vol 4: Glossary
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Tensor product
The stage for the quantum mechanical description of the world is known as Hilbert space. In elementary quantum mechanics, every isolated quantum system is imagnined to evolve in its own Hilbert space. In order to construct a universe from isolated systems, however, we must learn how to particles interact to form more complex systems.
Complex systems made from more elementary systems operate in Hilbert spaces which are formed by taking the 'tensor product' of the elementary Hilbert Spaces
Hilbert space, named for David Hilbert, is a mathematical construction derived from ordinary three dimensional Euclidean space. It is the mathematical stage upon which we play the model of the physical world known as quantum mechanics.
Cartesian coordinates enable us to give each point in Euclidean space a name which we represent by a set of numbers called a vector. In one dimensional space (that is on a line) each point is represented by just one number, its distance from an arbitrary point called the origin, whose coordinate is 0. A point one unit to the right of the origin is called 1, a point two units to the left, -2 and so on. Points in two dimensional space ("the cartesian plane") are specified by two numbers, and so on. Points in a space with a countable number of dimensions are specified by ordered sets or vectors with a countable number of elements or components. As the number of dimensions of space increases, the amount of information encoded in each point of the space also increases.
Each such vector may be understood as representing a series or discrete function, that is a mapping from the natural numbers 0, 1, 2, ... to another set of numbers, which may be natural, rational, real or complex. The number ex, for instance, is represented by the series 1 + x + x2/2 + ... + xn/n! + ... . Since each point in Hilbert space may be used to represent an infinite series, we may think of Hilbert space as a space of functions, or function space.
Like Euclidean space, Hilbert space is a metric space. In a metric space, there exists a function which, when applied to two points a and b, yields a number which is called the distance between the points. In Hilbert space, the distance between two points is always positive, and may only be zero if the two poitns are i identical.
Hilbert spaces are members of the class of 'inner product' spaces.
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