## Particle mixing, two level systems and gauge theory

##### Abstract

In this thesis I will discuss the theory of two level systems and the theory
of the oscillating particles in quantum eld theory.
In the rst chapter I will consider the time evolution of a two level system,
a qubit, to show that it has inside a local in time gauge invariant evolution
equation. I construct the covariant derivative operator and show that it is
related to the free energy. The gauge invariance of the time evolution of the
two level system is analogous to the phenomenon of birefringence.I also show
that the two level systems present a Berry-like and an Anandan-Aharonov
phase. Finally, I discuss entropy environment e ects and the distance in
projective Hilbert space between two level states to show that the last one
is properly related to the Aharonov - Anandan phase.
In the second chapter I review the result obtained in QFT for particle mix-
ing, analyzing the theoretical construction and the oscillation formula in the
fermion case.I will emphasize the di erences between the quantum mechan-
ics formulas and the QFT formulas.The unitary inequivalence between the
avor and the mass eigenstates is also shown and the structure of the current
for charged elds is nally discussed. I found a non - perturbative vacuum
structure for the mixing particles that, among the other things, will lead to
a non zero contribution to the value of the cosmological constant (chapter
3).
Several links between rst and second chapter will arise from this thesis and
will shed the light on the fact that it is possible to construct a generic two
level quantum eld theory, that is an extension of the quantum mechanics
bit theory in a quantum eld theory framework. [edited by author]