# Question about the “wave function” on Relativistic Quantum Mechanics (RQM) and Quantum Field Theory (QFT)

I'm enrolled on a short and conceptual couse on RQM and QFT and the professor made a distinction about the Klein-Gordon (K-G) equation on RQM and the K-G equation on QFT. Roughly speaking, he said that RQM is about particles and QFT in about fields, and the object which captures that distinction is in fact the wave function. Then he said that the wave function on K-G of RQM is the common wave function from non-relativistic quantum mechanics, and the "wave function" from QFT is, also, a scalar field. He said that in RQM the wave function $$\psi$$ is for one electron and in QFT in for the field $$\Psi$$ of electrons, but since the wave function $$\psi$$ is already an scalar field, in the sense of pure mathematics, how can describe one electron and not a field?

• "the "wave function" from QFT" is not a spinor field, it is a scalar field. If your question is not about spinors, why mention them at all, especially when the statement is wrong? May I suggest to remove all references to spinors? – AccidentalFourierTransform Jul 27 '19 at 17:32
• @AccidentalFourierTransform Dirac spinors still satisfy the Klein-Gordon equation, so I don’t understand your objection. – Bob Knighton Jul 27 '19 at 17:44

Relativistic quantum mechanics (RQM) constructs single-particle wave equations that are consistent with special relativity (SR). However SR allows to create particles out of energy, while quantum mechanics is based on the conservation of probability. Nevertheless at energies low compared to the masses involved a single-particle description is a good approximation to nature.

Quantum field theory (QFT) enables the creation and annihilation of particles. The classical fields are promoted to operators that create and destroy particles. These operators commute with each other if they have integer spin, or anticommute for half-integer spin. A scalar field has spin zero, it commutes. Relativistic covariance is built in. This approach is known as second quantization.

Therefore there is a conceptual difference between the wave function in RQM and the wave function in QFT.

• "The classical fields are promoted to operators that create and destroy particles", so what you saying is that for every field we have this "property"? I mean, suppose that I invent some new kind of interaction, then I should expect that in QFT the solutions of the field equation like (K-G or Dirac), will occur in "pairs"? – M.N.Raia Jul 28 '19 at 16:40
• You start from the Lagrangian, then apply the principle of least action and get the equation of motion (Euler-Lagrange equation). The solutions are expressed as a superposition of waves with wave vector $\vec k$ and angular frequency $\omega$. Then you work out the conjugate momenta of the fields. To go from classical to quantum mechanics you apply the commutation relations to the fields and their conjugate momenta. So you get creation and annihilation operators in the momentum space, that is $a^\dagger(\vec k)$ and $a(\vec k)$. A field is a superposition of both the operators. – Michele Grosso Jul 29 '19 at 21:42