Interpretation of direct product of Hilbert spaces

[IanBerkman]

Dear all,

I know how to interpret a vector, inner product etcetera in one Hilbert space. However, I can not get my head around how the direct product of two (or more) Hilbert spaces can be interpreted.
For instance, the Hilbert space $W$ of a larger system is spanned by the direct product of the Hilbert spaces of the subsystem, $W = A \otimes B$. If the states $| n_A\rangle$ denote the basis states in $A$ and $|m_B\rangle$ those in $B$, then any state in $W$ can be written as $|\psi_W\rangle = \sum_{n,m} c_{nm}|n_A\rangle \otimes |m_B\rangle$.

How do you interpret the direct product of two Hilbert spaces?

 

[micromass]

This is the tensor product, not the direct product.

And what do you mean with interpret? What exactly is it that you want?

 

[IanBerkman]

Hmm, the book talks about the direct product between more Hilbert spaces.

I know how to "visualize" one Hilbert space i.e. a vector with eigenstates as orthogonal axes. However, is there also an intuitive way to visualize a Hilbert space which consists of the tensor product of two other Hilbert spaces? Which values are on the axes? How is a vector in the smaller system living in the Hilbert space represented in the larger system? Or is the tensor product purely mathematical?

If the question is unclear, please let me know.

Thanks.

 

[fresh_42]

As you mentioned coordinates (which are the scalars at basis vectors, their coefficients), do you know what $|n_A\rangle \otimes |m_B\rangle$ is according to your basis?

 

[chiro]

Hey IanBerkman.

If it is a tensor product (as micromass alluded to) then you should understand that the tensor product does what a Cartesian product does on sets but maintains the consistency and intuition of a vector space.

Basically vector spaces behave like "arrows" and when you take a tensor product you find out how to keep the space acting that way (like "arrows") with the intuition of a linear space but with the extra attribute of a Cartesian product (where you find all kinds of combinations of elements when you consider all permutations of each set).

So the points in the space become all combinations but the thing that separates this is that you have the vector space part and this is algebraic.

When you take the tensor product you can use the Cartesian product of two sets to get an idea of the points that remain but you have to make the space consistent with the organization that a vector space has (i.e. those ten axioms that every one has along with the inner product, norm, and metric space axioms it may additionally have) and what that means is that you need ways of identifying how to define the addition and scalar multiplication of the new space along with how a change of basis can occur and how the spaces are composed - which is what the tensor product is doing (it in "essence" is "composing" the "spaces").

It's like how you compose two mappings together except that you have the vector space axiom constraints to think about and make sure the new space is consistent with these new ones.