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A revision note by Phil dated 5.11.16 (with a 5.17.16 remark) for the tensor wedge document's Appendix A. It restates the tensor product definition for generic functions, tensors and tensor functions, then uses Dirac notation with products of generic kets to show why the outer product rule and the tensor-function product follow from the tensor space structure. It refers to Sections 2.8, 6.6 and Appendix D.
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App A added items PhL 5.11.16
This is a 5/11/16 update to Section A.10 where I include item (A.10.3) and explain why it works. I have a similar update do the tensor part of this appendix. // 5.17.16
A.10 Tensor Products in Generic Notation
Recall the following generic space definition of a tensor product of two functions made above in (A.2.19),
Definition: Tensor product: (fg)(1,2,....k+k') ≡ f(1,2...k) g(k+1,k+2....k+k') (A.2.19) (A.10.1)
If we translate this definition in the same way we translated everything else, we arrive at this corresponding statement in the tensor world,
Definition: Tensor product: (TS)ii...i ≡ Tii...i Sii...i,
where the ranks of tensors T,S are k,k'. (A.10.2)
Since (A.10.1) is already defined to be true using the "outer product definition" of the symbol in Section 2.8, it seems that here we are just lucky to obtain a consistent result. We put this issue on hold for a moment, and consider next the way (A.10.1) translates into the tensor function world,
Definition: Tensor product: (TS)(vi,vi....vi) ≡ T(vi,vi....vi) S(vi,vi....vi),
where the ranks of dual tensors T,S are k,k'. (A.10.3)
In Section 6.6 it seemed that we had to do quite a bit of work to obtain (A.10.3) as stated in (6.6.13), and here suddenly this same result just drops out as some kind of definition.
Both these issues can be clarified by use of the Dirac notation which reveals the underlying spaces. First, we can write (A.10.1) in the generic world as
<fg | 1,2,.....k+k'> = <f | 1,2,.....k> <g | k+1,k+2.....k+k'> (A.10.4)
where for example
| 1,2,.....k> = |1> |2> ... |k> (A.10.5)
<fg| = <f| <g| . (A.10.6)
Suddenly we are interpreting the generic argument set (1,2,...k) as if it were a tensor product of "generic kets". By itself, this does not really make much sense, but when we think of the generic description as being a stand-in for our tensor and tensor-function cases, then it does make sense. We show in Appendix D (D.1.2) and (D.1.5), and also in the main text (6.2.7) and (5.2.3a), that
T(vi,vi, .... vi) = <T | vi,vi, .... vi >
where | vi,vi, .... vi > = |vi> |vi> ..... |vi> (A.10.7)
Tii....i = <T | ui,ui, .... ui >
where | ui,ui, .... ui > = |ui> |ui> ..... |ui> (A.10.8)
and here we see the real meanings of the generic stand-in tensor product |1> |2> ... |k>.
For the tensor case we then write
(TS)ii...i = < TS| ui,ui, .... ui >
= [ k<T| k'<S| ] [ | ui,ui, .... ui >k | ui,ui, .... ui >k' ]
= k< T | ui,ui, .... ui >k k'< S | ui,ui, .... ui >k'
= Tii...i Sii...i (A.10.9)
and the tensor space structure then directly implies the "outer product" rule defined in Chapter 2. The subscript k labels a ket in Vk and a bra in V*k, for example.
In the tensor-function case we do exactly the same thing but with u→v,
(TS)(vi,vi....vi) = < TS | vi,vi....vi>
= [ k<T| k'<S| ] [ | vi,vi, .... vi >k | vi,vi, .... vi >k' ]
= k< T | vi,vi, .... vi >k k'< S | vi,vi, .... vi >k'
= T(vi,vi....vi) S(vi,vi....vi) (A.10.10)
and again the tensor space structure forces this tensor function result. In fact, this is exactly how we derived this result in (6.6.12) for a more general case.