Products in QM

[kleinwolf]

We have several products in Matrix Mechanics :

 

[math]C=A\otimes B \approx A_{ij}B[/math] Tensor product

Here A acts on a particle, and B on another particle.

 

[math]C=A\cdot B=\sum_k A_{ik}B_{kj}[/math] Matrix product

Here B acts on a particle, A acts after it on the same particle.

 

what about

 

[math]C:=A\times B=A_{ij}B_{ij}[/math] Element product ?

[ajb]

The tensor product of operators gives an operator on the tensor product of Hilbert spaces. So indeed your first operator acts on two particles (or two copies of the same particle) separately.

 

The matrix product of two operators acting on the same space gives another operator on that space. So again I agree with you so far.

 

I think your last product also known as the Hadamard or Schur product.

 

It can come in useful via Schur's theorem that states that for positive matrices, the Hadamard product is a positive matrix. I don't know much about this application so all I can suggest is hunting through the literature.

 

Charles R. Johnson's book Matrix theory and applications discusses the Hadamard product in detail.

 

One problem with the Hadamard product is that it depends explicitly on the basis chosen. This makes it quite unnatural.

[kleinwolf]

Thanks for the reference. I do not have looked at it yet. Does this book gives the physical interpretation to algebraic operation ?

[ajb]

Does this book gives the physical interpretation to algebraic operation ?

 

I do not know.

 

The Hadamard product does not seem to be common place in standard text books.

[kleinwolf]

It's a very interesting point I never thought about, that a product could be basis dependent, it's because it does not represent an intrisic operation (?)

 

By the way, do we know how many basis independent (intrinsic) product there are ?

[ajb]

It's a very interesting point I never thought about, that a product could be basis dependent, it's because it does not represent an intrisic operation (?)

 

By the way, do we know how many basis independent (intrinsic) product there are ?

 

 

For finite dimensional matrices in particular?

 

I have not thought much about it before. One can symmetrise and antisymmetrise tensor products. That can be very useful.

 

What you could do is write down a matrix in index notation [math]A_{a}^{\:\: b}[/math] (there is some conventions here about rows and columns, it should make no difference here if we restrict attention to matrices over complex or real numbers and not consider anything more "exotic" ) and then see what products you can come up with.

 

I expect we have only the standard matrix product [math]A_{a}^{\:\:c}B_{c}^{\:\:b}[/math]. (also known as inner product)

 

It has indices in the right places and by the tensor contraction theorem it is a genuine matrix.

 

Then there is the tensor or outer product [math]A_{a}^{\:\: b } B_{c}^{\: \: d} = C_{a \: \: \: \: c}^{\:\: b \: \: \: d}[/math] is not a matrix but a more general tensor.

[kleinwolf]

What is the difference between a matrix and a tensor ?

 

By natural products, do we mean operations X such that : [math] (P\times Q)^{-1}\cdot(A\times B)\cdot (P\times Q)=(P^{-1}\cdot A\cdot P)\times(Q^{-1}\cdot B\cdot Q), \cdot [/math] being the matrix product ?

 

i.e. the basis change of the product of 2 matrices should equals the product of their basis changed matrices.

 

---In fact I don't know if this thread should go to Algebra.

 

For example If I wrote it intrisically, is it true that the matrix product of the representation of the n-dimensional linear applications x->f(x) and x->g(x), should be

 

x->f(g(x)),

 

and the tensor product should (x,y)->f(x)*g(y) ? But I don't know what the * product here is.

Edited May 31, 2009 by kleinwolf

[ajb]

What is the difference between a matrix and a tensor ?

 

Have a look at the Wiki article on tensors.

 

The PlanetMath article is better.