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Hamiltonian Approach to QCD at Finite Temperature^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Finite Temperature from Compactification of a Spatial Dimension

- $O(4)$ invariance of the Euclidean Lagrange density, which should hold for any relativistically invariant theory. It does, however, not hold for a non-relativistic many-body system.
- The absence of massless modes, i.e., the existence of a mass gap on the manifold ${S}^{1}(L)\times {\mathbb{R}}^{2}$. This is the generic case. For instance, QCD in the confining phase is known to develop such a gap, and this follows also self-consistently from the results in the next section. Extra caution may be required at temperatures above the deconfinement phase transition, where we can check our findings against thermal perturbation theory.

## 3. Hamiltonian Approach to QCD in Coulomb Gauge

#### 3.1. Yang–Mills Sector

#### 3.2. Quark Sector

## 4. QCD at Finite Temperatures

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1. | We thank the anonymous referee for pointing out this implicit assumption. |

2. | Although we only consider a single massless quark flavor in this section, the results easily generalize to the case of ${N}_{f}$ massless flavors. In fact, the only change in this case would be an additional factor ${N}_{f}$ in the chiral condensate, which drops out when forming the ratios in Figure 7 and Figure 8. The results of this section are therefore independent of ${N}_{f}$ (as long as the quark flavors are massless), and it is sufficient to study ${N}_{f}=1$. |

**Figure 2.**The static gluon propagator in Coulomb gauge calculated on the lattice for SU(2) gauge theory (crosses). The dashed and the full curves show the result of the variational calculation using, respectively, a Gaussian ansatz and a non-Gaussian ansatz for the vacuum wave functional.

**Figure 4.**Mass function obtained from the (quenched) solution of the quark gap equation. Results are presented for $g\simeq 2.1$ (full curve) and $g=0$ (dashed curve).

**Figure 5.**Comparison between the full mass function $M({p}^{2})$ in Landau gauge (continuous line) and the mass function ${M}_{3}({\mathit{p}}^{2})$ of the equal-time propagator (dashed line) (see [35]).

**Figure 6.**(

**Left**) Mass function $M(k,{\xi}_{k})$ at $T=80\phantom{\rule{0.166667em}{0ex}}\mathrm{MeV}$ with the momentum $\mathit{k}$ pointing in various directions relative to the heat bath. (

**Right**) Mass function $M(k,1)$ for small temperatures compared to the $T=0$ limit.

**Figure 7.**Chiral condensate as a function of the temperature, from both the Matsubara and Poisson formulations. To guide the eye, a dashed line was added through the Poisson data from which the critical temperature is determined.

**Figure 8.**The quark condensate as function of the temperature calculated from the zero- and finite-temperature solutions of the quark gap equation.

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**MDPI and ACS Style**

Reinhardt, H.; Campagnari, D.; Quandt, M.
Hamiltonian Approach to QCD at Finite Temperature. *Universe* **2019**, *5*, 40.
https://doi.org/10.3390/universe5020040

**AMA Style**

Reinhardt H, Campagnari D, Quandt M.
Hamiltonian Approach to QCD at Finite Temperature. *Universe*. 2019; 5(2):40.
https://doi.org/10.3390/universe5020040

**Chicago/Turabian Style**

Reinhardt, Hugo, Davide Campagnari, and Markus Quandt.
2019. "Hamiltonian Approach to QCD at Finite Temperature" *Universe* 5, no. 2: 40.
https://doi.org/10.3390/universe5020040