Almost ideal clocks in quantum cosmology: a brief derivation of time

The factorization H0 = Hp ⊗ Hc is not independent of the parameter time t.

“In general, the decomposition H 0 = H p ⊗ t H c will depend on a choice of parameter time t.”

Except near a turning point, q_c(t) is roughly q_c(0)+n, with n the integer count of left-turning-point visits by q since t=0.

“Thus, unless q is near a turning point, q c (t) ∼ = q c (0)+n where n is the (integer) number of times that q has reached its left turning point since t = 0.”

• AssumptionThe authors state this as an assumption: The ambiguity of the pendulum as a clock is removed by coupling it to a counter that increments each complete oscillation.
• AssumptionThe authors state this as an assumption: V(q) has no more than a single critical point, and if it exists it is a global minimum.

To leading order in p_c, the A±τ,n operators are symmetric on H_aux.

“Thus, the A ± τ,n are symmetric on H aux to lowest order in p c .”

The expectation value of Aτ in a physical state equals 2πħ times an auxiliary inner product involving θ(P)√P χ0 τ A √P θ(P).

“ψ|A τ |ψ phys = 2πh ψ|θ(P ) √ P χ 0 τ A √ P θ (P )|ψ aux (4.5)”

When A is constructed from non-clock variables, A_{τ+δ} is e^{iδPτ} Aτ e^{-iδPτ} multiplied by (1+O(ε)+O(h)).

“A τ +δ = e iδP τ A τ e -iδP τ (1 + O ( ) + O (h))”

• AssumptionThe authors state this as an assumption: The clock coordinate is far from classical turning points so that classical q(t) is single-valued.
• AssumptionThe authors state this as an assumption: |p_c| is small in the sense that only states with spectral support in P_c concentrated where p_c << E are considered, E being the splitting between energy levels of H_p.

[Aτ,Bτ] equals ([A,B])τ when evaluated between states in which the clocks behave semiclassically.

“Since, by construction, (4.4) is independent of the choice of n(t), it follows that [A τ , B τ ] = ([A, B]) τ when evaluated between two states in which the clocks behave semiclassically.”

The formalism and WKB approximation give equation (4.3) for the matrix elements of -ih∂Aτ/∂τ.

“-ih n , p c , -E(n , p c ), a ∂ ∂τ A τ n, p c , -E(n, p c ), a phys = (E -E) n , p c, -E(n , p c ), a |A τ |n, p c , -E(n, p c ), a phys (4.3)”

• AssumptionThe authors state this as an assumption: The clock coordinate is far from classical turning points so that classical q(t) is single-valued.
• AssumptionThe authors state this as an assumption: |p_c| is small in the sense that only states with spectral support in P_c concentrated where p_c << E are considered, E being the splitting between energy levels of H_p.

Corrections to (4.12) vanish when V(Q)=0.

“From section 4.2 of [1], it follows that any corrections to (4.12) vanish when V (Q) = 0.”

• cites [1] Marolf D, 1995, Class. Quantum Grav. sourceNot checked yet

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A homogeneous comoving fleet of noninteracting metric-based clocks in a homogeneous spacetime is described by the constraint e^{3α}P+H1, not by the form (3.1).

“In particular, we show in this appendix that a homogeneous comoving fleet of noninteracting such clocks in a homogeneous spacetime is described by a constraint of the form e 3α P + H 1 and not of the form (3.1).”

• AssumptionThe authors state this as an assumption: The clocks in the fleet are evenly distributed with respect to the metric on each spatial slice.
• AssumptionThe authors state this as an assumption: The clock fleet is comoving, with its 4-velocity proportional to the normal to the family of slices.

φ|1lτ|ψ_phys = <φ;τ|ψ;τ>_H1.

“Note that (4.12) may also be written as φ|1l τ |ψ phys = φ; τ|ψ; τ H 1 , (4.13)”

The commutator [Pτ,1lτ] vanishes.

“We may calculate the commutator of the existence operator with the time translation generator as [P τ , 1l τ ] = √ -2H 1 -2V (τ ) τ , 1l τ and, since √ -2H 1 -2V (τ ), 1l = 0, this expression vanishes by the result just derived.”

If the state has a reasonably well-defined pendulum momentum p, then ψ|[-H1,A]|ψ = p ψ|sqrt(-2H1-2V(τ)),A|ψ.

“so that, if the state |ψ has a reasonably well defined value p of the pendulum momentum P , this is just ψ|[-H 1 , A]|ψ = p ψ| √ -2H 1 -2V (τ ), A |ψ .”

Equation (4.6) would describe a pure state except for the factor p_c|χ0|p_c.

“In fact, (4.6) would describe a pure state except for the factor p c |χ 0 |p c .”

In the small counter-coupling limit, p_c|χ0|p_c is essentially one where ψ is nonzero, so ρ(τ) effectively describes a pure state.

“Since p c |χ 0 |p c is a distribution of unit height and width 2h/ , this factor is essentially one in (4.6) when ψ(n, E, a) is nonzero and ρ(τ ) effectively describes a pure state.”

The final Hamiltonian is sqrt(-2H1-2V(τ)) rather than -H1.

“Indeed, where we would expect to find -H 1 as the Hamiltonian in (4.11), we instead find √ -2H 1 -2V (τ ).”

When H1=g_ij P⊥i P⊥j with a constant negative definite metric, (4.10) holds exactly for A=Q_i⊥.

“When H 1 = g ij P ⊥ i P ⊥ j with g ij a constant negative definite D Marolf metric, direct calculation as in section 4.2 of [1] shows that (4.10) holds exactly for A = Q i ⊥ as well.”

• cites [1] Marolf D, 1995, Class. Quantum Grav. sourceNot checked yet

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At lowest order in h, the matrix element τ|P^k θ(P)|n;p_c is (2E(n,p_c)-2V(τ))^{k/2} times τ|n;p_c^+.

“To lowest order in h, we have τ |P k θ(P )|n; p c = (2E(n, p c ) -2V (τ )) k/2 τ |n; p c + from (3.9).”

• AssumptionThe authors state this as an assumption: The clock coordinate is far from classical turning points so that classical q(t) is single-valued.

The map A→Aτ preserves the equal-time algebraic structure of external time quantum mechanics to lowest order in ħ.

“Thus, to this accuracy the map τ : A → A τ preserves the entire equal time algebraic structure of external time quantum mechanics.”

Evolution in the parameter τ is unitary, apart from convergence and domain issues for Pτ and Aτ.

“Since P τ is symmetric on H aux , we have shown that evolution in the parameter τ is unitary, modulo issues of convergence of the integrals defining P τ and A τ and of the domain of definition of P τ .”

|ψ;τ satisfies the Schrödinger equation with Hamiltonian sqrt(-2H1-2V(τ)).

“The state |ψ; τ satisfies ih ∂ ∂τ |ψ; τ = -2H 1 -2V (τ ) |ψ; τ (4.11)”

For functions A and B of non-clock canonical variables and states satisfying the approximation scheme, the matrix elements of [Aτ,Bτ] equal those of ([A,B])τ times (1+O(ħ)).

“In general, we verify that φ|[A τ , B τ ]|ψ = φ|([A, B]) τ |ψ (1 + O (h)) for A = A(Q i ⊥ , P ⊥ j ) and B = B(Q i ⊥ , P ⊥ j ) and any two states |φ and |ψ that satisfy the requirements of our approximation scheme.”

The non-clock canonical coordinates and momenta satisfy the canonical commutation relations to lowest order in ħ.

“Here, we consider the algebraic structure of the operators A τ and, in particular, show that if Q i ⊥ and P ⊥ j are the canonical coordinates and momenta of the non-clock degrees of freedom, then (Q i ⊥ ) τ and (P ⊥ j ) τ satisfy the canonical commutation relations to lowest order in h.”

For two such states, φ|Aτ|ψ = <φ;τ|A|ψ;τ>.

“Similarly, for two such states |φ and |ψ , we have φ|A τ |ψ = φ; τ|A|ψ; τ .(4.10)”

Weak self-interaction and weak interaction with other degrees of freedom are expected to make a generic degree of freedom approximate an ideal clock.

“In general, we expect a generic degree of freedom to approximate an ideal clock when it is weakly interacting, both with other degrees of freedom and with itself.”

A classical clock that directly tracks metric proper time at arbitrarily small spatial volumes also fails to have the needed properties.

“even classically, a clock which explicitly measures metric-based proper time for arbitrarily small spatial volumes does not possess the properties we require.”

Provided the coupling is weak, how the counter is built and coupled to the pendulum is expected not to matter.

“we expect the details of its construction and coupling to the pendulum to be irrelevant so long as this coupling is weak.”

Metric-defined proper-time clocks make the present approximations break down close to singularities in homogeneous models.

“for clocks that measure metric-defined proper time, our approximations always fail near a singularity in homogeneous models.”

Because (3.9) is ε-independent, it remains usable at ε=0, as when V(q) has at most one turning point.

“On the other hand, (3.9) is independent of .As a result, (3.9) may be used even in the case where we may take = 0, such as when the potential V (q) generically has only one turning point (or none at all).”

In several cosmological models, with q interpreted as the volume of a spatial hypersurface, H0 has the form H_p = 1/2 P^2 + V(Q).

“From, e.g.[20] we see that this is the case for a number of cosmological models if q represents the volume of a spatial hypersurface.”

• cites [20] Jantzen R, 1984Not checked yet

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The approach reproduces the classical recollapse of suitable cosmological models at the quantum level.

“In addition, this approach captures in a quantum sense the classical recollapsing behaviour of the appropriate cosmological models.”

The systems under study generally admit two separate proper-time notions: one from the Hamiltonian constraint and lapse, and one from a metric.

“there will, in general, be two distinct notions of proper time.One such notion is the proper time t 2 t 1 N(t) dt associated with any system described by a Hamiltonian constraint H and lapse N while the other is whatever concept of proper time may be defined by a metric for the system.”

Suitable adiabatic conditions are expected to give the homogeneous mode of a real scalar field and many noninteracting clocks dynamics of the H_p = 1/2 P^2 + V(Q) form.

“Similarly, under appropriately 'adiabatic' conditions (see section 5 and the appendix) we expect that the dynamics of the homogeneous mode of a real scalar field and a fleet of noninteracting clocks should be of this form.”

The almost local formalism has no basic concept of time or evolution.

“the approach of [1] (which we will refer to as the 'almost local formalism') contains no fundamental notion of time or evolution.”

• cites [1] Marolf D, 1995, Class. Quantum Grav. sourceNot checked yet

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The almost local formalism was successfully applied to the complicated mixmaster cosmology.

“In particular, it was successfully applied [22] to the complicated mixmaster cosmology [23].”

• cites [22] Ryan M, 1975Not checked yet

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The external-time formalism is already known to follow exactly for a reparametrization-invariant system containing an ideal clock.

“For such a system, it has already been shown in [1] that the external time formalism can be derived exactly.”

• cites [1] Marolf D, 1995, Class. Quantum Grav. sourceNot checked yet

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Almost-ideal-clock notions for the two proper-time definitions match except at very small spatial-slice volume.

“that the notions of 'almost ideal clocks' associated with the two kinds of proper time agree unless the volume of the spatial slices is very small”

When H_p has discrete spectrum, the link between H_phys and H_aux makes the counter essential.

“The relation of H phys to H aux will imply that the presence of the counter is crucial to this process when H p (the pendulum Hamiltonian) has a discrete spectrum.”

• AssumptionThe authors state this as an assumption: The auxiliary Hilbert space has a basis for a spectral interval [a,b] of H containing 0, and H has continuous spectrum with no point spectrum in that interval.

The need for H to have a continuous spectrum near zero forces homogeneous cosmological applications of the almost local formalism to use a constraint and lapse nontrivially linked to the ADM ones.

“the requirement that H has only a continuous spectrum near zero means (see [1,22]) that the almost local formalism can only be applied to homogeneous cosmological models by using a constraint H and lapse N which are nontrivially related to the ADM constraint and lapse.”

• cites [1] Marolf D, 1995, Class. Quantum Grav. sourceNot checked yet

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• cites [22] Ryan M, 1975Not checked yet

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• AssumptionThe authors state this as an assumption: The auxiliary Hilbert space has a basis for a spectral interval [a,b] of H containing 0, and H has continuous spectrum with no point spectrum in that interval.

The almost local formalism gives a complete quantization of Hamiltonian vacuum Bianchi models and similar finite-dimensional time reparametrization invariant systems.

“the almost local approach was shown in [1] to provide a complete quantization of Hamiltonian vacuum Bianchi models (see, e.g.[19][20][21]) and of similar finite-dimensional time reparametrization invariant systems.”

• cites [1] Marolf D, 1995, Class. Quantum Grav. sourceNot checked yet

  Not checked in the free run.

If the clock measured metric-defined proper time and is coupled as in [1] or [22], the system would be described by 0=e^{3α}H0+H1 instead of (3.2).

“Had the clock instead measured metric-defined proper time (assuming that it is coupled to the description of the gravitational field used in [1] or [22]), our system would be described (see the appendix) by a constraint of the form 0 = e 3α H 0 + H 1 (5.2) instead of by (3.2).”

• cites [1] Marolf D, 1995, Class. Quantum Grav. sourceNot checked yet

  Not checked in the free run.

• cites [22] Ryan M, 1975Not checked yet

  Not checked in the free run.

• AssumptionThe authors state this as an assumption: The clock is coupled to the description of the gravitational field used in [1] or [22] when considering metric-defined proper time.

Together with unitary evolution, this preservation extends to all Heisenberg operators and the Hamiltonian sqrt(-2H1-2V(τ)) to this order.

“Together with the unitary evolution derived in section 4.1, this in fact implies preservation to this order of the algebraic structure associated with all Heisenberg operators and the Hamiltonian √ -2H 1 -2V (τ ).”

The states |ψ;τ = σ_τ θ(P)√P η|ψ have the usual interpretation as probability-density amplitudes.

“It is the states |ψ; τ = σ τ θ(P ) √ P η|ψ that have the usual interpretation as amplitudes for the probability”

A clock measuring proper time in a state-independent way is not fundamental to deriving unitary evolution, the canonical commutation relations, or the Hilbert space structure.

“Thus, from the above results and the facts that our approximations are exact for a system of free particles and that section 4.1 and (4.12) are exact whenever the clock is a free particle, we may conclude that the use of a clock which measures proper time in a state-independent way is not fundamental to a derivation of unitary evolution, the canonical commutation relations, or the well-known Hilb”

The symmetry of Pτ on H_phys makes (4.1) a unitary evolution in the clock time τ.

“Since, as was noted in section 3.1, P τ defines a symmetric operator on H phys , it follows that (4.1) describes unitary evolution in the clock time τ .”

• AssumptionThe authors state this as an assumption: The clock coordinate is far from classical turning points so that classical q(t) is single-valued.
• AssumptionThe authors state this as an assumption: |p_c| is small in the sense that only states with spectral support in P_c concentrated where p_c << E are considered, E being the splitting between energy levels of H_p.

Repeated occurrences of b(t)=τ make convergence of (2.5) delicate, requiring the almost-ideal-clock definition to account for the classical trajectories.

“However, when b(t) takes this value many times the convergence of (2.5) becomes a subtle issue so that our definition of an almost ideal clock below must pay due attention to the corresponding classical solutions.”

Wheeler-DeWitt wavefunctions solving H|ψ>=0 are not the usual probability amplitudes; they require a Newton-Wigner/Klein-Gordon interpretation.

“The main consequence of (4.12) is to show that, in contrast to the suggestions of [33,34] the wavefunctions q ⊥ , q, q c |ψ which solve the 'Wheeler-DeWitt equation' q ⊥ , q, q c |H |ψ = 0 do not correspond to the usual probability amplitudes, but must instead be interpreted using constructions like those associated with the Newton-Wigner operators and the Klein-Gordon inner product.”

• cites [33] sourceNot checked yet

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• cites [34] Brown J D, 1994Not checked yet

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The difference from the usual Hamiltonian results from dispersion.

“This is a result of the above mentioned dispersion.”

|ψ;τ can be interpreted as a Schrödinger-picture state in H_ext with Hamiltonian sqrt(-2H1-2V(τ)).

“so that we may interpret |ψ; τ as a Schrödinger picture state in H ext ∼ = H 1 with Hamiltonian √ -2H 1 -2V (τ ).”

Equation (4.6) defines a map from physical states to Schrödinger-picture density matrices in external time quantum mechanics.

“Thus, (4.6) defines a map from states in H phys to density matrices in the Schrödinger picture of external time quantum mechanics.”

The metric-based and lapse-based notions of an almost ideal clock agree when e^{3α} can be treated as an adiabatic factor.

“However, the metric-based and lapse-based notions of an almost ideal clock will agree when e 3α may be treated as an adiabatic factor.”

Up to constant factors, the physical inner product in the ideal clock limit is the positive-frequency part of the Klein-Gordon inner product.

“Note that we arrived at a Schrödinger picture description due to the implicit mapping of the operator A τ to the operator A(t = 0) in (4.5) and ( 4.10 and, since 1l τ = 1l in our approximation, we see that up to constant factors the inner product on H phys in the ideal clock limit is the positive frequency part of the familiar Klein-Gordon inner product †.”

The operators from [1] represent an average of A over times where B=τ, including cases where b(t) hits τ repeatedly.

“The associated physical operators describe a certain sort of average over 'the values of the quantity A(t) at the times for which B(t) = τ ', even when the corresponding classical quantity b(t) may take on the given value τ more than once along a trajectory.”

• cites [1] Marolf D, 1995, Class. Quantum Grav. sourceNot checked yet

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Conservation of existence does not follow directly from unitarity alone.

“Note that while the same calculation (4.2) which showed the unitarity of evolution allowed us to derive conservation of existence (∂1l τ /∂τ = 0), this result does not follow directly from unitarity.”

Conservation of probability is an explicit consequence of conservation of existence.

“Finally, by differentiating both sides of (4.13) with respect to τ , we see that conservation of probability (that is, of the right-hand side) is an explicit consequence of conservation of existence on the left.”

Conservation of existence follows from a reasonable time-translation generator and the usual algebraic structure of Marolf theory.

“That is, conservation of existence follows from the presence of a reasonable generator of time translations and from the usual algebraic structure of the D Marolf theory.”

The two notions of an almost ideal clock agree for large spatial volumes.

“Thus, these two notions of an almost ideal clock agree for large spatial volumes.”

The usual external time formalism should not be expected to hold for a metric-based clock near a spacetime singularity.

“In this way it follows that we should not expect the usual external time formalism to hold for a metric-based clock near a spacetime singularity.”

The condition implies that the 3-volume V of a homogeneous slice should satisfy V ≫ l_p^3 m_p/M.

“Inserting the appropriate dimensional factors, this condition says that the 3-volume V of a homogeneous slice should satisfy V ℓ 3 p m p /M where m p and ℓ p”