarily. We study linear transformations, acting on the space of spatial and proper-time velocities rather than on coordinates. While ensuring the relativistic in- variance, they avoid these two exceptions: they describe the spin degree of freedom of a pointlike particle yet at a classical level and form a compact group hence with unitary finite-dimensional rep- resentations. Within this approach changes of the velocity modulus and direction can be accounted for by rotations of two independent unit vectors. Dirac spinors just provide the quantum description of these rotations.
The diffculties of relativistic particle theories formulated my means of canonical quantization, such as Klein-Gordon and Dirac theories, ultimately led theoretical physicists to turn on quantum field theory to model elementary particle physics. The aim of the present work is to pursue a method alternative to canonical quantization that avoids these dfficulties. In order to guarantee this result, the present approach is constrained to be developed deductively from physical principles. The physical principles assumed for a free particle consist of the symmetry properties of the particle with respect to the Poincaré group and of the transformation properties of the position observable, expressed by means of a suitably conceived notion of quantum transformation. In so doing, the effectiveness of group theoretical methods is exploited. Our work has pointed out the necessity of new classes of irreducible representations of the Poincaré group the theory can be based on. For spin 0 particle, four inequivalent theories are completely determined, that differ from Klein-Gordon theory for the physical interpretation but also for the mathematical structures. In particular, two of these theories requires irreducible representations in the new classes. The present lack of the explicit transformation properties of position with respect to boosts forbids the complete determination of non zero spin particle theories. A particular form of these transformation properties was in past adopted by Jordan and Mukunda. We check its consistency within the present approach; it is found that for spin 1/2 particles there is only one consistent theory; it is unitarily related to Dirac theory, but, once again, it requires the new classes of irreducible representations.
The Dirac equation provides a description of spin 1/2 particles, consistent with both the principles of quantum mechanics and of special relativity. Often its presentation to students is based on mathematical propositions that may hide the physical meaning of its contents. Here we show that Dirac spinors provide the quantum description of two unit classical vectors: one whose components are the speed of an elementary particle and the rate of change of its proper time and a second vector which fixes the velocity direction. In this context both the spin degree of freedom and antiparticles can be understood from the rotation symmetry of these unit vectors. Within this approach the Dirac Lagrangian acquires a direct physical meaning as the quantum operator describing the total time-derivative.