Noncommutative spacetime

As I explained in a previous post, it is only because one drops commutativity that, in the calculus, variables with continuous range can coexist with variables with countable range. In the classical formulation of variables, as maps from a set X to the real numbers, we saw above that discrete variables cannot coexist with continuous variables.
The uniqueness of the separable infinite dimensional Hilbert space cures that problem, and variables with continuous range coexist happily with variables with countable range, such as the infinitesimal ones. The only new fact is that they do not commute.
One way to understand the transition from the commutative to the noncommutative is that in the latter case one needs to care about the ordering of the letters when one is writing.
As an example, use the “commutative rule” to simplify the following cryptic message I received from a friend :”Je suis alençonnais, et non alsacien. Si t’as besoin d’un conseil nana, je t’attends au coin annales. Qui suis-je?”
It is Heisenberg who discovered that such care was needed when dealing with the coordinates on the phase space of microscopic systems.
At the philosophical level there is something quite satisfactory in the variability of the quantum mechanical observables. Usually when pressed to explain what is the cause of the variability in the external world, the answer that comes naturally to the mind is just: the passing of time. But precisely the quantum world provides a more subtle answer since the reduction of the wave packet which happens in any quantum measurement is nothing else but the replacement of a “q-number” by an actual number which is chosen among the elements in its spectrum. Thus there is an intrinsic variability in the quantum world which is so far not reducible to anything classical. The results of observations are intrinsically variable quantities, and this to the point that their values cannot be reproduced from one experiment to the next, but which, when taken altogether, form a q-number.

Heisenberg’s discovery shows that the phase-space of microscopic systems is noncommutative inasmuch as the coordinates on that space no longer satisfy the commutative rule of ordinary algebra. This example of the phase space can be regarded as the historic origin of noncommutative geometry. But what about spacetime itself ? We now show why it is a natural step to pass from a commutative spacetime to a noncommutative one.
The full action of gravity coupled with matter admits a huge natural group of symmetries. The group of invariance for the Einstein-Hilbert action is the group of diffeomorphisms of the manifold and the invariance of the action is simply the manifestation of its geometric nature. A diffeomorphism acts by permutations of the points so that points have no absolute meaning.
The full group of invariance of the action of gravity coupled with matter is however richer than the group of diffeomorphisms of the manifold since one needs to include something called “the group of gauge transformations” which physicists have identified as the symmetry of the matter part.
This is defined as the group of maps from the manifold to some fixed other group, G, called the `gauge group’, which as far as we known is: G=U(1).SU(2).SU(3). The group of diffeomorphisms acts on the group of gauge transformations by permutations of the points of the manifold and the full group of symmetries of the action is the semi-direct product of the two groups (in the same way, the Poincaré group which is the invariance group of special relativity, is the semi-direct product of the group of translations by the group of Lorentz transformations). In particular it is not a simple group (a simple group is one which cannot be decomposed into smaller pieces, a bit like a prime number cannot be factorized into a product of smaller numbers) but is a “composite” and contains a huge normal subgroup.
Now that we know the invariance group of the action, it is natural to try and find a space X whose group of diffeomorphisms is simply that group, so that we could hope to interpret the full action as pure gravity on X. This is the old Kaluza-Klein idea. Unfortunately this search is bound to fail if one looks for an ordinary manifold since by a mathematical result, the connected component of the identity in the group of diffeomorphisms is always a simple group, excluding a semi-direct product structure as that of the above invariance group of the full action of gravity coupled with matter.
But noncommutative spaces of the simplest kind readily give the answer, modulo a few subtle points. To understand what happens note that for ordinary manifolds the algebraic object corresponding to a diffeomorphism is just an automorphism of the algebra of coordinates i.e. a transformation of the coordinates that does not destroy their algebraic relations. When an involutive algebra A is not commutative there is an easy way to construct automorphisms.
One takes a unitary element u of the algebra i.e. such that u u*=u*u=1. Using u one obtains an automorphism called inner, by the formula x -> uxu*.
Note that in the commutative case this formula just gives the identity automorphism (since one could then permute x and u*). Thus this construction is interesting only in the noncommutative case. Moreover the inner automorphisms form a subgroup denoted Int(A) which is always a normal subgroup of the group of automorphisms of A.
In the simplest example, where we take for A the algebra of smooth maps from a manifold M to the algebra of matrices of complex numbers, one shows that the group Int(A) in that case is (locally) isomorphic to the group of gauge transformations i.e. of smooth maps from M to the gauge group G= PSU(n) (quotient of SU(n) by its center). Moreover the relation between inner automorphisms and all automorphisms becomes identical to the exact sequence governing the structure of the above invariance group of the full action of gravity coupled with matter.

It is quite striking that the terminology coming from physics: internal symmetries agrees so well with the mathematical one of inner automorphisms. In the general case only automorphisms that are unitarily implemented in Hilbert space will be relevant but modulo this subtlety one can see at once from the above example the advantage of treating noncommutative spaces on the same footing as the ordinary ones. The next step is to properly define the notion of metric for such spaces and we shall indulge, in the next post, in a short historical description of the evolution of the definition of the “unit of length” in physics. This will prepare the ground for the introduction to the spectral paradigm of noncommutative geometry.

14 thoughts on “Noncommutative spacetime

  1. Fabien Besnard

    Dear Alain, thank you for this post.
    I’m going to ask a question which is probably terribly naive and possibly a bit crazy. As I understand, classical Kaluza-Klein theory suffers form the drawback of instability of the extra compact dimensions which would tend to shrink down to singularities. Now, correct me if i’m wrong, but the finite NC part A_F of the algebra C(M)tensor A_F can be seen as the algebra of functions on a virtual discrete space. Could this virtual space be interpreted as the remnant of a shrunk down ancestral continuous space now in the quantum regime ? If so, could the particular structure of A_F relfect the topology of this continuous space ?

  2. AC

    Guy on the street, just try to permute some letters and get 4 times a name which is not so difficult to guess…….what can you come up with starting with “non alsacien” for instance?

  3. AC

    Dear Fabien

    Your question is pertinent. The role of the finite space is now much better understood from the very recent papers with A. Chamseddine: “why the Standard Model” and “A dress for SM the beggar” which are on the hep-th arXiv. My intention is to use this blog, this summer holidays, to explain their content in details, but one step at a time. So far I just wanted to explain why it is natural to consider NC spacetimes and not be so dependent on the “point-set” commutative view of spaces. So even if one cannot exclude that the finite space F is, as you suggest, a “remnant of a shrunk down ancestral continuous space” it will give us a lot more freedom to drop the dependence on the commutative view.

  4. Unknown

    Dear Alain,

    thank you for your inspiring post.
    Let me briefly outline a recent understanding about gravity based on noncommutative geometry.
    Recent developments from string theory imply that gravity may be emergent from gauge theories in noncommutative spacetime or large N gauge theories like as the AdS/CFT duality.
    As your motto, geometry is emergent from (noncommutative) algebra. In deep spaces, algebra seems to be more fundamental than geometry. Geometry is simply a consequence of coarse-graining approximations of an underlying algebra.

    An important point is that noncommutative spacetime is also a basically (noncommutative) phase space.
    So a noncommutative space endows the group of canonical transformations -symplectomorphism- which preserves the algebra of noncommutative spacetime. This symmetry is infinite dimensional, therefore there is a radical enhancement of spacetime symmetry compared to commutative one. More interestingly, it turns out that this symplectomorphism can be identified with a gauge symmetry in NC spacetime. This fact already implies that the transition from the commutative to the noncommutative spacetime goes with a radical change of physics since there is an infinite dimensional symmetry.
    Since the inner automorphism (its infinitesimal version) in noncommutative spacetime as you mentioned in your post defines a (generalized) derivation, as a result, gauge fields in noncommutative spacetime actually define vector fields on some manifold M (as an emergent geometry) and these vector fields form an orthonomal basis of M and so define a metric on M.

    A whole point for the emergent gravity is due to the Darboux theorem (as noticed above, noncommutative spacetime is a basically phase space). The curvature F=dA of U(1) bundle – electromagnetic fields – in noncommutative spacetime appears as a deformation of the original symplectic structure of background noncommutative spacetime. The Darboux theorem says that this dynamical deformation of symplectic structure in terms of gauge fields can always be (locally) translated into coordinate transformations, i.e., diffeomorphisms. In a sense, the Darboux theorem in noncommutative spacetime precisely plays a role of equivalence principle in Einstein relativity.
    One can make this argument be more precise in the context of deformation quantization a la Kontsevich. The Darboux theorem appears as an automorphism of (noncommutative) C*-algebra which can be identified with the diffeomorphism between gauge equivalent star products.
    Further musing about noncommutative field theory may lead to beautiful and surprising picutres about Einstein gravity as an emergent phenomenon from noncommuative spacetime.

  5. Anonymous

    This is a question for Hyun Seok. In your comment you mention NC `phase space. ‘ Do you know any precise definition of a NC phase space, e.g. a NC symplectic manifold? Can anyone comment on this?

  6. Unknown

    Dear anonymous,

    A famous example of NC phase space is quantum mechanics. Quantum mechanics is by definition the formulation of mechanics in “NC phase space”. So a paricle phase space in quantum mechanics is an example of NC symplectic manifold.

  7. Unknown

    Dear kea,

    unfortunately, this picture has not been condensed yet into a compact form although it is ubiquitous in recent string theory papers. But you may consult the following papers: arXiv:0704.0929 [hep-th]; hep-th/0612231; hep-th/0611174. Sorry, they are mine -.-.

  8. Anonymous

    Dear anonymous and Hyun Seok. Here is what I think about your question and answer. In Connes’ notion of `spectral triple’ we have a remarkable extension of the idea of spin Riemannian manifolds to a noncommutative setting. It is based on a non-trivial spectral realization of the distance in Riemannian geometry using the Dirac operator. The definition of a NC symplectic manifold, whatever it is, would similarly be based on a spectral realization of some key features of symplectic manifolds. I am not sure what it is, but a simple minded approach, e.g. by trying to mimic the definition of an alternating 2-form on the tangent bundle I don’t think would take us anywhere here.

  9. AC

    I believe that the extension of the “symplectic” framework to the NC world is simply the notion of the first order term in a deformation of the NC-algebra. This is quite clear in the commutative case where a symplectic structure (or more generally Poisson structure) is just the first term in the expansion of the deformed product. Thus it is a semi-classical form of the deformation. In the NC case there are many examples where it is natural to use a similar starting point for deformations (for instance in Rankin-Cohen brackets generalized to deformations of NC projective structures).

  10. Alejandro Rivero

    About the finite space of the Standard Model, and while we wait to hear more from it in this blog, I have been reading a bit more on the history of extra dimensional spaces of string theory. Gliozzi-Scherk-Olive, in 1977 “Supersymmetry, Supergravity theories and the dual spinor model” stress their need of Majorana-Weil spinors and then “The requeriment … gives thus the following condition on D: D=2[mod8]. The first non trivial dimension (2 is somewhat trivial) is thus D=10, which is a nice way to recover the critical dimension of the NSR model

    This is to be compared with Green-Schwarz 1984, “Covariant description of superstrings”, which focuses in triality and then gives a non periodic analisis: “This is precisely the identity that arises in the proof of supersymmetry for super Yang-Mills theories. Therefore, at the classical level, the possibilities for superstring actions are in one to one correspondence with super Yang-Mill theories: they are D=10 with Majorana-Weyl spinors, D=6 with Weyl…, D=4 … Majorana, and D=3 Majorana, as well as their dimensionally reduced forms. However, quantum mechanically only the D=10 theory is consistent, unless a Polyakov-type interpretation is possible in the other cases“. This view of super, but non-quantum, strings was reinforced with the advent of branes (d-dimensional objects in D-dimensional space), where codimension (D-d) works more or less in the same way, generating four “ladders” of admissible branes. The equivalent result for super yang-mils appears in 1977 (again!) by Brink-Schwarz-Scherk.

    It is not straightforward (to me) how the usual critical dimension, due to quantization of the [super]string, relates to the above considerations. It is interesting to note that some researchers developed the concept of “W3-gravity” and “W3-strings”, where the corresponding dimensions were not k+2, but 3k+2 (for k=1,2,4,8): 5,8,14,26 and then it marks the bosonic critical dimension. Also G. Sierra, in 1987, tried to see this jump as a move from “Jordan algebras” to “Freudenthal triple systems”. Perhaps “stringers” moved away from mod 8 periodicity because there is not a clear meaning available for the intermediate D=18. Moreover, the mod 8 game depends on fermions, and the other critical dimension, D=26, is supposedly a bosonic object.

  11. Fabien Besnard

    Dear Alain,
    you alluded in your text to Thurston’s theorem about the simplicity of the identity component of diffeo groups. As far as I know, this works for closed manifolds. With a noncompact Lorentzian manifold M as a base space, I guess one could still imagine to cook up a bundle X over M such that diff(X) is the semi-direct product of diff(M) with the SM gauge group G. Or is there a way to prevent this also ?

  12. Unknown


    I see that in the fourth comment above Alain Connes said that

    "My intention is to use this blog, this summer holidays, to explain [the] content [of articles on the NCG standard model] in details".

    Maybe these discussions haven't appeared on the blog so far, or did I miss them? (just asking)

    Over at the n-category Café I am having a discussion with Jacques Distler on the noncommutative standard model. He raised a couple of questions that would be good to hear the answer to from an expert.

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