Derivation of Sum Rule and Product Rule from FUNC

The three principles, in full detail, are:

FUNC: Let A be a self-adjoint operator associated with observable A, let f: R→R be an arbitrary function, such that f(A) is a self-adjoint operator, and let | φ> be an arbitrary state; then f(A) is associated uniquely with an observable f(A) such that:

v(f(A))^|φ> = f(v(A))^|φ>

Sum Rule: If A and B are commuting self-adjoint operators corresponding to observables A and B, respectively, then A + B is the unique observable corresponding to the self-adjoint operator A + B and

v(A + B)^|φ> = v(A)^|φ> + v(B)^|φ>

Product Rule: If A and B are commuting self-adjoint operators corresponding to observables A and B, respectively, then if A · B is the unique observable corresponding to the self-adjoint operator A · B and

v(AB)^|φ> = v(A)^|φ> · v(B)^|φ>

In order to derive Sum Rule and Product Rule from FUNC, we use the following mathematical fact: Let A and B be commuting operators, then there is a maximal operator C and there are functions f, g such that A = f(C) and B = g(C).

So, for two commuting operators A, B:

Since A = f(C) and B = g(C), there is a function h = f+g, such that A + B = h(C).

Therefore:

v(A + B)|φ>	=	h(v(C)|φ>)		(by FUNC)
	=	f(v(C)|φ>) + g(v(C)|φ>)
	=	v(f(C))|φ> + v(g(C))|φ>		(by FUNC)
	=	v(A)|φ> + v(B)|φ>		(Sum Rule)

Similarly:

Since A = f(C) and B = g(C), there is a function k = f·g, such that A·B = k(C).

Therefore:

v(A · B)|φ>	=	k(v(C)|φ>)		(by FUNC)
	=	f(v(C)|φ>) · g(v(C)|φ>)
	=	v(f(C))|φ> · v(g(C))|φ>		(by FUNC)
	=	v(A)|φ> · v(B)|φ>		(Product Rule)

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