The precise equivalence between discretized Euclidean field theories and a certain class of probabilistic graphical models, namely the mathematical framework of Markov random fields, opens up the opportunity to investigate machine learning from the p
erspective of quantum field theory. In this contribution we will demonstrate, through the Hammersley-Clifford theorem, that the $phi^{4}$ scalar field theory on a square lattice satisfies the local Markov property and can therefore be recast as a Markov random field. We will then derive from the $phi^{4}$ theory machine learning algorithms and neural networks which can be viewed as generalizations of conventional neural network architectures. Finally, we will conclude by presenting applications based on the minimization of an asymmetric distance between the probability distribution of the $phi^{4}$ machine learning algorithms and target probability distributions.
These lecture notes contain an elementary introduction to lattice QCD at nonzero chemical potential. Topics discussed include chemical potential in the continuum and on the lattice; the sign, overlap and Silver Blaze problems; the phase boundary at s
mall chemical potential; imaginary chemical potential; and complex Langevin dynamics. An incomplete overview of other approaches is presented as well. These lectures are meant for postgraduate students and postdocs with an interest in extreme QCD. A basic knowledge of lattice QCD is assumed but not essential. Some exercises are included at the end.
Some recent developments to handle the numerical sign problem in QCD and related theories at nonzero density are reviewed. In this contribution I focus on changing the integration order to soften the severity of the sign problem, the density of state
s, and the extension into the complex plane (complex Langevin dynamics and Lefshetz thimbles).
A brief overview of the QCD phase diagram at nonzero temperature and density is provided. It is explained why standard lattice QCD techniques are not immediately applicable for its determination, due to the sign problem. We then discuss a selection o
f recent lattice approaches that attempt to evade the sign problem and classify them according to the underlying principle: constrained simulations (density of states, histograms), holomorphicity (complex Langevin, Lefschetz thimbles), partial summations (clusters, subsets, bags) and change in integration order (strong coupling, dual formulations).
Lattice QCD at finite chemical potential is difficult due to the sign problem. We use stochastic quantization and complex Langevin dynamics to study this issue. First results for QCD in the hopping expansion are encouraging. U(1) and SU(3) one link m
odels are used to gain further insight into why the method appears to be successful.
The 1/N expansion of the two-particle irreducible effective action offers a powerful approach to study quantum field dynamics far from equilibrium. We investigate the effective convergence of the 1/N expansion in the O(N) model by comparing results o
btained numerically in 1+1 dimensions at leading, next-to-leading and next-to-next-to-leading order in 1/N as well as in the weak coupling limit. A comparison in classical statistical field theory, where exact numerical results are available, is made as well. We focus on early-time dynamics and quasi-particle properties far from equilibrium and observe rapid effective convergence already for moderate values of 1/N or the coupling.
A nonperturbative lattice study of QCD at finite chemical potential is complicated due to the complex fermion determinant and the sign problem. Here we apply the method of stochastic quantization and complex Langevin dynamics to this problem. We pres
ent results for U(1) and SU(3) one link models and QCD at finite chemical potential using the hopping expansion. The phase of the determinant is studied in detail. Even in the region where the sign problem is severe, we find excellent agreement between the Langevin results and exact expressions, if available. We give a partial understanding of this in terms of classical flow diagrams and eigenvalues of the Fokker-Planck equation.
We compute charmonium spectral functions in 2-flavor QCD on anisotropic lattices using the maximum entropy method. Our results suggest that the S-waves (J/psi and eta_c) survive up to temperatures close to 2Tc, while the P-waves (chi_c0 and chi_c1) melt away below 1.2Tc.
