Canonical Quantisation of Yang-Mills

2.2.1 Canonical Quantisation of Yang-Mills

Ultimately, we want to see how the ✓ term affects the quantisation of Yang-Mills.

But we can see the essence of the issue already in the classical theory where, as we will now show, the ✓ term results in a shift to the canonical momentum. The full Lagrangian is

(2.25)

To start, we make use of the gauge redundancy to set

With this ansatz, the Lagrangian becomes

(2.26)

Here Bi = 12 ✏ijk Fjk is the non-Abelian magnetic field (sometimes called the chromomagnetic field). Meanwhile, the non-Abelian electric field is Ei = Ȧi .

I’ve chosen not to use the electric field notation in (2.26) as the Ȧ terms highlight the canonical structure.

Note that the ✓ term is linear in time derivatives; this is reminiscent of the effect of a magnetic field in Newtonian particle mechanics and we will see some similarities below.

The Lagrangian (2.26) is not quite equivalent to (2.25); it should be supplemented by the equation of motion for A0 . In analogy with electromagnetism, we refer to this as Gauss’ law. It is

(2.27)

This is a constraint which should be imposed on all physical field configurations.

The momentum conjugate to A is

From this we can build the Hamiltonian

(2.28)

We see that, when written in terms of the electric field E, neither the constraint (2.27) nor the Hamiltonian (2.28) depend on ✓; all of the dependence is buried in the Poisson bracket structure.

When written in terms of the canonical momentum ⇡, the constraint becomes

where the would-be extra term Di Bi = 0 by virtue of the Bianchi identity (2.10).

Meanwhile the Hamiltonian becomes

It is this ✓-dependent shift in the canonical momentum which affects the quantum theory.