Holonomy Observables

[evanberkowitz]

Do the holonomies in the two formulations mean identical things? In other words does

$\left\langle \frac{1}{N_x} \sum_x n _t \right\rangle = \left\langle \frac{1}{N_x} \sum_x m_t \right\rangle$ and $\left\langle \frac{1}{N_t} \sum_t n _x \right\rangle = \left\langle \frac{1}{N_t} \sum_t m_x \right\rangle$

so that

class Holonomies(Observable):

    @staticmethod
    def Villain(S, phi, n):
        L = S.Lattice
        return np.array([ n[0].sum(), n[1].sum() ]) / np.array([L.nx, L.nt])

    @staticmethod
    def Worldline(S, m):
        L = S.Lattice
        return np.array([ m[0].sum(), m[1].sum() ]) / np.array([L.nx, L.nt])

make sense as a single observable?

[evanberkowitz]

The expectation values need not match. In fact, they have very different physical interpretations. In the Villain formulation the quantities are true (gauge field) holonomies. But in the worldline formulation it's particle trajectories wrapping around the world.

[evanberkowitz]

We should incorporate these as one observable because we expect them both to be quite slow to decorrelate. However, we should name it something that doesn't have suggestion of a meaningful physical quantity and give the user a big warning the documentation that the expectation values won't match between formulations.