Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1203.3147 · International Journal of Quantum Information

QUANTUM STATE OF A FREE SPIN-½ PARTICLE AND THE INEXTRICABLE DEPENDENCE OF SPIN AND MOMENTUM UNDER LORENTZ TRANSFORMATIONS

Tiago Debarba, Reinaldo O. Vianna

Published Jun 1, 2012• 15 citations
arXivHTMLPDF

Abstract

We revise the Dirac equation for a free particle and investigate Lorentz transformations on spinors. We study how the spin quantization axis changes under Lorentz transformations, and evince the interplay between spin and momentum in this context.
DOI
10.1142/s0219749912300033

Related papers

Quantum information in inertial frames

Marco O. Lanzagorta

Mathjax On | Off Abstract In this chapter we will explore some issues related to relativistic quantum information in the context of inertial frames of reference. We will begin by analysing how a qubit state is affected by a Lorentz transformation and we will limit most of our discussion to the case of spin-1/2 particles described by momentum eigenstates of the Dirac equation.

Morgan & Claypool Publishers eBooks•Jun 1, 2014•1 citation

Maps for Lorentz transformations of spin

Thomas F. Jordan, Anil Shaji, E. C. G. Sudarshan

Lorentz transformations of spin density matrices for a particle with positive mass and spin $1∕2$ are described by maps of the kind used in open quantum dynamics. They show how the Lorentz transformations of the spin depend on the momentum. Since the spin and momentum generally are not independent, the maps generally are not completely positive and act in limited domains. States with two momentum values are considered, so the maps are for the spin qubit correlated with the qubit made from the two momentum values, and results from the open quantum dynamics of two coupled qubits can be applied. Inverses are used to show that every Lorentz transformation completely removes the spin polarization, and so completely removes the information, from a number of spin density matrices. The size of the spin polarization that is removed is calculated for particular cases.

arXiv:quant-ph/0511067•Physical Review A•Mar 13, 2006•33 citations

The Connection Between Spin and Statistics

Wolfgang Ernst Friederich Pauli

In the following paper we conclude for the relativistically invariant wave equation for free particles: From postulate (I), according to which the energy must be positive, the necessity of Fermi-Dirac statistics for particles with arbitrary half-integral spin; from postulate (II), according to which observables on different space-time points with a space-like distance are commutable, the necessity of Einstein-Bose statistics for particles with arbitrary integral spin. It has been found useful to divide the quantities which are irreducible against Lorentz transformations into four symmetry classes which have a commutable multiplication like $+1,\ensuremath{-}1,+\ensuremath{\epsilon},\ensuremath{-}\ensuremath{\epsilon}$ with ${\ensuremath{\epsilon}}^{2}=1$.

Physical Review•Oct 15, 1940•813 citations

Relationship of the Foldy-Wouthuysen Transformation to Lorentz Transformations

Herbert Jehle, William C. Parke

The covariant spinor formalism of Infeld and van der Waerden and of Uhlenbeck and Laporte permits a simple interpretation of the relationship between a Foldy-Wouthuysen transformation for a free particle and an associated pair of Lorentz transformations.

Physical Review•Feb 8, 1965•14 citations

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

Taeseung Choi

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.

arXiv:1209.4440•Journal of the Korean Physical Society•Apr 1, 2013•23 citations

Dirac Equation

Lukong Cornelius Fai

We re-examine the square of the Hamiltonian, https://www.w3.org/1998/Math/MathML"> Η https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , in Equation 15.7 and also the Dirac approach in finding the relativistic Hamiltonian. The classical Hamiltonian for the free particle should be the positive square root of Equation 15.7 which is not a linear function of the momenta. Applying this to the problem, say for example in Equation 15.10, we have the resultant wave equation that is asymmetrical with respect to space and time derivatives and so the necessity to linearize the equation in the momentum operators. So for Equation 15.10, it is impossible to include the Pauli spin matrices without destroying the invariance of the theory since the spin matrices transform the components of a three-dimensional rather than a four-dimensional vector. In addition, the function, https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , has one component rather than two components like spinors and so the Schrödinger relativistic equation represents a particle with no spin. So for a potential to be added in Equation 15.10, the Lorentz-transformation properties of such an addition should be guaranteed by matrix algebra which is the requirement for the Hamiltonian https://www.w3.org/1998/Math/MathML"> Η ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> to be linear in the momentum operators and guarantees relativistic invariance. This matrix algebra which permits the linearization of the Hamiltonian in the momentum operator was introduced by Dirac with the help of the Pauli matrices.

CRC Press eBooks•Apr 19, 2022•0 citations
Lorentz transformations of spin density matrices for a particle with positive mass and spin 1∕21∕21∕2 are described by maps of the kind used in open quantum dynamics. They show how the Lorentz transformations of the spin depend on the momentum. Since the spin and momentum generally are not independent, the maps generally are not completely positive and act in limited domains. States with two momentum values are considered, so the maps are for the spin qubit correlated with the qubit made from the two momentum values, and results from the open quantum dynamics of two coupled qubits can be applied. Inverses are used to show that every Lorentz transformation completely removes the spin polarization, and so completely removes the information, from a number of spin density matrices. The size of the spin polarization that is removed is calculated for particular cases.
In the following paper we conclude for the relativistically invariant wave equation for free particles: From postulate (I), according to which the energy must be positive, the necessity of Fermi-Dirac statistics for particles with arbitrary half-integral spin; from postulate (II), according to which observables on different space-time points with a space-like distance are commutable, the necessity of Einstein-Bose statistics for particles with arbitrary integral spin. It has been found useful to divide the quantities which are irreducible against Lorentz transformations into four symmetry classes which have a commutable multiplication like $+1,\ensuremath{-}1,+\ensuremath{\epsilon},\ensuremath{-}\ensuremath{\epsilon}$ with ${\ensuremath{\epsilon}}^{2}=1$.
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.
We re-examine the square of the Hamiltonian, https://www.w3.org/1998/Math/MathML"> Η https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , in Equation 15.7 and also the Dirac approach in finding the relativistic Hamiltonian. The classical Hamiltonian for the free particle should be the positive square root of Equation 15.7 which is not a linear function of the momenta. Applying this to the problem, say for example in Equation 15.10, we have the resultant wave equation that is asymmetrical with respect to space and time derivatives and so the necessity to linearize the equation in the momentum operators. So for Equation 15.10, it is impossible to include the Pauli spin matrices without destroying the invariance of the theory since the spin matrices transform the components of a three-dimensional rather than a four-dimensional vector. In addition, the function, https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , has one component rather than two components like spinors and so the Schrödinger relativistic equation represents a particle with no spin. So for a potential to be added in Equation 15.10, the Lorentz-transformation properties of such an addition should be guaranteed by matrix algebra which is the requirement for the Hamiltonian https://www.w3.org/1998/Math/MathML"> Η ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> to be linear in the momentum operators and guarantees relativistic invariance. This matrix algebra which permits the linearization of the Hamiltonian in the momentum operator was introduced by Dirac with the help of the Pauli matrices.