## Wednesday, May 13, 2009

### Diffeomorphism equivalence and permutation equivalence

I've written a little (4-page) paper about diffeomorphism equivalence and permutation equivalence. It is on my university homepage, here. The argument, though a bit technical, is quite short. A diffeomorphism of a manifold M to itself is a permutation of the points that is an automorphism. A spacetime S is something of the form (M, g, phi_1, ..., phi_n). Suppose h : M -> M is a diffeomorphism of M (i.e., an automorphism). Then we can drag all other fields along by h, and obtain another spacetime, call it h[S]. By construction, S and h[S] are isomorphic. The "gauge equivalence" claim is that S and h[S] are "physically equivalent" solutions. However, a diffeomorphism h is just a special case of permuting the elements of the base set of the manifold. And one can in principle use any permutation one likes. Given a spacetime S and a permutation p, one can drag the topology along too, as well as the metric and other matter fields. Then one gets a spacetime p[S] isomorphic to the first, by construction. The equivalence claim now is that S and p[S] are "physically equivalent" solutions. If so, then perhaps the "gauge symmetry" of GR is not restricted to permutations of the points which are diffeomorphisms. Rather, it involves all permutations of the points: a kind of general permutation symmetry. (I relate this in the paper to Quine's idea that permutations of the interpretation of an interpreted language (using proxy functions) yields an interpreted language which is in some sense indistinguishable from the first.)
Anyway, the paper is short and comments by anyone who works on this kind of thing would be welcome.

UPDATE (May 18th).
After several days checking for any other related literature, I see that John Stachel seems to have made a somewhat similar suggestion in:

- Stachel, J. 2002. "‘The relations between things’ versus ‘the things between relations’: The deeper meaning of the hole argument." In D. Malament (ed.), Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics. Chicago and LaSalle, IL: Open Court.

I haven't read this paper yet. Stachel's proposal is discussed in a very interesting 2006 article by Oliver Pooley,

- Pooley, O. 2006. "Points, Particles and Structural Realism", in D. Rickles, S. French & J. Saatsi (eds), The Structural Foundations of Quantum Gravity, Oxford University Press.

This is available here

[Pooley's paper contains a nice discussion of the way to categorically axiomatize a structure M by, in effect, taking the logical conjunction of its diagram (set of atomic sentences true in M, in a language L_M with a name for each domain element), adding all inequation clauses saying all the names denote distinct things and that they exhaust the domain, and then existentially quantifying the result. For an infinite structure, clearly this is an infinitary formula.]

1. I am curious to know what the application of this result is or might be. That's not a rhetorical question. While I don't claim to have gotten my mind all the way around it, I get the sense that it is proposing or justifying an analytical shortcut, a way of simplifying a problem that is normally taken to be very complex. A way of cancelling out variables in a particular application of a calculus.

My question is: does it simplify a linguistic analysis or a physical one? Or does asking that question just more fully show I'm nowhere near "this kind of thing"?

2. Thanks for your comment and question, Thomas. No, it doesn't simplify a linguistic analysis or a physical one. The usual gauge principle associated with General Relativity focuses on diffeomorphisms of the underlying manifold M. I suggest that this principle be generalized to arbitrary permutations of the spacetime points.

3. Anytime. Normally, I'd think that a generalization simplifies our understanding of what were previously isolated cases. Bringing different things under the same generalization gives us one instead of several ways of understanding them.

(Like I say, this really isn't my kind of thing. But the challenge of understanding it intrigues me.)

4. Thomas, yes, now I think of it a bit more, I think you're right. There's a sense in which it does simplify how we understand these symmetries by, as you say, bringing different things under the same generalization. Maybe I'd prefer to say that it unifies our understanding of several different phenomena - diffeomorphism equivalence (Leibniz equivalence), permutation equivalence and the point I mention briefly, about Quine, proxy functions, etc.

5. Yes, I was driving at something like unification. But does it apply in all cases, or only to special cases?

6. "Special cases" could mean just certain spacetime theories, those for which spacetime has to be a differentiable manifold (with a metric plus some matter fields). If so, then I think no: for maybe spacetime turns out not be a differentiable manifold. Maybe it's some other fancy structure - say a bongifold. Then one could still maintain that permutation equivalent bongifolds "represent" the same physics.

7. I must say that, as a student of mathematics, i'm rather confused by your argument.

In mathematics, virtually every object is defined in terms of some structure on a set. It seems to me that what you're saying is: if we permute the elements of this set, and then we consider the structure (topology, differentiable structure, etc.) induced by this permutation, we get a new object which is "permutation equivalent" to the first one.

(Actually, it seems to me that in the paper you're "dragging along" only the topology with the permutation. If you only drag along the topology, what you get is not a differentiable manifold but a topological space, so you cannot say if it "represents the same physical situation" because you cannot do any GR on a topological space (you need a Riemannian manifold structure). But I'm considering this a minor technical issue, as you can drag all the structure you want.)

This is, of course, true, but it seems to me to be rather trivial and not related in any way to GR and diffeomorphism equivalence. You could do this to every mathematical object: a group, a vector space, a field, a manifold, and get the same object with the same structure and the same properties. So, in a definite sense, every mathematical object defined in terms of structure on a set is "permutation equivalent".

So "permutation equivalence" is something that belongs to set theory. Diffeomorphism equivalence is a geometric concept, and it tells us that all objects in GR transform in a certain way i.e. they are scalar or tensorial objects. This is not true in other physical theories (basically every theory except GR), which are defined in terms of objects that do not transform covariantly under arbitrary diffeomorphisms.

Maybe i didn't understand something. As I told you, I'm a bit confused.

8. Dave, thanks for the comments.
I think I need to answer your questions separately.

"Actually, it seems to me that in the paper you're "dragging along" only the topology with the permutation. If you only drag along the topology, what you get is not a differentiable manifold but a topological space, so you cannot say if it "represents the same physical situation" because you cannot do any GR on a topological space (you need a Riemannian manifold structure)."

No, I drag along everything - topology, charts, metric tensor and matter fields. Writing it more explicitly, we begin with a spacetime M:

(A, T, g_{ab}, phi_1, ..., \phi_n),

were A is the base set and T is the topology, g_{ab} is the metric tensor and phi_i are the matter fields. We use any permutation pi : A -> A to permute everything. The result is again another Riemannian manifold M^{pi}.

(A, T^{pi}, g^{pi}_{ab}, phi^{pi}_1, ..., phi^{pi}_n)

Then M and M^{pi} are isomorphic under pi, by construction.

Restricting our attention to the underlying differentiable manifold (A, T), we see that pi: (A, T) -> (A, T^{pi}) is a diffeomorphism.

[In model theory jargon, (A, T) is called the reduct of the original spacetime (A, T, g_{ab}, ...). We just "forget" the additional structure given by the metric and matter fields.]

As David Malament pointed out to me, my description is a bit unusual, as a differentiable manifold is usually given by a base set A, plus an atlas C of n-charts on A, satisfying compatibility conditions and differentiability conditions on the transition maps. In this scenario, the manifold is given as (A, C). [The topology T above is the one determined by the atlas C.]

So, I've now changed the formulation to one with the atlas as basic. So, the permutation pi takes us from (A, C) to (A, C^{pi}).

The important point is that given a Riemannian manifold (A, C, g_{ab}) and a permutation pi: A -> A of the base set, the Quine transform (A, C^{pi}, g^{pi}_{ab}) under pi is isomorphic to (A, C, g_{ab}). It's clear that if the first solves Einstein's field equations, then so does the second.

The less clear suggestion I make is that these permuted structures, in some sense, "represent the same physical world". The usual principle that physicists have given is that the result of applying a diffeomorphism to the base manifold yields a physically equivalent structure. I agree with this. But my suggestion is to generalize the group of permutations from diffeomorphisms of the spacetime (i.e., automorphisms of the base manifold) to any permutation whatsoever.

9. "This is, of course, true, but it seems to me to be rather trivial and not related in any way to GR and diffeomorphism equivalence. You could do this to every mathematical object: a group, a vector space, a field, a manifold, and get the same object with the same structure and the same properties. So, in a definite sense, every mathematical object defined in terms of structure on a set is "permutation equivalent"."

Yes, exactly. All of these are structures in the usual model-theory sense: a base set A with additional structure. Then one considers any permutation pi : A -> A of the base set, and drags everything along by pi.

For example, suppose (A, <) is a linear ordering. Let pi : A -> A be a permutation. Then (A, <^{pi}) is isomorphic to the original. I agree: it's mathematically trivial.

[There's an analogy here with structuralism in the philosophy of mathematics (Shapiro, Resnik). This maintains that all isomorphic structures exemplify the same abstract structure. E.g., arithmetic is taken to be the study of "the abstract omega-sequence", that all particular omega-sequences exemplify. If one considers any specifc omega-sequence and permutes it, then the result is isomorphic to the first.]

10. "So "permutation equivalence" is something that belongs to set theory. Diffeomorphism equivalence is a geometric concept, and it tells us that all objects in GR transform in a certain way i.e. they are scalar or tensorial objects. This is not true in other physical theories (basically every theory except GR), which are defined in terms of objects that do not transform covariantly under arbitrary diffeomorphisms."

This relates to a famous objection - Kretschmann's objection - to Einstein's requirement of general covariance. Kretschmann argued that any theory can be covariantly formulated. So, even Newtonian gravitational theory can be covariantly formulated (in doing this, one has to add additional structure).

So, what I'm saying is quite general. It will apply to any theory if the models are given in the standard way. I have to stress that given a differentiable manifold M = (A, C) and a permutation pi : A -> A, the permuted structure M^{pi}) is a differentiable manifold diffeomorphic to M. From an abstract point of view, M and M^{pi} exemplify the same abstract structure.

A permutation pi need not be a diffeomorphism from M to itself (i.e., an automorphism of M). However, any permutation pi is a diffeomorphism from M to M^{pi}.

11. Dave, I thought I'd add that there's a discussion of the Kretschmann point in section 5 of John Norton's paper "General Covariance and the Foundations of GR: Eight Decades of Dispute", available here: