# ADM formalism in canonical quantum gravity

1. Aug 7, 2010

### tom.stoer

I have a discussion witrh a colleague regarding the ADM formalism and foliation independence.

My argument goes as follows:
once I have chosen one (arbitrary) foliation the theory has two constraints H and D. H ~ 0 guarantuess that physical states do not change in coordinate-time, so physics essentially stays the same along the direction perpendicular to the foliation. Because the initially chosen foliation was arbitrary, the Hilbert space does not depend on the initial foliation and it does not change as the foliation changes in time (nothing changes in time as H ~ 0). Therefore the theory (Wheeler-deWitt, LQG) is foliation-independent.
My argument says that this is similar to the relativistic particle with h = p²-m² ~ 0 which guarantuees that the theory is reparametrization-invariant (parameterization of the world line of the particle). A specific foliation (= a specific parameterization) does not appear in the final theory, therefore the Hilbert space and the whole theory do not depend on it.

His argument is different:
H ~ 0 does not change the foliation at all. Reparameterization changes only the time coordinate t' = f(t) for each 3-space slice, but not the slice itself. Looking at the theory with (g=metric and A = wave functional A[g]) each foliation creates a different sequence of (g,A). These different sequences will lead to different paths in superspace. My argument only involves symmetries along one specific path. Therefore one specific (g,A) can be contained in different paths which describe different physics. The physics is not only in the state but in the paths which may be different, even if one specific state = slice are identical.
He says that my argument with the relativistic particle fails as there is nothing in this picture which corresponds to the foliation.

I am sure that my argument is correct (or at least that my conclusion regarding foliation independence is correct even if the argument itself is flawed). Can anybody explain this from a different point of view?

2. Aug 7, 2010

### sheaf

I didn't really understand this statement. I'm just thinking in terms of the classical picture:

I thought that the ADM formalism was designed to describe "the physics" in terms of the hypersurface data. Once you do that then the time evolution is "just" enforced by the constraints, but isn't said to contribute to the physical degrees of freedom.

I'm trying to understand what your friend is arguing. Is it this:

I have one foliation $$\Sigma_t$$. Suppose it specifies a given hypersurface $$\Sigma_0$$ at t=0.

I then consider another, different foliation $$\Delta_{t}$$ which agrees at t=0
$$\Sigma_{0}$$ = $$\Delta_{0}$$ but is different for t not equal to zero

Then is he claiming that these two foliations lead to different physics ?

3. Aug 7, 2010

### atyy

Long discussion (I understand this very poorly) in 4.4 of Wuthrich's http://philosophy.ucsd.edu/faculty/wuthrich/pub/WuthrichChristianPhD2006Final.pdf [Broken]

Last edited by a moderator: May 4, 2017
4. Aug 22, 2010

### atyy

Another small discussion, apparently not even Noui knows the answer, but like Wuthrich, he doesn't straight away buy the analogy that canonical special relativistic QFT doesn't break Poincare invariance.

"In standard Quantum Field Theories (QFT), this aspect is not problematical, even if we make an explicit choice of a preferred time, because one recovers at the end of the quantization that the Quantum Theory is invariant under the Poincare group. In General Relativity, the situation is more subtle because making a preferred time choice breaks a local symmetry whereas the Poincare symmetry is a global one in standard QFT." http://arxiv.org/abs/1003.6019

BTW, isn't this related to Nicolai's question "By contrast, we will here argue that this ‘on shell closure’ is not sufficient for a full proof of quantum spacetime covariance, but that a proper theory of quantum gravity requires a constraint algebra that closes ‘off shell’, i.e. without prior imposition of a subset of the constraints." http://arxiv.org/abs/hep-th/0501114

To which Thiemann seems to be thinking along interesting old fashioned lines that Wuthrich mentions "Thus Diff(M) is not the group of gauge symmetries in the canonical GR. A direct consequence is that the first-class constraints (the spatial diffeomorphism constraint and the Hamiltonian constraint) generate a constraint algebra which is not a Lie algebra, i.e. the structure functions appears. Therefore the gauge transformations doesn’t form a group in the canonical theory of gravity, since they are generated by the first-class constraints. The collection of the gauge transformations is at most an enveloping algebra, whose generic element is a product of infinitesimal gauge transformations. We refer to this enveloping algebra as the “Bergmann-Komar group” BK(M) [10]. It coincides with Diff(M) only when the equation of motion is imposed. This Bergmann-Komar group essentially determines the dynamical symmetry of canonical GR, while Diff(M) is only the kinematical symmetry of the theory." http://arxiv.org/abs/0911.3436

Last edited: Aug 22, 2010
5. Aug 23, 2010

### tom.stoer

I don't know if these two discussions are related. Regarding the latter one I am with Nicolai. Thiemann has to explain why on-shell closure is sufficient. I don't like his explanation that "it may generate a consistent theory and that's all what counts". But - even in SUGRA - and Nicolai is an expert - derivations regarding local SUSY closure are valid on-shell only (afaik).

Perhaps the starting point of the discussion should be purely classical. In standard field theory one can show the following: Chosing a foliation (Cauchy surface A) plus initial conditions (Cauchy conditions) does not change the physical content of the theory: one can chose a different foliation but describe the same physics. All what one has to do is to propagate the original initial condition (via the e.o.m.) from the first Cauchy surface A to the second one (B).

The main question is whether this is possible in GR (ADM) as well. I guess the answer is "yes", but I don't know how to prove it.

6. Aug 26, 2010

I had promised myself not to spend more time on online discussions, but this is a topic that I have thought quite a bit about, so here goes.

In gravity, a background foliation is about as bad as a background metric. In fact, the foliation is not independent of the metric, since every timeslice must be spacelike, and the notion of spacelikeness involves the metric. This is a problem already classically: we start by fixing a foliation, which is spacelike w.r.t. the metric that we solve for. In particular, I believe that the ADM formalism assumes that spacetime is globally space x time. If the solution to the Einstein equations is a more complicated manifold than that, problems must arise, and quantization can only make the problems worse.

According to an old observation, which goes back to Lagrange, phase space is really a (special-)covariant concept. It can be identified with the space of histories which solve the Euler-Lagrange equations; such a history is uniquely defined by position and velocity at time t = 0. However, also this formuation depends on the metric. Phase space can only be identified with the space of histories to the extent that velocities and momenta can be identified, which again requires a metric: p_u = g_uv v^v. I spent half a decade thinking about covariant canonical quantization, and in the end I concluded that it does not quite work.

AFAIU, the only way to resolve this paradox is with my pet idea: formulate physics in terms of data available to a physical observer. Since the observer always moves along a timelike curve, foliation is trivial.

7. Aug 26, 2010

### tom.stoer

Thanks :-)

You are right. There are some physical reasons why this global topology Time * Space = R * M³ should hold, but of course there are mathematical counter examples.

Yes, but of fourse in quantum gravity there is no observer :-)

The question in classical GR (ADM formalism) is whether it's possible to prove foliation-independence. This requires to propagate initial conditions between arbitrary spacelike 3-manifolds. Afaik this is trivial provided that global hyperbolicity holds which automatically solves the Time * Space = R * M³ issue.

The question in QG (again ADM, regardless whether old-fashioned or Ashtekar variables) is whether it's possible to translate this into an operator algebra plus physical states and show that the algebra has no anomalies.

I doub't that a pure on-shell formalism is sufficient.

8. Aug 26, 2010

I disagree. Every real experiment is done by someone or something, call it an observer or detector or test particle or whatever. If QG is unable to describe any real experiment, we have identified a problem.
Locally, spacelikeness means that g_uv dx^u dx^v < 0. After quantization, this turns into an operator equation. I don't really understand what it means to define the initial hypersurface in terms of operators, but it seems dubious to me.
Anomaly freedom is a subtle issue. Having discovered the Virasoro-like extensions of the diffeomorphism algebra in several dimensions, this is something that is of great interest to me.

9. Aug 26, 2010

### tom.stoer

It is not the case that one defines the 3-manifold or its spacelikeness in terms of operators.

One constructs a classical theory based on spacelike 3-manifolds. On constructs classical objects living on such a manifold and translates them into operators. From now on spacelikeness and manifolds are gone.

My problem is rather simple: the constraint H~0 says that whatever this 3-manifold will look like and whatever we may have constructed to live on it, we could just do that for any other 3-manifold. Does the equation H~0 or any other property of the theory guarantuee that the initial slicing which already deals with a metric is nothing else but the choice of an initial condition?

The equation H~0 in quantum gravity means that the theory is frozen; no propagation, nothing. But classically different slicings were possible and of course they were related by timelike propagation.