Classical spin and its quantization

Philip D. Mannheim

doi:10.1103/PhysRevD.32.898

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Classical spin and its quantization

Philip D. Mannheim

Department of Physics, University of Connecticut, Storrs, Connecticut 06268

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Phys. Rev. D 32, 898 – Published 15 August, 1985

DOI: https://doi.org/10.1103/PhysRevD.32.898

Abstract

We develop a new formulation of the quantum mechanics of a relativistic fermion based on the existence of an extended set of underlying classical coordinates. We enlarge the classical theory to a superspace built out of ordinary space-time coordinates and a set of Grassmann coordinates which transform like Majorana spinors and describe a classical spin. We introduce a natural supersymmetry between the Grassmann and the space-time coordinates. We quantize the theory both canonically and via a relativistic particle-mechanics path-integral prescription. The quantization of the motion of the coordinates in the superspace gives rise to the standard Dirac structure of a relativistic quantum-mechanical particle with spin one-half.

References (12)

There is a fairly extensive literature on the use of $ξ_{mu}$ , which includes F.A. Berezin and M. S. Marinov, Pis'ma Zh. Eksp. Teor. Fiz. 21, 678 (1975) [JETP Lett. 21, 320 (1975)]; A. Barducci, R. Casalbuoni and L. Lusanna, Nuovo Cimento 35A, 377 (1976); F. A. Berezin and M. S. Marinov, Ann. Phys. (N.Y.) 104, 336 (1977); P. DiVecchia and F. Ravndal, Phys. Lett. 73A, 371 (1979); A. Barducci, F. Bordi and R. Casalbuoni, Nuovo Cimento 64B, 287 (1981), and references therein.
With the Majorana spinor $ψ_{alpha}$ thus being another coordinate, the coordinates $x_{mu}$ and $ψ_{alpha}$ (which is usually called $θ_{alpha}$ in the superspace literature) form a classical superspace of the type introduced by Salam and Strathdee [A. Salam and J. Strathdee, Nucl. Phys. B76, 477 (1974)]. Our work differs from theirs and its subsequent applications in supersymmetry and supergravity theories [see, e.g., P. van Nieuwenhuizen, Phys. Rep. 68, 189 (1981) for a recent review] in two essential ways. First, for Salam and Strathdee the superspace coordinates are purely kinematical classical coordinates with the superfields (which are functions of the superspace coordinates) then describing the dynamics; for us the classical superspace coordinates are the dynamical variables themselves with the quantization of their motion in superspace giving rise to the quantum theory. Second, as will be seen below in Sec. V, even though we will use the same set of superspace coordinates as Salam and Strathdee we will impose a different superalgebra on the dynamics.
P. D. Mannheim, Phys. Lett. 137B, 385 (1984).
R. P. Feynman, Phys. Rev. 80, 440 (1950); see especially Appendix A.
Y. Nambu, Prog. Theor. Phys. 5, 82 (1950).
For an approach which is partway between ours and Feynman's see R. Casalbuoni, J. Gomis and G. Longhi, Nuovo Cimento 24A, 249 (1974), which uses the classical action of Eq. (2.4) and Nambu's quantum projection condition of Eqs. (2.24) and (2.25) below.
A recent pedagogical review of these aspects of Majorana spinors may be found in P. D. Mannheim, Int. J. Theor. Phys. 23, 643 (1984).
There is an analog to this result in standard Dirac theory when restricted to real space. The only real solutions to the real Dirac equation (i hbar $γ^{μ}$ ${partial}_{μ}$ - m) ψ (x) = 0 [in the Majorana basis of Eq. (3.2) each i $γ^{mu}$ is purely real] are of the form exp (p cdot x/ hbar ). These solutions satisfy (ip - m) ψ = 0 and consequently have a $p^{2}$ + $m^{2}$ = 0 mass shell.
A. Salam and J. Strathdee, Nucl. Phys. B80, 499 (1974).
R. Haag, J. T. Lopuszanski and M. Sohnius, Nucl. Phys. B88, 257 (1975).
Thus just as the inverse of i hbar partial / partial T is used as the integrating T factor for the measure in Eq. (5.41), Eq. (5.46) implies that we should use -i int d θ int d θ bar as the analogous dimensionless θ , θ bar measure for the θ , θ bar projection in Eq. (5.41).
Omitted end note.

[c1] There is a fairly extensive literature on the use of $ξ_{mu}$ , which includes F.A. Berezin and M. S. Marinov, Pis'ma Zh. Eksp. Teor. Fiz. 21, 678 (1975) [JETP Lett. 21, 320 (1975)]; A. Barducci, R. Casalbuoni and L. Lusanna, Nuovo Cimento 35A, 377 (1976); F. A. Berezin and M. S. Marinov, Ann. Phys. (N.Y.) 104, 336 (1977); P. DiVecchia and F. Ravndal, Phys. Lett. 73A, 371 (1979); A. Barducci, F. Bordi and R. Casalbuoni, Nuovo Cimento 64B, 287 (1981), and references therein.

[c2] With the Majorana spinor $ψ_{alpha}$ thus being another coordinate, the coordinates $x_{mu}$ and $ψ_{alpha}$ (which is usually called $θ_{alpha}$ in the superspace literature) form a classical superspace of the type introduced by Salam and Strathdee [A. Salam and J. Strathdee, Nucl. Phys. B76, 477 (1974)]. Our work differs from theirs and its subsequent applications in supersymmetry and supergravity theories [see, e.g., P. van Nieuwenhuizen, Phys. Rep. 68, 189 (1981) for a recent review] in two essential ways. First, for Salam and Strathdee the superspace coordinates are purely kinematical classical coordinates with the superfields (which are functions of the superspace coordinates) then describing the dynamics; for us the classical superspace coordinates are the dynamical variables themselves with the quantization of their motion in superspace giving rise to the quantum theory. Second, as will be seen below in Sec. V, even though we will use the same set of superspace coordinates as Salam and Strathdee we will impose a different superalgebra on the dynamics.

[c3] P. D. Mannheim, Phys. Lett. 137B, 385 (1984).

[c4] R. P. Feynman, Phys. Rev. 80, 440 (1950); see especially Appendix A.

[c5] Y. Nambu, Prog. Theor. Phys. 5, 82 (1950).

[c6] For an approach which is partway between ours and Feynman's see R. Casalbuoni, J. Gomis and G. Longhi, Nuovo Cimento 24A, 249 (1974), which uses the classical action of Eq. (2.4) and Nambu's quantum projection condition of Eqs. (2.24) and (2.25) below.

[c7] A recent pedagogical review of these aspects of Majorana spinors may be found in P. D. Mannheim, Int. J. Theor. Phys. 23, 643 (1984).

[c8] There is an analog to this result in standard Dirac theory when restricted to real space. The only real solutions to the real Dirac equation (i hbar $γ^{μ}$ ${partial}_{μ}$ - m) ψ (x) = 0 [in the Majorana basis of Eq. (3.2) each i $γ^{mu}$ is purely real] are of the form exp (p cdot x/ hbar ). These solutions satisfy (ip - m) ψ = 0 and consequently have a $p^{2}$ + $m^{2}$ = 0 mass shell.

[c9] A. Salam and J. Strathdee, Nucl. Phys. B80, 499 (1974).

[c10] R. Haag, J. T. Lopuszanski and M. Sohnius, Nucl. Phys. B88, 257 (1975).

[c11] Thus just as the inverse of i hbar partial / partial T is used as the integrating T factor for the measure in Eq. (5.41), Eq. (5.46) implies that we should use -i int d θ int d θ bar as the analogous dimensionless θ , θ bar measure for the θ , θ bar projection in Eq. (5.41).

[c12] Omitted end note.

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