9.1 Tensor products

9.1 Tensor products

Concept links · terms present in this machine draft; source roles are unverified: vector space · linear map · complexification

Given two vector spaces $V$ and W (over $R$ or C), the direct sum vector space $V\oplus W$ is constructed by taking pairs of elements $(v,w)$ for $v\in V,w\in W$ , and giving them a vector space structure by the obvious addition and multiplication by scalars. This space will have dimension

$$\dim (V\oplus W)=\dim V+\dim W$$

If $\left\{{\mathbf{e}}_1,{\mathbf{e}}_2,\dots ,{\mathbf{e}}_{\dim V}\right\}$ is a basis of $V$ , and $\left\{{\mathbf{f}}_1,{\mathbf{f}}_2,\dots ,{\mathbf{f}}_{\dim W}\right\}$ a basis of $W$ , the

$$\left\{{\mathbf{e}}_1,{\mathbf{e}}_2,\dots ,{\mathbf{e}}_{\dim V},{\mathbf{f}}_1,{\mathbf{f}}_2,\dots ,{\mathbf{f}}_{\dim W}\right\}$$

will be a basis of $V\oplus W$

A less trivial construction is the tensor product of the vector spaces $V$ and W. This will be a new vector space called $V\otimes W$ , of dimension

$$\dim (V\otimes W)=(\dim V)(\dim W)$$

One way to motivate the tensor product is to think of vector spaces as vector spaces of functions. Elements

$$v=v_1{\mathbf{e}}_1+v_2{\mathbf{e}}_2+\cdots +v_{\dim V}{\mathbf{e}}_{\dim V}\in V$$

can be thought of as functions on the dim V points ${\mathbf{e}}_j,$ taking values $v_j$ at ${\mathbf{e}}_j$ . If one takes functions on the union of the sets $\{{\mathbf{e}}_j\}$ and $\left\{{\mathbf{f}}_k\right\}$ one gets elements of $V\oplus W$ . The tensor product $V\otimes W$ will be what one gets by taking all functions on not the union, but the product of the sets $\{{\mathbf{e}}_j\}$ and $\left\{{\mathbf{f}}_k\right\}$ . This will be the set with (dim V)(dim W) elements, which we will write ${\mathbf{e}}_j\otimes {\mathbf{f}}_k$ , and elements of $V\otimes W$ will be functions on this set, or equivalently, linear combinations of these basis vectors.

This sort of definition is less than satisfactory, since it is tied to an explicit choice of bases for V and W. We won’t however pursue more details of this question or a better definition here. For this, one can consult pretty much any advanced undergraduate text in abstract algebra, but here we will take as given the following properties of the tensor product that we will need:

• Given vectors $v\in V,w\in W$ we get an element $v\otimes w\in V\otimes W$ , satisfying bilinearity conditions (for $c_1,c_2$ constants)

$$v\otimes (c_1{w}_1+c_2{w}_2)=c_1(v\otimes w_1)+c_2(v\otimes w_2)$$ $$\left(c_1{v}_1+c_2{v}_2\right)\otimes w=c_1(v_1\otimes w)+c_2(v_2\otimes w)$$

• There are natural isomorphisms

$$V\otimes W\simeq W\otimes V$$

and

$$U\otimes (V\otimes W)\simeq (U\otimes V)\otimes W$$

for vector spaces $U,V,W$

• Given a linear operator $A$ on $V$ and another linear operator $B$ on $W$, we can define a linear operator $A\otimes B$ on $V\otimes W$ by

$$(A\otimes B)(v\otimes w)=Av\otimes Bw$$

for $v\in V,w\in W.$

With respect to the bases ${\mathbf{e}}_j,{\mathbf{f}}_k$ of V and W, $A$ will be a (dim V) by (dim V ) matrix, $B$ will be a (dim W) by (dim W) matrix and $A\otimes B$ will be a (dim V)(dim W) by (dim V)(dim W) matrix (which can be thought of as a (dim V) by (dim V) matrix of blocks of size (dim W)).

• One often wants to consider tensor products of vector spaces and dual vector spaces. An important fact is that there is an isomorphism between the tensor product $V^{\ast }\otimes W$ and linear maps from V to W. This is given by identifying $l\otimes w(l\in V^{\ast },w\in W)$ with the linear map

$$v\in V\to l(v)w\in W$$

• Given the motivation in terms of functions on a product of sets, for function spaces in general we should have an identification of the tensor product of function spaces with functions on the product set. For instance, for square-integrable functions on $R$ we expect

$$\underset{\text{n times}}{\underbrace{L^2(\mathbf{R})\otimes L^2(\mathbf{R})\otimes \cdots \otimes L^2(\mathbf{R})}}=L^2({\mathbf{R}}^n)\text{\hspace{1em}(9.1)}$$

For V a real vector space, its complexification $V_{\mathbf{C}}$ (see section 5.5) can be identified with the tensor product

$$V_{\mathbf{C}}=V{\otimes }_{\mathbf{R}}\mathbf{C}$$

Here the notation ${\otimes }_{\mathbf{R}}$ indicates a tensor product of two real vector spaces: $V$ of dimension dim V with basis $\left\{{\mathbf{e}}_1,{\mathbf{e}}_2,\dots ,{\mathbf{e}}_{\dim V}\right\}$ and $\mathbf{C}={\mathbf{R}}^2$ of dimension 2 with basis $\{1,i\}$

Chapter contents · Next section: 9.2 Composite quantum systems and tensor products · 中文版本

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