Saturday, January 4, 2014

Solving Hilbert’s sixth problem

Invariances of the laws of nature

How is special theory of relativity derived? One starts with the invariance of the laws of nature to changing inertial reference frames. From this you get either the Lorentz transformation or the Galilean transformation, and use the second postulate, that of the constant value of the speed of light, to select between the two choices.

But are the laws of nature invariant only to changes in inertial reference frames? How about a trivial invariance: the laws of nature do not change during time evolution?

And how about another: the laws of nature do not change if we partition in our mind a larger system into smaller ones (the composability principle, or the invariance of the laws of Nature under tensor composition)?

Those statements are completely obvious, but their mathematical consequences are far from trivial. We need one more ingredient of a technical nature: a continuity property: if we represent the state of a physical system with a point in a state/phase/configuration space manifold we want to be able to compute derivatives, meaning we should be able to define a tangent plane (after all we will recover the Hamiltonian formalism and the cotangent bundle).

So now we are ready to begin the journey of deriving the two products we talked about last time. It will turn out that the symmetric product describes the ontology and the skew-symmetric one the dynamic. Their compatibility condition (which will be derived too) will help recover the quantum and classical mechanics. All this will come out of the two new invariance laws stated above!!! If you want to follow along in a technical paper (and peek into the future post contents) use as guidance. Let us begin…

First we consider a set of (abstract) unspecified operations {o} which include local laws of nature. As concrete examples of such operations in quantum mechanics, we think of the commutator understood as a product, and the Jordan product. At this point we do not assume any properties for those products, or even that they should exist. How can we mathematically state that the set {o} remains invariant under time evolution? The idea is that of an isomorphism: “o“ at time t must be isomorphic with “o” at time t+delta t because the isomorphism preserves the algebraic relationships. If T is a time translation operator (T A(t) = A(t+ delta t)) the isomorphism can be written as:

T [G o H] = [T(G) o T(H)]

Or equivalently:

(G  o H)(t +delta t) = G (t +delta t) o H(t+delta t)

This is the only consequence we can derive from the invariance of the laws of nature under time evolution.

Now consider infinitesimal time evolutions and use the ability to derivate. This introduces a tangent plane at “t” and a (particular) vector field associated with (a particular) time evolution.

If T_epsilon is a particular infinitesimal (time) translation operator we have:

T_epsilon [G(t) o H(t)] = [T_epsilon G(t) o T_epsilon H(t)]

If rho is one of the products in the set {o} corresponding to the time translation transformation T_epsilon, we can construct the following product between a distinguished f and any g:

f ρ g = [T_epsilon g – fg]/ ε in the limit of ε -> 0 (we pick this as the definition)


f (I + ε ρ) g = T_epsilon g

(Here T_epsilon and “f” are not arbitrary, but depend on each other. “f” in general corresponds to a particular Hamiltonian, and ρ corresponds to the Poisson bracket in classical mechanics and the commutator in quantum mechanics.)

T_epsilon can be uniquely represented as f (I + ε ρ) because if f (I + ε ρ) = T_epsilon = r (I + ε η), in the zero order: f=r. In turn this means that f (I + ε ρ) = T_epsilon = f (I + ε η) and therefore: ρ = η.

We generalize the product ρ for all f’s and g’s by repeating the argument for all conceivable dynamics. To make sure the domains of f and g are identical and well behaved, in case of pathologies, we can restrict the domain of g’s to the span of all possible f’s.

From the invariance of the laws of Nature under time evolution we have:

f (I + ε ρ) (g o h) = T_epsilon (g o h) = (T_epsilon g) o (T_epsilon h) =
= [f (I + ε ρ) g] o [f (I + ε ρ) h]

In first order in epsilon we have a left Leibnitz identity:

f ρ (g o h) = (f ρ g) o h + g o (f ρ h)

for any product “o” in the set {o} and for any f, g, h.

Similarly one derives a right Leibnitz identity:

(g o h) ρ f = (g ρ f) o h + g o (h ρ f)

The Leibnitz identities turn out to be of critical importance as we will see in subsequent posts.

In summary, the fact that the laws of nature do not change during time evolution along with differentiation demands the existence of left and right Leibnitz identities for an abstract product ρ.

As an important note, we do not know at this time that ρ is necessarily skew-symmetric. If we assume this from the very beginning the whole derivation is much shorter.

Next time we’ll show that dynamic alone is not enough and we will need a supporting product which in the end will show that it corresponds to observables (ontology). Then we’ll establish the most general relations between dynamic and ontology.