1. An Introduction to Field and Gauge Theories
Alec Tewsley-Booth
Physics 129
2 Dec. 2010
If they can see this, we have a problem!
Make sure you have notes for path integrals

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
Degrees of freedom = dimension
EM field as an example (4 degrees of freedom)

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
Mention Andre’s presentation about monopoles

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
Try not to take too long here

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 – Glashow unifies electromagnetism and weak interactions 1960s and 70s – Propagation of the Standard Model as a unified gauge theory
Try not to take too long here

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
Emphasize that gauge theories are field theories

8. Formulation of QFT – Path Integrals
Analogy to double slit experiments (draw on board)
Generalize not just spatial states, but over any continuum
Feynman Diagrams

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
Show BRIEF derivation on board

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)
Dressed v bare values
Loop infinities in QED
-Self energy (emit then reabsorb)
-vacuum polarization (virtual pair annihilates)
-Penguin

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
Draw and derive this on the board

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
Gauge trans. Changes phase as function of position
-doesn’t matter if start and end are the same
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
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
Pictures on board may be helpful

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, Cheng, T., Li, L. Gauge Theory of Elementary Particle Physics. Oxford University Press, Leader, E., Predazzi, E. An Introduction to Gauge Theories and Modern Particle Physics, Vol 1. Cambridge University Press, Srednicki, M. Quantum Field Theory. Cambridge University Press, Zee, A. Quantum Field Theory in a Nutshell. Princeton University Press, 2003.
Thank Dima for books