Cartesian Sums and Tensor Products

In this section we introduce two different ways of building a new vector space by combining old ones. 

One way to combine vector spaces is to take a Cartesian sum. (Mathematicians sometimes call this a Cartesian product. Another common term is direct sum.) 

Definition 2.11 Suppose Vi,. ..,Vn are vector spaces over the same scalar field. The Cartesian sum of these vector spaces, denoted Vi V or 

We will often use this isomorphism implicitly, letting V denote the subspace of W spanned by U =i writing Ui -P -H u instead of 

for example, C is equal (as a complex vector space) to the Cartesian sum of n copies of C  

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We will use Cartesian sums and tensor products to build and decompose representations in Chapters 5 and 7. Tensor products are useful in combining different aspects of one particle. For instance, when we consider both the mobile and the spin properties of an electron (in Section 11.4) the state space is the tensor product of the mobile state space defined in Chapter 3)... 

As an example, consider the product of two arbitrary first-rank tensor operators 0 and It is nine-dimensional and can be reduced to a sum of compound irreducible tensor operators of ranks 2, 1, and 0, respectively. Operators of this type play a role in spin-spin coupling Hamiltonians. In terms of spherical and Cartesian components of 0 and J2, the resulting irreducible tensors are given in Tables 8 and 9, respectively.70... 

Tensor notation is central to the theory of both linear and nonlinear light scattering, so a brief summary is appropriate here. A Greek subscript denotes a vector or tensor component and can be x,y or z. A repeated Greek subscript within a term denotes a sum over all three cartesian components this is the tensor equivalent of a scalar product so that, for example. 

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