This one is not written for a broad audience. Sorry. At some times, I need to get straight to the frontier, without providing much background. This one is like a personal journal entry.
I meditated a little on two mathematical questions this morning: (1) what is the easiest way to truly prove (or disprove) the claim that variational solitons exist in models such as the system which gives the famous BPS monopole, or the four new systems I discuss in a new paper under submission?
(i.e. to prove that such models do explain why stable particles can exist at all in some universes);
(2) how could we rationally look for "dice" -- underlying stochastic disturbance -- in the universe?
("Does God throw dice with the universe?")
Re the first question -- the topological arguments which have been widely accepted as "proofs"
in physics (e.g. by folks citing E. Weinberg, not to be confused with S. Weinberg) do show that
there is no path of small perturbations which go from a state with topological charge to
the vacuum. In other words, particles with topological charge simply cannot decay away to vacuum. The same can be shown easily for the four new systems, as I show in the new paper.
That's enough to justify working with these systems, but we really should try to know for sure, to develop proofs which meet a higher standard.
One thing which worries me is evidence to SUGGEST that there may be no "state" of minimum energy in the BPS system -- no well defined, C infinity function which minimizes energy. Could there be a whole family of states, starting with the usual BPS monopole state, of ever decreasing energy,
converging not to a function but to a "distribution" like the Dirac delta function? The "proofs" based on the work of Bogomolnyi ignore the energy in the Higgs term, which could conceivably be reduced
by states which reduce that term by bringing the asymptotic solutions ever closer to the origin,
approaching a limit which is not a "function" -- but perhaps is meaningful as some kind of distribution. I would guess, for good reason, that ALL five of these systems (BPS and my four)
actually have variational solitons (basically just states of local minimum energy) which are c infinity functions, but it may be much easier to prove the weaker claim that they have variational solitons within the larger space of distributions. It's a matter of exploiting the fact that sets of distributions are closed under limits, in a way that ordinary functions are not. Probably not so hard.
And then, the dice.
I am very grateful that the first text I studied on quantum field theory long ago
was Mandl's old book, Introduction to Quantum Field Theory. Lately I often cite the
newer classic Mandl and Shaw, which is mainly an expanded version.
Mandl began with the empirical, scientific approach. He noted that there were three great confirmations of quantum electrodynamics (QED), similar to the three great confirmations
of general relativity which everyone studying that subject knows about. For QED, the Lamb shift and the anomalous magnetic moment of the electron stand out in my memory. The key to getting the Lamb shift right was to calculate the "vacuum fluctuations" and "vacuum polarization" diagrams.
Many people have interpreted those words to mean that there is a huge amount of zero point energy out there, based on the noise terms in quantum field theory. But if you read Mandl's simple
presentation closely, you see that the correct predictions are based on the normal form Hamiltonian with that scalar noise term removed. The correct predictions are based on the "normal form Hamiltonian." It is normal form Hamiltonian (plus renormalization) which explained what we see, not the songs and dances which preceded it. And -- it turns out that the normal form Hamiltonian exactly describes statistics of unknown fields with dynamics which do not involve random disturbance other than the usual thermodynamic background fluctuations one would have in any nonlinear dynamical system.
And so, I see no evidence out there in the empirical world to justify believing that
"God throws dice with the universe." Unlike Einstein, I have absolutely no religious objections to the ideas of the universe being governed by a stochastic model, if it is mathematically meaningful
(realistic). But I don't see any evidence for it either. Certainly the usual "quantum noise terms" postulated by the ZPE people have no empirical basis.
But... if there MIGHT be dice out there (which is mathematically and logically quite a reasonable possibility to consider), how would we find evidence for them?
It seems to me that the first step would be to study/develop mathematically meaningful theories
in that class, building more concretely on what we actually know from centuries of empirical
One would start from the kind of real mathematics developed for stochastic systems.
I think immediately of the old work on stochastic differential equations, on mixed forward-backwards differential equations, and even of the book by Jacobsen and Mayne on
Differential Dynamic Programming which builds a bridge to concrete algorithms like much
of my work on RLADP.
This supports a kind of crude image... a stochastic model which is formulated by
assuming the existence of a kind of exogenous random number field, e(x-mu),
where e can be a random scalar, a random vector, whatever. The idea of a random field is
not new (e.g. see Markov Random Fields). The question would be how to
hook it up with physics.
There are lots of other formulations one might imagine, which entail asymmetries (e.g. with respect to time) for which there is no empirical support. Considering the possibility of dice is just one speculation, albeit a somewhat natural speculation; imposing new asymmetries at this time is much
more grossly speculative, and a detour from the core question about dice.
And so... it is interesting to ask about the family of models generated by ordinary Lagrange-Euler equations (as with classical PDE) EXCEPT by the addition of the exogenous random e field.
This is a time-symmetric formulation.
What do we know about the properties of such systems?
For ordinary time-forwards dynamical systems, introducing noise typically would introduce
drift. Energy conservation would not be exact at the microscopic level. Laws of large numbers
might reduce the implied violation of energy conservation at the measured, macroscopic level,
but things similar to roundoff error can accumulate and cause instability. Would we really see
the rigid degree of energy conservation we see in low-level bubble chamber events
if there were that kind of disturbance?
And yet -- in a time-symmetric situation it is not quite as clear. The perfect "optimization"
performed by Lagrange-Euler equations might possibly result in a much tighter kind
of law of large numbers, preventing such large excursions in energy at the macroscopic
level. Even at the bubble chamber level.
But at this point, this is just a theoretical, hypothetical kind of possibility.
I don't see any real reason to believe that there are "dice" out there in the basic laws of the universe.
I see that it COULD be possible, but I do not yet even see a path to testing the idea.
I did a paper once on "Q" versus "P" which is another way to get at the same issue,
leading to the same outcome for now.
There is some relation between this and the "stochastic chaos" work I discussed with Freeman
and Kozma years ago, in Biosystems and elsewhere. But on the physics side,
there are certainly more urgent areas for mathematical exploration at this time
(such as exploration of the four new systems which contain no dice).
An alternative formulation for "dice" might use the modern Hamilton-Jacobi-Bellman
equation in place of the old Hamilton-Jacobi or Lagrange-Euler equations, or -- given that
we seem to be seeing a mix-max system -- the Hamilton-Jacobi-Isaacs equation REVISED
for time symmetry. They sound entertaining, but perhaps the time-symmetric stochastic Lagrange-Euler system has more of the right properties. All speculative for now anyway.
Best of luck,