November-02-12 8:36 PM Part III Interacting Field Theory Quantum Electrodynamics (QED) M. Gericke Physics 7560, Relativistic QM 183
III.A Introduction December-08-12 9:10 PM At this point, we have the following: 1. 2. 3. We have a hamiltonian and equations of motion for integral spin particles and photons: The Klein-Gordon equation (eqn. 1.22 - without mass term this just becomes the wave equation for photons). Particles belonging to this group are referred to as bosons, because they obey Bose-Einstein statistics. We have a hamiltonian and equations of motion for half-odd-integral spin particles: The Dirac equation (eqn. 1.57). Particles belonging to this group are referred to as fermions, because they obey Fermi-Dirac statistics. We have quantized versions of the fields, as solutions to both of these equations. The "only" missing piece is in our picture is now a description of the interaction between particles. We already dealt with interaction of the K-G and Dirac solutions with external sources. But those treatments amounted to solving the motion of the particles in the presence of a static potential, like it is done for the standard non-relativistic Schrödinger equation hydrogen atom problem, for example. See sections I.C.4, I.D.11, II.B.3, and II.C.5, and II.D.6. These may in some cases be considered as the lowest order approximations for the "real" interaction. To go further, we need a better description, and now that we have fields, we can get it. This new description of the interaction will allow us to do perturbation theory (in all but one case) and therefore either work out the interactions exactly (if the series converges) or get better and better approximations. In my experience, the best way to go on from here, and the best way to motivate what happens next, is to start from gauge field theory. We will get the interaction terms in the lagrangians (e.g. the Dirac lagrangian) from this procedure without adding it in, in an ad-hoc way, as we did up to now, and we'll be able to connect gauge invariance to conservation theorems in a transparent way. We go from there to a discussion of formal scattering theory, but first deriving propagators (fundamental solutions) for bosons and fermions. The we'll do perturbation theory (Feynman diagrams) and work out some specific processes. M. Gericke Physics 7560, Relativistic QM 184
III.B Interacting Fields and Gauge Theory March-01-12 11:46 AM III.B.1: U(1) Gauge Invariance and Electro-Dynamics: Postulate a lagrangian as a function of the fields and their derivatives: A unitary 1x1 matrix transformation of the fields is given by: The quantity may be constant or depend on the position and time (x ). These two possibilities define the so-called global and local gauge transformations respectively. For infinitesimal transformations, we have Variation of the Lagrangian with respect to the fields and their derivatives is given by: Then, using Lagrange's equations and the fact that we find M. Gericke Physics 7560, Relativistic QM 185
March-01-12 11:58 AM The current density J is given by (you can check using the Dirac Hamiltonian that this is consistent with the usual definition of the current) then Global U(1) Gauge Transformations: For a global transformation, in which the lagrangian is varied uniformly for all points in space and time, we would have so that and current conservation is equivalent to invariance of the lagrangian under global U(1) gauge transformation. If one uses the Dirac lagrangian (eqn. 2.59) one obtains: and Thus, the basic Dirac lagrangian is invariant under global U(1) gauge transformations. That's good, but not particularly useful, except that it ensures that charge is conserved... M. Gericke Physics 7560, Relativistic QM 186
March-01-12 12:09 PM Local U(1) Gauge Transformations: Keeping with the Dirac Lagrangian for now, we now consider its properties under local U(1) gauge transformations. Since we had ( J = 0) for the Dirac lagrangian, we now have Which is generally not zero, since ( x 0), and we see that eqn. 2.59 is then not invariant under local gauge transformations as it stands. We may restore invariance though, by adding an interaction term to the lagrangian: Here, e is a constant and A is a four-vector field, satisfying the condition that when the particle fields transform as it transforms as Then, the new part of the lagrangian transforms as M. Gericke Physics 7560, Relativistic QM 187
March-01-12 12:23 PM So that the variation of the overall Lagrangian is zero again: And we restored U(1) gauge invariance, for any kind of U(1) transformation. Equation 3.4 is called the "minimal interaction of electrodynamics". It couples two fermion fields with a third field ( A ), which (as well will see later) corresponds to a Lorentz-vector, integral spin boson! The A are (in general) solutions to the Klein- Gordon equation (note that the K-G equation reduces to the wave equation of electro-dynamics when the mass of the particle is taken to be zero). Since there are no time derivatives in the interaction part of the lagrangian, the corresponding interaction hamiltonian is simply the negative of the lagrangian (see Energy-Momentum tensor discussion): This is the basic first order form of the Hamiltonian we want to use when evaluating the processes of QED! There is not much work involved in obtaining the full lagrangian for electrodynamics, so we might as well do it here. To do so, we have to add a gauge invariant term describing the radiation itself (the photon). Let's try this: where This is the anti-symmetric electromagnetic field strength tensor and we employ here the Heaviside-Lorentz units. Then with eqn. 3.5 we get the transformed form of the term as: M. Gericke Physics 7560, Relativistic QM 188
March-06-12 10:55 AM So that the term in the lagrangian describing the proton field is invariant: The new interaction term in the full lagrangian is consistently introduced via the so called covariant derivative: The full lagrangian of electrodynamics is: or Note that the interaction is hidden in the covariant derivative and involves the terms with the vector field A. M. Gericke Physics 7560, Relativistic QM 189
January-08-13 2:59 PM III.B.2: Equations of Motion for QED We can vary eqn. 3.8 with respect to the fields, to obtain the equations of motion: If we vary with respect to These equations are both (Lorentz) covariant and gauge invariant. Note that follows from these. Thus, we see that the fields are coupled together through the equations of motion and are no longer said to be "eternal". M. Gericke Physics 7560, Relativistic QM 190
January-08-13 3:10 PM III.B.3: The Energy-Momentum Tensor The energy momentum tensor is given by: So for the QED lagrangian this gives This result is obtained since the Dirac part of the lagrangian is zero, as seen from eqns. 3.9. We find the momentum density to be and the energy density is where we added and subtracted with M. Gericke Physics 7560, Relativistic QM 191