Abstract:
CPT invariance in neutrino physics has attracted attention after the revival of the hypothetical idea that neutrino and antineutrino might have nonequal masses ($m_{\bar\nu} \neq m_{\nu}$) when realizing neutrino oscillations as a new sensitive phenomenon to search for the violation of this fundamental symmetry. Moreover, the profound relation between the CPT and Lorentz symmetries turns the studies of CPT and Lorentz invariance violations into the {\bf one two-sided} problem. We present a guide for non-experts through the literature on neutrino physics. The basic works are reviewed thoroughly while for the other papers only current results or discussion issues are quoted. The review covers, mostly, oscillations of neutrinos, resonant change of their flavors and cosmic neutrino physics to systematize possible evidences of CPT/Lorentz violation in this sector of the Standard

Abstract:
I introduce environment - assisted invariance -- a symmetry related to causality that is exhibited by correlated quantum states -- and describe how it can be used to understand the nature of ignorance and, hence, the origin of probabilities in quantum physics.

Abstract:
Simplicity of fundamental physical laws manifests itself in fundamental symmetries. While systems with an infinity of strongly interacting degrees of freedom (in particle physics and critical phenomena) are hard to describe, they often demonstrate symmetries, in particular scale invariance. In two dimensions (2d) locality often promotes scale invariance to a wider class of conformal transformations which allow for nonuniform re-scaling. Conformal invariance allows a thorough classification of universality classes of critical phenomena in 2d. Is there conformal invariance in 2d turbulence, a paradigmatic example of strongly-interacting non-equilibrium system? Here, using numerical experiment, we show that some features of 2d inverse turbulent cascade display conformal invariance. We observe that the statistics of vorticity clusters is remarkably close to that of critical percolation, one of the simplest universality classes of critical phenomena. These results represent a new step in the unification of 2d physics within the framework of conformal symmetry.

Abstract:
It is shown that neither the wave picture nor the ordinary particle picture offers a satisfactory explanation of the double-slit experiment. The Physicists who have been successful in formulating theories in the Newtonian Paradigm with its corresponding ontology find it difficult to interpret Quantum Physics which deals with particles that are not sensory perceptible. A different interpretation of Quantum Physics based in a different ontology is presented in what follows. According to the new interpretation Quantum particles have different properties from those of Classical Newtonian particles. The interference patterns are explained in terms of particles each of which passes through both slits.

Abstract:
Since the particles such as molecules, atoms and nuclei are composite particles, it is important to recognize that physics must be invariant for the composite particles and their constituent particles, this requirement is called particle invariance in this paper. But difficulties arise immediately because for fermion we use Dirac equation, for meson we use Klein-Gordon equation and for classical particle we use Newtonian mechanics, while the connections between these equations are quite indirect. Thus if the particle invariance is held in physics, i.e., only one physical formalism exists for any particle, we can expect to find out the differences between these equations by employing the particle invariance. As the results, several new relationships between them are found, the most important result is that the obstacles that cluttered the path from classical mechanics to quantum mechanics are found, it becomes possible to derive the quantum wave equations from relativistic mechanics after the obstacles are removed. An improved model is proposed to gain a better understanding on elementary particle interactions. This approach offers enormous advantages, not only for giving the first physically reasonable interpretation of quantum mechanics, but also for improving quark model.

Abstract:
Reparametrization invariance being treated as a gauge symmetry shows some specific peculiarities. We study these peculiarities both from a general point of view and on concrete examples. We consider the canonical treatment of reparametrization invariant systems in which one fixes the gauge on the classical level by means of time-dependent gauge conditions. In such an approach one can interpret different gauges as different reference frames. We discuss the relations between different gauges and the problem of gauge invariance in this case. Finally, we establish a general structure of reparametrizations and its connection with the zero-Hamiltonian phenomenon.

Abstract:
\noindent In our contribution to this volume we deal with \emph{discrete} symmetries: these are symmetries based upon groups with a discrete set of elements (generally a set of elements that can be enumerated by the positive integers). In physics we find that discrete symmetries frequently arise as `internal', non-spacetime symmetries. Permutation symmetry is such a discrete symmetry arising as the mathematical basis underlying the statistical behaviour of ensembles of certain types of indistinguishable quantum particle (e.g., fermions and bosons). Roughly speaking, if such an ensemble is invariant under a permutation of its constituent particles (i.e., permutation symmetric) then one doesn't `count' those permutations which merely `exchange' indistinguishable particles; rather, the exchanged state is identified with the original state. This principle of invariance is generally called the `indistinguishability postulate' [IP], but we prefer to use the term `permutation invariance' [PI]. It is this symmetry principle that is typically taken to underpin and explain the nature of (fermionic and bosonic) quantum statistics (although, as we shall see, this characterisation is not uncontentious), and it is this principle that has important consequences regarding the metaphysics of identity and individuality for particles exhibiting such statistical behaviour.

Abstract:
The fundamental laws of physics are required to be invariant under local spatial scale change. In 3-dimensional space, this leads to a variation in Planck constant \hbar and speed of light c. They vary as \hbar ~ a^(1/2) and c ~ a^(-1/2), a is the local scale. A direct consequence is that the expanding universe progressively alters the values of c which in turn affects the evolution of the universe itself. Friedmann eqns violate scale invariance and neglect to account for the scale dependency of c. We build a cosmological model which is fully consistent with scale invariance and respects Lorentz invariance. This model leads to a universe different from the ones depicted in Friedmann model. We apply our model to resolve a series of observational and theoretical difficulties in modern cosmology: "runaway density parameter" problem, budgetary shortfall, horizon problem, eg. Our model does not resort to inflation hypothesis. We derive a modification to the Hubble law and Hubble constant, and a new brightness-redshift relationship for Type Ia supernovae: (a) Hubble constant has been inadvertently overestimated by a factor of 9/5; so has the critical density by (9/5)^2; (b) Our new photometric distance-redshift relationship d_L=2c/(3H0)*(1+z)^(5/6)\ln(1+z) with one parameter H0=37 fits to the high-$z$ objects as equally well as the traditional relationship does with three parameters H0=70.5, OmegaM=0.27, OmegaL=0.73. We draw two conclusions: (i) With H0=37, the critical density is only 0.28 time the value previously thought; dark energy is absent. (ii) H0=37 restores the age estimate to 17.6 Gy (via t_0=2/(3H0)). They also raise the possibility that the universe expansion is not accelerating, but rather a result of c progressively adapting to new spatial scale as the universe expands. Finally, we discuss an array of implications of scale invariance in the larger context of physics.

Abstract:
A basic principle of physics is the freedom to locally choose any unit system when describing physical quantities. Its implementation amounts to treating Weyl invariance as a fundamental symmetry of all physical theories. In this thesis, we study the consequences of this "unit invariance" principle and find that it is a unifying one. Unit invariance is achieved by introducing a gauge field called the scale, designed to measure how unit systems vary from point to point. In fact, by a uniform and simple Weyl invariant coupling of scale and matter fields, we unify massless, massive, and partially massless excitations. As a consequence, masses now dictate the response of physical quantities to changes of scale. This response is calibrated by certain "tractor Weyl weights". Reality of these weights yield Breitenlohner-Freedman stability bounds in anti de Sitter spaces. Another valuable outcome of our approach is a general mechanism for constructing conformally invariant theories. In particular, we provide direct derivations of the novel Weyl invariant Deser--Nepomechie vector and spin three-half theories as well as new higher spin generalizations thereof. To construct these theories, a "tractor calculus" coming from conformal geometry is employed, which keeps manifest Weyl invariance at all stages. In fact, our approach replaces the usual Riemannian geometry description of physics with a conformal geometry one. Within the same framework, we also give a description of fermionic and interacting supersymmetric theories which again unifies massless and massive excitations.

Abstract:
This article extends results described in a recent article detailing a structural scale invariance property of the simulated annealing (SA) algorithm. These extensions are based on generalizations of the SA algorithm based on Tsallis statistics and a non-extensive form of entropy. These scale invariance properties show how arbitrary aggregations of energy levels retain certain mathematical characteristics. In applying the non-extensive forms of statistical mechanics to illuminate these scale invariance properties, an interesting energy transformation is revealed that has a number of potentially useful applications. This energy transformation function also reveals a number of symmetry properties. Further extensions of this research indicate how this energy transformation function relates to power law distributions and potential application for overcoming the so-called ``broken ergodicity'' problem prevalent in many computer simulations of critical phenomena.