It is usually denoted by and defined by: where In the language of exterior algebra, it can be written as the Hodge dual of a trivector, Note, and evidently satisfies as well as the following commutator relations, Consequently, The scalar 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 for the mass of all states in a representation.
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.
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. 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 describes the total spin of the particle, with eigenvalues where is the spin quantum number of the particle and 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 ; hence and, 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 to describe the spin projection along the third direction in the rest frame. In moving frames, decomposing into components, with and orthogonal to, and parallel to, the Pauli–Lubanski vector may be expressed in terms of the spin vector = as where is the energy–momentum relation. The transverse components, along with, satisfy the following commutator relations, For particles with non-zero mass, and the fields associated with such particles,
Massless fields
In general, in the case of non-massive representations, two cases may be distinguished. For massless particles, where is the dynamic mass moment vector. So, mathematically, 2 = 0 does not imply 2 = 0.
Continuous spin representations
In the more general case, the components of transverse to may be non-zero, thus yielding the family of representations referred to as the cylindrical luxons, their identifying property being that the components of form a Lie subalgebra isomorphic to the 2-dimensional Euclidean group, with the longitudinal component of playing the role of the rotation generator, and the transverse components the role of translation generators. This amounts to a group contraction of, 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 proved that continuous spin states are unphysical.
Helicity representations
In a special case, is parallel to ; or equivalently. For non-zero, this constraint can only be consistently imposed for luxons, since the commutator of the two transverse components of is proportional to . For this family, and ; the invariant is, instead,, 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 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 continues to fall into this class, although the other mass eigenstates of the carriers of the electroweak force 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.
