### From Electromagnetism to Gauge Theory

09May15

Got myself invited to the 60 Years of Yang-Mills Gauge Theory at NTU. I thought it would be nice to introduce briefly the theory and do myself a revision.

Before jumping into Yang-Mills theory, which is a non-Abelian gauge theory, one should know about the simpler Abelian gauge theory. Those who have studied modern electrodynamics, have indeed studied an Abelian gauge theory. It is perhaps the first place one uses the word gauge (Lorentz gauge, for instance) in a context beyond experimental devices. For a nice discussion of why the word gauge is used, please refer here.

Recall Maxwell’s equations in electrodynamics:

$\underline{\nabla} \cdot \underline{B}=0\qquad;\qquad\cfrac{\partial\underline{B}}{\partial t} +\underline{\nabla}\times\underline{E}=\underline{0}$;

$\underline{\nabla}\cdot \underline{E}=\cfrac{\rho}{\epsilon_0}\qquad;\qquad\cfrac{\partial \underline{E}}{\partial t} - c^2\underline{\nabla}\times\underline{B} = - \cfrac{\underline{j}}{\epsilon_0}$.

One can get solutions to the equations by expressing the magnetic and electric fields in terms of (gauge) potentials:

$\underline{B}=\underline{\nabla}\times\underline{A}\qquad;\qquad\underline{E} =-\underline{\nabla}\phi - \cfrac{\partial\underline{A}}{\partial t}$.

These potentials are however not unique since the following replacements/gauge transformations yield the same fields:

$\phi \rightarrow \phi - \cfrac{\partial\chi}{\partial t}\qquad;\qquad\underline{A}\rightarrow \underline{A}+\underline{\nabla}\chi$.

It is natural to use this freedom to choose a particular condition in specifying a solution and this leads to the known Lorentz condition/gauge:

$\underline{\nabla}\cdot\underline{A}+\cfrac{1}{c^2}\cfrac{\partial\phi}{\partial t}= 0$.

With such gauge, one can still further add to the solution, solutions to the homogeneous wave equation:

$\Delta\chi - \cfrac{1}{c^2}\cfrac{\partial^2\chi}{\partial t^2}=0$.

It is instructive to observe that in performing the gauge transformations, the potentials are added with total (time and space) derivatives:

$(\phi,\underline{A})\rightarrow \left(\phi - \cfrac{\partial\chi}{\partial t}, \underline{A} + \underline{\nabla}\chi\right)$.

This suggests that the potentials are components of a four-vector $A^\mu\equiv (\phi, \underline{A}),\ (\mu=0,1,2,3)$ and the gauge transformation is simply $A_\mu + \partial_\mu\chi$. Note the metric signature taken is $(-,+,+,+)$.

In the four-dimensional description of the electric and magnetic fields, they involve a richer structure than a vector since they include derivatives of the potentials. They are indeed given by a second-rank tensor, namely

$F_{\mu\nu}=\partial_\mu A_\nu -\partial_\nu A_\mu$.

In matrix representation, this is given as

$F_{\mu\nu}=\begin{pmatrix} 0&-E_x&-E_y&-E_z\\E_x&0&B_z&-B_y\\ E_y&-B_z&0&B_x\\ E_z&B_y&-B_x&0 \end{pmatrix}$ .

The Maxwell equations are rewritten as

$\partial_\alpha F_{\beta\gamma} + \partial_\beta F_{\gamma\alpha} + \partial_\gamma F_{\alpha\beta} = 0$,  (trivially true);

$\partial_\beta F^{\alpha\beta} = J^{\alpha}$,

where $J^\alpha\equiv(\rho,\underline{j})$ is the current 4-vector.

In the above, it does seem that the potentials are merely introduced as calculational tools for the electric and magnetic fields and perhaps deemed artificial given that there are ambiguities in defining the potentials. It is however known from Aharonov-Bohm effect that the gauge potentials may have observable effect on the phase factor of an electron’s wavefunction even if its field strength is zero. For instance, had the electron traverse the region of $A_\mu\neq 0$ with path $\Gamma$, then its wavefunction acquire the phase factor $\psi \rightarrow \psi\exp ie\int_\Gamma A_\mu dx^\mu$. Over a closed loop (say from path difference), we have the Dirac phase factor

$\exp ie\oint A_\mu dx^\mu = \exp ie\Phi$

where $\Phi$ is the magnetic flux through area between the paths, causing an observable shift in interference pattern.

It is interesting now to see what happens to the phase factor under the ambiguity of the gauge transformation $A_\mu \rightarrow A_\mu +\partial_\mu\chi$:

$\exp ie\oint A_\mu dx^\mu \rightarrow \exp ie \oint (A_\mu +\partial_\mu\chi) dx^\mu=\exp ie\oint A_\mu dx^\mu$,

i.e. it is left unchanged, which is desirable for an observable effect. Such quantity is then said to be gauge invariant or a gauge scalar.

Consider now a general transformation of the wavefunction

$\psi(x) \rightarrow \psi'(x)= g(x)\psi(x)$,

where $g(x)= \exp ie \chi(x)$ which is an element of the $U(1)$ group. Conjugate of the wavefunction transforms as $\bar{\psi}(x) \rightarrow \bar{g}(x)\bar{\psi}(x)=g^{-1}(x)\bar{\psi}(x)$. Hence $\bar{\psi}\psi$ (probability density) is a gauge scalar. We need also to consider the property of the derivative $\partial_\mu\psi(x)$ under gauge transformation:

$\partial_\mu\psi'(x) =\partial_\mu \left(g(x)\psi(x)\right)=g(x)(\partial_\mu\psi(x))+(\partial_\mu g(x))\psi(x)$.

Notice the undesirable extra term containing the derivative of $g(x)$. We can cancel this term by noting that the gauge potential can be made to transform like

$A_\mu\rightarrow A'_\mu + \cfrac{1}{e}(\partial_\mu g(x)) g^{-1}(x)$

and hence

$A'_\mu\psi'(x)=A_\mu g(x)\psi(x) +\cfrac{i}{e}\left((\partial_\mu g(x))g^{-1}(x)\right)g(x)\psi(x)$,

also has the similar undesriable term. This is why sometimes the gauge potential or the gauge field is sometimes called a compensating field. We now define the covariant derivative of the wavefunction as

$D_\mu\psi(x)=\partial_\mu\psi(x) -ie A_\mu$.

The gauge covariant derivative transforms like

$D'_\mu\psi'(x) = g(x) D_\mu\psi(x)$.

This would help terms like $\bar{\psi}(x) D_\mu \psi(x)$ to be a gauge scalar.

Gauge theories are essentially ‘field theories’ whose Lagrangians are gauge invariant and this also includes kinetic terms of the gauge fields themselves for instance $F^{\mu\nu} F_{\mu\nu}$ (please check this is gauge scalar). Electromagnetism for instance is a gauge theory whose gauge group is the Abelian group $U(1)$; in other words an Abelian gauge theory. The next step is to actually generalize this to a theory with a non-Abelian gauge group (hope to do this in next blog post),

So far, all of the above are still classical and the challenge is to quantize these classical theories. the redundancy in the gauge freedom actually pose a problem in the process of quantization. This however can be dealt with and must be discussed in the context of quantum field theory.

References

1. Bjorn Felsager, “Geometry, Particles and Fields“, (Springer, 1998)
2. Chan Hong-Mo & Tshou Sheung Tsun, “Some Elementary Gauge Theory Concepts“, (World Scientific, 1993)

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