From: http://en.wikipedia.org/wiki/Quantum_mechanics#Mathematical_formulation ... In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). ... Let a three dimensional analog of a one dimensional string completely occupy the space S^3. Movement only occurs in some tangential space, let this space be S^1. This system has a type of global phase invariance, an equal global change of the coordinate in the space S^1. Any point P in S^3 labels a point of our "3D string". An additional coordinate gives the position of point P in our space S^1. If all points in S^3 have the same coordinate in S^1 then the system will be at rest and will be in a minimum energy configuration. The state of this system does not change with a global coordinate transformation. Now give the 3D string some energy and then freeze time. The physics is contained in the 3 dimensional surface that occupies the space S^3 U S^1, the union of S^3 and S^1. Again a global phase change does not change the physics. Thanks for any thoughts.