An Introduction to Field and Gauge Theories

1. An Introduction to Field and Gauge Theories Alec Tewsley-Booth Physics 129 2 Dec. 2010

2. Summary • Introduction to the concepts of a field and gauge theory • History of the development of modern field theories • Significance of QFTs • Path integral formulation • Renormalization • Examples of applications of gauge theories: conservation of charge, Aharonov-Bohm effect, gauge bosons

3. What is a field theory? • Field - a set of parameters (degrees of freedom) indexed over every point in space • Since, in principle, we can vary any of these parameters at any point in space, observables of the system form an infinite set • Ordinary QM does not have these large sets of degrees of freedom • Important that we can index repeated degrees of freedom over space, but not necessarily over spatial coordinates

4. What is a gauge theory? • Gauge theories are a subset of field theories where the Lagrangian of a system is invariant over a continuous group of gauge transformations • Gauge transformations involve switching between equivalent system expressed in different sets of degrees of freedom  symmetry groups which represent physical situations • Global gauge transformations – transformations applied to each set of degrees of freedom equivalently • Local gauge transformations – transformations applied to degrees of freedom as a function of index

5. Brief history lesson 1920 – Heisenberg, Born, and Jordan create free field theory by expressing field degrees of freedom as infinite set of quantum oscillators 1927 – Dirac constructs an early version of QED which could model creation and annihilation of photons 1927 – Jordan extends quantization of fields to many-body wavefunctions – Second Quantization 1928 – Jordan and Pauli combine special relativity and quantum mechanics by showing field commutators could be made Lorentz invariant 1928 – Dirac equation, which obeys rules of quantum mechanics and is inherently Lorentz invariant

6. Brief history lesson 1930s and early 40s – QFT plagued by divergences in perturbative approaches Late 40s early 50s – Bethe, Tomonaga, Schwinger, Feynman and Dyson develop the procedure of renormalization to solve divergence issues in QED 1954 – Yang and Mills postulate a non-abelian theory for strong interactions 1958 – QED well understood and divergences addressed and accepted 1958-1960 – Glashow unifies electromagnetism and weak interactions 1960s and 70s – Propagation of the Standard Model as a unified gauge theory

7. Why do they matter? • Motivations for QFT • Describe processes where particles are created/ annihilated • Unify quantum mechanics and special relativity • Address statistics of many-particle systems • Generalization to gauge theories • Very successful in unifying QFTs • Gauge bosons understood in terms of gauge theory • Produce 3 of the 4 forces

8. Formulation of QFT – Path Integrals

9. Formulation of QFT – Path Integrals • Amplitude to propagate from state A to B in time T governed by exp[-i H T], specifically the matrix element between A and B • Path integral breaks down the propagation into infinitesimal elements between complete sets of states • Feynman diagrams handy for keeping track of path integrals contributing to a system overall

10. Renormalization • Infinities arise in calculated quantities • Infinities in QED are results of closed loops of virtual particles – must integrate over all possible values of momentum around loop and momentum is not uniquely defined (off-the-shell) • Closely related to failures of classical EM, like infinite self-energy of electron , and to vacuum polarization • Solution: problem is purely mathematical, create a cutoff for high energy quanta (quantize space, forbidding short distances), then take limit as quantized space goes to zer0 • In essence, pay close attention of definitions of mass and charge in a field context (bare mass/ charge vs. shifted mass/ charge)

11. Conservation of Charge • Conservation of energy and gauge invariance necessitate conservation of charge • Consider an electric potential, break conservation of charge by creating a charge into the potential, propagating the charge to another point, annihilate the charge • This process conserves energy, but also gives us a way to measure ABSOLUTE potential, forbidden by gauge invariance • If gauge symmetry holds and energy is conserved, charge is conserved

12. Aharonov-Bohm Effect • Are fields or potentials fundamental? • Applying a varying magnetic field to an area where the particle does not pass will change the phase difference • Potential is fundamental, however the vector potential still shows gauge invariance • Gauge transformation will change phase of two paths by same amount, and only difference in phase matters

13. Gauge Bosons • Suppose two identical particles (ignoring spins), fix a gauge such that at a given point energy is distributed 50-50 • To measure momentum, measure wavelength of wavefunctions, measure at a nearby point • Changes in waves could be caused by oscillatory nature of the waves (trivial case) • Changes could also be attributed to a local gauge function changing distribution to 51-49

14. Gauge Bosons • If we ignore the second option then theory fail, momentum is no longer conserved • If the gauge function oscillates in time then it behaves as a wave with its own momentum  fixes conservation laws • In the case of electrons gauge function is represented as 4-vector (due to complications of spin), the EM field • Electromagnetic interactions required to maintain consistency of theory • Gauge function wave behaves like particle, hence we have photons, gluons, W and Z bosons

15. References Wikipedia Becher, P., Bohm, M., Joos, H. Gauge Theories of Strong and Electroweak Interactions. John Wiley and Sons, 1984. Cheng, T., Li, L. Gauge Theory of Elementary Particle Physics. Oxford University Press, 1984. Leader, E., Predazzi, E. An Introduction to Gauge Theories and Modern Particle Physics, Vol 1. Cambridge University Press, 1996. Srednicki, M. Quantum Field Theory. Cambridge University Press, 2007. Zee, A. Quantum Field Theory in a Nutshell. Princeton University Press, 2003.