Pauli–Lubanski pseudovector

For the notation, see Ricci calculus.

In physics, the Pauli–Lubanski pseudovector is an operator defined from the momentum and angular momentum, used in the quantum-relativistic description of angular momentum. It is named after Wolfgang Pauli and Józef Lubański.[1]

It describes the spin states of moving particles.[2] It is the generator of the little group of the Poincaré group, that is the maximal subgroup (with four generators) leaving the eigenvalues of the four-momentum vector P_μ invariant.[3]

Definition

It is usually denoted by W (or less often by S) and defined by:[4][5][6]

${\displaystyle W_{\mu }\mathrel {\stackrel {\text{def}}{=}} {\tfrac {1}{2}}\varepsilon _{\mu \nu \rho \sigma }J^{\nu \rho }P^{\sigma },}$

where

• ${\displaystyle \varepsilon _{\mu \nu \rho \sigma }}$ is the four-dimensional totally antisymmetric Levi-Civita symbol;
• ${\displaystyle J^{\nu \rho }}$ is the relativistic angular momentum tensor operator (${\displaystyle M^{\nu \rho }}$);
• ${\displaystyle P^{\sigma }}$ is the four-momentum operator.

In the language of exterior algebra, it can be written as the Hodge dual of a trivector,[7]

$${\displaystyle \mathbf {W} =\star (\mathbf {J} \wedge \mathbf {p} ).}$$

Note ${\displaystyle W_{0}={\vec {J}}\cdot {\vec {P}}}$, and ${\displaystyle {\vec {W}}=E{\vec {J}}-{\vec {P}}\times {\vec {K}}}$ where ${\displaystyle {\vec {J}}}$ is the generator of rotations and ${\displaystyle {\vec {K}}}$ is the generator of boosts.

W_μ evidently satisfies $${\displaystyle P^{\mu }W_{\mu }=0,}$$

as well as the following commutator relations, $${\displaystyle {\begin{aligned}\left[P^{\mu },W^{\nu }\right]&=0,\\\left[J^{\mu \nu },W^{\rho }\right]&=i\left(g^{\rho \nu }W^{\mu }-g^{\rho \mu }W^{\nu }\right),\end{aligned}}}$$

Consequently, $${\displaystyle \left[W_{\mu },W_{\nu }\right]=-i\epsilon _{\mu \nu \rho \sigma }W^{\rho }P^{\sigma }.}$$

The scalar W_μW^μ is a Lorentz-invariant operator, and commutes with the four-momentum, and can thus serve as a label for irreducible unitary representations of the Poincaré group. That is, it can serve as the label for the spin, a feature of the spacetime structure of the representation, over and above the relativistically invariant label P_μP^μ for the mass of all states in a representation.

Little group

On an eigenspace ${\displaystyle S}$ of the 4-momentum operator ${\displaystyle P}$ with 4-momentum eigenvalue ${\displaystyle k}$ of the Hilbert space of a quantum system (or for that matter the standard representation with ℝ^4 interpreted as momentum space acted on by 5×5 matrices with the upper left 4×4 block an ordinary Lorentz transformation, the last column reserved for translations and the action effected on elements ${\displaystyle p}$ (column vectors) of momentum space with 1 appended as a fifth row, see standard texts[8][9]) the following holds:[10]

• The components of ${\displaystyle W}$ with ${\displaystyle P^{\mu }}$ replaced by ${\displaystyle k^{\mu }}$ form a Lie algebra. It is the Lie algebra of the Little group ${\displaystyle L_{k}}$of ${\displaystyle k}$, i.e. the subgroup of the homogeneous Lorentz group that leaves ${\displaystyle k}$ invariant.
• For every irreducible unitary representation of ${\displaystyle L_{k}}$ there is an irreducible unitary representation of the full Poincaré group called an induced representation.
• A representation space of the induced representation can be obtained by successive application of elements of the full Poincaré group to a non-zero element of ${\displaystyle S}$ and extending by linearity.

The irreducible unitary representation of the Poincaré group are characterized by the eigenvalues of the two Casimir operators ${\displaystyle P^{2}}$ and ${\displaystyle W^{2}}$. The best way to see that an irreducible unitary representation actually is obtained is to exhibit its action on an element with arbitrary 4-momentum eigenvalue ${\displaystyle p}$ in the representation space thus obtained.[11] : 62–74 Irreducibility follows from the construction of the representation space.

Massive fields

In quantum field theory, in the case of a massive field, the Casimir invariant W_μW^μ describes the total spin of the particle, with eigenvalues $${\displaystyle W^{2}=W_{\mu }W^{\mu }=-m^{2}s(s+1),}$$ where s is the spin quantum number of the particle and m is its rest mass.

It is straightforward to see this in the rest frame of the particle, the above commutator acting on the particle's state amounts to [W_j , W_k] = i ε_jkl W_l m; hence W→ = mJ→ and W^0 = 0, so that the little group amounts to the rotation group, $${\displaystyle W_{\mu }W^{\mu }=-m^{2}{\vec {J}}\cdot {\vec {J}}.}$$ Since this is a Lorentz invariant quantity, it will be the same in all other reference frames.

It is also customary to take W^3 to describe the spin projection along the third direction in the rest frame.

In moving frames, decomposing W = (W₀, W→) into components (W₁, W₂, W₃), with W₁ and W₂ orthogonal to P→, and W₃ parallel to P→, the Pauli–Lubanski vector may be expressed in terms of the spin vector S→ = (S₁, S₂, S₃) (similarly decomposed) as $${\displaystyle {\begin{aligned}W_{0}&=PS_{3},&W_{1}&=mS_{1},&W_{2}&=mS_{2},&W_{3}&={\frac {E}{c^{2}}}S_{3},\end{aligned}}}$$

where $${\displaystyle E^{2}=P^{2}c^{2}+m^{2}c^{4}}$$ is the energy–momentum relation.

The transverse components W₁, W₂, along with S₃, satisfy the following commutator relations (which apply generally, not just to non-zero mass representations), $${\displaystyle {\begin{aligned}[][W_{1},W_{2}]&={\frac {ih}{2\pi }}\left(\left({\frac {E}{c^{2}}}\right)^{2}-\left({\frac {P}{c}}\right)^{2}\right)S_{3},&[W_{2},S_{3}]&={\frac {ih}{2\pi }}W_{1},&[S_{3},W_{1}]&={\frac {ih}{2\pi }}W_{2}.\end{aligned}}}$$

For particles with non-zero mass, and the fields associated with such particles, $${\displaystyle [W_{1},W_{2}]={\frac {ih}{2\pi }}m^{2}S_{3}.}$$

Massless fields

Main article: massless particle

In general, in the case of non-massive representations, two cases may be distinguished. For massless particles,[11]: 71–72  $${\displaystyle W^{2}=W_{\mu }W^{\mu }=-E^{2}\left((K_{2}-J_{1})^{2}+(K_{1}+J_{2})^{2}\right)\mathrel {\stackrel {\mathrm {def} }{=}} -E^{2}\left(A^{2}+B^{2}\right),}$$ where K→ is the dynamic mass moment vector. So, mathematically, P^2 = 0 does not imply W^2 = 0.

Continuous spin representations

In the more general case, the components of W→ transverse to P→ may be non-zero, thus yielding the family of representations referred to as the cylindrical luxons ("luxon" is another term for "massless particle"), their identifying property being that the components of W→ form a Lie subalgebra isomorphic to the 2-dimensional Euclidean group ISO(2), with the longitudinal component of W→ playing the role of the rotation generator, and the transverse components the role of translation generators. This amounts to a group contraction of SO(3), and leads to what are known as the continuous spin representations. However, there are no known physical cases of fundamental particles or fields in this family. It can be argued that continuous spin states possess an internal degree of freedom not seen in observed massless particles.[11]: 69–74 

Helicity representations

In a special case, ${\displaystyle {\vec {W}}}$ is parallel to ${\displaystyle {\vec {P}};}$ or equivalently ${\displaystyle {\vec {W}}\times {\vec {P}}={\vec {\boldsymbol {0}}}.}$ For non-zero ${\displaystyle {\vec {W}}}$ this constraint can only be consistently imposed for luxons (massless particles), since the commutator of the two transverse components of ${\displaystyle {\vec {W}}}$ is proportional to ${\displaystyle m^{2}{\vec {J}}\cdot {\vec {P}}\,.}$ For this family, ${\displaystyle W^{2}=0}$ and ${\displaystyle W^{\mu }=\lambda \,P^{\mu }}$ the invariant is, instead given by $${\displaystyle \left(W^{0}\right)^{2}=\left(W^{3}\right)^{2},}$$ where $${\displaystyle W^{0}=-{\vec {J}}\cdot {\vec {P}},}$$ so the invariant is represented by the helicity operator $${\displaystyle W^{0}/P.}$$

All particles that interact with the weak nuclear force, for instance, fall into this family, since the definition of weak nuclear charge (weak isospin) involves helicity, which, by above, must be an invariant. The appearance of non-zero mass in such cases must then be explained by other means, such as the Higgs mechanism. Even after accounting for such mass-generating mechanisms, however, the photon (and therefore the electromagnetic field) continues to fall into this class, although the other mass eigenstates of the carriers of the electroweak force (the
W^±
boson and anti-boson and
Z^0
boson) acquire non-zero mass.

Neutrinos were formerly considered to fall into this class as well. However, because neutrinos have been observed to oscillate in flavour, it is now known that at least two of the three mass eigenstates of the left-helicity neutrinos and right-helicity anti-neutrinos each must have non-zero mass.

See also

• Center of mass (relativistic)
• Wigner's classification
• Angular momentum operator
• Casimir operator
• Chirality
• Pseudovector
• Pseudotensor
• Induced representation

Notes

1. Lubański 1942a, pp. 310–324, Lubański 1942b, pp. 325–338
2. Brown 1994, pp. 180–181
3. Wigner 1939, pp. 149–204
4. Ryder 1996, p. 62
5. Bogolyubov 1989, p. 273
6. Ohlsson 2011, p. 11
7. Penrose 2005, p. 568
8. Hall 2015, Formula 1.12.
9. Rossmann 2002, Chapter 2.
10. Tung 1985, Theorem 10.13, Chapter 10.
11. Weinberg, Steven (1995). The Quantum Theory of Fields. Vol. 1. Cambridge University Press. ISBN 978-0521550017.

References

• Bogolyubov, N.N. (1989). General Principles of Quantum Field Theory (2nd ed.). Springer Verlag. ISBN 0-7923-0540-X.
• Brown, L. S. (1994). Quantum Field Theory. Cambridge University Press. ISBN 978-0-521-46946-3.
• Hall, Brian C. (2015), Lie groups, Lie algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, doi:10.1007/978-3-319-13467-3, ISBN 978-3319134666, ISSN 0072-5285
• Lubański, J. K. (1942a). "Sur la theorie des particules élémentaires de spin quelconque. I". Physica (in French). 9 (3): 310–324. Bibcode:1942Phy.....9..310L. doi:10.1016/S0031-8914(42)90113-7.
• Lubański, J. K. (1942b). "Sur la théorie des particules élémentaires de spin quelconque. II". Physica (in French). 9 (3): 325–338. Bibcode:1942Phy.....9..325L. doi:10.1016/S0031-8914(42)90114-9.
• Ohlsson, T. (2011). Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory. Cambridge University Press. ISBN 978-1-139-50432-4.
• Penrose, R. (2005). The Road to Reality. Vintage books. ISBN 978-0-09-944068-0.
• Rossmann, Wulf (2002), Lie Groups - An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0-19-859683-9
• Ryder, L.H. (1996). Quantum Field Theory (2nd ed.). Cambridge University Press. ISBN 0-521-47814-6.
• Tung, Wu-Ki (1985). Group Theory in Physics (1st ed.). New Jersey·London·Singapore·Hong Kong: World Scientific. ISBN 978-9971966577.
• Weinberg, S. (2002) [1995], Foundations, The Quantum Theory of Fields, vol. 1, Cambridge: Cambridge University Press, ISBN 0-521-55001-7
• Wigner, E. P. (1939). "On unitary representations of the inhomogeneous Lorentz group". Annals of Mathematics. 40 (1): 149–204. Bibcode:1939AnMat..40..149W. doi:10.2307/1968551. JSTOR 1968551. MR 1503456. S2CID 121773411.