Quantum Numbers as Representations of Gauge Groups

As emphasized by Klein and Weyl, a group is a collection of operations leaving a certain “object” unchanged. This amounts to classifying the symmetries of the object. When the “object” in question is the laws of Physics in a space-time with negligible gravitation-induced curvature, the symmetries can be classified as follows: (i) No point in four- dimensional space-time is privileged, hence one can shift or translate the origin of space-time arbitrarily in four directions. Noether’s theorem then implies there are four associated conserved quantities, namely the three components of space momentum and the energy. These four quantities naturally constitute the components of a 4-vector Pμ, μ = 0, 1, 2, 3. (ii) No direction is special in space; leading to three conserved quantities Ji, i = 1, 2, 3. (iii) There is no special inertial frame; the same laws of Physics hold in inertial frames moving with constant speed in any one of the three independent directions. What is generally known as Noether’s Theorem states that if the Lagrangian function for a physical system is not affected by a continuous change (transformation) in the coordinate system used to describe it, then there will be a corresponding conservation law; i.e. there is a quantity that is constant. For example, if the Lagrangian is independent of the location of the origin then the system will preserve (or conserve) linear momentum. If it is independent of the base time then energy is conserved. If it is independent of the angle of measurement then angular momentum is conserved.

As we suggested above, it is possible to get a non-mathematical insight into Noether’s theorem relating symmetries to conserved quantities. Consider a single particle moving in a completely homogeneous space. It cannot come to a stop or change its velocity because this would have to happen at some particular point, but all points being equal, it is impossible to choose one. Hence the particle has no choice but to move at constant velocity or, in other words, to conserve its linear momentum, which was anciently called “impetus”. It is easy to extend the argument to a rotating object in an isotropic space and conclude that it cannot come to a stop at any particular angle since there is no special angle; hence its angular momentum is conserved.

(ii) and (iii) amount to covariance of the laws of physics under rotations in a four-dimensional space with a metric that is not positive-definite. The squared length of a 4-vector defined via this metric must then be an important invariant independent of the orientation or the velocity of the frame. Indeed, for the 4-vector Pμ this is the squared mass m2 of the particle, and it is one of the two invariant labels used in specifying the representation. The other label is the squared length of another 4-vector called the Pauli-Lubanski vector. It then follows from the algebra of the group that this squared length takes on values s(s + 1) and that in contrast to m2, which assumes continuous values, s can only be zero, or a positive integer, or half a positive odd integer. The unitary representation of the Poincaré group for a particle of mass m and spin s provides its relativistic quantum mechanical wave function. The equation of motion the wave function must obey also comes with the representation; it is the Bargmann-Wigner equation for that spin and mass. In relativistic quantum mechanics and quantum field theory, the Bargmann–Wigner equations describe free particles of arbitrary spin j, an integer for bosons (j = 1, 2, 3 …) or half-integer for fermions (j = 123252 …). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields. The procedure of second quantization then naturally promotes the wave functions to quantized field operators, and in a sense demotes the particles to quanta created or destroyed by these operators. Pauli’s spin-statistics theorem, based on a set of very general requirements such as the existence of a lowest energy vacuum state, the positivity of energy and probability, microcausality, and the invariance of the laws of Physics under the Poincaré group, leads to the result that the only acceptable quantum conditions for field operators of integer-spin particles are commutation relations, while those corresponding to half-integer spin must obey anticommutation relations. The standard terms for the two families of particles are bosons and fermions, respectively. The Pauli’s Exclusion Principle, or the impossibility of putting two electrons into the same state, is now seen to be the result of the anticommutation relation between electron creation operators: to place two fermions in the same state, the same creation operator has to be applied twice. The result must vanish, since the operator anticommutes with itself.

The symmetries of space-time are reflected in the fields which are representations of the symmetry groups; a quantum mechanical recipe called quantization then turns these fields into operators capable of creating and destroying quanta (or particles, in more common parlance) at all space-time points. Actually, the framework we have described only suffices to describe “Free fields” which do not interact with each other. In order to incorporate interactions, one has to resort to another kind of symmetry called gauge symmetry, which operates in an “internal” space attached to each point of space-time. While the identity of masses and spins of, say, electrons can be attributed to space-time symmetries, the identities of additional quantum numbers such as charge, isospin and “color” can only be explained in terms of the representations of these gauge groups.