Syllabus & References for Classical Field Theory

Rather than give a conventional syllabus,1 I will discuss four levels (in increasing order of complexity) to quantify the goals of potential students. Periodically visiting this page while doing the lectures will help you see your gradual progression.

Level 1: Given a Lagrangian density, you should be able to obtain the Euler-Lagrange Equations Of Motion (EOM). You should also be able to verify that a given solution indeed solves the EOM. Further, if you are given an ansatz for the fields, you should be able to substitute the ansatz and obtain equation(s) for the undetermined function(s) in the ansatz. For instance, consider ($a,b,c=1,2,3$)

(1)
\begin{align} {\cal L} = -{1\over4}F^{\mu\nu a} F_{{\mu\nu}}^a +{1\over2}\pi^{\mu a} \pi_\mu^a + {1\over2} \mu^2 \phi^a \phi^a -{1\over4} \lambda (\phi^a\phi^a)^2\quad, \end{align}

where $F^a_{\mu\nu}=(\partial_\mu A^a_\nu - \partial_\nu A^a_\mu + e\epsilon^{abc} A_\mu^b A_\nu^c)$; and $\pi_\mu^a = \partial_\mu \phi^a +e \epsilon^{abc} A_\mu^b \phi^c$. Consider the following time-independent ansatz

(2)
\begin{eqnarray} A_0^a &=& 0 \quad, \nonumber \\ A_i^a &=& \epsilon_{aij}x_j [1-K(r)]/er^2 \quad, \nonumber \\ \phi^a &=& x_a H(r)/er^2\quad, \nonumber \end{eqnarray}

where $r=\sqrt{x_1^2 + x_2^2 + x_3^2}$. Starting from the equations of motion, obtain the equations satisfied by the functions $H(r)$ and $K(r$).

Level 2: You can do everything mentioned in Level 1. Further, you can do more such as identifying at least some of the symmetries from a Lagrangian density and a given solution to the EOM by inspection. Then you should be able to obtain the conserved currents and charges using Noether's theorem. After obtaining the Hamiltonian density, you should be able to impose boundary conditions on the fields in order to get finite energy.

Level 3: You can do everything mentioned in Levels 1-2. You are comfortable with Lie algebras and Lie groups. You understand the Lagrangian density for non-abelian gauge theories. You understand the statements for Goldstone's theorem and the Higgs mechanism in the language of groups and their cosets. The relationship of vortices in the abelian Higgs model and vortices in type II superconductors is obvious.

Level 4: You can do everything mentioned in Levels 1-3. You are able to fully comprehend the Prasad-Sommerfield paper, the 't Hooft ansatz for the monopole and why the 't Hooft-Polyakov monopole does not carry the minimum value allowed by the Dirac quantization condition. You are unhappy that fermions and supersymmetry were not discussed in this course2. You are desperate to learn the ADHM_construction of instantons. You are beginning to observe inaccuracies in some of my statements in the lectures.

## References

1. S. Coleman, Aspects of Symmetry, Cambridge Univ. Press. (This book has had an enormous influence on my teaching.)
2. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 4th edition, Pergamon (1975). (An all-time classic)
3. M. Carmeli, Classical Fields, Wiley (1982).
4. A. O. Barut, Electrodynamics and Classical Theory of Fields, Chap. 1, Macmillan (1986).
5. C. Itzykson and J. B. Zuber, Quantum Field Theory, International Student Edition, Chap. 1, McGraw-Hill (1986).
6. R. Rajaraman, Solitons and Instantons, North-Holland
7. N. Manton and P. Sutcliffe, Topological Solitons, Cambridge Univ. Press. (2004)
8. D. Tong, TASI Lectures on Solitons

The last three references are at a level higher than the level of this course.