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I can’t believe I don’t know the answer to this question. What is the classical analog of the state vector of quantum mechanics?

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The state vector represents a complete (as complete as quantum theory allows) specification of the system's physical state. So the classical analog of the state vector would be a point in phase space, or any other type of data that suffices to specify a unique solution of the classical equations of motion.

By the way, classical physics can be regarded as a special case of quantum physics in which all observables commute with each other and in which the state is always an eigenstate of all of the observables. In this sense, states in classical physics are always orthogonal to each other (cf What makes a theory "Quantum"?).$^\dagger$

$^\dagger$ Minor technicality: In quantum physics, we normally require that the Hilbert space be separable, which means that it must admit a countable basis of mutually orthogonal state vectors. An obstacle to treating classical physics as a special case of quantum physics is that the Hilbert space (state space) of classical physics would typically be non-separable. The significance of this is emphasized, in a different context, in the paper String quantization: Fock vs. LQG Representations.

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