Classical Field Theory

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Review of Classical Mechanics : Hamiltonians and Lagrangians. Legendre transforms and their properties. Euler-Lagrange equations. Principle of least action. (Functional calculus.) What is classical field theory – Vectors and Tensors : Group theory from invariances of classical equations – Newton’s equations and the Galilean group. Maxwell’s equations. Special Relativity and the Lorentz group. Vectors and tensors of the rotation and Lorentz groups. (Basics of group theory: definition, discrete groups and matrix groups.) – Basics of CFT : Systems with infinite degrees of freedom. Locality in space and time. Lagrangian densities for real and complex scalar fields. Euler-Lagrange (EL) equations. Functional calculus revisited. Hamiltonian density. The energy-momentum tensor – Solutions to the EL equations : Finite-energy time-independent solutions — classical vacua. Kinks in the Sine-Gordon and φ4 theories. Green functions as singular solutions. Boundary conditions.

Symmetries and Conservation Laws : Discrete and continuous symmetries. Noether’s theorem: the energy momentum tensor, the generalized angular momentum and the electromagnetic current. (Lie groups and Lie algebras. Representations of groups.) – The massless vector field : The Lagrangian density. Gauge invariance and the electromagnetic field strength. Maxwell’s equations. Lorentz invariants of the field strength. The symmetrized energy-momentum tensor. The generalized angular momentum and the spin of the photon – Secret Symmetry : Global symmetries. Spontaneous breakdown of symmetry. Goldstone’s theorem. (Coset spaces in group theory.) – Solitons : Solitons as finite-energy solutions. Derrick’s theorem. Getting around Derrick’s theorem. Local symmetries and gauge fields. Abelian vortices. The Dirac monopole as a singular solution of Maxwell’s equations. Dirac quantization – Local Symmetries : Abelian gauge fields. Covariant derivatives and minimal coupling. The abelian Higgs model. Vortex solutions (in type II superconductors). Topological conservation laws. The abelian Higgs mechanism.

Non-abelian gauge theories : Covariant derivatives; The Yang–Mills field strength; Coupling matter to non-abelian gauge fields; Higgs mechanism – SU(2)->U(1) and SO(3)->U(1) ; Weinberg’s theorem. The ‘t Hooft-Polyakov monopole as a non-singular solution. Julia-Zee dyons; the Bogomolnyi-Prasad- Sommerfield(BPS) limit. Dirac quantization for dyons – The Standard Model of Particle Physics as a CFT : Basic forces in nature. Symmetry breaking in the electroweak sector. Quantum Chromodynamics and the quark model – Instantons : Instantons as finite action solutions to the EL equations.The ‘t Hooft solution. Nahm and Bogomolnyi equations from dimensional reduction.

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