A fresh look at the tensor product
(the very first lesson in category theory)
Recently I was reviewing Hopf algebras and their applications in physics. This is a very interesting and straightforward topic on par with linear algebra which students learn in first year in college, but unfortunately not well known in the physics community. Starting with this post I will present a gentle introduction and motivation and we'll get all the way to the application in renormalization theory for quantum field theory.
The place to start is to understand what a tensor product really is. In physics one encounters tensors every step of the way and the usual drill is about covariant and contravariant tensors, but this is not what tensors are about.
We want to start with two vector spaces V and W over the real numbers and attempt to combine them, The easiest way to do that is to have the cartesian product VxW which are the pairs of elements (v, w) each of them in their vector space. If those spaces are finite dimensional, say of dimensions m and n, what is the dimension of VxW? The dimension is m+n but we want to combine them in a tighter way such that the resulting object dimension is m*n not m+n. How can we get from the Cartesian product to the tensor product?
The mathematical answer is a bit dry so let's simply state it. We start with a free vector space over out field or real numbers F(V) and this is nothing but formal sums of elements in V such as:
\(\alpha_1 v_1 + \alpha_2 v_2 + \cdots \alpha_n v_n\)
with \(\alpha_i \in R\) and \(v_i\) in V.
Then of course we can consider F(VxW) and now let's ask what the dimension of this object is? Its dimension equals the number of elements in VxW which is infinite and so we constructed a big monstrosity. We want the dimension of the tensor product to be m*n so to get from \( F(V \times W)\) to \(V\otimes W\) we want to cut down the dimension of \( F(V \times W)\) by using appropriate equivalence relations which capture the usual behavior of tensor products.
To recap, we started with \(V \times W\) but this is too small. We expand it to \(F(V\times W)\) but this is too big, and now we'll cut it down to "Goldilocks" size by equivalence relations.
What are the properties of \(v\otimes w\)? Not too many:
- \(\lambda (v\otimes w) = (\lambda v)\otimes w\)
- \(\lambda (v\otimes w) = v\otimes (\lambda w)\)
- \(v_1\otimes w + v_2\otimes w = (v_1 + v_2)\otimes w\)
- \(v\otimes w_1 + v\otimes w_2 = v\otimes (w_1 + w_2 )\)
Then \(F(V\times W) \) modulo the equivalence relationship above is the one and only tensor product: \(V\otimes W\) with dimension m*n.
So what? What is the big deal? Stay tuned for next time when this humble tensor product will transform the way we look at products in general.