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Setup:

Let us assume we are in $D$-dimensional Minkowski space-time where $D=d+1$. Consider a free Abelian gauge theory. Then the electromagnetic field will satisfy $$\partial_{\mu}F^{\mu \nu}=0 \tag{1}$$ and this means that $\partial_{\mu} \partial_{\nu} F^{\mu \nu}$ is identically zero.

Question:

  1. The gauge field $A_{\mu}$ out of which we construct the field strength comprises $D$ independent components (the indices take a value for each dimension). Then, why people say that the field strength comprises $D-1$ independent components? How is this counting being done?

  2. And why these $D-1$ independent components are the off-shell degrees of freedom? In what sense?

  3. Finally, what happens with spinors? How many independent components has a spinor in $D$-dimensions and how do we find its off-shell degrees of freedom?

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The Maxwell Lagrangian for the gauge field $A_\mu$ is given by,

$$\mathcal L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$

The equations of motion are $\partial_\mu F^{\mu\nu} = 0$, supplemented by the Bianchi identity. Taking as an example the case $D=4$, naively one would think we have $4$ degrees of freedom. However, we know photons have only two degrees of freedom, so it must be reduced somehow.


It turns out the component $A_0$ is not dynamical, since we have no kinetic term for it in the Lagrangian. In other words, $A_0$ is fully determined by the other components through,

$$\nabla^2 A_0 + \nabla \cdot \frac{\partial \vec A}{\partial t}=0$$

which has the solution,

$$A_0 (\vec x) = \int d^3 y \frac{1}{4\pi |\vec x - \vec y|}\frac{\partial \vec A (\vec y)}{\partial t}.$$

Just like in general relativity where we have constraints on what initial data surface to specify, we cannot choose our initial $A_0$. So, in general we've gone from $D \to D-1$ independent components. For the case $D=4$, the degrees of freedom are two due to the additional constraint,

$$\nabla \cdot \vec A = 0$$

known as Coulomb gauge. (One should not think of it per se as a constraint, but rather as exploiting a redundancy in the description of the system due to the gauge symmetry.) This bumps us down to two.

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  • $\begingroup$ So in 4 dimensions we have 3 independent components and by analogy in $D$-dimensions we have $D-1$ independent components. Why are they called off-shell though? Also, how did you go from the first equation to the second equation? $\endgroup$ – Marion Jan 9 '15 at 12:38
  • $\begingroup$ @Marion Don't forget you lose another degree of freedom from gauge symmetry to get two in four dimensions. $\endgroup$ – JamalS Jan 9 '15 at 12:39
  • $\begingroup$ @JamalS, you probably should include that in the answer because you mentioned photons having 2 d.o.f in 4D but you only explain how to get rid of one of them :) $\endgroup$ – Constandinos Damalas Jan 9 '15 at 12:42
  • $\begingroup$ @PhotonicBoom Updated. $\endgroup$ – JamalS Jan 9 '15 at 13:20

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