### Alec Booth Gauge Theories

```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
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
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
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
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
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
Formulation of QFT – Path
Integrals
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
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 selfenergy 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)
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
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
Gauge Bosons
 Suppose two identical particles (ignoring spins), fix a
gauge such that at a given point energy is distributed 5050
 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
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 4vector (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
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.
```