# Why is a critical quantum system described by a conformal theory in one higher dimension of space?

These questions are linked, so I've asked them in a single post:

Why is a critical one-dimensional many-body system a two-dimensional conformal field theory?- Why the switch from 1D to 2D?

What does 2+1 dimensional mean? Two dimensions of space and one of time? Or is it a strange way of saying three dimensional?

Can a critical many-body system in thermal equilibrium be a 2+1 dimensional system? - Assuming the "+1" is time, considering a Wick's rotation, does the +1 become a measure of temperature?

• Are you familiar with path integral approach to thermal/statistical field theory? If not, that would be why this is puzzling to you --- and an answer can focus on that. Commented Dec 12, 2011 at 11:51
• @genneth I am familiar with path integrals - I thought that was the answer to the 2nd and third questions, but nowhere seems to say it explicitly. But it's the first question I'm really stuck on, and I don't know enough about CFT to know if it's somehow related to path integrals too. Commented Dec 12, 2011 at 13:05
• a CFT is just a field theory with some special symmetries. As Ron alludes to in his answer, the equilibrium properties of a quantum field theory in $d$ dimensions at finite temperature is given by a $d+1$-dimensional classical field theory (if this is not clear, please say so). Commented Dec 12, 2011 at 17:00
• @genneth So a finite temperature d-dimensional many-body system is not necessarily a field theory, and this is why a (d+1)-dimensional classical field theory describes it in path integral form? Commented Dec 13, 2011 at 21:12
• I'm not sure I understood what you said; but assuming I did: yes --- a quantum field theory in $d$-dimensions has the same partition function as a classical field theory in $d+1$-dimensions. Commented Dec 13, 2011 at 23:57

The field theory doesn't have to be conformal--- a quantum theory is always defined by a path integral in one dimension more, which is the time variable. The reason is that the partition function is $\mathrm{tr}(e^{-\beta H})$, which is the imaginary time periodic boundary conditions for a path integral in one dimension more, namely the time dimension. The notation "2+1" means 2 space and 1 time dimensions.