Tensor product over 3 (or more) vector spaces

In summary, the problem is to write the product ##D_{\beta}R_{\beta\alpha 1}R_{\beta\alpha 2}##, which makes sense on the space ##V_{\beta} \otimes V_{\alpha 1} \otimes V_{\alpha 2}##.
  • #1
Maybe_Memorie
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Homework Statement



I have the operators

##D_{\beta}:V_{\beta}\rightarrow V_{\beta}##
##R_{\beta\alpha 1}: V_{\beta} \otimes V_{\alpha 1} \rightarrow V_{\beta}\otimes V_{\alpha 1}##
##R_{\beta\alpha 2}: V_{\beta} \otimes V_{\alpha 2} \rightarrow V_{\beta}\otimes V_{\alpha 2}##

where each of the vector spaces are copies of ##\mathbb{C}^2##

Homework Equations

The Attempt at a Solution



I want to write the product ##D_{\beta}R_{\beta\alpha 1}R_{\beta\alpha 2}##, which makes sense on the space ##V_{\beta} \otimes V_{\alpha 1} \otimes V_{\alpha 2}##.

So I order to act on the full space I write ##D## as ##D_{\beta}\otimes I_{\alpha 1} \otimes I_{\alpha 2}##
and write ##R_{\beta\alpha 1}## as ##R_{\beta\alpha 1} \otimes I_{\alpha 2}##, where ##I_{\alpha 1}## and ##I_{\alpha 2}## are the identity operators in ##V_{\alpha 1}## and ##V_{\alpha 2}##.

My problem is that I don't know how to write ##R_{\beta\alpha 2}## since I'd basically have to stick and identity "in the middle".
 
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  • #2
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  • #3
Maybe_Memorie said:

Homework Statement



I have the operators

##D_{\beta}:V_{\beta}\rightarrow V_{\beta}##
##R_{\beta\alpha 1}: V_{\beta} \otimes V_{\alpha 1} \rightarrow V_{\beta}\otimes V_{\alpha 1}##
##R_{\beta\alpha 2}: V_{\beta} \otimes V_{\alpha 2} \rightarrow V_{\beta}\otimes V_{\alpha 2}##

where each of the vector spaces are copies of ##\mathbb{C}^2##

What is the statement of the problem?
 
  • #4
There isn't any statement of the problem - it's something that I need for my Bachelor's thesis. However, I can provide some more detail. The operator ##T## is defined as ##D_{\beta}R_{\beta\alpha n}\dots R_{\beta\alpha 2}R_{\beta\alpha 1}##, where the D and R operators are defined as above.

It's equation (44) on page 8 of this paper http://arxiv.org/pdf/1204.2089v3.pdf
 
  • #5
Maybe_Memorie said:
I want to write the product ##D_{\beta}R_{\beta\alpha 1}R_{\beta\alpha 2}##, which makes sense on the space ##V_{\beta} \otimes V_{\alpha 1} \otimes V_{\alpha 2}##.

Do you mean "I want to define a product..."?

I don't know what "makes sense" implies with respect to physics (because I don't understand the physics). Thinking only of mathematics, an attempt to refine the question could begin:

Let [itex] V_{\beta},V_{\alpha_1},V_{\alpha_2} [/itex] be vector spaces.
Let [itex] R_{\beta \alpha_2} [/itex] be an operator [itex] R_{\beta \alpha_2}: V_\beta \otimes V_{\alpha_2} \rightarrow V_\beta \otimes V_{\alpha_2} [/itex].

Construct an operator [itex] R: V_{\beta} \otimes V_{\alpha_1} \otimes V_{\alpha_2} \rightarrow V_\beta \otimes V_{\alpha_1} \otimes V_{\alpha_2} [/itex] such that ...

Roughly speaking, we want what [itex] R [/itex] does on [itex] V_\beta \otimes V_{\alpha_1} \otimes V_{\alpha_2} [/itex] to project to what [itex] R_{\beta \alpha_2} [/itex] does on [itex] V_\beta \otimes V_{\alpha_2} [/itex].

I don't know whether we are dealing with finite dimensional vectors spaces. Perhaps we are only dealing with linear operators. Perhaps you need to express the operator [itex] R [/itex] explicitly as a product of matrices or tensor product of operators.

In finite dimensions, you can find an embedding operation [itex] E: V_\beta \otimes V_{\alpha_2} \rightarrow V_\beta \otimes V_{\alpha_1} \otimes V_{\alpha 2} [/itex] and a projection operation [itex] P: V_\beta \otimes V_{\alpha_1} \otimes V_{\alpha 2} \rightarrow V_\beta \otimes V_{\alpha_2}[/itex] and define [itex] R = E R_{\beta \alpha_2} P [/itex]If the mathematical requirements aren't specific then I'd say this isn't a "homework type problem". You'd could ask about it in the physics sections of the forum.
 

FAQ: Tensor product over 3 (or more) vector spaces

What is a tensor product over 3 (or more) vector spaces?

A tensor product over 3 (or more) vector spaces is a mathematical operation that combines vectors from multiple vector spaces to create a new vector space. It is denoted by the symbol ⊗ and is used in various fields of mathematics and physics, such as linear algebra, quantum mechanics, and general relativity.

How is the tensor product over 3 (or more) vector spaces different from the tensor product over 2 vector spaces?

The main difference between tensor product over 3 (or more) vector spaces and the tensor product over 2 vector spaces is the number of vector spaces involved. The tensor product over 3 (or more) vector spaces combines vectors from three or more vector spaces, while the tensor product over 2 vector spaces combines vectors from only two vector spaces.

What are some applications of the tensor product over 3 (or more) vector spaces?

The tensor product over 3 (or more) vector spaces has many applications in mathematics and physics. It is used in quantum mechanics to describe the state of a system, in general relativity to describe space-time curvature, and in computer graphics to represent 3D objects.

How is the tensor product over 3 (or more) vector spaces calculated?

The tensor product over 3 (or more) vector spaces is calculated by taking the tensor product of individual vectors from each vector space. This involves multiplying the components of each vector and then combining them in a specific way to create a new vector.

What are some properties of the tensor product over 3 (or more) vector spaces?

The tensor product over 3 (or more) vector spaces has several important properties, including associativity, distributivity, and bilinearity. It also follows the commutative property, meaning the order in which the tensor product is taken does not affect the result. Additionally, the tensor product is also multiplicative with respect to scalar multiplication.

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