Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 1 PHYS 5326 – Lecture #13 Wednesday, Mar. 5, 2003 Dr. Jae Yu Local Gauge Invariance and Introduction of Massless Vector Gauge Field

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 2 Announcements Remember the mid-term exam next Friday, Mar. 14, between 1-3pm in room 200 –Written exam Mostly on concepts to gauge the level of your understanding on the subjects Some simple computations might be necessary –Constitutes 20% of the total credit, if final exam will be administered otherwise it will be 30% of the total Strongly urge you to go into the colloquium today.

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 3 Prologue Motion of a particle is express in terms of the position of the particle at any given time in classical mechanics. A state (or a motion) of particle is expressed in terms of wave functions that represent probability of the particle occupying certain position at any given time in Quantum mechanics.  Operators provide means for obtaining observables, such as momentum, energy, etc A state or motion in relativistic quantum field theory is expressed in space and time. Equation of motion in any framework starts with Lagrangians.

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 4 Non-relativistic Equation of Motion for Spin 0 Particle Energy-momentum relation in classical mechanics give Provides non-relativistic equation of motion for field, , Schrodinger Equation Quantum prescriptions;. represents the probability of finding the pacticle of mass m at the position (x,y,z)

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 5 Relativistic Equation of Motion for Spin 0 Particle Relativistic energy-momentum relation With four vector notation of quantum prescriptions; Relativistic equation of motion for field, , Klein-Gordon Equation 2 nd order in time

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 6 Relativistic Equation of Motion (Direct Equation) for Spin 1/2 Particle To avoid 2 nd order time derivative term, Direct attempted to factor relativistic energy-momentum relation This works for the case with 0 three momentum But not for the case with non-0 three momentum The terms linear to momentum should disappear, so To make it work, we must find coefficients  k to satisfy:

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 7 Dirac Equation Continued… The coefficients like  0 =1 and  1 =  2 =  3 = i do not work since they do not eliminate the cross terms. It would work if these coefficients are matrices that satisfy the conditions Using Pauli matrix as components in coefficient matrices whose smallest size is 4x4 The energy-momentum relation can be factored Or using Minkowski metric, g  w/ a solution By applying quantum prescription of momentum Acting it on a wave function , we obtain Dirac equation

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 8 Euler-Lagrange Equation For conservative force, it can be expressed as the gradient of a scalar potential, U, as Therefore the Newton’s law can be written. The Euler-Lagrange fundamental equation of motion Starting from Lagrangian In 1D Cartesian Coordinate system

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 9 Euler-Lagrange equation in QFT Unlike particles, field occupies regions of space. Therefore in field theory motion is expressed in space and time. Euler-Larange equation for relativistic fields is, therefore, Note the four vector form

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 10 Klein-Gordon Largangian for scalar (S=0) Field For a single, scalar field variable , the lagrangian is From the Euler-Largange equation, we obtain Sinceand This equation is the Klein-Gordon equation describing a free, scalar particle (spin 0) of mass m.

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 11 Dirac Largangian for Spinor (S=1/2) Field For a spinor field , the lagrangian From the Euler-Largange equation for , we obtain Sinceand Dirac equation for a particle of spin ½ and mass m. How’s Euler Lagrangian equation looks like for 

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 12 Proca Largangian for Vector (S=1) Field For a vector field A , the lagrangian From the Euler-Largange equation for A , we obtain Sinceand For m =0, this equation is for an electromagnetic field. Proca equation for a particle of spin 1 and mass m.

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 13 Local Gauge Invariance - I Dirac Lagrangian for free particle of spin ½ and mass m ; Is invariant under a global phase transformation (global gauge transformation) since. However, if the phase, , varies as a function of space-time coordinate, x , is L still invariant under the local gauge transformation, ? No, because it adds an extra term from derivative of .

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 14 Local Gauge Invariance - II The derivative becomes So the Lagrangian becomes Since the original L is L’ is Thus, this Lagrangian is not invariant under local gauge transformation!!

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 15 Local Gauge Invariance - III Defining a local gauge phase, (x), as where q is the charge of the particle involved, L becomes Under the local gauge transformation:

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 16 Local Gauge Invariance - IV Requiring the complete Lagrangian be invariant under (x) local gauge transformation will require additional terms to free Dirac Lagrangian to cancel the extra term Where A  is a new vector gauge field that transforms under local gauge transformation as follows: Addition of this vector field to L keeps L invariant under local gauge transformation, but…

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 17 Local Gauge Invariance - V The new vector field couples with spinor through the last term. In addition, the full Lagrangian must include a “free” term for the gauge field. Thus, Proca Largangian needs to be added. This Lagrangian is not invariant under the local gauge transformation,, because

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 18 Local Gauge Invariance - VI The requirement of local gauge invariance forces the introduction of massless vector field into the free Dirac Lagrangian.

Wednesday, Mar. 5, 2003PHYS 5326, Spring 2003 Jae Yu 19 Homework Prove that the new Dirac Lagrangian with an addition of a vector field A , as shown on page 12, is invariant under local gauge transformation. Describe the reason why the local gauge invariance forces the vector field to be massless.