Pauli–Lubanski pseudovector

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

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 Pμ invariant.[3]

Definition[edit]

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

where

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

Note , and

Wμ evidently satisfies

as well as the following commutator relations,

Consequently,

The scalar WμWμ 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 PμPμ for the mass of all states in a representation.

Little group[edit]

On an eigenspace of the 4-momentum operator with 4-momentum eigenvalue of the Hilbert space of a quantum system (or for that matter the standard representation with 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 (column vectors) of momentum space with 1 appended as a fifth row, see standard texts[8][9]) the following holds:[10]

  • The components of with replaced by form a Lie algebra. It is the Lie algebra of the Little group of , i.e. the subgroup of the homogeneous Lorentz group that leaves invariant.
  • For every irreducible unitary representation of 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 and extending by linearity.

The irreducible unitary representation of the Poincaré group are characterized by the eigenvalues of the two Casimir operators and . 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 in the representation space thus obtained.[11] Irreducibility follows from the construction of the representation space.

Massive fields[edit]

In quantum field theory, in the case of a massive field, the Casimir invariant WμWμ describes the total spin of the particle, with eigenvalues

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 [Wj , Wk] = i εjkl Wl m; hence W = mJ and W0 = 0, so that the little group amounts to the rotation group,

Since this is a Lorentz invariant quantity, it will be the same in all other reference frames.

It is also customary to take W3 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

where

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),

For particles with non-zero mass, and the fields associated with such particles,

Massless fields[edit]

In general, in the case of non-massive representations, two cases may be distinguished. For massless particles,

where K is the dynamic mass moment vector. So, mathematically, P2 = 0 does not imply W2 = 0.

Continuous spin representations[edit]

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.

Helicity representations[edit]

In a special case, W is parallel to P; or equivalently W × P = 0.   For non-zero W, this constraint can only be consistently imposed for luxons, since the commutator of the two transverse components of W is proportional to m2 J · P. For this family, W 2 = 0 and Wμ = λPμ; the invariant is, instead, (W0)2 = (W3)2, where

so the invariant is represented by the helicity operator

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 particle and anti-particle and Z particle) acquire non-zero mass.

Neutrinos were formerly considered to fall into this class as well. However, through neutrino oscillations, it is now known that at least two of the three mass eigenstates of the left-helicity neutrino and right-helicity anti-neutrino each must have non-zero mass.

See also[edit]

Notes[edit]

  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 2002, Chapter 2.

References[edit]

  • Rossmann, Wulf (2002), Lie Groups - An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0 19 859683 9