### The Master constraint program in LQG

#### Posted by Aaron

I’m a little reluctant to post much on the master constraint program because I haven’t read much on it. But I thought I’d post this if others want to comment on the subject.

My initial question is how does the master constraint program work in classical mechanics? In particular, say we are given some symplectic manifold and some set of constraints. The master constraint is$$M={K}^{\mathrm{IJ}}{C}_{I}{C}_{J}.$$Using this, how does one obtain the constrained phase space?

Or is this the wrong question to ask?

Posted at September 2, 2006 5:20 AM UTC
## Re: The Master constraint program in LQG

To formalize the question a bit, let’s review how things normally work. Consider a phase space $M$, and a set of (for simplicity, first-class) constraints, ${C}_{i}$. The construction of the reduced phase space, $\tilde{M}$ can be described either algebraically or geometrically as follows

The geometrical characterization is a little simpler to understand. The algebraic one is closer to what we need to do in quantum mechanics.

But, if you refrain from availing yourself of the ${C}_{i}$ and, instead, work

onlywith $\mathcal{M}={K}^{\mathrm{ij}}{C}_{i}{C}_{j}$, is there any analogue ofeitherof the classical constructions above?If not, why should there be a quantum-mechanical construction, which uses only $\mathcal{M}$ and not the ${C}_{i}$ themselves, which gives the “right” answer?