Pauli-Lubanski pseudovector ( $W_{μ}$ )

The Pauli-Lubanski pseudovector is

W_{μ} = ϵ_{μν ρ σ} M^{ν ρ} P^{σ} /2

, up to an overall sign convention. Its square is a Casimir of the Poincare algebra and measures spin in a massive irreducible representation.

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Pauli-Lubanski orthogonality ( $W^{μ} P_{μ} = 0$ )

In

W^{μ} = - ϵ^{μν ρ σ} P_{ν} M_{ρ σ} /2

, contracting with

P_{μ}

gives zero because the commuting momentum products are symmetric in

μ, ν

while the alternating tensor is antisymmetric. Reordering

P

and

M

makes no difference because the resulting metric contractions with the alternating tensor vanish.

// Target: special-relativity.bigb

Massive Pauli-Lubanski Casimir ( $W^{2} = - m^{2} s (s + 1)$ )

For a massive positive-energy unitary irreducible representation of the Poincaré group with signature

(+ - - -)

, choose a rest frame. The Pauli-Lubanski pseudovector has zero time component and spatial components

m J_{i}

, up to a common sign convention. Its square is therefore

- m^{2} J^{2}

. The SU(2) little-group representation has spin

s

and

J^{2} = s (s + 1)

, proving the invariant value. The mass and spin label the massive induced representation of the Poincare double cover.

Massless longitudinal Pauli-Lubanski eigenvalues ( $(W_{0}, W_{3}) = (k j_{3}, - k j_{3})$ )

With signature

(+ - - -)

and

ϵ_{0123} = + 1

(k, 0, 0, k)

j_{3}

give lower components

W_{0} = k j_{3}

and

W_{3} = - k j_{3}

. They obey

W_{μ} P^{μ} = k (W_{0} + W_{3}) = 0

. These longitudinal identities require only the momentum and rotation-projection eigenvalues, not a statement about transverse little-group translations.

Massive rest-frame Pauli-Lubanski eigenvalues ( $(W_{0}, W_{3}) = (0, - m j_{3})$ )

With signature

(+ - - -)

,

ϵ_{0123} = + 1

,

J_{3} = M^{12}

and the lower-index definition of

W_{μ}

, a rest momentum

(m, 0, 0, 0)

gives

W_{0} = 0

and

W_{3} = - m J_{3}

on that momentum eigenspace. A Hermitian spin projection

j_{3}

consequently gives eigenvalues

0

and

- m j_{3}

. Raising the third index changes its sign.

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Pauli–Lubanski pseudovector by

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The Pauli-Lubanski pseudovector is an important concept in theoretical physics, particularly in the context of relativistic quantum mechanics and the study of angular momentum and symmetry in particle physics. It serves as a relativistic generalization of angular momentum. In the realm of special relativity, the total angular momentum \( J^{\mu} \) of a system can be expressed in terms of the orbital angular momentum and the intrinsic spin of the particles involved.

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