Notes on Tensor Products Meta

Meta Notes on Tensor Products PhL 1.19.15 First of all, I think my tensor doc makes use of the outer product, which is a special case of a tensor product. I think the term direct product there is incorrect, but people do use it. I review the tensor doc sections on such products in the raw notes. There seem to be two major theoretical approaches to the Tensor Product. 1. The first approach starts with a free vector space, defines bilinear-like equivalence classes, and then concludes that VW= F(VxW)/N where F is the free vector space defined over the Cartesian product Vx W, and N is the subspace of elements in F(VxW) which are declared equivalent to 0. This basically has the effect of imposing what I call the bilinearity relations on the operator. 2. The second approach starts off with bilinear functions in a category triangle diagram, Since a unique and linear τ can be constructed to satisfy the triangle (that is, to make it "commute"), the pair (T,t) is a universal pair, and then by definition the space T is the tensor product space T = UV. This approach emphasizes that the tensor product is a more or less unique construct, up to isomorphism via the linear function τ. One has t : UxV → UV so t(u,v) → uv The bilinearity rules then just move from function t to the tensor product uv. For example t(αu+βu',v) = α t(u,v) + βt(u',v) maps into (αu + βu')v = α uv + β u'v 3. Both approaches arrive at the same notion of a tensor product. The main aspects are bilinearity and uniqueness of the tensor product. 4. One can generalize to a tensor product ABC..... . In this case, the triangle diagram has at its left vertex A x B x C.... . The bilinearity rules become multilinearity rules. 5. The outer product is a tensor product in which an extra ingredient is added, resulting in no conflict with the tensor product structure as outlined above. One writes for example, (ABC ...)abc... ≡ AaBbCc..... (MN)ab,AB ≡ MaANbB (MN)ab,AB,αβ = MaAαNbBβ In all these cases, the expressions on the right have the correct multilinearity form. For example ([αM+βM']N)ab,AB,αβ = [αM+βM']aAαNbBβ = α MaAαNbBβ + β M'aAαNbBβ 6. One can write a most general element of VV in this manner T ≡ ΣijTij eiej but then with the extra ingredient, we can write TIJ ≡ ΣijTij [eiej]IJ = ΣijTij (ei)I (ej)J = Σij Tij δiIδjJ = TIJ and so the rank-2 tensor components agree left and right end. 7. When the outer product construction has involves tensors of rank 2 or more (as shown in the last two examples above), it is called a Kronecker product, but again it is just a special version of the generic tensor product.