Hamiltonian constraint in polymer parametrized field theory
- Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park, Pennsylvania 16802-6300 (United States)
- Raman Research Institute, Bangalore-560 080 (India)
Recently, a generally covariant reformulation of two-dimensional flat spacetime free scalar field theory known as parametrized field theory was quantized using loop quantum gravity (LQG) type ''polymer'' representations. Physical states were constructed, without intermediate regularization structures, by averaging over the group of gauge transformations generated by the constraints, the constraint algebra being a Lie algebra. We consider classically equivalent combinations of these constraints corresponding to a diffeomorphism and a Hamiltonian constraint, which, as in gravity, define a Dirac algebra. Our treatment of the quantum constraints parallels that of LQG and obtains the following results, expected to be of use in the construction of the quantum dynamics of LQG: (i) the (triangulated) Hamiltonian constraint acts only on vertices, its construction involves some of the same ambiguities as in LQG and its action on diffeomorphism invariant states admits a continuum limit, (ii) if the regulating holonomies are in representations tailored to the edge labels of the state, all previously obtained physical states lie in the kernel of the Hamiltonian constraint, (iii) the commutator of two (density weight 1) Hamiltonian constraints as well as the operator correspondent of their classical Poisson bracket converge to zero in the continuum limit defined by diffeomorphism invariant states, and vanish on the Lewandowski-Marolf habitat, (iv) the rescaled density 2 Hamiltonian constraints and their commutator are ill-defined on the Lewandowski-Marolf habitat despite the well-definedness of the operator correspondent of their classical Poisson bracket there, (v) there is a new habitat which supports a nontrivial representation of the Poisson-Lie algebra of density 2 constraints.
- OSTI ID:
- 21503992
- Journal Information:
- Physical Review. D, Particles Fields, Vol. 83, Issue 2; Other Information: DOI: 10.1103/PhysRevD.83.025019; (c) 2011 American Institute of Physics; ISSN 0556-2821
- Country of Publication:
- United States
- Language:
- English
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ALGEBRA
CONSTRUCTION
DENSITY
FIELD THEORIES
GAUGE INVARIANCE
GRAVITATION
HAMILTONIANS
LIE GROUPS
LIMITING VALUES
POLYMERS
QUANTUM GRAVITY
SCALAR FIELDS
SPACE-TIME
TWO-DIMENSIONAL CALCULATIONS
INVARIANCE PRINCIPLES
MATHEMATICAL OPERATORS
MATHEMATICS
PHYSICAL PROPERTIES
QUANTUM FIELD THEORY
QUANTUM OPERATORS
SYMMETRY GROUPS