When Canonically Quantising the Klein-Gordon Field you usually start with the Klein-Gordon Equation, from which you can guess a corresponding Lagrangian Density. Then utilising this information along with assumed Commutation relations for the field and it’s conjugate momentum you “Quantise” the field. From what I’ve read, upgrading the Klein Gordon Field (which is a scalar field) to an operator (or I guess “operator-valued distribution” or so I’m told) is what “Quantises” what I assume to be the classical Klein Gordon Equation.

But it’s also trivially easy to prove that the Klein Gordon Equation can be derived from the relativistic Energy-Momentum Relation by substituting in Energy and Momentum Operators. It would seem then that the Klein Gordon Equation is already “Quantum”. This seems to be a contradiction to me (and I’m guessing may be the historical motivation behind the term “Second Quantisation”?).

If anybody can answer, I’d like to know what the orthodox viewpoint on this. Is the Klein Gordon Equation already “Quantum” or does it only become Quantum when you go through the process of Canonical Quantization? Is it both somehow? If the Klein Gordon Equation is really a classical field then why can I derive it by substituting Quantum Operators into a classical expression from Special Relativity?


There are several issues tangled here:

  • One should distinguish between math and physics: physics heavily uses math to describe natural phenomena, but an existence of a mathematical equation itself does not imply the existence of a physical interpretation or that such a physical interpretation should be unique. The wave equation (i.e. Klein-Gordon equation with $m=0$) is the most notorious example.
  • As follows from the above, Klein-Gordon can describe several different phenomena. In particular, when performing the first quantization for particles, replacing energy and momentum by the corresponding operators, one obtained the Klein-Gordon (although for spin-1/2 it turns out to be inappropriate). However, when we are dealing with a classical field, its first quantization coincides in technical terms with the second quantization for particles. Electromagnetic field is the most obvious example. In other words: a classical wave field does not automatically become quantized just because it is described by Klein-Gordon equation.

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