Pauli-Lubanski pseudo vector

The Pauli-Lubanski pseudo-vector is named after Wolfgang Pauli and Józef Lubański . It occurs in the special theory of relativity and the associated quantum theory . For massive particles, its square is the (negative) square of their spin times the square of their mass. For massless particles it is proportional to the four-momentum with a factor which is the helicity of the particle. ${\displaystyle W^{\mu }}$

The Pauli-Lubanski pseudo-vector is defined as

${\displaystyle W^{\mu }={\frac {1}{2}}\varepsilon ^{\mu \nu \rho \sigma }P_{\nu }M_{\rho \sigma }\ ,}$

in which

• ${\displaystyle \varepsilon }$the Levi Civita symbol ,
• ${\displaystyle P}$the momentum operator and
• ${\displaystyle M}$denote the angular momentum tensor.

The components of the Pauli-Lubanski pseudo-vector can also be used as

${\displaystyle W^{0}={\vec {P}}\cdot {\vec {J}}\ ,\ {\vec {W}}=P^{0}{\vec {J}}-{\vec {P}}\times {\vec {N}}\ ,}$

where the angular momentum operator is and . ${\displaystyle {\vec {J}}}$ ${\displaystyle N_{k}=-M_{0k}}$

properties

Because the Levi-Civita symbol is totally antisymmetric, the Pauli-Lubanski pseudo-vector is perpendicular to the four-pulse

${\displaystyle P_{\mu }W^{\mu }=0}$

and exchanged with him

${\displaystyle [P^{\mu },W^{\nu }]=0}$.

With the angular momentum tensor, the Pauli-Lubanski pseudo-vector has the commutator relation

${\displaystyle [M^{\mu \nu },W^{\rho }]=-\mathrm {i} \left(g^{\mu \rho }W^{\nu }-g^{\nu \rho }W^{\mu }\right)}$,

where is the metric tensor , and with itself ${\displaystyle g}$

${\displaystyle [W^{\mu },W^{\nu }]=-\mathrm {i} \varepsilon ^{\mu \nu \rho \sigma }W_{\rho }P_{\sigma }}$.

Therefore the square of the Pauli-Lubanski pseudo-vector swaps with all generators and the Poincaré group . So this generator is a Casimir operator of algebra . In particular, all momentum wave functions of a particle are eigenfunctions of having the same eigenvalue. Likewise, the square of its momentum is a Casimir operator. The eigenvalues ​​of both determine the particle's mass and spin, or, if the mass disappears, its helicity. ${\displaystyle W^{2}=W^{\mu }W_{\mu }}$ ${\displaystyle P^{\mu }}$ ${\displaystyle M^{\mu \nu }}$ ${\displaystyle W^{2}}$ ${\displaystyle \Psi :p\mapsto \Psi (p)}$ ${\displaystyle W^{2}}$ ${\displaystyle P^{2}}$

Effect on single-particle states

Massive particles

For a massive particle with mass , there are states whose pulse wave function at not disappear. There applies ${\displaystyle m>0}$ ${\displaystyle \Psi }$ ${\displaystyle {\underline {p}}={\bigl (}m,0,0,0{\bigr )}}$

${\displaystyle (W^{2}\Psi )({\underline {p}})=-m^{2}({\vec {J}}^{2}\Psi )({\underline {p}})=-m^{2}s(s+1)\Psi ({\underline {p}})\ ,}$

where is the spin of the particle. ${\displaystyle s}$

As a Casimir operator acts on every irreducible representation of the Poincaré group according to Schur's lemma as a multiple of . ${\displaystyle W^{2}}$ ${\displaystyle 1}$

Consequently, applies not only to but for all impulses and for every wave function of the particle. ${\displaystyle W^{2}\Psi =-m^{2}s(s+1)\Psi }$ ${\displaystyle {\underline {p}}}$ ${\displaystyle p}$

Hence the square of the spin. ${\displaystyle -W^{2}/m^{2}}$

Massless particles

For a massless particle with there are states whose pulse wave functions at not disappear. There applies ${\displaystyle m=0}$ ${\displaystyle \Psi }$ ${\displaystyle {\underline {p}}={\bigl (}E,0,0,E{\bigr )}}$

${\displaystyle (W^{2}\Psi )({\underline {p}})=-E^{2}{\Bigl (}(J^{1}+N^{2})^{2}\Psi +(J^{2}-N^{1})^{2}\Psi {\Bigr )}({\underline {p}})}$.

The Casimir operator is therefore non-positive for all momenta and for all massless states. ${\displaystyle W^{2}}$ ${\displaystyle p}$

However, contain massless representations of the Poincaré group with an infinite number of helicities , or . Such particles (irreducible representations of the Poincaré group) have never been observed and would give an infinite heat capacity of each cavity. So it is on physical, massless particles , as it applies in the limit case of massive particles with fixed spin. From the explicit form of the generator it follows for all wave functions ${\displaystyle W^{2}<0}$ ${\displaystyle h}$ ${\displaystyle \{h\}=\mathbb {Z} \ }$ ${\displaystyle \{h\}=\mathbb {Z} +1/2}$ ${\displaystyle W^{2}=0}$ ${\displaystyle m\rightarrow 0}$ ${\displaystyle M^{\mu \nu }}$

${\displaystyle W^{\mu }=hP^{\mu }}$

initially at . Because of the Lorentz invariance, this also applies to every non-vanishing momentum of the forward light cone. ${\displaystyle {\underline {p}}}$ ${\displaystyle p}$

The factor is the helicity of the particle. ${\displaystyle h={\vec {P}}{\vec {J}}/|{\vec {P}}|}$

literature

• Edouard B. Manoukian: Quantum Field Theory I: Foundations and Abelian and Non-Abelian Gauge Theories . Springer, 2016, ISBN 978-3-319-30938-5 , pp. 141-146 .

Individual evidence