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Classical and Quantum Relativistic Mechanics of a Spinning Particle

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Abstract

Search for the relativistic equations that describe evolution of rotational degrees of freedom and their influence on the trajectory of a spinning body, represents a problem with a long and fascinating history. Closely related problem consists in establishing of classical equations that could mimic quantum mechanics of an elementary particle with spin in a semiclassical approximation. The relationship among classical and quantum descriptions has an important bearing, providing interpretation of results of quantum-field-theory computations in usual terms: particles and their interactions. In this Chapter we develop the Lagrangian and Hamiltonian formulations of a particle with rotational degrees of freedom. Taking a variational problem as the starting point, we avoid the ambiguities and confusion, otherwise arising in the passage from Lagrangian to Hamiltonian description and vice-versa. Besides, it essentially fixes the possible form of interaction with external fields. We show that so called vector model of spin represents a unified conceptual framework, allowing to collect and tie together a lot of remarkable ideas, observations and results accumulated over almost a century of studying this subject. On the classical level, the vector model adequately describes spinning particle in an arbitrary gravitational and electromagnetic fields. Moreover, taking into account the leading relativistic corrections it explains the famous one-half factor in non-relativistic Hamiltonian. Canonical quantization of the model yields one-particle relativistic quantum mechanics with positive-energy states.

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Notes

  1. 1.

    Quantum electrodynamics gives g = 2. 002322… due to radiative corrections.

  2. 2.

    There is an elegant formalism developed by Berezin and Marinov [42] based on using of anticommuting (Grassmann) variables for the description of spin. We present here another formulation based on commuting variables, without appealing to a rather formal methods of the Grassmann mechanics.

  3. 3.

    Note that this coincides with S appeared in (9.15).

  4. 4.

    On-shell symmetries considered as trivial symmetries, see [43].

  5. 5.

    With the factor \(-\frac{i} {4}\) in (9.63) and with the standard transformation law for a vector, δ v μ  = ω μ ν v ν , the function \(v_{\mu }\bar{\varPsi }\gamma ^{\mu }\varPsi\) is a scalar function.

  6. 6.

    For an electron interacting with electromagnetic field this analysis has been repeated by Feynman in [65].

  7. 7.

    In discussing this factor often refer to Thomas precession [52]. We will not touch this delicate and controversial issue [72, 73] because of the covariant formalism automatically accounts the Thomas precession [15].

  8. 8.

    Some models of doubly special relativity predict rainbow geometry at Planck scale [83].

  9. 9.

    Our S is twice of that of Dixon.

  10. 10.

    Besides S μ ν P ν  = 0, there are known others supplementary spin conditions. In this respect we point out that the MPTD theory implies this condition with certain P ν written in Eq. (9.258). Introducing κ, we effectively changed P ν and hence changed the supplementary spin condition. For instance, when κ = 1 and in the space with ∇R = 0, we have \(P^{\mu } = \frac{\tilde{m}c} {\sqrt{-\dot{x}g\dot{x}}}\dot{x}^{\mu }\) instead of (9.258).

  11. 11.

    Under the finite transformations x′μ = Λ μ ν (ω)x ν, Ψ′ = D(ω)Ψ and Ψ′† = Ψ † D †(ω), where \(D = e^{\frac{i} {4} \omega _{\mu \nu }\sigma ^{\mu \nu } }\), the \(\bar{\sigma }^{\mu }\) is an invariant tensor, that is \((D^{\dag }\bar{\sigma }_{\mu }D)\varLambda ^{\mu }_{\nu } =\bar{\sigma } _{\nu }\). For the proof, see [59, 61].

  12. 12.

    Note that \(\bar{\xi }\) can be considered as conjugated momentum for ψ, then the passage from (9.374) to (9.377) is just the passage from a Lagrangian to Hamiltonian formulation. A similar interpretation was developed for the Schrodinger equation in Sect. 2.9.1

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Authors and Affiliations

  1. Universidade Federal de Juiz de Fora Dept. of Mathematics, Juiz de Fora, Minas Gerais, Brazil

    Alexei Deriglazov

  2. Laboratory of Mathematical Physics, Tomsk Polytechnic University, Tomsk, Russian Federation

    Alexei Deriglazov

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Deriglazov, A. (2017). Classical and Quantum Relativistic Mechanics of a Spinning Particle. In: Classical Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-44147-4_9

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