Discrete Models for Arguments and Continuous Variables in Quantum Mechanics

[Derek P]

Arguments can often be presented using a discrete model on the assumption that continuous variables can be accommodated by taking the limit as the resolution is increased.

I would have thought that this would be just fine in QM where functions are continuous. But maybe mathematicians here can say where it breaks down. For instance there might be a problem with irrational numbers. We could specify an operator which works differently with different classes of numbers but could it have any physical meaning?

Any hints, other than "take a degree in mathematics and come back when you can formulate your question in better language" would be appreciated.

 

[stevendaryl]

Arguments can often be presented using a discrete model on the assumption that continuous variables can be accommodated by taking the limit as the resolution is increased.

I would have thought that this would be just fine in QM where functions are continuous. But maybe mathematicians here can say where it breaks down. For instance there might be a problem with irrational numbers. We could specify an operator which works differently with different classes of numbers but could it have any physical meaning?

Any hints, other than "take a degree in mathematics and come back when you can formulate your question in better language" would be appreciated.

Taking the continuum limit of discrete theories is a little problematic. An example is the "Fermion doubling" problem that comes up when you try to do quantum field theory on a lattice. I'm not qualified to talk about that, but it's described in Wikipedia:

https://en.wikipedia.org/wiki/Fermion_doubling

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Daryl