Posted: Jul 22, 2010 6:57 am
newolder wrote:Rovelli, your ref. wrote:In this paper we consider the quantum theory corresponding to this dynamical system. In particular, we consider the quantum operators corresponding to the physically observable quantities A and A′, we show that (1) is true, and that the operator A′ can be obtained (under certain assumptions) from a unitary transformation that implements a local Lorentz transformation in the Hilbert space of the theory.
Edit: Where (1) is from earlier:
Now, our main point is the technical observation that A and A′ do not commute:
[A,A′ ] ≠ 0. (1)
A 'unitary transformation' preserves all constant ratios, doesn't it?What if nature is not unitary?
Unitarity in this context is referring to the representation of the Lorentz group on the Hilbert space of quantum states. The transformations need to be represented in terms of unitary operators so that Hilbert space scalar products <| |> are preserved. If they weren't then it would be difficult to proceed, since stuff like normalization wouldn't stay constant in time, and states which are orthogonal to each other wouldn't stay orthogonal.
Until I saw it, I'd never even thought of the question the paper was addressing - "how can the Planck length be a minimum length if you can make it aribitrarily small for different observers by Lorentz contraction ?"
I haven't read the whole paper yet, but for me the key thing was this paragraph:
Clever People wrote:It follows that a generic eigenstate of A is not an eigenstate
of A′. If the observer O measures the area and obtains
the minimal value A0, the state of the gravitational
field will be projected on an eigenstate of A. This, in
turn, is not going to be an eigenstate of A′. If then the
observer O′ measures the area, he will therefore find the
state in a superposition of eigenstates of A′. That is to
say, the theory predicts that, for him, the surface does
not have a sharp area. If the experiment is repeated several
times, O′ will observe a probability distribution of
area values. The mean value of the area can be Lorentz
contracted, while the minimal nonzero value of the area
can remain A0.