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arXiv:1703.02557

Pauli Lubanski Vector Operator, Eigenvalue Problem and Entanglement

Garreth Kemp, Willi-Hans Steeb

Published Mar 7, 2017• 0 citations
arXivHTMLPDF

Abstract

We study an eigenvalue problem for a spin matrix arising in the Pauli Lubanski vector operator. Entanglement of the eigenvectors and its connection with degeneracy is discussed.
DOI
10.48550/arxiv.1703.02557

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The interpretation of quantum mechanics due to Lande' is applied to the connection between wave mechanics and matrix mechanics. The connection between the differential eigenvalue equation and the matrix eigenvalue equation for an operator is elucidated. In particular, we show that the elements of a matrix vector state are probability amplitudes with a structure rather than being mere constants. We obtain the most general expressions for the probability amplitudes for the description of spin-1/2 measurements. As a result, we derive spin-1/2 operators and vectors from first principles. The procedure used is analogous to that by which orbital angular momentum wavefunctions and operators are transformed to matrix mechanics vectors and matrices. The most generalized forms of the spin operators and their eigenvectors for spin-1/2 are derived and shown to reduce to the Pauli spin matrices and vectors in an appropriate limit.
In Part I, we argued that the square root of a positive real number bb has a degeneracy, i.e. -b and +b are solutions. Usually one adds +/- by hand, but we argued that there may exist a mathematical formalism which automatically carries this degeneracy. We suggested this formalism involves the three 2x2 Pauli matrices, each of which has eigenvalues -1 and +1 associated with this degeneracy for 3-vector components. Here we suggest that a positive number bb upon which the square root acts, may be thought of as a vector dot product. In other words, b is the vector magnitude, but in a rotated frame one would have (bx,by,bz) and each of these carry a degeneracy of +/-. A dot product is formally defined as: Sum i,j bi bj delta(i,j), but this equation forces one to add +/- by hand when taking the square root. It simply ensures that there is no mixing of i with j (for i not equal to j). The relationship for Pauli matrices: { s(i), s(j)} = 2 delta(i,j) I ((A)) (I = identity matrix and {} means anticommutation) may be used to show that the dot product has removed information related to the +1/-1 degeneracy..In other words, information inherent to a Pauli matrix, namely its eigenvalues of +1 and -1 disappear in ((A)) and so I the 2x2 identity matrix has 2 eigenvectors with the same +1 eigenvalue. This is related to the product of two identical Pauli matrices so +1 * +1 = -1*-1=1. In other words, delta(i,j) used in the dot product explicitly loses degeneracy (+/-) information.
In non-relativistic mechanics, the total (orbital) angular momentum (AM) of a spatially-distributed system can be decomposed into intrinsic and extrinsic contributions. In relativistic quantum mechanics, intrinsic AM is typically associated with spin, which can be described using the Pauli-Lubanski four-vector. Here, we develop a unified formalism that combines the main features of both approaches and describes the intrinsic AM of a relativistic wavepacket, including both spin and orbital contributions. Our approach is based on the "expectation Pauli-Lubanski vector" constructed from the expectation values of the wavepacket's momentum and AM. Equivalently, it defines the intrinsic AM relative to the wavepacket's energy centroid. In contrast to the conventional Pauli-Lubanski formalism, the zero-mass singularity does not occur for the expectation Pauli-Lubanski vector. Consequently, the intrinsic AM of a wavepacket may have an arbitrary orientation with respect to its momentum, even for massless particles. We illustrate the general theory with a number of examples of relativistic wave beams and packets carrying spin and orbital AM.