From Abelian to Non-Abelian Groups: Yang-Mills Theory

28May15

I have just left a conference celebrating 60 years of Yang-Mills gauge field theories. I would like to continue from my previous post on Abelian gauge theories and introduce Yang-Mills gauge theories.

First, we recall how Abelian gauge fields are introduced to make the following Lagrangian

\mathcal{L}=\bar{\psi}i\gamma^\mu(\partial_\mu -ieA_\mu)\psi - \bar{\psi}m\psi- \frac{1}{4}F_{\mu\nu}F^{\mu\nu}

gauge invariant. The gauge transformation for the gauge field is

A'_\mu=A_\mu + \partial_\mu\Lambda (x),

while the ‘wavefunction’\matter field transforms like

\psi'(x)= e^{ie\Lambda(x)}\psi(x);

\bar{\psi}'(x) = \bar{\psi}(x) e^{-ie\Lambda(x)}.

Thus applying to first term in Lagrangian gives

\bar{\psi}(x) e^{-ie\Lambda(x)} e^{ie\Lambda(x)} i\gamma^\mu \left(ie(\partial_\mu\Lambda(x))+\partial_\mu-ieA_\mu-ie(\partial_\mu\Lambda(x)\right),

which reduces back to the first term. The modification of the derivative of the matter field \partial_\mu\psi(x) to its covariant derivative D_\mu\psi(x)=(\partial_\mu -ieA_\mu)\psi(x) is crucial in ensuring the gauge covariance (D_\mu\psi(x))'=e^{ie\Lambda(x)}D_\mu\psi(x). If one thinks the gauge (phase factor) transformation as performing transformations on the internal degrees of freedom of \psi(x), it also says that one need not necessarily perform the same transformation (global) of these degrees of freedom throughout the whole space-time. Suffice to do the local ones at the expense of introducing the gauge fields. The field strength tensor F_{\mu\nu} can be expressed as commutator of the covariant derivative operators:

[D_\mu, D_\nu]=-ieF_{\mu\nu}.

In fact one can g0 on to show that F_{\mu\nu} is gauge invariant:

F'_{\mu\nu}\psi'(x)=e^{ie\Lambda(x)}F_{\mu\nu}\psi(x)=F_{\mu\nu}\psi'(x).

Thus the Lagrangian given earlier is gauge invariant.

Now how could one generalise this theory? Recall how we begin by introducing the gauge transformation through the phase factor multiplying the wavefunction: \psi'(x)=e^{i\Lambda(x)}\psi(x), where \alpha(x) is a real (scalar) function generating the phase factor  e^{i\Lambda (x)}\in U(1). One can now replace this phase factor by a  SU(n) element instead:

\psi'_i(x) = U_{ij}\psi_j(x)\quad;\qquad U\in SU(n)

where

U= \exp (igA^a_\mu \hat{T}_a dx^\mu),

with  \hat{T}^a are the generators of  SU(n) and \psi'(x) is considered parallel to the wavefunction at x+dx (parallel transporting in the presence of a gauge field). Note also that we have allowed  \psi to have components due to the fact that  U_{ij} are no longer one-dimensional. The generators form its Lie algebra u(n):

[\hat{T}_a,\hat{T}_b] =i f_{abc}\hat{T}_c

(assume Euclidean metric in algebra indices). We still need to know how to handle gauge equivalence in this new theory. Note that the phase factor is no longer commutative (belonging to a non-Abelian group) and hence one has to be careful of the ordering. Suppose at x, the wavefunction is gauge transformed to

\tilde{\psi}(x) =S(x)\psi(x)\quad;\qquad S(x)\in SU(n),

then the wavefunction at x+dx will transform analogously:

\tilde{\psi}'(x)=S(x+dx)\psi'(x).

We now introduce the gauge equivalent gauge field by writing

\tilde{\psi}'(x)=\exp (igA'^a_\mu \hat{T}_a dx^\mu) \psi'(x).

On equating and combining the results, we have

\exp (igA'^a_\mu \hat{T}_a dx^\mu) S(x) \psi(x) = S(x+dx) \exp (igA^a_\mu \hat{T}_a dx^\mu)\psi(x) .

Expanding to first order in dx^\mu gives

A'_\mu(x)=S(x)A_\mu(x) S^{-1}(x)- (i/g)(\partial_\mu S(x)) S^{-1}(x).

If one further infinitesimally expand S(x)\approx (1+ig\Lambda(x)) then

A'_\mu (x) = A_\mu (x) +\partial_\mu\Lambda(x)+ig[\Lambda(x),A_\mu (x)].

Note that it has the familiar derivative term but it is augmented by the commutator term which reminds us the non-Abelian nature. To get the field strength tensor, we follow analogously the Abelian case by taking the commutator of the covariant derivative:

F_{\mu\nu}=\partial_mu A_\nu -\partial_\nu A_\mu + ig[A_\mu,A_\nu].

Under the gauge transformation, the field strength tensor transforms like

F'_{\mu\nu}=S(x) F_{\mu\nu} S^{-1}(x).

All this is now pretty standard for students of theoretical physics and it can be traced back to the original paper of Yang-Mills back in 1954. This paper was not intended as it is used today but yet it evolved to a point (which is not really known precisely when as Robert Crease pointed out in the conference) that it gets gradually accepted by physicists as the correct structure underlying the standard model. It is now accepted for a renormalizable field theory, it has to be a gauge theory (see here and here).

References:

  1. Chan Hong-Mo & Tshou Sheung Tsun, “Some Elementary Gauge Theory Concepts“, (World Scientific, 1993)
  2. Walter Dittrich & Martin Reuter, “Selected Topics in Gauge Theories“, (Springer, 1986)
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