Connes' way in (the quantum theater)

From a geometric stand point it is natural to specify our position x in space by giving curvature invariants at x, under the standard hypothesis that space is well modeled as a Riemannian space X of dimension three. Of course there is no way to distinguish between points which are obtained from each other by an isometry of X. A very related problem is the problem of giving observable quantities in the theory of gravity. An observable should be an invariant of the geometry. Besides the above geometric point of view there is a “dual” one which is based on spectral invariants and whose relation to the geometric one is through the heat kernel expansion of the trace of operators in Hilbert space. Our thesis is that, since much of the information we have about the nature of space-time is of spectral nature, one needs to understand carefully the process which transforms this spectral information into a geometric one. At the mathematical level one knows that the spectrum of the Dirac operator D of a compact Riemannian space gives a sequence of invariants of the geometry: the list of the eigenvalues. It is also known from Milnor’s one page paper [34], that this invariant is not complete. The missing information is given by the relative position in Hilbert space of two commutative algebras A and B. The first is the algebra of measurable bounded functions on X acting by multiplication in the Hilbert space H of L2-spinors. It is an old result of von Neumann that, once the dimension of X is fixed, the pair (A, H) does not depend upon the manifold X. Said in simple terms what it means is that, from the point of view of measure theory, the “bunches of points” coming from different manifolds of the same dimension can be identified and say nothing about the geometry. Similarly the spectral theory of operators tells us that the spectrum of the Dirac operator D i.e. its list of eigenvalues as a subset with multiplicity inside R gives us the full information about the pair (H,D) of the Hilbert space of L2-spinors and the Dirac operator acting in H. The missing invariant (cf. [20]) that one needs in order to assemble together the pieces (A,H) and (H,D) and obtain the spectral triple (A,H,D) is given by the relative position of the algebra A with the algebra B of functions f(D) of the Dirac operator. The invariant defined in [20] is an infinite dimensional analogue of an invariant which is familiar to physicists and which measures the relative position of the mass eigenstates for the upper quarks with respect to the mass eigenstates for the lower ones lifted up using the action of the weak isospin group. This is the CKM matrix which measures the generalized “angle” of two basis (whose elements are given up to phase). Once the spectral triple (A,H,D) is assembled from its two pieces one recovers the points of the space and the full geometric information. It is interesting that the invariant manner of encoding a point x in this process can be understood by specifying an infinite hermitian matrix Hλµ with complex entries. The labels λ, µ are the eigenvalues of the Dirac operator D. The matrix elements Hλµ are the inner products Hλµ = <ψλ(x), ψµ(x)> of the eigenspinors ψλ evaluated at the point x. While these eigenspinors are globally orthogonal, they are not so when evaluated at a point x since the spinor space Sx at x is finite dimensional. Thus the matrix Hλµ expresses the correlations between different frequencies at the point x. Modulo the obvious gauge ambiguity this matrix characterizes the point x (cf. [4][20]) in a way which is dual to the local expansion of curvature invariants of the metric. This suggests that in order to specify “where we are” we should first give the spectrum of the Dirac operator and then the matrix Hλµ which, in essence, gives the correlations between the various frequencies. It is not quite what happens concretely in physics since most of the observations which are done involve light (bosons), rather than fermions such as neutrinos, but we are nevertheless quite used to the need for labeling the information in a directional manner (as what would happen using spinor space at x) and for establishing correlations between observations at different frequencies such as infrared and ultraviolet ones. 
The... notion... of “variable”... played a central role at the very beginning of the calculus. According to Newton: “In a certain problem, a variable is the quantity that takes an infinite number of values which are quite determined by this problem and are arranged in a definite order”... In our modern language we are used to think of a real variable as a map f : X → R from a set X to the real line R. Let us start from the remark that discrete and continuous variables cannot coexist in this formalism. The simple point is that if a variable is continuous the set X is necessarily at least of the cardinality of the continuum and this precludes the existence of a variable with countable range such that each value is reached only a finite number of times. This problem of treating continuous and discrete variables on the same footing is solved using the formalism of quantum mechanics. In this formalism a “real variable” is not given in the above classical manner but is a self-adjoint operator in Hilbert space. As such it has a “spectrum” which is its set of values, each being reached with some (spectral) multiplicity. A continuous variable is an operator with continuous spectrum and a discrete variable an operator with discrete spectrum (and of course the mixed case occurs). The uniqueness of the separable infinite dimensional Hilbert space shows that the Hilbert space L2[0,1] of square integrable functions on the unit interval, is the same as the Hilbert space of square integrable sequences ℓ2(N). This shows that variables with continuous range, such as T where (T ξ)(x) = xξ(x) for ξ ∈ L2[0,1], coexist with variables with countable range such as S, (Sa)n = 1/nan for (an) ∈ ℓ2(N). The only new fact is that they just cannot commute. If they would it would have been possible to model them in a classical manner but this is not the case. In classical physics the basic variability is due to the passing of time so that “t” is the paradigm of the “variable”. But in quantum physics there is a more profound inherent variability which is a basic experimental physics fact. It prevents one from reproducing experimental results of quantum physics which display the choice of an eigenvalue by reduction of the wave packet, as in the diffraction of an electron by a narrow slit. This spontaneous variability of the quantum world far surpasses in originality the simple time variation of the classical world. 
The Riemannian paradigm is based on the Taylor expansion in local coordinates of the square of the line element and in order to measure the distance between two points one minimizes the length of a path joining the two points... Thus in this paradigm of geometry only the square of the line element makes sense, and the formula for the geodesic distance involves the extraction of a square root. This extraction of a square root is in fact hiding a deeper understanding of the line element and the choice of a square root is associated to a global structure which is that of a spin structure. The Dirac operator D is canonically associated to a Riemannian metric and a spin structure by a formula which can be traced back to Hamilton (who wrote down the operator i∂x + j∂y + k∂z in terms of his generators i, j, k for quaternions). Paul Dirac showed, in the flat case, how to extract the square root of the Laplacian in order to obtain a first order version of the Maxwell equation and Atiyah and Singer gave the general canonical definition of the Dirac operator on a Riemannian manifold endowed with a spin structure. This provides a direct connection with the quantum formalism: the line element is now upgraded to this formalism as the propagator ds = D−1 and the same geodesic distance d(a, b) can be computed in a dual manner as
d(a, b) = Sup {|f(a) − f(b)| ; f ∈ A , ||[D, f]|| ≤ 1} 
where, as above, A is the algebra of measurable functions acting by multiplication in the Hilbert space H of L2-spinors. In other words one measures distances not by taking the shortest continuous path between the two points a, b, but by sending a wave f(x) whose frequency is limited from above, and measuring its maximal variation |f(a)−f(b)| from a to b. The operator norm ||[D, f]|| controls from above the frequency of f since it is given by the supremum of the gradient of f measured using the Riemannian metric. It is quite important to understand at this point how one reconstructs the underlying space X. A point a of X is a character f → f(a) of the subalgebra of A given by the condition f ∈ A , ||[D, f]|| ≤ ∞. This condition involves the choice of D and determines the Lipschitz functions inside the algebra of measurable functions. Thus in particular the operator D determines what it means to be continuous, and hence the topology of X. In fact it also determines smoothness and the latter comes from the one parameter group eit|D| which plays the role of the geodesic flow in the operator theoretic framework. The algebra A is fixed once and for all thanks to [a] Theorem of von Neumann ... [that] should be thought of as the operator theoretic version of the uniqueness of the continuum as a set. It shows that if we are interested in sifting through all Riemannian geometries of a given dimension, we can fix the algebra A and the way it is represented as operators in the Hilbert space H (which is unique itself also). Thus the only remaining “variable” is the operator D. This operator has two invariants which are also invariants of the geometry:
  • Its spectrum Spec D.
  • The relative spectrum SpecN(A) where N = {f(D)} is the algebra of functions of D.
We refer to [20] for the definition of the relative spectrum, which as explained in the introduction measures the relative position of the two algebras A and N.
Ali H. Chamseddine, Alain Connes, August 5th 2010