Two-sided higher Heisenberg quantization of four-dimensional geometry: particle-picture dynamics and noncommutative representation theory beyond manifolds

The paper 'Geometry and the Quantum: Basics' (Chamseddine, Connes, Mukhanov, JHEP 12 (2014) 098) introduced a higher-degree analogue of the Heisenberg commutation relation in which the Dirac operator $D$ plays the momentum and the Feynman slash $Y$ of real scalar fields plays the position. Its two-sided version, built with the real structure $J$, the chirality $\gamma$, and two Clifford algebras $C_\pm$, quantizes the volume, $\int -D^{-n}\in\mathbb{N}$, and in dimension four selects exactly $C_+=M_2(\mathbb{H})$ and $C_-=M_4(\mathbb{C})$, the algebraic constituents of the Standard Model; every connected spin 4-manifold with quantized volume $\ge 5$ appears as an irreducible representation, motivating a tentative 'particle picture' of quantum gravity in which irreducible representations of the relation replace Wigner's Poincaré representations. The announced physical applications have so far been realized only in the gravity/cosmology sector, in the companion paper 'Quanta of Geometry: Noncommutative Aspects' (quantized cosmological constant, mimetic dark matter, black-hole area quantization). Two linked components of the announced development remain open: the spectral action and dynamics evaluated on the two-sided quantization data, giving the particle picture equations of motion, observables or scattering amplitudes and phenomenological predictions; and the representation theory of the two-sided relation in the noncommutative spectral-triple category beyond manifold examples — classification of admissible irreducible representations, their moduli, and their stability under perturbations of $D$ and $Y$.

Background

In noncommutative geometry, a 'space' is described not by its points but by an algebra of operators playing the role of the algebra of functions. A compact Riemannian spin manifold $M$ is encoded by a spectral triple $(\mathcal{A},\mathcal{H},D)$: the commutative algebra $\mathcal{A}=C^\infty(M)$ acting by multiplication on the Hilbert space $\mathcal{H}=L^2(M,S)$ of square-integrable spinors, together with the Dirac operator $D$, a self-adjoint first-order operator whose inverse plays the role of the line element; the growth of the eigenvalues of $D$ (the Weyl asymptotics) fixes the metric dimension, and commutators $[D,f]$ recover the geodesic distance. Even-dimensional triples carry a chirality operator $\gamma$ (the grading that splits spinors into left- and right-handed parts), and real triples carry an antiunitary charge-conjugation operator $J$ fulfilling commutation relations with $D$ and $\gamma$ that depend on the dimension modulo 8 — together these are the operator-algebraic encoding of a spin geometry. Tensoring $C^\infty(M)$ with a finite-dimensional noncommutative algebra ('almost-commutative' geometry) yields, through inner fluctuations of $D$ — the canonical deformation of $D$ by one-forms of the larger algebra — Yang–Mills gauge fields and a Higgs field. Dynamics is supplied by the spectral action principle of Chamseddine and Connes: the action $\mathrm{Tr}\, f(D/\Lambda)$, for a positive cutoff function $f$ and scale $\Lambda$, has a heat-kernel expansion (the asymptotic expansion of $\mathrm{Tr}\, f(D/\Lambda)$ in powers of $\Lambda$) whose leading terms reproduce the Einstein–Hilbert gravitational action plus the bosonic Lagrangian of the Standard Model when the finite algebra is $M_2(\mathbb{H})\oplus M_4(\mathbb{C})$, with $\mathbb{H}$ the quaternions and $M_n$ the $n\times n$ matrices; imposing a first-order condition truncates this to the Standard Model, and without it one obtains the Pati–Salam grand-unified extension. Why nature should select precisely this algebra was long an open question in the program.

'Geometry and the Quantum: Basics' (2014) proposed an answer through a quantization condition. One takes the 'position' variable to be $Y=Y^A\Gamma_A\in\mathcal{A}\otimes C_\kappa$, the Feynman slash of $n+1$ real scalar fields $Y^A$ (their contraction with gamma matrices $\Gamma_A$ spanning a real Clifford algebra $C_\kappa$, $\kappa=\pm 1$, with $Y^2=\kappa$ and $Y^*=\kappa Y$). The one-sided higher Heisenberg equation is

$$\frac{1}{n!}\langle Y[D,Y]\cdots[D,Y]\rangle = \sqrt{\kappa}\,\gamma$$

with $n$ commutators and $\langle\cdot\rangle$ the normalized trace over the Clifford representation. For the spectral triple of a spin manifold, solutions exist if and only if $M$ is a disjoint union of unit-volume spheres, viewed as 'quanta of geometry'. The two-sided refinement uses the real structure: with the double slash $Y=Y_+\oplus Y_-$ valued in $C^\infty(M,C_+\oplus C_-)$, its second slash $Y'=iJY_-J^{-1}$ (acting on the chirality subspace of $\mathcal{H}$ on which $C_+$ acts, and commuting with $Y$), and $E$ the spectral projection of $\{1,i\}$, one sets $Z=2EJEJ^{-1}-1$ and requires

$$\frac{1}{n!}\langle Z[D,Z]\cdots[D,Z]\rangle = \gamma.$$

This still quantizes the volume but admits connected manifolds: solutions exist exactly when the volume of $M$ lies in the set $q(M)$ of integers $\deg(\varphi_+)+\deg(\varphi_-)$ realized by pairs of maps $\varphi_\pm:M\to S^n$ (the degree, or winding number, counts how many times a map wraps the sphere around itself) whose pulled-back volume forms never cancel; for spin 4-manifolds, $q(M)$ contains every integer $\ge 5$. Here the volume is the noncommutative integral $\int -D^{-n}$, the Dixmier-trace functional whose value on $D^{-n}$ gives the leading Weyl term of the volume. The paper proves, in the abstract operator setting without any manifold, that the quantization equation implies $\int -D^{-n}\in\mathbb{N}$ up to normalization, using the local index formula and the determination of the Hochschild class of the Chern character in K-homology (the pairing of the Chern character of the idempotent $e=\frac{1}{2}(1+Y)$ with the index cocycle is a Fredholm index, hence an integer). In dimension four, the two Clifford algebras are exactly $C_+=M_2(\mathbb{H})$ and $C_-=M_4(\mathbb{C})$ — the internal algebra of the Standard Model — so that, as the paper puts it, together with the algebra of functions this allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model at the level of the almost-commutative framework of prior work. Moreover, every connected Riemannian spin 4-manifold with quantized volume $\ge 5$ (in units of the Planck sphere) occurs as an irreducible representation of the two-sided relations, with algebra the double commutant of the components of $Y,Y'$. The authors read this as a tentative 'particle picture': in Euclidean quantum gravity, irreducible representations of the two-sided higher Heisenberg relation — rather than irreducible unitary representations of the Poincaré group, as in Wigner's classification of relativistic particles — would be the fundamental excitations, with the four-geometry emerging from the joint spectrum of $Y,Y'$ inside $S^4\times S^4$. The paper closed by announcing consequences and physical applications in a separate publication and left the particle picture tentative; the dynamical framework for it, and the theory of the relation beyond manifold examples, were not developed.

Problem Statement

The problem is to complete the theory of the two-sided higher Heisenberg quantization of geometry announced in 'Geometry and the Quantum: Basics' (JHEP 12 (2014) 098). Two linked components constitute the announced development; both remain open.

(A) Dynamics of the particle picture. Formulate and evaluate the spectral action on the two-sided quantization data — the operators $D$, $Y$, $Y'$, $J$, $\gamma$ together with the Clifford algebras $C_+=M_2(\mathbb{H})$ and $C_-=M_4(\mathbb{C})$ obeying the two-sided relation $\frac{1}{4!}\langle Z[D,Z]\cdots[D,Z]\rangle=\gamma$ (four commutators, with $Z=2EJEJ^{-1}-1$ as defined above) — in the Hilbert-space setting without an underlying manifold, and derive from it a genuine dynamics for the particle picture: equations of motion, observables or transition amplitudes in which the irreducible representations of the relation play the role that Wigner's Poincaré representations play for particles, together with phenomenological consequences going beyond the gravity-sector consequences of volume quantization already derived in the literature (the cosmological constant as an integration constant, mimetic dark matter, black-hole area quantization).

(B) Representation theory beyond manifolds. Develop the two-sided quantization condition in the general noncommutative spectral-triple category, with no base manifold: classify the admissible irreducible operator representations (with the eigenvalues of $D$ growing as in dimension four), describe their moduli, and establish their stability — or instability — under perturbations of $D$ and $Y$. The 2014 paper established the quantization $\int -D^{-n}\in\mathbb{N}$ in this abstract setting and classified only the commutative (manifold) representations; no classification, moduli description, or stability result is known for genuinely noncommutative representations, and no systematic study of noncommutative examples exists.

The two components are jointly the announced program, each with independent standing: (A) without (B) rests on an uncharacterized class of representations, and (B) without (A) reproduces the kinematical situation of the 2014 paper. The full problem is to make the particle picture dynamical — equations of motion together with observables or transition amplitudes evaluated on the two-sided quantization data — and to ground the representation theory in a characterized class, namely a classification of the admissible irreducible representations together with their moduli and their stability under perturbations of $D$ and $Y$.

References

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