Where does statistical physics/mechanics fit in with QFT,GR?

[Indiana]

We have two theories namely,Quantum Field Theory which works very well at sub-atomic scales, and the General Relativity which works very well at very large scales.So, my question is where does statistical physics/mechanics fit in? What role statistical physics/mechanics play in today's modern physics? Did it play any role in the development of QFT or GR?

And concepts like Renormalization group and regularization were discovered in the literature of statistical physics/mechanics?

 

[vanhees71]

Statistical physics matches very well with QT and GR. In fact it's the link to understand why macroscopic objects very often obey the classical laws and how to connect to the classical theory of gravity, i.e., GR. Unfortunately we don't have a full quantum description of everything. Only for three (or two) of the four (or three) fundamental interactions are described satisfactorily by the Standard model (including QCD for the strong and QFD for the weak and electromagnetic interactions) in terms of a local gauge theory based on the compact Lie group $\mathrm{SU}(3)_{\text{color}} \times \mathrm{SU}(2)_{\text{wiso}} \times \mathrm{U}(1)_{\text{Y}}$ which is "Higgsed" to $\mathrm{SU}(3)_{\text{color}} \times \mathrm{U}(1)_{\text{em}}$.

Also there are various cross relations between condensed-matter and HEP physics. The most famous one is the discovery of the Higgs mechanism by Anderson, i.e., the description of superconductivity as Higgsing the electromagnetic gauge symmetry through the formation of couper-pair condensates (BCS theory). The only difference between the Higgs mechanism in the electroweak sector of the SM and superconductivity is that in the latter case one understands the dynamical reason for the Anderson-Higgs mechanism from an underlying more microscopic theory (BCS) while in the case of the Standard Model it's still introduced by hand and in some sense can be seen as an effective description (except one accepts the somewhat unsatisfactory fine-tuning feature due to the introduction of a fundamental scalar Higgs field).

In this picture GR is a gauge theory gauging the Poincare (or Lorentz) symmetry of SR, and this is a non-compact group and thus leads to hitherto unsolved trouble in quantizing the theory.

The renormalization group has been discovered in HEP first with the seminal works by Stückelberg and Petermann, Callen, Symanzik, and Gell-Mann. In this version, however, it looked like rather mathematical formal tool to resum large logarithms in perturbation theory in a clever way. This changed with the work by Kadanoff and K. Wilson, who discovered the physical meaning in terms of many-body theory. It's about coarse-graining and effective low-energy theories built from underlying theories "integrating out" the high-energy degrees of freedom which are not resolved in a coarse-grained picture leading to an effective description of the relevant degrees of freedom. Nowadays RG (particularly in form of the exact functional renormalization group) plays a prominent role in describing both condensed-matter systems and high-energy particle/nuclear physics, particularly in the understanding of phase transitions (again in both condensed-matter physics and in high-energy physics, e.g., to understand the phase diagram of QCD matter, which is the prime interest of the heavy-ion research programs at RHIC, LHC, and future FAIR and NICA facillities).