Noncommutative geometry, its spectral calculus and functional action principle

(This is the second entry of my Fragments of a lover's dictionary on spectral physics)

Noncommutative Geometry

Plato derives the knowledge of ideas from body by abstraction and cutting away, leading us by various steps in mathematical discipline from arithmetic to geometry, thence to astronomy, and setting harmony above them all. For things become geometrical by the accession of magnitude to quantity; solid, by the accession of profundity to magnitude; astronomical, by the accession of motion to solidity; harmonical, by the accession of sound to motion. 

Platonicae quaestiones, Plutarch

Plato alleges that God forever geometrizes, Einstein & Hilbert did show.

If the devil of algebra fools us with mimetic dark matter,

Chamseddine, Connes & Mukhanov might know

  S.T.R.I.N.G.S. Folklore

The geometric concepts have first been formulated and exploited in the Framework of Euclidean geometry. This framework is best described using Euclid’s axioms (in their modern form by Hilbert’). These axioms involve the set X of points p∈X of the geometric space as well as families of subsets: the lines and the planes for 3-dimensional geometry. Besides incidence and order axioms one assumes that an equivalence relation (called congruence) is given between segments, i.e., pairs of points (p,q),p,q ∈X and also between angles, i.e., triples of points (a,O,b);a,O,b ∈X. These relations eventually allow us to define the length |(p.q)| of a segment and the size ∢(a,O,b) of an angle. The geometry is uniquely specified once these two congruence relations are given. They of course have to satisfy a compatibility axiom: up to congruence a triangle with vertices a,O,b ∈X is uniquely specified by the angle ∢(a,O,b) and the lengths of (a,O) and (0,b) ... Besides the completeness or continuity axiom, the crucial one is the axiom of unique parallel. The efforts of many mathematicians trying to deduce this last axiom from the others led to the discovery of non-Euclidean geometry...  

The introduction by Descartes of coordinates in geometry was at first an act of violence (cf. Ref. 2). In the hands of Gauss and Riemann it allowed one to extend considerably the domain of validity of geometric ideas. In Riemannian geometry the space Xⁿ is an n-dimensional manifold. Locally in X a point p is uniquely specified by giving n real numbers x¹,...,xⁿ which are the coordinates of p. The various coordinate patches are related by diffeomorphisms. The geometric structure on X is prescribed by a (positive definite) quadratic form, g_µν dx^µdx^ν, (1.4) which specifies the length of tangent vectors... This allows, using integration, to define the length of a path γ... The analog of the lines of Euclidean or non-Euclidean geometry are the geodesics. The analog of the distance between two points p,q ∈X is given by the formula, d(p,q)=Inf Length(γ)... where γ varies among all paths with γ(0)=p, γ(l)=q ... The obtained notion of “Riemannian space” has been so successful that it has become the paradigm of geometric space. There are two main reasons behind this success. On the one hand this notion of Riemannian space is general enough to cover the above examples of Euclidean and non-Euclidean geometries and also the fundamental example given by space-time in general relativity (relaxing the positivity condition Of (1.4)). On the other hand it is special enough to still deserve the name of geometry, the point being that through the use of local coordinates all the tools of the differential and integral calculus can be brought to bear ...   

Besides its success in physics as a model of space-time, Riemannian geometry plays a key role in the understanding of the topology of manifolds, starting with the Gauss Bonnet theorem, the theory of characteristic classes, index theory, and the Yang Mills theory. 

Thanks to the recent experimental confirmations of general relativity from the data given by binary pulsars there is little doubt that Riemannian geometry provides the right framework to understand the large scale structure of space-time. 

The situation is quite different if one wants to consider the short scale structure of space-time. We refer to Refs. 5 and 6 for an analysis of the problem of the coordinates of an event when the scale is below the Planck length. In particular there is no good reason to presume that the texture of space-time will still be the 4-dimensional continuum at such scales.  

In this paper we shall propose a new paradigm of geometric space which allows us to incorporate completely different small scale structures. It will be clear from the start that our framework is general enough. It will of course include ordinary Riemannian spaces but it will treat the discrete spaces on the same footing as the continuum, thus allowing for a mixture of the two. It also will allow for the possibility of noncommuting coordinates. Finally it is quite different from the geometry arising in string theory but is not incompatible with the latter since supersymmetric conformal field theory gives a geometric structure in our sense whose low energy part can be defined in our framework and compared to the target space geometry. 

It will require the most work to show that our new paradigm still deserves the name of geometry. We shall need for that purpose to adapt the tools of the differential and integral calculus to our new framework. This will be done by building a long dictionary which relates the usual calculus (done with local differentiation of functions) with the new calculus which will be done with operators in Hilbert space and spectral analysis, commutators.... The first two lines of the dictionary give the usual interpretation of variable quantities in quantum mechanics as operators in Hilbert space. For this reason and many others (which include integrality results) the new calculus can be called the quantized calculus’ but the reader who has seen the word “quantized” overused so many times may as well drop it and use “spectral calculus” instead. 

Noncommutative geometry and reality 

Alain Connes 

Received 4 April 1995; accepted for publication 7 June 1995

... we shall build our notion of geometry, in a very similar but somehow dual manner [to the Riemann's concept], on the pair (A, ds) of the algebra A of coordinates and the infinitesimal length element ds. For the start we only consider ds as a symbol, which together with A generates an algebra (A, ds). The length element ds does not commute with the coordinates, i.e. with the functions f on our space, f ∈ A. But it does satisfy non trivial relations. 

... we shall write down the axioms of geometry as the presentation of the algebraic relations between A and ds and the representation of those relations in Hilbert space. In order to compare different geometries, i.e. different representations of the algebra (A, ds) generated by A and ds, we shall use the following action functional,

(14) Trace(ϕ(ds/ℓ_p))

where ℓ_p is the Planck length and ϕ is a suitable cutoff function which will cut off all eigenvalues of ds larger than ℓ_p. We shall show in [CC] that for a suitable choice of the algebra A, the above action will give Einstein gravity coupled with the Lagrangian of the standard U(1)×SU(2)×SU(3) model of Glashow Weinberg Salam. The algebra will not be C^∞(M) with M a (compact) 4-manifold but a non commutative refinement of it which has to do with the quantum group refinement of the Spin covering of SO(4). 

1 → Z/2 → Spin(4) → SO(4) → 1.  

Amazingly, in this description the group of gauge transformations of the matter fields arises spontaneously as a normal subgroup of the generalized diffeomorphism group Aut(A). It is the non commutativity of the algebra A which gives for free the group of gauge transformations of matter fields as a (normal) subgroup of the group of diffeomorphisms.

What the present paper shows is that one should consider the internal gauge symmetries as part of the diffeomorphism group of the non commutative geometry, and the gauge bosons as the internal fluctuations of the metric. It follows then that the action functional should be of purely gravitational nature. We state the principle of spectral invariance, stronger than the invariance under diffeomorphisms, which requires that the action functional only depends on the spectral properties of D=ds^-1 in H. This is verified by the action,

I =Trace (ϕ(ds/ℓ_p))+<Dψ, ψ>

for any nice function ϕ from R^*₊ to R. We shall show in [CC] that this action gives the SM Lagrangian coupled with gravity. It would seem at first sight that the algebra A has disappeared from the scene when one writes down the above action, the point is that it is still there because it imposes the constraints [[D, a], b₀]=0 ∀ a, b ∈ A and Σa_0i[D, a_1i]...[D, a_4i]= γ coming from axioms [required to provide with the spectral calculus and the volume form]. It is important at this point to note that the integrality, n ∈ N of the dimension of a non commutative geometry appears to be essential to define the [algebraic formulation of a differential form called a] Hochschild cycle c∈Z_n and in turns the chirality γ. This is very similar to the obstruction which appears when one tries to apply dimensional regularization to chiral gauge theories.

Gravity coupled with matter and foundation of non-commutative geometry

A. Connes

(Submitted on 8 Mar 1996)

This leads us to the postulate that: 

The symmetry principle in noncommutative geometry is invariance under the group Aut(A). 

We now apply these ideas to derive a noncommutative geometric action unifying gravity with the standard model. The algebra is taken to be A=C^∞(M)⊗A_F where the algebra A_F is finite dimensional, A_F=ℂ⊕ℍ⊕M₃(ℂ) and ℍ ⊂ M₂(ℂ) is the algebra of quaternions, ℍ ...  

A is a tensor product which geometrically corresponds to a product space, an instance of spectral geometry for A is given by the product rule, 

H = L²(M, S)⊗ H_F , D = ∂_M ⊗ 1 + γ₅ ⊗ D_F  

where (H_F, D_F) is a spectral geometry on A_F, while both L²(M, S) and the Dirac operator ∂_M on M are as above. The group Aut(A) of diffeomorphisms falls in equivalence classes under the normal subgroup Int(A) of inner automorphisms. In the same way the space of metrics has a natural foliation into equivalence classes. The internal fluctuations of a given metric are given by the formula, 

D = D₀ + A + JAJ ^-1,   A = Σa_i[D₀, b_i] , a_i, b_i ∈ A and A = A^*... 

For Riemannian geometry these fluctuations are trivial. 

The hypothesis which we shall test in this letter is that there exist an energy scale Λ in the range 10¹⁵−10¹⁹ Gev at which we have a geometric action given by the spectral action...   

We now describe the internal geometry. The choice of the Dirac operator and the action of AF in HF comes from the restrictions that these must satisfy: 

J² = 1 , [J, D] = 0,        [a, Jb^*J ^-1]=0 ,        [[D, a], Jb^*J ^-1]=0 ∀ a, b. (4) 

We can now compute the inner fluctuations of the metric and thus operators of the form:   A = Σa_i[D, b_i]. This with the self-adjointness condition A = A^* gives a U(1), SU(2) and U(3) gauge fields as well as a Higgs field... 

It is a simple exercise to compute the square of the Dirac operator ... This can be cast into the elliptic operator form [7]: 

P = D² = −(g^µν∂_µ∂_ν · 1I + Aγ^µ∂_µ + B) 

where 1I, A µ and B are matrices of the same dimensions as D. Using the heat kernel expansion for Tr(e^-tP) ... we can show that ... a very lengthy but straightforward calculation ... gives for the bosonic action ... {the standard model action coupled to Einstein and Weyl gravity} plus higher order non-renormalizable interactions suppressed by powers of the inverse of the mass scale in the theory}... 

We ... adopt Wilson’s view point of the renormalization group approach to field theory [9] where the spectral action is taken to give the bare action with bare quantities ... at a cutoff scale Λ which regularizes the action the theory is assumed to take a geometrical form.

The renormalized action receives counterterms of the same form as the bare action but with physical parameters  ... The renormalization group equations ... yield relations between the bare quantities and the physical quantities with the addition of the cutoff scale Λ. Conditions on the bare quantities would translate into conditions on the physical quantities. The renormalization group equations of this system were studied by Fradkin and Tseytlin [10] and is known to be renormalizable, but non-unitary [11] due to the presence of spin-two ghost (tachyon) pole near the Planck mass. We shall not worry about non-unitarity (see, however, reference 12), because in our view at the Planck energy the manifold structure of space-time will break down and must be replaced with a geniunely noncommutative structure.

Relations between the bare gauge coupling constants as well as equations (3.19) have to be imposed as boundary conditions on the renormalization group equations [9]. The bare mass of the Higgs field is related to the bare value of Newton’s constant, and both have quadratic divergences in the limit of infinite cutoff Λ... 

There are some relations between the bare quantities. The renormalized action will have the same form as the bare action but with physical quantities replacing the bare ones. The relations among the bare quantities must be taken as boundary conditions on the renormalization group equations governing the scale dependence of the physical quantities. These boundary condition imply that the cutoff scale is of order ∼ 10¹⁵ Gev and sin²θw∼0.21 which is off by ten percent from the true value. We also have a prediction of the Higgs mass in the interval 170 − 180 Gev. There is ... a stronger disagreement where Newton’s constant comes out to be too large... Incidentally the problem that Newton’s constant is coming out to be too large is also present in string theory where also has unification of gauge couplings and Newton’s constant occurs [15]. These results must be taken as an indication that the spectrum of the standard model has to be altered as we climb up in energy. The change may happen at low energies (just as in supersymmetry ...) or at some intermediate scale. This also could be taken as an indication that the concept of space-time as a manifold breaks down and the noncommutativity of the algebra must be extended to include the manifold part.

A Universal Action Formula

Ali Chamseddine, Alain Connes 

(Submitted on 11 Jun 1996)

The notion of spectral geometry has... to do with the understanding of the notion of (smooth) manifold. While this notion is simple to define in terms of local charts i.e. by gluing together open pieces of finite dimensional vector spaces, it is much more difficult and instructive to arrive at a global understanding ... What one does is to detect global properties of the underlying space with the goal of characterizing manifolds... At the beginning of the 80’s, motivated by numerous examples of noncommutative spaces arising naturally in geometry from foliations or in physics from the Brillouin zone in the work of Bellissard on the quantum Hall effect, I realized that specifying an unbounded representative of the Fredholm operator was giving the right framework for spectral geometry ...

Over the years this new [noncommutative geometric paradigm of spectral nature] has been considerably refined ... The noncommutative geometry dictated by physics is the product of the ordinary 4-dimensional continuum by a finite noncommutative geometry which appears naturally from the classification of finite geometries of KO-dimension equal to 6 modulo 8 (cf. [15, 18]). The compatibility of the model with the measured value of the Higgs mass was demonstrated in [20] due to the role in the renormalization of the scalar field already present in [19].

In [21, 22], with Chamseddine and Mukhanov we gave the conceptual explanation of the finite noncommutative geometry from Clifford algebras and obtained a higher form of the Heisenberg commutation relations between p and q, whose irreducible Hilbert space representations correspond to 4-dimensional spin geometries. The role of p is played by the Dirac operator and the role of q by the Feynman slash of coordinates using Clifford algebras. The proof that all spin geometries are obtained relies on deep results of immersion theory and ramified coverings of the sphere. The volume of the 4-dimensional geometry is automatically quantized by the index theorem and the spectral model, taking into account the inner automorphisms due to the noncommutative nature of the Clifford algebras, gives Einstein gravity coupled with the slight extension of the standard model which is a Pati-Salam model. This model was shown in our joint work with A. Chamseddine and W. van Suijlekom [24, 25] to yield unification of coupling constants.

Th{e} quantization of the volume implies that the bothering cosmological leading term of the spectral action is now quantized and thus it no longer appears in the variation of the spectral action. Thus provided one understands how to reinstate all the ne details of the nite geometry (the one encoded by the Clifford algebras) such as the nuance on the grading and the number of generations, the variation of the spectral action will reproduce the Einstein equations coupled with matter.

Geometry and the Quantum 

Alain Connes 

Draft version from February 21, 2017