I will try to describe in loose terms the steps that lead to the emergence of time from noncommutativity in operator algebras. This hopefully will answer the questions of Paul and Sirix (at least in parts) and of Urs.

First I’ll explain the basic formula due to Tomita that associates to a state L a one parameter group of automorphisms. The basic fact is that one can make sense of the map x –> **s**(x)= L x L^{-1} as an (unbounded) map from the algebra to itself and then take its complex powers **s**^{it}.

To define this map one just compares the two bilinear forms on the algebra given by L(xy) and L(yx) . Under suitable non-degeneracy conditions on L both give an isomorphism of the algebra with its dual linear space and thus one can find a linear map s from the algebra to itself such that

L(yx)=L(x**s**(y)) for all x and y.

One can check at this very formal level that s fulfills **s**(ab)=**s**(a)**s**(b) :

L(abx)=L(bx**s**(a))=L(x**s**(a)**s**(b))

Thus still at this very formal level s is an automorphism of the algebra, and the best way to think about it is as x –> L xL^{-1} where one respects the cyclic ordering of terms in writing Lyx=LyL^{-1}Lx=LxLyL^{-1}. Now all this is formal and to make it “real” one only needs the most basic structure of a noncommutative space, namely the measure theory. This means that the algebra one is dealing with is a von-Neumann algebra, and that one needs very little structure to proceed since the von-Neumann algebra of an NC-space only embodies its measure theory, which is very little structure. Thus the main result of Tomita (which was first met with lots of skepticism by the specialists of the subject, was then succesfully expounded by Takesaki in his lecture notes and is known as the Tomita-Takesaki theory) is that when L is a faithful normal state on a von-Neumann algebra M, the complex powers of the associated map **s**(x)= L x L^{-1} make sense and define a one parameter group of automorphism **s**_L of M.

There are many faithful normal states on a von-Neumann algebra and thus many corresponding one parameter groups of automorphism **s**_L . It is here that the two by two matrix trick (Groupe modulaire d’une algèbre de von Neumann, C. R. Acad. Sci. Paris, Sér. A-B, 274, 1972) enters the scene and shows that in fact the groups of automorphism **s**_L are all the same modulo inner automorphisms!

Thus if one lets Out(M) be the quotient of the group of automorphisms of M by the normal subgroup of inner automorphisms one gets a completely canonical group homomorphism from the additive group **R** of real numbers

delta: **R**–> Out(M)

and it is this group that I always viewed as a tantalizing candidate for “emerging time” in physics. Of course it immediately gives invariants of von-Neumann algebras such as the group T(M) of “periods” of M which is the kernel of the above group morphism. It is at the basis of the classification of factors and reduction from type III to type II + automorphisms which I did in June 1972 and published in my thesis (with the missing III _1 case later completed by Takesaki).

This “emerging time” is non-trivial when the noncommutative space is far enough from “classical” spaces. This is the case for instance for the leaf space of foliations such as the Anosov foliations for Riemann surfaces and also for the space of Q-lattices modulo scaling in our joint work with Matilde Marcolli.

The real issue then is to make the connection with time in quantum physics. By the computation of Bisognano-Wichmann one knows that the **s**_L for the restriction of the vacuum state to the local algebra in free quantum field theory associated to a Rindler wedge region (defined by x_1 > + – x_0) is in fact the evolution of that algebra according to the “proper time” of the region. This relates to the thermodynamics of black holes and to the Unruh temperature. There is a whole literature on what happens for conformal field theory in dimension two. I’ll discuss the above real issue of the connection with time in quantum physics in another post.