# Independent components in a 4-vector representing massless fields

In Ryder Page141, it is written "the electromagnetic field, like any massless field, possesses only two independent components, but is covariantly described by a 4-vector $A_{\mu}$".

Why are there only two independent components? Shouldn't all four components be independent from one another?

## 2 Answers

At the beginning, you may start with four (component) functions $A_{\mu}(x,y,z,t)$ with $\mu=0,1,2,3$.

But then you may choose one function $\lambda(x,y,z,t)$ (pronounce "lambda") and gauge transform $A_\mu$ to $$A'_\mu (x,y,z,t) = A_\mu (x,y,z,t) +\frac{\partial}{\partial x^\mu} \lambda$$ which produces a new configuration $A'_\mu$ that we must consider physically identical. So we have 4 numbers per point up to 1 number that doesn't matter, leaving us with three.

However, there is one more reduction because one of the Maxwell's equations, Gauss' law $${\rm div}(\vec D) = \rho$$ doesn't really contain time derivatives, so it restricts the initial states. Equivalently, the time derivative of Gauss' law may be derived from the other Maxwell equations, so there is another number at each point that is "redundant". This leaves us with two physical off-shell fields per point.

It's perhaps clearer to count the "on-shell" degrees of freedom, the number of actual waves and their polarizations.

Considering waves moving in the vacuum with wave vector $\vec k$ and frequency $\omega$. They are proportional to $$\exp(i\vec k \cdot \vec x - i\omega t)$$ where $\omega=c |\vec k|$. The full $A_\mu$ is $$A_\mu = \epsilon_\mu \exp(i\vec k \cdot \vec x - i\omega t)$$ and there are a priori four independent solutions. However, because of the $\lambda$-related gauge symmetry, if you change $\epsilon^\mu$ by a multiple of $k^\mu$, you get a physically identical electromagnetic wave. These $\epsilon\sim k^\mu$ waves are "pure gauge".

The other degree of freedom is lost because we may impose the condition $$\epsilon_\mu k^\mu = 0$$ the Lorentz gauge, without a loss of generality. This reduces "another" non-transverse degree of freedom because the null vectors $\epsilon^\mu$ that are orthogonal to the null $k^\mu$ (and therefore allowed) are proportional to it, while these values of $\epsilon^\mu$ proportional to $k^\mu$ are exactly those that are unphysical by the gauge symmetry.

To summarize, only the two transverse polarizations are physical. For example, if the wave is moving in the $z$-direction, it's only the $x$- and $y$- linearly polarized waves (or the two circular polarizations which are their combinations) that are physical. Both the longitudinal and the timelike wave are unphysical because of the "two applications" of the gauge symmetry above.

QFT is strongly based irreducible representations of the Poincare group: by having mass $m$ and spin (helicity) $s$ we can construct field and corresponding equations for it. It can be shown that massless representations of the Poincare group which is not invariant under spatial inversion are characterized only by one value - helicity $\lambda$, so there is only one independent component for corresponding field. If you want to construct theory of field helicity $\lambda$ ($( \lambda , 0)$ for helicity $\lambda$ , $(0, \lambda )$ for helicity $-\lambda$) which is invariant under spatial inversion than you must take the direct sum $\left( \lambda , 0 \right) \oplus \left( 0 , \lambda\right)$ of representations. So in conclusion you have two independent components and hence two values of helicity $\lambda , -\lambda$. It is not hard to get equations for corresponding field.

The "true" massless Poincare covariant field for helicity 1 (invariant under spatial inversion) is $F_{\mu \nu}$, for helicity 2 it's $C_{\mu \nu \alpha \beta}$ (linearized Weyl tensor) etc. But if we then introduce $A_{\mu}, g_{\mu \nu}$ (respectively) for these theories, we will get fields which have more independent components than it must be. But gauge invariance which is appeared in these theories after "replacing" $C_{\mu \nu \alpha \beta} \to g_{\mu \nu}, F_{\mu \nu} \to A_{\mu}$ $$A_{\mu} \to A_{\mu}^{\omega} = A_{\mu} + \partial_{\mu}\omega, \quad g_{\mu \nu} \to g_{\mu \nu}^{\omega} = g_{\mu \nu} + \partial_{\mu}c_{\nu} + \partial_{\nu}d_{\mu} : L(\varphi ) \to L(\varphi^{\omega}) = L (\varphi )$$ will help us to reduce the number of components to 2.

Unfortunately, fields $A_{\mu}, g_{\mu \nu}$ can't represent massless particles and be Poincare-covarint ones at the same time. The requirement that they represent massless particles (I mean representation of Poincare transformations in terms of little group) leads to the fact that they are transformed under Lorentz group transformations uncorrectly. But we must to introduce these fields beacuse they provide interaction $\frac{1}{r^{2}}$ law.

Fortunately it can be shown that for realistic theories of interaction S-matrix is invariant under some transformations of polarization vectors (tensors) of these massless fields so the non-poincare-covariant part of fields transformations does not contribute to physical processes.