Tensor product and representations

[JonnyMaddox]

Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of a group with a matrix representation as:
\hat{G_{i}}=q^{I}(G_{i})_{I}^{J}p_{j} where q and p are the coordinates and the conjugate momenta.

Can someone give me an easy example? :)

 

[Nugatory]

I'm reading a book

Which book?

 

[JonnyMaddox]

Which book?

The bible...no jk I'm reading "Fields" from Warren Siegel. It's free on arxiv.

 

[samalkhaiat]

Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of a group with a matrix representation as:
\hat{G_{i}}=q^{I}(G_{i})_{I}^{J}p_{j} where q and p are the coordinates and the conjugate momenta.

Can someone give me an easy example? :)

Consider an abstract, n-dimensional linear vector space spanned by complete and ortho-normal basis vectors, i.e. \langle I | J \rangle = \delta_{I}^{J} , \ \ \ \ \ (1) \sum_{K}^{n} | K \rangle \langle K | = \mathbb{I}_{n} . \ \ \ \ (2) In this n-dimensional (index) space, a n \times n matrix M acts as operator: M | J \rangle = | M J \rangle = \sum_{K}^{n} M^{J}_{K} \ | K \rangle , \ \ \ (3) where the numbers M^{J}_{I} , i.e. the expansion coefficients, are determined from the orthonomality condition (1): \langle I | M | J \rangle = \sum_{K} M^{J}_{K} \ \delta^{K}_{I} = M^{J}_{I} . To construct a matrix representation of operator M, we use the completeness relation (2) and the expansion (3): M \sum_{J} | J \rangle \langle J | = M_{n \times n} = \sum_{I , J} M^{J}_{I} \ | I \rangle \langle J | . \ \ \ (4) As an example, consider 2-dimensional index space with basis vectors | 1 \rangle = \langle 1 |^{ \dagger } = ( 1 \ , \ 0 )^{T} , | 2 \rangle = \langle 2 |^{ \dagger } = ( 0 \ , \ 1 )^{T} . In this case, equation (4) should give a 2 \times 2 matrix representation for the operator M: M_{ 2 \times 2 } = \sum_{I , J}^{2} M^{J}_{I} \ | I \rangle \langle J | . \ \ \ (5) Indeed, since | 1 \rangle \langle 1 | = \left( \begin {array} {c c} 1 & 0 \\ 0 & 0 \end {array} \right) , \ \ | 1 \rangle \langle 2 | = \left( \begin{array} {c c} 0 & 1 \\ 0 & 0 \end{array} \right) , | 2 \rangle \langle 1 | = \left( \begin{array} {c c} 0 & 0 \\ 1 & 0 \end{array} \right) , \ \ | 2 \rangle \langle 2 | = \left( \begin{array} {c c} 0 & 0 \\ 0 & 1 \end{array} \right) , equation (5) gives M_{2 \times 2} = \left( \begin{array} {c c} M^{1}_{1} & M^{2}_{1} \\ M^{1}_{2} & M^{2}_{2} \end{array} \right) .

Sam

 

[JonnyMaddox]

Ok thanks Sam. Could you also help me with coordinate representations? How do I apply this formula
\hat{G_{i}}=q^{I}(G_{i})_{I}^{J}p_{j}
Is this like an inner product of the coordinates and the conjugate momentum?
Like this
\hat{G_{1}}=q^{1}(G_{1})_{1}^{1}p_{1}+q^{2}(G_{1})_{2}^{2}p_{2}+...

 

[Demystifier]

The bible...no jk I'm reading "Fields" from Warren Siegel. It's free on arxiv.

It's too advanced for a beginner. You need something which presents all the elementary calculational details, and for that purpose I recommend McMahon
https://www.amazon.com/dp/0071765638/?tag=pfamazon01-20

 

[JonnyMaddox]

It's too advanced for a beginner. You need something which presents all the elementary calculational details, and for that purpose I recommend McMahon
https://www.amazon.com/dp/0071765638/?tag=pfamazon01-20

Sry but that's not answer to my question.

 

[Demystifier]

Sry but that's not answer to my question.

I know, but unless I misjudged you, I thought it might be an answer to many similar questions you might have in the future:
http://en.wiktionary.org/wiki/give_...a_man_to_fish_and_you_feed_him_for_a_lifetime

 

[samalkhaiat]

Ok thanks Sam. Could you also help me with coordinate representations? How do I apply this formula
\hat{G_{i}}=q^{I}(G_{i})_{I}^{J}p_{j}
Is this like an inner product of the coordinates and the conjugate momentum?
Like this
\hat{G_{1}}=q^{1}(G_{1})_{1}^{1}p_{1}+q^{2}(G_{1})_{2}^{2}p_{2}+...

First, you need to understand the connection between this and equation (4) in my previous post. In here, ( q^{I} , p_{I} ) are continuous coordinate numbers, i.e. they don’t (directly) span the index space of matrices. However, if we write |Q \rangle = \sum_{I}^{n} q_{I} \ | I \rangle , \ \ \Rightarrow \ q^{I} = ( q_{I} )^{T} = \langle Q | I \rangle , | P \rangle = \sum_{J}^{n} p_{J} \ | J \rangle , \ \ \Rightarrow \ p_{J} = \langle J | P \rangle , then we can transform equation (4) into equation similar to the one you wrote: M ( p , q ) \equiv \langle Q | M | P \rangle = \sum_{I , J} M^{J}_{I} \ \langle Q | I \rangle \langle J | P \rangle = \sum_{I , J} M^{J}_{I} \ q^{I} \ p_{J} . In fact (see the exercise below) G^{I}_{J} \equiv i | I \rangle \langle J | and J^{I}_{J} \equiv q^{I} \ p_{J} generate “isomorphic” Lie algebras.
Now, let us talk about coordinate representation. Let G_{a} , a = 1 , 2 , \cdots , m be basis in an m-dimensional Lie algebra \mathcal{L}^{m} with the following Lie bracket relations [ G_{a} , G_{b} ] = C_{a b}{}^{c} \ G_{c} . \ \ \ \ \ (1) Since every Lie algebra has a faithful matrix representation, we may the G_{a}’s to be a set of (m) matrices (n \times n) and, therefore, realizing the Lie brackets by commutation relations [ G_{a} , G_{b} ]^{J}_{I} = C_{a b}{}^{c} \ ( G_{c} )^{J}_{I} , \ \ I , J = 1 , 2 , \cdots , n . \ \ \ (2) Now, we take n-pairs of real munbers ( q^{I} , p_{I} ) and define m numbers (functionals) J_{a} ( q , p ) by J_{a} = ( G_{a} )^{J}_{I} \ q^{I} \ p_{J} , \ \ \ a = 1 , 2 , \cdots , m . \ \ \ \ (3) Clearly, the set of numbers J_{a} forms a representation of \mathcal{L}^{m} , i.e. they satisfy a Lie bracket relation. To see this, take ( q^{I} , p_{I} ) to be local coordinates on Poisson manifold \mathcal{P}^{2 n} and evaluate the Poisson bracket \{ J_{a} , J_{b} \} = \sum_{K}^{n} \left( \frac{ \delta J_{a} }{ \delta q^{K} } \frac{ \delta J_{b} }{ \delta p_{K} } - \frac{ \delta J_{a} }{ \delta p_{K} } \frac{ \delta J_{b} }{ \delta q^{K} } \right) . Using (2) and (3), we find \{ J_{a} , J_{b} \} = C_{a b}{}^{c} \ J_{c} . \ \ \ \ \ (4) Since the structure constants of \mathcal{L}^{m} appear on the RHS of (4), then \{ J_{a} , J_{b} \} is a Lie bracket on \mathcal{L}^{m}. Mathematically speaking, to every (associative) Lie algebra there corresponds a Poisson structure, i.e., the universal enveloping algebra of \mathcal{L}^{m} is a Poisson-Lie algebra.

Okay, now I leave you with the following exercise. Define M^{I}_{J} = i \ | I \rangle \langle J | , \ \ \mbox{and} \ \ G^{I}_{J} = q^{I} \ p_{J} , then prove the following Lie brackets [ M^{I}_{J} , M^{L}_{K} ] = \delta^{I}_{K} \ M^{L}_{J} - \delta^{L}_{J} \ M^{I}_{K} , \{ G^{I}_{J} , G^{L}_{K} \} = \delta^{I}_{K} \ G^{L}_{J} - \delta^{L}_{J} \ G^{I}_{K} . What is the corresponding Lie group?


Sam

 

[JonnyMaddox]

Ok I see. Now I understood this:
A_{ij}|e_{i}><e_{j}|= A_{11}|e_{1}><e_{1}|+A_{21}|e_{2}><e_{1}|+A_{12}|e_{1}><e_{2}|+A_{22}|e_{2}><e_{2}|=A_{11}\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}+A_{21}\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}+A_{12}\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}+A_{22}\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}=\begin{pmatrix} A_{11} & A_{12} \\ A_{12} & A_{22} \end{pmatrix}
I think I have to wrap my head a little around how he (in his book) uses this in all the different ways it can be used. So when he uses |^{I}> in the context of a vector V it just means |^{I}>=|e_{i}>. Then in the context of matrices or generators he uses it as |^{I}>=|e_{i}><e_{j}|? Same with the conjugate representations and so on.
And I'm right that this equation I asked you about is analogous to this equation with which you can define a regular representation of a finite group:
[D(g)]_{ij}=<e_{i}|D(g)|e_{j}>?
Ok I think I have to play a little around with all this and then I will try to solve your exercise. Thank you Sam !

 

[JonnyMaddox]

Ok I think I'm confused about what q^{I} and p_{J} are. Coordinates and conjugate momentum ok. But are they the basis of the space or what? If not how does this make sense \hat{G_{i}}= q^{1}(G_{1})_{1}^{1}p_{1}+...? So it's just a number or what?

 

[samalkhaiat]

Ok I think I'm confused about what q^{I} and p_{J} are. Coordinates and conjugate momentum ok. But are they the basis of the space or what?

Did you read the first two lines in post #9 ?

If not how does this make sense \hat{G_{i}}= q^{1}(G_{1})_{1}^{1}p_{1}+...? So it's just a number or what?

Did you read the line just before equation (3) in post #9 ?