Functional methods in quantum and classical mechanics
A mathematical framework that uses rigged Hilbert spaces to unify the standard formalisms of classical mechanics, relativity and quantum theory will be introduced. In the framework states of a classical particle are identified with Dirac delta functions. The classical space (or space-time) is "made" of these functions and is a submanifold in a Hilbert space of quantum states of the particle. The resulting embedding of the classical space into the space of states is highly non-trivial and accounts for numerous deep relations between classical and quantum physics and relativity. One of the most striking results is the proof that the normal probability distribution of position of a macroscopic particle (equivalently, position of the corresponding delta state within the classical space submanifold) yields the Born rule for probability of transitions between arbitrary quantum states.