Phenomenology

Abstract

That completes this introductory look at quantum field theory. Although we did not get as far as some of the more relevant physical applications of QFT, we have looked in detail at what a QFT is, and how the description of scattering amplitudes leads to Feynman diagrams. To recap how we did this: 1. We reviewed the Lagrangian formalism for classical field theory, and also the canonical quantisation approach to quantum mechanics. 2. We constructed the Lagrangian for a relativistic field theory (the free Klein-Gordon field), and applied the techniques of canonical quantisation to this field theory. 3. States in this theory were found to represent particle excitations, such that a particle of momentum p was found to be a quantum of excitation in the relevant Fourier mode of the field. 4. We then studied the interacting theory, arguing that at initial and final times (when the interaction dies away) we can work with free fields. These were related by an operator S, whose matrix elements represented the transition probability to go from a given initial to a given final state. 5. Using the interaction picture for time evolution, we found an expression for the S matrix in terms of an evolution operator U, describing how the fields at general time t deviate from the initial free fields. 6. We also found a formula which related S matrix elements to n-particle Green's functions (vacuum expectation values of time-ordered fields). This was the LSZ formula of eq. (199). 7. We related the Green's functions involving Heisenberg fields to those involving the "in" fields at time t→-∞ (eq. (212)). 8. We then found how to compute these Green's functions in perturbation theory, valid when the strength of the interaction is weak. This involved having to calculate vacuum expectation values of time-ordered products, for which we could use Wick's theorem. 9. We developed a graphical representation of Wick's theorem, which led to simple rules (Feynman rules) for the calculation of Green's functions in position or momentum space. 10. These can easily be converted to S matrix elements by truncating the free propagators associated with the external lines. Needless to say, there are many things we did not have time to talk about. Some of these will be explored by the other courses at this school: Here we calculated S-matrix elements without explaining how to turn these into decay rates or cross-sections, which are the measurable quantities. This is dealt with in the QED/ QCD course. The Klein-Gordon field involves particles of spin zero, which are bosons. One may also construct field theories for fermions of spin 1/2 , and vector bosons (spin 1). Physical examples include QED and QCD. Fields may have internal symmetries (e.g. local gauge invariance). Again, see the QED /QCD and Standard Model courses. Diagrams involving loops are divergent, ultimately leading to infinite renormalisation of the couplings and masses. The renormalisation procedure can only be carried out in certain theories. The Standard Model is one example, but other well-known physical theories (e.g. general relativity) fail this criterion. There is an alternative formulation of QFT in terms of path integrals (i.e sums over all possible configurations of fields). This alternative formulation involves some extra conceptual overhead, but allows a much more straightforward derivation of the Feynman rules. More than this, the path integral approach makes many aspects of field theory manifest i.e. is central to our understanding of what a quantum field theory is. This will not be covered at all in this school, but the interested student will find many excellent textbooks on the subject. There are other areas which are not covered at this school, but nonetheless are indicative of the fact that field theory is still very much an active research area, with many exciting new developments: Calculating Feynman diagrams at higher orders is itself a highly complicated subject, and there are a variety of interesting mathematical ideas (e.g. from number theory and complex analysis) involved in current research. Sometimes perturbation theory is not well-behaved, in that there are large coefficients at each order of the expansion in the coupling constant. Often the physics of these large contributions can be understood, and summed up to all orders in the coupling. This is known as resummation, and is crucial to obtaining sensible results for many cross-sections, especially in QCD. Here we have "solved" for scattering probabilities using a perturbation expansion. It is sometimes possible to numerically solve the theory fully non-perturbatively. Such approaches are known as lattice field theory, due to the fact that one discretizes space and time into a lattice of points. It is then possible (with enough supercomputing power!) to calculate things like hadron masses, which are completely incalculable in perturbation theory. Here we set up QFT in Minkowski (flat space). If one attempts to do the same thing in curved space (i.e. a strong gravitational field), many weird things happen that give us tantalising hints of what a quantum field of gravity should look like. There are some very interesting recent correspondences between certain limits of certain string theories, and a particular quantum field theory in the strong coupling limit. This has allowed us to gain new insights into nonperturbative field theory from an analytic point of view, and there have been applications in heavy ion physics and even condensed matter systems. I could go on of course, and many of the more formal developments of current QFT research are perhaps not so interesting to a student in experimental particle physics. However, at the present time some of the more remarkable and novel extensions to the Standard Model (SUSY, extra dimensions) are not only testable, but are actively being looked for. Thus QFT, despite its age, is very much at the forefront of current research efforts and may yet surprise us!