Gauge Interactions

The principle of gauge symmetry is introduced as a consequence of the invariance of the equations of motion under local transformations. We apply it to Abelian, as well as non-Abelian, internal symmetry groups. We derive in this way the Lagrangian of quantum electrodynamics and that of Yang–Mills theories. We quantise the latter using the path integral method and show the need for unphysical Faddeev–Popov ghost fields. We exhibit the geometric properties of the theory by formulating it on a discrete space-time lattice. We show that matter fields live on lattice sites and gauge fields on oriented lattice links. The Yang–Mills field strength is related to the curvature in field space.

Matter Waves in the Talbot-Lau Interferometry

Particle paths, emitted from distributed sources and passing out through slits of two gratings, G0 and G1, up to detectors, have been computed in detail by the path integral method. The particles under consideration are fullerene molecules with a De Broglie wavelength equal to 5 pm. The slits are Gaussian functions that simulate fuzzy edges of the slits. Waves of the matter computed by this method show perfect interference patterns both between the gratings and behind the second grating. Coherent and non-coherent distributed particle sources reproducing the interference patterns are discussed in detail. Paraxial approximation results from removing the distributed sources onto innity. This approximation gives a wave function reproducing an exact copy of the Talbot carpet. PACS numbers: 03.75.-b, 03.75.Dg, 42.25.Hz

Note on entropy dynamics in the Brownian SYK model

We study the time evolution of Rényi entropy in a system of two coupled Brownian SYK clusters evolving from an initial product state. The Rényi entropy of one cluster grows linearly and then saturates to the coarse grained entropy. This Page curve is obtained by two different methods, a path integral saddle point analysis and an operator dynamics analysis. Using the Brownian character of the dynamics, we derive a master equation which controls the operator dynamics and gives the Page curve for purity. Insight into the physics of this complicated master equation is provided by a complementary path integral method: replica diagonal and non-diagonal saddles are responsible for the linear growth and saturation of Ŕenyi entropy, respectively.

Newtonian dynamics of imaginary time-dependent mean field theory

A Time Dependent Hartree-Fock (TDHF) based classical model is applied to sub-barrier fusion reactions using the Feynman Path Integral Method (FPIM). The fusion cross-sections and modified astrophysical S*-factors are calculated for the 12C+12C reactions and compared to direct and indirect experimental results. Different channels cross-sections are estimated from the statistical decay of the compound nucleus. A good agreement with the direct data is found. We suggest a complementary observable given by the (imaginary) action A easily derived from theory and experiments. When properly normalized by the action in the Gamow limit it has an upper value of 1 at zero beam energies. It becomes negative at the Coulomb barrier which is Vcb=5.05±0.05MeV from direct data and Vcb=5.5MeV from model calculations.

Path Integral Methods for the Probabilistic Analysis of Nonlinear Systems Under a White-Noise Process

In this paper, the widely known path integral method, derived from the application of the Chapman–Kolmogorov equation, is described in details and discussed with reference to the main results available in literature in several decades of contributions. The most simple application of the method is related to the solution of Fokker–Planck type equations. In this paper, the solution in the presence of normal, α-stable, and Poissonian white noises is first discussed. Then, application to barrier problems, such as first passage problems and vibroimpact problems is described. Further, the extension of the path integral method to problems involving multi-degrees-of-freedom systems is analyzed. Lastly, an alternative approach to the path integration method, that is the Wiener Path integration (WPI), also based on the Chapman–Komogorov equation, is discussed. The main advantages and the drawbacks in using these two methods are deeply analyzed and the main results available in literature are highlighted.