Video Lecture Series from IIT Professors :
Classical Field Theory by Prof. Suresh Govindarajan sirDr. Suresh Govindarajan – Suresh Govindarajan is undoubtedly one of the most brilliant String Theorists in India. He has been at the forefront of String Theory research and has a lot of publications in Superstring Theory and related fields. He had completed his bachelor's in Electrical Engineering from IIT Madras and went on to get a PhD in Physics from University of Pennsylvania. He has worked in many Institutions which are leaders in the field of research including CERN, TIFR and IIT Madras. Currently, he is an Associate Professor in the Dept. Of Physics, IIT Madras. People who have attended his classes will vouch for the fact that he is a wonderful and enthusiastic teacher
 High School (1982, Atomic Energy Central School, Hyderabad)
 B.Tech. in Electrical Engineering (1986, Indian Institute of Technology Madras)
 Ph. D. in Theoretical Physics (1991, University of Pennsylvania)
The course introduces the student to
relativistic classical field theory. The basic object is a field (such
as the electromagnetic field) which possesses infinite degrees of
freedom. The use of local and global symmetries (such as rotations)
forms an underlying theme in the discussion.
Concepts such as conservation laws, spontaneous breakdown of symmetry, Higgs mechanism etc. are discussed in this context. Several interesting solutions to the EulerLagrange equations of motion such as kinks, vortices, monopoles and instantons are discussed along with their applications. The Standard Model of particle physics is used to illustrate how the various concepts discussed in this course are combined in real applications. All necessary mathematical background is provided to make the course selfcontained. This course may also be considered as a prelude to Quantum Field Theory. 
 Classical Mechanics, Electromagnetism (and possibly the special theory of relativity).

L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Pergamon (1975).

M. R. Spiegel, Vector Analysis, Schaum Outline Series, McGrawHill (1974).

M. Carmeli, Classical Fields, Wiley (1982).

A. O. Barut, Electrodynamics and Classical Theory of Fields, Chap. 1,Macmillan (1986).

C. Itzykson and J. B. Zuber, Quantum Field Theory, Chap. 1, McGrawHill (1986).

S. Coleman, Aspects of Symmetry, Cambridge Univ. Press.

R. Rajaraman, Solitons and Instantons, NorthHolland.
Module
1 Introduction to Classical Field Theory (1 Lecture)
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Lecture 1: What is
Classical Field Theory?

Review of classical mechanics,
Particle Trajectories and the Principle of least action, Feynman's
description of QM, Classical Mechanics to Classical Fields.

Do Problem Set 1 before viewing Lecture 2!

Module
2 Symmetries and Group Theory (6 Lectures)
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Lecture 2: Symmetries and
Invariances  I

Symmetries, Invariances of
Newton's EOM vs Maxwell's Equations, The Galilean Group.


Lecture 3: Symmetries and
Invariances  II

Invariances of Maxwell's Equations
continued, Common Four Vectors, Covariant Formulation of Maxwell's
Equations,Lorentz and Poincare Groups, Rotation Group and vectors under
rotation.

Attempt Problem Set 2 after viewing Lecs. 2 and 3.


Lecture 4: Group Theory
in Physics  I

Definition of a Group,
Antisymmetric Matrices and SO(d),Vectors and Tensors of SO(d), Parity: Polar and Axial Vectors.

Solve Problem Set 3 while/after viewing lecs. 4 and
5.


Lecture 5 Group Theory in
Physics  II

Generalizations of SO(d) (specifically the Lorentz Group),Simple Boost Matrices
and Rapidity, SO(p,q) with general signatures in metric,The Symplectic Group.

40:30 The matrix should be
symmetric and nondegenerate. Symplectic matrices have det=1


Lecture 6: Finite Groups
 I

Finite Groups of low order :
Cyclic and Coxeter(specifically Dihedral) Groups,Definition of a Subgroup,
Equivalence relation and Cosets.

49:19 Left coset wrongly called
right coset. Corrected at the start of lec. 7


Lecture 7: Finite Groups
 II

Left and Right Cosets, Permutation
Group,Normal Subgroups, Classification of Finite Simple Groups, Monstrous
moonshine.

Solve Problem Set 4 while/after viewing lecs. 6 and
7. Normal Subgroups

Module
3 Actions for Classical Field Theory (3 Lectures)
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Lecture 8: Basics of CFT
 I

Classical Mechanics of Fields,
Structure of the KE term in the Lagrangian density, the ultralocal term, and
Lorentz invariance of the Lagrangian.

Solve Problem Set 5 after viewing lecs. 8 and 9.


Lecture 9: Basics of CFT
 II

Action Principle for fields,
Conditions on Lagrangian density for no surface contribution, Conserved
Currents,Hamiltonian density, Conditions for Finite Energy.

at 50:46 and 51:33 min[finite
energy cond — wrong power]


Lecture 10: Basics of CFT
 III

Definition of Vacuum and examples,
Vacuum Solutions for quartic potential, Topological Currents and Charges,
Noether's Theorem, Application to translational invariance.

Solve Problem Set 6 after viewing lec. 10 but before
lec. 15 where it is discussed.

Module
4 Green Functions for the KleinGordon Operator (2 Lectures)
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Lecture 11 Green
Functions  I

Inhomogenous KleinGordon
Equation, Method of the Green functions, Advanced and Retarded Green
Functions.


Lecture 12 Green
Functions  II

Green Functions of the KG
operator, Closing the contour,The Feynman propagator.

Module
5 Symmetries and Conserved quantities (2 Lectures)
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Lecture 13 Noether's
Theorem  I

Types of Symmetries, Internal
Symmetries, Notion of "small", Transformations to first order (for
Lorentz Group), Formulation to derive the Master formula.


Lecture 14 Noether's
Theorem  II

Derivation of the Master formula
for the Noether current, The energymomentum tensor and the generalized angular
momentum tensor as examples.

Solve Problem Set 7 after viewing lec. 14 but before
lec. 20 where it is discussed.

Module
6 Solitons  I (Kink soliton) (1 lecture)
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Lecture 15 Kink Soliton

Studying timeindependent, finite
energy solutions to the EulerLagrange equations of motion,the kink soliton,
Derrick's theorem and its proof.

Module
7 Hidden Symmetry (Spontaneous Symmetry Breaking) & the abelian Higgs
mechanism (3 Lectures)
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Lecture 16: Hidden
Symmetry

Spontaneous symmetry breaking and
statement of Goldstone's theorem.


Lecture 17: Local
Symmetries

Symmetry breaking continued, The
MerminWagnerColeman theorem,The ideas of global and local symmetries, the
covariant derivative,Minimal prescription for the covariant derivative.

23:13 Should change the location
of the minus sign in the SO(2) matrix or equivalently take q to −q.


Lecture 18 The Abelian
Higgs model

Definition of field strength using
the covariant derivative, Small fluctuations about the vacuum solution,The
Higgs mechanism in the U(1) case

29:25: Index mismatch in cov.
current Î¼/Î½ on LHS/RHS.

Module
8 Lie algebras, symmetry breaking and Noether's theorem for Maxwell Equations
(2 Lectures)
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Lecture 19: Lie Algebras
 I

Recap of symmetries and Noether's
theorem, Lie algebras and finitedimensional representations, su(2) Lie Algebra.

Solve Problem Set 8 while viewing lectures 19/20.


Lecture 20 Lie Algebras 
II

su(3) Lie Algebra ; Symmetry breaking in terms of Lie algebras,
Conserved currents for the Proca action: energymomentum, generalized angular
momentum and the symmetric energymomentum tensors.

Module
9 Solitons — II (Magnetic Vortices) (2 Lectures)
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Lecture 21: Magnetic
Vortices  I

Finite energy, timeindependent
solutions in the Abelian Higgs model in 2+1 dimensions, Topological charge ==
Magnetic flux, Quantization of magnetic flux, The
Bogomol'nyiPrasadSommerfield(BPS) bound for energy, Saturation of the BPS
bound.

Solve Problem Set 9 before viewing lecture 30.


Lecture 22: Magnetic
Vortices  II

Vortices in the Abelian Higgs
model applied to superconducting materials,characteristic lengths in the
problem, "size" of a vortex, Description of vortex number using the
fundamental group of the gauge group U(1), or the circle.

35:38 and 36:03 min[finite energy
cond — wrong power]

Module
10 Towards Nonabelian gauge theories (2 Lectures)
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Lecture 23: Nonabelian
gauge theories  I

Nonabelian gauge symmetry with
SU(2) as an example, Covariant derivative in the nonabelian case,
Construction of a locally SU(2) invariant Lagrangian, Transformation of the
gauge fields under local gauge transformations.

Solve Problem Set 10 while viewing lectures 23/24.


Lecture 24: Nonabelian
gauge theories  II

Transformation of the gauge
fields(continued), Derivation of the field strength for the gauge field,
Symmetry breaking in the nonabelian case, Goldstone's theorem in terms of
Lie Algebras.

Module
11 Representation theory of Lie Algebras (2 Lectures)
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Lecture 25: Irreps of Lie
algebras  I

Representation theory of su(2) and su(3), the Cartan subalgebra, the adjoint representation.

3:00  Misleading statement: Map
from G to GL(N). While GL(N) is set of all linear maps on V, the map from G
to GL(N) is not linear.28:00  Blocks "0" and "*" in the
matrix have been interchanged.


Lecture 26: Irreps of Lie
algebras  II

Representation theory continued,
Ferrer's diagrams.

Quiz
(Test yourself)
If you have gotten this far, you can
test you understanding by taking this Quiz. It is meant to be an open notes (i.e., your
own notes) examination and the expected duration is one and half hours. I don't
intend to post the solutions online but they will be provided on request. This
is to counter the natural human tendency to look at solutions if they are
available! What is a good score? I would say anything over 50% is acceptable.
Module
12 The Standard Model of Particle Physics (2 Lectures)
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Lecture 27 The Standard
Model  I

su(3) multiplets, Motivation for the Standard Model, Colour
confinement, GellMann—Nishijima relation.


Lecture 28 The Standard
Model  II

Electroweak symmetry breaking — An
application of symmetry breaking in the nonabelian case.

Module
13 The Lorentz and Poincare Lie Algebras (1 Lecture)
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Lecture 29: Irreps of the
Lorentz/Poincare algebras

The Lorentz and Poincare algebras
and their representations.

Module
14 Solitons — III (Monopoles and Dyons) (3 Lectures)
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Lecture 30: The Dirac
mononpole

Magnetically charged solutions:
The Dirac monopole, Flux quantization.

Solve Problem Set 11 while viewing lectures 30/31/32.


Lecture 31: The 't
HooftPolaykov monopole

Magnetically charged solutions:
The ‘t HooftPolyakov monopole, The PrasadSommerfield limit.


Lecture 32: Revisiting
Derrick’s Theorem

Revisiting Derrick's theorem, BPS
solution


Lecture 33: The JuliaZee
dyon

Constructing dyonic solutions,
Dirac quantization for dyons; Dimensional reduction.

Module
15 Instantons and their physical interpretation (4 Lectures)
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Lecture 34: Instantons 
I

Quantum mechanical tunnelling and
Instantons.

Solve Problem Set 12 while viewing lectures 34/37.


Lecture 35: Instantons 
II

Kink soliton and tunnelling,
Instantons in pure YangMills theories(SU(2)).


Lecture 36:  Instantons
 III

More on instantons, The BPS bound.


Lecture 37: Instantons 
IV

Free parameters in instanton
solutions, moduli space, Complexified YangMills and theta vacua.

Module
16 An introduction to some advanced topics (2 Lectures)
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Lecture 38: Dualities

Dualities in Field Theory: Ising
Model; SineGordon / Massive Thirring; SU(2) YangMills in 3+1 dimensions.


Lecture 39:
Geometrization of Field Theory

General relativity as a gauge
theory; Geometrization of Field Theory; Glimpse into String theory and
branes.

The
Final
If you have gotten this far, you can
test your understanding (of the course material) by taking this Final Examination. It is meant to be an open
notes (i.e., your own notes) examination and the expected duration is three
hours. I don't intend to post the solutions online but they will be provided on
request.