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Geometry and the Quantum: Basics

Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume>4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the"particle picture"for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics. Physical applications of this quantization scheme will follow in a separate publication.


Introduction
The goal of this paper is to reconcile Quantum Mechanics and General Relativity by showing that the latter naturally arises from a higher degree version of the Heisenberg commutation relations. One great virtue of the standard Hilbert space formalism of quantum mechanics is that it incorporates in a natural manner the essential "variability" which is the characteristic feature of the Quantum: repeating twice the same experiment will generally give different outcome, only the probability of such outcome is predicted, the various possibilities form the spectrum of a self-adjoint operator in Hilbert space. We have discovered a geometric analogue of the Heisenberg commutation relations [p, q] = i . The role of the momentum p is played by the Dirac operator. It takes the role of a measuring rod and at an intuitive level it represents the inverse of the line element ds familiar in Riemannian geometry, in which only its square is specified in local coordinates. In more physical terms this inverse is the propagator for Euclidean Fermions and is akin to an infinitesimal as seen already in its symbolic representation in Feynman diagrams where it appears as a solid (very) short line •−−−−−− • . The role of the position variable q was the most difficult to uncover. It has been known for quite some time that in order to encode a geometric space one can encode it by the algebra of functions (real or complex) acting in the same Hilbert space as the above line element, in short one is dealing with "spectral triples". Spectral for obvious reasons and triples because there are three ingredients: the algebra A of functions, the Hilbert space H and the above Dirac operator D. It is easy to explain why the algebra encodes a topological space. This follows because the points of the space are just the characters of the algebra, evaluating a function at a point P ∈ X respects the algebraic operations of sum and product of functions. The fact that one can measure distances between points using the inverse line element D is in the line of the Kantorovich duality in the theory of optimal transport. It takes here a very simple form. Instead of looking for the shortest path from point P to point P as in Riemannian Geometry, which only can treat path-wise connected spaces, one instead takes the supremum of |f (P ) − f (P )| where the function f is only constrained not to vary too fast, and this is expressed by asking that the norm of the commutator [D, f ] be ≤ 1. In the usual case where D is the Dirac operator the norm of [D, f ] is the supremum of the gradient of f so that the above control of the norm of the commutator [D, f ] means that f is a Lipschitz function with constant 1, and one recovers the usual geodesic distance. But a spectral triple has more information than just a topological space and a metric, as can be already guessed from the need of a spin structure to define the Dirac operator (due to Atiyah and Singer in that context) on a Riemannian manifold. This additional information is the needed extra choice involved in taking the square root of the Riemannian ds 2 in the operator theoretic framework. The general theory is K-homology and it naturally introduces decorations for a spectral triple such as a chirality operator γ in the case of even dimension and a charge conjugation operator J which is an antilinear isometry of H fulfilling commutation relations with D and γ which depend upon the dimension only modulo 8. All this has been known for quite some time as well as the natural occurrence of gravity coupled to matter using the spectral action applied to the tensor product A ⊗ A of the algebra A of functions by a finite dimensional algebra A corresponding to internal structure. In fact it was shown in [4] that one gets pretty close to zooming on the Standard Model of particle physics when running through the list of irreducible spectral triples for which the algebra A is finite dimensional. The algebra that is both conceptual and works for that purpose is where H is the algebra of quaternions and M k the matrices. However it is fair to say that even if the above algebra is one of the first in the list, it was not uniquely singled out by our classification and moreover presents the strange feature that the real dimensions of the two pieces are not the same, it is 16 for M 2 (H) and 32 for M 4 (C). One of the byproducts of the present paper is a full understanding of this strange choice, as we shall see shortly. Now what should one beg for in a quest of reconciling gravity with quantum mechanics? In our view such a reconciliation should not only produce gravity but it should also naturally produce the other known forces, and they should appear on the same footing as the gravitational force. This is asking a lot and, in the minds of many, the incorporation of matter in the Lagrangian of gravity has been seen as an unnecessary complication that can be postponed and hidden under the rug for a while. As we shall now explain this is hiding the message of the gauge sector which in its simplest algebraic understanding is encoded by the above algebra A = M 2 (H) ⊕ M 4 (C). The answer that we discovered is that the package formed of the 4-dimensional geometry together with the above algebra appears from a very simple idea: to encode the analogue of the position variable q in the same way as the Dirac operator encodes the components of the momenta, just using the Feynman slash. To be more precise we let Y ∈ A ⊗ C κ be of the Feynman slashed form Y = Y A Γ A , and fulfill the equations Here κ = ±1 and C κ ⊂ M s (C), s = 2 n/2 , is the real algebra generated by n + 1 gamma matrices Γ A , 1 ≤ a ≤ n + 1 1 The one-sided higher analogue of the Heisenberg commutation relations is where the notation T means the normalized trace of T = T ij with respect to the above matrix algebra M s (C) (1/s times the sum of the s diagonal terms T ii ). We shall show below in Theorem 1 that a solution of this equation exists for the spectral triple (A, H, D) associated to a Spin compact Riemannian manifold M (and with the components Y A ∈ A) if and only if the manifold M breaks as the disjoint sum of spheres of unit volume. This breaking into disjoint connected components corresponds to the decomposition of the spectral triple into irreducible components and we view these irreducible pieces as quanta of geometry. The corresponding picture, with these disjoint quanta of Planck size is of course quite remote from the standard geometry and the next step is to show that connected geometries of arbitrarily large size are obtained by combining the two different kinds of geometric quanta. This is done by refining the one-sided equation (2) using the fundamental ingredient which is the real structure of spectral triples, and is the mathematical incarnation of charge conjugation in physics. It is encoded by an anti-unitary isometry J of the Hilbert space H fulfilling suitable commutation relations with D and γ and having the main property that it sends the algebra A into its commutant as encoded by the order zero condition : [a, JbJ −1 ] = 0 for any a, b ∈ A. This commutation relation allows one to view the Hilbert space H as a bimodule over the algebra A by making use of the additional representation a → Ja * J −1 . This leads to refine the quantization condition by taking J into account as the two-sided equation 2 where E is the spectral projection for It is the classification of finite geometries of [4] which suggested to use the direct sum C + ⊕C − of two Clifford algebras and the algebra C ∞ (M, C + ⊕C − ). As we shall show below in Theorem 6 this condition still implies that the volume of M is quantized but no longer that M breaks into small disjoint connected components. More precisely let M be a smooth connected oriented compact manifold of dimension n. Let α be the volume form (of unit volume) of the sphere S n . One considers the (possibly empty) set D(M ) of pairs of smooth maps φ ± : M → S n such that the differential form 3 does not vanish anywhere on M (ω(x) = 0 ∀x ∈ M ). One introduces an invariant q(M ) ⊂ Z defined as the subset of Z: where degree(φ) is the topological degree of the smooth map φ. Then a solution of (3) exists if and only if the volume of M belongs to q(M ) ⊂ Z.
We first check (Theorem 10) that q(M ) contains arbitrarily large numbers in the two relevant cases M = S 4 and M = N × S 1 where N is an arbitrary connected compact oriented smooth three manifold. We then give the proof (Theorem 12) that the set q(M ) contains all integers m ≥ 5 for any smooth connected compact spin 4-manifold, which shows that our approach encodes all the relevant geometries.
In the above formulation of the two-sided quantization equation the algebra C ∞ (M, C + ⊕ C − ) appears as a byproduct of the use of the Feynman slash.
It is precisely at this point that the connection with our previous work on the noncommutative geometry (NCG) understanding of the Standard Model appears. Indeed as explained above we determined in [4] the algebra A = M 2 (H) ⊕ M 4 (C) as the right one to obtain the Standard Model coupled to gravity from the spectral action applied to the product space of a 4-manifold M by the finite space encoded by the algebra A. Thus the full algebra is the algebra C ∞ (M, A) of A-valued functions on M . Now the remarkable fact is that in dimension 4 one has More precisely, the Clifford algebra Cliff(+, +, +, +, +) is the direct sum of two copies of M 2 (H) and thus in an irreducible representation, only one copy of M 2 (H) survives and gives the algebra over R generated by the gamma matrices Γ A . The Clifford algebra Cliff(−, −, −, −, −) is M 4 (C) and it also admits two irreducible representations (acting in a complex Hilbert space) according to the linearity or anti-linearity of the way C is acting. In both the algebra over R generated by the gamma matrices Γ A is M 4 (C). This fact clearly indicates that one is on the right track and in fact together with the above two-sided equation it unveils the following tentative "particle picture" of gravity coupled with matter, emerging naturally from the quantum world. First we now forget completely about the manifold M that was used above and take as our framework a fixed Hilbert space in which C = C + ⊕ C − acts, as well as the grading γ, and the anti-unitary J all fulfilling suitable algebraic relations. So far there is no variability but the stage is set. Now one introduces two "variables" D and Y = Y + ⊕ Y − both selfadjoint operators in Hilbert space. One assumes simple algebraic relations such as the commutation of C and JCJ −1 , of Y and JY J −1 , the fact that Y ± = Y ± A Γ A ± with the Y A commuting with C, and that Y 2 = 1 + ⊕ (−1) − and also that the commutator [D, Y ] is bounded and its square again commutes with both C ± and the components Y A , etc... One also assumes that the eigenvalues of the operator D grow as in dimension 4. One can then write the two-sided quantization equation (3) and show that solutions of this equation give an emergent geometry. The geometric space appears from the joint spectrum of the components Y ± A . This would a priori yield an 8-dimensional space but the control of the commutators with D allows one to show that it is in fact a subspace of dimension 4 of the product of two 4-spheres. The fundamental fact that the leading term in the Weyl asymptotics of eigenvalues is quantized remains true in this generality due to already developed mathematical results on the Hochschild class of the Chern character in Khomology. Moreover the strong embedding theorem of Whitney shows that there is no a-priori obstruction to view the (Euclidean) space-time manifold as encoded in the 8-dimensional product of two 4-spheres. The action functional only uses the spectrum of D, it is the spectral action which, since its leading term is now quantized, will give gravity coupled to matter from its infinitesimal variation.

Geometric quanta and the one-sided equation
We recall that given a smooth compact oriented spin manifold M , the asso- where γ µ = e µ a γ a and ω µ is the spin-connection. Proof. We can assume that κ = 1 since the other case follows by multiplication by i = √ −1. Equation (1) shows that a solution Y of the above equations gives a map Y : M → S n from the manifold M to the n-sphere. Given n operators T j ∈ C in an algebra C the multiple commutator

One sided equation and spheres of unit volume
(where σ runs through all permutations of {1, . . . , n}) is a multilinear totally antisymmetric function of the T j ∈ C. In particular, if the T i = a j i S j are linear combinations of n elements S j ∈ C one gets Let us compute the left hand side of (2). The normalized trace of the product of n + 1 Gamma matrices is the totally antisymmetric tensor where we let ∇f be the Clifford multiplication by the gradient of f . Thus one gets at any x ∈ M the equality For fixed A, and x ∈ M the sum over the other indices where all other indices are = A. At x ∈ M one has ∇Y j = γ µ ∂ µ Y j and by (6) the multi-commutator (with ∇Y A missing) gives Since γ µ = e µ a γ a and i n/2 [γ 1 , . . . , γ n ] = n!γ one thus gets by (6), where so that ωdx 1 ∧ · · · ∧ dx n is the pullback Y # (ρ) by the map Y : M → S n of the rotation invariant volume form ρ on the unit sphere S n given by Thus, using the inverse vierbein, the one-sided equation (2) is equivalent to This equation (9) implies that the Jacobian of the map Y : M → S n cannot vanish anywhere, and hence that the map Y is a covering. Since the sphere S n is simply connected for n > 1, this implies that on each connected component Moreover equation (9) shows that the volume of each component M j is the same as the volume S n ρ of the sphere. Conversely it was shown in [8] that, for n = 2, 4, each Riemannian metric on S n whose volume form is the same as for the unit sphere gives a solution to the above equation. In fact the above discussion gives a direct proof of this fact for all (even) n. Since all volume forms with same total volume are diffeomorphic ( [17]) one gets the required result.
The spectral triple (A, H, D) is then the direct sum of the irreducible spectral triples associated to the components. Moreover one can reconstruct the original algebra A as the algebra generated by the components Y A of Y together with the commutant of the operators D, Y, Γ A . This implies that a posteriori one recovers the algebra A just from the representation of the D, Y, Γ A in Hilbert space. As mentioned above the operator theoretic equation (2) implies the integrality of the volume when the latter is expressed from the growth of the eigenvalues of the operator D. Theorem 1 gives a concrete realization of this quantization of the volume by interpreting the integer k as the number of geometric quantas forming the Riemannian geometry M . Each geometric quantum is a sphere of arbitrary shape and unit volume (in Planck units).

The degree and the index formula
In fact the proof of Theorem 1 gives a statement valid for any Y not necessarily fulfilling the one-sided equation (2). We use the non-commutative integral as the operator theoretic expression of the integration against the volume form det e a µ dx 1 ∧ · · · ∧ dx n of the oriented Riemannian manifold M . The factor 2 n/2+1 on the right comes from the factor 2 in Y = 2e − 1 and from the normalization (by 2 −n/2 ) of the trace. The − is taken in the Hilbert space of the canonical spectral triple of the Riemannian manifold.
Proof. This follows from (8) for any scalar function f on M one has (see [7], Chapter IV,2,β, Proposition 5), with Ω n = 2π n/2 /Γ(n/2) the volume of the unit sphere S n−1 , Thus the left hand side of (10) gives 3 Quantization of volume and the real structure J We consider the two sided equation (3). The action of the algebra C + ⊕ C − in the Hilbert space H splits H as a direct sum H = H (+) ⊕ H (−) of two subspaces corresponding to the range of the projections 1 ⊕ 0 ∈ C + ⊕ C − and 0⊕1 ∈ C + ⊕C − . The real structure J interchanges these two subspaces. The algebra C + acts in H (+) and the formula x → Jx * J −1 gives a right action of This allows us to reduce to the following simplified situation occurring in H (+) . We take M of dimension n = 2m and consider two sets of gamma matrices Γ A and Γ B which commute with each other. We consider two fields The condition Let e = 1 2 (Y + 1) , e = 1 2 (Y + 1) , E = ee = 1 2 (Z + 1) then Z = 2ee − 1 and thus This means that Z 2 = 1 and we can use it to write the quantization condition in the form where is the normalized trace relative to the matrix algebra generated by all the gamma matrices Γ A and Γ B .

The normalized traces
More precisely we let Mat + be the matrix algebra generated by all the gamma matrices Γ A and Mat − be the matrix algebra generated by all the gamma matrices Γ B . We define T ± as above as the normalized trace, which is 2 −m times the trace relative to the algebras Mat ± of an operator T in H. It is best expressed as an integral of the form where Spin ± ⊂ Mat ± is the spin group and dg the Haar measure of total mass 1. Proof. 1) follows from (16) since gST U g −1 = SgT g −1 U for S, U commuting with Mat + and g ∈ Spin + .
2) The representation of the product group G = Spin + × Spin − given by (g, g ) → gg ∈ Mat + Mat − is irreducible, and thus parallel to (16) one has using the fact that any g commutes with any g .
3) This follows from (17) since one has The proof is the same as for 3).

Case of dimension 2
This is the simplest case, one has: The condition (15) implies that the (2-dimensional) volume of M is quantized. If M is a smooth connected compact oriented 2-dimensional manifold with quantized volume there exists a solution of (15).
Proof. We shall compute the left hand side of (15) and show that Thus as above we see that (15) is equivalent to the quantization condition which gives the volume of M as the sum of the degrees of the two maps Y : M → S 2 and Y : M → S 2 . This shows that the volume is quantized (up to normalization). Conversely let M be a compact oriented 2-dimensional manifold with quantized volume. Choose two smooth maps Y : M → S 2 and Y : M → S 2 such that when you add the pull back of the oriented volume form ω of S 2 by Y and Y you get the volume form of M . This will be discussed in great details in §4. However, it is simple in dimension 2 mostly because, on a connected compact smooth manifold, all smooth nowhere-vanishing differential forms of top degree with the same integral are equivalent by a diffeomorphism ( [17]). This solves equation (19). It remains to show (18). We use the properties

The two sided equation in dimension 4
This calculation will now be done for the four dimensional case: Lemma 5 In the 4-dimensional case one has The condition 15 implies that the (4-dimensional) volume of M is quantized.
Proof. Now Γ A and Γ A will have A = 1, · · · , 5. We now compute, using (23) We multiply by Z = 2ee −1 on the left and treat the various terms as follows. Thus combining with (26), we get that the total contribution of the quadratic terms is 0.
Finally the second statement of Lemma 5 follows from Lemma 2.

Algebraic relations
It is important to make the list of the algebraic relations which have been used and do not follow from the definition of Y and Y . Note first that for Y = Y A Γ A with the hypothesis that the components Y A belong to the commutant of the algebra generated by the Γ B , one has Indeed the matrices Γ A Γ B for A < B, are linearly independent and the coef-

The Quantization Theorem
In the next theorem the algebraic relations between Y ± , D, J, C ± , γ are assumed to hold. We shall not detail these relations but they are exactly those discussed in §3.4 and which make the proof of Lemma 5 possible. As in the introduction we adopt the following definitions. Let M be a connected smooth oriented compact manifold of dimension n. Let α be the volume form of the sphere S n . One considers the (possibly empty) set D(M ) of pairs of smooth maps φ, ψ : M → S n such that the differential form does not vanish anywhere on M (ω(x) = 0 ∀x ∈ M ). One defines an invariant which is the subset of Z:  Proof. (i) By Lemma 5 one has, as in the two dimensional case that the left hand side of (15) is up to normalization, so that (15) implies that the volume of M is (up to sign) the sum of the degrees of the two maps. This is enough to give the proof in the case of the spectral triple of a manifold, and we shall see in Theorem 17 that it also holds in the abstract framework.
(ii) Using Lemma 5 the proof is the same as in the two dimensional case. Note that the connectedness hypothesis is crucial in order to apply the result of [17].

Differential geometry and the two sided equation
The invariant q M makes sense in any dimension. For n = 2, 3, and any connected M , it contains all sufficiently large integers. The case n = 4 is more difficult but we shall prove below in Theorem 12 that it contains all integers m > 4 as soon as the connected 4-manifold M is a Spin manifold, an hypothesis which is automatic in our context.

Case of dimension n < 4
Lemma 7 Let M be a compact connected smooth oriented manifold of dimension n < 4. Then for any differential form ω ∈ Ω n (M ) which vanishes nowhere, agrees with the orientation, and fulfills the quantization M ω ∈ Z, | M ω| > 3, one can find two smooth maps φ, φ such that where α is the volume form of the sphere of unit volume.

up to normalization
Proof. By [1] as refined in [20], any Whitehead triangulation of M provides (after a barycentric subdivision) a ramified covering of the sphere S n obtained by gluing two copies ∆ n ± of the standard simplex ∆ n along their boundary. One uses the labeling of the vertices of each n-simplex by {0, 1, . . . , n} where each vertex is labeled by the dimension of the face of which it is the barycenter. The bi-coloring corresponds to affecting each n-simplex of the triangulation with a sign depending on wether the orientation of the simplex agrees or not with the orientation given by the labeling of the vertices. One then gets a PL-map M → S n by mapping each simplex with a ± sign to ∆ n ± respecting the labeling of the vertices. This gives a covering which is ramified only on the (n − 2)-skeleton of ∆ n ± . After smoothing one then gets a smooth map φ : M → S n whose Jacobian will be > 0 outside a subset K of dimension n − 2 of M . Using the hypothesis n < 4 (which gives (n − 2) + (n − 2) < n), the set of orientation preserving diffeomorphisms ψ ∈ Diff + (M ) such that ψ(K) ∩ K = ∅ is a dense subset of Diff(M ) + , thus one finds ψ ∈ Diff + (M ) such that the Jacobian of φ and the Jacobian of φ = φ • ψ never vanish simultaneously. This shows that the differential form ρ = φ # (α) + φ # (α) does not vanish anywhere and by the result of [17] there exists an orientation preserving diffeomorphism of M which transforms this form into ω provided they have the same integral. But the integral of ρ is twice the integral of φ # (α) which in turns is the degree of the map φ and thus the number of simplices of a given color. As performed the above construction only gives even numbers, since the integral of ρ is twice the degree of the map φ, but we shall see shortly in Lemma 9 that in fact the degree of the map φ is in q(M ) from a fairly general argument.

Preliminaries in dimension 4
Let us first give simple examples in dimension 4 of varieties where one can obtain arbitrarily large quantized volumes. First for the sphere S 4 itself one can construct by the same procedure as in the proof of Lemma 7 a smooth map φ : S 4 → S 4 whose Jacobian is ≥ 0 everywhere and whose degree is a given integer N . One can then simply take the sum ω = φ # (α) + α which does not vanish and has integral N + 1. Next, let us take M = S 3 ×S 1 . Then one can construct by the same procedure as in the proof of Lemma 7 a smooth map φ : M → S 4 whose Jacobian is ≥ 0 everywhere and which vanishes only on a two dimensional subset K ⊂ M . Let p : M → S 3 be the first projection using the product M = S 3 × S 1 .  Then p(K) is a two dimensional subset of S 3 and hence there exists x ∈ S 3 , x / ∈ p(K). One can thus find a diffeomorphism ψ ∈ Diff + (S 3 ) such that ψ(p(K)) ∩ p(K) = ∅. Then the diffeomorphism ψ ∈ Diff + (M ) which acts as (x, y) → ψ (x, y) = (ψ(x), y) is such that ψ (K)∩K = ∅. Thus it follows that the Jacobian of φ and the Jacobian of φ = φ•ψ never vanish simultaneously and the proof of Lemma 7 applies. Note moreover that in this case M = S 3 × S 1 is not simply connected and one gets smooth covers of arbitrary degree which can be combined with the maps (φ, φ ).

Necessary condition
Jean-Claude Sikorav and Bruno Sevennec found the following obstruction which implies for instance that D(CP 2 ) = ∅. In general

Reduction to a single map
Here is a first lemma which reduces to properties of a single map. Proof. Let first φ be such that φ # (α) + φ # (α) does not vanish anywhere. Then φ # (α) does not vanish on the closed set R and hence in a neighborhood V ⊃ R. Its restriction to V gives the desired immersion. Conversely let f : V → R 4 be an immersion of a neighborhood V of R. We can assume by changing the orientation of R 4 for the various connected components of V that f # (v) > 0 where v is the standard volume form on R 4 . We first extend f to a smooth mapf : M → R 4 by extending the coordinate functions. We then can assume that f (M ) ⊂ B 4 ⊂ R 4 where B 4 is the unit ball which we identify with the half sphere so that B 4 ⊂ S 4 . We denote by β = α|B 4 the restriction of α to B 4 . We have f # (β) > 0 on V but not on M since the map f : M → S 4 is of degree zero. Let ρ > 0 be a fixed volume form (nowhere vanishing) on M . Let > 0 be such that For y ∈ B 4 and 0 < λ ≤ 1 we let λy be the rescaled element (using rescaling in R 4 ). Then for λ small enough one has where the absolute value is on the ratio of (λf ) # (α) with ρ. One then gets that with φ = λf one has (φ # (α) + φ # (α))(x) = 0, ∀x ∈ M.

Products
Let N be a smooth oriented compact three manifold. Then N is Spin, thus the condition w 2 2 = 0 is automatically fulfilled by M = N × S 1 . In fact: Theorem 10 Let N be a smooth oriented connected compact three manifold. Let M = N × S 1 , then the set q(M ) is non-empty, and contains all integers m ≥ r for some r > 0.
Proof. Let g : S 3 × S 1 → S 4 be a ramified cover of degree m and singular set Σ g . Let N be described as a ramified cover f : N → S 3 ramified over a knot K ⊂ S 3 ([16], [13]). One may, using the two dimensionality of Σ g , assume that Let h = f × id : N × S 1 → S 3 × S 1 . Let Σ f ⊂ N be the singular set of f . one has f (Σ f ) ⊂ K and thus, with Σ h ⊂ N × S 1 the singular set of h, The singular set Σ φ of φ is the union of Σ h with h −1 (Σ g ). This two closed sets are disjoint since h(Σ h ) ∩ Σ g = ∅. By Lemma 9 it is enough to find immersions in R 4 of neighborhoods V ⊃ Σ h and W ⊃ h −1 (Σ g ). By construction Σ h = Σ f × S 1 is a union of tori with trivial normal bundle, since their normal bundle is the pullback by the projection of the normal bundle to Σ f which is a union of circles. This gives the required immersion V → R 4 . Moreover the restriction of h to a suitable neighborhood W of h −1 (Σ g ) is a smooth covering of an open set of S 3 × S 1 . On each of the components W j of this covering, the local situation is the same as for the inclusion of Σ g in S 3 × S 1 . Thus one gets the required immersion W → R 4 . This shows that the hypothesis of Lemma 9 is fulfilled and one gets that D(M ) = ∅ and that degree(f ) + degree(g) ∈ q(M )

Remark 11
Here is a variant, due to Jean-Claude Sikorav, of the above proof, also using Lemma 9. The 4-manifold M = N × S 1 is parallelizable since any oriented 3-manifold is parallelizable (see for instance [13] for a direct proof ), and by [18] Theorem 5, there is an immersion ψ : M \{p} → R 4 of the complement of a single point p ∈ M so that it is easy to verify the hypothesis of Lemma 9 and show that for any ramified cover φ : M → S 4 one has degree(φ) ∈ q(M ).

Spin manifolds
Theorem 12 Let M be a smooth connected oriented compact spin 4-manifold.
Then the set q(M ) contains all integers m ≥ 5.
Proof. We proceed as in the proof of Lemma 7 and get from any Whitehead triangulation of M (after a barycentric subdivision) a ramified covering γ of the sphere S 4 obtained by gluing two copies ∆ 4 ± of the standard simplex ∆ 4 along their boundary. Let then V be a neighborhood of the 2-skeleton of the triangulation which retracts on the 2-skeleton. Then the restriction of the tangent bundle of M to V is trivial since the spin hypothesis allows one to view T M as induced from a Spin(4) principal bundle while the classifying space BSpin(4) is 3-connected. Thus the extension by Poenaru [18], Theorem 5, (see also [19]), of the Hirsch-Smale immersion theory ( [14], [22]) to the case of codimension zero yields an immersion V → R 4 . After smoothing γ while keeping its singular set inside V one gets that the hypothesis of Lemma 9 is fulfilled and this gives that m ∈ q(M ) where 2m is the number of simplices of the triangulation. For the finer result involving the small values of m one can use the theorem 5 of M. Iori and R. Piergallini [15], which gives a smooth ramified cover φ : M → S 4 of any degree m ≥ 5 whose singular set R ⊂ M is a disjoint union of smooth surfaces S j ⊂ M . As above, when M is a Spin manifold, the condition of Lemma 9 is fulfilled so that m ∈ q(M ). Indeed as above, this shows that there exists an immersion of a neighborhood of each S j in R 4 . Thus q(M ) contains any integer m ≥ 5 for any Spin 4-manifold.

Remark 13
In fact in the above proof one needs to use immersion theory only when S j is non-orientable. If S j is orientable, then by Whitney's theorem ( [23], §6.b)) the Euler class χ(ν) of the normal bundle of φ(S j ) ⊂ S 4 is χ(ν) = 0, while one has the proportionality with the Euler class of the normal bundle ν of S j ⊂ M . Thus χ(ν ) = 0 and it follows that there is an embedding of a tubular neighborhood of S j in R 4 .

Remark 14
As a countercheck it is important to note why the above proof does not apply in the case of CP 2 seen as a double cover of the 4-sphere which is the quotient of CP 2 by complex conjugation and gives a ramified cover with ramification on RP 2 . It is an exercice for the reader to compute directly the second Stiefel-Whitney class of the tangent space of CP 2 restricted to the submanifold RP 2 and check that it does not vanish.
and it follows that the representation of the two-sided quantization relation is irreducible. The formulas for the Hilbert space and the operator are straightforward.

A tentative particle picture in Quantum Gravity
One of the basic conceptual ingredients of Quantum Field Theory is the notion of particle which Wigner formulated as irreducible representations of the Poincaré group. When dealing with general relativity we shall see that (in the Euclidean = imaginary time formulation) there is a natural corresponding particle picture in which the irreducible representations of the two-sided higher Heisenberg relation play the role of "particles". Thus the role of the Poincaré group is now played by the algebra of relations existing between the line element and the slash of scalar fields. We shall first explain why it is natural from the point of view of differential geometry also, to consider the two sets of Γ-matrices and then take the operators Y and Y as being the correct variables for a first shot at a theory of quantum gravity. Once we have the Y and Y we can use them and get a map (Y, Y ) : M → S n × S n from the manifold M to the product of two nspheres. The first question which comes in this respect is if, given a compact n-dimensional manifold M one can find a map (Y, Y ) : M → S n × S n which embeds M as a submanifold of S n × S n . Fortunately this is a known result, the strong embedding theorem of Whitney, [24], which asserts that any smooth real n-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2n-space. Of course R 2n = R n × R n ⊂ S n × S n so that one gets the required embedding. This result shows that there is no restriction by viewing the pair (Y, Y ) as the correct "coordinate" variables. Thus we simply view Y and Y as operators in Hilbert space and we shall write algebraic relations which they fulfill relative to the two Clifford algebras C κ , κ = ±1 and to the self-adjoint operator D. We should also involve the J and the γ. The metric dimension will be governed by the growth of the spectrum of D. The next questions are: assuming that we now no-longer use a base manifold M , A: Why is it true that the joint spectrum of the Y A and Y B is of dimension n while one has 2n variables.
B: Why is it true that the non-commutative integrals

Why is the joint spectrum of dimension 4
The reason why A holds in the case of classical manifolds is that in that case the joint spectrum of the Y A and Y B is the subset of S n × S n which is the image of the manifold M by the map x ∈ M → (Y (x), Y (x)) and thus its dimension is at most n.

Why is the volume quantized
The reason why B holds in the case of classical manifolds is that this is a winding number, as shown in Lemma 2.
The reason why B holds in the general case is that all the lower components of the operator theoretic Chern character of the idempotent e = 1 2 (1 + Y ) vanish and this allows one to apply the operator theoretic index formula which in that case gives (up to suitable normalization) This follows from the local index formula of [9] but in fact one does not need the technical hypothesis of [9] since, when the lower components of the operator theoretic Chern character all vanish, one can use the non-local index formula in cyclic cohomology and the determination in [7] Theorem 8, IV.2.γ of the Hochschild class of the index cyclic cocycle.
To be more precise one introduces the following trace operation, given an algebra A over R (not assumed commutative) and the algebra M n (A) of matrices of elements of A, one defines tr : M n (A) ⊗ M n (A) ⊗ · · · ⊗ M n (A) → A ⊗ A ⊗ · · · ⊗ A by the rule, using M n (A) = M n (R) ⊗ A tr ((a 0 ⊗ µ 0 ) ⊗ (a 1 ⊗ µ 1 ) ⊗ · · · ⊗ (a m ⊗ µ m )) = Trace(µ 0 · · · µ m )a 0 ⊗a 1 ⊗· · ·⊗a m where Trace is the ordinary trace of matrices. Let us denote by ι k the operation which inserts a 1 in a tensor at the k-th place. So for instance ι 0 (a 0 ⊗ a 1 ⊗ · · · ⊗ a m ) = 1 ⊗ a 0 ⊗ a 1 ⊗ · · · ⊗ a m One has tr • ι k = ι k • tr since (taking k = 0) tr • ι 0 ((a 0 ⊗ µ 0 ) ⊗ (a 1 ⊗ µ 1 ) ⊗ · · · ⊗ (a m ⊗ µ m )) = = tr ((1 ⊗ 1) ⊗ (a 0 ⊗ µ 0 ) ⊗ (a 1 ⊗ µ 1 ) ⊗ · · · ⊗ (a m ⊗ µ m )) = Trace(1µ 0 · · · µ m )1 ⊗ a 0 ⊗ a 1 ⊗ · · · ⊗ a m = = ι 0 (tr ((a 0 ⊗ µ 0 ) ⊗ (a 1 ⊗ µ 1 ) ⊗ · · · ⊗ (a m ⊗ µ m ))) The components of the Chern character of an idempotent e ∈ M s (A) are then given up to normalization by Ch m (e) := tr ((2e − 1) ⊗ e ⊗ e ⊗ · · · ⊗ e) ∈ A ⊗ A ⊗ . . . ⊗ A with m even and equal to the number of terms e in the right hand side. Now the main point in our context is the following general fact Lemma 16 Let A be an algebra (over R) and Y = Y A Γ A with Y A ∈ A and Γ A ∈ C + ⊂ M w (C) as above, n + 1 gamma matrices. Assume that Y 2 = 1. Then for any even integer m < n one has Ch m (e) = 0 where e = 1 2 (1 + Y ). Proof. This follows since the trace of a product of m + 1 gamma matrices is always 0. It follows that the component Ch n (e) is a Hochschild cycle and that for any cyclic n-cocycle φ n the pairing < φ n , e > is the same as < I(φ n ), Ch n (e) > where I(φ n ) is the Hochschild class of φ n . This applies to the cyclic ncocycle φ n which is the Chern character φ n in K-homology of the spectral triple (A, H, D) with grading γ where A is the algebra generated by the components Y A of Y and Y A of Y . By [7] Theorem 8, IV.2.γ, (see also [11] Theorem 10.32 and [2] for recent optimal results), the Hochschild class of φ n is given, up to a normalization factor, by the Hochschild n-cocycle: τ (a 0 , a 1 , . . . , a n ) = −γa 0 [D, a 1 ] · · · [D, a n ]D −n , ∀a j ∈ A.
Thus one gets that, by the index formula, for any idempotent e ∈ M s (A) < τ, Ch n (e) >=< φ n , e >= Index (D e ) ∈ Z Now by (29) for m = n and the fact that D commutes with the two Clifford algebras C ± , one gets, with Y = 2e − 1 as above, the formula

Conclusions
In this paper we have uncovered a higher analogue of the Heisenberg commutation relation whose irreducible representations provide a tentative picture for quanta of geometry. We have shown that 4-dimensional Spin geometries with quantized volume give such irreducible representations of the two-sided relation involving the Dirac operator and the Feynman slash of scalar fields and the two possibilities for the Clifford algebras which provide the gamma matrices with which the scalar fields are contracted. These instantonic fields provide maps Y, Y from the four-dimensional manifold M 4 to S 4 . The intuitive picture using the two maps from M 4 to S 4 is that the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. The volume of space-time is quantized in terms of the sum of the two winding numbers of the two maps. More suggestively the Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis. Moreover, amazingly, in dimension 4 the algebras of Clifford valued functions which appear naturally from the Feynman slash of scalar fields coincide exactly with the algebras that were singled out in our algebraic understanding of the standard model using noncommutative geometry thus yielding the natural guess that the spectral action will give the unification of gravity with the Standard Model (more precisely of its asymptotically free extension as a Pati-Salam model as explained in [5]).
Having established the mathematical foundation for the quantization of geometry, we shall present consequences and physical applications of these results in a forthcoming publication [6].