Unitarity, Locality and Spacetime Geometry: Foundations That Are Not Foundations by Lawrence B Crowell

To be honest I would prefer that my essay page not be filled with posts over this imagined controversy.

LC

There have been some developments along these lines which may give support for my thesis here. The paper Black Holes: Complementarity or Firewalls? by Almheiri, Marolf, Polchinski, Sully raises an important point. This points out an inconsistency with the holographic principle. They focus on the suggestion that postulate #2; Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations, is to be "relaxed." I take it that this relaxation focuses on the issue of "massive" as the mass approaches around 10^3 to 10^4 Planck units of mass. This still makes the black hole massive when compared to the masses of elementary particles.

In discussions with Stoica on singularities I suggested the following metric with 1 - 2m/r = e^u. so then

ds^2 = e^udt^2 - e^{-u)dr^2 dΩ^2.

We now have to get dr from

dr = -2me^u/(1 - e^u)^2du.

Now the metric is

ds^2 = e^udt^2 -2m[e^u/(1 - e^u)^4]du^2 dΩ^2.

The singularity is at u = ∞, where the dt term blows up, and the horizon coordinate singularity at u = 0 is obvious in the du term. My rational was that the singularity had been removed "to infinity" in these coordinates. This makes the black hole metric similar to the Rindler wedge coordinates, which does not contain a singularity. In the accelerated frame or Rindler wedge there is singularity. The treatment of the Schwarzschild metric in the near horizon approximation Susskind uses is one where the singularity is sufficiently removed so that field in the Rindler wedge may be continued across the horizon without concerns. In this metric of mine the singularity is at infinity so the analytic functions for fields in the Rindler wedge are replaced with meremorphic functions with a pole at infinity.

Stoica made the observation that this runs into trouble with Hawking radiation. The singularity at infinity causes trouble with the end point of the radiance process for it has to "move in" from infinity. The final quantum process of a black hole is a problem not well known in any coordinates. Your objection does have a certain classical logic to it. However, by the time the black hole is down to its last 10^4 or 10^3 Planck mass units the black hole itself is probably quantum mechanical. In my coordinates (assuming they are unique to me, which is not likely) the singularity at infinity may not have to "move" from infinity. There may be some nonlocal physics which causes its disappearance without having to move at all. This nonlocality is a correspondence between states interior to a black hole and those on the stretched horizon. The Susskind approach does not consider the interior, and he raises this as a question towards the end of his book "The Holographic Principle."

This nonlocaity would be a relaxation of the postulate #2. The issue of unitarity comes into play. If the theory is replaced with meremorphic functions, say analytic in a portion of the complex plane, then fundamentally quantum fields in curved spacetime or quantum gravity is not unitary but modular.

Unitarity is represented by a complex function e^{-iHt} and so forth, which is analytic. The loss of unitarity does not mean there is a complete loss of everything; in particular quantum information can still be conserved. A simple analytic function of this sort describes standard quantum physics. Gravity as we know is given by a hyperbolic group, such as SO(3, 1) ~ SL(2,C), where the latter has a map to SL(2,R)^2. The functions over these groups have posed difficulties for quantum gravity, for they are explicitly nonunitary. The trick of performing a Wick rotation on time or with τ = it is a way of recovering the compact groups we know in quantum physics.

It does turn out I think that we can think directly about quantum gravity by realizing that the SL(2,R) is related to a braid group with Z --- > B --- > PSL(2,Z), and that the braid group is contained in SL(2,R). Braid groups have correspondence with Yang-Baxter relations and quantum groups. The group SL(2,Z) is the linear fractional group, which is an elementary modular form. An elementary modular function is

f(z) = sum_{n=-∞}^{n=∞}c(n)e^{-2πi nz}

which in this case is a Fourier transform. In this case we are safely in the domain of standard QM and QFT. In general modular functions are meromorphic (analytic everywhere but infinity) and analytic condition is held on the upper half of the complex plane.

Of particular interest to me are the Eisenstein series of modular functions or forms. These define an integer partition function, which is an acceptable partition function or path integral for a stringy black hole. I include a graphic here illustrating an Eisenstein function. This has a certain self-similar structure to it, or what might be called an elementary form of a fractal. In this picture unitarity is replaced with modularity. In this more general setting the transformation do no promote a field through time by some operator, but that the operator simply computes the number of states or degrees of freedom in a way that is consistent. Unitarity is then a special case of this, which happens to fit into our standard ideas of causality.

The Eisenstein series describes a partition function or path integral for a black hole. The theory is not one of unitary evolution, but simply one of counting states or degrees of freedom on the horizon. In effect physics is more general than unitarity, where unitarity is a necessary condition to describe the semi-classical states in postulate #2.

Cheers LC

The gravitational constant in naturalized units is "area," so it is sqrt{G} that is ~ length or reciprocal of mass.

Verlinde has a proposal that the work done by gravity W = ∫F•dr is equal to entropy S = nkT. There has been some controversy over this. However if we look at an increment of work done through some increment of time δt, I will not worry about the relativistic issues with this definition of time for now, then the increment of work is

δW = F•(dr/dt)δt = Pδt.

Power has in natural units reciprocal length squared L^{-2}, or ~ 1/G. Consequently, this increment in work can be written as δW = δt/G ~ n/G. We interpret G as the fundamental Planck unit of area, and n = # of Planck units of area generated by this work. This would then correspond (roughly) to the Bekenstein bound or entropy S = kA/4L_p^2. This is why I think his entropy force of gravity pertains to moving the holographic screen.

Cheers LC

Thanks guys for the response. I think the situation we face with quantum gravity may mirror something in the past. The solution might in part be under our noses.

LC

It is not difficult to quantize weak gravity. This is usually written as a bimetric theory g_{ab} = η_{ab} h_{ab}, where η_{ab} is a flat spacetime (Minkowski) metric and h_{ab} is a perturbation on to of flat spacetime. We may write a theory of the sort g_{ab} = (e^{ω})_a^c η_{cb}, where the bimetric theory is to O(ω) in a series expansion

g_{ab} =~ (δ_a^c ω_a^c) η_{cb}.

Gravitons enter in if you write the perturbing metric term as h_{ab} = φ_aφ_b, or ω_a^c = φ_aφ^c. The Ricci curvature in this weak field approximation is

R_{ab} - (1/2)Tg_{ab} = □h^t_{ab},

with h^t_{ab} the traceless part of the metric, and □ the d'Alembertian operator. Which in a sourceless region this computes plane waves. The two polarization directions of the graviton may then be interpreted as a form of diphoton, or two photons in an entanglement or a "bunching" as in Hanbury Brown-Twiss quantum optical physics.

If we now think of extending this to a strong field limit there are the square of connection terms Γ^a_{bc} in the Ricci curvature, or cryptically written as R ~ ∂Γ ΓΓ where there is the appearance of the nonlinear quadratic term in the connection. This nonlinear term indicates the group structure is nonabelian, so the photon interpretation breaks down. The graviton in this case is a form of di-gluon, or gluons in a state entanglement or chain that has no net QCD color charge. This connects with the AdS_n ~ CFT_{n-1} correspondence, where for n = 4 the conformal field theory is quark-gluon QCD physics. Further D-branes have QCD correspondences and this takes one into the general theory I lay out. One does need to look at the references to learn more of the specifics. The quantum phase transition to entanglement states is given in the paper I write in ref 11 L. B. Crowell

The simple fact is that as physics develops it will invoke new mathematics. I don't think I am overly mathematical in this essay, and I leave most of those details in the references. A theoretical physicist I think is wise to have a decent toolbox of mathematical knowledge and thinking. Physics invokes ideas of symmetries, remember Noether: symmetry corresponds to conservation law, and invariant quantities can also have connections with topology and number theory. I think the more one is familiar with advanced mathematics the more capable one is of thinking deeply about these matters.

It is true that my work is commensurate with P. Gibbs'. If field theoretic locality and spacetime are emergent structures then so is causality. This emergence is connected with a quantum phase transition, or a quantum critical point (tricritical point of Landau), and something occurring on a scale much larger than the string length.

Cheers LC

Large symmetries are clearly important. The more general a symmetry group is, say with a larger Lie group, the transformations of that group can maintain a more general vacuum as a vacuum. In other words, symmetry preserves the ground state (vacuum), and broken symmetry does not, or maintains a more restricted ground state. There may of course be other elements to the foundations of physics than simply using ever larger Lie groups, such as removing certain postulates like locality of field data.

A lot of this about mathematics and logic relies upon the philosophy of mathematics, which I have read about and find somewhat interesting. However, I am not that steeped in the subject, nor does it concern me that deeply. Some mathematical subjects have no reference to geometry, such as most of number theory. Of course we humans have to exist with all our causal structure in spacetime to study it. However, a mathematical realist would say that number theoretic proofs are true whether we know them or not.

Cheers LC

The matter over Leibniz is not in my essay. I appear to be wrong with respect to Spinoza in my blog post.

The sentence is missing the word be, which is unfortunate. it should read, "This would not be quantum mechanics in any natural or realistic

way."

I discuss modularity towards the end. I was planning on breaking this out further, and in fact did so more. However, I exceeded the word/page limits for the essay. So this got rather scant mention at the end.

The word unity is supposed to be unitarity.

The emergence of unitarity is complicated. The existence of singularities means that quantum wave functions are not analytic functions. They are meremorphic, which define modular function or forms. I wrote a bit more on this in a post above on Aug. 13, 2012 @ 14:44 GMT. This is a deep subject, which gets into Borel groups, Leech lattices and so forth. The length restriction on this essay prevents me from breaking this out. Besides most of this that I have worked out is on notebook paper and not published. Yet analytic functions or unitarity occurs in the special case the singularity is removed or has minimal nonlocal connection to a region outside the event horizon in the case of a black hole. This occurs for large black holes.

I am not sure what the main objection is you want to raise. I am not likely to respond much if your objection is about the foundations of mathematics or set theory.

Cheers LC

My response is below. I didn't put it in a response.

LC

I am not particularly interested in getting further into math foundations issues.. My point is there are plenty of mathematical topics which are independent of geometry. The question of mathematical realism, which is related to Platonism, is something I am not interested in debating much. There are various schools of math foundations, intutionism, constructivism and so forth, and I am not particularly a partisan to one over the other. The mathematics I work with is not terribly dependent on set theory subtleties, and most mathematics used in physics is not directly dependent on these matters.

This extends to Blumschein as well. If I recall he has some big alt-math idea that "upends" the foundations of mathematics. I am not terribly interested in going there.

Cheers LC