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arXiv:2607.21724

How to calculate the Wigner angle

C. J. McKinstrie, M. V. Kozlov

physics.opticsphysics.class-phquant-phSubmitted Jul 23, 2026• Announced Jul 27, 2026
arXivHTMLPDF

Abstract

Lorentz transformations in time and two space dimensions consist of boosts and rotations, and combinations thereof. In general, the combination of two boosts is not another boost: It is a boost followed by a rotation. The rotation angle is called the Wigner angle. Although it is straightforward to determine the energy and direction of the combined boost, it is difficult to determine the Wigner angle. In this article, the vector, matrix and spinor derivations of formulas for the Wigner angle are reviewed, and the underlying mathematics and physics are discussed briefly. Although the derivations are different, the results they produce are equivalent, as they should be. Like many physics problems, if one looks at the problem in the right way, it is not difficult to solve.

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Lorentz transformations in time and two space dimensions consist of boosts and rotations, and combinations thereof. In general, the combination of two boosts is not another boost: It is a boost followed by a rotation. The rotation angle is called the Wigner angle. Although it is straightforward to determine the energy and direction of the combined boost, it is difficult to determine the Wigner angle. In this article, the vector, matrix and spinor derivations of formulas for the Wigner angle are reviewed, and the underlying mathematics and physics are discussed briefly. Although the derivations are different, the results they produce are equivalent, as they should be. Like many physics problems, if one looks at the problem in the right way, it is not difficult to solve.
In this paper, we calculate the helicity rotation angle induced by Lorentz boosts. This is relevant for the study of Lorentz boost effects on quantum entanglement encoded in pairs of massive fermions, which are described in terms of positive energy solutions of the Dirac equation with definite helicity. A Lorentz boost describing the change to an inertial frame moving at uniform speed will in general rotate the particle’s helicity. We obtain the coefficients of the helicity superposition in the boosted frame and specialize our results for a perpendicular boost geometry. We verify that the helicity rotation angle can be obtained in terms of the Wigner rotation angle for spin [Formula: see text] states, bridging the framework considered in our previous works to the one of the Wigner rotations. Finally, we calculate the boost-induced spin-parity entanglement for a single particle.
This note summarizes the physics and mathematics of Lorentz transformations for massless particles, specifically for photons. We provide a complete analytical derivation of Wigner's little group matrix and a closed formula for the calculation of Wigner's angle for arbitrary Lorentz transformations. Our derivation highlights the geometrical content of the sequence of little group transformations leading to Wigner's matrix and links it to classical theorems in spherical trigonometry.
An account is given of the Wigner concept of particle spin and velocity rotations and of the variation of the angle between them under Lorentz transformations with noncollinear velocities. It is shown that Moller's description of spin rotation can be reduced to the Wigner rotation, and Moller's formula for the angle of spin rotation in the curvilinear motion of a particle is corrected. The permutation asymmetry of the relativistic velocity addition law distinguishes the Wigner sequence of Lorentzian boosts by its applicability to the description of spin and velocity rotations in curvilinear motion.
This paper is concerned with the spin–momentum correlation in single-particle quantum states, which is described by the mixed states under Lorentz transformations. For convenience, instead of using the superposition of momenta we use only two momentum eigenstates (p1 and p2) that are perpendicular to the Lorentz boost direction. Consequently, in 2D momentum subspace we show that the entanglement of spin and momentum in the moving frame depends on the angle between them. Therefore, when spin and momentum are perpendicular the measure of entanglement is not an observer-dependent quantity in the inertial frame. Likewise, we have calculated the measure of entanglement (by using the concurrence) and have shown that entanglement decreases with respect to the increase in observer velocity. Finally, we argue that Wigner rotation is induced by Lorentz transformations and can be realized as a controlling operator.
Recall the two-component spinor, whose behavior under space rotations of angle θ about unit vector n was described by the operator exp (i θ n · M), where M = (1/2)σ. In Problem 21-3 you were concerned with the behavior of this spinor under Lorentz transformation. As was the case in space rotations it is sufficient to consider infinitesimal transformations. We write the corresponding operator as 1 + i ( v / c ) ⋅ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq413.tif"/>
The effect of a canonical Lorentz boost on the Newton-Wigner-Pryce position operator of a spin zero system is calculated. The resulting equations differ significantly from those corresponding to the Lorentz boost of a position coordinate in classical special relativity. The components of the boosted Newton-Wigner-Pryce position operator are interpreted in terms of a classical analog. It is seen that the transformed operator corresponds not to the coordinates of the same event in two reference frames, but to the coordinates of the intersection of the worldline of the system's center of momentum with hyperplanes of the same coordinate time in the two frames.