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PHYS 5326 – Lecture #13

PHYS 5326 – Lecture #13. Wednesday, Mar. 5, 2003 Dr. Jae Yu. Local Gauge Invariance and Introduction of Massless Vector Gauge Field. Announcements. Remember the mid-term exam next Friday, Mar. 14, between 1-3pm in room 200 Written exam

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PHYS 5326 – Lecture #13

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

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

  5. 2nd order in time 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, y, Klein-Gordon Equation PHYS 5326, Spring 2003 Jae Yu

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

  7. By applying quantum prescription of momentum Dirac Equation Continued… It would work if these coefficients are matrices that satisfy the conditions The coefficients like g0=1 and g1= g2= g3=i do not work since they do not eliminate the cross terms. Or using Minkowski metric, gmn Using Pauli matrix as components in coefficient matrices whose smallest size is 4x4 The energy-momentum relation can be factored w/ a solution Acting it on a wave function y, we obtain Dirac equation PHYS 5326, Spring 2003 Jae Yu

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

  9. Note the four vector form 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, PHYS 5326, Spring 2003 Jae Yu

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

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

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

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

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

  16. Local Gauge Invariance - IV Requiring the complete Lagrangian be invariant under l(x) local gauge transformation will require additional terms to free Dirac Lagrangian to cancel the extra term Where Am 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… PHYS 5326, Spring 2003 Jae Yu

  17. This Lagrangian is not invariant under the local gauge transformation, , because 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. PHYS 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. PHYS 5326, Spring 2003 Jae Yu

  19. Homework • Prove that the new Dirac Lagrangian with an addition of a vector field Am, 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. PHYS 5326, Spring 2003 Jae Yu

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