Friday, August 7, 2015

From time to quantum mechanics

Last time I presented the case for canonical time evolution stemming from the non-commutativity of operator algebras, and today I'll start talking about the reverse implication: obtaining quantum mechanics from the existence of time. 

First, let me expand on the brief statement from the prior post that the transition from classical to quantum mechanics is NOT as simple as replacing the Poisson bracket with the commutator. We need to put this in rigorous mathematical formalism and this is known as the Dirac problem:

"Does  there exists two matrices P and Q and a correspondence \(\phi\) which, to every polynomial g in the classical variables p and q, associates a matrix \(\phi (g)\) in such a manner that:
(i) \(\phi (p) = P\) and \(\phi (q) = Q\)
(ii) to the unit function \(1 : (p, q) \in R^2 \rightarrow 1\), \(\phi\) associates the unit matrix I
(iii) \(\phi\) is linear
(iv) \((i/\hbar) [\phi (f), \phi(g)] = \phi(-\{f,g\})\) for every polynomials f and g in p and q
(v) the matrices P and Q form an irreducible system, i.e. the only matrices A satisfying [X,P] = 0 = [X,Q] are of the form \(X = \lambda I\) where \(\lambda \in C\) and I is the unit matrix?"

It turns out that the Dirac problem has no solution and the proof is actually not very complicated. What are really complicated are the solutions to the quantization problem, and different approaches reject different assumptions in Dirac's problem. 

What is important for our purposes is that in classical mechanics one encounters the Poisson bracket and the quantum mechanics we have the commutator. Moreover both of them obey Leibniz identity:

\(\{H, fg\} = \{H, f\} g + f \{H, g\}\)
\([H, AB] = [H,A] B + A [H, B]\)

In the case of the Poisson bracket this is a trivial consequence of the partial differential operators, while in the case of the commutator this is a simple algebraic identity:

\([H, AB] = HAB-ABH = HAB-AHB + AHB -ABH = [H,A] B + A [H, B]\)

We can understand the commutator and the Poisson bracket as a product "\(\alpha\)":

\(A \alpha B = [A, B]\)
\(f \alpha g = \{f,g\}\)

and this is related to time evolution. If we call T a time translation operator and o any algebraic product used in physics, the invariance of the laws of Nature under time evolution implies the following commutative diagram:

T(A) o T(B) = T(A o B) which shows that [T, o] = 0 (T after o is the same as o after T) or that time translation preserves algebraic relations.

In the infinitesimal case in natural units (ignoring the usual factors of h bar and such): \(T = I + \epsilon H \alpha\) 

Substitution in T(A) o T(B) = T(A o B) yields:

\(((I + \epsilon H \alpha) A)  o ((I + \epsilon H \alpha) B) = ((I + \epsilon H \alpha )(A o B))\)

which to first order in \(\epsilon\) is:

\(H \alpha (A o B) = (H \alpha A) o B + A o (H \alpha B) \)

which is known as Leibniz identity. (A trivial observation is that when \(o = \alpha\), the Leibniz identity becomes the Jacobi identity and this gives rise to a Lie algebra.)

Now the heavy mathematical lifting follows using category theory arguments (invariance under composition = universality of the theory) to completely recover the algebraic structure of quantum and classical mechanics. Then using geometric or deformation quantization (to avoid the lack of the solutions for Dirac's problem) one obtains the usual Hilbert space formalism for quantum mechanics. Therefore the Poisson bracket and commutator are the only mathematical realizations of Leibniz identity for theories of nature obeying invariance under composition.

The starting point of quantum mechanics reconstruction using category theory arguments is Leibniz identity and this follows from infinitesimal time translations of the commutative diagram from above.

Moreover, in the usual Hilbert space formulation Leibniz identity corresponds to unitarity. This cuts both ways and violation of unitarity implies violation of Leibniz identity. And so now we have a big mathematical problem:

collapse postulate -> unitarity violation -> violation of Leibniz identity -> no Hilbert space formalism for quantum mechanics!!! 

It makes no more sense to talk about Hilbert space, operators, etc and this is clearly impossible. Something must give. Is it that there is no real collapse (MWI)? Or do we need to add contextual protection (Bohmian, QBism)? Or maybe there is an extension of quantum mechanics (GRW)?

How can this be solved? The so-called measurement problem just got much more serious than a simple problem of philosophical interpretation. There is good news however: the same categorical arguments which highlighted the problem in the first place, point the way to a most natural solution: unitary dynamical generation of superselection rules similar with spontaneous symmetry breaking. To be continued...

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