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Tensor product over 3 (or more) vector spaces

  • #1

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".
 

Answers and Replies

  • #2
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  • #3
Stephen Tashi
Science Advisor
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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##
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
Stephen Tashi
Science Advisor
7,017
1,237
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.
 

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