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(xs(y)) for all x and y.

One can check at this very formal level that s fulfills s(ab)=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.

9 thoughts on “Time

  1. Anonymous

    Thanks a lot for this detailed reply!
    I have taken the liberty of forwarding it to this discussion. Hope you don’t mind.

    Can you point me to any literature where the action by outer derivations that you describe is explcitly identified in a concrete field theoretical context?

    I gather that all local nets of algebras of observables appearing in AQFT are type III factors and should therefore come equipped with the canonical 1-parameter action by outer automorphisms, as you discuss.

    At the same time, usually for these field theories we have a notion of time evolution that is derived from the physical specification of the system.

    How is this “standard” time evolution related to the canonical one by outer automorphisms?

    Has this been discussed anywhere, possibly with concrete examples?

  2. Kea

    I’m just a physicist. In our spectral higher operad approach to mass generation, which is linked to this time emergence, there is no Higgs boson, because the concept of vacuum is unnecessary. Could you see a ‘no Higgs boson’ SM within your framework?

  3. Matti Pitkänen

    Thanks a lot for the explanation for the origin of the outer automorphism. There are fascinating interpretational problems and questions to ponder even with the shaky mathematical background of a physicist. Just one question. Could inner automorphisms be identified as universal local gauge symmetries? In the case of HFFs of type II_1 one can also ask whether outer automorphisms could be seen as analogs of global gauge transformations.

  4. Anonymous

    Hi Urs and thanks for your comments. Now we are in a higher pitch of course
    (thanks to Alain’s recent post!), but I still want to use your earlier
    comments to discuss some elementary aspects of time evolution and a
    non-example! I sense that this may be useful for our larger audience so to speak. As for algebras arising from quantizing symplectic
    manifold, e.g., in the simplest case, the Weyl algebra of a symplectic
    vector spaces (as in your earlier comment), if by this we mean the
    algebra say generated by p and q with pq-qp=ih (and its multidimensional
    analogues), then as is well known this algebra is not an operator algebra in the technical
    sense that we use in this blog: it is not realizable as an algebra of
    bounded operators in a Hilbert space and in fact does not even have a
    Banach algebra representation. So there seems to be no obvious method of inducing a one parameter group of automorphisms on the Weyl algebra using v N algebras and the modular theory. At least not directly (Note however that Dixmier was very interested in Weyl algebras specially their irreps and automorphisms and wrote a a couple of papers on them in the 1960’s). The Weyl algebra however has an “integrated”
    or “exponential” version which, once is properly completed, is nothing but the `algebra of functions’ on the noncommutative torus A_{theta} with theta = e^{2 pi ih}
    (or if you want to consider h not as a constant but as a central generator, it is
    going to be the group C* algebra of the integer Heisenberg group).
    Now A_{theta} comes canonically equipped with a faithful tracial state and the
    corresponding v N algebra, assuming theta is irrational, in the GNS rep. is a type II factor (in fact the unique hyperfinite II_{infty} factor).
    In fact I have a feeling that v N algebras obtained by quantizing a calssical system with a finite number of degrees of freedom (and a symplectic phase space) won’t be type III. There may be interesting examples coming from Poisson manifolds through their symplectic foliations but I am not sure and leave this to others to comment. If you look at chpaters 1 and 5 of Alain’s 1994 book (available for download at http://www.alainconnes.org) you will find a nice dictionary between the type of the von Neuman algebra of a foliation and its modular group (= time evolution) and the geometry and ergodic properties of the foliation. There is also a nice duality between type II and type III
    v N algebras that has an analogue in the foliation picture. This is all Alain’s work and is very well explained in the book.

  5. Anonymous

    Dear Masoud,

    many thanks for your reply!

    I did indeed have the exponentiated version in mind when I was referring to the “Weyl algebra” obtained from a symplectic manifold. I should have made that more explicit.

    As far as I understand, the C*-algebra generated from such a Weyl algebra is always all of L(H).

    Ordinary quantum mechanical time evolution acts on that by inner automorphisms given by the ordinary “Heisenberg-picture” propagation equation

    x |-> x(t) := e^(itH) x e^(-itH)

    Over at the n-Cafe an anonymous commenter suggested (but wasn’t sure) that the difference to higher dimensional QFT (i.e. with at least one spatial dimension) is that there the exponentiated Hamiltonian


    is an element of the inductive limit over all causal diamonds of all local algebras, hence conjugating by it, as above, is not entirely inner with respect to these local algebras (which are all type III factors, I think), but only with respect ot the “global” algebra obtained as their inductive limit.

    That would actually seem to make good sense. But I still need to confirm this.

    So there would be a local part of the Hamiltonian, coming from conjugating with an operator in the local algebra we are acting on, as well as a global part, acting by outer automorphisms.

    It would, apparently, be (only) that global part which might be related to the canonical 1-parameter family of automorphisms that Alain Connes described.

  6. AC

    Urs: Yes, what happens in fact is that for any quantum system with infinitely many degrees of freedom the hamiltonian H does not belong to the algebra of observables. Thus the corresponding automorphisms are not inner. To see what happens it is simplest to take the case of a system of spins on a lattice. The algebra of observables is the inductive limit of the finite tensor products of matrix algebras one for each lattice site. The hamiltonian H is, even in the simplest non-interacting case, an infinite sum of the hamiltonians associated to each lattice site. Thus it does not belong to the algebra of observables and the corresponding one parameter group is not inner(both in the norm closure ie the C*-algebra, and in the weak closure)… In QFT the situation is entirely similar and has of course infinitely many degrees of freedom from the start…

  7. Doug

    Whenever I see infinitely many DoF as in the ac last sentence, “… In QFT the situation is entirely similar and has of course infinitely many degrees of freedom from the start …“, I have difficulty visualizing infinitely many spatial DoF.

    Rather, I suspect that such a number of DoF, I suspect that this is more consistent with strategy DoF as successfully used by applied mathematicians and engineers [electrical and mechanical] in Max Plus Algebra, referred to in my previous comment of March 21, 2007 9:51 PM.

    Such DoF might be EM, concentration gradients, pH, temperature or other non-spatial DoF.

    An example of such may be an experiment published in Science 23 FEB 2007, v315, n5816, p1116-1119 by these physicists:

    Yale Klein, Efi Efrati, Eran Sharon [Racah IoP, Hebrew U, Jerusalem, Israel]
    Shaping of Elastic Sheets by Prescription of Non-Euclidean Metrics
    Abstract: The connection between a surface’s metric and its Gaussian curvature (Gauss theorem) provides the base for a shaping principle of locally growing or shrinking elastic sheets. We constructed thin gel sheets that undergo laterally nonuniform shrinkage. This differential shrinkage prescribes non-Euclidean metrics on the sheets. To minimize their elastic energy, the free sheets form three-dimensional structures that follow the imposed metric. We show how both large-scale buckling and multiscale wrinkling structures appeared, depending on the nature of possible embeddings of the prescribed metrics. We further suggest guidelines for how to generate each type of feature.

    The experiment is described in “Fig. 1. The experimental system. High (~ 30%) and low (~ 10%) monomer concentration solutions are mixed in a programmable mixer and injected into a Hele-Shaw cell (left). Polymerization leads to the generation of a flat disc having internal lateral gradients in monomer concentration (center). Once this programmed disc is activated in a hot bath of temperature T > Tc = 33″C, it shrinks differentially, adopting a new, non-Euclidean target metric (right). As a result, it attains a 3D configuration. Illustrated is a surface of positive Gaussian curvature, generated by increasing monomer concentration during the injection.”

    The authors concluding sentences:
    “Such new materials are being developed to respond to different external stimuli, such as light, pH, glucose level, and oilier chemical signals. Further study of the principles of shaping by metric prescription can extend the types and variety of structures that can be formed by using thin sheets, as well as improve our understanding of developmental processes.”

  8. Anonymous

    I have found a concrete discussion of the relation of the modular operator Delta, that induces the twist sigma that is discussed in the above entry, and the ordinary notion of time evolution of quantum field theory, in the appendix of

    K.-H. Rehren, On Local Boundary CFT and Non-Local CFT on the Boundary


    By the Bisognano-Wichner theorem (1976), the conjugation with the operator Delta^it relative to the algebra of observables associated by a local QFT to a wedge region of Minkowski space generates the group of Lorentz boosts that leaves that wedge region invariant.

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