The discussion revolves around the quantum description of a single photon, exploring the appropriate equations and interpretations within the framework of quantum mechanics and quantum field theory. Participants examine the nature of wave functions, vector states in Hilbert space, and the implications of position observables for photons.
Participants express multiple competing views regarding the existence and interpretation of the wave function for photons, the nature of position observables, and the applicability of certain mathematical operators. The discussion remains unresolved with no consensus reached.
Limitations include the dependence on specific interpretations of quantum mechanics and quantum field theory, as well as unresolved mathematical steps regarding the completeness of certain operators.
fxdung said:The wave function of photon does not exist,but the vector state in Hilbert space still exists
Read again my post above! It exists, but it just has a different physical interpretation.fxdung said:The wave function of photon does not exist
That frequent statement is also, strictly speaking, not correct. There is a position observable for a photon, but it is not Lorentz covariant. Despite non-covariance it has a sensible physical interpretation in terms of position measurements with apparatus at rest in a specific Lorentz frame.bhobba said:That's because position is not an observable for photons hence they can't be expanded in terms of eigenfunctions of position which a wave-function is.
What interpretation? Can you give a reference?Demystifier said:Mathematically, one can introduce a wave function of a single photon, and this wave function satisfies Maxwell equations. Physically, however, such a wave function has a different interpretation than wave function in non-relativistic quantum mechanics (QM). For that reason people sometimes say that "wave function of a single photon does not exist", which really means that photon wave function with the same physical interpretation as in non-relativistic QM does not exist.
An example of a position operator in QFT, is the Newton-Wigner operator which, IIRC, its eigenvectors aren't a complete set, i.e. ## \int |x\rangle \langle x| dx \neq 1 ##, which doesn't make sense!Demystifier said:That frequent statement is also, strictly speaking, not correct. There is a position observable for a photon, but it is not Lorentz covariant. Despite non-covariance it has a sensible physical interpretation in terms of position measurements with apparatus at rest in a specific Lorentz frame.
Another correct statement is that there is no Lorentz covariant photon-position observable in the physical Hilbert space. It is possible to introduce a Lorentz covariant position observable, but in an extended Hilbert space containing non-physical states.
E.g. Fourier transform gives probability in the momentum space. For instance, Bjorken Drell 1.Shyan said:What interpretation? Can you give a reference?
I don't think it's true for the NW operator. Can you give a reference?Shyan said:Newton-Wigner operator which, IIRC, its eigenvectors aren't a complete set, i.e. ## \int |x\rangle \langle x| dx \neq 1 ##,
Sorry about that. I confused the things I read before!Demystifier said:I don't think it's true for the NW operator. Can you give a reference?
The wave function of the photon is the complex Riemann-Silberstein vector field. Unlike Schroedinger wave functions, it doesn't have a probability interpreation. But it provides a massless spin 1 representation of the Poincare group, hence has all the properties needed in quantum field theory.fxdung said:The wave function of photon does not exist,but the vector state in Hilbert space still exists.Then how does the vector state envolve in time?