The Dirac Lagrangian for a free fermion, L= ¯ψ(i∂/−m)ψ is invariant under the global U(1) gauge transformationψ→eiαψ. This global symmetry, by Noether’s theorem, implies
6This statement may seem peculiar in light of the fact that the Lorentz group was first discovered as the symmetry of the Maxwell equations of classical electrodynamics. But those equations are written in terms of the fieldsEandB. The scalar and vector potentials (A0andArespectively) enter classical electrodynamics only as computational aids. It is quantum mechanics whichrequiresa formulation in terms ofAµ.
7This irresistibly brings to my mind a scene from the Woody Allen movie comedy Bananas in which victorious rebel commander Esposito announces from the Presidential Palace that “from this day on, the official language of San Marcos will be Swedish... Furthermore, all children under 16 years old are now 16 years old.”
conservation of the current:
jµ= ¯ψγµψ . (2.61)
In the established model of quantum electrodynamics, this Lagrangian is transformed into an interacting theory by making the gauge invariance local: The phaseαis allowed to be a function of the space-time pointx. This requires the introduction of a gauge fieldAµ with the the transformation property
Aµ→Aµ+∂µα (2.62)
and the use of a covariant derivative Dµ=∂µ−iAµ instead of the usual derivative ∂µ. This procedure automatically couplesAµto the conserved current in Eq. (2.61) so that the coupling is invariant under transformations of the form Eq (2.62). We than add a Lorentz- invariant kinetic term −Fµν2 /4 for the field Aµ. The generalization to non-abelian gauge groups is well known, as is the Higgs mechanism to break the gauge invariance spontaneously and give the fieldAµ a mass.
This is what we are taught in elementary courses on QFT, but the question remains:
Why do we promote a global symmetry of the free fermion Lagrangian to a local symmetry?
Equation (2.58) provides a deeper insight into the physical meaning of local gauge invariance:
a massless particle, having no rest frame, cannot have its spin point along any axis other than that of its motion. Therefore, it can have only two polarizations. By describing it as a 4-vector, spin-1 field Aµ (which has three polarizations) a mathematical redundancy is introduced.
This redundancy is local gauge invariance. A field with local gauge symmetry is coupled to the conserved current of the corresponding global gauge symmetry in order to make the coupling locally gauge-invariant. The procedure described of promoting the global gauge symmetry to a local gauge invariance is therefore required in order to couple fermions in a Lorentz-invariant way via a long-range, spin-1 force.
2.4.1 Expecting the Higgs
Remarkably, local gauge invariance also comes to our aid in writing sensible QFT’s for the short-range weak nuclear interaction. At low energies, this interaction is naturally described as being mediated by massive, spin-1 vector fields. The Lagrangian for such a mediator must
look like
L=−1
4Fµν2 +1
2m2A2−AµJµ, (2.63)
whereJµ is the current to which it couples. But in the case of the weak nuclear interaction this current is not conserved. At energy scales much higher than the m in Eq. (2.63), we therefore expect the same problem we found in Subsection 2.3.2 of a divergent emission rate for the longitudinal polarization, unless other higher-derivative operators, which were not relevant at low energies, have come to our rescue.
In the standard model of particle physics, the resolution of this problem is to make the mediators of the weak nuclear interaction gauge bosons, and then to break that gauge invariance spontaneously by introducing a scalar Higgs field with a non-zero VEV, thus giving the bosons the mass that accounts for the short range of the force they mediate. At high energies the gauge invariance is restored. The problematic longitudinal polarization disappears and is transmuted into the Goldstone boson of the spontaneously broken sym- metry. Since the Goldstone boson has no spin, it does not have the problem of a divergent rate of emission. This is the reason why many billions of dollars have been spent in the search for that yet-unseen Higgs boson, a search soon to come to a head with the turning on of the Large Hadron Collider (LHC) at CERN next year.
2.4.2 Further successes of gauge theories
Gauge theories as descriptions of the fundamental particle interactions have other very attractive attributes. It was shown by ’t Hooft that these theories are always renormalizable, i.e., that the infinities that plague QFT’s can all be absorbed into a redefinition of the bare parameters of the theory, namely the masses and the coupling constants ([5]). Politzer ([6]) and, independently, Gross and Wilczek ([7]), showed that the renormalization flow of the coupling constants in non-abelian gauge theories provides a natural explanation of the observed phenomenon of asymptotic freedom, whereby the nuclear interactions become more feeble at higher energies.
It is also widely believed, though not strictly demonstrated, that QCD, the theory in which the strong nuclear force is mediated by the bosons of an SU(3) gauge theory, accounts for confinement, i.e., for the fact that the strongly interacting fermions (quarks) never occur alone and can appear only in bound states that are singlets of SU(3). These
successes illustrate what we meant when we said in Subsection 2.3.3 that having to accept local gauge symmetry was a disaster with a rich silver lining. For interesting accounts of the history of local gauge invariance in classical and quantum physics, see [8, 9].