Quantization of Dynamical Symplectic Reduction

  title={Quantization of Dynamical Symplectic Reduction},
  author={Martin Bojowald and Artur Tsobanjan},
  journal={arXiv: Mathematical Physics},
A long-standing problem in quantum gravity and cosmology is the quantization of systems in which time evolution is generated by a constraint that must vanish on solutions. Here, an algebraic formulation of this problem is presented, together with new structures and results, which show that specific conditions need to be satisfied in order for well-defined evolution to be possible. 
Comment on “Towards a quantum notion of covariance in spherically symmetric loop quantum gravity”
The recent paper [1] by Gambini, Olmedo and Pullin (GOP) promises new constructions that may make it possible to achieve covariance in spherically symmetric models of loop quantum gravity. This claim
Trinity of relational quantum dynamics
The problem of time in quantum gravity calls for a relational solution. Using quantum reduction maps, we establish a previously unknown equivalence between three approaches to relational quantum
Equivalence of Approaches to Relational Quantum Dynamics in Relativistic Settings
We have previously shown that three approaches to relational quantum dynamics—relational Dirac observables, the Page-Wootters formalism and quantum deparametrizations—are equivalent. Here we show
Relational evolution with oscillating clocks
A fundamental description of time can be consistent not only with the usual monotonic behavior but also with a periodic physical clock variable, coupled to the degrees of freedom of a system evolving


In canonical quantization of gravity, the state functional does not seem to depend on time. This hampers the physical interpretation of quantum gravity. I critically examine ten major attempts to
Generalized Hamiltonian Dynamics
  • P. Dirac
  • Physics
    Canadian Journal of Mathematics
  • 1950
The author’s procedure for passing from the Lagrangian to the Hamiltonian when the momenta are not independent functions of the velocities is put into a simpler and more practical form, the main
Quantization of a Friedmann universe filled with a scalar field
The problem of quantizing a Robertson-Walker metric with a scalar field as source is discussed within the framework of a true canonical theory. A discussion is given of the role played by different
Magnetic charge and non-associative algebras
We consider the possibility that the quantum mechanics of a nonrelativistic electron in the magnetic field of a magnetic charge distribution can be described in terms of a non-associative algebra of
Time in quantum gravity: An hypothesis.
  • Rovelli
  • Physics
    Physical review. D, Particles and fields
  • 1991
It is shown that a natural extension of canonical Heisenberg-picture quantum mechanics is well defined and can be used to describe the "non-Schr\"odinger regime," in which a fundamental time variable is not defined.
Three-cocycle in mathematics and physics.
  • Jackiw
  • Mathematics
    Physical review letters
  • 1985
It is shown that the three-cocycle arises when a representation of a transformation group is nonassociative, so that the Jacobi identity fails. A physical setting is given: When the translation group
Deformation Quantization of Poisson Manifolds
I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the