Weak Correspondence Principle

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Abstract
The weak correspondence principle (WCP) for a scalar field states that the diagonal matrix elements G(f,g)≡〈f,g|G|f,g〉of a quantum generator G necessarily have the form of the appropriate classical generator G in which f(x) and g(x) are interpreted as the classical momentum and field, respectively. For a field operator φ(x) and its canonically conjugate momentum π(x) the states in question are given by |f,g〉≡exp {i ∫ [φ(x)f(x)−π(x)g(x)] dx} |0〉,where |0〉 denotes the vacuum. The validity of the WCP is established for the six Euclidean generators (plus the Hamiltonian) of a Euclidean-invariant theory, and for the ten Poincaré generators of a Lorentz-invariant theory. Only general properties and certain operator domain conditions are essential to our argument. The WCP holds whether the representation of π and φ is irreducible or reducible; in the latter case, the WCP holds even if the vectors |f, g〉 do not span the Hilbert space, or even if the generator G is not a function solely of π and φ. Thus, the WCP is an exceedingly general and completely representation-independent connection between a classical theory and its quantum generators which is especially useful in the formulation of nontrivial, Euclidean-invariant quantum field theories.