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?
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.