I Is Non-Commutative Geometry the Key to a Discrete Model of Spacetime?

[waves and change]

Physics could be fundamentally discrete. Are their any notable theories that have discrete mathematics at its core and have QM, GR and differential equations in general as emergent features?

 

[fresh_42]

Physics could be fundamentally discrete. Are their any notable theories that have discrete mathematics at its core..,

Here's a discrete part: https://en.wikipedia.org/wiki/Lattice_gauge_theory

... and have QM, GR and differential equations...

Here's one for that: https://en.wikipedia.org/wiki/Finite_difference_method

... in general as emergent features?

I didn't know there is one in a continusous set-up.

 

[waves and change]

Thank you !

 

[waves and change]

Thank you !

Finite difference method is just a discrete method to approximate a diff. Equation. That doesn’t really give any insight into my question as posing discreteness as the underlying fundamental feature. Same can be said for Lattice gauge theory which just serves as a tool to approximation continuity

 

[fresh_42]

Finite difference method is just a discrete method to approximate a diff. Equation.

I knew a professor who saw it the other way around: "Since there is nothing really continuous in the world out there, differentiabilty is an approximation of the discrete things which happen!" O.k. he mainly meant it to motivate the drawings of vector fields, but anyhow, there is some truth in it.

P.s.: I mentioned this as a contradictory possible statement, not to start a discussion upon. This belongs into philosophy and will not be dealt with on PF. However, I couldn't let stand this extreme claim as only possible truth. It is not.

 

[phyzguy]

Physics could be fundamentally discrete. Are their any notable theories that have discrete mathematics at its core and have QM, GR and differential equations in general as emergent features?

Many people are working along these lines, but it is not simple. If spacetime is discrete, for example, it cannot simply be pixelized like the pixels on your display screen, because this would not be Lorentz invariant and would be observer dependent. An attempt to formulate a discrete model of spacetime, which I personally believe is in the right direction, is to use non-commutative geometry. This draws on the idea of quantum phase space, where the coordinates (x and px, ...) do not commute and the volume of quantum phase space is basically discrete in units of (2 π ħ)^3. An example that helped me to understand these ideas is that of the "fuzzy sphere", where the area of the sphere is discretized into N units (where N can be any integer from 2 on up) in a way that is observer independent. As N goes to infinity, one recovers the usual continuous spherical surface.

 

[waves and change]

Many people are working along these lines, but it is not simple. If spacetime is discrete, for example, it cannot simply be pixelized like the pixels on your display screen, because this would not be Lorentz invariant and would be observer dependent. An attempt to formulate a discrete model of spacetime, which I personally believe is in the right direction, is to use non-commutative geometry. This draws on the idea of quantum phase space, where the coordinates (x and px, ...) do not commute and the volume of quantum phase space is basically discrete in units of (2 π ħ)^3. An example that helped me to understand these ideas is that of the "fuzzy sphere", where the area of the sphere is discretized into N units (where N can be any integer from 2 on up) in a way that is observer independent. As N goes to infinity, one recovers the usual continuous spherical surface.

Thank you. I will look into this!