on the light front Sergei Alexandrov Laboratoire Charles Coulomb - - PowerPoint PPT Presentation

First order gravity

• n the light front

Sergei Alexandrov

Laboratoire Charles Coulomb Montpellier work in progress with Simone Speziale

Light Front

Light cone coordinates:

Light Front

Light cone coordinates: Main features:

• Triviality of the vacuum

– energy – momentum

Light Front

Light cone coordinates: Main features:

• Triviality of the vacuum

– energy – momentum

• Non-trivial physics of zero modes

Light Front

Light cone coordinates: Main features:

• Triviality of the vacuum

– energy – momentum

• Non-trivial physics of zero modes
• Importance of boundary conditions at

Light Front

Light cone coordinates: Main features:

• Triviality of the vacuum

– energy – momentum linear in velocities

• Presence of second class constraints
• Non-trivial physics of zero modes
• Importance of boundary conditions at

Null surfaces are natural in gravity (Penrose,…)

Gravity on the light front

null vectors

• Sachs(1962) – constraint free formulation
• conformal metrics on
• intrinsic geometry of
• extrinsic curvature of
• Reisenberger – symplectic structure on

the constraint free data null foliation

Null surfaces are natural in gravity (Penrose,…)

Gravity on the light front

null vectors

• Torre(1986) – canonical formulation in the metric formalism
• Goldberg,Robinson,Soteriou(1991) – canonical formulation

in the complex Ashtekar variables

• Inverno,Vickers(1991) – canonical formulation in the complex

Ashtekar variables adapted to the double null foliation Constraint algebra becomes a Lie algebra

• Sachs(1962) – constraint free formulation
• conformal metrics on
• intrinsic geometry of
• extrinsic curvature of
• Reisenberger – symplectic structure on

the constraint free data null foliation

Null surfaces are natural in gravity (Penrose,…)

Gravity on the light front

null vectors

• Torre(1986) – canonical formulation in the metric formalism
• Goldberg,Robinson,Soteriou(1991) – canonical formulation

in the complex Ashtekar variables

• Inverno,Vickers(1991) – canonical formulation in the complex

Ashtekar variables adapted to the double null foliation Constraint algebra becomes a Lie algebra

• Veneziano et al. (recent) – light-cone averaging in cosmology
• Sachs(1962) – constraint free formulation
• conformal metrics on
• intrinsic geometry of
• extrinsic curvature of
• Reisenberger – symplectic structure on

the constraint free data null foliation

Motivation

Can the light front formulation be useful in quantum gravity (black holes, spin foams…)?

Motivation

– was not analyzed yet

• One needs the real first order formulation
• n the light front

Can the light front formulation be useful in quantum gravity (black holes, spin foams…)?

Motivation

Exact path integral (still to be defined) – was not analyzed yet

• One needs the real first order formulation
• n the light front

Can the light front formulation be useful in quantum gravity (black holes, spin foams…)?

• Can one find constraint free data in the first order formulation?

(preferably without using double null foliation)

Motivation

Exact path integral (still to be defined) – was not analyzed yet

• One needs the real first order formulation
• n the light front
• The issue of zero modes in gravity was not studied yet

Can the light front formulation be useful in quantum gravity (black holes, spin foams…)?

• Can one find constraint free data in the first order formulation?

(preferably without using double null foliation)

Motivation

Exact path integral (still to be defined) – was not analyzed yet

• One needs the real first order formulation
• n the light front
• The issue of zero modes in gravity was not studied yet

Can the light front formulation be useful in quantum gravity (black holes, spin foams…)?

• Can one find constraint free data in the first order formulation?

(preferably without using double null foliation)

• In the first order formalism the null condition can be controlled

by fields in the tangent space

Technical motivation

3+1 decomposition of the tetrad Used in various approaches to quantum gravity (covariant LQG, spin foams…)

Technical motivation

3+1 decomposition of the tetrad Used in various approaches to quantum gravity (covariant LQG, spin foams…) lapse shift spatial metric

Technical motivation

3+1 decomposition of the tetrad determines the nature of the foliation spacelike spacelike lightlike timelike Used in various approaches to quantum gravity (covariant LQG, spin foams…) lapse shift spatial metric

Technical motivation

3+1 decomposition of the tetrad determines the nature of the foliation spacelike spacelike lightlike timelike light front formulation Used in various approaches to quantum gravity (covariant LQG, spin foams…) What happens at ? lapse shift spatial metric

Technical motivation

3+1 decomposition of the tetrad Perform canonical analysis for the real first order formulation of general relativity on a lightlike foliation determines the nature of the foliation spacelike spacelike lightlike timelike light front formulation Used in various approaches to quantum gravity (covariant LQG, spin foams…) What happens at ? lapse shift spatial metric

1. Canonical formulation of field theories on the light front 2. A review of the canonical structure of first order gravity 3. Canonical analysis of first order gravity on the light front 4. The issue of zero modes

Plan of the talk

Massless scalar field in 2d

Solution:

Massless scalar field in 2d

Solution:

Primary constraint Hamiltonian

Light front formulation

Massless scalar field in 2d

Solution:

Primary constraint Hamiltonian

Stability condition: is of second class Light front formulation

Massless scalar field in 2d

Solution:

Primary constraint Hamiltonian

Stability condition: is of second class zero mode first class Identification: Light front formulation

Massless scalar field in 2d

Solution:

Primary constraint Hamiltonian

Stability condition: is of second class zero mode first class Identification:

• the phase space is one-dimensional
• the lost dimension is encoded in the Lagrange multiplier

Conclusions: Light front formulation

Massive theories

One generates the same constraint but different Hamiltonian

Massive theories

One generates the same constraint but different Hamiltonian inhomogeneous equation Stability condition:

Massive theories

One generates the same constraint but different Hamiltonian inhomogeneous equation Stability condition: The existence of the zero mode contradicts to the natural boundary conditions

Massive theories

One generates the same constraint but different Hamiltonian In massive theories the light front constraints do not have first class zero modes inhomogeneous equation Stability condition: The existence of the zero mode contradicts to the natural boundary conditions

Massive theories

One generates the same constraint but different Hamiltonian In massive theories the light front constraints do not have first class zero modes inhomogeneous equation Stability condition: The existence of the zero mode contradicts to the natural boundary conditions In higher dimensions: behave like massive 2d case

Dimensionality of the phase space

On the light front

• dim. phase space = num. of deg. of freedom

Dimensionality of the phase space

On the light front

• dim. phase space = num. of deg. of freedom

Fourier decompositions

Dimensionality of the phase space

On the light front

• dim. phase space = num. of deg. of freedom

Second class constraint Symplectic structure is non-degenerate Dirac bracket Fourier decompositions

First order gravity (spacelike case)

Fix – normal to the foliation

Canonical variables:

First order gravity (spacelike case)

Fix – normal to the foliation

Canonical variables: Linear simplicity constraints

First order gravity (spacelike case)

Fix – normal to the foliation

Canonical variables: Linear simplicity constraints

Hamiltonian is a linear combination of constraints

First order gravity (spacelike case)

Fix – normal to the foliation

Canonical variables: Linear simplicity constraints

Hamiltonian is a linear combination of constraints

3 6

First order gravity (spacelike case)

Fix – normal to the foliation

Canonical variables: Linear simplicity constraints

Hamiltonian is a linear combination of constraints

3 6 secondary constraints

First order gravity (spacelike case)

Fix – normal to the foliation

Canonical variables: Linear simplicity constraints

Hamiltonian is a linear combination of constraints

3 6 secondary constraints 1st class 2d class

First order gravity (spacelike case)

Fix – normal to the foliation

Canonical variables: Linear simplicity constraints

Hamiltonian is a linear combination of constraints

3 6 secondary constraints

• dim. of phase space = 2×18 – 2(3+3+1)-(3+9+6)=4

1st class 2d class

Cartan equations

Cartan equations

Cartan equations

Cartan equations

do not contain time derivatives and Lagrange multipliers

12 constraints

Cartan equations

Cartan equations

do not contain time derivatives and Lagrange multipliers

12 constraints

for fixed do not contain time derivatives

Cartan equations

Cartan equations

do not contain time derivatives and Lagrange multipliers

12 constraints

for fixed do not contain time derivatives

fix 3 components

• f

( – 2d class)

Cartan equations

Cartan equations

do not contain time derivatives and Lagrange multipliers

12 constraints

for fixed do not contain time derivatives

fix 3 components

• f

( – 2d class) fix 2 components

• f and the lapse

Cartan equations

Cartan equations

do not contain time derivatives and Lagrange multipliers

12 constraints

for fixed do not contain time derivatives

We expect that the Hamiltonian constraint becomes second class fix 3 components

• f

( – 2d class) fix 2 components

• f and the lapse

Hamiltonian analysis on the light front

Light front condition Canonical variables: Linear simplicity constraints

Hamiltonian analysis on the light front

Light front condition Canonical variables: Linear simplicity constraints

Hamiltonian analysis on the light front

Light front condition Canonical variables: Linear simplicity constraints

Hamiltonian is a linear combination of constraints where

Hamiltonian analysis on the light front

Light front condition Canonical variables: Linear simplicity constraints 2 7

Hamiltonian is a linear combination of constraints where

Hamiltonian analysis on the light front

Light front condition Canonical variables: Linear simplicity constraints 2 7

Hamiltonian is a linear combination of constraints where

secondary constraints equation fixing the lapse +

Tertiary constraints

The crucial observation: has 2 null eigenvectors

induced metric

• n the foliation

and

Tertiary constraints

The crucial observation: has 2 null eigenvectors

induced metric

• n the foliation

and There are two tertiary constraints

Projector on the null eigenvectors

Tertiary constraints

The crucial observation: has 2 null eigenvectors

induced metric

• n the foliation

and Stabilization procedure stops due to There are two tertiary constraints

Projector on the null eigenvectors

Summary

List of constraints:

Gauss preserving Gauss rotating spatial diffeos Hamiltonian primary simplicity secondary simplicity tertiary

Summary

List of constraints:

Gauss preserving

4

Gauss rotating spatial diffeos Hamiltonian primary simplicity secondary simplicity tertiary

First class Second class 3 2 1 9 6 2

2 1 4 2 2 4

Summary

List of constraints:

Gauss preserving

4

Gauss rotating spatial diffeos Hamiltonian primary simplicity secondary simplicity tertiary

First class Second class 3 2 1 2 Lie algebra

2 1 4 2 2 4

Summary

List of constraints:

Gauss preserving

4

Gauss rotating spatial diffeos Hamiltonian primary simplicity secondary simplicity tertiary

First class Second class 3 2 1 2

• dim. of phase space = 2×18 – 2(4+3)-(2+1+9+6+2)=2

as it should be on the light front Lie algebra

Zero modes

The zero modes of constraints are determined by equations fixing Lagrange multipliers

Zero modes

The zero modes of constraints are determined by equations fixing Lagrange multipliers Potential first class constraints:

Zero modes

The zero modes of constraints are determined by equations fixing Lagrange multipliers Potential first class constraints: homogeneous equations

Zero modes

The zero modes of constraints are determined by equations fixing Lagrange multipliers Potential first class constraints: homogeneous equations

Conjecture

A zero mode can exist only if the corresponding Lagrange multiplier satisfies a homogeneous equation One may expect only two zero modes In our case there are two homogenous equations for and

Conjecture

A zero mode can exist only if the corresponding Lagrange multiplier satisfies a homogeneous equation One may expect only two zero modes In our case there are two homogenous equations for and

Gravity behaves like 2d massless theory

exist

Conjecture

A zero mode can exist only if the corresponding Lagrange multiplier satisfies a homogeneous equation One may expect only two zero modes In our case there are two homogenous equations for and

Gravity behaves like 2d massless theory

exist Initial data on one null hypersurface fix the solution do not exist

Open problems

• Do the zero modes in gravity really exist?

If yes, what is the geometric meaning of the zero modes?

• What are the appropriate boundary conditions along ?
• How do singularities appear in this formalism?
• Can one solve (at least formally) all constraints?
• What is the right symplectic structure (Dirac bracket)?
• Can this formulation be applied to quantum gravity problems?