Product Spaces

Prerequisites: Linear Algebra.

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

A product space is a way to build a bigger space from smaller ones by describing a point with one entry from each. A point in the plane is given by two numbers, one for the horizontal line and one for the vertical line, so the plane is the product of two lines. In the same way, the state of a system of two independent parts is given by listing the state of the first part and the state of the second part. If a bead can sit anywhere on a ring and a second bead anywhere on a second ring, the possible arrangements of the pair form the product of two circles, which is the surface of a doughnut.

The product inherits its structure from the factors. Distances, notions of "nearby", and smoothness are defined entry by entry. The size also combines in a simple way: if one factor has three possible values and the other has four, the product has twelve; if one factor is a line and the other a plane, the product is three-dimensional.

Physicists use this construction all the time without naming it, for example when they say that the configuration space of two particles in space has six coordinates.

• Multiplying Non-Numbers

Concrete

The basic construction is the Cartesian product of two sets,

$$X\times Y = \{(x,y) \mid x\in X,\ y\in Y\},$$

together with the projections $\pi_X(x,y)=x$ and $\pi_Y(x,y)=y$. If $X$ and $Y$ carry more structure, the product carries the same kind of structure, defined componentwise:

• Vector spaces. For vector spaces $V$ and $W$, $V\times W$ with $(v,w)+(v',w') = (v+v',w+w')$ and $\lambda(v,w) = (\lambda v,\lambda w)$ is again a vector space, usually written $V\oplus W$ (direct sum). Its dimension is $\dim V + \dim W$; a basis is $\{(e_i,0)\}\cup\{(0,f_j)\}$. For example $\mathbb{R}^n\times\mathbb{R}^m = \mathbb{R}^{n+m}$.
• Manifolds. If $M$ has dimension $m$ and $N$ has dimension $n$, then $M\times N$ is a manifold of dimension $m+n$, with charts $\phi\times\psi$. Its tangent spaces split as $T_{(p,q)}(M\times N) = T_pM\oplus T_qN$. The torus is $T^2 = S^1\times S^1$, and the cylinder is $S^1\times\mathbb{R}$.
• Groups. For groups $G$ and $H$ the direct product $G\times H$ has the multiplication $(g,h)(g',h') = (gg',hh')$. The gauge group of the Standard Model, $SU(3)\times SU(2)\times U(1)$ (up to a finite quotient), is of this form.
• Metrics. If $M$ and $N$ carry metrics $g_M$ and $g_N$, then $g = g_M + g_N$ is the product metric. Flat space-time $\mathbb{R}\times\mathbb{R}^3$ with $ds^2 = -dt^2 + d\vec x^{\,2}$ is an example (with signature $(-,+,+,+)$).

Example: configuration space. One particle in space has configuration space $\mathbb{R}^3$. Two distinguishable particles have configuration space $\mathbb{R}^3\times\mathbb{R}^3 = \mathbb{R}^6$, with coordinates $(\vec x_1,\vec x_2)$. A double pendulum in a plane has configuration space $S^1\times S^1 = T^2$, parametrised by two angles. The phase space of a particle on a line is $\mathbb{R}\times\mathbb{R}$, with coordinates $(q,p)$.

A common confusion. The state space of a composite quantum system is the tensor product $\mathcal{H}_1\otimes\mathcal{H}_2$ of the Hilbert spaces, not their product $\mathcal{H}_1\times\mathcal{H}_2 = \mathcal{H}_1\oplus\mathcal{H}_2$. The dimensions differ: for $n$ qubits the tensor product $(\mathbb{C}^2)^{\otimes n}$ has dimension $2^n$, while the direct sum of $n$ copies of $\mathbb{C}^2$ has dimension $2n$. The classical configuration spaces still combine by the Cartesian product, and the wave function of two particles is a function on $\mathbb{R}^3\times\mathbb{R}^3$. This is consistent, since $L^2(\mathbb{R}^3\times\mathbb{R}^3) \cong L^2(\mathbb{R}^3)\otimes L^2(\mathbb{R}^3)$.

Abstract

In category theory the product of two objects $X$ and $Y$ is an object $X\times Y$ with morphisms $\pi_X: X\times Y\to X$ and $\pi_Y: X\times Y \to Y$ such that for every object $Z$ and every pair of morphisms $f:Z\to X$, $g:Z\to Y$ there is exactly one morphism $\langle f,g\rangle: Z\to X\times Y$ with $\pi_X\circ\langle f,g\rangle = f$ and $\pi_Y\circ\langle f,g\rangle = g$. This universal property determines the product up to unique isomorphism. The constructions of the Concrete layer are the products in the categories of sets, vector spaces, groups, topological spaces and smooth manifolds.

In topological spaces the product topology on $X\times Y$ is the coarsest topology for which both projections are continuous; a basis is given by the sets $U\times V$ with $U$, $V$ open. For an arbitrary family $\{X_i\}_{i\in I}$ the product topology on $\prod_i X_i$ has as basis the sets $\prod_i U_i$ with $U_i = X_i$ for all but finitely many $i$. This is the topology for which the universal property holds; the box topology, in which all $U_i$ may be proper open subsets, is finer for infinite $I$ and does not have it. Tychonoff's theorem states that any product of compact spaces is compact in the product topology.

In the category of vector spaces, finite products and coproducts coincide, $V\times W\cong V\oplus W$, with inclusions $v\mapsto(v,0)$ and $w\mapsto(0,w)$. The tensor product $V\otimes W$ is a different object: it is characterised by the universal property for bilinear maps, $\mathrm{Bil}(V\times W, U)\cong\mathrm{Hom}(V\otimes W,U)$, and it is the monoidal structure used for composite quantum systems.

Product spaces are the local model for fiber bundles. A bundle $E\to M$ with fibre $F$ is locally of the form $U\times F$ with the projection onto $U$, but in general not globally. The bundle is trivial exactly when $E\cong M\times F$ over $M$. The Möbius strip is the standard example of a bundle over $S^1$ with fibre $\mathbb{R}$ that is not a product.

Why is it interesting?

Product spaces are an important tool, for example, to construct configuration spaces and in considering fiber bundles.

Leads to: Topology.