The document discusses the Hamiltonian constraint and its potential deformations related to the hypersurface-deformation algebra in general relativity. It explores the implications of these deformations for the formulation of deformed special relativity, loop quantum gravity, and the interplay between classical and quantum descriptions of spacetime. The analysis includes mathematical formulations and possible modifications to accommodate matter fields and various symmetries of the theory.

1. The Hamiltonian constraint and its possible
deformations
O.M. Lecian2
July 18, 2015
2
Sapienza University of Rome, Physics Department and ICRA,
Piazzale Aldo Moro,5- 00185 Rome, Italy
O.M. Lecian3
The Hamiltonian constraint and its possible deformations

2. Summary
• Hypersurface-deformation algebra for GR
• Deformations of the hypersurface-deformation algebra
• Composition of Hamiltonian constraints
• Deformed Special Relativity, Loop Quantum Gravity and
Quantum Deformations of General Relativity
O.M. Lecian4
The Hamiltonian constraint and its possible deformations

3. Hypersurface-deformation algebra
Lie algebra of the Hamiltonian constraint H and of the
diffeomorphism constraint D evaluated on suitable test functionals,
in general M(xµ) ∈ Cn, as
{ D[Ma
], D[Nb
]} = D[La
Nb
]
{ H[M], D[Na
]} = H[La
N]
{ H[M], H[N]} = D[hab
(M∂bN − N∂bM)]
L Lie derivative
hab spatial metric
O.M. Lecian5
The Hamiltonian constraint and its possible deformations

4. • the presence of matter fields outlines the symmetry groups,
other than those characterizing a vacuum solution for the
Einstein field equations, under which the theory must be
invariant
• once general covariance is ensured, any modification of the
action of three-dimensional transformations and of four
dimensional ones is therefore directly connected to
modifications of the structure of the spacetime
O.M. Lecian6
The Hamiltonian constraint and its possible deformations

5. Deformed hypersurface-deformation algebra
{ D[Ma
], D[Nb
]} = D[La
Nb
]
{ H[M], D[Na
]} = H[La
N]
{ H[M], H[N]} = βD[hab
(M∂bN − N∂bM)]
β generic function
O.M. Lecian7
The Hamiltonian constraint and its possible deformations

6. Deformed Special Relativity
• in Deformed-special relativity, a curved momentum space
renders the complete phase space of a Minkowski
4-dimensional spacetime non-trivial
• any such modifications must be ensured by both a Poincar´e
algebra for the classical properties of the spacetime, and a
Heisenberg algebra, for the quantum description
O.M. Lecian8
The Hamiltonian constraint and its possible deformations

7. A modification for the Dirac hypersurface-deformation algebra was
proposed with the aim of investigating the compatibility the
implementation of a proposal for a curved-momentum-space theory
on curved spacetime, so-called Deformed General Relativity, and
Loop Quantum Gravity in 9, with β a generic function of the
extrinsic curvature for perturbed FRW universes. Indeed, with β a
function of the spatial metric as well, an investigation of less
symmetric universes is possible.
9
M. Bojowald, G. M. Paily, Deformed General Relativity, Phys. Rev. D 87
(2013) 044044
O.M. Lecian10
The Hamiltonian constraint and its possible deformations

8. Deformed Hamiltonian Constraints
Deformed Hamiltonian constraints resulting from this deformations
of the Hypersurface deformation algebra are the same order in β
{ H[M]H[N], H[P]} ∼ βH[Q]. (1)
but defined on a different (with respect to the LQG generalized
connections by which the directional derivative is calculated) test
functional Q, Q ∈ Cn−2
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The Hamiltonian constraint and its possible deformations

9. Such deformations of the Hamiltonian constraint do not imply, at
this order, any (conformal) deformations of the Hamiltonian itself.
the Poisson brackets between the smeared Hamiltonian and
diffeomorphism constraints of the theory have always the form,
independently form the definition of β.
The composition of Hamiltonian constraints defines the (proper)
time at which a diffeomorphism is generated, i.e. the choice of a
particular hypersurface of the (ADM) foliation to which the
transformation generated by the constraint is orthogonal
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The Hamiltonian constraint and its possible deformations

10. Minkowskian limit
The action of a composition of diffeomorphsisms, both
three-dimensional and four-dimensional, allow one to express the
hypothesized non-commutativity of the spacetime coordinates of
Deformed General Relativity, for which the expression is further
complicated by the choice of a suitable factor ordering, due to the
presence of directional derivatives and of the three-metric
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The Hamiltonian constraint and its possible deformations

11. Above the Planck scale: the quantum counterpart is also
investigated in its 1 + 1 dimensional version, where the action of
the spatial position operator is defined as the dimensionally-reduced
version of the action of a ’spherically-symmetric’ position operator
The physical settings and the mathematical mechanisms able to
reduce or eliminate the quantum λ effects should be regarded to as
suitable semiclassical scenarios.
Pµ, Xν = iηµν + iλP0ηµν + O(λ2)
The Heisenberg algebra able to reconduct the deformed
Poincar´algebra to the Heisenberg algebra of a model exhibiting
non-trivial structures in the momentum space has to be analyzed
with respect to its Minkowskian limit above the Planck scale
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The Hamiltonian constraint and its possible deformations

12. References
- M. Bojowald, G. M. Paily, Deformed General Relativity, Phys. Rev. D
87 (2013) 044044;
- D. Kovacevic, S. Meljanac, A. Pachol and R. Strajn, Generalized
Poincare algebras, Hopf algebras and kappa-Minkowski spacetime, Phys.
Lett. B 711 (2012) 122 [arXiv:1202.3305 [hepth]];
J. Mielczarek, Loop-deformed Poincar´algebra, EPL 108 40003
(2014)arXiv:1304.2208 [gr-qc];
- G. Amelino-Camelia, L. Cesarini, O.M. Lecian, in preparation.
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The Hamiltonian constraint and its possible deformations