Proca action/equation describes massive spin-1 particle, but I was unable to find an equation that describes massless spin-1 particle.
Can anyone tell me what the name of this equation is?
[Physics] An equation that describes massless spin-1 particle
quantum mechanicsquantum-field-theoryquantum-spin
Related Solutions
There is no quantum mechanics of a photon, only a quantum field theory of electromagnetic radiation. The reason is that photons are never non-relativistic and they can be freely emitted and absorbed, hence no photon number conservation.
Still, there exists a direction of research where people try to reinterpret certain quantities of electromagnetic field in terms of the photon wave function, see for example this paper.
In general, quantum numbers are labels of irreducible representations of the relevant symmetry group, not primarily eigenvalues of an otherwise simply defined operator.
But for every label that has a meaningful numerical value in every irreducible representation, one can define a Hermitian operator having it as an eigenvalue, simply by defining it as the sum of the projections to the irreducible subspaces multiplied by the label of this representation. It is not clear whether such an operator has any practical use.
This also holds for the spin. However, one can define the spin in a representation independent way, though not via eigenvalues.
The spin of an irreducible positive energy representation of the Poincare group is $s=(n-1)/2$, where $s$ is the smallest integer such that the representation occurs as part of the Foldy representation in $L^2(R^3,C^n)$
with inner product defined by
$~~~\langle \phi|\psi \rangle:= \displaystyle \int \frac{dp}{\sqrt{p^2+m^2}} \phi(p)^*\psi(p)$.
The Poincare algebra is generated by $p_0,p,J,K$ and acts on this space
as follows (units are such that $c=1$):
with the position operator $q := i \hbar \partial_p$ and the spin vector $S$ in a unitary irreducible representation of $so(3)$ on the vector space $C^n$ of complex vectors of length $n$, with the same commutation relations as the angular momentum vector.
The Poincare algebra is generated by $p_0,p,J,K$ and acts on this space irreducibly if $m>0$ (thus givning the spin $s$ representation), while it is reducible for $m=0$. Indeed, in the massless case, the helicity
$~~~\lambda := \displaystyle\frac{p\cdot S}{p_0}$,
is central in the universal envelope of the Lie algebra, and the possible eigenvalues of the helicity are $s,s-1,...,-s$, where $s=(n-1)/2$. Therefore, the eigenspaces of the helicity operator carry by restriction unitary
representations of the Poincare algebra (of spin $s,s-1,...,0$), which are easily seen to be irreducible.
The Foldy representation also exhibits the massless limit of the massive representations.
Edit: In the massless limit, the formerly irreducible representation becomes reducible. In a gauge theory, the form of the interaction (multiplication by a conserved current) ensures that only the irreducible representation with the highest helicity couples to the other degrees of freedom, so that the lower helicity parts have no influence on the dynamics, are therefore unobservable, and are therefore ignored.
Best Answer
It's called Maxwell's equations.
A spin-1 relativistic particle has to have a 4-vector $A_\mu$ and the equation may essentially be written as $\Box A_\mu=0$, like always. However, when we quantize it, we find out that the squared norm of the states created by the time-like components has the opposite sign than the spacelike components. This would make the Hilbert space indefinite - probabilities could be negative.
So the single timelike mode has to be made unphysical. The only way to do so is to impose a gauge symmetry. So configurations related by $A'_\mu = A_\mu +\partial_\mu \lambda$ – and that's essentially the only way how the gauge invariance with 1 scalar parameter may act – must correspond to the same physical situations. Consequently, we may rewrite the equations as $\partial_\mu F^{\mu\nu}=0$, the usual equations of electromagnetism, which differ from the previous box-equation by a term that can be set to zero by a gauge choice. There can't be consistent spin-1 massless equations without a gauge invariance.
We never learn Maxwell's equations as a single-particle quantum mechanical equation because single-particle equations assume that the number of particles is approximately conserved. That's true in the non-relativistic limit. However, when the speeds approach the speed of light, we're heavily in the relativistic realm and the particle production and annihilation is important; quantum field theory is paramount. Obviously, that's also the case of the photons that move by the speed of light. After all, we know that the number of photons is changing all the time.