# What happens to the dynamical critical exponent in the quantum-classical mapping?

It is well-known that one can, e.g., map the classical 2D Ising model to the 1D quantum Ising chain. Moreover, their critical points are related. Hence, if one is interested in critical exponents describing, for example, the non-analytic behavior of the order parameter in the 2D classical model, one can answer the equivalent question in the $1+1D$ quantum system. Moreover, since the latter is dual to a 1D system of free fermions, one can actually analytically solve such questions.

However, is the dynamical critical exponent of the 2D classical problem somehow encoded in the 1D quantum problem? Indeed, this seems problematic: in the quantum-classical correspondence, one of the two spatial directions of the classical problem is re-interpreted as the timelike direction in the quantum set-up. Hence, for example, the dynamical critical exponent in the quantum formulation tells us how to the two spatial directions of the classical set-up are related. But is the dynamical critical exponent of the classical problem completely lost in the quantum picture? (This does seem to be expected based on a dimensional-counting argument...)