Quantum Mechanics: Defining Dimensions

[Chemist@]

How is a dimension defined in quantum mechanics?

 

[Simon Bridge]

Usually the same as anywhere else.
Can you provide an example of what you are thinking of?

 

[tom.stoer]

To define dimension for an object like a manifold or a vector space just construct (locally!) a bijective map between this object a standard n-dim. Euclidean space.

Examples:
1) for the spin states |+> and |-> and their linear combinations you need a 2-dim. Euclidean space spanned by e₁ and e₂, so the state space is 2-dim.
2) for the simple harmonic oscillator you have to map |n> to an Euclidean space; b/c that doesn't work for finite n we say that the associated Hilbert space is ∞-dim.

 

[Fredrik]

I like to define the dimension of a vector space V as the positive integer n such that V contains a linearly independent set with n members, but no linearly independent set with n+1 members. If there's no such n, V is said to be infinite dimensional.

 

[tom.stoer]

That's a very elegant definition

 

[Chemist@]

What do you mean by the positive integer?

What are dimensions in the string theory then? I think there are 11.

 

[Fredrik]

What do you mean by the positive integer?

I'm not sure I understand the question. The positive integers are the numbers 1,2,3,... and so on. Does that answer your question?

What are dimensions in the string theory then? I think there are 11.

That refers to the dimension of spacetime. Spacetime is defined as a smooth manifold, not as a vector space. The simplest way to explain the dimension of smooth manifold is to say that it's the number of coordinates that a coordinate system assigns to each point in its domain.

The technical explanation goes like this: A smooth manifold is a topological manifold with additional structure. The dimension of the smooth manifold is defined as the definition of the underlying topological manifold. A topological manifold is a topological space that's locally homeomorphic to $\mathbb R^n$ for some positive integer n. The number n is called the dimension of the topological manifold.

 

[Chemist@]

Tough. Anyway, thanks for the explanation.

 

[Fredrik]

I should have included the simplified explanation: It's the number of real numbers that it takes to identify a specific event, i.e. a location in "space" at a moment in "time".

The tricky part is to understand how that number can be anything but four (three for space, one for time). The simplest possibility is that there are some directions in space with the property that if you move in one of those directions, you will end up traveling around the universe and come back from the opposite direction. If the distance required to complete a full "lap" around the universe this way is extremely short, much shorter than e.g. the distance an atom extends in the directions that don't have this property, then it's plausible that these extra dimensions (=independent special directions) are unnoticeable.

The book "The elegant universe" by Brian Greene makes an attempt to explain these things to readers who don't know math.

 

[ArtistIC]

Isn't a dimension meant to be an independent vector that cannot be described or calculated by another one (i.e. being vertical to another vector(s) ?

 

[Fredrik]

Isn't a dimension meant to be an independent vector that cannot be described or calculated by another one (i.e. being vertical to another vector(s) ?

I suppose so, but this notion is never given a precise meaning in math books. There's no definition that appears in math books that allows you to pick a specific line, vector or direction and say "this is a dimension". The precise definitions look like the statements I've made above. In the context of vector spaces, it has to do with linear independence. (How many members can a linearly independent set have?) In the context of manifolds, it has to to with how many real numbers a coordinate system associates with a point.

 

[tom.stoer]

Isn't a dimension meant to be an independent vector that cannot be described or calculated by another one (i.e. being vertical to another vector(s) ?

That's what Fredrik described in #4.

 

[ttakacs]

I have got 2 questions, please.

In microphysics one can met 11 or even more dimensions while macrospace is of 3(+1) dimensional. Is there any transition between them and is this transition is continuous or discrete?

At quantum level we are facing fluctuations of physical variables. What is the official contemporary standpoint: could -in principle- the number of space dimensions itself fluctuate? (I guess the calculation from uncertainty relations is not adequate here.)

Thank you.

 

[Fredrik]

In microphysics one can met 11 or even more dimensions while macrospace is of 3(+1) dimensional. Is there any transition between them and is this transition is continuous or discrete?

They just get more noticeable the smaller region of space you're looking at, like how the thickness of a wire get more noticeable when you zoom in on it.

At quantum level we are facing fluctuations of physical variables. What is the official contemporary standpoint: could -in principle- the number of space dimensions itself fluctuate? (I guess the calculation from uncertainty relations is not adequate here.)

As far as I know, the number is fixed.

 

[ttakacs]

Thank you.

I guess if there were fluctuations in dimension within our measurement range we could notice somehow. (Probably this is a fairy tale like e, pi or the universal constants of physics can change in time.)