M. O. Katanaev

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• M. O. Katanaev
• Steklov Mathematical Institute, Moscow

Dirac (1958) Arnowitt, Deser, Misner (ADM)
(1960) DeWitt (1967) Regge, Teitelboim
(1974) .............................
- metric formulation
Dirac (1962) Schwinger (1963) ....................
.....
- vielbein formulation (time gauge)
Deser, Isham (1976) Nelson, Teiltelboim
(1978) Henneaux (1983) Charap, Nelson
(1986) .........................
- vielbein formulation
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ADM parameterization of the metric
- n-dimensional space-time
- local coordinates
- metric
Pseudo-Riemannian manifold
The rule
- subsets
- ADM parameterization
- lapse function
- shift function
- the inverse to
- one-to-one correspondence
For
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ADM parameterization of the metric (continued)
- time
Theorem. The metric has Lorentzian
signature if and only if the metric
is negative definite.
- is negative definite
- the Hilbert Einstein action
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- ADM parameterization of the metric
- the induced metric on hypersurfaces
- the induced connection
- the internal curvature
- the extrinsic curvature
- normal to a hypersurface
- the trace of extrinsic curvature
here
and
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- the Lagrangian
- primary constraints
- the canonical momenta
- the Hamiltonian density
where
- the Hamiltonian
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- the Hamiltonian
- Poisson brackets
- primary constraints
secondary constraints
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- the Hamiltonian
- phase space variables
- Lagrange multipliers
- constraints
- generator of space diffeomorphisms
Dirac (1951) DeWitt (1967)
where
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- irreducible decomposition
additional constraints
- generating functional depending on new
coordinates and old momenta
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- scalar curvature
A. Peres, Nuovo Cimento (1963)
- polynomial of degree
- totally antisymmetric tensor density
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- Poisson manifold
Basic Poisson brackets
- degenerate
- algebra of the constraints
Submanifold defined by the equations
is the phase space
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- Hamiltonian
- independent variables
- scalar curvature (fifth order)
- quadratic polynomial