Quantum gravity, probabilities and general boundariesrelativity.phys.lsu.edu/ilqgs/oeckl101706.pdf · 1 Interpretational problems in quantum gravity The problem of time The quantum

Quantum gravity, probabilities and generalboundaries

Robert Oeckl

Instituto de MatemáticasUNAM, Morelia

International Loop Quantum Gravity Seminar17 October 2006

Outline

1 Interpretational problems in quantum gravityThe problem of timeThe quantum cosmology problem

2 The general boundary formulationOverviewProbability interpretation

3 Scattering probabilitiesA hypercylinder in QFTGravitons and quantum gravity

Robert Oeckl (IM-UNAM) quantum gravity, probabilities, boundaries ILQGS 20061017 2 / 23

Interpretational problems in quantum gravity

The most popular approaches to quantum gravity assume thatsomehow standard quantum theory holds.The "output" of these approaches is usually either:

I an S-matrix (e.g. string theory)I or transition amplitudes (e.g. LQG)

In QFT such quantities can be directly converted to observableprobabilities.In QG several problems appear when trying to relate suchquantities to observable probabilities. I will focus on transitionamplitudes here and consider:

I the problem of timeI the quantum cosmology problem


The problem of time (I)

Consider transition amplitudes in a background spacetime.

We prepare a state ψ at t1,wait for a time ∆t , thenmeasure if we obtain thestate η at t2. The probabilityfor this depends on ∆t :

P = |〈η|U(∆t)|ψ〉|2

Recall properties of U:

Composition: U(∆t)U(∆t ′) = U(∆t + ∆t ′)

Unitarity: U† = U−1


The problem of time (II)

In quantum gravity no background time is available on which the“evolution operator” U can depend. By composition, we expectU2 = U. Then unitarity yields U = 1. The transition probability from ψto η should hence be merely the inner product:

P = |〈η|ψ〉|2

But, it is then unclear how the operational information of the timedifference ∆t can be encoded into the expression for the transitionprobability.


The problem of time (III)

(Semi)classical regime:

Thick sandwichconjectureGiven an initial 3-metric hand a (similar) final 3-metrich′ there is generically oneinterpolating 4-metric g (upto equivalence).

Let ψh and ψh′ be semiclassical states associated to h and h′. Thetentative transition probability P = |〈ψh′ |ψh〉|2 contains the informationabout ∆t in the states.But, we should have |〈ψh′ |ψh〉|2 = 1 and also |〈ψh|ψh〉|2 = 1. Henceψh = ψh′ up to a phase. However, h and h′ are generally physicallydifferent states (not related by a 3-diffeomorphism).


The quantum cosmology problem

States are associated with spacelike hypersurface. These extendover the whole universe. Hence a state is a priori a state of theuniverse.

In quantum field theory we can avoid talking about the holeuniverse by noticing that distant systems (with respect to thebackground metric) are independent (causality, clusterdecomposition etc.). We can thus successfully describe a localsystem as if it was alone in a Minkowski universe.

In quantum gravity there is no background to separate systems.Worse, diffeomorphism symmetry makes any kind of localizationdifficult. Hence, we need to worry about the global structure ofspace(time) and it seems hard to avoid having to do quantumcosmology.


General boundary formulation: Basic idea


Basic structures

Basic spacetime structures:

Basic algebraic structures:

To each hypersurface Σ associate a Hilbert space HΣ of states.

To each region M with boundary Σ associate a linear amplitudemap ρM : HΣ → C.

The structures are subject to a number of rules. For example:

Σ is Σ with opposite orientation. Then HΣ = H∗Σ.

Σ = Σ1 ∪ Σ2 is a disjoint union of hypersurfaces. ThenHΣ = HΣ1 ⊗HΣ2 .


Recovering standard quantum mechanicsConsider the geometry of a standard transition amplitude.

region: M = [t1, t2]× R3

boundary: ∂M = Σ1 ∪ Σ2

state space:H∂M = HΣ1⊗HΣ2

= HΣ1⊗H∗Σ2

Via time-translation symmetry identify HΣ1∼= HΣ2

∼= H, where His the state space of standard quantum mechanics.

Write the amplitude map as ρM : H⊗H∗ → C.

The relation to the standard amplitude is:

ρM(ψ ⊗ η) = 〈η|U(t2 − t1)|ψ〉


The need for a generalized interpretation

So far we have only extended the technical part of quantumtheory. The physically interpretable quantities are the same asbefore. This alone is not sufficient to address the conceptualproblems.

We want to use local regions of spacetime to address thequantum cosmology problem. Hence, we need a physicalinterpretation of amplitudes for local regions.

In general, the boundary state space does not split into a tensorproduct of initial and final state spaces. Hence, we cannot applythe standard probability formula, but need something new.

This will also allow us to avoid the problem of time.


Generalized probability interpretation

Consider the context of a general spacetimeregion M with boundary Σ.

Probabilities in quantum theory are generally conditional probabilities.They depend on two pieces of information. Here these are:

S ⊂ HΣ representing preparation or knowledgeA ⊂ HΣ representing observation or the question

The probability that the system is described by A given that it isdescribed by S is:

P(A|S) =|ρM ◦ PS ◦ PA|2

|ρM ◦ PS |2

PS and PA are the orthogonal projectors onto the subspaces.Let α : H → C be a bounded linear map. Then there exists ξ ∈ Hsuch that α(ψ) = 〈ξ, ψ〉 ∀ψ ∈ H. Define |α| := |ξ|.


Recovering standard probabilities

Recall the geometry for standard transitionamplitudes with H∂M = H⊗H∗ andρM(ψ ⊗ η) = 〈η|U(t2 − t1)|ψ〉.

We want to compute the probability of measuring η at t2 given that weprepared ψ at t1. This is encoded via

S = ψ ⊗H∗, A = H⊗ η.

The resulting expression yields correctly

P(A|S) =|ρM ◦ PS ◦ PA|2

|ρM ◦ PS |2=|ρM(ψ ⊗ η)|2

1= |〈η|U(t2 − t1)|ψ〉|2.


Probability conservation

Probability conservation in time is generalized to probabilityconservation in spacetime.

Consider a region M and aregion N “deforming” it. CallΣ the boundary of M ∪ Nand Σ′ the boundary of M.

The amplitude map for N induces a unitary map ρ̃ : HΣ → HΣ′ .

Let S ⊂ HΣ and A ⊂ HΣ. Define S ′ := ρ̃(S) and A′ := ρ̃(A).

Then, probability is conserved, P(A|S) = P(A′|S ′).


Scattering on a hypercylinder in QFT (I)

To go beyond the realm of standard quantum theory, we consider anexample with a connected boundary.

Consider a region with the shape of a solidhypercylinder in Minkowski space, M = R× B3.Its boundary is Σ = ∂M = R× S2.In Klein-Gordon theory the state space HΣ turnsout to be a Fock space. Particles can becharacterized by energy-momentum quantumnumbers. Unusually, these also determine if aparticle is in- or out-going.

We want to describe a 1-1 scattering process. Say a particle withquantum numbers pin goes in and we want to test if a particle withquantum numbers qout comes out.


Scattering on a hypercylinder in QFT (II)

To obtain a probability, we need to specify:

The knowledge about the experiment is that exactly one particlepin goes in, but we don’t know what comes out. S is the subspaceof HΣ of states with this property.

The question about the experiment is if one particle qout comesout, while we do not care about what goes in. A is the subspaceof HΣ of states with this property.

We getP(A|S) = |ρM(|pin,qout〉)|2,

where |pin,qout〉 is the state with one in-particle pin and oneout-particle qout. As we should expect in a free theory this probability isa kind of delta function δ(p − q).


On graviton scattering in quantum gravityTo see how the problem of time is avoided we consider a gravitonscattering problem.

Consider a ball shaped region in spacetime,M = B4, with boundary Σ = ∂M = S3. Supposethe state space HΣ has a sector Hlin that(approximately) describes gravitons onMinkowski spacetime, i.e., HΣ = Hlin ⊕Hrest.

Again, we want to describe a 1-1 scattering process, with pin thein-particle (prepared) and qout the out-particle (to be measured). Thesubspaces S and A are set up in analogy to the previous example.However, they are now subspaces not only of HΣ but of Hlin.Supposing that the dynamics in Hlin is near to that of a free field theorywe would get similarly as before:

P(A|S) ∼ |ρM(|pin,qout〉)|2.


Graviton scattering and the problem of time

This result cannot be obtained as a standard transition probability.

The key to avoiding the problem of time is that the subspace Sfixes the time “∆t”.

We could apply the same reasoning to a region with spacelikeboundaries. Then, crucially, S and A would not factorize assubspaces of HΣ = Hinitial ⊗Hfinal. Neither would thedecomposition HΣ = Hlin ⊕Hrest.

The fact that we consider a region of spacetime with connectedboundary is useful (to avoid using the Thick Sandwich conjecture)but not essential to avoiding the problem of time.


Relation to recent LQG/SF calculations

Using creation and annihilation operators in Hlin on a “Minkowskivacuum state” ψ0 we get

P(A|S) ∼ |ρM(a†(pin)a†(qout)ψ0)|2 = |ρM(φ(pin)φ(qout)ψ0)|2

In this sense the recent computations of a graviton propagator inLQG/SF could be interpreted as yielding 1-1 graviton scatteringprobabilities.

It is important to remember that |ρM(ψ)|2 for some ψ does not ingeneral have the interpretation of a probability. This is true aboveonly due to special circumstances and only approximately.

The details will depend on how exactly we choose Hlin in HΣ, inwhich way it approximates a Fock space, up to which energies,etc. These ambiguities might be related to the renormalizationambiguities of perturbative quantum gravity.


Conclusions

The probability formula enables us to encode the informationabout “∆t”, thus avoiding the problem of time.

State spaces, amplitudes and probabilities for local regions inspacetime allow us to describe their physics independent of thephysics outside, thus avoiding the quantum cosmology problem.We do not need to invoke such principles as causality, clusterdecomposition etc.

The generalized probability formula may be used to interpretrecent computations of a graviton propagator from LQG/SF interms of scattering probabilities.


Outlook

The general boundary formulation is still work in progress, developby application to known physics (QFT):

I more general QFTs and more general geometriesI derive the S-matrix of QFTI develop suitable quantization prescriptions

Application to LQG/SF:I cast spin foam models in general boundary formI LQG is based on spacelike hypersurfaces, generalize this

Apply to other quantum gravity approaches (e.g. string theory).

Develop understanding of non-standard probabilities, especially innon-perturbative contexts.

Understand implications for interpretational issues, e.g. “collapseof the wavefunction”.


References on the general boundary formulation I

Initial ideas, partial formalism:I R. O., Schrödinger’s cat and the clock: Lessons for quantum

gravity, CQG 20 (2003) 5371-5380, gr-qc/0306007.I R. O., A “general boundary” formulation for quantum mechanics

and quantum gravity, PLB 575 (2003) 318-324, hep-th/0306025.I F. Conrady, L. Doplicher, R. O., C. Rovelli, M. Testa, Minkowski

vacuum in background independent quantum gravity, PRD 69(2004) 064019, gr-qc/0307118.

Probability interpretation, formalism:I R. O., General boundary quantum field theory: Foundations and

probability interpretation, hep-th/0509122.

Application to QFT:I R. O., States on timelike hypersurfaces in quantum field theory,

PLB 622 (2005) 172-177, hep-th/0505267.I R. O., General boundary quantum field theory: Timelike

hypersurfaces in Klein-Gordon theory, PRD 73 (2006) 065017,hep-th/0509123.


References on the general boundary formulation II

I R. O., Two-dimensional quantum Yang-Mills theory with corners,hep-th/0608218.

Generalized Tomonaga-Schwinger quantization:I F. Conrady, C. Rovelli, Generalized Schroedinger equation in

Euclidean field theory, IJMPA 19 (2004) 4037-4068,hep-th/0310246.

I L. Doplicher, Generalized Tomonaga-Schwinger equation from theHadamard formula, PRD 70 (2004) 064037, gr-qc/0405006.

LQG/SF graviton propagator:→ see recent talks by C. Rovelli and S. Speziale and references

therein.


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Quantum gravity, probabilities and general boundariesrelativity.phys.lsu.edu/ilqgs/oeckl101706.pdf · 1 Interpretational problems in quantum gravity The problem of time The quantum