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

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• waves and change
In summary, the conversation discusses the idea of physics being fundamentally discrete and whether there are any notable theories that have discrete mathematics at its core and have quantum mechanics, general relativity, and differential equations as emergent features. The conversation also mentions the concept of non-commutative geometry and the "fuzzy sphere" as potential models for discrete 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?

waves and change
Thank you !

waves and change said:
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

waves and change said:
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.

waves and change
waves and change said:
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.

phyzguy said:
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!

1. What is discrete theory of physics?

Discrete theory of physics, also known as discrete physics or digital physics, is a scientific theory that suggests that the universe is fundamentally made up of discrete, digital units rather than being continuous.

2. How does discrete theory of physics differ from traditional physics?

Traditional physics is based on the idea that the universe is continuous, meaning that it is composed of infinitely divisible elements. Discrete theory of physics, on the other hand, proposes that the universe is made up of discrete, indivisible units.

3. What evidence supports discrete theory of physics?

There are several pieces of evidence that support discrete theory of physics, including the quantization of energy levels in atoms, the discrete nature of quantum mechanics, and the discreteness of space and time at the Planck scale.

4. Are there any current theories or experiments that contradict discrete theory of physics?

While there are some theories, such as string theory, that suggest a continuous universe, there is currently no experimental evidence that directly contradicts discrete theory of physics. However, further research and experiments are needed to fully understand the nature of the universe.

5. How does discrete theory of physics impact our understanding of the universe?

Discrete theory of physics challenges our traditional understanding of the universe and raises questions about the nature of reality and the fundamental laws of physics. It has the potential to provide a deeper understanding of the universe and may have practical applications in fields such as quantum computing and cosmology.

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