manifestly gauge -- invariant relativistic perturbation...
TRANSCRIPT
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
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & 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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
Relational Formalism
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 )
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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|
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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 ′
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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’
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Problem of Time in GRRelational formalismClocks for GRBrown-Kuchar-MechanismPhysical HamiltonianSummary: Classical Setup
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Specialisation to FRW spacetimesLinear Cosmological Perturbation Theory
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)
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Specialisation to FRW spacetimesLinear Cosmological Perturbation Theory
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Specialisation to FRW spacetimesLinear Cosmological Perturbation Theory
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Specialisation to FRW spacetimesLinear Cosmological Perturbation Theory
Linear Perturbation around FRW
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Specialisation to FRW spacetimesLinear Cosmological Perturbation Theory
Linear Perturbation around FRW
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Specialisation to FRW spacetimesLinear Cosmological Perturbation Theory
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Specialisation to FRW spacetimesLinear Cosmological 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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Specialisation to FRW spacetimesLinear Cosmological Perturbation Theory
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Specialisation to FRW spacetimesLinear Cosmological Perturbation Theory
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Specialisation to FRW spacetimesLinear Cosmological Perturbation Theory
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Reduced Phase Space QuantisationQuantisation in LQGQuantisation in AQG
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Reduced Phase Space QuantisationQuantisation in LQGQuantisation in AQG
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
Conclusions & Outlook
Conclusions & Outlook
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory
Observables in General RelativityApplication to Cosmology
QuantisationConclusions & Outlook
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
Kristina Giesel Manifestly Gauge – Invariant Relativistic Perturbation Theory