What is Trace Dynamics?

 Trace dynamics is quantisation, without the Heisenberg algebra.

1. Quantisation Step 1 is to raise classical degrees of freedom, the q and p, to the status of operators / matrices. A very reasonable thing to do.

2. Quantisation Step 2 is very unreasonable! Impose the Heisenberg algebra [q, p] = i \hbar Its only claim to fame is that the theory it gives rise to is extremely successful.

In classical dynamics, the initial values of q and p are independently prescribed. There is NO relation between the initial q and p. Once prescribed initially, their evolution is determined by the dynamics. Whereas, in quantum mechanics, a theory supposedly more general than classical mechanics, the initial values of the operators q and p must also obey the constraint [q, p] = i \har. This is highly restrictive!

3. It would be more reasonable if there were to be a dynamics based only on Quantisation Step 1. And then Step 2 emerges from this underlying dynamics in some approximation. This is precisely what Trace Dynamics is. Only step 1 is applied to classical mechanics. q and p are matrices, and the Lagrangian is the trace of a matrix polynomial made from q and its velocity. The matrix valued equations of motion follow from variation of the Lagrangian. They describe dynamics.

4. This matrix valued dynamics, i.e. trace dynamics, is more general than quantum field theory, and assumed to hold at the Planck scale. The Heisenberg algebra is shown to emerge at lower energies, after coarse-graining the trace dynamics over length scales much larger than Planck length scale. Thus, quantum theory is midway between trace dynamics and classical dynamics.

5. The moral of the story is that quantum field theory does not hold at the Planck scale. Trace dynamics does. QFT is emergent.

6. The other assumption one makes at the Planck scale is to replace the 4-D classical spacetime manifold by an 8D octonionic spacetime manifold, so as to obtain a canonical definition of spin. This in turn allows for a Kaluza-Klein type unification of gravity and the standard model. Also, an 8D octonionic spacetime is equivalent to a 10-D Minkowski space-time. It is very rewarding to work with 8D octonionic, rather than 10D Minkowski - the symmetries manifest much more easily.

7. Trace dynamics plus octonionic spacetime together give rise to a highly promising avenue for constructing a theory of quantum gravity, and of unification. 4D classical spacetime obeying GR emerges as an approximation at lower energies, alongside the emergent quantum theory.

8. How is this different from string theory? In many ways it IS like string theory, but *without* the Heisenberg algebra! The gains coming from dropping [q,p]=i\hbar at the Planck scale are enormous. One now has a non-perturbative description of space-time at the Planck scale.

The symmetry principle behind the unification is very beautiful: physical laws are invariant under algebra automorphisms of the octonions. This unifies the internal gauge transformations of the standard model with the 4D spacetime diffeomorphisms of general relativity. The automorphism group of the octonions, the Lie group G2, which is the smallest of the five exceptional Lie groups, contains within itself the symmetries SU(3)xSU(2)xU(1) of the standard model, along with the Lorentz symmetry. The free parameters of the standard model are determined by the characteristic equation of the exceptional Jordan algebra J_3(O), whose automorphism group F4 is the exceptional Lie group after G2.