B What is quantum equation of single photon?

[fxdung]

Equation of matter particles are Schrodinger,Klein-Gordon and Dirac equation.But the state of photons can not be represented by positions,then what is quantum equation of a single photon?Also what is the equation of single gluon?
(quantum equation means the evolving of the state in time)

 

[Ssnow]

Photons are never non-relativistic, there is no quantum mechanics. Mathematically we don't know what is the appropriate procedure in order to quantize the General Relativity theory.

 

[Demystifier]

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.

 

[fxdung]

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?(Not existing wave function then are we not able derive the equation?)

 

[bhobba]

The wave function of photon does not exist,but the vector state in Hilbert space still exists

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.

Thanks
Bill

 

[Demystifier]

The wave function of photon does not exist

Read again my post above! It exists, but it just has a different physical interpretation.

 

[Demystifier]

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.

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.

 

[ShayanJ]

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.

What interpretation? Can you give a reference?

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.

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!
It seems defining a position operator is a problem in QFT even for massive fields, let alone for massless ones!

 

[Demystifier]

What interpretation? Can you give a reference?

E.g. Fourier transform gives probability in the momentum space. For instance, Bjorken Drell 1.

Newton-Wigner operator which, IIRC, its eigenvectors aren't a complete set, i.e. $\int |x\rangle \langle x| dx \neq 1$,

I don't think it's true for the NW operator. Can you give a reference?

 

[ShayanJ]

I don't think it's true for the NW operator. Can you give a reference?

Sorry about that. I confused the things I read before!

 

[A. Neumaier]

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?

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.

Full details are given in arXiv:quant-ph/0508202.

Note that if one wants to see explicitly the Lorentz covariance one needs a different but isomorphic representation by gauge orbits of solutions of the free Maxwell equations. (See, e.g., Weinberg's QFT book.)