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The Hamiltonian constraint in Polymer Parametrized Field Theory | | Alok Laddha
; Madhavan Varadarajan
; | | Date: |
10 Nov 2010 | | Abstract: | Recently, a generally covariant reformulation of 2 dimensional flat spacetime
free scalar field theory known as Parameterised 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 (LM)
habitat (iv) the rescaled density 2 Hamiltonian constraints and their
commutator are ill defined on the LM habitat despite the well defined- ness of
the operator correspondent of their classical Poisson bracket there (v) there
is a new habitat which supports a non-trivial representation of the Poisson-
Lie algebra of density 2 constraints | | Source: | arXiv, 1011.2463 | | Services: | Forum | Review | PDF | Favorites | |
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