Quantum observables and recollapsing dynamics

All (convergent) observables of the form 3.9 are symmetric on H_phys with the inner product 3.23.

“We conclude that - All (convergent) observables of the form 3.9 are symmetric on H phys with the inner product 3.23”

• AssumptionThe authors state this as an assumption: The integrals defining the observables converge for the results of section III.

Equation 3.14 is sufficient to determine the physical inner product.

“Equation 3.14 is also sufficient for us to determine the physical inner product.”

The methods of III lead to a familiar Hilbert space for the relativistic free particle.

“and lead to a familiar Hilbert space for the relativistic free particle (IV B).”

If [A(t), ∂/∂t θ(Q(t)-τ)]=0, then [A]_{Q=τ}=[A]^L_{Q=τ}=[A]^R_{Q=τ}.

“and, if [ A ( t ) , ∂ ∂t θ ( Q ( t ) -τ )] = 0, we have [ A ] Q = τ = [ A ] L Q = τ = [ A ] R Q = τ .”

In a proper parametrization for which N(t)>0 and proper time always flows forward, [sign(1/N dq/dt)]_{q=τ} is just the number of times for which q(t)=τ.

“In a 'proper parametrization' for which N ( t ) > 0 and proper time always flows forward, this is just the number of times for which q ( t ) = τ .”

For gravity coupled to scalar fields φk, the integral 1.1 defines a well-behaved function on any part of the classical space of solutions for which the fields φk vanish simultaneously only at a finite number of spacetime points and for which the metric is smooth whenever φk=0.

“This object defines a well-behaved function on any part of the classical space of solutions for which the fields φ k vanish simultaneously only at a finite number of spacetime points and for which the metric is smooth whenever φ k = 0.”

Expression 2.1 is invariant under the more general reparametrizations (2.2).

“Because its integrand is a one-form, expression 2.1 is in fact invariant under the more general reparametrizations (2.2).”

Within a simple quantization scheme, observables for a large class of finite dimensional time reparametrization invariant systems can be constructed by integration over the manifold of time labels.

“Within a simple quantization scheme, observables for a large class of finite dimensional time reparametrization invariant systems may be constructed by integration over the manifold of time labels.”

Many of these observables have a clear interpretation in the classical limit.

“Furthermore, many of these observables have a clear interpretation in the classical limit”

The approach may be applied to a large class of minisuperspace models, including the LRS Bianchi IX and Kantowski-Sachs cosmologies.

“that it may be applied to a large class of minisuperspace models - which includes the LRS Bianchi IX and Kantowski-Sachs cosmologies -”

The approach does not require identifying an internal time function.

“This approach has the advantage that it is not necessary to identify an 'internal time function' [1]”

• cites [1] Kuchar K, 1992Not checked yet

  Not checked in the free run.

The ordering 3.15 has the property that [sign(P0)]_{X0=τ} is the identity operator for the free relativistic particle.

“it has the reassuring property (see IV B) that [sign( P 0 )] X 0 = τ is the identity operator for the free relativistic particle”

Reparametrization invariant operators do commute with the Hamiltonian.

“However, we have seen that reparametrization invariant operators do commute with the Hamiltonian.”

The methods of III give the usual results for already deparametrized systems.

“These test cases show that the methods of III give the usual results for 'already deparametrized systems' (IV A)”

Any symmetric ordering of the factors ∂/∂t Q(t), Q(t), and A(t) will produce a Hermitian gauge invariant that may be taken to be a quantization of [a]_{q=τ}.

“From 3.22 and 3.24, we see that any symmetric ordering of the factors ∂ ∂t Q ( t ), Q ( t ), and A ( t ) will produce a Hermitian gauge invariant that we may take to be a 'quantization' of [ a ] q = τ .”

Certain properties of the resulting operators can be studied even though the models may not be exactly solvable.

“and that certain properties of the resulting operators can be studied even though the models may not be exactly solvable.”

The procedure produces a complete set of densely defined operators on a physical Hilbert space for which an inner product is identified.

“This procedure is shown to produce a complete set of densely defined operators on a physical Hilbert space for which an inner product is identified and to provide reasonable results for simple test cases.”

The approach provides an overcomplete set of quantum observables, meaning a set whose classical counterparts are overcomplete on the constrained phase space.

“In the examples we will see that our approach in fact provides an 'overcomplete set' of quantum observables. By this we mean a set of observables whose classical counterparts are overcomplete (see. for example [13]) on the (constrained) phase space.”

• cites [13] Higuchi A, 1991, Class. Quantum Grav. sourceNot checked yet

  Not checked in the free run.

Quantum analogues of these observables exist as densely defined operators on a space of physical states for interesting models.

“we will see that quantum analogues of these observables exist as densely defined operators on a space of physical states for interesting models”

The inner product 3.23 is unique, provided that a simple set of observables are required to be symmetric.

“We will also see that the inner product 3.23 is unique, provided that a simple set of observables are required to be symmetric.”

The case A=X^i does not correspond to a Newton-Wigner operator (which would leave the positive and negative frequency subspaces invariant).

“Thus, the case of A = X i does not correspond to a Newton-Wigner operator (which would leave the positive and negative frequency subspaces invariant).”

[sign(P0)]_{X0=τ}=1 as claimed.

“A similar calculation verifies that [sign( P 0 )] X 0 = τ = 1 1 as claimed.”

Hermiticity of a few interesting operators guarantees that the inner product 3.23 is unique.

“It now follows that Hermiticity of a few interesting operators guarantees that the inner product 3.23 is unique.”

[θ(P0)signP0 X^i]_{X0=τ} is just the Newton-Wigner operator X^i_NW(τ) associated with the coordinate X^i.

“On the other hand, it may be checked that [ θ ( P 0 )signP 0 X i ] X 0 = τ is just the Newton-Wigner operator X i NW ( τ ) associated with the coordinate X i .”

The ̃[B]_{Q0=τ} form a complete set of bounded quantum observables.

“The ˜ [ B ] Q 0 = τ form a complete set of bounded quantum observables.”

• AssumptionThe authors state this as an assumption: For each E0 and E1, the label k takes only a finite set of values.

We have constructed an overcomplete set of observables as densely defined operators on H_phys.

“Finally, we verify that we have constructed an overcomplete set of observables as densely defined operators on H phys .”

The matrix elements are of the form 〈E,k|[A]_{Q0=τ}|E′,k′〉 = δ(E−E′) f(E,E′ k,k′) for eigenstates |E,k〉 of H with eigenvalue E and some function f.

“We have seen that the matrix elements are of the form 〈 E,k | [ A ] Q 0 = τ | E ′ , k ′ 〉 = δ ( E -E ′ ) f ( E,E ′ k, k ′ ) for eigenstates | E,k 〉 of H with eigenvalue E and some function f”

If we also ask that [X^i]_{X0=τ} and [P^j]_{X0=τ} be symmetric, then the inner product is fixed up to an overall normalization on the positive frequency subspace and a similar normalization on the negative frequency subspace.

“If we also ask that [ X i ] X 0 = τ and [ P j ] X 0 = τ ( i, j ∈ { 1 , 2 , 3 } ) be symmetric, then the inner product is fixed up to an overall normalization on the positive frequency subspace and a similar normalization on the negative frequency subspace.”

If we require [P0 sign(P0)]_{X0=τ} to be Hermitian on the physical space, it follows that the positive and negative frequency subspaces are orthogonal.

“If we require [ P 0 sign( P 0 )] X 0 = τ to be Hermitian on the physical space (spanned by | ǫ | p | , p ; t ′ 〉 ), it follows that the positive and negative frequency subspaces are orthogonal.”

The approach of III gives the usual quantization of nonrelativistic particles when applied to systems of the form 4.1.

“We will see that the approach of III gives the usual quantization of nonrelativistic particles when applied to systems of this form.”

The most general rigorous results were obtained for already deparametrized systems (IV A), the free relativistic particle (IV B), and the separable semibound models of V.

“The most general rigorous results were obtained for already deparametrized systems (IV A), the free relativistic particle (IV B), and the separable semibound models of V.”

The construction of gauge invariants through integration over manifolds can be combined with Hamiltonian methods for constrained systems to build a complete set of gauge invariant operators that are densely defined in a physical Hilbert space for a number of time reparametrization invariant models.

“We have seen that the construction of gauge invariants through integration over manifolds can be combined with Hamiltonian methods for constrained systems to build a complete set of gauge invariant operators that are densely defined in a physical Hilbert space for a number of time reparametrization invariant models.”

Result 5: The ̃[B]_{Q0=τ} of Result 5 converge strongly to zero as τ→∞.

“Result 5: The ˜ [ B ] Q 0 = τ of Result 5 converge strongly to zero as τ →∞ .”

• AssumptionThe authors state this as an assumption: For each E0 and E1, the label k takes only a finite set of values.

Result 2: For A as in Result 1, [A]_{Q0=τ}→0 as τ→∞ in the sense of convergence on a dense set of states; that is, [A]_{Q0=τ} converges strongly to the zero operator.

“Result 2: For A as in Result 1, [ A ] Q 0 = τ → 0 as τ →∞ in the sense of convergence on a dense set of states; that is, [ A ] Q 0 = τ converges strongly to the zero operator.”

The canonical idea that an observable is an object that commutes with the constraint (the Hamiltonian, H) does not capture the entire notion of gauge invariant.

“Within our scheme, the canonical idea that an 'observable' is an object that commutes with the constraint (the Hamiltonian, H ) does not capture the entire notion of 'gauge invariant'”

We interpret this as a demonstration that our quantization of these models describes recollapsing cosmologies.

“We interpret this as a demonstration that our quantization of these models describes recollapsing cosmologies.”

Such observables can be used to show that, in correspondence with the classical result, our quantization of appropriate minisuperspace models describes spacetimes whose homogeneous slices expand to some maximal size and then recollapse.

“In particular, such observables can be used to show that, in correspondence with the classical result, our quantization of appropriate minisuperspace models describes spacetimes whose homogeneous slices expand to some maximal size and then recollapse.”

If the density ω is distributional, the resulting gauge invariant may be effectively local on M.

“If the density ω is distributional, the resulting gauge invariant may be effectively local on M .”

• cites [2] DeWitt B, 1962Not checked yet

  Not checked in the free run.

For a class of minisuperspace models including LRS Bianchi IX and the Kantowski-Sachs model, this quantization agrees with classical physics in predicting that such spacetimes recollapse.

“we use this to demonstrate that, for a class of minisuperspace models including LRS Bianchi IX and the KantowskiSachs model this quantization agrees with classical physics in predicting that such spacetimes recollapse.”

• Also stated in: § I. INTRODUCTION · p. 2 · plain

  “for the models just mentioned it predicts that the corresponding spacetimes recollapse, in accord with the classical result.”

Procedures based on a Klein-Gordon type inner product instead describe the universe as expanding forever.

“procedures based on a Klein-Gordon type inner product instead describe the universe as expanding 'forever' [4].”

• cites [4] Wald R, 1993, Phys. Rev. sourceNot checked yet

  Not checked in the free run.

Given any density ω on an n-manifold M, ∫M ω is a gauge invariant observable for classical diffeomorphism invariant theories.

“Given any density ω on an n-manifold M , ∫ M ω is such an observable.”

• cites [2] DeWitt B, 1962Not checked yet

  Not checked in the free run.

It is highly nontrivial for a quantization scheme to satisfy the recollapse condition.

“Note that it is highly nontrivial for a quantization scheme to satisfy this last condition”

If zero lies in the point spectrum of H and a solution to the constraint is normalizable in the auxiliary space, then the action of [A]_{Q0=τ} on |E,q0> contains a divergent factor.

“if zero lies in the point spectrum of H and a solution to the constraint is actually normalizable in the auxiliary space, the action of [ A ] Q 0 = τ on | E,q 0 〉 contains a divergent factor.”

For separable semi-bound systems, H has a purely continuous spectrum with σc(H)=(-∞,∞).

“From this and the above conditions, it follows that H has a purely continuous spectrum with σ c ( H ) = ( -∞ , ∞ ).”

The set of physical states that are delta-function normalizable in the auxiliary inner product and satisfy H|ψ>=0 is not empty.

“Since the spectrum of H includes zero, the set of physical states that are delta-function normalizable in the auxiliary inner product and satisfy H | ψ 〉 = 0 is not empty.”

The physical states satisfying H|ψ>=0 are of the form |E0=E1,E1,k;t'> where E1 ranges over the spectrum of H1 and k over its full set of values.

“They are of the form | E 0 = E 1 , E 1 , k ; t ′ 〉 where E 1 ranges over the spectrum of H 1 and k ranges over its full set of values.”

The LRS Bianchi IX and Kantowski-Sachs minisuperspaces are separable semi-bound models when described by appropriate variables and a rescaled Hamiltonian.

“The LRS Bianchi IX and Kantowski-Sachs minisuperspaces fall into the class of models that we refer to as 'separable and semi-bound.'”

• Also stated in: § A. The System and Observables · p. 20 · hedged

  “From [15], we see that when described by the appropriate variables and with a rescaled Hamiltonian, the LRS Bianchi IX and Kantowski-Sachs models may be placed in this form with q 0 = ln(det g ) where g is the 3-metric of a homogeneous slice.”

• cites [15] Ashtekar A, 1991Not checked yet

  Not checked in the free run.

When Q0 has the interpretation of a scale factor for a cosmology, the quantization predicts that such spacetimes recollapse.

“For this reason, we feel justified in saying that when Q 0 has the interpretation of a scale factor for a cosmology, our quantization predicts that such spacetimes recollapse.”

Quantization of separable semi-bound models captures the classical notion of recollapse expressed through 2.4 for a(t) independent of q0(t) and p0(t).

“It now follows that quantization of 'separable semi-bound models' captures the classical notion of 'recollapse' expressed through 2.4 for a ( t ) independent of q 0 ( t ) and p 0 ( t ).”

There is no exact unitary evolution in τ-time for semi-bound separable systems.

“There is no exact unitary evolution in τ -time for semi-bound separable systems, just as there is no corresponding exact Hamiltonian evolution at the classical level.”

Discrete spectra are associated with confining potentials.

“However, discrete spectra are associated with confining potentials as in [19,20]”

• cites [19] Haag R, 1992Not checked yet

  Not checked in the free run.

• cites [20] Glimm J, 1987Not checked yet

  Not checked in the free run.

For separable semi-bound systems, the spectrum of H0 is purely continuous in the q0-coordinate representation on L2(R).

“These conditions imply that, when viewed as an operator on the Hilbert space L 2 ( R ) in a ' q 0 - coordinate representation,' the spectrum of H 0 is purely continuous.”