# Finite temperature quantum field theory

In a QFT at finite temperature we consider the Euclidean time to be periodic, i.e. we consider a theory on the manifold $$\mathbb{R}^{d - 1} \times S^1$$, where the spatial coordinates are in $$\mathbb{R}^{d - 1}$$ and the time coordinate in $$S^1$$. The temperature of the system is then the inverse of the circumstance $$T = \frac{1}{\beta} \ , \tau\in[0,\beta)$$. Thus the zero temperature limit is $$\beta\rightarrow\infty$$ where $$S^1\rightarrow\mathbb{R}$$. Now what I'm wondering is whether the limit $$S^1\rightarrow\mathbb{R}$$ is single-valued or multi-valued?

Single-valued = one point in $$S^1$$ goes to one point in $$\mathbb{R}$$. In this case the entire $$\mathbb{R}$$ becomes the $$S^1$$ in the finite temperature limit.

Multi-valued = one point in $$S^1$$ maps to several points in $$\mathbb{R}$$. In this case the $$\mathbb{R}$$ wraps around the $$S^1$$ several times in the finite temperature limit.

• Is there a mistake at $T = \frac{1}{\beta} \ , \tau\in[0,\beta)$? Should it be $T = \frac{1}{\tau} \ , \tau\in[0,\beta)$?
– Ozz
Commented Aug 29, 2018 at 9:54
• There is not a mistake. $T$ is the temperature, $\beta$ is the circumstance of the $S^1$ and $\tau\in S^1$ is the Euclidean time. This gives $\tau\in[0,\beta)$ where $\tau$ is periodic $\tau + \beta \equiv \tau$, and a temperature of $T = \frac{1}{\beta}$. This formalism of finite temperature in QFT is btw called imaginary time formalism. Commented Aug 29, 2018 at 11:46

Saying that the theory is defined on $S^1$ is equivalent to saying it is defined on $\mathbb{R}$ but with all functions being periodic with period $2 \pi \beta$. This way you see that the $\beta \rightarrow \infty$ limit just corresponds to sending the periodicity to infinity, or in other words not having periodicity at all.
That being said, I think your alternative is not very well defined. You have the identification $S^1 = \mathbb{R}/(2 \pi \beta \mathbb{Z})$ so there are always, by definition, an infinite number of points in $\mathbb{R}$ corresponding to a single point in $S^1$. But these points are very far apart in the large $\beta$ limit, and as far as local physics is concerned, you can cover any bounded region of $\mathbb{R}$ by a single patch of $S^1$ in such a way that the correspondence be bijective -- which I believe is what you want to express with your "single-valued" possibility.