Euclidean geometry doesn't exist?

[K1NS]

As a newbie, I apologize if this topic has been discussed before.

It seems to me that one result of quantum physics is that Euclidean geometry is artificial and cannot be represented in real space. For example, there can be no such thing as a straight line in granular quantum space.

And Euclid's fifth postulate, that parallel lines never meet, becomes false. Why? Because of random variations in space, lines are not straight, but they would "wobble," albeit at tiny Planck distances. But over an infinite distance, they would meet and diverge an infinite number of times.

I believe we need a "quantum geometry" to describe space and time, where measurements, lines and angles are replaced by probabilities.

If this topic has been discussed here or elsewhere, I would appreciate a reference so I could read more.

 

[xepma]

It's called non-commutative geometry. Basically it assumes that the elements of a manifold are non-commuting objects, i.e. the different components of the coordinates satsify some commutation relation:

[ x_i, x_j] = i \theta_{ij}

where the object on the right needs to be specified. Real coordinates would ofcourse commute.

 

[mathman]

Euclidean geometry is a branch of mathematics, not physics. How closely the real world (universe) comes to Euclidean geometry is a separate question.

 

[Daverz]

I think Euclidean geometry is physics. It models the objects of everyday experience extremely well. Of course it fails outside of its domain of validity.

 

[pythagoras88]

Sorry, but why does quantum mechanics need a non-commutative geometry? Is it because it is dealing complex space?

 

[K1NS]

Sorry, but why does quantum mechanics need a non-commutative geometry? Is it because it is dealing complex space?

OP here.

I'm not sure if quantum mechanics needs a non-commutative geometry. That was xepma who said that. xepma may be right that non-commutative and stochastic geometries are the same, but I am not enough of a mathematician to know.

But I believe it needs a stochastic geometry, based on probabilities. It is not because we are dealing with complex space (although I'm not sure what you mean--multidimensional, with imaginary dimensions?), but because we are dealing with granular space, at least at very small dimensions. If spacetime is indeed granular, then straight lines would randomly "wobble" and measurements would be replaced by probabilities. It seems to me that Euclidean geometry may be sufficient for macro-space and for approximations of micro-space, but for small dimensions "quantum" geometry would have to supplant Euclidean geometry much as quantum mechanics supplanted Newtonian and Einsteinian physics at small dimensions.

 

[pythagoras88]

I see... thanks for the explanation. I will go and do some read up on this!

 

[D H]

I think Euclidean geometry is physics. It models the objects of everyday experience extremely well. Of course it fails outside of its domain of validity.

Just because accountants use addition, subtraction, multiplication, and division does not mean that those operations are accounting rather than mathematics. Those concepts are mathematical concepts that happen to be very useful to accountants. Similarly, just because physicists use Euclidean geometry does not mean that Euclidean geometry is physics.

Euclidean geometry is mathematics, not physics. Whether Euclidean geometry does or does not exist in the "real world" is not necessarily relevant to mathematicians. Mathematics does not have to have any connection to the "real world" at all. In fact, a rather famous mathematician, G. H. Hardy, argued that the best mathematics has nothing to do with the "real world."

 

[comote]

You are trying to think about Quantum mechanics in a classical way. Quantum mechanics is done differently, it is all about manipulating information. Quantum mechanics does not imply that the world is granular at some level rather it implies that our knowledge about the world has certain limits.

When you transfer to Hilbert space you can in fact think this way that you have said.