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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?

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    $\begingroup$ "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? $\endgroup$ Commented Jul 27, 2019 at 17:32
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    $\begingroup$ @AccidentalFourierTransform Dirac spinors still satisfy the Klein-Gordon equation, so I don’t understand your objection. $\endgroup$ Commented Jul 27, 2019 at 17:44

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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.

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  • $\begingroup$ "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"? $\endgroup$
    – M.N.Raia
    Commented Jul 28, 2019 at 16:40
  • $\begingroup$ 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. $\endgroup$ Commented Jul 29, 2019 at 21:42
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In similar questions, some high-energy people tend to repeat definitions from books as if they are explaining. A person who understands the question should focus on the relation between these ideas. It is obvious anyone who had this puzzle has understood the difference.

Here is a small explanation attributed to Prof.Littlejohn in the Berkeley QM book which you can find online. The book is really excellent and I quote here.

Short answer: some particular wave functions (Dirac or Schrodinger) has the following properties. When treating the wave function as fields, you come back to yourself except for many-particle physics.

"The idea of second quantizing the Dirac wave field did not come out of the blue. In 1928 Wigner and Jordan explored the question of what would happen if the usual Schrodinger wave function were reinterpreted as a quantum field. This was in the days in which the interpretation of the wave function was not settled, and different possibilities were being tried. Wigner and Jordan were slightly disappointed in their results, because they found that second quantizing the Schr¨odinger theory led to a theory physically equivalent to the first quantized version, but one in which the symmetry or antisymmetry of multiparticle wave functions of identical particles was handled in a natural manner. To achieve this result in the case of fermions, it turned out to be necessary to modify the process of Dirac quantization, which we described in Notes 41 in application to the electromagnetic field. Nowadays the second quantized approach of Wigner and Jordan is the preferred method for handling many body problems in atomic, nuclear and condensed matter physics, often for nonrelativistic processes in which massive particles are not actually created or destroyed. This is mainly because of the convenience with which the second quantized formalism handles the requirements of the symmetrization postulate, and because of the modes of thinking and physical insight that follow from the use of quantum field theory."

"This suggests that we might be able to deal with the many-particle aspects of Dirac’s hole theory, including processes of creation and annihilation of electrons and positrons, by reinterpreting the Dirac wave function ψ, which up to this point has been a c-number field, as a quantum field. The process of passing from a c-number field to a quantum field is called “quantization,” but since the Dirac wave function ψ already describes the quantum mechanics of a single particle, when this process is applied to the Dirac wave function, it is called “second quantization.” In this process the Dirac equation, as we have dealt with it up to this point, may be called the “first quantized” version of the theory, that is, the quantum theory of a single particle. In the process of second quantizing the Dirac equation, the first quantized Dirac theory will be treated as a “classical” theory, insofar as the process of quantization is concerned. That is, the “classical” theory concerns a c-number field that satisfies certain equations of motion. When used in this context, we will put the word “classical” in quotes, to acknowledge that the Dirac equation actually describes a single-particle quantum system."

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