Abstract
Dynamical symmetries are of considerable importance in elucidating the complex behaviour of strongly interacting systems with many degrees of freedom. Paradigmatic examples are cooperative phenomena as they arise in phase transitions, where conformal invariance has led to enormous progress in equilibrium phase transitions, especially in two dimensions. Non-equilibrium phase transitions can arise in much larger portions of the parameter space than equilibrium phase transitions. The state of the art of recent attempts to generalise conformal invariance to a new generic symmetry, taking into account the different scaling behaviour of space and time, will be reviewed. Particular attention will be given to the causality properties as they follow for co-variant n-point functions. These are important for the physical identification of n-point functions as responses or correlators.
Highlights
Improving our understanding of the collective behaviour of strongly interacting systems consisting of a large number of strongly interacting degrees of freedom is an ongoing challenge
Phase transitions naturally acquire some kind of scale-invariance, and it becomes a natural question whether further dynamical symmetries can be present
We list several results relevant for the extension of the representations discussed in the introduction
Summary
Improving our understanding of the collective behaviour of strongly interacting systems consisting of a large number of strongly interacting degrees of freedom is an ongoing challenge. From the point of view of the statistical physicist, paradigmatic examples are provided by systems undergoing a continuous phase transition, where fluctuation effects render traditional methods such as mean-field approximations inapplicable [1,2]. It turns out that these systems can be effectively characterised in terms of a small number of “relevant” scaling operators, such that the net effect of all other physical quantities, the “irrelevant” ones, merely amounts to the generation of corrections to the Symmetry 2015, 7 leading scaling behaviour. Phase transitions naturally acquire some kind of scale-invariance, and it becomes a natural question whether further dynamical symmetries can be present
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