Unification of QM and relativity theory

[jfy4]

Hi everybody,

I have been looking over a number of papers by Alexey Kryukov and I have some questions. I have been in some brief correspondence with him about some of my questions but I wanted to ask the forums.

Consider this paper

http://depts.uwc.edu/math/faculty/kryukov/files/IARD2008.pdf"

one of my main questions is that in his approach he identifies quantum states with members of $\mathbb{R}^4$. This formalism is generalized to curved manifolds. However, $x_\alpha$ does not transform covariantly for arbitrary curved manifolds. I'm worried that the identification will imply the state vectors do not transform correctly either.

Thanks,

 

[jonnyfish]

I took a glance at this paper, and there are definitely some issues being swept under the rug, so if you get some feedback from the author I would like to hear it.

The first issue is that I see no mention of Poisson brackets or commutation relations. We know that classical mechanics is described canonically conjugate pairs $(x,p)$ satisfying the Poisson bracket relation $\{x,p\} = 1$, and that in order to recover the correct classical limit, Poisson brackets should be taken to commutators (up to a factor of $i\hbar$). However, the Stone-von Neumann theorem guarantees that any quantization of flat space is isomorphic to the standard one. So if you somehow produce a quantization that is not isomorphic to the standard one, you have to be in violation of at least one of the axioms of quantum mechanics (or doing something which is mathematically ill-defined).

A second related issue is that canonical quantization of flat space inherits a unitary action of the symplectic group. This roughly says that quantization doesn't depend on your choice of canonically conjugate coordinates. This doesn't seem to be mentioned in the paper at all, and it's not obvious (to me anyway) that you get an action of the symplectic group for free.

A third problem is that we know that even if we did have a "good" theory of N-particle relativistic quantum mechanics, so what? We know that virtual particle-antiparticle pair creation is real (see e.g. Casimir effect), and to my knowledge this can only be explained by QFT (you need something that creates and annihilates in the first place!).

So the author gives a construction of a hilbert space in which the original spacetime manifold embeds, but it is not at all clear that the quantum mechanics on this hilbert space looks anything like the good old quantum mechanics we know and love (and has been experimentally tested beyond any reasonable doubt). It would certainly be more convincing if he gave an explicit calculation.

Putting physics aside, the actual mathematics of the construction is vaguely reminiscent of the philosophy of the Gelfand-Naimark theorem. In the GN theorem you start with a nice topological space $X$ (e.g. a manifold), and consider the algebra $\mathcal{A}$ of all continuous functions $f: X \to \mathbb{C}$. Any point $x \in X$ defines an algebra map $ev_x: \mathcal{A} \to \mathbb{C}$ given by $ev_x(f) = f(x)$ (called the evaluation map). The trick to GN theorem is that if you consider the set $S$ of all such algebra maps, then (suitably topologized) it contains a copy of the original space $X$ as the subset $\{ ev_x: x \in X\}$. Thus the algebra of functions is sufficient to reconstruct the space $X$.

The relation with the author's construction is as follows. He encodes the original space $X$ as the space of delta functions $\delta(x-a)$. For a general space $X$, the evaluation map $ev_x$ is exactly the right generalization of the delta function, since on flat space
$$ev_x(f) = f(x) = \int \delta(y-x)f(y) dy.$$
So it seems that whatever it is the author is doing, it is probably closely related to Gelfand-Naimark.

Just some thoughts.

 

[Chronos]

The fatal flaw in most promising new 'theory of everything' ideas usually resides in the assumptions. Part of the problem is we have few means to validate fundamental assumptions aside from their interactions with other assumptions. When we rigorously cross check assumptions, something unexpected often results - which reminds us to never assume any assumption is inviolable.

 

[Demystifier]

one of my main questions is that in his approach he identifies quantum states with members of $\mathbb{R}^4$. This formalism is generalized to curved manifolds. However, $x_\alpha$ does not transform covariantly for arbitrary curved manifolds. I'm worried that the identification will imply the state vectors do not transform correctly either.

I don't think that it would be a problem. To understand why do I think so, consider an analogy: Consider standard nonrelativistic QM. As you know from solving the hydrogen atom, QM and Schrödinger equation are consistent in the polar coordinates as well. Yet, curvilinear coordinates (such as the polar ones) do not transform covariantly.

Anyway, you might be interested in an approach to relativistic QM
http://xxx.lanl.gov/abs/0811.1905
similar to that in the paper you mentioned