QFT on quantum spacetime: a compatible classical framework (Tux ...

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

arXiv:1302.3038

**on** **quantum** **spacetime**:

a **compatible** **classical** **framework**

A Dapor, J Lewandowski, J Puchta

University of Warsaw

**Tux**, 28 February 2013

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

1 motivati**on**

2 full theory, full reducti**on**

3 again full

4 c**on**straints up to 1st order (aka no l**on**ger full)

5 physical phase space

outline

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

1 motivati**on**

2 full theory, full reducti**on**

3 again full

4 c**on**straints up to 1st order (aka no l**on**ger full)

5 physical phase space

outline

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the goal:

motivati**on**

**classical** **framework** for **on** **quantum** (cosmological) **spacetime**

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the goal:

motivati**on**

**classical** **framework** for **on** **quantum** (cosmological) **spacetime**

Is the current **framework** not good? Let’s recall it.

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the goal:

motivati**on**

**classical** **framework** for **on** **quantum** (cosmological) **spacetime**

Is the current **framework** not good? Let’s recall it.

Each dynamical varialbe γ (a coordinate in full phase space) is expanded

γ = γ (0) + ɛδγ (1) + 1

2 ɛ2 δγ (2) + ...

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the goal:

motivati**on**

**classical** **framework** for **on** **quantum** (cosmological) **spacetime**

Is the current **framework** not good? Let’s recall it.

Each dynamical varialbe γ (a coordinate in full phase space) is expanded

This is a ”dynamical expansi**on**”:

γ = γ (0) + ɛδγ (1) + 1

2 ɛ2 δγ (2) + ...

• γ (0) is soluti**on** of the 0th order equati**on**s (wait for it)

• δγ (1) is the 1st order correcti**on**, so γ (0) + ɛδγ (1) is soluti**on** to 1st order equati**on**s

• and so **on**

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the goal:

motivati**on**

**classical** **framework** for **on** **quantum** (cosmological) **spacetime**

Is the current **framework** not good? Let’s recall it.

Each dynamical varialbe γ (a coordinate in full phase space) is expanded

This is a ”dynamical expansi**on**”:

γ = γ (0) + ɛδγ (1) + 1

2 ɛ2 δγ (2) + ...

• γ (0) is soluti**on** of the 0th order equati**on**s (wait for it)

• δγ (1) is the 1st order correcti**on**, so γ (0) + ɛδγ (1) is soluti**on** to 1st order equati**on**s

• and so **on**

⇒ the kinematics is messed up: if I expand around homogeneous isotropic ST q (0)

ab ,

then part of δq (1)

ab

will be correcti**on** to the homogeneous isotropic sector, so that

{δq (1)

ab , πcd (0)

} 0

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Two possibilities to avoid this problem:

motivati**on**

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Two possibilities to avoid this problem:

1 fix the background **on**ce and for all

motivati**on**

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Two possibilities to avoid this problem:

1 fix the background **on**ce and for all

2 forget about the ”dynamical expansi**on**” given above

motivati**on**

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Two possibilities to avoid this problem:

1 fix the background **on**ce and for all

2 forget about the ”dynamical expansi**on**” given above

motivati**on**

first possibility: what is d**on**e in standard cosmological perturbati**on** theory

1a Fix FRW soluti**on** as a **classical** curved **spacetime**

1b Perturbati**on**s δγ (1) are c**on**sidered as the **on**ly dynamical variables

1c **on** curved **spacetime** c**on**structi**on** is employed

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Two possibilities to avoid this problem:

1 fix the background **on**ce and for all

2 forget about the ”dynamical expansi**on**” given above

motivati**on**

first possibility: what is d**on**e in standard cosmological perturbati**on** theory

1a Fix FRW soluti**on** as a **classical** curved **spacetime**

1b Perturbati**on**s δγ (1) are c**on**sidered as the **on**ly dynamical variables

1c **on** curved **spacetime** c**on**structi**on** is employed

sec**on**d possibility: not an expansi**on**, but an exact splitting

γ = γ (0) + δγ

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Two possibilities to avoid this problem:

1 fix the background **on**ce and for all

2 forget about the ”dynamical expansi**on**” given above

motivati**on**

first possibility: what is d**on**e in standard cosmological perturbati**on** theory

1a Fix FRW soluti**on** as a **classical** curved **spacetime**

1b Perturbati**on**s δγ (1) are c**on**sidered as the **on**ly dynamical variables

1c **on** curved **spacetime** c**on**structi**on** is employed

sec**on**d possibility: not an expansi**on**, but an exact splitting

This is a ”kinematical splitting”:

γ = γ (0) + δγ

• γ (0) is the homogeneous isotropic comp**on**ent (not a soluti**on** to 0th equati**on**s)

• δγ is the rest (inhomogeneous and anisotropic degrees of freedom)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Two possibilities to avoid this problem:

1 fix the background **on**ce and for all

2 forget about the ”dynamical expansi**on**” given above

motivati**on**

first possibility: what is d**on**e in standard cosmological perturbati**on** theory

1a Fix FRW soluti**on** as a **classical** curved **spacetime**

1b Perturbati**on**s δγ (1) are c**on**sidered as the **on**ly dynamical variables

1c **on** curved **spacetime** c**on**structi**on** is employed

sec**on**d possibility: not an expansi**on**, but an exact splitting

This is a ”kinematical splitting”:

γ = γ (0) + δγ

• γ (0) is the homogeneous isotropic comp**on**ent (not a soluti**on** to 0th equati**on**s)

• δγ is the rest (inhomogeneous and anisotropic degrees of freedom)

⇒ valid **on** the full phase space: it simply gives rise to a coordinate system **on** it.

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

motivati**on**

We will follow the sec**on**d soluti**on**, as it treats homogeneous and inhomogeneous

dof’s **on** the same footing:

you agree it’s a better **framework** for quantizing perturbati**on**s and background

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

1 motivati**on**

2 full theory, full reducti**on**

3 again full

4 c**on**straints up to 1st order (aka no l**on**ger full)

5 physical phase space

outline

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

full theory

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Acti**on**:

where κ = 8πG.

S =

full theory

d 4 x √

1 1

−g R −

2κ 2 gµν∂µT∂νT − 1

2 gµν

∂µφ∂νφ

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Acti**on**:

S =

where κ = 8πG. Three sectors:

full theory

d 4 x √

1 1

−g R −

2κ 2 gµν∂µT∂νT − 1

2 gµν

∂µφ∂νφ

• the geometric (G) sector, associated to the metric gµν

• the time (T) sector, a K-G ”clock field” T

• the matter (M) sector, a K-G ”matter field” φ

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Acti**on**:

S =

where κ = 8πG. Three sectors:

full theory

d 4 x √

1 1

−g R −

2κ 2 gµν∂µT∂νT − 1

2 gµν

∂µφ∂νφ

• the geometric (G) sector, associated to the metric gµν

• the time (T) sector, a K-G ”clock field” T

• the matter (M) sector, a K-G ”matter field” φ

Hamilt**on**ian analysis (ADM): kinematical phase space Γ = ΓG × ΓT × ΓM with

{qab(x), π cd (y)} = δ (c

a δ d)

b δ(3) (x, y), {T(x), pT(y)} = δ (3) (x, y), {φ(x), πφ(y)} = δ (3) (x, y)

and c**on**straints

⎧

C =

⎪⎨

2κ

√

q

πabπ ab − 1

+ 1

2 √ q p2 T +

2 (qabπab ) 2

√

q

−

2κ R(3) +

√

q

2 qab∂aT∂bT + 1

2 √ q π2 φ +

⎪⎩

Ca = −2qac∇bπbc + pT∂aT + πφ∂aφ

√ q

2 qab ∂aφ∂bφ

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

full reducti**on**: kinematics

We perform here the full reducti**on** to homogeneous and isotropic sector. This is

d**on**e to introduce notati**on**, and to show how not to do in our **framework**.

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

full reducti**on**: kinematics

We perform here the full reducti**on** to homogeneous and isotropic sector. This is

d**on**e to introduce notati**on**, and to show how not to do in our **framework**.

Cauchy surface Σ = T 3 , fiducial coordinates (x a ) ∈ [0, 1) 3 .

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

full reducti**on**: kinematics

We perform here the full reducti**on** to homogeneous and isotropic sector. This is

d**on**e to introduce notati**on**, and to show how not to do in our **framework**.

Cauchy surface Σ = T 3 , fiducial coordinates (x a ) ∈ [0, 1) 3 . We c**on**sider the sector

where

Γ (0) = Γ (0)

G

× Γ(0)

T × Γ(0)

M ⊂ ΓG × ΓT × ΓM

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

full reducti**on**: kinematics

We perform here the full reducti**on** to homogeneous and isotropic sector. This is

d**on**e to introduce notati**on**, and to show how not to do in our **framework**.

Cauchy surface Σ = T 3 , fiducial coordinates (x a ) ∈ [0, 1) 3 . We c**on**sider the sector

where

• Γ (0)

G

Γ (0) = Γ (0)

G

c**on**sists of (q(0)

ab , πab

(0)

q (0)

× Γ(0)

T × Γ(0)

M ⊂ ΓG × ΓT × ΓM

) with

ab (x) = e2αδab, π ab

−2α παe

(0) (x) =

6

δ ab

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

full reducti**on**: kinematics

We perform here the full reducti**on** to homogeneous and isotropic sector. This is

d**on**e to introduce notati**on**, and to show how not to do in our **framework**.

Cauchy surface Σ = T 3 , fiducial coordinates (x a ) ∈ [0, 1) 3 . We c**on**sider the sector

where

• Γ (0)

G

Γ (0) = Γ (0)

G

c**on**sists of (q(0)

ab , πab

(0)

q (0)

× Γ(0)

T × Γ(0)

M ⊂ ΓG × ΓT × ΓM

) with

ab (x) = e2αδab, π ab

−2α παe

(0) (x) =

6

• Γ (0)

T is parametrized by hom. scalar field and its momentum, (T(0) , p (0)

T ).

δ ab

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

full reducti**on**: kinematics

We perform here the full reducti**on** to homogeneous and isotropic sector. This is

d**on**e to introduce notati**on**, and to show how not to do in our **framework**.

Cauchy surface Σ = T 3 , fiducial coordinates (x a ) ∈ [0, 1) 3 . We c**on**sider the sector

where

• Γ (0)

G

Γ (0) = Γ (0)

G

c**on**sists of (q(0)

ab , πab

(0)

q (0)

× Γ(0)

T × Γ(0)

M ⊂ ΓG × ΓT × ΓM

) with

ab (x) = e2αδab, π ab

−2α παe

(0) (x) =

6

• Γ (0)

T is parametrized by hom. scalar field and its momentum, (T(0) , p (0)

T ).

• φ is test field, i.e. at 0th order vanishes: φ (0) = 0, π (0)

φ

δ ab

= 0 ⇒ Γ(0)

M

= (0, 0).

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

full reducti**on**: dynamics

Only n**on**-vanishing c**on**straint is the homogeneous part of C:

C (0)

(N) = d 3 xN(x)C (0) (x) = e −3α

1

2 (p(0)

T )2 − κ

12 π2

α d 3 xN(x)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

full reducti**on**: dynamics

Only n**on**-vanishing c**on**straint is the homogeneous part of C:

C (0)

(N) = d 3 xN(x)C (0) (x) = e −3α

1

2 (p(0)

T )2 − κ

12 π2

α d 3 xN(x)

The intersecti**on** Γ (0) ∩ ΓC c**on**sists of points representing ST of the FRW type

g (0)

µν dx µ dx ν = −dt 2 + e 2α(t) δabdx a dx b

filled with a homogeneous scalar field T = T (0) (t).

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

full reducti**on**: dynamics

Only n**on**-vanishing c**on**straint is the homogeneous part of C:

C (0)

(N) = d 3 xN(x)C (0) (x) = e −3α

1

2 (p(0)

T )2 − κ

12 π2

α d 3 xN(x)

The intersecti**on** Γ (0) ∩ ΓC c**on**sists of points representing ST of the FRW type

g (0)

µν dx µ dx ν = −dt 2 + e 2α(t) δabdx a dx b

filled with a homogeneous scalar field T = T (0) (t).

Hamilt**on** eq. with C (0) (0) ⇒ t-dependence:

⎧

⎪⎨

⎪⎩

˙α = − κ

6 e−3α πα

˙πα = 3

2 e−3α (p (0)

T )2 − κ

4 e−3απ2 α

˙T (0) = e−3αp (0)

T

˙p (0)

T = 0

In other words: g (0)

µν satisfies Einstein eq sourced by a homogeneous field T (0) , which

satisfies K-G eq **on** a **spacetime** of the FRW type g (0)

µν .

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

1 motivati**on**

2 full theory, full reducti**on**

3 again full

4 c**on**straints up to 1st order (aka no l**on**ger full)

5 physical phase space

outline

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

In our **framework**, we c**on**sider the full phase space, with coordinates defined as

⎧

1

α = 2

⎪⎨

⎪⎩

ln

1

δab

3 Σ d3

xqab

πα = 2e2α

δab Σ d3xπab T (0)

=

Σ d3

xT

=

p (0)

T

Σ d3 xpT

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

In our **framework**, we c**on**sider the full phase space, with coordinates defined as

⎧

1

α = 2

⎪⎨

⎪⎩

ln

1

δab

3 Σ d3

xqab

πα = 2e2α

δab Σ d3xπab T (0)

=

Σ d3

xT

=

and

p (0)

T

Σ d3 xpT

⎧

δqab(x) = qab(x) − e

⎪⎨

⎪⎩

2αδab δπab (x) = πab (x) − πα

6 e−2αδab δT(x) = T(x) − T (0)

δpT(x) = pT(x) − p (0)

T

δφ(x) = φ(x)

δπφ(x) = πφ(x)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

In our **framework**, we c**on**sider the full phase space, with coordinates defined as

⎧

1

α = 2

⎪⎨

⎪⎩

ln

1

δab

3 Σ d3

xqab

πα = 2e2α

δab Σ d3xπab T (0)

=

Σ d3

xT

=

and

In this coordinates we:

p (0)

T

Σ d3 xpT

⎧

δqab(x) = qab(x) − e

⎪⎨

⎪⎩

2αδab δπab (x) = πab (x) − πα

6 e−2αδab δT(x) = T(x) − T (0)

δpT(x) = pT(x) − p (0)

T

δφ(x) = φ(x)

δπφ(x) = πφ(x)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

In our **framework**, we c**on**sider the full phase space, with coordinates defined as

⎧

1

α = 2

⎪⎨

⎪⎩

ln

1

δab

3 Σ d3

xqab

πα = 2e2α

δab Σ d3xπab T (0)

=

Σ d3

xT

=

and

In this coordinates we:

p (0)

T

Σ d3 xpT

⎧

δqab(x) = qab(x) − e

⎪⎨

⎪⎩

2αδab δπab (x) = πab (x) − πα

6 e−2αδab δT(x) = T(x) − T (0)

δpT(x) = pT(x) − p (0)

T

δφ(x) = φ(x)

δπφ(x) = πφ(x)

1 solve the full c**on**straints of the theory

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

In our **framework**, we c**on**sider the full phase space, with coordinates defined as

⎧

1

α = 2

⎪⎨

⎪⎩

ln

1

δab

3 Σ d3

xqab

πα = 2e2α

δab Σ d3xπab T (0)

=

Σ d3

xT

=

and

In this coordinates we:

p (0)

T

Σ d3 xpT

⎧

δqab(x) = qab(x) − e

⎪⎨

⎪⎩

2αδab δπab (x) = πab (x) − πα

6 e−2αδab δT(x) = T(x) − T (0)

δpT(x) = pT(x) − p (0)

T

δφ(x) = φ(x)

δπφ(x) = πφ(x)

1 solve the full c**on**straints of the theory

2 reduce to the c**on**straint surface ΓC

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

In our **framework**, we c**on**sider the full phase space, with coordinates defined as

⎧

1

α = 2

⎪⎨

⎪⎩

ln

1

δab

3 Σ d3

xqab

πα = 2e2α

δab Σ d3xπab T (0)

=

Σ d3

xT

=

and

In this coordinates we:

p (0)

T

Σ d3 xpT

⎧

δqab(x) = qab(x) − e

⎪⎨

⎪⎩

2αδab δπab (x) = πab (x) − πα

6 e−2αδab δT(x) = T(x) − T (0)

δpT(x) = pT(x) − p (0)

T

δφ(x) = φ(x)

δπφ(x) = πφ(x)

1 solve the full c**on**straints of the theory

2 reduce to the c**on**straint surface ΓC

3 study the gauge transformati**on**s generated by the c**on**straints and restrict to ΓP

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

In our **framework**, we c**on**sider the full phase space, with coordinates defined as

⎧

1

α = 2

⎪⎨

⎪⎩

ln

1

δab

3 Σ d3

xqab

πα = 2e2α

δab Σ d3xπab T (0)

=

Σ d3

xT

=

and

In this coordinates we:

p (0)

T

Σ d3 xpT

⎧

δqab(x) = qab(x) − e

⎪⎨

⎪⎩

2αδab δπab (x) = πab (x) − πα

6 e−2αδab δT(x) = T(x) − T (0)

δpT(x) = pT(x) − p (0)

T

δφ(x) = φ(x)

δπφ(x) = πφ(x)

1 solve the full c**on**straints of the theory

2 reduce to the c**on**straint surface ΓC

3 study the gauge transformati**on**s generated by the c**on**straints and restrict to ΓP

4 c**on**struct the algebra of basic Dirac observables

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

In our **framework**, we c**on**sider the full phase space, with coordinates defined as

⎧

1

α = 2

⎪⎨

⎪⎩

ln

1

δab

3 Σ d3

xqab

πα = 2e2α

δab Σ d3xπab T (0)

=

Σ d3

xT

=

and

In this coordinates we:

p (0)

T

Σ d3 xpT

⎧

δqab(x) = qab(x) − e

⎪⎨

⎪⎩

2αδab δπab (x) = πab (x) − πα

6 e−2αδab δT(x) = T(x) − T (0)

δpT(x) = pT(x) − p (0)

T

δφ(x) = φ(x)

δπφ(x) = πφ(x)

1 solve the full c**on**straints of the theory

2 reduce to the c**on**straint surface ΓC

3 study the gauge transformati**on**s generated by the c**on**straints and restrict to ΓP

4 c**on**struct the algebra of basic Dirac observables

5 study the 1-dimensi**on**al group of automorphisms **on** it parametrized by T (0)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

In our **framework**, we c**on**sider the full phase space, with coordinates defined as

⎧

1

α = 2

⎪⎨

⎪⎩

ln

1

δab

3 Σ d3

xqab

πα = 2e2α

δab Σ d3xπab T (0)

=

Σ d3

xT

=

and

In this coordinates we:

p (0)

T

Σ d3 xpT

⎧

δqab(x) = qab(x) − e

⎪⎨

⎪⎩

2αδab δπab (x) = πab (x) − πα

6 e−2αδab δT(x) = T(x) − T (0)

δpT(x) = pT(x) − p (0)

T

δφ(x) = φ(x)

δπφ(x) = πφ(x)

1 solve the full c**on**straints of the theory

2 reduce to the c**on**straint surface ΓC

3 study the gauge transformati**on**s generated by the c**on**straints and restrict to ΓP

4 c**on**struct the algebra of basic Dirac observables

5 study the 1-dimensi**on**al group of automorphisms **on** it parametrized by T (0)

6 find the physical Hamilt**on**ian (generator of such group)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

the programme

In our **framework**, we c**on**sider the full phase space, with coordinates defined as

⎧

1

α = 2

⎪⎨

⎪⎩

ln

1

δab

3 Σ d3

xqab

πα = 2e2α

δab Σ d3xπab T (0)

=

Σ d3

xT

=

and

In this coordinates we:

p (0)

T

Σ d3 xpT

⎧

δqab(x) = qab(x) − e

⎪⎨

⎪⎩

2αδab δπab (x) = πab (x) − πα

6 e−2αδab δT(x) = T(x) − T (0)

δpT(x) = pT(x) − p (0)

T

δφ(x) = φ(x)

δπφ(x) = πφ(x)

1 solve the full c**on**straints of the theory

2 reduce to the c**on**straint surface ΓC

3 study the gauge transformati**on**s generated by the c**on**straints and restrict to ΓP

4 c**on**struct the algebra of basic Dirac observables

5 study the 1-dimensi**on**al group of automorphisms **on** it parametrized by T (0)

6 find the physical Hamilt**on**ian (generator of such group)

7 go home and find a real job

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

except that...

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

... we do it up to linear order.

except that...

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

... we do it up to linear order.

except that...

More precisely: in order to see explicit expressi**on**s, we implement the can**on**ical

programme expanding the full c**on**straints in δγ and keeping track of the

results of the calculati**on**s up to the 1st order **on**ly.

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

... we do it up to linear order.

except that...

More precisely: in order to see explicit expressi**on**s, we implement the can**on**ical

programme expanding the full c**on**straints in δγ and keeping track of the

results of the calculati**on**s up to the 1st order **on**ly.

⇒ background variables α, πα, T (0) , p (0)

T

standard **framework**.

do obey the 0th order dynamics, as in the

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

... we do it up to linear order.

except that...

More precisely: in order to see explicit expressi**on**s, we implement the can**on**ical

programme expanding the full c**on**straints in δγ and keeping track of the

results of the calculati**on**s up to the 1st order **on**ly.

⇒ background variables α, πα, T (0) , p (0)

T

standard **framework**.

However, our **framework**

• allows to c**on**tinue to higher orders

do obey the 0th order dynamics, as in the

• shows that the usual gauge-invariant dof’s are observables **on**ly up to 1st order

and have n**on**-trivial Poiss**on** algebra with the background

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

first thing to do: Fourier-transform

Flying over the subtleties, we associate to each field γ(x) a real Fourier transformed

field ˘γ(k) where k ∈ L = (2πZ) 3 .

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

first thing to do: Fourier-transform

Flying over the subtleties, we associate to each field γ(x) a real Fourier transformed

field ˘γ(k) where k ∈ L = (2πZ) 3 .

k = 0 comp**on**ents comprise the homogeneous dof’s:

˘T(0) = T (0) , ˘pT(0) = p (0)

T

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

first thing to do: Fourier-transform

Flying over the subtleties, we associate to each field γ(x) a real Fourier transformed

field ˘γ(k) where k ∈ L = (2πZ) 3 .

k = 0 comp**on**ents comprise the homogeneous dof’s:

˘T(0) = T (0) , ˘pT(0) = p (0)

T

As for the metric dof’s, we separate the isotropic part from the rest:

such that δ ab δ˘qab(0) = δabδ˘π ab (0) = 0.

α, πα, δ˘qab(0), δ˘π ab (0)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

first thing to do: Fourier-transform

Flying over the subtleties, we associate to each field γ(x) a real Fourier transformed

field ˘γ(k) where k ∈ L = (2πZ) 3 .

k = 0 comp**on**ents comprise the homogeneous dof’s:

˘T(0) = T (0) , ˘pT(0) = p (0)

T

As for the metric dof’s, we separate the isotropic part from the rest:

α, πα, δ˘qab(0), δ˘π ab (0)

such that δabδ˘qab(0) = δabδ˘π ab (0) = 0.

k 0 comp**on**ents of δ˘qab and δ˘π ab are traceless symmetric 3 × 3 matrices:

expanding them **on** the basis {Am ab } with m = 1, 2, ..., 6, we get

qm(k) := A ab

m (k)δ˘qab(k), p m (k) := A m

ab (k)δ˘πab (k)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

first thing to do: Fourier-transform

Flying over the subtleties, we associate to each field γ(x) a real Fourier transformed

field ˘γ(k) where k ∈ L = (2πZ) 3 .

k = 0 comp**on**ents comprise the homogeneous dof’s:

˘T(0) = T (0) , ˘pT(0) = p (0)

T

As for the metric dof’s, we separate the isotropic part from the rest:

α, πα, δ˘qab(0), δ˘π ab (0)

such that δabδ˘qab(0) = δabδ˘π ab (0) = 0.

k 0 comp**on**ents of δ˘qab and δ˘π ab are traceless symmetric 3 × 3 matrices:

expanding them **on** the basis {Am ab } with m = 1, 2, ..., 6, we get

qm(k) := A ab

m (k)δ˘qab(k), p m (k) := A m

ab (k)δ˘πab (k)

Can**on**ical Poiss**on** algebra induces the following algebra **on** the new variables:

{α, πα} = 1, {T (0) , p (0)

T } = 1, {δ˘qab(0), δ˘π cd (0)} = δc (aδd 1

b) − 3 δcdδab {qm(k), p n (k ′ )} = δ n mδk,k ′, {δ ˘T(k), δ˘pT(k ′ )} = δk,k ′, {δ ˘φ(k), δ˘πφ(k ′ )} = δk,k ′

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

1 motivati**on**

2 full theory, full reducti**on**

3 again full

4 c**on**straints up to 1st order (aka no l**on**ger full)

5 physical phase space

outline

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

linearized c**on**straints

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

linearized c**on**straints

Replace the splitting γ = γ (0) + δγ in the c**on**straints, and expand for small δγ:

C(x) = C (0) (x) + C (1) (x) + C (2) (x) + ..., Ca(x) = C (0)

a (x) + C (1)

a (x) + C (2)

a (x) + ...

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

linearized c**on**straints

Replace the splitting γ = γ (0) + δγ in the c**on**straints, and expand for small δγ:

C(x) = C (0) (x) + C (1) (x) + C (2) (x) + ..., Ca(x) = C (0)

a (x) + C (1)

a (x) + C (2)

a (x) + ...

Now, make a real Fourier-transform of the c**on**straints. Using some facts, we get

˘C(0) = C (0) + d3xC (2) (x) + O(δγ3 ), ˘Ca(0) = O(δγ2 )

˘C(k) = i(1−sgn(k))/2

3 ik·x −ik·x

√ d x e + sgn(k)e

2

C (1) (x) + O(δγ2 )

˘Ca(k) = i(1−sgn(k))/2

3 ik·x −ik·x

√ d x e + sgn(k)e

2

C (1)

a (x) + O(δγ2 )

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

linearized c**on**straints

Replace the splitting γ = γ (0) + δγ in the c**on**straints, and expand for small δγ:

C(x) = C (0) (x) + C (1) (x) + C (2) (x) + ..., Ca(x) = C (0)

a (x) + C (1)

a (x) + C (2)

a (x) + ...

Now, make a real Fourier-transform of the c**on**straints. Using some facts, we get

˘C(0) = C (0) + d3xC (2) (x) + O(δγ3 ), ˘Ca(0) = O(δγ2 )

˘C(k) = i(1−sgn(k))/2

3 ik·x −ik·x

√ d x e + sgn(k)e

2

C (1) (x) + O(δγ2 )

˘Ca(k) = i(1−sgn(k))/2

3 ik·x −ik·x

√ d x e + sgn(k)e

2

C (1)

a (x) + O(δγ2 )

⇒ at 1st order we **on**ly have

C (0) , E(k) := ˘C (1) (k), M(k) := k a ˘C (1)

a (k), V(k) := v a ˘C (1)

a (k), W(k) := w a ˘C (1)

a (k)

where (k, v, w) form an orthog**on**al basis for the momentum space R 3 .

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Explicitely:

analysis of true dof’s

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Explicitely:

C (0) = e−3α 1

2 (p(0)

T )2 − κ

12 π2

α

E(k) = − 3e−5α

4

κπ 2 α

18

+ (p(0)

T )2

− κπαe−α

3 p1 (k) + e −3α p (0)

T δ˘pT(k)

analysis of true dof’s

q1(k) − e−α

κ k2 q1(k) + e−α

3κ k2 q2(k)−

M(k) = παe−2α

6 q1(k) − 2παe−2α

q2(k) − 9

2e2α

3 p1 (k) − 2e2αp2 (k) + p (0)

T δ ˘T(k)

V(k) = παe−2α

3 q3(k) + 2e 2α p 3 (k)

W(k) = παe−2α

3 q4(k) + 2e 2α p 4 (k)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Explicitely:

C (0) = e−3α 1

2 (p(0)

T )2 − κ

12 π2

α

So:

E(k) = − 3e−5α

4

κπ 2 α

18

+ (p(0)

T )2

− κπαe−α

3 p1 (k) + e −3α p (0)

T δ˘pT(k)

analysis of true dof’s

q1(k) − e−α

κ k2 q1(k) + e−α

3κ k2 q2(k)−

M(k) = παe−2α

6 q1(k) − 2παe−2α

q2(k) − 9

2e2α

3 p1 (k) − 2e2αp2 (k) + p (0)

T δ ˘T(k)

V(k) = παe−2α

3 q3(k) + 2e 2α p 3 (k)

W(k) = παe−2α

3 q4(k) + 2e 2α p 4 (k)

• k = 0 sector (α, πα, T (0) , p (0)

T , δ˘qab(0), δ˘π ab (0)) is c**on**strained by C (0)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Explicitely:

C (0) = e−3α 1

2 (p(0)

T )2 − κ

12 π2

α

So:

E(k) = − 3e−5α

4

κπ 2 α

18

+ (p(0)

T )2

− κπαe−α

3 p1 (k) + e −3α p (0)

T δ˘pT(k)

analysis of true dof’s

q1(k) − e−α

κ k2 q1(k) + e−α

3κ k2 q2(k)−

M(k) = παe−2α

6 q1(k) − 2παe−2α

q2(k) − 9

2e2α

3 p1 (k) − 2e2αp2 (k) + p (0)

T δ ˘T(k)

V(k) = παe−2α

3 q3(k) + 2e 2α p 3 (k)

W(k) = παe−2α

3 q4(k) + 2e 2α p 4 (k)

• k = 0 sector (α, πα, T (0) , p (0)

T , δ˘qab(0), δ˘π ab (0)) is c**on**strained by C (0)

• E and M c**on**straint the scalar sector (δ ˘T, δ˘pT, δ ˘φ, δ˘πφ, q1, p 1 , q2, p 2 )

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Explicitely:

C (0) = e−3α 1

2 (p(0)

T )2 − κ

12 π2

α

So:

E(k) = − 3e−5α

4

κπ 2 α

18

+ (p(0)

T )2

− κπαe−α

3 p1 (k) + e −3α p (0)

T δ˘pT(k)

analysis of true dof’s

q1(k) − e−α

κ k2 q1(k) + e−α

3κ k2 q2(k)−

M(k) = παe−2α

6 q1(k) − 2παe−2α

q2(k) − 9

2e2α

3 p1 (k) − 2e2αp2 (k) + p (0)

T δ ˘T(k)

V(k) = παe−2α

3 q3(k) + 2e 2α p 3 (k)

W(k) = παe−2α

3 q4(k) + 2e 2α p 4 (k)

• k = 0 sector (α, πα, T (0) , p (0)

T , δ˘qab(0), δ˘π ab (0)) is c**on**strained by C (0)

• E and M c**on**straint the scalar sector (δ ˘T, δ˘pT, δ ˘φ, δ˘πφ, q1, p 1 , q2, p 2 )

• V and W c**on**straint the vector sector (q3, p 3 , q4, p 4 )

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Explicitely:

C (0) = e−3α 1

2 (p(0)

T )2 − κ

12 π2

α

So:

E(k) = − 3e−5α

4

κπ 2 α

18

+ (p(0)

T )2

− κπαe−α

3 p1 (k) + e −3α p (0)

T δ˘pT(k)

analysis of true dof’s

q1(k) − e−α

κ k2 q1(k) + e−α

3κ k2 q2(k)−

M(k) = παe−2α

6 q1(k) − 2παe−2α

q2(k) − 9

2e2α

3 p1 (k) − 2e2αp2 (k) + p (0)

T δ ˘T(k)

V(k) = παe−2α

3 q3(k) + 2e 2α p 3 (k)

W(k) = παe−2α

3 q4(k) + 2e 2α p 4 (k)

• k = 0 sector (α, πα, T (0) , p (0)

T , δ˘qab(0), δ˘π ab (0)) is c**on**strained by C (0)

• E and M c**on**straint the scalar sector (δ ˘T, δ˘pT, δ ˘φ, δ˘πφ, q1, p 1 , q2, p 2 )

• V and W c**on**straint the vector sector (q3, p 3 , q4, p 4 )

• the tensor sector (q5, p 5 , q6, p 6 ) is unc**on**strained

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

gauge-fixing

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Example:

C (0) = O(δγ 2 ) ⇒ p (0)

T

gauge-fixing

= ±

κ

6 πα + O(δγ 2 )

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Example:

C (0) = O(δγ 2 ) ⇒ p (0)

T

G-fixing: T (0) − τ = 0 for τ ∈ R.

gauge-fixing

κ

= ±

6 πα + O(δγ 2 )

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Example:

C (0) = O(δγ 2 ) ⇒ p (0)

T

G-fixing: T (0) − τ = 0 for τ ∈ R.

Similarly:

gauge-fixing

κ

= ±

6 πα + O(δγ 2 )

E(k) ≈ 0, M(k) ≈ 0 ⇒ p 1 = p 1 (γfree), p 2 = p 2 (γfree)

V(k) ≈ 0, W(k) ≈ 0 ⇒ p 3 = p 3 (γfree), p 4 = p 4 (γfree)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Example:

C (0) = O(δγ 2 ) ⇒ p (0)

T

G-fixing: T (0) − τ = 0 for τ ∈ R.

Similarly:

gauge-fixing

κ

= ±

6 πα + O(δγ 2 )

E(k) ≈ 0, M(k) ≈ 0 ⇒ p 1 = p 1 (γfree), p 2 = p 2 (γfree)

V(k) ≈ 0, W(k) ≈ 0 ⇒ p 3 = p 3 (γfree), p 4 = p 4 (γfree)

G-fixing: q1 = q2 = q3 = q4 = 0.

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Example:

C (0) = O(δγ 2 ) ⇒ p (0)

T

G-fixing: T (0) − τ = 0 for τ ∈ R.

Similarly:

gauge-fixing

κ

= ±

6 πα + O(δγ 2 )

E(k) ≈ 0, M(k) ≈ 0 ⇒ p 1 = p 1 (γfree), p 2 = p 2 (γfree)

V(k) ≈ 0, W(k) ≈ 0 ⇒ p 3 = p 3 (γfree), p 4 = p 4 (γfree)

G-fixing: q1 = q2 = q3 = q4 = 0.

So we have a τ-dependent embedding of the physical phase space ΓP in ΓC ⊂ Γ:

ΓP → Γ τ P ⊂ ΓC ⊂ Γ

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Example:

C (0) = O(δγ 2 ) ⇒ p (0)

T

G-fixing: T (0) − τ = 0 for τ ∈ R.

Similarly:

gauge-fixing

κ

= ±

6 πα + O(δγ 2 )

E(k) ≈ 0, M(k) ≈ 0 ⇒ p 1 = p 1 (γfree), p 2 = p 2 (γfree)

V(k) ≈ 0, W(k) ≈ 0 ⇒ p 3 = p 3 (γfree), p 4 = p 4 (γfree)

G-fixing: q1 = q2 = q3 = q4 = 0.

So we have a τ-dependent embedding of the physical phase space ΓP in ΓC ⊂ Γ:

Γ τ P

is parametrized by

ΓP → Γ τ P ⊂ ΓC ⊂ Γ

(γfree) = α, πα, δ˘qab(0), δ˘π ab (0), δ ˘T(k), δ˘pT(k), δ ˘φ(k), δ˘πφ(k), q5(k), p 5 (k), q6(k), p 6 (k)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Example:

C (0) = O(δγ 2 ) ⇒ p (0)

T

G-fixing: T (0) − τ = 0 for τ ∈ R.

Similarly:

gauge-fixing

κ

= ±

6 πα + O(δγ 2 )

E(k) ≈ 0, M(k) ≈ 0 ⇒ p 1 = p 1 (γfree), p 2 = p 2 (γfree)

V(k) ≈ 0, W(k) ≈ 0 ⇒ p 3 = p 3 (γfree), p 4 = p 4 (γfree)

G-fixing: q1 = q2 = q3 = q4 = 0.

So we have a τ-dependent embedding of the physical phase space ΓP in ΓC ⊂ Γ:

Γ τ P

is parametrized by

ΓP → Γ τ P ⊂ ΓC ⊂ Γ

(γfree) = α, πα, δ˘qab(0), δ˘π ab (0), δ ˘T(k), δ˘pT(k), δ ˘φ(k), δ˘πφ(k), q5(k), p 5 (k), q6(k), p 6 (k)

simplicity of g-fixings ⇒ symplectic structure reduced to Γτ P is simple:

{α, πα} = 1, {δ˘qab(0), δ˘π cd (0)} = δc (aδd 1

b) − 3 δcdδab, {δ ˘φ(k), δ˘πφ(k ′ )} = δk,k ′

{q5(k), p 5 (k ′ )} = δk,k ′, {q6(k), p 6 (k ′ )} = δk,k ′, {δ ˘T(k), δ˘pT(k ′ )} = δk,k ′

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

1 motivati**on**

2 full theory, full reducti**on**

3 again full

4 c**on**straints up to 1st order (aka no l**on**ger full)

5 physical phase space

outline

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

observables and dynamics

Pull-back (γfree) from Γ τ P ⊂ Γ to the physical phase space ΓP al**on**g the τ-dependent

embedding

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

observables and dynamics

Pull-back (γfree) from Γ τ P ⊂ Γ to the physical phase space ΓP al**on**g the τ-dependent

embedding

⇒ τ-dependent coordinates (γ τ

free ) for ΓP, satisfying can**on**ical Poiss**on** algebra

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

observables and dynamics

Pull-back (γfree) from Γ τ P ⊂ Γ to the physical phase space ΓP al**on**g the τ-dependent

embedding

⇒ τ-dependent coordinates (γ τ

free ) for ΓP, satisfying can**on**ical Poiss**on** algebra

Purpose of dynamics: find a functi**on** **on** ΓP that generates this change

d

dτ γτ

free = γ τ

free , hτ

P

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

observables and dynamics

Pull-back (γfree) from Γ τ P ⊂ Γ to the physical phase space ΓP al**on**g the τ-dependent

embedding

⇒ τ-dependent coordinates (γ τ

free ) for ΓP, satisfying can**on**ical Poiss**on** algebra

Purpose of dynamics: find a functi**on** **on** ΓP that generates this change

d

dτ γτ

free = γ τ

free , hτ

P

We call such a functi**on** hτ P the physical Hamilt**on**ian.

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

observables and dynamics

Pull-back (γfree) from Γ τ P ⊂ Γ to the physical phase space ΓP al**on**g the τ-dependent

embedding

⇒ τ-dependent coordinates (γ τ

free ) for ΓP, satisfying can**on**ical Poiss**on** algebra

Purpose of dynamics: find a functi**on** **on** ΓP that generates this change

d

dτ γτ

free = γ τ

free , hτ

P

We call such a functi**on** hτ P the physical Hamilt**on**ian.

Relati**on**al observables: f → Of . In particular Oγ τ free

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

observables and dynamics

Pull-back (γfree) from Γ τ P ⊂ Γ to the physical phase space ΓP al**on**g the τ-dependent

embedding

⇒ τ-dependent coordinates (γ τ

free ) for ΓP, satisfying can**on**ical Poiss**on** algebra

Purpose of dynamics: find a functi**on** **on** ΓP that generates this change

d

dτ γτ

free = γ τ

free , hτ

P

We call such a functi**on** hτ P the physical Hamilt**on**ian.

Relati**on**al observables: f → Of . In particular Oγ τ free

If H = p (0)

T − ˜h, then

d

dτ Oγτ ∂

= −

free ∂T (0) Oγτ free = −{Oγτ , p(0)

free T } = −{Oγτ free , ˜h} = −{Oγτ , OhP }

free

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

where

and H = p (0)

T − ˜h.

observables and dynamics

d

dτ Oγ τ free = −O{γ τ free ,hP}

hP(γ τ

free ) = ˜h γ τ

free ; qn = 0, T (0) = τ; p n = p n (γ τ

free ), p(0)

T = p(0)

T (γτ free )

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

where

and H = p (0)

T − ˜h.

observables and dynamics

d

dτ Oγ τ free = −O{γ τ free ,hP}

hP(γ τ

free ) = ˜h γ τ

free ; qn = 0, T (0) = τ; p n = p n (γ τ

free ), p(0)

T = p(0)

T (γτ free )

H = d 3 x [N(x)C(x) + N a (x)Ca(x)] with C(x) = (pT(x) + h(x))(pT(x) − h(x))/2 √ q

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

where

and H = p (0)

T − ˜h.

observables and dynamics

d

dτ Oγ τ free = −O{γ τ free ,hP}

hP(γ τ

free ) = ˜h γ τ

free ; qn = 0, T (0) = τ; p n = p n (γ τ

free ), p(0)

T = p(0)

T (γτ free )

H = d3x [N(x)C(x) + Na (x)Ca(x)] with C(x) = (pT(x) + h(x))(pT(x) − h(x))/2 √ q

⇒ N(x) = 2 q(x)/(pT(x) + h(x)) and Na (x) = 0 gives

H = d 3

xpT(x) − d 3 xh(x) = p (0)

T − ˜h

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Explicitely:

observables and dynamics

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Explicitely:

hP = Hhom + Hk=0 +

k0,m=5,6

observables and dynamics

H G

m,k +

k0

H T

k +

k

H M

k

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Explicitely:

with

Hhom =

H G

H T

m,k = −

k = −

H M

k = −

hP = Hhom + Hk=0 +

κ(π τ α) 2

6

6

κ(πτ α) 2 2κe4ατ

6

κ(πτ α) 2

1

2

6

κ(πτ α) 2

k0,m=5,6

p m (k) τ + πτ αe −4ατ

12

observables and dynamics

H G

m,k +

k0

H T

k +

k

H M

k

2 τ qm(k) + 1

κ(πτ α)

2

2e−4ατ 12

δ˘pT(k) τ − κπτα 2 δ ˘T(k) τ2 + 1

2 e4ατk2

(δ ˘T(k) τ ) 2

1

2 (δ˘π(k)τ ) 2 + 1

2 e4ατk2

(δ ˘φ(k) τ ) 2

can be thought of as various Hamilt**on**ians for the different sectors.

+ k2

(qm(k) 4κ

τ ) 2

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

comment **on** M-S variables

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Define

comment **on** M-S variables

Q(k) := δ ˘T(k), P(k) := δ˘pT(k) − κπα

2 δ ˘T(k)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Define

comment **on** M-S variables

Q(k) := δ ˘T(k), P(k) := δ˘pT(k) − κπα

2 δ ˘T(k)

In terms of these variables, HT k looks like

H T

k

= −

6

κ(π τ α) 2

1

2 P(k)2 + 1

2 e4ατk

2 Q(k) 2

Q and P are nothing but M-S variables: commute with linearized c**on**straints

E, M, V, W ⇒ are called the gauge-invariant dof’s of the scalar sector.

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Define

comment **on** M-S variables

Q(k) := δ ˘T(k), P(k) := δ˘pT(k) − κπα

2 δ ˘T(k)

In terms of these variables, HT k looks like

H T

k

= −

6

κ(π τ α) 2

1

2 P(k)2 + 1

2 e4ατk

2 Q(k) 2

Q and P are nothing but M-S variables: commute with linearized c**on**straints

E, M, V, W ⇒ are called the gauge-invariant dof’s of the scalar sector.

But:

• as so**on** as you c**on**sider higher orders, they stop being D-obs

• can**on**ical pair **on**ly restricted to perturbati**on**s phase space:

{α, P(k)} = − κ Q(k) 0

2

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Define

comment **on** M-S variables

Q(k) := δ ˘T(k), P(k) := δ˘pT(k) − κπα

2 δ ˘T(k)

In terms of these variables, HT k looks like

H T

k

= −

6

κ(π τ α) 2

1

2 P(k)2 + 1

2 e4ατk

2 Q(k) 2

Q and P are nothing but M-S variables: commute with linearized c**on**straints

E, M, V, W ⇒ are called the gauge-invariant dof’s of the scalar sector.

But:

• as so**on** as you c**on**sider higher orders, they stop being D-obs

• can**on**ical pair **on**ly restricted to perturbati**on**s phase space:

{α, P(k)} = − κ Q(k) 0

2

nice dynamics ⇒ n**on**-can**on**ical kinematics, nice kinematics ⇒ n**on**-h.o. dynamics

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

c**on**clusi**on**s

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

c**on**clusi**on**s

• new **framework** for cosmological perturbati**on**s

• at least as good as the standard **on**e in the **classical** background case

• arguably better in the **quantum** background case

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

c**on**clusi**on**s

• new **framework** for cosmological perturbati**on**s

• at least as good as the standard **on**e in the **classical** background case

• arguably better in the **quantum** background case

• next steps:

• quantizati**on** of perturbati**on**s and background

• generalizati**on** to other matter fields (inflat**on**, maxwell, ...)

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

c**on**clusi**on**s

• new **framework** for cosmological perturbati**on**s

• at least as good as the standard **on**e in the **classical** background case

• arguably better in the **quantum** background case

• next steps:

• quantizati**on** of perturbati**on**s and background

• generalizati**on** to other matter fields (inflat**on**, maxwell, ...)

• more technical points:

• role of the 2nd order diffeomorphism c**on**straint (relati**on** to global symmetries?)

• meaning of the gauge-fixing q1 = q2 = q3 = q4 = 0

**on**

**quantum**

**spacetime**

AD, JL, JP

motivati**on**

full theory,

full reducti**on**

again full

c**on**straints to

1st order

physical

phase space

Vielen Dank!