Manifestly Gauge -- Invariant Relativistic Perturbation Theoryrelativity.phys.lsu.edu/ilqgs/giesel032508.pdfObservables in General Relativity Application to Cosmology Quantisation

Observables in General RelativityApplication to Cosmology

QuantisationConclusions & Outlook

Manifestly Gauge – Invariant Relativistic

Perturbation Theory

Kristina Giesel

Albert – Einstein – InstituteILQGS

25.03.2008

References:K.G., S. Hofmann, T. Thiemann, O.Winkler, arXiv:0711.0115, arXiv:0711.0117

K.G., T. Thiemann, arXiv:0711.0119

Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory



Plan of the Talk

Content

Application of Relational framework to General Relativity

Special Case of Deparametrisation: Two examples

Manifestly gauge-invariant framework for General Relativity

Application to Cosmology (FRW and perturbation around FRW)

Quantisation: Reduced Phase Space Quantisation

Conclusions & Outlook




Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup

Problem of Time in General Relativity

Observables in General Relativity

Observables are by definition gauge invariant quantities

The gauge group of GR is Diff(M)

Canonical picture:

Constraints c,~c generate spatial and ’time’ gauge transformations

O gauge invariant ⇔ {c,O} = {~c,O} = 0

’Hamiltonian’ hcan for Einstein Equations is linear combination ofconstraints and thus constrained to vanish

Consequently: O gauge invariant ⇔ {hcan,O} = 0

Frozen picture, contradicts experiments, problem of time in GR





Relational Formalism

Basic Idea [Bergmann ’60, Rovelli ’90]

Einstein Equations are no physical evolution equations

Rather describe flow of unphysical quantities under gauge transf.

Relational formalism:

Take two gauge variant f , g and choose T := g as a clock

Define gauge invariant extension of f denoted by Ff ,T in relation tovalues T takes

Ff ,T : Values of f when clock T = g takes values 5, 17, 23, 42, ...

Solve αt(T ) = τ for t, then use solution tT (τ) for Ff ,T whichbecomes a function of τ





Relational Formalism: Idea

f , g move along gauge orbit

PSfrag replacements

gauge orbit f gauge orbit g

f (t1)

f (t3)

g(t2)

g(t4)






Basic Idea [Bergmann ’60, Rovelli ’90]

Einstein Equations are no physical evolution equations

Rather describe flow of unphysical quantities under gauge transf.

Relational formalism:

Take two gauge variant f , g and choose T := g as a clock

Define gauge invariant extension of f denoted by Ff ,T in relation tovalues T takes

Ff ,T : Values of f when clock T = g takes values 5, 17, 23, 42, ...

Solve αt(T ) = τ for t, then use solution tT (τ) for Ff ,T whichbecomes a function of τ






Explicit Form for Ff ,T [Dittrich ’04]

Take as many clocks TI as they are CI then Ff ,T (τ) can beexpressed as powers series in T I with coefficients involving multiplePoisson brackets of CI and f .

Explicit form in general quite complicated

But: One has explicit strategy how to construct observables

Analysed in several examples, application to cosmology andcosmological perturbations [Dittrich, Dittrich & Tambornino]

Automorphism property

{Ff ,T (τ),Ff ′,T (τ)} = F{f ,f ′},T (τ),

If f (q, p) then Ff ,T = f (Fq,T ,Fp,T )





Strategy of the Formalism

Steps to obtain EOM for observables

Consider a physical System for instance gravity & some standardmatter

We would like to derive EOM for the observables associated to(qa, p

a) of gravity & matter

Add additional action to the system which become clocks T

We havectot = cgeo + cmatter + cclock =: c + cclock = 0ctota = cgeo

a + cmattera + cclock

a =: ca + cclocka = 0

Construct observables wrt to these constraints: Fqa,T (τ) & Fpa,T (τ)

Construct so called physical Hamiltonian Hphys which generatestrue evolution of Fqa,T (τ), Fpa ,T (τ)





Special Case of Deparametrisation

Steps technically simplify

Deparametrisation: c tot and ctota can be solved for pclock

Expressions for Fqa/pa,T (τ) and Hphys simplify

Note: Hphys is in general different for each chosen clock system

Evolution of observables is generated by Hphys

EOM for observables are clock – dependent

Consider two examples for clarification:

scalar field without potentialk-essence





Scalar field as a Clock (LQC-Model)

Deparametrisation for scalar field φ

Constraints:ctot = c(qa, p

a) + 12λ ( π2

√q

+ qabφ,aφ,b)

ctota = ca(qa, p

a) + πφ,a

Using ctota = 0 we get qabφ,aφ,b = 1/π2qabcacb (more details later)

Using c tot = 0 we get

π =√

| − √qλc −√

q√

λ2c2 − qabcacb| =: hφ(qa, pa)

Equivalent Hamiltonian constraint: c tot = π − hφ(qa, pa)

Construct observables Qa(τφ) := Fqa,φ(τ) and Pa(τφ) := Fpa,φ(τ)

Evolution:Qa(τφ) = {Hphys,Qa(τφ)} and Pa(τφ) = {Hphys,P

a(τφ)}

Hφphys :=

∫

d3σ

√

| −√

QλC −√

Q√

λ2C 2 − QabCaCb|





K-essence (Thiemann ’06)

Deparametrisation for k-essence field ϕ: Case I

Constraints:ctot = c(qa, p

a) −√

[1 + qabϕ,aϕ,b][π2 + α2√q], α > 0ctota = ca(qa, p

a) + πϕ,a

Using ctota = 0 we get again qabϕ,aϕ,b = 1/π2qabcacb

Using c tot = 0 we get π = −hϕ(qa, pa)

hϕ(qa, pa) :=

√

12 (c2 − qabcacb − α2q) +

√

14 (c2 − qabcacb − α2q)2 − α2qabcacbq

Equivalent Hamiltonian constraint: c tot = π + hϕ(qa, pa)

Construct observables Qa(τϕ) := Fqa,ϕ(τ) and Pa(τϕ) := Fpa,ϕ(τ)

Qa(τφ) = {Hphys,Qa(τφ)} and Pa(τϕ) = {Hphys,Pa(τφ)}

Hϕphys :=

∫

d3σhϕ(Qa,Pa)





K-essence (Thiemann ’06)

Deparametrisation for k-essence field ϕ: Case II

Constraints:ctot = c ′(qa, p

a) −√

[1 + qabϕ,aϕ,b][π2 + α2√q], α > 0ctota = c ′

a(qa, pa) + πϕ,a

Using ctota = 0 we get again qabϕ,aϕ,b = 1/π2qabc ′

ac′b

Using c tot = 0 we get π = −h′ϕ(qa, p

a)h′(qa, p

a) :=√

12 ((c ′)2 − qabc ′

ac′b − α2q) +

√

14 ((c ′)2 − qabc ′

ac′b − α2q)2 − α2qabc ′

ac′bq

Equivalent Hamiltonian constraint: c tot = π + h′ϕ(qa, p

a)

Construct observables Qa(τϕ) := Fqa,ϕ(τ) and Pa(τϕ) := Fpa,ϕ(τ)

Qa(τφ) = {Hphys,Qa(τφ)} and Pa(τϕ) = {Hphys,Pa(τφ)}

Hϕphys :=

∫

d3σh′ϕ(Qa,P

a)





Comparison of both physical Hamiltonian

Comparison of Hφphys and H

ϕphys

Physical Hamiltonians: (D2 := QabCaCb), (D′)2 := QabC ′

aC′b)

Hφphys =

∫

d3σ√

| −√

QλC −√

Q√λ2C 2 − D2|

Hϕphys =

∫

d3σ

√

12 ((C ′)2 − (D ′)2 − α2Q) +

√

14 ((C ′)2 − (D ′)2 − α2Q)2 − α(D ′)2Q

Specialise both Hphys to cosmology (FRW – symmetry)

Then D2 = (D ′)2 = 0 and

Hφphys =

∫

d3σ√

| − 2λ√

QCFRW| and Hϕphys =

∫

d3σC ′FRW

Note that CFRW 6= 0 here only C totFRW

= 0





Clocks for General Relativity

Choose Clock and Ruler for GR

Choose clock and ruler to give time & space physical meaning

We need 1 ×∞ clocks and 3 ×∞ rulers: 4 scalar fields

Chosen clocks & rulers such that good for cosmology:

Free falling observerStandard cosmology CFRW as true Hamiltonian





Brown-Kuchar-Mechanism

Dust Lagrangian

Add dust Lagrangian to Gravity & Standard Model

Sdust = −1

2

∫

M

d4X√

| det(g)|ρ(gµνUµUν + 1)

where Uµ = −T,µ + WjSj,µ, ρ energy density

Uµ = gµνUν is a geodesic, fields Wj , Sj are constant along

geodesics, T defines proper time along each geodesic

Tµν of a pressureless perfect fluid

αt(T ) = τ becomes clock, αx (Sj ) = σj becomes ruler

Dust serves as a physical reference system





A few Words on Notation

Canonical (3+1) split of Gravity + Standard Model + Dust

Dust variables time αt(T ) = τ and space αx (Sj ) = σj : Conjugate

momenta P and Pj , j = 1, 2, 3

Remaining gravity & matter degrees of freedom qab, pab and φ, π

are denoted by qa, pa

Gauge variant quantities: Lower case letters qa, pa

Gauge invariant quantities: Capital letters Qa,Pa





Brown-Kuchar-Mechanism

Deparametrisation of the Constraints in GR

Canonical 3+1 split: (P,T ),(Pj ,Sj ) & remaining non dust (pa, q

a)

Detailed constraints analysis, then 1st class constraintsc tot = c + cdust

with cdust = −√

P2 + qab(PT,b + PjSj,b)(PT,b + PjS

j,b)

c tota = ca + cdust

a with cdusta = PT,a + PjS

j,a

Brown-Kuchar-Mechanism:

cdust = −√

P2 + qabcdusta cdust

b

Use c tota = 0 and replace cdust

a by −ca in cdust

Then solve c tot for P and c tota for Pj

Need to assume S j,a is invertible with inverse S a

j





Deparametrisation of the Constraints in GR

(Partial) Deparametrisation of the Constraints in GR

Constraints in (partial) deparametrised form

c tot = P + h with h(pa, qa) :=

√

c2 − qabcacb

c tota = Pj + hj with hj (T , S

j , pa, qa) = Sa

j (ca − hT,a)

c tot , c tota mutually commute

Here Ff ,T simplifies a lot

Construction of Ff ,T in two steps

1.) Reduction wrt to c tota : qab(x , t) −→ qij(σ, t)

2.) Reduction wrt to c tot : qij (σ, t) −→ Qij (σ, τ)





Observables with respect to Dust Clock & Rulers

Space time points are labled by τ and σ j

PSfrag replacements

(σ1=1,σ2=4,σ3=35)

(σ1=8,σ2=0.3,σ3=44)

x

τ proper time on each geodesic

x ′





Construction of Observables

Explicit Form of Observables

1.) Spatial diff’-invariant quantities

qij (σ, t) =∫

d3x | det(∂S(x) ∂x)|δ(S(x), σ)qab(x)Sai (x)Sb

j (x)

local in σ but ultra – non – local in x

2.) Full Observables

Qij (σ, τ) =∞∑

n=0

1n!{h(τ), qij (σ)}(n)

where {f , g}(0) = g , {f , g}(n) := {f , {f , g}(n−1)}}and h(τ) :=

∫

Sd3σ(τ − T (σ))h(σ)

S=range(σ) so called ’dust space’





Physical Hamiltonian for GR

Physical Hamiltonian Hphys

We have a strategy to construct gauge invariant extension for allpa, q

a and get Pi ,Qi

Due to automorphism property of Ff ,T , we can extend this tofunctions of pa, q

a which just become functions of Pi ,Qi

However, we would like to have so called physical HamiltonianHphys for GR that generates evolution of observables

Recall: We cannot use canonical Hamiltonian hcan from Einsteinequations because {hcan,P

i} = {hcan,Qi} = 0

Hphys should itself be gauge invariant

Hphys can be derived from deparametrised constraints





Reduced Phase Space & Physical Hamiltonian

Physical Hamiltonian

We have c tot = P + h(pa, qa) with h =

√

c2 − qabcacb

H(σ, τ) := Fh,T =√

C 2(τ, σ) − Q ij (τ, σ)Ci (σ)Cj (σ)

Physical Hamiltonian is given by Hphys =∫

S d3σH(σ, τ)(S dust space)

Physical Physical time evolution:dFf ,T (σ,τ)

dτ = {Hphys,Ff ,T (σ, τ)}

Symmetries of Hphys: {Hphys,Cj(σ)} = 0, {Hphys,H(σ)} = 0

Hphys no τ dependence: conservative system





Reduced Phase space of Gravity + Scalar field and Dust

Comparison with Unreduced Phase Space

Standard unreduced framework: Gravity & scalar field

Einstein Equations: EOM for qab, pab and matter dof

’Hamiltonian’ hcan =∫

Σ d3x (n(x)c(x) + na(x)ca(x))

Constraints c := cgeo + cmatter = 0 and ca := cgeoa + cmatter

a = 0

Reduced framework with additional Dust: [K.G., Hofmann, Thiemann, Winkler]

Manifestly gauge invariant EOM for Qij , P ij and matter dof

Physical Hamiltonian Hphys =∫

S d3σ√

C 2 − Q ijCiCj (σ)

Energy & momentum conservation H = −ε, Cj = −εjLapse & Shift dynamical: N = C/H ,N j = −Q ijCi/H





Equation of Motion for Unreduced Case

Second Order Time Derivative Equation of Motion for qab

qab =[ n

n− (

√

det(q))˙√

det(q)+

n√

det(q)

(

L~n

√

det(q)

n

)

]

(

qab −(

L~nq)

ab

)

+qcd(

qac −(

L~nq)

ac

)(

qbd −(

L~nq)

bd

)

+qab

[

− n2κ

2√

det(q)C + n2

(

2Λ +κ

2λv(ξ)

)

]

+ n2[κ

λξ,aξ,b − 2Rab

]

+2n(

DaDbn)

+ 2(

L~nq)

ab+

(

L~nq)

ab−

(

L~n

(

L~nq))

ab












a = 0




S d3σ√







Equation of Motion for Reduced Case

Second Order Time Derivative Equation of Motion for Qjk

Qjk =[ N

N− (

√detQ)˙√det Q

+N√

det Q

(

L~N

√detQ

N

)

]

(

Qjk −(

L~NQ)

jk

)

+Qmn(

Qmj −(

L~NQ)

mj

)(

Qnk −(

L~NQ)

nk

)

+Qjk

[

− N2κ

2√

det QC + N2

(

2Λ +κ

2λv(Ξ)

)

]

+ N2[κ

λΞ,jΞ,k − 2Rjk

]

+2N(

DjDkN)

+ 2(

L~NQ)

jk+

(

L~NQ

)

jk−

(

L~N

(

L~NQ))

jk

− NH√det Q

GjkmnNmNn

Qjk refers to derivative with respect to dust time τ here












a = 0




S d3σ√







Equation of Motion for Reduced Case

Second Order Time Derivative Equation of Motion for Qjk

Qjk =[ N

N− (

√detQ)˙√det Q

+N√

det Q

(

L~N

√detQ

N

)

]

(

Qjk −(

L~NQ)

jk

)

+Qmn(

Qmj −(

L~NQ)

mj

)(

Qnk −(

L~NQ)

nk

)

+Qjk

[

− N2κ

2√

det QC + N2

(

2Λ +κ

2λv(Ξ)

)

]

+ N2[κ

λΞ,jΞ,k − 2Rjk

]

+2N(

DjDkN)

+ 2(

L~NQ)

jk+

(

L~NQ

)

jk−

(

L~N

(

L~NQ))

jk

− NH√det Q

GjkmnNmNn




Specialisation to FRW spacetimesLinear Cosmological Perturbation Theory

Application to Cosmology

Apply Manifestly Gauge Invariant Framework to FRW

1.) Specialise Qij equations to FRW spacetime

2.) Consider linear perturbations around FRW spacetime

3.) Compare with standard results and check that dust clocks do notcontradict current experiments





Check Manifestly Gauge Invariant Equations for FRW Case

Standard Framework: FRW Spacetime

ds2 = −dt2 + a(t)2δabdxadxb = a(x0)2ηµνdxµdxν

Metric qab = a2(t)δab, Momenta pab = −2aδab, ca = 0,

FRW eqn from qab = {hcan, qab} and pab = {hcan, pab}, c(q, p) = 0

FRW equation

3 aa

= Λ − κ4 (ρmatter + 3pmatter)

Reduced Framework: FRW Spacetime

FRW equation

3 AA

= Λ− κ4 (ρmatter+ρdust+3pmatter)





Standard Cosmological Perturbation Theory: Lagrange Formalism

Einstein Equations

Gµν + Λgµν = Rµν − 12gµν + Λgµν = κ

2 Tµν

Specialisation to FRW for (gravity + scalar field ξ) with k=0, (-,+,+,+)

G 00 = 3H, H = a′

a, G ab = −(2H′ + H2)δab

T 00 = a2ρ, T ab = a2pδab

FRW – background quantities are indicated by a bar on the top





Linear Perturbation around FRW

Linear Perturbation [Mukhanov, Feldman, Brandenberger 1992]

Consider perturbations δgµν := gµν − gµν , δξ := ξ − ξ

Any F (g , ξ) is expanded up to linear order in δg and δξ

δF denotes linear term in Taylor expansion F (g , ξ) − F (g , ξ)

One obtains equations for δGµν and δTµν

One decomposes these equations into scalar, vector and tensormodes in order to extract physical dof

4 scalar fields φ, ψ,B ,E , two transversal covector fields Sa,Fa and atraceless, symmetric,transversal tensor hab and Z for scalar fieldcontribution






Perturbed metric

δg00 = 2a2φ, δg0a = a2(Sa +B,a), δgab = a2[2(ψδab +E,ab +F(a,b))+hab]

Metric is not invariant under gauge transformations xµ 7→ xµ + uµ

One can construct seven invariants out of the 11 by using B − E ′,Fa in order to compensate gauge shift up to linear order

The seven invariants

Φ = φ− 1a[a(B − E ′)]′, Ψ = ψ + H(B − E ′), Va = Sa − F ′

a, hab,

Z = δξ + ξ′(B − E )

Ten perturbed Einstein equation can be expressed in terms of theseseven invariants






Physical Degrees of Freedom

Four of these equations do not contain second order time derivativeof four of the seven fields

These are constraints −→ Four of the seven can be expressed interms of the other three: Va = 0 and Φ,Z in terms of Ψ

Finally: 3 physical dof: hab,Ψ, for these evolution equations

Usually in standard cosmological perturbation theory, gauge –invariance is constructed order by order

Repeat similar analysis in Hamiltonian framework [Langlois 1994]

Additional aim: Use relational formalism to treat gauge invariancenon – perturbatively





Reduced Phase space of GR with Dust

Manifestly Gauge Invariant Cosmological Perturbation Theory[K.G., Hofmann, Thiemann, Winkler]

We have EOM for Qij and Ξ

Specialise to FRW background: Equation formally agree (A → a)

Consider perturbation around FRW: δQij = Qij − Q ij , δΞ = Ξ − Ξ

δQij and δΞ are automatically gauge invariant

Any power (δQij )n and (δΞ)n will be also gauge invariant!

Interesting for higher order perturbation theory





Reduced Phase space of GR with Dust: Results

Manifestly Gauge Invariant Cosmological Perturbation Theory[K.G., Hofmann, Thiemann, Winkler]

Results for Linear Order Perturbation Theory

Perturbed eqn for δQjk , δΞ agree up to one term which showsinfluence of the dust clock

This has to be expected because we consider a gravitationallyinteracting observer

Not an idealised observer as one has usually in cosmology

Mukhanov et. al: Gravity + scalar field

Here: Gravity + scalar field + dust

Difference in physical degrees of freedom





Comparison MFB and Dust framework

Counting physical degrees of freedom

Start with 15 dof (gravity+scalar field + 4 dust fields)

Lapse function and shift vector are pure gauge: reduction by 4 dof

Hamiltonian & diffeomorphism constraint: reduction by 4 dof

We end up with 7 physical dof

Potentially dangerous, because 4 more than usual might contradictexperiment

Reason: We use dust fields to construct gauge invariant quantities,all components in three metric become physical

Can we still match with the results obtained by Mukhanov, Feldmanand Brandenberger?

We need to show that these additional modes are zero or decay





Comparison MFB and Dust framework

Constants of motion in Dust framework

Energy density H(σ) =: ε(σ) is constant of motion

Momentum density C j (σ) =: −εj (σ) is constant of motion

Perturbations δε, δεj are again constant of motion wrt perturbedHamiltonian

Additional modes decay:

MFB: Constraints

∆Va = 0, f||mom(Ψ,Φ,Z ),a = 0

fenergy(Ψ,Φ,Z ) = 0

Dust: Energy & momentum conservation laws

∆Vj = −κ δε⊥jA2 , f

||mom(Ψ,Φ,Z ),j = κ

4A(− 1

Aδε

||j + ε[B − E ′],j )

fenergy(Ψ,Φ,Z ) = 1A(δε− εg(ψ,B ,E ))

In the limit of vanishing ε, δε and εj , δεj exact agreement





Summary: Application to Cosmology

Comparison with Standard Framework

Background Equations agree formally

Linear cosmological perturbation Theory: Results are in agreementwith the one of Mukhanov et al.

Dust seems to be appropriate clock for cosmological situations

So far everything was purely classical..

Reduced phase space approach also of advantage when quantisationis considered




Reduced Phase Space QuantisationQuantisation in LQGQuantisation in AQG

Why is such a Framework Useful for Quantisation?

Advantages when Quantisation is Considered

Constraints have completely disappeared from the picture

No Constraint – Equations

Constraints have been reduced classically

Only algebra of observables of interest:Includes all physical degrees of freedom

Direct access to physical Hilbert space





Reduced Phase Space Quantisation

Reduced Phase Space Quantisation for LQG [K.G., Thiemann]

Algebra of observables simple

{Qij ,Pkl} ' {qab(x), pcd (y)} = δc

(aδdb)δ

3(x , y)

Easy to find representations of this algebra, even Fock possible

However, apart from algebra representations need to support Hphys

Choose standard LQG representation used for Hkin

Physical Hilbert space where volume spectrum discrete!

Problematic to preserve classical symmetries of Hphys

Recall: {Hphys,H(σ)} = 0, {Hphys,Cj (σ)} = 0

This leads to infinitely number of conservation laws in LQG

Moreover, physical Hilbert space is non-separable





Reduced Phase Space for AQG

Reduced Phase Space Quantisation for AQG [K.G.,Thiemann]

AQG:One fundamental algebraic graph, subgraphs are not preserved

No additional infinitely many conservation laws

Quantisation can be performed using the techniques of AQG

AQG is formulated as (background independent) HamiltonianLattice Gauge Theory

Anomalies of Hphys: Notion of Diff(S) is meaningless

Idea: ”M-like functional”∫

S d3σaH2(σ)+bQ jkCjCk (σ)√

det(Q)

Hphys has no anomalies ⇔ [Hphys, [Hphys,M]]M=0 = 0






Problem of time in GR has been circumvented by dust clocks

Results agree with standard cosmological perturbation theory

Next Step: Second order and quantisation of perturbation

Beyond linear order manifestly gauge – invariant quantities shouldbe of advantage compared to standard framework

Improve (possible) anomaly issue of Hphys

Scattering Theory

Relation of Hphys with SM – Hamiltonian on Minkowski space

Vacuum problem in QFT on curved background




Conclusion & Outlook

Choosing dust as a clock...

One could think of the dust as NIMP-particles (non – interacting –massless particles)

It could be interpreted as the ’gravitational Higgs’

Hope for the future

Extract some physics out of LQG such that working at an interfaceof a fundamental theory & (cosmological) observations becomespossible


Documents

Manifestly Gauge -- Invariant Relativistic Perturbation Theoryrelativity.phys.lsu.edu/ilqgs/giesel032508.pdfObservables in General Relativity Application to Cosmology Quantisation