Quantum Geometry

[Ryan Miller]

Hello everyone,

I originally intended to write an essay on Quantum Geometry for an undergraduate Geometry class, but I ran into some problems. Although there is plenty of information available on thebasics of the Heisenberg Uncertainty Principle and Planck's Constant, there doesn't seem to be any information available on how this affects the behavior of points, lines, plains, and half planes (the undefined terms of neutral geometry) on a quantum scale. Would it even be possible to use the basic theorems of Neutral Geometry? Does a Euclidean model fit Quantum Geometry? Is a completely different set of axioms required to work with Quantum Geometry? Do I need a good understanding of calculus even to work with Quantum Geometry? I tried asking my professor, but this seems to require knowledge of both theoretical particle physics and Geometry.

[Ryan Miller]

I'm not sure how long I'll have to wait. The project must be submitted by January 8, 2019.

[Ryan Miller]

I'm not sure where else to look....

[mathematic]

Quantum theory is a branch of physics.  Geometry is a branch of mathematics.  It would be better to address your question in a physics forum.  The question you seem to be asking is what is the geometry of (physical) space at a Planck level.

[Ryan Miller]

Before I repost this thread in the Physics section, is it possible for a moderator to move the thread over there first?

[studiot]

Linear algebra is as good a place as any, but you do need to know the Physics of what you are applying it to.

Remember that modern mathematical geometry is algebraic in nature so higher algebra underlies the application.

I don't know what level your Physics or Maths is at, only that it is presumably up to undergraduate standards?

So read this extract first and then come back with some details. I have highlighted the relevant section.

It boils down to the connection between the triangle inequality and algebra, as applied to (quantum) Physics.

The article is about Emmy Noether, of Noether's Theorem fame, which theorem you will need.

Some good books to approach this from a mathematical point of view are

An older book

Mathematical Foundations of Quantum Mechanics

G W Mackey

Benjamin

 

A Modern Book

The Mathematical Principles of Quantum Mechanics

D F Lawden

There is a Dover version of this.

Drawing together the maths background is a mammouth task (but very worthwhile) and will entail a great deal of background reading.

 

 

[MigL]

Not even sure if this is a valid approach.

Math, and geometry, are 'tools' we use to build the model ( in physics ).
If you are asking how the probabilistic and non-commuting nature of Heisenberg's UP affects various types of geometries,  I'm not sure it does. You may as well ask, how do numbers change at the quantum level.
The application doesn't affect how a hammer works. Sometimes you need a different tool, like a saw, for different circumstances.

If on the other hand, you're asking about the geometry of space-time at the quantum level, some of the best minds in Physics are wrestling with variations of that question, as it leads to Quantum Gravity.

Edited December 22, 2018 by MigL