Suppose we are asked to `quantise' classical dynamics, and are given the following two choices for how to do it. Which one should we choose, given that the second choice agrees with all experiments done so far, and the first one is untested because it is a Planck scale theory :
1. Trace Dynamics [Stephen Adler, 1996]
Starting from classical dynamics, raise all dynamical variables to the status of matrices / operators, and hence arrive at a Lagrangian which is a matrix polynomial. Take its matrix trace, and use this trace Lagrangian (a scalar) as the new Lagrangian in the action principle. You can now develop a matrix-valued Lagrangian dynamics, derive its matrix-valued equations of motion, and also the corresponding Hamiltonian dynamics. Everything proceeds as in conventional classical dynamics,
(i) except that
The dynamical variables, now being matrices, do not commute with each other. The commutator [q,p] evolves with time and is determined by the dynamics.
(ii) and except that
The new matrix dynamics has a conserved Noether charge, absent in classical dynamics, which is a result of the invariance of the trace Hamiltonian under global unitary transformations. This is coming about because we are now working with matrices and with a Hamiltonian which is a trace over matrices. The conserved charge is
Sum over all degrees of freedom i of the commutators
[q_i, p_i]
That is, whereas each [q_i. p_i] evolves with time, the sum of all such commutators is conserved. It is as if the d.o.f. exchange [q,p] with each other dynamically. This conserved quantity known as the Adler-Millard charge has the dimensions of action. Its existence is what makes trace dynamics into a pre-quantum theory. One never quantises trace dynamics; rather quantum theory emerges from it, as follows.
It is assumed that trace dynamics holds at some energy scale, not yet tested in the laboratory, say the Planck scale. We then ask what is the emergent dynamics at a lower energy scale, such as at the LHC, if one is not observing at the Planck scale. Techniques of statistical thermodynamics are employed to answer this question, and it is shown that in the emergent low energy theory, the Adler-Millard charge is equipartitioned over all d.o.f. As a result, for all coarse-grained d.o.f. the averaged commutator < [q,p] > takes the same value, and it is set equal to i\hbar. This is how one gets [q,p]=\ihbar, the Heisenberg algebra.
The averaged Hamilton's equations of motion of the underlying theory become Heisenberg equations of motion, and quantum field theory is recovered as a low energy emergent approximation to trace dynamics.
2. The second choice: Quantum Theory
Raise all classical dynamical variables to the status of matrices / operators, and impose by hand in an ad hoc way the Heisenberg algebra
[q, p] = i\hbar
The resulting quantum field theory agrees with all experiments done so far. But from a theoretical viewpoint imposing the Heisenberg algebra seems ad hoc. q and p do not commute once they are matrices. Shouldn't the dynamics determine the commutator [q,p] as in trace dynamics, with [q,p]=i\hbar emerging in an approximation?
With trace dynamics as a benchmark, one can now view quantum theory as a special case of trace dynamics. Using trace dynamics along with a spacetime described by the octonions opens up new possibilities for better understanding of the standard model of particle physics, and its unification with general relativity.
So which one do we choose: 1 or 2? Should the Heisenberg algebra be imposed a priori, or allowed to emerge from a more general theory which does not constrain the commutator [q,p] but lets it evolve dynamically?
