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Announcements. Reading quiz shifted to tomorrow. Lectures through Monday posted. Homework solutions 1, 2a, 2b posted. 9/12. The Dirac Equation with E & M. Schrodinger Equation. Solutions of H 0 are plane waves H int allows electrons to scatter off of EM source. Initial and Final States.

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  1. Announcements Reading quiz shifted to tomorrow Lectures through Monday posted Homework solutions 1, 2a, 2b posted 9/12

  2. The Dirac Equation with E & M Schrodinger Equation • Solutions of H0 are plane waves • Hintallows electrons to scatter off of EM source

  3. Initial and Final States • Need to get the normalization right

  4. The Electromagnetic Field • We need to know the vector potential • Any function can be written as a sum of Fourier modes • We will work with just one mode at a time • Suppose two people used different gauges, • How would their Fourier components differ?

  5. Setting it up . . . • Fermi’s Golden Rule: • Needs to be modified because it assumed a constant Hamiltonian • Ignore all time dependence because it is already handled in Fermi’s Golden rule.

  6. … And Calculating …

  7. … And Finishing

  8. Announcements Homework returned in Boxes 9/12

  9. “I'm also confused on the physical meaning of gauge. I realize the invariance and how to show that they are equivalent, but I'm lacking a way of comparing them to something I'm familiar with physically.” Consider a coordinate transformation: Rewrite as a matrix: Physics should be invariant under this Rotation is a symmetry of physics

  10. Phase Invariance Change the phase of a wave function: Break it into real and imaginary parts: Write it out explicitly: This is also a symmetry of physics It is called an internal symmetry

  11. SO(2) and U(1) The set of (proper) rotations in 2D space is called SO(2) The set of 11 matrices which are unitary is called U(1) Unitary means: These two groups are mathematically equivalent

  12. U(1) gauge invariance Not surprisingly, physics is invariant if we take: Surprisingly, we can also make it invariant if we take: The derivatives change: To fix it, need to switch to covariant derivative: Gauge field must also change:

  13. A Relativistic Hamiltonian Fock Space – 0 or 1 Particle states • Want a notation that allows arbitrary numbers of particles • Not a wave function • The ground state will be noted as: • Assume properly normalized • One particle states will be denoted by: • t tells us the type of particle (e-, for example) • p tells us its four-momentum • s tells us any spin information • Normalization: • Dimension of one-particle states:

  14. Fock Space – Many Particle states • Two particles will look like • Normalization: • Dimension of two-particle states: • Order (almost) doesn’t matter:

  15. “I'm a little confused on how lorentz invariance … affects the states we use.” Our Normalization: What happens when we Lorentz transform these states? Why is the factor of 2E there? What if we worked with these states anyway?

  16. Our Normalization: Why is the factor of V there? Another way to see it: take V  Despite appearances, this expression is Lorentz invariant

  17. “I am still a little unclear regarding the usage and meaning of the spectator property. It seems that its function is minimal if it can simply be factored out, yet I'm curious as to why it is mentioned as a property.” Can the following scattering occur in the restricted* theory? Draw a diagram of how this can occur

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