Talk presented at the Quantum Gravity Meeting Rome 2015 20-23 July 2015. Sapienza University of Rome, Italy

1. Deformations of the Hamiltonian Constraints
in
General Relativity
O.M. Lecian2
July 29, 2015
2
Sapienza University of Rome, Physics Department and ICRA,
Piazzale Aldo Moro,5- 00185 Rome, Italy
O.M. Lecian3
Deformations of the Hamiltonian Constraints in General Relativ

2. Summary
• Hypersurface-deformation algebra for General Relativity
• Deformations of the hypersurface-deformation algebra
• Composition of constraints
• Minkowskian Limit: Deformed Special Relativity, Loop
Quantum Gravity and Quantum Deformations of General
Relativity
• Phenomenological implications
O.M. Lecian4
Deformations of the Hamiltonian Constraints in General Relativ

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
Deformations of the Hamiltonian Constraints in General Relativ

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
Deformations of the Hamiltonian Constraints in General Relativ

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
Deformations of the Hamiltonian Constraints in General Relativ

6. A modification for the Dirac hypersurface-deformation algebra was
proposed possibly 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 P phenomenological low-energy limits of quantum effects in
curved spacetime 8, 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.
8
such as LQG, cfr, i.e., M. Bojowald, G. M. Paily, Deformed General
Relativity, Phys. Rev. D 87 (2013) 044044
O.M. Lecian9
Deformations of the Hamiltonian Constraints in General Relativ

7. 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[LQ]. (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−1
O.M. Lecian10
Deformations of the Hamiltonian Constraints in General Relativ

8. other deformed Poisson brackets might have a different form, for
example
{ H[M], H[N]D[P]} = Dβ[Q]H + HD[P] (2)
nevertheless with an analogous physical interpretation:
O.M. Lecian11
Deformations of the Hamiltonian Constraints in General Relativ

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
O.M. Lecian12
Deformations of the Hamiltonian Constraints in General Relativ

10. Minkowskian limit
The action of a composition of diffeomorphsisms, both
three-dimensional and four-dimensional, allow one to express the
possible implications on the commutativity of spacetime
coordinates for deformations of 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
O.M. Lecian13
Deformations of the Hamiltonian Constraints in General Relativ

11. Deformations of 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. Lecian14
Deformations of the Hamiltonian Constraints in General Relativ

12. Phenomenological implications
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
O.M. Lecian15
Deformations of the Hamiltonian Constraints in General Relativ

13. References
- J. Mielczarek, Loop-deformed Poincar´algebra, EPL 108 40003
(2014)arXiv:1304.2208 [gr-qc]
- O.M. Lecian, L. Cesarini, G. Amelino-Camelia, M. Arzano, in
preparation
- 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]]
O.M. Lecian16
Deformations of the Hamiltonian Constraints in General Relativ