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Information geometry in quantum field theory: lessons from simple examples
by Johanna Erdmenger, Kevin T. Grosvenor, Ro Jefferson
This is not the current version.
|As Contributors:||Ro Jefferson|
|Date submitted:||2020-02-03 01:00|
|Submitted by:||Jefferson, Ro|
|Submitted to:||SciPost Physics|
Motivated by the increasing connections between information theory and high-energy physics, particularly in the context of the AdS/CFT correspondence, we explore the information geometry associated to a variety of simple systems. By studying their Fisher metrics, we derive some general lessons that may have important implications for the application of information geometry in holography. We begin by demonstrating that the symmetries of the physical theory under study play a strong role in the resulting geometry, and that the appearance of an AdS metric is a relatively general feature. We then investigate what information the Fisher metric retains about the physics of the underlying theory by studying the geometry for both the classical 2d Ising model and the corresponding 1d free fermion theory, and find that the curvature diverges precisely at the phase transition on both sides. We discuss the differences that result from placing a metric on the space of theories vs. states, using the example of coherent free fermion states. We also clarify some misconceptions in the literature pertaining to different notions of flatness associated to metric and non-metric connections, with implications for how one interprets the curvature of the geometry. Our results indicate that in general, caution is needed when connecting the AdS geometry arising from certain models with the AdS/CFT correspondence, and seek to provide a useful collection of guidelines for future progress in this exciting area.
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Reports on this Submission
Anonymous Report 1 on 2020-3-8 Invited Report
- Cite as: Anonymous, Report on arXiv:scipost_202002_00001v1, delivered 2020-03-08, doi: 10.21468/SciPost.Report.1557
1) A good overall discussion of information geometry within quantum field theories.
2) Clearly presented calculations.
3) Clarifications of common misunderstandings.
1) Not highly novel
2) Somewhat speculative
In this paper, Erdmenger et al. give an overview of the applications of information geometry within quantum field theories. After providing a clear explanation of the statistical origins of the Fisher Metric they discuss the exponential family of distributions and show how they can give rise to hyperbolic information manifolds. They show how symmetries of the distribution give rise to symmetries of the metric but that this is not always the case.
They then explain how instabilities in a field theory can show up in the Fisher metric as complex components. This is a novel application of information geometry in this context. There seem to be a number of issues with this calculation however. The Hitchin procedure takes the Lagrangian and defines it as the probability density. However, in this example such a direct mapping is not clear as the Lagrangian is not positive-definite on the solution of the instanton. This needs clarification and I believe that a more general proof, or at least another example of an instability showing up in this relationship between instability and complexity is needed.
The next two sections are split into studying the information geometry of the theory space and of the state space respectively. In theory space they show how the Ising model information metric encodes aspects of the phase structure of the theory, and then that parts of this are also encoded in the dual one dimensional fermionic theory. In state space, the Bures metric is used to study the information space of coherent fermions showing how a spherical metric can come about.
In the last main section the fact that notions of curvature differ between the physics and information theory community is explained in detail.
Finally the authors summarise and conclude with some speculations about the use of information geometry in holographic RG and complexity within a gauge/gravity context.
Overall this paper gives a good overview of the potential uses of information geometry in quantum field theories with some interesting examples and some clarifications of common misconceptions. I believe that the paper could be strengthened with more general statements about the correspondence, particularly between unstable configurations and the complexity of the metric.
1) Page 5, below equation 2. The $n$ in the size of the space of variables and that in the parameter space should be different
2) Page 7, below equation 20. This is not AdS as AdS is Lorentzian and this is not. This is a hyperbolic metric.
3) Page 7, "yields and" -> "yields an"
4) The terms Fisher matrix and Fisher metric are used interchangeably. This should be made uniform.
5) Page 9. It is not clear where the $a$ comes from in equation 29, nor where the $zeta$ goes. This should be clarified.
6) Page 9. That the Lagrangian density in this case isn't a probability density needs to be made clear.
7) Page 9. A more general statement about the relationship between complexity and instability needs to be made.
8) Page 10. "in thermodynamic"->"in the thermodynamic".
9) Page 10, equation 36. $k$ should be replaced with another letter as this equation contains $k$ and $K$.
10) Page 13, Figure 1. It's not clear why the imaginary part line needs to be there when it's clearly zero from the equation.
11) Page 14, "the one can"->"one can".
12) Page 15, equation 58. The Bures distance is not correct here.
13) I believe that section 7 should come earlier in the text as it does not follow on naturally from section 6.