However, at this point we are still dealing with a classical field. If you then try and search for traveling wave solutions to this equation, and look at the Fourier transform of the solutions, you find that we can express the field φ as an integral over all momenta p, but crucially that each mode of the field looks a lot like a QHO for. If we then apply this corresponding operator to a classical field φ we would find the KG equation, which is the equation that any field (classical or quantised) must obey. If we were to replace the quantities E and p with the corresponding quantum mechanical operators, and moved everything to one side,we would find the Klein Gordon operator (which I’m sure you have at least seen before). We start with the relativistic energy-momentum relation in natural units: E 2 =p 2 + m 2. I’ll only consider scalar fields (fields that don’t transform under a Lorentz transformation) for ease. The answer to your question somewhat lies in understanding canonical quantisation, so let me try and kill two birds with one stone for you.
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