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IMPROPER TRANSFORMATIONS OF NON RELATIVISTIC SPIN 1 2 SPINORS

D. Cervantes, Sandra L. Quiroga, Luis José Perissinotti, M. Socolovsky, Circuito Exterior Ciudad Universitaria

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

We first give a geometrical description of the action of the parity operator ( ˆ P) on non relativistic spin 1 2 Pauli spinors in terms of bundle theory. The relevant bundle, SU(2) ⊙Z2 → O(3), is a non trivial extension of the universal covering group SU(2) → SO(3). ˆ P is the non relativistic limit of the corresponding Dirac matrix operator P = iγ0 and obeys ˆP 2 = −1. Then, from the direct product of O(3) by Z2, naturally induced by the structure of the galilean group, we identify, in its double cover, the time reversal operator ( ˆ T) acting on spinors, and its product with ˆ P. Both, ˆ P and ˆ T, generate the group Z4 × Z2. As in the case of parity, ˆT is the non relativistic limit of the corresponding Dirac matrix operator T = γ 3 γ 1, and obeys ˆ T 2 = −1. Key words: parity transformation; time reversal; non relativistic spinors; group theory. 1.

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We first give a geometrical description of the action of the parity operator ( ˆ P) on non relativistic spin 1 2 Pauli spinors in terms of bundle theory. The relevant bundle, SU(2) ⊙Z2 → O(3), is a non trivial extension of the universal covering group SU(2) → SO(3). ˆ P is the non relativistic limit of the corresponding Dirac matrix operator P = iγ0 and obeys ˆP 2 = −1. Then, from the direct product of O(3) by Z2, naturally induced by the structure of the galilean group, we identify, in its double cover, the time reversal operator ( ˆ T) acting on spinors, and its product with ˆ P. Both, ˆ P and ˆ T, generate the group Z4 × Z2. As in the case of parity, ˆT is the non relativistic limit of the corresponding Dirac matrix operator T = γ 3 γ 1, and obeys ˆ T 2 = −1. Key words: parity transformation; time reversal; non relativistic spinors; group theory. 1.
By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
In this chapter we study the application of geometric algebra to both nonrelativistic and relativistic quantum mechanics. We concentrate on the quantum theory of spin-1/2 particles, whose dynamics is described by the Pauli and Dirac equations. For interactions where spin and relativity are not important the dynamics reduces to that of the Schrödinger equation. There are many good textbooks describing this topic and we will make no attempt to cover it here. We assume, furthermore, that most readers have a basic understanding of quantum mechanics, and are familiar with the concepts of states and operators. Both the Pauli and Dirac matrices arise naturally as representations of the geometric algebras of space and spacetime. It is no surprise, then, that much of quantum theory finds a natural expression within geometric algebra. To achieve this, however, one must reconsider the standard interpretation of the quantum spin operators. Like much discussion of the interpretation of quantum theory, certain issues raised here are controversial. There is no question about the validity of our algebraic approach, however, and little doubt about its advantages. Whether the algebraic simplifications obtained here are indicative of a deeper structure embedded in quantum mechanics is an open question. In this chapter we only consider the quantum theory of single particles in background fields. Multiparticle systems are considered in the following chapter.
We consider a massive particle of arbitrary spin and the basis vectors that carry the unitary, irreducible representations of the Poincaré group. From the complex coefficients in normalizable superpositions of these basis vectors, we identify momentum/spin-component probability amplitudes with the same interpretation as in the nonrelativistic theory. We find the relativistic transformations of these amplitudes, which are unitary in that they preserve the modulus-squared of scalar products from frame to frame. Space inversion and time reversal are also treated. We reconsider the Newton- Wigner construction of eigenvectors of position and the position operator. Position/spin-component probability amplitudes are also identified and their relativistic, unitary, transformations derived. Again, space inversion and time reversal are considered. For reference, we show how to construct positive energy solutions of the Klein-Gordon and Dirac equations in terms of probability amplitudes. We find the boost transformation of the position operator in the spinless case and present some results on the relativity of position measurements. We consider issues surrounding the classical concept of causality as it applies in quantum mechanics. We briefly examine the relevance of the results presented here for theories of interaction.
The foundations of quantum theory are closely tied to a formulation of classical relativistic physics. In Clifford's geometric algebra classical relativistic physics has a spinorial formulation that is closely related to the standard Dirac equation. The algebra of physical space, APS, gives clear insight into the quantum/classical interface. Here, APS is compared to other formulations of relativistic quantum theory, especially the Dirac equation. These formulations are shown to be effectively equivalent to each other and to the standard theory, as demonstrated by establishing several isomorphisms. Dirac spinors are four-component complex entities, and so must be represented by objects containing 8 real degrees of freedom in the standard treatment (or 7 if a normalization constant is added). The relation [special characters omitted]indicates that the 8-dimensional even subalgebra [special characters omitted] of the Space-time algebra, STA is isomorphic to APS Cl 3, which is isomorphic to complex quaternions [special characters omitted]. The complex quaternions should not be confused with the biquaternions, a name sometimes used for them. The biquaternions are more generally elements of the algebra [special characters omitted]. The algebras Cl1,3 and Cl 3,1 are not isomorphic but their even sub-algebras are[1]. The Klein paradox is resolved in APS by considering Feynman's picture of antiparticles as negative energy solutions traveling backward in time. It is also shown that the algebra of physical space can naturally describe an extended version of the De Broglie-Bohm approach to quantum theory. A relativistic causal account of a spin measurement in APS is given. The Stern-Gerlach magnet acts on the eigenspinor Λ field of a charged particle in a way that is analogous to the interaction of a birefringent medium acts on a beam of light. Then we introduce a covariant interpretation of complex algebra of physical space, CAPS, the complex extension of APS. This is done to solve a problem in that the space-time inversion, PT transformation, when P is parity inversion and T time reversal, although it is a proper transformation is not physical, yet, it has the form of a physical rotation in the traditional Dirac theory. The CAPS form of the PT transformation does not have the form of a physical rotation. A further problem is that the explicit form of the time inversion, T and charge conjugation, C transformations depends on the matrix representation chosen. A representation-independent form would be more fundamental. In CAPS, the complex extension of APS these problems are resolved.
Abstract The Pryce (e) spin and position operators of the quantum theory of Dirac's free field were re-defined and studied recently with the help of a new spin symmetry and suitable spectral representations [Eur. Phys. J. C 82, 1073 (2022)]. This approach is generalized here, associating a pair of integral operators acting directly on particle and antiparticle wave spinors in momentum representation to any integral operator in configuration representation, acting on mode spinors. This framework allows an effective quantization procedure, giving a large set of one-particle operators with physical meaning as the spin and orbital parts of the isometry generators, the Pauli-Lubanski and position operators, or other spin-type operators proposed to date. Special attention is paid to the operators that mix the particle and antiparticle sectors whose off-diagonal associated operators have oscillating terms producing Zitterbevegung. The principal operators of this type, including the usual coordinate operator, are derived here for the first time. As an application, it is shown that an apparatus measuring these new observables may prepare and detect one-particle wave packets moving uniformly without Zitterbewegung or spin dynamics, spreading in time normally as any other relativistic or even non-relativistic wave packet.
The authors study a family of Hilbert spaces with positive definite invariant scalar product for the quantum mechanical description of relativistic spin- 1 / 2 particles. The Hermitian and anti-Hermitian parts of the Dirac operator gamma mu p mu are related to energy and helicity. Replacing p mu by p mu -eA mu , the difference between the squares of these operators provides a minimal coupling evolution operator, similar to the second-order Dirac operator, for which F mu nu is coupled to a covariant form of the Pauli spin matrices which generate SU(2) in a space-like surface. No Dirac sea is required for the consistency of the theory.