We explore the existence of time reparameterization symmetry in p-spin models. Using the Martin–Siggia–Rose generating functional, we analytically probe the long time dynamics. We perform a renormalization group analysis where we systematically integrate over short timescale fluctuations. We find three families of stable fixed points and study the symmetry of those fixed points with respect to time reparameterizations. One of those families is composed entirely of symmetric fixed points, which are associated with the low temperature dynamics. The other two families are composed entirely of non-symmetric fixed points. One of these two non-symmetric families corresponds to the high temperature dynamics. Time reparameterization symmetry is a continuous symmetry that is spontaneously broken in the glass state and we argue that this gives rise to the presence of Goldstone modes. We expect the Goldstone modes to determine the properties of fluctuations in the glass state, in particular predicting the presence of dynamical heterogeneity.

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The presence of fluctuating local relaxation times, τ r (t) has been used for some time as a conceptual tool to describe dynamical heterogeneities in glass-forming systems. However, until now no general method is known to extract the full space and time dependent τ r (t) from experimental or numerical data. Here we report on a new method for determining the local phase field, φ r (t) ≡from snapshots { r(ti)}i=1...M of the positions of the particles in a system, and we apply it to extract φ r (t) and τ r (t) from numerical simulations. By studying how the phase field depends on the number of snapshots, we find that it is a well defined quantity. By studying fluctuations of the phase field, we find that they describe heterogeneities well at long distance scales.