A question about Dirac equation.

[ndung200790]

It seems that notions of quantum field and wave function are utterly different from each other.Then is Dirac equation being equation for field or for relativistic wave function or for the both?

 

[tom.stoer]

The Dirac equation has been derived as a "relativistic Schroedinger equation". Later it became clear that it is problematic to interpret the spinors as a kind of wave functions in relativistic quantum mechanics. Today the Dirac equation is understood as a operator equation for field operators in quantum field theory.

 

[andrien]

Then is Dirac equation being equation for field or for relativistic wave function or for the both?

In the early days it was believed that dirac eqn is the only one to receive as the true relativistic wave eqn(klein gordon was rejected because of negative probabilities) and dirac eqn correctly predicted the energy levels of hydrogen as a wave function eqn(single particle).There was however difficulties associated with it.Now we know that second quantized form of dirac eqn is really the most correct one.

 

[ndung200790]

What was the difficulties for wave function Dirac eqn?

 

[VantagePoint72]

It seems that notions of quantum field and wave function are utterly different from each other.Then is Dirac equation being equation for field or for relativistic wave function or for the both?

You are correct that they are very different notions. Since we take quantum field theory to be the more fundamental theory of matter, the Dirac equation being the equation for the field is the standard interpretation. As the others have pointed out, attempts at interpreting as a wavefunction didn't work so well.

However, we get the wavefunction of quantum mechanics as a non-relativistic limit of quantum field theory. If you start with the equation for a complex scalar quantum field (the complex Klein-Gordon equation) and take the non-relativistic limit, you get an approximate field equation identical to the non-relativistic Schrödinger equation. However, this is still a quantum field, not a wavefunction. But now you can construct a wavefunction by constructing a position operator for the field and building up a superposition of particles in its eigenstates. This is a true wavefunction in the usual sense of squaring to the probability density of finding your particle at a particular place. You can determine how the wavefunction evolves since we know how the field whose quantization is the particle in question evolves—and it turns out the wavefunction obeys the Schrödinger equation too!

So, both the non-relativistic scalar field and the wavefunction for one particle states consisting of the scalar field's particles obey the Schrödinger equation—but with a very different interpretation for each. This originally caused a lot of confusion and people thought that they were quantizing the wavefunction itelf. They dubbed this procedure "second quantization", a name which has stuck even though now we know better.

The key point of the Dirac equation is that each spinor component obeys the Klein-Gordon equation separately so the same sort of argument works. In fact, if you work it through, you find that two of the four spinor components are negligible and so along the way we recover the usual two component spinors of non-relativistic QM. The resulting field equation is the Pauli equation for two component spinors (with each component separately obeying the Schrödinger equation). You can then follow the same procedure for determining the evolution of a wavefunction for the theory's particles and, as in the complex scalar case, find that the wavefunction obeys the same equation as the field.

This whole procedure of constructing the wavefunction based on the field equation doesn't really work until you take the non-relativistic limit. In the fully relativistic theory, particle number isn't conserved, along with a whole bunch of other things that don't really make much sense in the context of a wavefunction. So, we generally just stick with the quantum fields.