In Quantum Field Theory the positive frequency solutions to the classical field equations are quite important since they are the basis of the definition of particles.

In Minkowski spacetime we have a canonical choice because of Poincare symmetry: the solutions positive frequency with respect to the inertial reference frame. In general spacetimes this isn't the case, but still, a picture of particles with respect to one reference frame is usually constructed out of positive frequency solutions for him.

Nevertheless, most of the time this concept seems to be defined just for the scalar field solutions.

If $(M,g)$ is spacetime and $\phi$ is a solution to the Klein-Gordon equation, it is usual to say that $\phi$ is positive-frequency with respect to a timelike vector field $Z$ when

$$\mathfrak{L}_Z\phi=-i\omega \phi,\quad \omega > 0,$$

where $\mathfrak{L}_Z$ is the Lie derivative.

But I'm wondering in general how this is defined.

  1. For general tensor fields - in particular the Maxwell field which is a vector field - is it just the straightforward generalization saying that a solution $\Phi$ to the equations of motion is positive frequency when $$\mathfrak{L}_Z\Phi=-i\omega \Phi,\quad \omega>0?$$

    Since the Lie derivative naturally acts on tensors, I think this is a reasonable generalization.

  2. And what about spinor fields? I remember that a colleague which has more experience with them than me told me that the Lie derivative is not very well defined for spinor fields. So in that case how one defines positive frequency solutions?

  3. A third problem is that this defines monochromatic solutions. In Minkowski spacetime, one defines a general positive frequency solution by Fourier transformation. But this is in general ill-defined in a general spacetime. So the definition of non-monochromatic positive frequency seems to be hard to make sense in general even for the scalar field.

So for general fields, how positive frequency solutions are defined? Is my guess for tensor fields correct? For spinor fields how does one proceeds here? And how one defines general non-monochromatic positive frequency solutions?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.