1.4 Mathematical Definition of the Lorentz Group

where E is the energy of the considered particle. Indeed, so doing, we find:

kinetic energy

where we recognize the Newtonian kinetic energy plus an absolute normalization of the zeroth level of E, arbitrary in Newtonian mechanics and fixed to a precise value in the relativistic case, namely to the rest energy E0 = mc2.

The third principle of special relativity which concludes the construction is

system made of N -components is a conserved quantity, namely all possible physical processes will preserve its value throughout time evolution. The correct transformations that relate inertial systems to each other is the Lorentz group, namely that group of linear substitutions which leaves the quadratic form (1.3.21) invariant.

1.4 Mathematical Definition of the Lorentz Group

Let us define mathematically the Lorentz group, which, as we emphasized in previous sections can be introduced for any space-time dimension D = 1 + (D − 1), where one is the number of time-like directions and D − 1 is the number of spacelike directions. In the classification of Classical Lie Groups, the Lorentz group is just SO(1, D − 1), whose elements are all those D × D matrices Λ that satisfy the following defining relation:

The reason of the above definition and of the choice of the form of the invariant matrix η was discussed in the previous section. It is dictated by the notion of Minkowski space-time and by the choice of (1.3.21) as the invariant quadratic form of Special Relativity.

Let us consider two physical events that in one inertial frame are described by the D-vectors {xμ, yμ}. In another inertial frame the same events will be described

by new D-vectors, obtained from the former ones by means of a linear substitution:

The Minkowskian scalar product of the two events will be frame independent, namely:

if and only if the condition (1.4.1) is satisfied, as it is immediately evident by the transcription in matrix notation of the fundamental quadratic form:

So it is mandatory to study the structure of the Lie group SO(1, D − 1) and the properties of its representations. From a historical perspective it is worth recalling that by the end of the XIX century, the theory of Lie groups, namely of continuous groups whose product law has an analytic structure, had already reached perfection through the work of Killing and Cartan. As we discuss more extensively in Sect. 3.2.5, the classification of all simple Lie groups and the construction of their fundamental representations, including those of the exceptional ones was presented in Cartan’s doctoral thesis of 1894. Hence the study of the D-dimensional Lorentz group SO(1, D − 1) could be considered at the time of Minkowski just an application of a well established theory to a specific case. Yet the history of science is never so linear and the D = 4 Lorentz group was separately studied in all of his aspects and for his own sake by several authors in a large number of physical and mathematical papers.

1.4.1 The Lorentz Lie Algebra and Its Generators

Let us consider a Lorentz matrix Λ which is infinitesimally close to the identity, namely:

which states that the matrix ηM is antisymmetric. Hence the Lorentz Lie algebra so(1, D − 1) is composed by all those matrices that satisfy (1.4.6). We easily construct the solution of such a problem, since, as a matrix, the Minkowskian metric η

squares to unity η2 = 1. Hence it suffices to parameterize the space of antisymmetric matrices A and any matrix M satisfying condition (1.4.6) will be of the form:

The space A has dimension 12 D(D − 1) and it is customary to introduce a basis of

row with the νth column and at the intersection of the νth row with μth column. The entries μν and νμ of Jμν have both norm 1 and have the same sign if ημμ = 1 while they have opposite signs if ημμ = −1. This means that

generators J0i are non-compact and give rise to special Lorentz transformations, while the generators Jij span the compact Lie subalgebra so(D − 1) so(1, D − 1). Altogether the commutation relations of this standard basis generators are:

and a generic element of the Lorentz Lie algebra can be written as:

where the parameters εμν = −ενμ constitute an antisymmetric tensor.

If we focus on the physical relevant case of D = 4, the overall number of Lorentz generators is six, three non-compact and three compact. Specifically we have: