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arXiv:1209.4440 · Journal of the Korean Physical Society

Relativistic spin operator and Lorentz transformation of the spin state of a massive Dirac particle

Taeseung Choi

Published Apr 1, 2013• 23 citations
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Abstract

We have shown that the covariant relativistic spin operator is equivalent to the spin operator commuting with the free Dirac Hamiltonian. This implies that the covariant relativistic spin operator is a good quantum observable. The covariant relativistic spin operator has a pure quantum contribution that does not exist in the classical covariant spin operator. Based on this equivalence, reduced spin states can be clearly defined. We have shown that depending on the relative motion of an observer, the change in the entropy of a reduced spin density matrix sweeps through the whole range.

DOI
10.3938/jkps.62.1085

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We have shown that the covariant relativistic spin operator is equivalent to the spin operator commuting with the free Dirac Hamiltonian. This implies that the covariant relativistic spin operator is a good quantum observable. The covariant relativistic spin operator has a pure quantum contribution that does not exist in the classical covariant spin operator. Based on this equivalence, reduced spin states can be clearly defined. We have shown that depending on the relative motion of an observer, the change in the entropy of a reduced spin density matrix sweeps through the whole range.
The spin reduced matrix defined by taking partial trace with the momentum variable has been shown not to transform covariantly under Lorentz transformations. In this paper, I suggest another representation space of the Poincare group, the elements of which are in a sense apparent wave functions. In this new Hilbert space, the naive spin observable 12τ\frac{1}{2}\boldsymbol{\tau}21​τ from nonrelativistic quantum mechanics on the old Hilbert space is represented as the Newton-Wigner spin operator. Also on this space, one can naturally associate to each state a Lorentz covariant spin reduced density matrix, which is then given a new interpretation.
We have studied a relativistically covariant Stern-Gerlach (SG) experiment for a relativistic spin and a Dirac spin. We have obtained the relativistic spin in an arbitrary frame by using the classical spin dipole tensor, which gives the covariant spin dipole interactions, and the relation between a spin and a spin magnetic dipole moment. The relativistic spin is shown to have problems to become a proper spin operator for a massive relativistic particle because of two reasons. First, the relativistic spin three-vector operators cannot satisfy the spin algebra. Second, the SG experiment for the relativistic particle provides a paradox between two observers in the particle rest frame and the laboratory frame, in which the particle is moving. We have shown that the paradox in the SG experiment is resolved by the Dirac spin, which is covariantly defined by a Lorentz transformation in the Dirac spinor representation. The Dirac spin three-vector operators satisfy the spin algebra. It is shown that the SG experiment for the Dirac spin in the inertial frame, where there is only magnetic field, can determine the spin without the information of the momentum of the particle. This shows that the reduced spin density matrix for the Dirac particle can be well-defined by integrating out the momentum degrees of freedom.
Since the discovery a century ago, spin describing the intrinsic angular momentum of massive elementary particles has exposed its nature and significant roles in wide ranges of (relativistic) quantum phenomena and practical applications for future quantum technology. Emerging inconsistencies have also disclosed its telltale incomplete description. Finding relativistic spins (operators) of massive particles is a long-standing fundamental problem from the beginning of relativistic quantum mechanics. Here we present the rigorous derivation and the representation of spin operators from the spacetime symmetry. The covariant parity operation, defined by the spin operators, naturally leads to a fundamental equation equivalent to the covariant Dirac equation, which manifests existent relativistic spins. Proper understanding position operator in the Dirac theory on account of the spin operator through total angular momentum predicts no Zitterbewegung as well as conserving orbital and spin currents. The spin operators can be applicable for unraveling the inconsistencies and for exploring unveiled physics of massive particles.
We have studied the relativistic effect on the spin state of a massive Dirac particle under an arbitrary Lorentz transformation using the Foldy-Woutheysen representation. We have obtained the transformation matrix from the state of a massive Dirac particle with an initial momentum to a state with a transformed momentum under an arbitrary homogeneous Lorentz transformation. We have shown moving observers obtain the consistent monotonic behavior between the concurrence and the maximum value of Bell parameter for the spin state of the pair of particles.
Lorentz transformation of the reduced helicity matrix for a massive spin 1/2 particle is first investigated in the framwork of relativistic quantum information theory. The corresponding entangled helicity entropy is calculated and has no invariant meaning as that of spin. The variation of the entropy as the relative velocity of motion, however, differs significantly from that of spin due to their distinct transformation behaviors under the action of Lorentz group. The underlying physical explanations are also discussed.
Recently, in the context of Relativistic Quantum Information Theory (RQI) of massive spin-1/2 particles, it has been suggested that it is impossible to perform a momentum-independent spin measurement, showing the inadequacy of the spin reduced density matrix as a legitimate information resource. This is because there is an unavoidable ambiguity in the definition of the spin of a moving particle. In this paper, by introducing a vector bundle theoretic way to view the single-particle state space, we try to rule out this ambiguity. The discrete degree of freedom of the resulting representation space contains information about the Pauli–Lubanski four-vector of the particle instead of the ambiguous spin. Comparing this representation with the standard one used in the RQI literature, we show that the discrete degree of freedom of the standard representation space attains the meaning of the Newton–Wigner spin. In addition, using this viewpoint, we give a mathematical proof of why the spin reduced density matrix is meaningless, which is stronger than the previous claims in that it asserts that the matrix is void of any meaning at all, not just in terms of the impossibility of measurement or Lorentz non-covariance. We give a way (which turns out to be the only way) to modify it to obtain the Pauli–Lubanski reduced density matrix, which is covariant under Lorentz transformations.