It's about time

After discussing some well known results in quantum mechanics, I will start presenting my approach for solving the measurement problem which was my talk at the Vaxjo conference. This will take several posts, and today I start the discussion by presenting some ideas about time.

Time is essential in Hamiltonian mechanics which is the natural way to transition from classical to quantum mechanics. Usually one is introduced to the subject by the naive statement that we replace the Poisson bracket with the commutator. The story is much more subtle (and interesting) than that and today I want to explore only some parts of those issues with the remaining advanced topics to be discussed at a later time.

It turns out that there is a circularity problem: time is derived from quantum mechanics and quantum mechanics is derived from time. Let me state upfront that I do not yet know how to solve this very interesting and hard problem.

Today I'll present the claim that time has a quantum mechanical origin. The main proponents of this are Alain Connes and Carlo Rovelli in their Thermal Time Hypothesis. I have talked about this last year, but now I want to go in depth in the mathematical reasoning. As a pre-requisite the reader should understand the concept of short exact sequences and I have a review of this topic here.

I am not attempting to justify the canonical gravity point of view and oppose it to string theory, but I want to present the mathematical reasons of why time may have a non-commutative (quantum mechanical) origin. The mathematical underpinnings are von Neumann algebras and the Tomita-Takesaki theory.

The von Neumann algebras are the generalization of measure theory: they reduce to the study of measureable spaces when we restrict to the commutative case.

Suppose \(M\) is a von Neumann algebra in a Hilbert space H, \(a\) is an element of \(M\), and we have an operator S defined by:

\(S a \psi = a^* \psi\)

Then S admits a polar decomposition:

\(S = J \Delta^{1/2}\)

with \(\Delta\) a positive self-adjoined operator and J anti-unitary.

The Tomita-Takesaki theory proves that:

\(JMJ = M^{'}\): M has the same size as its commutant
\(\Delta^{-it} M \Delta^{it} = M\): there is an one-parameter group of automorphism  \(\sigma^\phi_t\) which gives the time flow in the Heisenberg picture.

So far so good, but stronger statements can be made. If we impose the KMS condition, then the automorphism is unique. Still, it depends on the choice of the state \(\phi\). The really powerful result is that in the exact sequence:

\(1\rightarrow Inn(M) \rightarrow Aut(M) \rightarrow Out(M) \rightarrow 1\)

where \(Inn(M)\) is the normal subgroup of inner automorphisms: \(T\rightarrow aTa^*\), the automorphism \(\sigma^\phi_t\) does not depend upon the choice of state \(\phi\) and hence it is a canonical time evolution.

This is the main mathematical result (by Connes) which needs to be applied to physics to understand what is going on. And this is what Connes and Rovelly do in their thermal time hypothesis paper. But there is more to the story. The short exact sequence from above was the starting point of Connes non-commutative geometry description of the Standard Model which led to the geometric unification of gauge theory weakly coupled with gravity in the non-commutative framework (no quantum gravity here: no background independence arguments to justify loop quantum gravity vs. string theory). In the simplest model of the noncommutative description of the \(SU(n)\) gauge group weakly coupled with non-quantized gravity, the short exact sequence from above is equivalent with:

\( 1 \rightarrow \mathcal{G} = Map(M, SU(n)) \rightarrow Diff(X) \rightarrow Diff(M) \rightarrow 1\)

\(1\rightarrow fiber\rightarrow total space\rightarrow base\rightarrow 1\)

where M here is a spin Riemannian manifold. The full group of invariance on a new space \( X = M \times M_n (C) \) is the semidirect product of the diffeomorphisms on M with the gauge group. The diffeomorphism shuffles (acts on) the group of gauge transformations.

If we take the main lesson of quantum mechanics to be that of non-commutativity of operators, we can construct a generalization of commutative mathematics into a non-commutative domain:

Commutative	Noncommutative
measure space	von Neumann algebra
locally compact space	C∗- algebra
vector bundle	finite projective module
complex variable	operator on a Hilbert space
real variable	sefadjoint operator infinitesimal compact operator
range of a function	spectrum of an operator
K-theory	K-theory
vector field	derivation
integral	trace
closed de Rham current	cyclic cocycle
de Rham complex	Hochschild homology
de Rham cohomology	cyclic homology
Chern character	Chern-Connes character
Chern-Weil theory	noncommutative Chern-Weil theory
elliptic operator	K-cycle
spin Riemannian manifold	spectral triple
index theorem	local index formula
group, Lie algebra	Hopf algebra, quantum group
Symmetry	action of Hopf algebra

In the non-commutative domain (quantum mechanics) one encounters a universal definition of a time flow which has no counterpart in commutative mathematics (or classical physics). In this sense time is a mathematical necessity of quantum mechanics arising out of operator non-commutativity.

But the implication works in the other way too. Starting with the necessity of time, we can consider infinitesimal time evolution and extract the Leibniz identity. And in the categorical approach or quantum mechanics reconstruction the Leibniz identity is the main starting point.

Next time I'll expand on this and the implication for the measurement problem.