Local Gauge Invariance and Existence of the Gauge Particles 1. Gauge transformations are like “rotations” 2. How do functions transform under “rotations”? 3. How can we generalize to rotations in “strange” spaces (spin space, , flavor space, color space)? 4. How are Lagrangians made invariant under these “rotations”? (Lagrangians “laws of physics” for particles interactions.) 5. Invariance of L requires the existence of the gauge boson! momentum operator x component momentum operator angular momentum operator The angular momentum operator, generates rotations in x,y,z space! One can generate the “rotation” of a spinor (like the u derived for the electron) using the “spin” operators: This takes a little work -- must expand and use z2 = [a]n /n! = 1 , z3 = z …. more later! e a This approach is used in the Standard Model to “rotate” a particle which has an “up” and a “down” kind of property -- like flavor! Gauge transformations are like the “rotations” we have just been considering Real function of space and time one has to find a Lagrangian which is invariant under this transformation. can be an operator -- as we have just seen. How are Lagrangians made invariant under these “rotations”? It won’t work! Constructing a gauge invariant Lagrangian: 1. Begin with the “old Lagrangian”: called the “covariant derivative” 2. Replace Aµ is the gauge boson (exchange particle) field! 3. “old” Lagrangian the interaction term. Showing L is invariant transformed L transformed A µ = Aµ - (1/e) transformed A Maxwell’s equations are invariant under this! First a simplifying expression: Use this simple result in L’ Summary of Local gauge symmetry Real function of space and time covariant derivative The final invariant L is given by: The correct, invariant Lagrangian density, includes the interaction between the electron (fermion) and the photon (the gauge particle). free electron Lagrangian interaction Lagrangian If the coupling, e, is turned off, L reverts to the free electron L. This use of the covariant derivative will be applied to all the interaction terms of the Standard Model. 1. Initial state 2. Rotate Aµ ’ 3. Transform A ’ Aµ 4. Final state invariance Note that the photon field must also be transformed. Comments: 1. There is no difference between changing the phase of the field operator of the fermion (by (r,t) at every point in space) and the effects of a gauge transformation [ -(1/e)µ (r,t) ] on the photon field! 2. Maxwell’s equations are invariant under A µ A µ - (1/e)µ (r,t) -- and, in particular, the gauge transformation has no effect on the free photon. 3. It is only because (r,t) depends on r and t that the above is possible. This is called a local gauge transformation. 4. Note that a global gauge transformation would require that is a constant! L transformed simple result!