In the phase space formulation, the terms "representation", "function, and "quasi-probability distribution" (as in Glauber–Sudarshan P representation, $P$-function) seem to be used interchangeably. I wonder about the semantics of these terms: Do they all really mean exactly the same thing, even mathematically, or might there be some subtleties to be understood?
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These three things all emphasize different properties:
- "Function": reminds you you're dealing with a function on phase space. Also the shortest one, so can be used for convenience.
- "Representation": reminds you this function represents the content of a particular quantum state.
- "Quasi-probability distribution": reminds you this function is like a probability distribution but isn't really one.
These adjectives end up being used interchangably not because they mean the same thing, but because the three most common things they're applied to ($P$-function, $Q$-function, Wigner function) have all three of these properties.
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$\begingroup$ A small tweak: in that article, "representation" is mostly used to contradistinguish functions derived by different operator-ordering maps. They are equivalent, but different rules for producing different functions (P, Q, and W, and... and...) corresponding to ("representing") the same operators and physical quantities. $\endgroup$ Commented Jan 22, 2020 at 1:36