It is evident from (1.5.16) that such a relation is not an isomorphism, rather it is a two-valued homorphism, since to the two matrices A and −A corresponds the same matrix Λ. In proper mathematical language this homorphism is a local isomorphism, since the corresponding Lie algebras sl(2, C) and so(1, 3) are isomorphic.

We conclude that the fundamental representation of the group SL(2, C) is actually a complex two-dimensional representation of the Lorentz group SO(1, 3). Which representation is it? The answer is easily given: it is that provided by a Weyl spinor. Indeed the Weyl condition halves the number of non-vanishing components of a Dirac spinor and from four we step down to two.

1.6 Lorentz Covariant Field Theories and the Little Group

Once the principles of special relativity have been accepted, the classical and quantum field theories one is led to consider are described, in D-dimensions, by an action principle of the form:

where {φ(x)} denotes collectively a set of fields, each of which belongs to some representation of the Lorentz group, either bosonic or fermionic and {∂φ(x)} denotes collectively the set of their derivatives with respect to the space-time coordinates:

The Lagrangian density L ({φ}, {∂φ}, x) is required to be invariant under Lorentz transformations. In addition we always assume that the full action is invariant under space-time translations, namely under transformations of the following form:

where cμ is a set of constant parameters. As an abstract group, the translation group in D dimensions T(D) is isomorphic to the Abelian non-compact Lie group RD . Its

generators are named P μ and can be identified with the total momentum operators which we declared to be constant in all physical processes (see Principle 1.3.3). This is automatically guaranteed by translation invariance of the action via Noether theorem that we recall later on in this chapter (see Sect. 1.7). Putting together spacetime translations and the Lorentz group, results in a semidirect product:

which is named the D-dimensional Poincaré group (see Fig. 1.7). The corresponding Lie algebra is described by the following commutations relations:

Fig. 1.7 Jules Henri Poincaré (1854–1912) was born near Nancy in a very influential French family. One of his cousins became President of the French Republic during the time of World War One, namely from 1913 to 1920. By that time, however, the great mathematician relative of the President was already dead. Henri Poincaré is often considered one of the last universal geniuses. His contributions to all branches of Mathematics are so extensive and profound that produce a sense of astonishment. Poincaré education was in Paris at the Ècole Polytechnique where he had such a teacher as Charles Hermite. After graduation he taught for some time at the University of Caen, but very young, in 1881 he was appointed professor at the Sorbonne and at the age of 32 he was already elected member of the French Academy of Sciences. In 1909, three years before his death he became member of the Academie Française. The major contributions of Poincaré to Mathematics are the complete solution of the three-body problem in Newtonian mechanics, the foundation of algebraic geometry and topology, where in 1894 he introduced the notion of the fundamental group and posed one of the most famous mathematical conjectures, the clear-cut formulation of non-Euclidian hyperbolic geometry and finally his controversial contribution to the birth special relativity [6]

which clearly expose the semidirect product structure. The momentum generators commute among themselves (1.6.5) but they transform in the fundamental representation of the Lorentz group as imposed by (1.6.6)–(1.6.7).

We quote a couple of examples of Poincaré invariant action functionals that we also use later on, while discussing Noether theorem (see Fig. 1.8). The first example is given by the free Dirac Lagrangian for an electron or another charged fermion which, utilizing the conventions and notations of Appendix A.4, takes the following form:

Fig. 1.8 Amalie Emmy Noether (1882–1935), together with Henrietta Leavitt and Madame Curie is one among the very few but very great woman-scientists who lived by the end of the XIX and the beginning of the XX century. German by nationality, she was born in a Jewish family in the Bavarian city of Erlangen, the same from where in 1872, ten years before her birth, Felix Klein had announced his famous programme, reducing the classification of possible geometries to the classification of Lie groups under which the geometric relations are invariant. Emmy’s father was also a mathematician and she studied at the University of Erlangen. After working several years as a voluntary assistant without salary, in 1915, just after the outbreak of world-war one she was invited by David Hilbert and Felix Klein to what was, by that time, the very center of the scientific world, namely the University of Göttingen. She had to suffer the prejudiced opposition of the faculty against women and obtained her habilitation only in 1919, after the defeat of Germany and the end of the war. Her algebraic Göttingen school became renowned around the world and she was described by David Hilbert and Albert Einstein as the most important woman in the history of mathematics. Although in theoretical physics Emmy Noether is mostly known for her theorem on the relation between symmetries and conserved currents, her major contributions were in pure mathematics and in abstract algebra in particular, which she contributed to refound. To this effect it suffices to recall the notion of Noetherian Rings. It must be noted that David Hilbert invited Miss Noether to Göttingen precisely because he was puzzled by the issue of energy conservation in Einstein’s theory of Gravitation. The fact that gravitational energy could gravitate seemed to him a violation of the energy conservation theorem. By means of her theorem, Emmy Noether solved the problem not only for General Relativity but for all systems endowed with a continuous group of symmetries. In 1932 in her plenary address to the International Congress of Mathematicians in Zürich, Emmy Noether was at the top of her mathematical career and a world-wide recognized authority. She had also worked, for the winter semester 1928–1929, at Moscow State University, where she collaborated with Lev Pontryagin and Nikolai Chebotaryov. The same year 1932, together with Emil Artin, she received a long-due recognition by means of the Ackermann-Teubner Memorial Award for Mathematics. In 1933, Hitler rose to power, Emy’s chair in Göttingen was revoked and she emigrated to the Unitated States of America where she obtained a chair in Bryn Mawr College in Pennsylvanya. Unfortunately two years later, in 1935, she died from cancer

The second example we mention is provided by the action functional for a scalar field, with a self-interaction encoded in a potential function W (φ):

28 1 Special Relativity: Setting the Stage

By means of Fourier transform (1.6.19) takes the same form as (1.6.13) with ϕ(k) substituted by Vν (k). The auxiliary condition (1.6.20 ) becomes

So when the momentum vector is rotated to the rest frame (1.6.21) implies V0 = 0 and what remains is Vi namely a vector representation of the little group SO(D − 1) which contains D − 1 states.

In the massless case one arrives at the same reduction to a representation of the little group SO(D − 2) but in a different way, namely using local gauge invariances. For instance let us consider the case of a massless vector field. The action is the same as that in (1.6.16) but with m = 0. Correspondingly the field equation is just:

In this case the condition ∂ · V = 0 cannot be derived from the equation, but it can be imposed as a gauge fixing condition since, at m = 0 the action is invariant under the following local symmetry:

A careful use of this symmetry allows to show that, at the end of the day, when kμ is reduced to the normal form (1.6.15) of a light-like vector, the only remaining degrees of freedom of Vν (k), are those of an SO(D − 2) vector living in the perpendicular space to the wave propagation. We do not dwell on the details of this derivation since we will address it for the graviton in comparison with the photon in Sect. 5.7.1. The important message to be remembered is that the degrees of freedom of a Lorentz field are given by the dimension of the corresponding representation of the little group, SO(D − 1) in the massive case SO(D − 2) in the massless one.

1.6.1 Representations of the Massless Little Group in D = 4

In view of the conclusions reached in the previous sections it is useful to consider the representations of the massless little group for the physically relevant case D = 4. In this case there are some peculiarities since all representations of SO(2) happen to be two-dimensional and characterized by a single number s that is the spin of the corresponding massless particles. Let us see how this happens.

To begin with, an irreducible representation bosonic representation of SO(2) is a traceless symmetric tensor with s-indices:

The number of independent components of such a tensor is easily calculated. In d = 2 an object with s indices has components. Yet the trace of such an