There is one concept which quantum theory shares with classical mechanics and classical electro-dynamics that is to say the concept of a mathematical Phase space. According to this concept any physical system G is at each instant associated with a point p in a Phase space, this point is supposed to represent the state of G and the state of G is supposed to be ascertainable by maximal observations. Furthermore, the point p0 associated with G at a time t0, together with a prescribed mathematical law of propagation, fix the point pt associated with G at any later time t, this assumption evidently embodies the principle of mathematical causation. Thus in classical mechanics, each point of Phase space corresponds to a choice of n position and n conjugate momentum coordinates and the law of propagation may be Newtons inverse-square law of attraction. Hence in this case phase-space is a region of ordinary 2n dimensional space. In electro-dynamics, the points of phase space can only be specified after certain functions such as the electromagnetic and electrostatic potential are known, hence E is a function-space of infinitely many dimensions. Similarly, in quantum theory the points of Phase space correspond to so called wave functions and hence phase-space is again a function space considered to be Hilbert.
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