March 23, 1993

This Week's Finds in Mathematical Physics (Week 11)

John Baez

I'm hitting the road again tomorrow and will be going to the Quantum Topology conference in Kansas until Sunday, so I thought I'd post this week's finds early. As a result they'll be pretty brief. Let me start with one that I mentioned in week9 but is now easier to get:

1) Unique determination of an inner product by adjointness relations in the algebra of quantum observables, by Alan D. Rendall, 10 pages, now available as gr-qc/9303026.

and then mention another thing I've gotten as a spinoff from the gravity conference at UCSB:

2) An algebraic approach to the quantization of constrained systems: finite dimensional examples, by Ranjeet S. Tate, Syracuse University physics department PhD dissertation, August 1992, SU-GP-92/8-1. (Tate is now at, but please don't ask him for copies unless you're pretty serious, because it's big.)

Both the technical problems of "canonical" quantum gravity and one of the main conceptual problems - the problem of time - stem from the fact that general relativity is a system in which the initial data have constraints. So improving our understanding of quantizing constrained classical systems is important in understanding quantum gravity.

Let me say a few words about these constraints and what I mean by "canonical" quantum gravity.

First consider the wave equation in 2 dimensions. This is an equation for a function from R^2 to R, say φ(t,x), where t is a timelike and x is a spacelike coordinate. The equation is simply

		d^2 φ/dt^2 - d^2φ/dx^2 = 0.

Now this equation can be rewritten as an evolutionary equation for initial data as follows. We consider pairs of functions (Q,P) on R - which we think of φ and dφ/dt on "space", that is, on a surface t = constant. And we rewrite the second-order equation above as a first-order equation:

              d/dt (Q,P) = (P, d^2Q/dx^2).			1)

This is a standard trick. We call the space of pairs (Q,P) the "phase space" of the theory. In canonical quantization, we treat this a lot like the space R^2 of pairs (q,p) describing the initial position and momentum of a particle. Note that for a harmonic oscillator we have an equation a whole lot like 1):

		  d/dt (q,p) = (p, -q).  

This is why when we quantize the wave equation it's a whole lot like the harmonic oscillator.

Now in general relativity things are similar but more complicated. The analog of the pairs (φ, dφ/dt) are pairs (Q,P) where Q is the metric on spacetime restricted to a spacelike hypersurface - that is, the "metric on space at a given time" - and P is concocted from the extrinsic curvature of that hypersurface as it sits in spacetime. Now the name of the game is to turn Einstein's equation for the metric into a first-order equation sort of like 1). The problem is, in general relativity there is no god-given notion of time. So we need to pick a "lapse function" on our hypersurface, and a "shift vector field" on our hypersurface, which say how we want to push our hypersurface forwards in time. The lapse function says at each point how much we push it in the normal direction, while the shift vector field says at each point how much we push it in some tangential direction. These are utterly arbitrary and give us complete flexibility in how we want to push the hypersurface forwards. Even if spacetime was flat, we could push the hypersurface forwards in a dull way like:

			-------------------- new
			____________________ old

or in a screwy way like

 			 /    \   /\
			/      ---  \
                                     ------- new
			____________________ old

Of course, in general relativity spacetime is usually not flat, which makes it ultimately impossible to decide what counts as a "dull way" and what counts as a "screwy way," which is why we simply allow all possible ways.

Anyway, having chosen a lapse function and shift vector field, we can rewrite Einstein's equations as an evolutionary equation. This is a bit of a mess, and it's called the ADM (Arnowitt-Deser-Misner) formalism. Schematically, it goes like

              d/dt (Q,P) = (stuff, stuff').			2)

where both "stuff" and "stuff'" depend on both Q and P in a pretty complex way.

But there is a catch. While the evolutionary equations are equivalent to 6 of Einstein's equations (Einstein's equation for general relativity is really 10 scalar equations packed into one tensor equation), there are 4 more of Einstein's equations which turn into constraints on Q and P. 1 of these constraints is called the Hamiltonian constraint and is closely related to the lapse function; the other 3 are called the momentum or diffeomorphism constraints and are closely related to the shift vector field.

For those of you who know Hamiltonian mechanics, the reason why the Hamiltonian constraint is called what it is is that we can write it as

		H(Q,P) = 0

for some combination of Q and P, and this H(Q,P) acts a lot like a Hamiltonian for general relativity in that we can rewrite 2) using the Poisson brackets on the "phase space" of all (Q,P) pairs as

		d/dt Q = {P,H(Q,P)}
		d/dt P = {Q,H(Q,P)}.

The funny thing is that H is not zero on the space of all (Q,P) pairs, so the equations above are nontrivial, but it does vanish on the submanifold of pairs satisfying the constraints, so that, in a sense, "the Hamiltonian of general relativity is zero". But one must be careful in saying this because it can be confusing! It has confused lots of people worrying about the problem of time in quantum gravity, where they naively think "What - the Hamiltonian is zero? That means there's no dynamics at all!"

The problem in quantizing general relativity in the "canonical" approach is largely figuring out what to do with the constraints. It was Dirac who first seriously tackled such problems, but the constraints in general relativity always seemed intractible (when quantizing) until Ashtekar invented his "new variables" for quantum gravity, that all of a sudden make the constraints look a lot simpler. Ashtekar also has certain generalizations of Dirac's general approach to quantizing systems with constraints, and part of what Tate (who was a student of Ashtekar) is doing is to study a number of toy models to see how Ashtekar's ideas work.

I should note that there are lots of other ways to handle problems with constraints, like BRST quantization, that aren't mentioned here at all.

Well, I'm off to Kansas and I hope to return with a bunch of goodies and some gossip about 4-manifold invariants, topological quantum field theories and the like. Lee Smolin will be talking there too so I will try to extract some information about quantum gravity from him.

© 1993 John Baez