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Journal of Physics A Mathematical and General

A new nonGrassmannian pseudoclassical action for spin-1/2 particles

Chris Belyea, Bruce H. J. McKellar, Roland Warner

Published Feb 21, 1990• 1 citation
Publisher

Abstract

The first-quantised 'spinor chain' path-integral description of spin-1/2 particles due to Jacobson (1984) is a novel non-Grassmannian approach to the problem of finding a path integral which yields the Dirac or Weyl equation. Unlike conventional path integrals, the spinor chain description does not explicitly use a path integral of the form integral exp iS, instead summing over all contributing spinor chains (paths) with equal weight, and hence the question of the existence of a canonical pseudoclassical action is left open. The authors examine the possibility of recasting the spinor chain system in the form of an action, and show that this involves subtleties concerning the weighting of unruly paths. They find the true action corresponding to the spinor chain which when canonically quantised yields the Weyl equation. Their action is extendable to the problem of the massive Dirac equation in a manner unlike the spinor chain. They discuss the classical solutions and compare these new actions with a similar non-Grassmannian action due to Barut and co-workers (1989).

DOI
10.1088/0305-4470/23/4/019

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The first-quantised 'spinor chain' path-integral description of spin-1/2 particles due to Jacobson (1984) is a novel non-Grassmannian approach to the problem of finding a path integral which yields the Dirac or Weyl equation. Unlike conventional path integrals, the spinor chain description does not explicitly use a path integral of the form integral exp iS, instead summing over all contributing spinor chains (paths) with equal weight, and hence the question of the existence of a canonical pseudoclassical action is left open. The authors examine the possibility of recasting the spinor chain system in the form of an action, and show that this involves subtleties concerning the weighting of unruly paths. They find the true action corresponding to the spinor chain which when canonically quantised yields the Weyl equation. Their action is extendable to the problem of the massive Dirac equation in a manner unlike the spinor chain. They discuss the classical solutions and compare these new actions with a similar non-Grassmannian action due to Barut and co-workers (1989).
The Feynman path integral plays a central role in quantum mechanics, linking classical action to propagators and relating quantum electrodynamics (QED) to Feynman diagrams. However, the path-integral formulations used in non-relativistic quantum mechanics and in QED are neither unified nor directly connected. This suggests the existence of a missing path integral that bridges relativistic action and the Dirac equation at the single-particle level. In this work, we analyze the consistency and completeness of existing path-integral theories and identify a spinor path integral that fills this gap. Starting from a relativistic action written in spinor form, we construct a spacetime path integral whose kernel reproduces the Dirac Hamiltonian. The resulting formulation provides a direct link between the relativistic classical action and the Dirac equation, and it naturally extends the scalar relativistic path integral developed in our earlier work. Beyond establishing this structural connection, the spinor path integral offers a new way to interpret the origin of classical mechanics for the Dirac equation and suggests a spacetime mechanism for spin and quantum nonlocal correlations. These features indicate that the spinor path integral can serve as a unifying framework for existing path-integral approaches and as a starting point for further investigations into the spacetime structure of quantum mechanics.
A path integral that reduces to Feynman's checkerboard rule in one space dimension is found for the retarded Dirac propagator in three space dimensions. The only variable are two-component spinors and a binary chirality variable. No action functional is employed. Each spinor together with a chirality corresponds to a spacetime displacement during a time epsilon . A sequence of spinors and chiralities determines a polygonal spacetime path, the first and last spinors specify the initial and final spin states. The transition amplitude for a sequence is given by ( nu N mod nu N-1 )( nu N-1 mod . . . mod nu 2 )( nu 2 mod nu 1 )(i epsilon m) R where nu i are the spinors, < mod > is the ordinary inner product in spin space, m is the electron mass and R is the number of times the chirality switches. Integrating over all sequences corresponding to a given displacement yields the Dirac propagator in the limit epsilon to 0. With epsilon >0 this formulation provides an alternative to the point particle model of the electron. In an external electromagnetic potential A mu the amplitude for a path C is multiplied by exp(-ie integral c A mu dx mu ), requiring spacetime coordinates to specify A mu (x). The usual perturbation expansion is derived from this rule. These results are extended to nonAbelian gauge potentials. Quantised interaction of particles is not treated.
A c-number path integral representation is constructed for the solution of the Dirac equation. The integration is over the real trajectories in the continuous three-space and other two canonical pairs of compact variables controlling the dynamics of the spin and the chirality flips. The problems to overcome in order to find a c-number path integral representation for the solution of the Dirac equation i¯h∂tΨ = HΨ = [ (−ic¯h ∂ + eA) · α + mc 2 β − eV] Ψ (1) is (i) to generate the first order derivatives in the space coordinates and (ii) to treat the spinor indices. The traditional method to tackle these problems is to use space-time lattice and sum over the lattice trajectories [1], and the application of Grassman variables [2]. The inherent limitation of this solution lies in the difficulty in arriving at the propagation in continuous space and in developing analytic non-perturbative approximations. We present
We discuss the formulation of spin observables associated to a non-relativistic spinning particles in terms of grassmanian differential operators. We use as configuration space variables for the pseudo-classical description of this system the positions xxx and a Grassmanian vector ε⃗\vecεε. We consider an explicit discretization procedure to obtain the quantum amplitudes as path integrals in this superspace. We compute the quantum action necessary for this description including an explicit expression for the boundary terms. Finally we shown how for simple examples, the path integral may be performed in the semi-classical approximation, leading to the correct quantum propagator.
Spinor structure is usually introduced as part of relativistic quantum theory. We show that its underlying algebra can instead emerge without quantisation from a covariant statistical description of a massive relativistic particle subject to an explicit finite-dimensional linear factorisation of the complete massive mass-shell constraint motivated by requiring the theory retain both sheets of the massive mass shell. A finite-dimensional factorisation of the complete mass-shell constraint, linear in all four momentum components, forces a Clifford algebra. In (3+1) dimensions its minimal complex representation is four-dimensional, while each on-shell factor selects a rank-two sector; in this representation statistical completeness requires an arbitrary 2×22\times22×2 matrix distribution within that sector before quantisation. Projecting the Weyl-ordered matrix Liouvillian gives relativistic transport while preserving all internal populations and coherences. For universal deformations on flat canonical phase space satisfying the stated covariance, matrix-neutrality, homogeneity and associativity assumptions, the resulting product is the matrix Weyl--Moyal product. Its parameter has dimensions of action and, when identified with ℏ\hbarℏ, supplies the scale of physical spin-12\tfrac1221​. Requiring all-order constraint preservation gives a vanishing star commutator, while preservation of the selected star-spectral sector yields the two-sided Dirac--Wigner equations. Our central conclusion is therefore that relativistic mass-shell factorisation supplies spinor structure, while completion of the constraint-preservation hierarchy within the stated deformation class leads to the Weyl--Moyal phase-space formulation of quantum mechanics.
The method of Weyl transforms is used to rigorously derive path integral forms for position and momentum transition amplitudes from the time-dependent Schrödinger equation for arbitrary Hermitian Hamiltonians. It is found that all paths in phase space contribute equally in magnitude, but that each path has a different phase, equal to 1/h/ times an ’’effective action’’ taken along it. The latter is the time integral of p⋅q̇−h (p,q), h (p,q) being the Weyl transform of the Hamiltonian operator H, which differs from the classical Hamiltonian function by terms of order h/2, vanishing in the classical limit. These terms, which can be explicitly computed, are zero for relatively simple Hamiltonians, such as (1/2M)[P−eA (Q)]2+V (Q), but appear when the coupling of the position and momentum operators is stronger, such as for a relativistic spinless particle in an electromagnetic field, or when configuration space is curved. They are always zero if one opts for Weyl’s rule for forming the quantum operator corresponding to a given classical Hamiltonian. The transition amplitude between two position states is found to be expressible as a path integral in configuration space alone only in very special cases, such as when the Hamiltonian is quadratic in the momenta.