Proca Equation

Prerequisites: Maxwell Equations, Klein-Gordon Equation. Full path (19 pages)

Intuitive
Concrete
Abstract
Why is it interesting?
Definitions

$m^2 A^\rho = \partial_\sigma F^{\sigma \rho}$ (signature $(-,+,+,+)$ )

Intuitive

The Maxwell equations describe the electromagnetic field, whose quantum is the photon, a particle without mass. The Proca equation describes what changes if the particle of a field like this has a mass. Examples of massive particles of this type (spin one) are the W and Z bosons of the weak interaction.

Two things change. First, the force carried by the field gets a finite range. The electric force between two charges falls off slowly with distance and reaches arbitrarily far. With a massive carrier, the force dies off exponentially beyond a distance set by the mass: the heavier the carrier, the shorter the range. This is why the weak interaction is so short ranged.

Second, the field has one more way to oscillate. A light wave can vibrate only in the two directions perpendicular to its direction of travel. A massive spin one particle can also vibrate along its direction of travel, so it has three polarization states. This is possible because a massive particle can be brought to rest, and at rest there is no preferred direction, so all three directions in space are equivalent.

The price is that the mass term breaks the gauge symmetry of electromagnetism. In the Maxwell theory, different potentials that describe the same physical situation are related by a gauge transformation. In the Proca theory the potential itself becomes physical.

Concrete

$\begin{align}m^2 A^\rho &= \partial_\sigma ( \partial^\sigma A^\rho - \partial^\rho A^\sigma) \\ &=\partial_\sigma F^{\sigma \rho} \end{align}$

The general solution for the Proca equation is

$\begin{align} A_\mu &= \int \frac{d^3 k}{\sqrt{ (2\pi)^3 2 \omega_k}} \left( \epsilon_{r,\mu}(k) a_r(k) {\mathrm{e}}^{-ikx} + \epsilon^*_{r,\mu}(k) a_r^\dagger(k) {\mathrm{e}}^{ikx} \right) \\ &\equiv A_\mu^+ + A_\mu^- \end{align}$ where $\epsilon_{r,\mu}(k)$ are basis vectors called polarization vectors.

Graphical Summary

The diagram below shows the Proca equation and its Lagrangian in various forms. For a more detailed explanation see Fun with Symmetry.

No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, section "Free Gauge Fields and the Proca/Maxwell equations"

Abstract

We use $\hbar = c = 1$ . In the signature $(+,-,-,-)$ the Proca Lagrangian and equation are

$\mathscr L = -\frac14 F_{\mu\nu}F^{\mu\nu} + \frac12 m^2 A_\mu A^\mu , \qquad \partial_\mu F^{\mu\nu} + m^2 A^\nu = 0 .$

In the signature $(-,+,+,+)$ the sign of the mass term relative to $\partial_\mu F^{\mu\nu}$ flips, and the equation takes the form $\partial_\sigma F^{\sigma\rho} = m^2 A^\rho$ written above. In both cases each component satisfies the Klein-Gordon equation with mass $m$ , as follows.

Constraint. Applying $\partial_\nu$ to the equation and using the antisymmetry of $F^{\mu\nu}$ gives $m^2\,\partial_\nu A^\nu = 0$ . For $m \neq 0$ the Lorenz condition $\partial_\nu A^\nu = 0$ is therefore a consequence of the equation of motion, not a gauge choice. With it, $\partial_\mu F^{\mu\nu} = \partial_\mu\partial^\mu A^\nu$ and (in $(+,-,-,-)$ )

$(\partial_\mu\partial^\mu + m^2)A^\nu = 0, \qquad \partial_\nu A^\nu = 0 .$

The second equation removes one of the four components. In the Hamiltonian formulation, the momentum conjugate to $A_0$ vanishes, and this primary constraint together with the secondary constraint from the $A_0$ equation forms a second-class pair. Of the eight phase space variables per point this leaves six, that is three degrees of freedom.

Representation theory. The solutions with $k^0 > 0$ realise the unitary irreducible representation of the Poincaré group with mass $m$ and spin $1$ . For $m > 0$ the little group of a timelike momentum is $SO(3)$ (or $SU(2)$ for the double cover), and the polarization vectors $\epsilon_{r,\mu}(k)$ , $r = 1,2,3$ , with $k^\mu\epsilon_{r,\mu} = 0$ , span its three-dimensional vector representation. They satisfy

$\sum_{r=1}^3 \epsilon_{r,\mu}(k)\,\epsilon^*_{r,\nu}(k) = -g_{\mu\nu} + \frac{k_\mu k_\nu}{m^2},$

and the Feynman propagator is

$\frac{i\left(-g_{\mu\nu} + k_\mu k_\nu/m^2\right)}{k^2 - m^2 + i\epsilon} .$

For $m = 0$ the little group is $ISO(2)$ and only two helicities $\pm 1$ remain; the missing state is pure gauge in the Maxwell theory.

Gauge invariance and the massless limit. The mass term breaks the $U(1)$ gauge symmetry $A_\mu\to A_\mu + \partial_\mu\alpha$ . It can be restored by the Stückelberg substitution $A_\mu \to A_\mu + \frac1m\partial_\mu B$ with a new scalar field $B$ that shifts as $B \to B - m\alpha$ . The longitudinal polarization then appears as the scalar $B$ . If $A_\mu$ couples to a conserved current $j^\mu$ , the $k_\mu k_\nu/m^2$ term in the propagator drops out of amplitudes, $B$ decouples, and the limit $m\to 0$ is smooth. In a non-abelian theory a mass term of this kind spoils renormalizability, which is why the masses of the W and Z bosons in the standard model come from the Higgs mechanism.

Why is it interesting?

The Proca equation is a generalization of the Maxwell equation for massive spin $1$ particles. Formulated differently, the Maxwell equation is only a special case of the Proca equation for massless particles/fields.

The Proca equation is important because it correctly describes massive spin $1$ particles/fields.

Definitions

$\partial_{\sigma}$ denotes the partial derivative,
$m$ denotes the mass of the particle,
$A^\rho$ is either the wave function of the spin $1$ particle if we use the Proca equation in a particle theory, or describes the spin $1$ field if we work in a field theory.
$F^{\sigma \rho}$ is the electromagnetic field tensor: $F^{\sigma \rho} \equiv \partial^\sigma A^\rho - \partial^\rho A^\sigma$ .

Path to this page

Leads to: Stueckelberg Trick.