1.5 Representations of the Lorentz Group

transformations can be reduced to the hyperbolic rotations contained in the group SO(1, D − 1).

1.5 Representations of the Lorentz Group

As we already stressed the general advances in Lie group theory and Lie algebras were already conspicuous by the time Special Relativity was introduced, but the study of the Lorentz group proceeded for some time on an independent track, related to physics, and peculiarities of the case so(1, 3) were widely used and incorporated into the treatment. From the point of view of Mathematics, Èlie Cartan discovered the representations of the so(D, C) Lie algebras that now we name spinorial in 1913 [7], namely in between the advent of Special Relativity and that of General Relativity. From the point of view of Physics, Pauli introduced the intrinsic spin of quantum particles in 1927 [8] and, by means of the three σ -matrices named after him, he constructed the spinor representation of the three-dimensional rotation group SO(3). Pauli’s construction and three-dimensional spinors are quite special since they appear as a manifestation of the sporadic isomorphism so(3) su(2). In 1928 Paul Dirac discovered the fully relativistic theory of the electrons by introducing the anti-commuting γ -matrices and, in this way, he was able to show the connection between spinors and the Lorentz group [9]. Actually what Dirac did was the construction of the spinor representation of so(1, 3). Dirac spinors in D = 4 are once again special, since they appear as a manifestation of another sporadic isomorphism of Lie algebras, namely so(1, 3) sl(2, C). Yet, as it was already implicitly contained in Cartan’s paper of 1913, the existence of spinor representations is an intrinsic property of all Lie algebras of type so(D) and the systematic way to construct them is via the study of the Clifford algebras of Γ -matrices, defined by the following anti-commutation relations:

An exhaustive study of Γ -matrices and spinors is contained in Appendix A, to which we also refer for conventions. In this chapter we will study all representations of the Lorentz group and for the physically relevant case D = 4 we will dwell on the special features provided by the sporadic isomorphisms mentioned above.

From a general point of view the irreducible representations of so(1, D − 1) divide into two classes that have a profound physical significance, since they match with the spin-statistics theorem of Quantum Field Theory:

Bosons The bosonic representations of so(1, D − 1) are obtained from all tensor products of the fundamental representation, in other words they are tensors tμ1μ2...μn with n-indices. These tensors can be split into irreducible representations by means of two subsequent operations. First one applies to tμ1μ2...μn one of the symmetrization-anti-symmetrization schemes codified in the Young

the Lorentz algebra so(1, D − 1), forming a set of D × D matrices which contains 12 D(D − 1) elements, satisfy the commutation relations (1.4.9). If we construct a representation of the D-dimensional Clifford algebra (1.5.1), then according to the notation introduced in the Appendix (see (A.3.1)) we can set:

and we can easily verify that these generators satisfy the same commutation relations (1.4.9) as Jμν . So doing we succeeded in constructing a representation of the Lorentz algebra in dimension 2[D/2], which is the dimension of the gamma matrices. Such a representation is the spinor representation. Fields valued in the carrier vector space of the latter are the Dirac spinor fields. They are usually denoted as follows:

ψ −1

ψ

The entries of ψ are generically complex. As we discuss in Sect. A.4 of the appendix, Dirac spinors are not necessarily irreducible. Depending on the dimension D we can impose the Majorana or the Weyl condition, which are Lorentz invariant, or even both of them and, in this way, we obtain irreducible spinors.

A spinor tensor Ξμα1μ2...μn that is irreducible both as a spinor and as a tensor can be further reduced by subtracting Lorentz invariant γ -traces. Consider for instance a spinor tensor Ξ(μν) which is symmetric and traceless as a rank two tensor:

In a Lorentz invariant way we can extract from Ξ(μν) a spinor vector by setting:

In order to obtain a fully irreducible representation of the Lorentz group we have to substract such γ -traces:

where α is an appropriate coefficient that can be calculated in each dimension D

corresponded to a fully irreducible representation of the D-dimensional Lorentz group.

1.5.2The Two-Valued Homomorphism SO(1, 3) SL(2, C) in the Four-Dimensional Case

Let us enlarge the set of Pauli matrices introducing also:

and let us define the following linear combination of the four sigmas:

Consider now a generic element A SL(2, C). By definition A is a complex unimodular 2 × 2 matrix:

Since σμ provide a complete basis set for 2 × 2 matrices it follows that X is some other linear combination of the same matrices with new coefficients x˜μ:

Necessarily the new coefficients must be linear combination of the old ones:

By virtue of its own definition the 4 × 4 matrix Λ SO(1, 3) is an element of the Lorentz group.

This simple construction shows that to each element of A SL(2, C) we can uniquely associate a Lorentz group element Λ. The explicit form of the latter is easily obtained using the trace orthogonality of the σ μ matrices, namely 12 Tr(σμσν ) = δμν . Relying on this we can write: