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

Thanks for the thumbs up. I seem to be falling downwards in the community rankings, though my paper has only been up about 36 hours. I am not sure what is going on there, for I know the physics I present is better than a whole lot of the papers ranking higher.

Cheers 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

I think that unitarity is a special case of modular transformations when there is no singularity, or if in the case of a black hole the singularity is hidden by a very classical event horizon that causes decoherence of nonlocal fields across it. If the event horizon is quantized, say with a very small black hole, then this breaks down. Further, the meaning of spacetime and light cones becomes lost. They are so to speak blurred out by quantum fluctuations.

If the definition of spacetime breaks down on a very small scale then the definition of time is lost. If there is no effective definition of time there is then no unitary time development of quantum states or observable by an operator, such as the Hamiltonian. This is a loss of unitarity.

The Wheeler-DeWitt equation tells us this to begin with. The Hamiltonian in classical gravity is zero, or NH = 0, for N the lapse function. This is a standard result of ADM general relativity. The reason for this is Gauss' law, where there is no boundary sphere around the universe by which one can integrate out the mass-energy contained within. This argument can be posed according to the nature of coordinate time in general relativity, where this is a frame dependent quantity and physics should not depend upon it. So the Schrodinger equation

i∂ψ/∂t = Hψ = 0

is seen to be zero on both the left and right hand side in a consistent manner. This is the Wheeler-DeWitt equation Hψ = 0, which is the quantum form of the Hamiltonian constrant NH = 0. There is in this case no Hamltonian which acts as a Hermitian operator that define a unitary time development operator. Unitarity is gone.

What takes the place of unitary transformations are modular transformations, in particular the Eisestein series and Jocobi functions. The Jacobi functions are realizations of the E_8 and Leech lattice Λ_{24} based sporadic groups --- in particular the Mathieu group of quantum error correction.

In that case my essay does propose the removal of or change in established physical postulates.

I am less concerned with mathematical foundations. The connection to matters such as Zermelo-Fraenkel (ZF) set theory is at best very subtle, and really could be nonexistent. ZF set theory has some strange features, such as the duplication of spheres with the Axiom of Choice. There are alternatives such as Polish sets. This goes double for philosophical issues over how or whether mathematics exists independent of physical reality or the mind of a mathematician. These questions simply go far beyond the scope of what I am concerned with.

Cheers LC

I did it again and pasted the response in as a fresh post.

LC

The duplication of a sphere is called the Banach-Tarski paradox. This is the result of "immeasurable measures" that can occur with the axiom of choice.

Cheers LC