Strictly speaking, you want a "tensor product" of the two single-particle spaces, notKFC said:For a system consists of two particle, says two spins, the wavefunction for that system is a direct product of each individual states. And the corresponding operator is also a direct product of each individual operators. I know this is a procedure to construct such a space, but can anybody told me why in this way? Why direct product?
A tensor product of 2 vector spaces yields another vector space. Anhow do we know the new space is still a Hilbert space?
strangerep said:Strictly speaking, you want a "tensor product" of the two single-particle spaces, not
the "direct product". See:
http://en.wikipedia.org/wiki/Direct_product
http://en.wikipedia.org/wiki/Tensor_product
The tensor product ensures the correct linearity properties (correct superposition
of 2-particle states).
A tensor product of 2 vector spaces yields another vector space. An
Hermitian inner product can be defined on the tensor product space
in this case easily enough. That almost gets you a Hilbert space.
For the final step (relating to the Hilbert space being "complete"),
one appeals to product topology (relating to the topologies of
the component spaces) to define "convergence" sensibly.
If it's not too much work for you, would you mind writing a short summary of when the different products are used and why? I actually still find these things a bit confusing. E.g. in the construction of a Fock space, the n-particle spaces of each species are supposed to be tensor products, right? And the Fock space itself is a direct sum of those? Is a "direct product" ever used at all?Haelfix said:Incidentally the confusion between tensor product, direct product, semidirect product, etc etc all applied to different mathematical objects (fields, rings, groups, spaces, modules etc etc), is one of the worst abuses of language I know off, and caused me a lot of grief back in the day.
Not only is it confusing to keep track off, in some cases there really is only surface level similarities between the concepts when applied.
In physics in particular, this can be particularly irratating, b/c its not always obvious upon inspection what various quantities actually are.
I'm guessing those were physics textbooks? Those tend to be insufficiently rigorousKFC said:Oh, this is really confusing. Because many textbook just said direct product.
I doubt that, but perhaps different authors have different conventions.I just look it up in some mathematics book, and I guess the 'direct product'
in some textbook is actually an abbreviation of 'tensor direct product'
strangerep said:Roughly, one could think of "direct product" of two vector spaces
V and W as the cartesian product [itex]V\times W[/itex]. If v,w are vectors
in V, W respectively, then the pair (v,w) is in [itex]V\times W[/itex]. However,
(2v, w) and (v, 2w) are distinct elements of [itex]V\times W[/itex], whereas
in the tensor product [itex]V\otimes W[/itex] these two are considered equivalent
(ie a single vector in the tp space). That's one of the properties
you want for a 2-particle Hilbert space: [itex]2(\psi_1\psi_2)[/itex] , [itex](2\psi_1)\psi_2[/itex]
and [itex]\psi_1 (2 \psi_2)[/itex] should all be the same state physically.
strangerep said:I'm guessing those were physics textbooks? Those tend to be insufficiently rigorous
in such matters.
A direct product space is a mathematical concept in which two or more mathematical structures, such as groups, vector spaces, or rings, are combined in a specific way to create a new structure. It is denoted by the symbol ⊗ or × and is used to represent the cartesian product of sets in a more general setting.
The purpose of a direct product space is to create a new mathematical structure that has properties of both of its components. It allows us to study multiple structures simultaneously and understand their interactions and relationships. Direct product spaces are also useful in applications such as coding theory and cryptography.
While both direct product and direct sum involve combining two or more structures, they differ in the way the components are combined. In a direct product, the elements are combined in a pairwise manner, whereas in a direct sum, the elements are combined by "stacking" them together. Additionally, a direct product space will inherit the operations and properties of its components, while a direct sum may have its own unique operations and properties.
Some examples of direct product spaces include the direct product of two groups, the direct product of two vector spaces, and the direct product of two rings. In group theory, the direct product of two groups A and B would be denoted by A × B and would consist of all ordered pairs (a, b) where a is an element of A and b is an element of B. In linear algebra, the direct product of two vector spaces V and W would be denoted by V ⊗ W and would consist of all possible linear combinations of elements from V and W.
Some key properties of direct product spaces include closure, associativity, distributivity, and commutativity. The direct product of two structures will also have a unique identity element and inverse element. Additionally, the direct product of two finite structures will have a finite order, which can be calculated by multiplying the orders of its components.