# On the relation between rigging inner product and master constraint direct integral decomposition

@article{Han2010OnTR, title={On the relation between rigging inner product and master constraint direct integral decomposition}, author={Muxin Han and Thomas Thiemann}, journal={Journal of Mathematical Physics}, year={2010}, volume={51}, pages={092501-092501} }

Canonical quantization of constrained systems with first-class constraints via Dirac’s operator constraint method proceeds by the theory of Rigged Hilbert spaces, sometimes also called refined algebraic quantization. This method can work when the constraints form a Lie algebra. When the constraints only close with nontrivial structure functions, the Rigging map can no longer be defined. To overcome this obstacle, the master constraint method has been proposed which replaces the individual…

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## References

SHOWING 1-10 OF 60 REFERENCES

On the relation between operator constraint, master constraint, reduced phase space and path integral quantization

- Physics
- 2010

Path integral formulations for gauge theories must start from the canonical formulation in order to obtain the correct measure. A possible avenue to derive it is to start from the reduced phase space…

Quantum spin dynamics: VIII. The master constraint

- Mathematics
- 2006

Recently the master constraint programme (MCP) for loop quantum gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single master constraint. The MCP is…

The Holst Spin Foam Model via Cubulations

- Physics
- 2012

Spin foam models are an attempt for a covariant, or path integral formulation of canonical loop quantum gravity. The construction of such models usually rely on the Plebanski formulation of general…

Canonical Path-Integral Measures for Holst and Plebanski Gravity. II. Gauge Invariance and Physical Inner Product

- Physics
- 2009

This paper serves as a continuation for the discussion in Engle et al (2010, Class. Quantum Grav. 27 245014). We analyze the invariance properties of the gravity path-integral measure derived from…

A Uniqueness theorem for constraint quantization

- Mathematics
- 1999

This work addresses certain ambiguities in the Dirac approach to constrained systems. Specifically, we investigate the space of so-called `rigging maps' associated with refined algebraic…

Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models

- Mathematics
- 2004

This is the third paper in our series of five in which we test the master constraint programme for solving the Hamiltonian constraint in loop quantum gravity. In this work, we analyse models which,…

On the generality of refined algebraic quantization

- Mathematics
- 1999

The Dirac quantization `procedure' for constrained systems is well known to have many subtleties and ambiguities. Within this ill-defined framework, we explore the generality of a particular…

Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework

- Computer Science
- 2004

It is shown that the master constraint programme has a wide range of applicability but that there are many, physically interesting subtleties that must be taken care of in doing so.

Canonical path integral measures for Holst and Plebanski gravity: I. Reduced phase space derivation

- Mathematics
- 2009

An important aspect in defining a path integral quantum theory is the determination of the correct measure. For interacting theories and theories with constraints, this is non-trivial, and is…

Testing the master constraint programme for loop quantum gravity : V. Interacting field theories

- Physics
- 2006

This is the fifth and final paper in our series of five in which we test the master constraint programme for solving the Hamiltonian constraint in loop quantum gravity. Here we consider interacting…