(No) Time for Quantum Theory? · Then I explore the consequences of taking seriously the proposal...

(No) Time for Quantum Theory?

Thomas Pashby

Department of History and Philosophy of ScienceUniversity of Pittsburgh

[email protected]

September 27, 2013

Introduction Evolving Constants Toy Models Event POVMs Extended QM & Beyond Conclusion

The Problem

Ordinary QM provides assignments of probabilities at an instant of time.

In (pure) state ψ, the expectation value of an observable A at time t is

〈A〉t = 〈ψt |Aψt〉 = 〈ψ|U†t AUtψ〉.

Yet quantum gravity seems to demand a quantum theory interpretedwithout dependence on a time parameter.

How do we provide a meaningful assignment of probabilities that doesn’tdependent on a classical time parameter t?

The Project:

Reconstruct ordinary (non-relativistic) QM such that explicit reference tot is avoided (in a way that lends itself to further generalization).


Kill or Cure?

Option 1: Eliminate (Gambini-Porto-Pullin)

• Avoid reference to a time parameter by avoiding reference to time.

• Predictions phrased so that they don’t refer to times at all.

• To be interpreted in terms of conditional probabilities.

Option 2: Promote (Brunetti-Fredenhagen-Hoge)

• Promote t to an operator—an observable T .

• Predictions referred to the values taken by this observable.

• To be interpreted in terms of conditional probabilities.

I will show how these two views lead to essentially the same place:Extended QM.

Then I explore the consequences of taking seriously the proposal ofBrunetti, Fredenhagen & Hoge (2010) which goes beyond this.


Relativistic Mechanics

Carlo Rovelli proposes that we approach quantum theory from areconstructed classical theory making no explicit reference to a timeparameter t.

In his Relativistic Mechanics (2002, 2004) a dynamical system isdescribed by (C,H), where C is extended configuration space andH : T ∗C → R is the Hamiltonian.

A point in the state space Γ corresponds to an orbit of the presymplecticform ω on the constraint surface Σ ⊂ T ∗C given by H = 0.

An orbit (i.e. a state) defines a subspace of C, an entire dynamicalmotion.


Evolving Constants (I)

For the free particle C = R×Q, with co-ords (t, q), T ∗C has co-ords(qa; pa) = (t, q; pt , p) with Hamiltonian

H = pt + H0 = pt +1

2mp2.

The constraint surface Σ = R× T ∗Q contains the dynamical motions,parameterized curves qa(τ) = (t(τ), q(τ)).

The variables t, q and p are partial observables and the theorydetermines correlations between them.

We have: t(τ) = τ ; q(t(τ)) = q(τ) = q0 + (p/m)τ ; p(τ) = p0.

The complete observables commute with the constraints, i.e.{O,H} = 0.

p (i.e. p0) is a complete observable, but so is q0 = q(τ)− (p/m)τ .Together, they pick out a dynamical motion.


Quantized Relativistic Mechanics

Rovelli’s quantization procedure first constructs an ‘auxiliary” Hilbertspace H+ = L2(C) and promotes the Hamiltonian constraint to anoperator H.

The ‘physical states’ are defined as those states annihilated by theHamiltonian constraint,

H|Ψ〉 = 0.

But as H has a continuous spectrum, this constraint can only be satisfiedin a distributional sense, i.e. |Ψ〉 /∈ H+.

The states |Ψ〉 ∈ S∗ lie in the dual of the Schwartz space,S ⊂ H+ ⊂ S∗.

Refined Algebraic Quantization (Ashtekar et al, 1995) provides a Hilbertspace H∗ for these ‘physical’ states on which the Dirac Observables (thatcommute with the constraint) can be defined.

Question: How does quantum mechanics look on extended Hilbert spaceH+ = L2(C)?


Evolving Constants (II)

The sort of question usually asked in QM takes the form:

What is the value of the partial observables q, p given that the time ist = τ?

The one-parameter family q(τ) has the form of the Heisenberg pictureobservable Q(t),

dQ(t)

dt= −i [Q,H] = −i [Q,P2/2m] = −i P

m[Q,P] = P/m

Integrating this expression we obtain:

Q(t) =

∫ t

−∞

(dQ

dt

)dt = Q(0) + (P/m)t

(= U†t QUt

).

But a generally covariant system with Γ 6= R×Q won’t have aconvenient parameterization of the configuration space.

To generalize we need to remove the dependence on the parameter τ .


Quantum Constants?

Gambini, Porto, Pullin & Torterolo (2009) suggest that we promote theparameterized partial observables qa(τ) to one-parameter families ofoperators Qa(τ).

To remove the dependence of these families on the unphysical parameterτ they integrate the expectation value over all τ .

We replace the condition ‘given that the time is τ ’ by ‘given that Qa

takes a value a ∈ ∆.’

In this reformulation of Ordinary QM we ask:

What is the probability that the partial observable Qb takes a valueb ∈ ∆a given that Qa has a value a ∈ ∆b?

The resulting conditional probabilities are time-independent.

(N.B. There is a lot more to say here. Inspired by Page and Wooters(1983).)


A Toy ModelTwo (non-interacting) free particles with mass m and phase spaceco-ords (t, qa, qb; pt , pa, pb)

H = pt +1

2m

(p2a + p2

b

).

We will use the location of particle a to predict the location of particle b.

The Ordinary QM system has Hilbert space Ha ⊗Hb, whereHa = Hb = L2[R], with state ρ = ρa ⊗ ρb.

The condition that Qa takes a value in ∆ at time t = τ is

PQa(∆)(τ)⊗ I

The outcome that Qb takes a value in ∆′ at time t = τ is

I⊗ PQb(∆′)(τ)

where PQa(∆)(τ) = U†τPQa(∆)Uτ =

∫∆d(U†τP

Qa

λ Uτ ).

(From Qa =∫∞−∞ λdPQa

λ .)


Time-Independent Conditional Probabilities

Conditional probabilities in QM are given by the Luders Rule:

Pr(R|P) = tr [R ρ]; ρ =PρP

tr [Pρ],

where R,P are projections s. t. R ≤ P.

Given that the time is τ , we have

Pr(b ∈ ∆(τ)|a ∈ ∆′(τ)) =tr [(PQa(∆)(τ)⊗ PQb(∆′)(τ))ρ]

tr [(PQa(∆)(τ)⊗ I)ρ].

To remove the time dependence, integrate over all τ

Pr(a ∈ ∆|b ∈ ∆′) = limt→∞

∫ t

−t tr [(PQa(∆)(τ)⊗ PQb(∆′)(τ))ρ]dτ∫ t

−t tr [(PQa(∆)(τ)⊗ I)ρ]dτ

How do we interpret this expression?


Conditioned On What? (I)

Suggestion 1: Particle A functions as a clock (Porto-Gambini-Pullin)

But shouldn’t a clock provide an indication of t? The usual expression ofthis is time shift covariance.

If T is an ideal clock observable then

T (t) = U†t TUt = T + t.

Measure T (t) at times t, t ′ to get expectation value〈T (t ′)〉 − 〈T (t)〉 = t ′ − t.

But Qa(τ) does not behave like this: Qa(τ) = Qa(0) + (p/m)Qa(τ).

Perhaps we need to measure a different operator, which does.


Clock Operators

Aharonov and Bohm (1961) suggest the following operator as a quantumclock

TAB =m

2

(QP−1 + P−1Q

).

It satisfies the (Heisenberg) CCR with H = P2/(2m) (formally, at least)

[TAB ,H] =1

2

(P−1[Q,P2] + [Q,P2]P−1

)= i ,

and we can see thatU†t TABUt = TAB + t.

But two problems:

1. TAB is not self-adjoint (Pauli’s Theorem)

2. TAB combines the Schrodinger and Heisenberg pictures in anawkward way.


TAB is no Clock Observable

1. Pauli’s Theorem: If (i) T (t) = U†t TUt = T + t and (ii) T isself-adjoint then (iii) sp(H) is unbounded.

But here sp(H) = sp(P2) > 0 (not iii) and U†t TABUt = TAB + t (i),therefore TAB is not self-adjoint (not ii).

TAB is maximally symmetric; it has no self-adjoint extensions.

2. Classically, q(t) = q0 + (p/m)t

⇒ t = (m/p)q(t) (when q0 = 0).

But tAB = (m/p)q0 which does not vary with time.

Measuring TAB(t) at time t involves Q(0) rather than Q(t).

TAB(t) is not an observable measured at time t in the usual way.


Conditioned on What? (II)

Suggestion 2: Qa corresponds to the location of an event involvingparticle a.

Imagine that we have two detectors set up: A is sensitive to a only; B issensitive to b only.

We prepare the system in state ρa ⊗ ρb and wait for both detectors tofire.

We use the location of the spot on detector A to condition ourpredictions of the location of detector spot B.

But these are by assumption independent events.

(This would not be the case in a less trivial model.)

But there is an even simpler system we can use as a toy model.


Time-Independent Conditional Probabilities (II)We can use the same framework to condition on events involving just asingle particle.

Consider a screen Σ ⊂ R at which the particle is certain to be detected,sooner or later.

Let PQ(Σ)(t) be the Heisenberg picture spatial projection onto thatscreen. Let PQ(∆)(t) be the projection onto a sub-region, ∆ ⊆ Σ.

Then PQ(∆)(t) ≤ PQ(Σ)(t) ≤ I.

At time t, in state ρ, we apply Luders’ Rule:

Pr(∆(t)|Σ(t)) =tr [PQ(Σ)(t)PQ(∆)(t)(t)PQ(Σ)(t)ρ]

tr [PQ(Σ)(t)ρ]=

tr [PQ(∆)(t)ρ]

tr [PQ(Σ)(t)ρ].

To remove the time dependence, integrate over all t:

Pr(∆|Σ) = limτ→∞

∫ τ−τ tr [PQ(∆)(t)ρa]dt∫ τ−τ tr [PQ(Σ)(t)ρa]dt

=:tr [B∆ρa]

tr [BΣρa].


Event POVMsBΣ (likewise B∆) is a positive operator defined by

〈ψ|BΣφ〉 := 〈ψ|∫ ∞−∞

PΣ(t)dt|ψ〉 = limτ→∞

∫ τ

−τdt

∫Σ

d〈ψ|U†t PQλ Utφ〉,

when the limit converges. (For the free particle this is all ψ, φ ∈ H).

Brunetti & Fredenhagen (2002) and Hoge (2008) consider BΣ as themaximal element of a net of operators defined on intervals I = [t1, t2],

EΣ(I ) =

∫ t2

t1

dt

∫Σ

d〈ψ|U†t PQλ Utφ〉.

They define a (normalized) time covariant POVM, an event time POVM

FΣ(I ) := B−1/2Σ EΣ(I )B

−1/2Σ ,

where B−1/2Σ is the inverse of the positive square root

(B

1/2Σ

)2

= BΣ.

(Normalized because FΣ(R) = B−1/2Σ

(B

1/2Σ

)2

B−1/2Σ = I.)


Time-Independent Conditional Probabilities (III)This suggests we use BΣ to define a normalized POVM

FΣ(∆) := B−1/2Σ B∆B

−1/2Σ .

Note that we have from before

Pr(∆|Σ) =tr [B∆ρ]

tr [BΣρ]=

tr [B1/2Σ B

−1/2Σ B∆B

−1/2Σ B

1/2Σ ρ]

tr [BΣρ].

That is,

Pr(∆|Σ) =tr [B

1/2Σ FΣ(∆)B

1/2Σ ρ]

tr [BΣρ]= tr [FΣ(∆)ρΣ],

where ρΣ is the Luders operation defined by BΣ,

ρ→ ρΣ =B

1/2Σ ρB

1/2Σ

tr [BΣρ].

This brings event POVMs close to the usual form for a conditionalprobability, but these are not projections B2 6= B.


Event Time POVMs

Note that ∆ 7→ FΣ(∆) is an event POVM that applies only under thecondition BΣ, i.e. given that the particle is detected in Σ at some t.

The expression Pr(∆|Σ) = tr [FΣ(∆)ρΣ] answers questions of the form:

What is the probability that the particle was detected in ∆ given that itwas detected in Σ?

The event time POVM I 7→ FΣ(I ) applies in the same way

Pr(I |Σ) = tr [FΣ(I )ρΣ],

but answers a different question:

What is the probability that the particle was detected in Σ duringI = [t1, t2] given that it was detected in Σ at some t?

Note that these expressions use the Heisenberg, not Schrodinger state,but consider many instants of time.

Can we get these conditional probabilities by Luders’ Rule proper?


Extended Hilbert Space (I)

Let us return to the ‘extended’ Hilbert space H+ = L2[C] = L2[R×Q].

Vectors in this space are functions of (configuration) space and time.

Let a vector Ψ(τ) ∈ H+ be given in the time-representation by a vectorvalued function of τ

Ψ(τ) = Uτψ where ψ ∈ L2[Q].

Instead of thinking of Uτψ as a family of states that follow a paththrough L2[Q], we think of Ψ(τ) as a vector in H+.

Each (pure) state Uτψ lies in an instantaneous space Hτ ∼= L2[Q].

Think of a vector Ψ(τ) as a representing the career of a system ofOrdinary QM.


Extended Hilbert Space (II)

An instantaneous state Uτψ lives in Hτ ∼= L2[Q], with inner product

〈φ|ψ〉τ =∑k

〈Uτφ|ek〉〈ek |Uτψ〉,

where {ek} is a fixed orthonormal basis for H0.

Construct extended H+ is as a continuous direct sum of instantaneousspaces

H+ =

∫⊕Hτdτ

with inner product

〈Φ|Ψ〉+ =

∫ ∞−∞〈φ|ψ〉τdτ = lim

t→∞

∫ t

−t

∑k

〈Uτφ|ek〉〈ek |Uτψ〉dτ.

(Ψ(τ) ∈ H+ only if 〈Ψ|Ψ〉+ =

∫ ∞−∞

∑k

|〈Uτψ|ek〉|2dτ <∞

)


Time-Independent Conditional Probabilities (IV)

Any bounded operator A ∈ B(H0) defines an operator A+ as follows

〈Φ|A+Ψ〉+ =

∫ ∞−∞〈φ|Aψ〉τdτ =

∫ ∞−∞

∑k

〈Uτφ|ek〉〈ek |UτAψ〉dτ.

Define the extended trace Tr+ as follows

Tr+[A+] :=

∫ ∞−∞〈ek |A+ek〉τdτ =

∫ ∞−∞

tr [U†τAUτ ]dτ.

Then we have

Pr(∆|Σ) =tr [B∆ρ]

tr [BΣρ]=

Tr+[P∆+ρ+]

Tr+[PΣ+ρ+]=

Tr+[PΣ+P∆+PΣ+ρ+]

Tr+[PΣ+ρ+]= Tr+[P∆+ρΣ+].

We have recovered Luders’ Rule (valid for projections R+ ≤ P+), with

ρ+ → ρ+ =P+ρ+P+

Tr+[P+ρ+].


Event Time Observables

In this formalism we can define a time PVM I 7→ ET (I ),

ET (I ) =

∫ t2

t1

dτ |Hτ 〉〈Hτ | .

These are orthogonal projections since

〈Ψ|ET (I )ET (I ′)Ψ〉+ = 0 when I ∩ I ′ = ∅, for all Ψ ∈ H+.

So the question:

What is the probability that the particle was detected in Σ duringI = [t1, t2] given that it was detected in Σ at some t?

Has the answer

Pr(I |Σ) = Tr+[ET (I )ρΣ+] =Tr+[PΣ+E

T (I )PΣ+ρ+]

Tr+[PΣ+ρ+]= tr [FΣ(I )ρΣ].


Ordinary QM as Instantaneous Limit

The expectation value of an operator B+ ∈ B(H+) given that the timeτ ∈ [t − ε, t + ε], ε > 0, is

〈B+〉[t−ε,t+ε] =Tr+[ET ([t − ε, t + ε])B+E

T ([t − ε, t + ε])ρ+]

Tr+[ET ([t − ε, t + ε])ρ+]

As Brunetti, Fredenhagen and Hoge (2010) claim, Ordinary(instantaneous) QM is returned in the limit ε→ 0. That is,

limε→0〈B+〉[t−ε,t+ε] = tr [U†t BUtρ] = 〈B〉t .

But note that we have not yet applied the Hamiltonian constraint!

All the predictions of Ordinary QM can be given in the Extended Hilbertspace H+ from vectors that explicitly don’t solve the “Wheeler-DeWitt”equation

H|ψ〉 = 0.


The Schrodinger Equation

In Ordinary QM all the dynamical information is provided by the group oftime translations generated by H0, i.e. Ut .

The time-dependent Schrodinger equation is just the instantaneousversion of this symmetry group.

If Utψ is in the domain of H0 then

id

dtUtψ = lim∆t→0

1

it(Ut+∆t − Ut)ψ = H0Utψ.

If we only wanted to recover Ordinary QM, we should stop here.

Proposal: To understand the equation H|Ψ〉 = 0 requires anprobabilistic interpretation that goes beyond Ordinary QM:

(a) a distinct notion of state;(b) a replacement of the trace.

(Gleason’s Theorem tells us these go together.)


The Extended Schrodinger Equation

Define a self-adjoint operator T s. t. τ ∈ sp(T ) = R

T =

∫ ∞−∞

τdPτ , where PTτ =

∫ τ

−∞dt |Ht〉〈Ht | .

This is a quantization of the partial observable t(τ).

T has a conjugate −W 6= H0 (with unbounded spectrum) defined in thetime-representation by

(WΨ)(τ) = id

dτΨ(τ).

This is the quantization of pt .

We can write the Extended Schrodinger equation as

(H0 −W )ψ(x , τ) = 0,

as Dirac first wrote it in 1926.


The Weight

Rather than escape H+ for the “physical” Hilbert space H∗, Brunetti,Fredenhagen and Hoge (2010) use Hψ(x , τ) = 0 to define a weight onthe ‘algebra of events’ L ⊂ B(H+).

An algebraic state A 7→ ω(A) is a normed positive linear functional, i.e.an expectation value functional. Here A ∈ B(H).

(A normal state admits a representation as a density operator, as inOrdinary QM.)

A weight B 7→ w(B) is an unbounded positive linear functional,i.e. w(B) ∈ R+ ∪ {∞}.


The Schrodinger Weight

The procedure is (roughly) as follows:

1. Linear functional 〈ψ(x , τ)|·〉+ defined on domain D ⊂ H+.

2. Define a weight wψ(B) on D for (B+ > 0) ∈ B(H+) on D.

3. Extend wψ to the subalgebra L = L∗ψLψ, where

B+ ∈ L ⇔ 0 < 〈Ψ|B+Ψ〉+ <∞, for all Ψ ∈ H+.

4. Define a inner product 〈C ,D〉 := wψ(C∗D) for C ,D ∈ Lψ, andconstruct a GNS representation (Hψ, πψ).

The events B+ ∈ L are vectors of Hψ, and the inner product provides anassignment of expectation values under the condition that the event doesin fact occur.

ωψ(A|B) :=wψ[B

1/2+ A+B

1/2+ ]

wψ[B+].

Note that the weight wψ plays the role of the trace and state combined.


Time Independent Conditional Probabilities (V)Now we use the extended Hamiltonian,

H = H0 −W = − 1

2m

∂

∂x2− i

∂

∂t,

which gives a solution of the Extended Schrodinger Equation the form

ψ(x , t) =

∫dp exp

(−i p

2

2mt + ipx

)ϕ(p).

We have

Pr(∆|Σ) =wψ[P∆+]

wψ[PΣ+],

where

wψ[PΣ+] =

∫ ∞−∞

dt

∫Σ

dx |ψ(x , t)|2

=

∫Σ

dx

∫dp

∫dp′2πδ

(p2 − p′2

2m

)e−i(p−p

′)xϕ(p)ϕ(p′).


Time-Dependent Probabilities

The conditional probability for the detector firing within ∆ during I is

Pr(I |Σ) =wψ[ET (I )P∆+E

T (I )]

wψ[PΣ+],

with

wψ[ET (I )P∆+ET (I )] = wψ[P∆+E

T (I )]

=

∫I

dt

∫∆

dx

∫dp

∫dp′e( it

2m (p2−p′2)−i(p−p′)x)ϕ(p)ϕ(p′).

We could have obtained this with the ‘Extended Luders Rule,’ but notehow in this formalism the following also makes sense:

Pr(∆(I )|∆′(J)) =wψ[P∆+E

T (I )P∆′+ET (J)]

wψ[P∆′+ET (J)],

where ∆(I ),∆′(J) are disjoint space-time regions.


Quantum Partial Observables

This proposal generalizes to systems with Γ 6= R×Q since we justcondition on the values taken by a partial observable (e.g.Brunetti-Fredenhagen-Hoge consider scalar field on FRW spacetime).

I would urge a further interpretation in terms of events: definiteoccurrences that correspond to a localization within phase space, i.e. arestriction of the spectra of several quantum partial observables.

Since probabilities are given by the weight not the trace, the occurrenceof multiple events corresponding to orthogonal subspaces of H+ isallowed.

Predictions require (i) specification of H (ii) specification of events onwhich to condition.

But this formalism does not require a vector state (density operator).


(Tentative) Conclusions

• Eliminating time (by integrating it out) and promoting it to anoperator result in the same theory: Extended QM.

• The space-time projections have a natural interpretation in terms ofconditional probabilities for event observables.

• If the aim is just to recover Ordinary QM, there is no need toconsider the Hamiltonian constraint; in making use of the constraint,one goes beyond Ordinary QM.

• The idea of conditioning on values taken by partial observables findsa natural expression in terms of the Schrodinger weight.

Some questions:

• Rather than constructing a distinct “physical” Hilbert space H∗,might we be able to do physics in the “auxiliary” space H+?

• Are Dirac observables really all that one can observe?

• Isn’t what we observe—what’s real in quantum theory—the discreteevents that occur as the outcomes of experiments?

(No) Time for Quantum Theory? · Then I explore the consequences of taking seriously the proposal...

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Transcript of (No) Time for Quantum Theory? · Then I explore the consequences of taking seriously the proposal...