Quantization Of Diffeomorphism Invariant Theories Research Articles

Quantization of Diffeomorphism Invariant Theories of Connections with a Non-Compact Structure Group—an Example

A simple diffeomorphism invariant theory of connections with the non-compact structure group R of real numbers is quantized. The theory is defined on a four-dimensional 'space-time' by an action resembling closely the self-dual Plebanski action for general relativity. The space of quantum states is constructed by means of projective techniques by Kijowski. Except this point the applied quantization procedure is based on Loop Quantum Gravity methods.

Combinatorial Space from Loop Quantum Gravity

The canonical quantization of diffeomorphism invariant theories of connections in terms of loop variables is revisited. Such theories include general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills fields (with or without fermions) coupled to gravity. It is argued that the operators induced by classical diffeomorphism invariant or covariant functions are respectably invariant or covariant under a suitable completion of the diffeomorphism group. The canonical quantization in terms of loop variables described here, yields a representation of the algebra of observables in a separable Hilbert space. Furthermore, the resulting quantum theory is equivalent to a model for diffeomorphism invariant gauge theories which replaces space with a manifestly combinatorial object.

Quantization of diffeomorphism-invariant theories with fermions

We extend ideas developed for the loop representation of quantum gravity to diffeomorphism-invariant gauge theories coupled to fermions. Let P→Σ be a principal G-bundle over space and let F be a vector bundle associated to P whose fiber is a sum of continuous unitary irreducible representations of the compact connected gauge group G, each representation appearing together with its dual. We consider theories whose classical configuration space is A×F, where A is the space of connections on P and F is the space of sections of F, regarded as a collection of Grassmann-valued fermionic fields. We construct the “quantum configuration space” Ā×F̄ as a completion of A×F. Using this, we construct a Hilbert space L2(Ā×F̄) for the quantum theory on which all automorphisms of P act as unitary operators, and determine an explicit “spin network basis” of the subspace L2((Ā×F̄)/Ḡ) consisting of gauge-invariant states. We represent observables constructed from holonomies of the connection along paths together with fermionic fields and their conjugate momenta as operators on L2((Ā×F̄)/Ḡ). We also construct a Hilbert space Hdiff of diffeomorphism-invariant states using the group averaging procedure of Ashtekar, Lewandowski, Marolf, Mourão and Thiemann.

Quantization of diffeomorphism invariant theories of connections with local degrees of freedom

Quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved. The space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions. This provides, in particular, a quantization of the Husain–Kuchař model. The main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity. In the Riemannian case (i.e., signature ++++), the approach appears to contain all the necessary ingredients already. In the Lorentzian case, it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections.