first order gravity on the light front
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First order gravity
on 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 on 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 on 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 on 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 on 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 of
( – 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 of
( – 2d class)
fix 2 components of 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 of
( – 2d class)
fix 2 components of 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
on the foliation
and
Tertiary constraints The crucial observation:
has 2 null eigenvectors
induced metric
on the foliation
and
There are two
tertiary constraints
Projector on the
null eigenvectors
Tertiary constraints The crucial observation:
has 2 null eigenvectors
induced metric
on 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?
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