IVD1. General Form of the Underlying Theory

Independent Variable Equation

We are looking for an underlying theory that has a representational form identical to the current theory of quantum mechanics.  To accommodate the structure of the current theory, the underlying pre-representational theory must be linear, it must be Lorentz invariant, it must have an internal symmetry group, and it must naturally allow for antisymmetry.  We conjecture that the underlying theory has the following form: 

 (1) All physically relevant states correspond to solutions of a linear equation

(IVD1-1)      .                                   

 is a linear operator in some currently unknown set of underlying independent variables, , with the interaction term, , being a symmetric function of (and/or operator on) the two sets of variables . 

The underlying Eq. (IVD1-1) is pre-representational in the sense that does not correspond to a particular representation.  (An analogous situation occurs in the hydrogen atom problem.  The wave function is “pre-representational” in the sense that it does not refer to a particular angular momentum state; different solutions of the Schrödinger equation correspond to different angular momenta—that is, to different representations of the rotational invariance group.)  In fact,  does not even correspond to a fixed number of particles; all the different possible solutions correspond to all the different possible physical states, including states containing both fermions and bosons.  For a particular , its representation in some (particlelike) basis will correspond to the total wave function of that state.  It is the representation of , rather than itself (which is a function of the underlying independent variables), that we deal with in current quantum mechanics.

We do not know the form of the linear operator O, or even the nature of the underlying independent variables.  In an exploratory example given in Sec. IVIE, each corresponds to a set of  complex variables, and the operator O is a second order differential operator.  Different solutions correspond to different masses and spins. 

Not a Hidden Variable Theory.

Note that the independent variables bear no relation to the “hidden” variables that are sometimes assumed to underlie quantum mechanics.¹  The conjectured hidden variables uniquely determine the outcome of a given experiment, whereas the underlying independent variables have nothing to do with which outcome is perceived.

Interpretation

How are we to interpret this theory?  First, from Sec. IIIB3, there are to be no actually existing particles or any other objective reality.  Second, from section IIIB4, there is to be no collapse.  Thus what exists is a function of the independent variables that has many branches, with each corresponding to a possible physical reality.

Third, what the “perceiving principle” within each of us perceives is one branch of the wave function.  Fourth, what we are physically aware of when we perceive that branch are its characteristics, specifically its space-time-matter characteristics.  It might be helpful to give an analogy here.  When we learn to read, we start out by learning the alphabet.  But after we have been reading for many years, we don’t notice the details of the letters.  We only notice the words and the meaning the words convey.  So we no longer notice the hidden-variableness of reality.  We only notice its “physical” characteristics because that is what is important to our functioning, and it is what we have become used to perceiving.

(2) The second requirement on the theory is that there be a group of transformations of each set of variables  that is homomorphic to the direct product of the inhomogeneous Lorentz group, , and the internal symmetry group , which we assume for convenience here to be SU(n), with the value of n not known.  The linear operator  is invariant under  transformations of , is invariant under simultaneous and identical   transformations of , and is therefore invariant under simultaneous and identical transformations of all the variables; that is, there is a global  invariance.

The linearity of Eq. (IVD1-1) plus assumption (2) guarantee that solutions of the equation can be labeled by the particlelike group representation labels of  mass, energy, momentum, spin (all from ), and charge (from SU(n)).  It also guarantees that when there are several “particles” (particlelike states) present, the usual addition and conservation laws for these quantities will hold.  Note that the kets  of quantum mechanics, denoted by the group theoretic labels, represent functions of the underlying independent variables in this approach. 

(3) Because ,  the linear operator is invariant under permutations of the sets of variables .  It is presumed that all solutions of physical interest belong to the completely antisymmetric, one dimensional representation of the permutation group.

This completes our specification of the general form of the underlying equation.  It will take considerable insight to discover the particular form of the equation.

Note:

1. David Bohm, Phys. Rev. 85, 166, 180 (1951).  Bohm, D. and Hiley, B. J. The Undivided Universe  (Routledge, New York, 1993)