This is a continuation of "Spin Chronicles Part 43: Feeding GNS doll", where we have treated the GNS construction as if it was a lovely sweet doll. We played with it, and we have discussed three different ways of looking at it. The third method, called there Feed 3 is the most satisfactory, and in the following we will use it for the discussion of "time" emerging from space a'la Connes-Rovelli and Heller-Sasin. Do not worry now, we will get there soon, and all will be clear.
What if there is always a little bit of imaginary space and imaginary time?
Like a boat that floats on the surface, but is always embedded in water.
Perhaps there is a kind of "Archimedes Principle" here?
Why do I connect GNS to spinors? To answer this question let us first
take a look how spinors are defined when we want to connect them to
geometry, so that, for instance, they can be used also in curved
spacetimes. There are several standard ways of doing so. There is no
problem with getting the Clifford algebra and spin group inside it. In
General Relativity it will be the Clifford algebra of the tangent space
at a given point. But then what? One way is: take an irreducible
representation of this algebra. Call vectors of this representation
"spinors". Problem solved. Yes and no. There is unanswered question:
where do I get this representation from? Which one? To ease this
uneasiness we give the representation space a name "Clifford module". We
are dealing now with named objects, we feel better.
The second approach is: introduce "spinor structure". We postulate
its existence together with the covariant 2:1 map from spin frames to
orthonormal frames. But where does it come from? The answer is:
"somehow, does it really matter?". And then it is added: for some
spacetimes there may exist several inequivalent spin structures!
The third approach is: take a minimal left ideal in the Clifford algebra, perhaps take two or four of them - you will fit different spinors to different Fermions. Isn't it nice? Even better, the right action will shuffle the ideals. Left action corresponds to spacetime rotations, right action corresponds to "internal symmetry operations". It can even fit the Standard Model. For all practical purposes it can even work, but the question remains: which ideal and why this and not another one? I am not completely happy with this. We are onto something, but what is it, this "something"?
GNS construction associates representations to "states". If we follow
this philosophy, spacetime (or just "space") at a given point may be in
some "state". It fits my intuition. Once We have "state", we have the
associated representation. Pure states lead to irreducible
representations. Mixed states lead to reducible ones (we will discuss
this issue in the next post). This opens my memory bank. Pure states are
"extreme points" of the convex set of all states. They are at the
boundary. It is hard to get dynamically exactly to the boundary. Rather,
when we experimentally attempt to get exactly pure state, we will
obtain "almost pure" state, never "exactly pure". Pure states is an
extreme idealization. No state is in the Nature is exactly pure.
This opens a whole new perspective with possible physical consequences. I
like it. It leads to another idea: we think that our space is "real",
time is "real". Imaginary space and imaginary time are simply
mathematical tricks. But are they? What if "impurities" ("defects") are
important? What if there is always a little bit of imaginary space and
imaginary time? Like a boat that floats on the surface, but is always
embedded in water. Perhaps there is a kind of "Archimedes Principle"
here? In recent years Peter Woit develops a similar vision.
That is just "talk". Next post will be just "math". We will discuss "states dominated by other states", representations, cyclic vectors and irreducibility.
