Quantum Geometry on Quantum Spacetime: Distance, Area and Volume Operators

Abstract

Abstract:We develop the first steps towards an analysis of geometry on the quantum spacetime proposed in Doplicher et al. (Commun Math Phys 172:187-220, 1995). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3-and 4-volumes.The Minkowski distance operator between two independent events is shown to have pure Lebesgue spectrum with infinite multiplicity. The Euclidean distance operator is shown to have spectrum bounded below by a constant of the order of the Planck length. The corresponding statement is proved also for both the space-space and space-time area operators, as well as for the Euclidean length of the vector representing the 3-volume operators. However, the space 3-volume operator (the time component of that vector) is shown to have spectrum equal to the whole complex plane. All these operators are normal, while the distance operators are also selfadjoint.The Lorentz invariant spacetime volume operator, representing the 4-volume spanned by five independent events, is shown to be normal. Its spectrum is pure point with a finite distance (of the order of the fourth power of the Planck length) away from the origin.The mathematical formalism apt to these problems is developed and its relation to a general formulation of Gauge Theories on Quantum Spaces is outlined. As a byproduct, a Hodge Duality between the absolute differential and the Hochschild boundary is pointed out.

Renormalization of noncommutative quantum field theories

“…Thus, it follows immediately from (94), that for φ 4 θ, * theory in d = 4 dimensions, which is the model we are presently interested in, there are divergences for E = 2 and E = 4. These correspond to the one particle irreducible (1PI) 2-point function Γ (2) θ and 4-point function Γ (4) θ respectively, implying that, Γ (2) θ and Γ (4) θ will be divergent. We need to renormalize them, resulting in corrections to propagators and vertices.…”

Minimal length in quantum space and integration of the line element in noncommutative geometry

“…However a careful examination of the representations of L 1 (H) indicates that C * (H) is too big: -First of all in (3.7) it is reasonable to take into account irreducible representations only, since the q µ 's are known as soon as one has determined their irreducible action on H . 4 Recall that given a symplectic vector space (V, σ ) of real dimension n, the Heisenberg algebra with central element c is the real central extension of the Lie algebra r n = R n of the additive group R n , characterized by the relations…”

“…= (ω, δ i ) where ω is a pure state of A and δ i=1,2 are the pure states of C 2 4) while the distance between the sheets is constant and non-zero [29,15],…”

“…have been made in [19,4] for the model of Doplicher, Fredenhagen and Roberts (DFR) in which the commutators Q µν 's are central operators with selfadjoint closure, covariant under the action of the Poincaré group; as well as in [1] for the canonical noncommutative space θ -Minkowski where the Q µν 's are constant. These results are recalled in section 5.1.…”

“…There are two difficulties: on the one hand the spectrum of this operator is not bounded below from zero and its physical interpretation as a quantum observable is not transparent (see [4]); on the other hand the comparison with the spectral distance formula would not be easy, since the latter makes sense only in the Riemannian case (although some generalization to the Minkovskian case have been investigated in [31,21]). …”