Information geometry in quantum field theory: lessons from simple examples
Erdmenger, J., Grosvenor, K., Jefferson, R.
Abstract
Motivated by the increasing connections between information theory and<br>high-energy physics, particularly in the context of the AdS/CFT correspondence,<br>we explore the information geometry associated to a variety of simple systems.<br>By studying their Fisher metrics, we derive some general lessons that may have<br>important implications for the application of information geometry in<br>holography. We begin by demonstrating that the symmetries of the physical<br>theory under study play a strong role in the resulting geometry, and that the<br>appearance of an AdS metric is a relatively general feature. We then<br>investigate what information the Fisher metric retains about the physics of the<br>underlying theory by studying the geometry for both the classical 2d Ising<br>model and the corresponding 1d free fermion theory, and find that the curvature<br>diverges precisely at the phase transition on both sides. We discuss the<br>differences that result from placing a metric on the space of theories vs.<br>states, using the example of coherent free fermion states. We also clarify some<br>misconceptions in the literature pertaining to different notions of flatness<br>associated to metric and non-metric connections, with implications for how one<br>interprets the curvature of the geometry. Our results indicate that in general,<br>caution is needed when connecting the AdS geometry arising from certain models<br>with the AdS/CFT correspondence, and seek to provide a useful collection of<br>guidelines for future progress in this exciting area.<br>