Friday, October 20, 2017

Misunderstanding can be constructive

During a discussion, the interlocutor may say something, and if it is not clear, you may come up with the wrong idea. When something is not clear or complete, we try to fill the gaps. Sometimes when I do this in discussions about physics I come up with a mathematical model completely different from the idea the interlocutor had in mind. Here are two examples, one about causality and general relativity, and the other about Maxwell equations independent on the space metric. Both happened to me when discussing with Tim Maudlin.

The first time it was back in 2005, at the FQXi essay contest, where we shared the third prize. I was attracted by his essay, where he describes his idea to reconstruct general relativity in a way emphasizing time as fundamental. I didn't understand what he was doing, so I asked some clarifications on the comment section of his essay. He said at some point
The idea is a bit like Causal sets, but the actual implementation is quite different. The most fundamental structure is light-like rather than time-like, and the Causal sets it is time-like.
I completely misunderstood that in a way that suggested me an idea to recover the causal structure of general relativistic spacetime, in a way that works for both continuous and discrete, and for any dimension and even variable dimension. By our exchange of emails his idea was completely different. He already published a first volume about this, but the part about relativity I understand it will be in the second one. So because my idea was completely different from what he said, I made it into a paper
and published it in Journal of Gravity, vol. 2016, Article ID 6151726, 6 pages, 2016.
It shows how you can start with a simple relation and what you get is a generalization of the causal structure of spacetime, which works for both discrete and continuum, and for any number of dimensions, even variable from point to point. I also know how to add gauge theory on top of this without complicating with a single bit the definition of the fundamental relation, maybe I will do this someday.

The second time was now. Tim wrote
And then it further turns out that there is an even simpler, less mathematically committed formalism in which you can see this same fact (by "less committed" I mean you don't need any metric). As far as I know, I just invented that formalism in the last month, but maybe someone else has already figured it out. So I do a lot of playing around with basic questions and basic formalisms, trying to be very clear about the fundamental physical postulates (the "ontology") being made. But since the very distinction between the physical postulates of a theory and the mathematical representations of those postulated items has been systematically ignored and confused and erased in contemporary practice, these questions tend not to even be asked, much less answered.
In my reply I suggested a way to get rid of the space metric:
I tried for long time to find a way to be completely metricless in 4D, but I only found partial commitments.

In 3D, E and B look like vector fields (well, B is not a vector field, but a pseudovector, or equivalenty a 3D 2-form, hence the right-hand rule), but in special relativity they are parts of the same 2-differential form F, which I like. But I prefer the U(1) fiber bundle formulation with a connection that corresponds to the potential, which I agree with you is more fundamental than the field.

If you have something written about this metric independence in 3D, I would be interested to read it. Here is how I see it, maybe it's the same as you.

In 3D this works indeed, if you write the equations in terms of the 3D Hodge duals of E and B, *E and *B and forget about the metric and Hodge. And it is not committed to the 3D metric. But if you want to use only the potential A, then the equations contain the 3D Hodge dual, which commits you to the 3D metric.

Now if you look at the *E and *B formulation in 4D, there is some partial commitment to the 4D metric in the Minkowski case. If you are interested in Galilean invariance, you are aware that Maxwell's equations are not invariant. However, let's see what we get. The Galilei group, considering the Galilean spacetime as a 4D vector space V, is the group of transformations preserving a degenerate metric of rank 3 on the dual of V, which gives the foliation and the space metric in each slice, and a 1-differential form whose annihilator or kernel is the space slice. Since you can eliminate the space metric, you will only need the 4D 1-form that gives you the time and space slices. This would extend the Galilei group. In the Minkowski case decomposed in space+time, you will need a vector field that gives you the time, and also a slicing, which can be given by a 1-form which doesn't annihilate the vector field defining the time. These encode the partial dependence of the 4D metric. In both cases you can get independence of the metric if you refer only to the 3D metric, having given the slicing and the time.

Here is what I know about the commitment to the metric in 4D, which interests me more. One thing you precisely know is that conformal symmetry of Maxwell's equations reduces the commitment to just the causal structure of the metric. This is broken back to the full metric if you plug the stress-energy tensor of the electromagnetic field in Einstein's equation. But this is regained in conformal gravity (Gerard got great results in this), where in the Einstein equation you replace the Einstein tensor with the Bach tensor. And this actually works for the full standard model without Higgs and masses, but there are results indicating that you can get a mechanism identical to Higgs by breaking conformal symmetry going back to the metric.

You can look at the differential formulation in 4D, dF=0 and d*F=J, in more ways. You see that * which is the Hodge dual operator in 4D this time, which is built out of the metric. But you can formulate it so that it is not committed to lengths. You can also avoid using the codiferential *d*, which depend on the metric, and consider instead that you have two fields F and F', where F' is actually *F. For more on this I recommend to look up "premetric electromagnetism". But I am not very attracted by premetric electromagnetism, because if you consider F' as independent also for other gauge fields F, they will lead to several independent metrics.
 But again, this was not what Tim had in mind:
There are spaces with the same dimensionality, in terms of degrees of freedom, but no one-to-one mapping from element to element. And where the expression in terms of the differential forms uses star-d-star, which requires the Hodge, I use my two different fundamental operations, Lift and Drop, each of which can be defined without any metric. The thing is defined for discrete spaces: what a continuum limit version would look like, if it is possible to even define one, I don't know.
He didn't show me how he does this in discrete spaces, but I was happy to find a way to do it for continuous space. Clearly the Maxwell equations in term of *E and *B is known, what I don't know if it is known is the full construction, which includes the 4D Galilean or Minkowski spacetime. If it is new, maybe somebody will find this useful.

A debate inside another one

Tim Maudlin debated Gerard 't Hooft about his cellular automaton interpretation of quantum mechanics in a series of Facebook posts, the fourth one being here Somewhere in the forest of comments I was engaged in a sort of sub-debate, with Tim, Hans, and others. Sabine was there too. The discussion was completely surrealistic, Tim and Hans completely misunderstood my point. This started by me intervening with a theoretical counterexample to a claim that all so called superdeterministic theories (in particular 't Hooft's) are not falsifiable, and of course it led to different topics. It is not known, but not a secret that the wavefunction collapse leads to violations of conservation laws, and that it is possible at least in principle to remove the collapse while remaining with a single world. But removing the collapse can be seen as superdeterministic (although I wouldn't call it like this, because it is based on spacetime, not on initial conditions), and I even proposed a principle to explain this, and experiments to test it. I paste here most of this *debate*, because there are some parts I am interested to keep. I skipped some parts in which I was not involved.

Thursday, May 11, 2017

Maudlin's "(Information) Paradox Lost" paper

Tim Maudlin has an interesting paper in which he criticizes the importance given to the black hole information paradox, and even brings arguments that it is not even a problem: (Information) Paradox Lost. I agree that the importance of the problem is perhaps exaggerated, but at the same time many consider it to be a useful benchmark to test quantum gravity solutions. This led to decades of research made by many physicists, and to many controversies. I wrote a bit about some of the proposed solutions to the problem in some older posts, for example [1,2,3]. Maudlin's paper is discussed by Sabine here.

One of the central arguments in Maudlin's paper is that the well-known spacetime illustrating the information loss can be foliated into some 3D spaces (which are Cauchy hypersurfaces that are discontinuous at the singularity). These hypersurfaces have a part outside the black hole, and another one inside it, which are not connected to one another. Cauchy hypersurfaces contain the Cauchy data necessary to solve the partial differential equations, so the information should be preserved if we consider both their part inside and their part outside the black hole.

I illustrate this with this animated gif:

I made this gif back in 2010, when I independently had the same idea and wanted to write about it, but I don't think I made it public. Probably the idea is older. The reason I didn't write about it was that I was more attracted* to another solution I found, which led to an analytic extension of the black hole spacetime, and has Cauchy hypersurfaces but no discontinuities. I reproduce a picture of the Penrose diagram from an older post in which I say more about this:

A. The standard Penrose diagram of an evaporating black hole.
B The diagram from the analytic solution  I proposed.

* The reason I preferred to work at the second solution is that it allows the information to become available after the evaporation to an external observer. The solution which relies on completing the Cauchy hypersurface with a part inside the black hole doesn't restore information and unitarity for an external observer. I don't know if this is a problem, but many physicists believe that information should be restored for an external observer, because otherwise we would observe violations of unitarity even in the most mundane cases, considering that micro black holes form and evaporate at very high energies. I don't think this argument, also given by Sabine, is very good, because there is no reason to believe that micro black  holes form at high energy under normal conditions. People arrive at high energies for normal situations because they use perturbative expansions, but this is just a method of approximation. And even so, I doubt anyone who sums over Feynman diagrams includes black holes. But nevertheless, I wouldn't like information to be lost for an outside observer after evaporation, but this is just personal taste, I don't claim that there is some experiment that proved this. And the solution I preferred to research allows recovery of information and unitarity for an external observer, and other things which I explained in the mentioned posts and my PhD thesis.

Friday, March 10, 2017

The Tablet of the Metalaw

This edition of the FQXi essay contest is called Wandering Towards a Goal. My entry is called The Tablet of the Metalaw. This is the abstract:

Reality presents to us in multiple forms, as a multiple level pyramid. Physics is the foundation, and should be made as solid and complete as possible. Suppose we will find the unified theory of the fundamental physical laws. Then what? Will we be able to deduce the higher levels, or they have their own life, not completely depending on the foundations? At the higher levels arise goals, life, and even consciousness, which seem to be more than mere constructs of the fundamental constituents. Are all these high level structures completely reducible to the basis, or by contrary, they also affect the lower levels? Are mathematics and logic enough to solve these puzzles? Are there questions objective science can't even define rigorously? Why is there something rather than nothing? What is the world made of?

At this time (2017-03-11 08.59 AM ET) my essay is in the top position, so I will immortalize this ephemeral moment in the picture below, since I expect the order will change dramatically, given that the votes will continue for nearly a month, and then the FQXi panel will add their choices:

Wednesday, February 15, 2017

The Standard Model Algebra

arXiv link:
A simple geometric algebra is shown to contain automatically the leptons and quarks of a generation of the Standard Model, and the electroweak and color gauge symmetries. The algebra is just the Clifford algebra of a complex six-dimensional vector space endowed with a preferred Witt decomposition, and it is already implicitly present in the mathematical structure of the Standard Model. The minimal left ideals determined by the Witt decomposition correspond naturally pairs of leptons or quarks whose left chiral components interact weakly. The Dirac algebra is a distinguished subalgebra acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle θW given by sin2(θW)=0.25.