quantum gravity

[granpa]

relativity says that all objects consider themselves to be at rest and no experiment that they can perform can indicate otherwise. but if space and time are quantized then a moving particle should 'see' spacetime differently than a stationary one.

 

imagine walking past a picket fence with evenly spaced pickets while a strobe flashes at exactly the same rate that the pickets are passing you. you would see the pickets as being stationary. now imagine that space and time are quantized into discrete units (arranged in a grid). if the minimum nonzero velocity (along one axis of the grid) for any object is one space quantum per one time quantum and all velocities are integer multiples of that then all objects should 'see' spacetime as being stationary with respect to themselves. (think of space quanta as the pickets and time as the strobe flashing once per time quantum).

 

the only trouble is that due to relativity a moving object would see space as being shrunk. I assume that this is where noncommutative geometry comes in. that means that xy doest equal yx. 4 groups of 3 isnt equal to 3 groups of 4. but I dont really understand this at all. could someone explain it to me.

 

http://en.wikipedia.org/wiki/Doubly_special_relativity

Edited April 14, 2009 by granpa

[ajb]

Are you having trouble understanding the details of doubly-special relativity as a deformation of the Poncaré group/algebra or noncommutative geometry more generally?

[granpa]

the latter. noncommutative geometry.

[ajb]

How about a lightning course in NCG? I'll try not to assume too much classical geometry, but I will assume you know some elementary topology.

 

It turns out that a topological space [math]X[/math] and the ring of continuous function to [math]\mathbb{R}[/math] on that space [math]C(X)[/math] are equivalent. That is once you have specified one, you have the other.

 

The important thing is that [math]C(X)[/math] is a commutating ring. Thus we have a "duality" between commutative rings and (commutative) geometry.

 

Now, the idea of noncommutative geometry is to replace [math]C(X)[/math] with a more general noncommutative ring and treat this ring as if it were the ring of functions on some space. This space is what people call a noncommutative geometry.

 

Noncommutative ring = Noncommutative geometry

 

Note that such geometries are not topological spaces in the classical sense. They are not specified by a collection of points. In fact, no where in the definition of a noncommutative geometry do we use the notion of a point.

 

I am not an expert in the field of NCG, but I do work with (in the category of) supermanifolds which are a mild form of NCG. Here one takes the ring of smooth functions on a manifold and extends it to include anticommuting generators. The resulting ring is not quite commutative, but rather supercommutative. (We have commutators and anticommutators). Due to this rather mild NCG almost all of the theory of smooth manifolds carries over to supermanifolds with little effort.

 

This can be done using "local coordinates". The structure ring on a supermaifold consists of functions of the variables [math]\{x^{a}, \theta^{\alpha}\}[/math]. ([math]x[/math] commute, [math]\theta[/math] anticommute).

 

For more general NCG we (can) have a similar situation. The ring of functions can be described by formal variables, lets say the functions are of the form [math]f(x)[/math], with [math]x[/math] belonging to some noncommutative algebra. (Think of the matrix algebra or a Lie algebra for example). One can then interpret [math]x[/math]'s as coordinates on some general space. Again, note no mention of a point.

 

 

I think it is worth stressing that at present there is no "theory of noncommutative geometries". There is much work to be done before one can claim that.

 

Hope that goes some way towards helping you.

[granpa]

thank you. I'll be digesting that for quite a while.