IVB. Technical Introduction.

Is quantum mechanics as it now stands the final theory of the physical universe?  Its highly unified mathematical structure and its successes in elementary particles, atomic and nuclear structure, and solid state physics certainly seem to imply it is on the right track. 

Quantum Mechanics and Representation Theory

On the other hand, the current theory of quantum mechanics is built entirely upon concepts from representation theory.  (Representation theory is reviewed in Secs. IIIA2.2 and IVC.) 

● The particlelike properties of mass, energy, momentum, spin and charge, along with their addition and conservation laws, all follow from group representation theory; they are group representational labels on the quantum mechanical states.

● The states of fermions correspond to spin ½ (or 3/2) representations of the inhomogeneous Lorentz group.

● The states of vector bosons correspond to spin 1 representations of the inhomogeneous Lorentz group.

● Both fermion and vector boson states correspond to representations of the internal symmetry group.

● Both fermion and boson states correspond to (antisymmetric and symmetric, resp.) representations of the permutation group.

● The free particle Dirac and Proca equations follow from group representation theory.

● The internal symmetry group representational properties of the interactions play a prominent role in elementary particle physics.

Thus the theory is constructed exactly as if it were a representational form of an underlying theory.

This would seem to imply with near certainty that there is a more fundamental, pre-representational form of quantum mechanics, with the current form being a representational form of the underlying, pre-representational theory.¹

The Underlying Independent Variable Theory

To conform to the basic principles of quantum mechanics, the underlying, pre-representational theory must be linear.  And it must be invariant under a set of transformations that is homomorphic to the direct product of the Lorentz group, an internal symmetry group, and a permutation group.  The general structure of this conjectured underlying theory, given in Sec. IVD1, takes the form of a linear (partial differential) equation in a currently unknown set of independent variables.  In it, space, time and matter are derivative, not primary, concepts.  An example of such a theory—a linear partial differential equation in complex variables—is given in Sec. IVE. 

Spin ½ Representational Form of the Theory.

To go from the general, pre-representational form of the theory to the conventional, representational form, one must choose a set of basis functions (functions of the independent variables).  We suppose these functions form the basis for a particlelike spin ½ representation of the inhomogeneous Lorentz group.  The four-fermion spin ½ representational formulation can then be viewed either as following from the underlying independent variable formulation, or as an underlying formulation of current quantum mechanics in its own right.

All states in the spin ½ formulation are sums of antisymmetrized products of the basis functions.  The vacuum state is presumed to be constructed from tightly bound “molecules” that are invariant under global transformations from the internal symmetry group.  Vector bosons states correspond to perturbed vacuum molecules.

Insights Offered by the Spin ½ Formulation.

The spin ½ formulation has several advantages.

● One explicitly sees from the molecular construction of the vacuum that gauge transformations produce vector bosons.

● One explicitly sees from the molecular construction of the vacuum that the boson fields produced are proportional to the derivatives of the gauge transformation.

● One explicitly sees from the molecular construction of the vacuum that the boson fields transform as a spin 1 representation of the Lorentz group.

● One sees why the number and transformation properties of the vector bosons are the same as those of the generators of the internal symmetry group.

● In an SU(6) model, one can explicitly construct a broken symmetry vacuum that leads to results very similar to the standard model  theory.  This includes an explicit, scalar symmetry-breaking, mass-generating term with the proper group transformation properties.  It also yields quarks, electrons and neutrinos with the proper  properties.

Note:

1. Our reasoning that there must be an underlying pre-representational form of the theory is based on thinking of quantum mechanics as a mathematical theory.  One could take a different tack here.  One could suppose the theory is relativistic because the underlying physical reality is relativistic (and has internal and permutation symmetries) by nature.  Then it is natural to assume its mathematical description would use representations of the inherent symmetry group(s).  This reasoning, however, effectively assumes there is a physical reality other than the wave function.  And we show in Sec. IIIB3 that it is apparently not possible to construct a theory of such a reality that meshes properly with quantum mechanics.  That is, there cannot be a physical reality other than the wave function.