Tuesday, 8 December 2009

Lorentz-Covariant and Gauge-Invariant Electrodynamics

Over a year ago I started a series of blog posts aimed at describing quantum field theory, beginning with Lagrangian mechanics and working all the way up. It's been a while since I made any posts in that series. Indeed, the last post was a roadmap of sorts; a list of the topics still to be covered.

The roadmap hinted that the next step is a relativistic description of electromagnetism. Actually, what we really want to do is describe electromagnetism as a gauge theory. That is, a theory which is gauge invariant. By the end of this post, what we mean by gauge invariance will become obvious.

So, here goes.

Maxwell's Equations
Any treatment of electromagnetism must start with Maxwell's equations, presented below.

(1)\,\,\,\,\,\nabla\cdot\mathbf{E} = \rho_{em}

(2)\,\,\,\,\,\nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}

(3)\,\,\,\,\,\nabla\cdot\mathbf{B} = 0

(4)\,\,\,\,\,\nabla\times\mathbf{B} = \mathbf{j}_{em}


Here,
\rho_{em}
and
\mathbf{j}_{em}
are the electric charge density and electric current density, respectively, and we are working in Heaviside-Lorentz units. The densities act as the sources of the electric and magnetic fields.

Taking the divergence of (4) leads to a problem; the continuity equation for electric charge states that
(5)\,\,\,\,\,\frac{\partial \rho}{\partial t} + \nabla\cdot\mathbf{j} = 0

(note that I will omit the "em" subscript on the densities from now on).

Since
\nabla\cdot\nabla\times\mathbf{B} = 0\Rightarrow\nabla\cdot\mathbf{j}=0

This is only true when the charge density does not vary with time. In general, Ampere's law must be modified to be
(6)\,\,\,\,\,\nabla\times\mathbf{B}=\mathbf{j}+\frac{\partial\mathbf{E}}{\partial t}

This is now consistent with (5), the continuity equation.

The continuity equation states that the rate of change of charge in some volume is due entirely to the flux of current through its surface. That is to say, electric charge is conserved. Since this volume can be made arbitrarily small, we can say that electric charge is locally conserved. This means that we can't balance things out by having a negative charge appear out of nowhere, as long as a positive charge is made the other side of the universe (or a negative charge disappears on the other side of the universe); this conservation law has to apply on an arbitrarily small scale.

We can introduce the vector potential
A_\mu
in place of the fields E and B:
(7)\,\,\,\,\,\mathbf{B}=\nabla\times\mathbf{A}

(8)\,\,\,\,\,\mathbf{E}=-\nabla V-\frac{\partial \mathbf{A}}{\partial t}


This defines the three-vector potential A, and the scalar potential V (sometimes called the electrostatic potential, and sometimes denoted
\phi
. Equations (2) and (3) are then satisified automatically.

The gauge invariance of electromagnetism is based on the fact that, for a given E and B, the equations above do not uniquely determine A and V. In fact, one can apply transformations to A and V which leave E and B completely unchanged; and therefore leave all of the physics unchanged. These are known as gauge transformations, and the invariance of Maxwell's equations under these transformations is known as gauge invariance

We are free to change A as follows;
(9)\,\,\,\,\,\mathbf{A}\rightarrow \mathbf{A}^\prime = \mathbf{A}+\nabla\chi

where
\chi
is an arbitrary scalar function. Since
\nabla\times(\nabla\chi) = 0
, this does not change B.

To preserve E when making the above transformation, we must simultaneously transform the scalar potential V:
(10)\,\,\,\,\,V\rightarrow V^\prime=V-\frac{\partial\chi}{\partial t}


The transformations (9) and (10) can be written in a single expression using the four-potential,
(11)\,\,\,\,\,A^\mu = (V, \mathbf{A})


Note now that the differential operators,
(12)\,\,\,\,\,\left(\frac{\partial}{\partial t}, -\nabla\right)

form a four-vector operator
\partial^\mu
. The gauge transformation can then be specified by:
(13)\,\,\,\,\,A^\mu \rightarrow A^{\prime\mu} = A^\mu - d^\mu\chi


Maxwell's equations can be written in a manifestly Lorentz covariant form (meaning, they have the same form under Lorentz transformations) using the four-current:
(14)\,\,\,\,\,\mathbf{j}^\mu = (\rho,\mathbf{j})


The continuity equation can then be written:
(15)\,\,\,\,\,\partial_\mu j^\mu = 0

which in turn allows us to write equations (1) and (6) as:
(16)\,\,\,\,\,\partial_\mu F^{\mu\nu} = j^\nu


The equation above introduced the Field Strength tensor (or the Faraday tensor),
(17)\,\,\,\,\,F^{\mu\nu}=\partial^\mu A^\nu - \partial^\nu A^\mu


Under a gauge transformation (13), the Faraday tensor (17) remains unchanged; it is gauge invariant, and so, therefore, are Maxwell's equations in the form (16). As I already said above, Maxwell's equations are Lorentz covariant in the form (16), leading to a Lorentz covariant and Gauge invariant theory.

Now, one can write
(18)\,\,\,\,\,\Box A^\nu - \partial^\nu(\partial_\mu A^\mu) = j^\nu


Equation (6) can be inferred (and indeed, was inferred by Maxwell) from the local charge conservation requirement expressed by the continuity equation (15). The field equations (16) automatically include the continuity equation. Mathematically speaking, the Faraday tensor (17) is a sort of "four-dimensional curl". This is unchanged by the gauge transformation (13), suggesting that gauge invariance in electromagnetism is related to charge conservation. The reality is a little more complicated.

It was shown in 1949 by Wigner that the principle that no physical quantity can depend on the absolute value of the electrostatic potential V, combined with conservation of energy, implies that charge conservation holds. This relates charge and energy conservation to an invariance under the transformation of the electrostatic potential by a constant. Charge conservation alone does not require the more general space-time dependent transformation of the gauge invariance we've discussed above.

Changing the electrostatic potential by a constant amount is an example of a global transformation. Invariance under this global transformation is related to the conservation of electric charge, but it is not sufficient to obtain all of electromagnetism. Instead, we must impose a local change in the electrostatic potential V (the time-derivative term in (10), which is compensated by a simultaneous change in the vector potential A, leaving Maxwell's equations ultimately unchanged. By including these magnetic effects, the global invariance under a change in V, related to conservation of electric charge, can be extended to a local invariance.

The concept of local gauge invariance is important for the development of quantum field theory in subsequent posts. Next time, a tour of gauge invariance in quantum mechanics, and the gauge principle.

Much of this post was based on Chapter 3 of Aitchison & Hey (Gauge Theories in Particle Physics, 3rd Edition, Vol. I - From Relativistic Quantum Mechanics to QED). It is a book I strongly recommend purchasing if you have an interest in quantum field theory. The second volume discusses QCD and Electroweak theory, and is another good purchase.