Composite systems and tensor products
Robert B. Griffiths
Introduction A composite system is one involving more than one particle, or a particle with internal degrees of freedom in addition to its center of mass. In classical mechanics the phase space of a composite system is a Cartesian product of the phase spaces of its constituents. The Cartesian product of two sets A and B is the set of (ordered) pairs {( a, b )}, where a is any element of A and b is any element of B . For three sets A, B , and C the Cartesian product consists of triples {( a, b, c )}, and so forth. Consider two classical particles in one dimension, with phase spaces x 1 , p 1 and x 2 , p 2 . The phase space for the composite system consists of pairs of points from the two phase spaces, that is, it is a collection of quadruples of the form x 1 , p 1 , x 2 , p 2 , which can equally well be written in the order x 1 , x 2 , p 1 , p 2 . This is formally the same as the phase space of a single particle in two dimensions, a collection of quadruples x, y, p x , p y . Similarly, the six-dimensional phase space of a particle in three dimensions is formally the same as that of three one-dimensional particles. In quantum theory the analog of a Cartesian product of classical phase spaces is a tensor product of Hilbert spaces. A particle in three dimensions has a Hilbert space which is the tensor product of three spaces, each corresponding to motion in one dimension. The Hilbert space for two particles, as long as they are not identical, is the tensor product of the two Hilbert spaces for the separate particles.
Cambridge University Press eBooks•Nov 29, 2001•1 citation