Quantum theory of geometry: III. Non-commutativity of Riemannian structures

The non-commutativity also occurs for gauge-invariant area operators: area operators for two surfaces intersecting along a line do not commute.

“In particular, if we have two surfaces S 1 and S 2 which intersect along a line, the corresponding area operators Â[S 1 ] and Â[S 2 ] fail to commute.”

The Hamiltonian vector fields generated by two-dimensionally smeared triads map Cyl to itself.

“It is then natural to use two-dimensionally smeared triads, 2 E[S, f ], as momenta because their Hamiltonian vector fields map Cyl to itself.”

Those states form an infinite-dimensional subspace of the gauge-invariant state space.

“Nonetheless, they constitute an infinite-dimensional subspace of the space of gauge-invariant states.”

The classical Poisson brackets between the momentum variables do not vanish.

“the classical brackets between momentum variables already fail to vanish.”

• Also stated in: § 3.2. Origin of non-commutativity · plain

  “although the three-dimensionally smeared triads do Poisson commute with one another, the 2 E[S, f ] do not.”

The quantum algebra is faithfully represented by operators on H = L^2(Ā, dμ).

“this quantum algebra is faithfully represented by operators on H = L 2 ( Ā, dµ • ).”

• cites paper INot checked yet

  Not checked in the free run.

Using distributional test fields with two-dimensional support, the Poisson bracket of the smeared triad with cylindrical functions closes on Cyl.

“Thus, if we use distributional test fields with two-dimensional support and evaluate the Poisson bracket of the resulting smeared triad with cylindrical functions via a limiting procedure, we find that the Poisson bracket closes on Cyl.”

The naive Poisson brackets, including the assumption that two-dimensionally smeared triads commute, fail to satisfy the Jacobi identity.

“the naive Poisson brackets fail to satisfy the Jacobi identity.”

• Also stated in: § 2.3. Problem with the naive Poisson algebra · plain

  “the brackets (2.9) and (2.10) are, in fact, inconsistent with the Jacobi identity.”

• Also stated in: § 2.3. Problem with the naive Poisson algebra · plain

  “Hence, the Jacobi identity will be violated if we demand that the Poisson bracket 2 E[S, f ], 2 E[S, g] between momenta must vanish.”

A generic derivation on Cyl need not correspond to any phase space function.

“a generic derivation need not correspond to any phase space function.”

The resulting quantum algebra is exactly the one used in the earlier papers [1-14].

“Furthermore, the resulting quantum algebra is precisely the one used in [1][2][3][4][5][6][7][8][9][10][11][12][13][14].”

• cites [1] Ashtekar A, 1997, Class. Quantum Grav. sourceNot checked yet

  Not checked in the free run.

• cites [2] Ashtekar A, 1998, Adv. Theor. Math. Phys. sourceNot checked yet

  Not checked in the free run.

• cites [3] Ashtekar A, 1992, Class. Quantum Grav. sourceNot checked yet

  Not checked in the free run.

Not all elements of this Lie algebra can be represented as functions on the phase space.

“However, it turns out that not all elements of this Lie algebra can be represented as functions on the phase space.”

The triad operators smeared by two different test fields on the same 2-surface do not commute.

“However, somewhat surprisingly, the resulting operators 2 Ê[S, f ] turn out not to commute.”

In this framework, the quantum operators for triads and areas do not commute.

“The quantum versions of Riemannian structures-such as triad and area operators-exhibit a non-commutativity.”

• Also stated in: § 1. Introduction · plain

  “However, somewhat surprisingly, the resulting operators 2 Ê[S, f ] turn out not to commute.”

All derivations on Cyl can be allowed and still yield a Lie algebra.

“Could we have allowed all derivations on Cyl and still obtained a Lie algebra?The answer is in the affirmative.”

For triad operators smeared along two distinct surfaces, their commutator is non-zero when the surfaces intersect and the Lie bracket of the test fields is non-zero on the intersection.

“If the two operators are smeared along two distinct surfaces, the commutator is again non-zero if the two surfaces intersect and the Lie bracket of the corresponding test fields is non-zero on the intersection.”

The commutator of area operators for intersecting surfaces is non-zero only on states that have a 4-or higher valent vertex on the intersection.

“The commutator [ Â[S 1 ], Â[S 2 ]] is non-zero only on those states which (in the terminology of paper I) have a 4-or higher valent vertex on the intersection.”

One can define brackets between the classical momentum variables 2E[S,f] using the cotangent bundle structure so that a true Lie algebra results.

“In section 3.1, we will see that one can extract this information by exploiting the fact that our phase space has a cotangent bundle structure; that is, one can define brackets between the classical momentum variables 2 E[S, f ] such that a true Lie algebra results.”

If a graph has edges intersecting two surfaces on the one-dimensional curve where the surfaces intersect, the commutator of the corresponding vector fields will generally fail to vanish.

“Thus, if γ has edges that intersect S and S on the one-dimensional curve on which S and S themselves intersect, the commutator will fail to vanish in general.”

For semiclassical states, the uncertainties associated with the non-commutativity of two area operators are completely negligible because their commutator is a distribution with only one-dimensional support.

“Thus, because the commutator of two area operators is a distribution with only one-dimensional support, the uncertainties associated with their non-commutativity are completely negligible on semiclassical states.”

The subspace of Cyl on which the commutator has a non-trivial action is infinite dimensional.

“Nonetheless, the subspace of Cyl on which is the commutator has a non-trivial action is infinite dimensional.”

The smeared triad operators do not necessarily commute.

“It is nonetheless striking that the smeared triad operators do not necessarily commute”

Assuming commutativity of the 2E[S,f] leads to a violation of the Jacobi identity.

“if we simply assume commutativity, we are led to a violation of the Jacobi identity!”

In this quantum theory, the commutators between momentum operators mirror the commutators between the vector fields X_S,f in the classical theory.

“In this quantum theory, the commutators between the momentum operators simply mirror the commutators between the vector fields X S,f in the classical theory.”

The vector fields 3X[f] commute.

“it is obvious from their definition that the vector fields 3 X[f ] commute.”

The framework has no triad representation.

“At first sight, this feature is surprising because it implies that the framework does not admit a triad representation.”

• Also stated in: § 1. Introduction · plain

  “the fact that the triad-and hence the metric-representation fails to exist in this approach is striking”

The quantization of the classical Lie algebra has no anomaly.

“there is no anomaly in quantization”

• Also stated in: § 3.1. Quantization strategy: exploiting the cotangent bundle structure · plain

  “In particular, there is no anomaly associated with this quantization.”

• Also stated in: § 3.2. Origin of non-commutativity · plain

  “Thus, there is no anomaly.”

The Lie algebra is nevertheless a viable point of departure for quantization.

“Nonetheless, we will see that the Lie algebra does serve as a viable point of departure for quantization.”

The quantum theory of references [1]-[14] results from quantizing this classical Lie algebra.

“the quantum theory of [1][2][3][4][5][6][7][8][9][10][11][12][13][14] in fact results from 'quantization of this classical Lie algebra'.”

• cites [1][2][3][4][5][6][7][8][9][10][11][12][13][14]Not checked yet

  Not checked in the free run.

The limiting procedure from three-dimensionally smeared triads to two-dimensionally smeared ones is technically subtle and is not well defined unless regularity conditions are imposed.

“However, the limiting procedure used to pass from threedimensionally smeared triads to the two-dimensionally smeared ones is technically subtle and the result is not well defined unless appropriate regularity conditions are imposed.”

• AssumptionThe authors state this as an assumption: The 3-manifold is analytic.

The phase space is a cotangent bundle over the configuration space A of connections.

“our phase space is a cotangent bundle over the configuration space A of connections.”

The uncertainties tied to this non-commutativity are negligible in the semiclassical regime.

“We also indicate why the uncertainties associated with this non-commutativity become negligible in the semiclassical regime.”

• Also stated in: § 1. Introduction · boosted

  “the quantum uncertainties turn out to be completely negligible in the semiclassical regime.”

The simple Poisson algebra (2.3) is not suitable as the starting point for quantization.

“Hence, the simple Poisson algebra (2.3) is no longer suitable as the starting point for quantization.”

The route using functions and vector fields on configuration space carries over directly to the present approach to quantum gravity.

“It is the second that carries over directly to the present approach to quantum gravity.”

For simple finite-dimensional systems, quantization can proceed equivalently from the Poisson algebra of configuration and momentum functions on phase space or from functions and vector fields on configuration space.

“For simple finite-dimensional systems, there are two equivalent routes to quantization, one starting from the Poisson algebra of configuration and momentum functions on the phase space and the other from functions and vector fields on the configuration space.”

• cites [25] Kolmogorov A N, 1936Not checked yet

  Not checked in the free run.

• cites [26] Ashtekar A, 1994Not checked yet

  Not checked in the free run.

The surprising features of quantum geometry-non-commutativity, polymerlike one-dimensional fundamental excitations, and discreteness of geometric operator spectra-can in essence be traced back to the assumption on configuration variables and to gauge and diffeomorphism invariance.

“Thus, it appears that the surprising features of quantum geometry-non-commutativity, the polymerlike, one-dimensional nature of fundamental excitations and the discreteness of spectra of geometric operators-can in essence be traced back to the assumption on the configuration variables and to the gauge and diffeomorphism invariance of the theory.”

• AssumptionThe authors state this as an assumption: The configuration variables are cylindrical functions, or in the gauge-invariant context traces of holonomies (Wilson loops).

The non-commutativity arises because, when acting on cylindrical functions, the action of the vector fields 3X does not commute with the operation of taking limits.

“By expanding out the two terms one can see explicitly that the non-commutativity arises because, while acting on cylindrical functions, the action of the vector fields 3 X does not commute with the operation of taking limits.”

Quantum Riemannian geometry is intrinsically noncommutative.

“Hence, quantum Riemannian geometry is now intrinsically noncommutative.”

The result of this quantization is precisely the quantum theory developed in references [1]-[14].

“The result is precisely the quantum theory that was developed in [1][2][3][4][5][6][7][8][9][10][11][12][13][14].”

• cites [1][2][3][4][5][6][7][8][9][10][11][12][13][14]Not checked yet

  Not checked in the free run.

The construction is background-independent because connection 1-forms are integrated on one-dimensional curves and triad 2-forms are integrated over two-dimensional surfaces.

“This construction is natural in the sense that it does not involve any background structures: the connection 1-forms are integrated on one-dimensional curves, while the triad 2-forms are integrated over two-dimensional surfaces.”

In this approach, the metric representation, at least the naive one, does not exist.

“this implies, in particular, that in this approach the metric representation (at least the naive one) does not exist.”

The elementary variables are singular in the sense that the smearing fields themselves are distributional, unlike the standard procedure in Minkowskian field theories.

“our elementary variables are 'singular' in a certain sense: compared to the standard procedure followed in Minkowskian field theories, we have let the smearing fields themselves be distributional.”

Even after extending the definitions of momentum functions and basic vector fields to piecewise-analytic surfaces, one cannot establish a one-to-one correspondence between the vector fields and momentum functions.

“Thus, even after extending the definitions of momentum functions and basic vector fields, one cannot establish a one-to-one correspondence between the vector fields and momentum functions.”

Because the generators of the classical Lie algebra are sufficiently large, the fact that some derivations do not correspond to phase-space functions does not obstruct quantization.

“Thus, because the set of generators of the classical Lie algebra is 'sufficiently large', the fact that some derivations do not correspond to phase space functions does not create an obstacle.”

Although both approaches use algebraic methods heavily, the two meanings of non-commutativity are quite different.

“Thus, although in both approaches algebraic methods are used heavily to introduce geometric notions, the two meanings of 'non-commutativity' are quite different from one another.”