I remember that when someone had started to teach me about creation and annihilation operators, that this operator creates an electron, I said ‘How do you create an electron? It disagrees with conservation of charge’.
R. P. Feynman Nobel lectureThe real Klein-Gordon field
We considered in Chapter 2 the simplest relativistic equation, the Klein- Gordon equation, as a single-particle equation, and found the following difficulties with it (i) the occurrence of negative energy solutions, (ii) the current jμ does not give a positive definite probability density ρ, as the Schrodinger equation does. For these reasons we must abandon the interpretation of the Klein-Gordon equation as a single-particle equation. (Historically, this was the motive which led Dirac to his equation.) Can any sense then be made out of the Klein-Gordon equation? After all, spin 0 particles do exist (π, K, η, etc.) so, surely, there must be some interpretation of the equation which makes sense.
What we shall do first is to consider the Klein-Gordon equation as describing a field φ(x). Since the equation has no classical analogue, φ(x) is a strictly quantum field, but nevertheless we shall begin by treating it as a classical field, as we did in the last chapter, and shall find that the negative energy problem does not then exist. We shall then take seriously the fact that φ(x) is a quantum field by recognising that it should be treated as an operator, which is subject to various commutation relations analogous to those in ordinary quantum mechanics.
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