Lagrangian formulation of the Klein Gordon equation Klein

Lagrangian formulation of the Klein Gordon equation Klein Gordon field Manifestly Lorentz invariant } } T V Classical path : Euler Lagrange equation Klein Gordon equation

New symmetries

New symmetries Is invariant under …an Abelian (U(1)) gauge symmetry

New symmetries Is not invariant under

New symmetries Is invariant under …an Abelian (U(1)) gauge symmetry

New symmetries Is invariant under …an Abelian (U(1)) gauge symmetry

New symmetries Is invariant under …an Abelian (U(1)) gauge symmetry A symmetry implies a conserved current and charge. e. g. Translation Rotation Momentum conservation Angular momentum conservation

New symmetries Is invariant under …an Abelian (U(1)) gauge symmetry A symmetry implies a conserved current and charge. e. g. Translation Rotation Momentum conservation Angular momentum conservation What conservation law does the U(1) invariance imply?

Noether current Is invariant under …an Abelian (U(1)) gauge symmetry

Noether current Is invariant under …an Abelian (U(1)) gauge symmetry

Noether current Is invariant under …an Abelian (U(1)) gauge symmetry

Noether current Is invariant under …an Abelian (U(1)) gauge symmetry

Noether current Is invariant under …an Abelian (U(1)) gauge symmetry 0 (Euler lagrange eqs. )

Noether current Is invariant under …an Abelian (U(1)) gauge symmetry 0 (Euler lagrange eqs. ) Noether current

The Klein Gordon current Is invariant under …an Abelian (U(1)) gauge symmetry

The Klein Gordon current Is invariant under …an Abelian (U(1)) gauge symmetry

The Klein Gordon current Is invariant under …an Abelian (U(1)) gauge symmetry This is of the form of the electromagnetic current we used for the KG field

The Klein Gordon current Is invariant under …an Abelian (U(1)) gauge symmetry This is of the form of the electromagnetic current we used for the KG field is the associated conserved charge

Suppose we have two fields with different U(1) charges : . . no cross terms possible (corresponding to charge conservation)

Additional terms

Additional terms } Renormalisable

Additional terms } Renormalisable

U(1) local gauge invariance and QED

U(1) local gauge invariance and QED

U(1) local gauge invariance and QED not invariant due to derivatives

U(1) local gauge invariance and QED not invariant due to derivatives To obtain invariant Lagrangian look for a modified derivative transforming covariantly

U(1) local gauge invariance and QED not invariant due to derivatives To obtain invariant Lagrangian look for a modified derivative transforming covariantly Need to introduce a new vector field

is invariant under local U(1)

is invariant under local U(1) Note : is equivalent to universal coupling of electromagnetism follows from local gauge invariance

is invariant under local U(1) Note : is equivalent to universal coupling of electromagnetism follows from local gauge invariance The Euler lagrange equation give the KG equation:

is invariant under local U(1) Note : is equivalent to universal coupling of electromagnetism follows from local gauge invariance

The electromagnetic Lagrangian

The electromagnetic Lagrangian

The electromagnetic Lagrangian

The electromagnetic Lagrangian

The electromagnetic Lagrangian Forbidden by gauge invariance

The electromagnetic Lagrangian Forbidden by gauge invariance The Euler-Lagrange equations give Maxwell equations !

The electromagnetic Lagrangian Forbidden by gauge invariance The Euler-Lagrange equations give Maxwell equations !

The electromagnetic Lagrangian Forbidden by gauge invariance The Euler-Lagrange equations give Maxwell equations ! EM dynamics follows from a local gauge symmetry!!

The photon propagator The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations.

The Klein Gordon propagator (reminder) In momentum space: With normalisation convention used in Feynman rules = inverse of momentum space operator multiplied by -i

The photon propagator The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations.

The photon propagator The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations. Gauge ambiguity

The photon propagator The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations. Gauge ambiguity

The photon propagator The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations. Gauge ambiguity i. e. with suitable “gauge” choice of α (“ξ” gauge) want to solve

The photon propagator The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations. Gauge ambiguity i. e. with suitable “gauge” choice of α (“ξ” gauge) want to solve In momentum space the photon propagator is (‘t Hooft Feynman gauge ξ=1)

Extension to non-Abelian symmetry

Extension to non-Abelian symmetry

Extension to non-Abelian symmetry where

Extension to non-Abelian symmetry where i. e. Need 3 gauge bosons

Weak Interactions Symmetry : SU(2) local gauge theory Local conservation of 2 weak isospin charges Weak coupling, α 2 u d Wa=1. . 3 Gauge boson (J=1) e Neutral currents A non-Abelian (SU(2)) local gauge field theory

Symmetry : Local conservation of 3 strong colour charges QCD : a non-Abelian (SU(3)) local gauge field theory

The strong interactions QCD Quantum Chromodynamics Symmetry : Local conservation of 3 strong colour charges SU(3) Strong coupling, α 3 q q Ga=1. . 8 Gauge boson (J=1) “Gluons” QCD : a non-Abelian (SU(3)) local gauge field theory

Partial Unification Matter Sector “chiral” Family Symmetry? Up Down Family Symmetry? Neutral

Partial Unification Matter Sector “chiral” } Family Symmetry? } Up Down Family Symmetry? Neutral

Partial Unification } Matter Sector “chiral” } } Family Symmetry? Up Down Family Symmetry? Neutral