So, the question is pretty much in the title: why are critical points of classical systems described by quantum conformal field theories? I get that schematically, conformal symmetry (or rather scale invariance) arises at a critical point due to divergence of the correlation length and disappearance of a preferred length scale (in low-energy theory). But why do quantum rather than classical conformal theories describe statistical correlations at the critical point?


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The Feynman path integral gives an isomorphism between quantum field theories in $d$ dimensions and classical statistical mechanics in $d+1$ dimensions. Basically the time trace of the quantum time evolution operator in imaginary time is evaluated as a path integral over classical field configurations $q[t,x]$ that are the same at time $0$ and $\beta $; $$ Z= {\rm tr} \left\{e^{-\beta \hat H}\right\}= \int_{\rm q(0,x )= q[\beta,x]} d[q] e^{-\frac 1 \hbar \int_0^\tau L[q] d^dxdt}. $$ The LHS is the partition function of the quantum system with inverse temperature $\beta=1/kT$, while the path integral on the RHS looks like the partition function of a $d+1$ dimensional classical system with inverse temperature $\hbar$.

  • $\begingroup$ I encountered this relation before, but now thinking about it makes it very puzzling. I can also regard your RHS as a quantum partition function of a Wick-rotated theory, right? Does this mean that classical statistical field theory $\approx$ Euclidian QFT? $\endgroup$ Jul 8, 2021 at 13:48
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    $\begingroup$ @Weather Report: Yes: statistical field theory is essentially euclidean field theory. $\endgroup$
    – mike stone
    Jul 8, 2021 at 15:21
  • $\begingroup$ Great, thanks! Maybe you can also comment on how this relates to the difference between classical and quantum correlations (say a la Bell's inequalities)? On the surface it is very surprising that quantum correlations can be precisely mimicked by classical ensemble. $\endgroup$ Jul 8, 2021 at 15:48
  • $\begingroup$ The mathematical isomomorphism is useful for calculating things such as correlators and wavefunctions, but it does not address any of the interpretational aspects of QM such as entanglement or the Born rule. $\endgroup$
    – mike stone
    Jul 8, 2021 at 19:55

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