3.1 Massless Phase

In the massless phase, the physical content of ``electrodynamics in $(1+1)$ dimensions, is encoded into the gauge invariant action :

$$S=\int d^2x \left[ {1\over 4} F^ {\mu\nu} F_ {\mu\nu}... ... \partial_ { [ \mu} A_ {\nu ]} -e J^\mu A_\mu \right]$$ (8)

so that the current density $J^\mu$, without further boundary conditions, is divergenceless: $\partial_\mu J^\mu=0$. The first order formulation of the action is not mandatory but makes it clear that, in two dimensions, the ``Maxwell tensor'' is assumed to be the covariant curl of the gauge potential, which is then treated as an independent variable.
Thus, variation of the action with respect to the potential $A_\mu$ leads to the Maxwell equation

$$\partial_\mu F^ {\mu\nu}= e J^\nu .$$ (9)

The general solution of Eq. (9) is the sum of the free equation solution ($e=0$), and a special solution of the inhomogeneous equation ($e\ne 0$). The complete equation can be formally solved by the Green function method. The final result is

$$F^ {\mu\nu}(x) = \sqrt\Lambda \epsilon^ {\mu\nu} + e \partial^{ [\mu} {1\over \Box} J^{\nu ]}$$

$$= \sqrt\Lambda \epsilon^ {\mu\nu} + e \int d^2y \partial_x^ { [\mu} G( x-y ) J^ {\nu ]}(y)$$ (10)

Inserting the above solution into the action (8), and neglecting surface terms, we obtain

$$S = - {1\over 2} \int d^2x \left[ \Lambda + e^2 J^ \nu {1\over \Box} J_ \nu \right]$$

$$= - {1\over 2} \int d^2x \Lambda + e^2 \int d^2x \int d^2y J^ \nu(x) G( x-y ) J_ \nu( y )$$ (11)

which we interpret as follows

$$S=- {1\over 2} \int d^2x \left[ \hbox{\lq\lq Cosmological Constant'' + Coulomb Potential} \right] .$$ (12)

The first term represents a constant energy background, or cosmological term, even though it can be ``renormalized away'' in the absence of gravity. The second term in (11) describes the long-range, ``Coulomb interaction'' in two spacetime dimensions. In reality, it represents the linear confining potential between point charges written in a manifestly covariant form. In such a covariant formulation, the existence of a boundary, even though not explicitly codified in the action (8), introduces a symmetry breaking condition since it implies that the world line of the ``charge'' has a free end-point through which the symmetry leaks out, so that $\partial_\mu J^\mu=J\ne 0$. In that case, gauge invariance is topologically broken and the current density is no longer divergence free. Under such circumstances, the field equation (9) needs to be modified since the left hand side is divergenceless, while the right hand side is not.