Dear Hans-Thomas,
I enjoyed reading your essay. I have some
questions/comments.
1. If I understand correctly, you have contributed the notion of "extensity" as
a way to implement matter within the causet framework. Previously this
framework existed only for spacetime, w.r.t the property of "causality".
2. Spacetime-matter duality follows. I liked the picture you presented in which
you criticise the notion that most people have (myself included), that
spacetime can be thought of as devoid of matter, while nobody can image matter
living outside, or independently of, being embedded within a spacetime. This is
very cute, I'd never thought about it, it shows how deeply prejudiced one can be
without even being aware of it.
3. Nature is however highly nonsymmetric w.r.t. the spacetime-matter duality. A
posteriori, this is kind of a justification for our prejudice as mentioned
above.
4. It follows that some symmetry-breaking mechanism must be present, some kind
of Higgs particle, that brings us into this nonsymmetric state.
5. Do I understand correctly, that the symmetry breaking is due to the presence
of interactions? This notion is appealing, but it raises some doubts. In the
continuum limit, many interactions (the gravitational one first and foremost)
can be mimicked by passing to an appropriate reference frame (this is the
principle of equivalence). This is, in a sense, a way of rendering interactions
"unphysical", because we can mimic them geometrically. Something similar (though
not exactly identical) happens in classical mechanics on phase space, when one
considers the Hamilton-Jacobi equation: one looks for new canonical coordinates
and momenta such that the system, however complicated in the old variables,
evolves freely in the new variables.
6. Which leads me to the point: at least in the continuum limit, interactions
will not do as a symmetry-breaking mechanism. This raises the possibility that
interactions will also not do before the continuum limit is taken. What makes
you think they will? And, in case they do serve as a Higgs mechanism, how
exactly do they do so? I think this is the crux of the matter.
I enjoy looking at things from a dual perspective, and I found the duality you
presented very interesting. I think it may be related to my old programme of
"rendering the notion of an elementary quantum relative to the observer", which
is my own, dual way of "quantising gravity", if you wish.
Just one more thought-
at least in the continuum limit, judging by the two examples given (gravitaion
and Hamilton-Jacobi), interactions are actually a statement of the duality
between spacetime and matter, rather than a statement of the symmetry breaking
thereof. Of course, in the discrete causet framework this might be otherwise.
But then one has to account for the fact that a symmetry-breaking mechanism in
the discrete picture actually becomes a manifestation of the symmetry itself in
the continuum limit. This I find somewhat bizarre, though of course not
impossible.
Jose.