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Video Lectures on "Classical Field Theory" by Prof. Suresh Govindarajan sir

Video Lecture Series from IIT Professors :
Classical Field Theory by Prof. Suresh Govindarajan sir

 
Prof. Suresh Govindarajan
Dr. 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 Euler-Lagrange 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 self-contained. This course may also be considered as a prelude to Quantum Field Theory.

 
  1. Classical Mechanics, Electromagnetism (and possibly the special theory of relativity).

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

  2. M. R. Spiegel, Vector Analysis, Schaum Outline Series, McGraw-Hill (1974).

  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, Chap. 1, McGraw-Hill (1986).

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

  7. R. Rajaraman, Solitons and Instantons, North-Holland.






Module 1 Introduction to Classical Field Theory (1 Lecture)

Lecture Number
Content of the Lecture
Additional Info

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)

Lecture Number
Content of the Lecture
Additional Info

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 non-degenerate. 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)

Lecture Number
Content of the Lecture
Additional Info

Lecture 8: Basics of CFT - I
Classical Mechanics of Fields, Structure of the KE term in the Lagrangian density, the ultra-local 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 Klein-Gordon Operator (2 Lectures)

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Content of the Lecture
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Lecture 11 Green Functions - I
Inhomogenous Klein-Gordon 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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Content of the Lecture
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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 energy-momentum 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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Content of the Lecture
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Lecture 15 Kink Soliton
Studying time-independent, finite energy solutions to the Euler-Lagrange 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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Content of the Lecture
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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 Mermin-Wagner-Coleman 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 μ/ν on LHS/RHS.

Module 8 Lie algebras, symmetry breaking and Noether's theorem for Maxwell Equations (2 Lectures)

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Content of the Lecture
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Lecture 19: Lie Algebras - I
Recap of symmetries and Noether's theorem, Lie algebras and finite-dimensional 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: energy-momentum, generalized angular momentum and the symmetric energy-momentum tensors.

Module 9 Solitons — II (Magnetic Vortices) (2 Lectures)

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Content of the Lecture
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Lecture 21: Magnetic Vortices - I
Finite energy, time-independent solutions in the Abelian Higgs model in 2+1 dimensions, Topological charge == Magnetic flux, Quantization of magnetic flux, The Bogomol'nyi-Prasad-Sommerfield(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 Non-abelian gauge theories (2 Lectures)

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Content of the Lecture
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Lecture 23: Non-abelian gauge theories - I
Non-abelian gauge symmetry with SU(2) as an example, Covariant derivative in the non-abelian 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: Non-abelian gauge theories - II
Transformation of the gauge fields(continued), Derivation of the field strength for the gauge field, Symmetry breaking in the non-abelian case, Goldstone's theorem in terms of Lie Algebras.


Module 11 Representation theory of Lie Algebras (2 Lectures)

Lecture Number
Content of the Lecture
Additional Info

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)

Lecture Number
Content of the Lecture
Additional Info

Lecture 27 The Standard Model - I
su(3) multiplets, Motivation for the Standard Model, Colour confinement, Gell-Mann—Nishijima relation.


Lecture 28 The Standard Model - II
Electroweak symmetry breaking — An application of symmetry breaking in the non-abelian case.


Module 13 The Lorentz and Poincare Lie Algebras (1 Lecture)

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Content of the 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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Content of the Lecture
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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 Hooft-Polaykov monopole
Magnetically charged solutions: The ‘t Hooft-Polyakov monopole, The Prasad-Sommerfield limit.


Lecture 32: Revisiting Derrick’s Theorem
Revisiting Derrick's theorem, BPS solution


Lecture 33: The Julia-Zee dyon
Constructing dyonic solutions, Dirac quantization for dyons; Dimensional reduction.


Module 15 Instantons and their physical interpretation (4 Lectures)

Lecture Number
Content of the Lecture
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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 Yang-Mills 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 Yang-Mills and theta vacua.


Module 16 An introduction to some advanced topics (2 Lectures)

Lecture Number
Content of the Lecture
Additional Info

Lecture 38: Dualities
Dualities in Field Theory: Ising Model; Sine-Gordon / Massive Thirring; SU(2) Yang-Mills 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.