Regolarizzazione di teorie quantistiche di campo su ditali di Lefschetz come soluzione del problema del segno

The introduction of a chemical potential in a quantum field theory in its Euclidean formulation is essential for many applications, both in solid state physics and in high energy physics. In particular, QCD simulations at finite density would provide an insight into heavy ion collisions occurring at LHC and they would also allow a study of the equation of state for nuclear matter inside neutron stars. Explicit introduction of chemical potential comes with the appearance of imaginary terms in the action, so that all Monte Carlo algorithms commonly used to compute expectation values of observables cannot be employed any more. So far this problem, called the †œsign problem†�, has prevented a study of most parts of the QCD phase diagram. Various approaches have been tried to overcome this problem, but a general and conclusive solution is still missing. An innovative proposal consists in complexifying the degrees of freedom and using Morse theory to decompose the original integrals that are to be computed into a combination of integrals on several Lefschetz thimbles, each attached to a critical point of the action. The greatest advantage of this deformation of integration contour is that, on each thimble, the imaginary part of the action stays constant, thus eliminating the sign problem. Being the method still young, it was necessary to test it on various toy-models. The first model which was studied is a simple one-dimensional integral, whose action is quartic in the only (real) field. This model, although in some sense trivial because of its low dimensionality, features a very non trivial solution within the framework of thimbles; by varying a parameter, one can have the relevance of either only one or three thimbles. The second model that was studied is a random matrix model. Here the degrees of freedom are 4 NxN real matrices: in this model the dimensionality can be varied by changing N. Studying the matrix model, a new algorithm to sample field configurations on a thimble was devised. This algorithm made it possible to recover correct results for the matrix model, even in regions of parameter space where the sign problem is severe. As the final purpose of this method is the study of gauge theories, the next step was to apply the approach to simple toy-models featuring gauge invariance, in order to test the effectiveness of the algorithmic solutions which had been introduced for the matrix model. As a first test, SU(N) one-link models were studied: they consist in a single SU(N) group integral of an action which is basically the trace of the group variable and the sign problem is introduced by hand by means of a complex coupling constant. The cases that were examined were successfully solved by combining multiple thimbles. The next step was 0+1-dimensional QCD. Despite the low dimensionality, in this context one faces a sign problem coming from a chemical potential, just as in QCD. The model was solved with the thimble approach at different numbers of quark flavours. The last model which was considered is Yang-Mills theory in 1+1 dimensions (the sign problem coming from a complex coupling constant). This theory features new aspects with regards to the decomposition in thimbles, namely local gauge invariance and the presence of toronic modes (zero modes due to periodic boundary conditions). These two aspects were discussed in detail in the present work.