The use of Grassmann variables to give a semi-classical description of quantum variables with a finite spectrum introduced by Berezin is described. Then pseudo-classical Lagrangians for the description of spin, of the electric charge, of the sign of the energy of a particle are described. This approach regularizes the divergences of the self-energies: (1) its quantization gives finite results; (2) a suitable mean gives the underlying finite classical theory.

In the chosen family of Einstein space-times one can give an ADM formulation of tetrad gravity and of its first-class constraints, so that fermions can be described in this framework. There are 16 configuration variables, 16 momenta, and 14 first-class constraints. Then one can define a new canonical basis adapted to 10 of the 14 constraints (not to the super-Hamiltonian and super-momentum ones) with a Shanmugadhasan canonical transformation. This allows identifying two pairs of canonical variables describing the tidal effects (the gravitational waves after linearization). However, they are not Dirac observables.

After a review of regular Lagrangians, their Hamiltonian formulation, and the first Noether theorem, there is the exposition of theory of singular Lagrangians and of Dirac–Bergmann theory of first- and second-class constraints. Also, the gauge transformations of field theory and general relativity are analyzed.

There is a description of the 3+1 approach allowing definition of global non-inertial frames in Minkowski space-time. One gives a time-like observer and a nice foliation with 3-spaces (namely a clock synchronization convention). Then one introduces Lorentz scalar radar 4-coordinates: the time is an increasing function of the proper time of the observer and the 3-coordinates live in the instantaneous 3-spaces. The connection of the radar coordinates with the standard ones defines the four embedding functions describing the foliation with 3-spaces. Then there is the definition of parametrized Minkowski theories for every kind of matter admitting a Lagrangian description. The new Lagrangian is a function of the matter and of the embedding, but is singular so that the embedding variables are gauge variables. As a consequence, the transition from a non-inertial frame to either an inertial or non-inertial frame is a gauge transformation not changing the physics but only the inertial forces.

The following family of Einstein space-times allows the use of the 3+1 approach: (1) globally hyperbolic (this allows the ADM Hamiltonian formulation); (2) asymptotically Minkowskian at spatial infinity (all the 3-spaces approach parallel space-like hyper-planes); (3) without super-translations (at spatial infinity there is the asymptotic ADM Poincaré algebra needed for particle physics). It turns out that the asymptotic ADM Poincaré 4-momentum is orthogonal to the asymptotic hyper-planes. Therefore, the 3+1 approach allows describing the Hamiltonian formulation of metric gravity and of its first-class constraints in the family of the non-inertial rest-frames.

In this chapter there is the description of fields and fluids in the rest-frame instant form of dynamics with the definition of their Wigner-covariant degrees of freedom inside the Wigner 3-spaces after the decoupling of the external center of mass. This is done for the Klein–Gordon, electromagnetic, Dirac, and Yang–Mills fields. In the case of the electromagnetic field there is the identification of the Wigner-covariant Dirac observables. This procedure can be applied also to Yang–Mills fields, but to get the Dirac observables one needs the knowledge of an explicit solution of the Gauss’s law constraints. Then there is the description of relativistic fluids in this framework. In particular a definition of the relativistic micro-canonical ensemble in the Wigner 3-spaces is given and it is shown which equations have to be solved to get a consistent relativistic statistical mechanics.