1. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 1 PHYS 5326 – Lecture #9 Wednesday, Feb. 28, 2007 Dr. Jae Yu 1.Quantum Electro-dynamics (QED) 2.Local Gauge Invariance 3.Introduction of Massless Vector Gauge Field

2. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 2 Announcements First term exam will be on Wednesday, Mar. 7 It will cover up to what we finish today The due for all homework up to last week’s is Monday, Mar. 19

3. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 3 Prologue How is a motion described? –Motion of a particle or a group of particles can be expressed 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 –With the operators provide means for obtaining values for 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.

4. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 4 Non-relativistic Equation of Motion for Spin 0 Particle Energy-momentum relation in classical mechanics give provides the non-relativistic equation of motion for field, ,, the Schrödinger Equation Quantum prescriptions;. represents the probability of finding the particle of mass m at the position (x,y,z)

5. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 5 Relativistic Equation of Motion for Spin 0 Particle Relativistic energy-momentum relationship With four vector notation of quantum prescriptions; Relativistic equation of motion for field, , the Klein-Gordon Equation 2 nd order in time

6. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 6 Relativistic Equation of Motion (Dirac Equation) for Spin 1/2 Particle To avoid 2 nd order time derivative term, Dirac attempted to factor relativistic energy-momentum relation This works for the case with zero three momentum This results in two first order equations

7. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 7 The previous prescription does not work 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 k to satisfy: Dirac Equation Continued… The coefficients like  0 =1 and  1 =  2 = 3=i 3=i do not work since they do not eliminate the cross terms.

8. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 8 Dirac Equation Continued… It would work if these coefficients are matrices that satisfy the conditions Using gamma matrices with the standard Bjorken and Drell convention Or using Minkowski metric, gg where Where i i are Pauli spin matrices

9. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 9 Dirac Equation Continued… Using Pauli matrix as components in coefficient matrices whose smallest size is 4x4, the energy-momentum relation can now be factored w/ a solution By applying quantum prescription of momentum Acting the 1-D solution on a wave function, , we obtain Dirac equation where Dirac spinor, 

10. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 10 Euler-Lagrange Equation For a conservative force, the force can be expressed as the gradient of the corresponding scalar potential, U Therefore the Newton’s law can be written. The 1-D Euler-Lagrange fundamental equation of motion Starting from Lagrangian In 1D Cartesian Coordinate system

11. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 11 Euler-Lagrange equation in QFT Unlike particles, field occupies regions of space. Therefore in field theory, the motion is expressed in terms of space and time. Euler-Larange equation for relativistic fields is, therefore, Note the four vector form

12. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 12 Klein-Gordon Largangian for scalar (S=0) Field For a single, scalar field , 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.m.

13. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 13 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.m. How’s Euler Lagrangian equation looks like for 

14. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 14 Proca Largangian for Vector (S=1) Field Suppose we take the Lagrangian for a vector field AA Where F F is the field strength tensor in relativistic notation, E and B in Maxwell’s equation form an anti-symmetic second-rank tensor

15. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 15 Proca Largangian for Vector (S=1) Field Suppose we take the Lagrangian for a vector field A  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.m.

16. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 16 Lagrangians we discussed are concocted to produce desired field equations –L –L derived (L=T-V) in classical mechanics –L –L taken as axiomatic in field theory The Lagrangian for a particular system is not unique –Can always multiply by a constant –Or add a divergence –Since these do not affect field equations due to cancellations

17. Wednesday, Feb. 28, 2007PHYS 5326, Spring 2007 Jae Yu 17 Homework Prove that Fmn can represent Maxwell’s equations, pg. 225 of Griffith’s book. Derive Eq. 11.17 in Griffith’s book Due is Wednesday, Mar. 7