IVC2. Representation Theory in Quantum Mechanics.

See also Sec. IIIA2.2

The Relevant Groups

The primary invariance group of interest in relativistic quantum mechanics is the inhomogeneous Lorentz group,  (or ISO(3,1) or ISL(2)).  It consists of all “rotations” that leave the form invariant, plus translations of x, y, z and t.  The invariant labels on representations are mass, M, and spin, S.  The labels on basis vectors within the (M,S) representation are energy, E, momentum, p and the z component of spin, . 

There is also an internal symmetry group in quantum mechanics, where “internal” means the group operations have nothing to do with space and time.  Labels on vectors for representations of the internal symmetry group include the charges Q—weak, electromagnetic and strong.  A state  in quantum mechanics can be expanded in terms of functions labeled by M, S, E, p,  and ,

(IVC2-1)             

with the vector representative  of the state being the wave function in the momentum representation, and the ket  representing, in our scheme, a function of the independent variables.  In general, the labels on the kets correspond to the observable characteristics of the state.  The independent variables are not observables.

Antisymmetry

There is one more twist to representation theory.  Suppose the linear operator is invariant under permutations and that one wishes to look only at totally antisymmetric representations, as in assumption (3) of Sec. IVD1.  Then all states can be expressed as sums of products of anticommuting creation operators acting on the vacuum state.  And all operators can be expressed as sums of products of creation and annihilation operators, with the creation and annihilation operators denoted by group theoretic labels.  For example, the state of Eq. (IVC2-1) becomes (dropping the M,S)

(IVC2-2)           

with  being a creation operator and being the vacuum state. 

Thus field operators are not the “basic mathematical elements” of the underlying theory proposed in this section, as they are in current quantum field theory.  Instead, they are a consequence of representation theory applied to the scheme outlined in Sec. IVD1.  The “basic mathematical elements” in our conjectured underlying theory are the particlelike functions of the underlying independent variables.