arXiv subjects
Publications and source records attributed to Leonardo Pedro.
We show that the unitary operator on a separable Hilbert space is a parametrization of any conditional probability measure in a standard measure space. We propose unitary inference, a generalization of Bayesian inference. We study implications for classical statistical mechanics.
One attractive interpretation of quantum mechanics is the ensemble interpretation, where Quantum Mechanics merely describes a statistical ensemble of objects and not individual objects. But this interpretation does not address why the wave-function plays a central role in the calculations of probabilities, unlike most other interpretations of quantum mechanics. On the other hand, Classical Statistical Field Theory suffers from severe mathematical inconsistencies (specially for Hamiltonians which are non-polynomial in the fields, e.g. General relativistic statistical field theory). We claim that both problems are related to each other and we propose a solution to both. We prove: 1) the wave-function is a parametrization of any probability distribution of a statistical ensemble: there is a surjective map from an hypersphere to the set of all probability distributions; 2) for a quantum system defined in a 2-dimensional real Hilbert space, the role of the (2-dimensional real) wave-function is identical to the role of the Euler's formula in engineering, while the collapse of the wave-function is identical to selecting the real part of a complex number; 3) the collapse of the wave-function of any quantum system is a recursion of collapses of 2-dimensional real wave-functions; 4) the wave-function parametrization is key in the mathematical definition of Classical Statistical Field Theory we propose here. The same formalism is applied to Quantum Yang-Mills theory and Quantum Gravity in another article.
Quantum Yang-Mills theory, Classical Statistical Field Theory (for Hamiltonians which are non-polynomial in the fields, e.g. General relativistic statistical mechanics) and Quantum Gravity all suffer from severe mathematical inconsistencies and produce unreliable predictions at best. We define with mathematical rigor, a class of statistical field theories in Minkowski space-time where the (classical) canonical coordinates when modified by a non-deterministic time evolution, verify the canonical commutation relations. We then extend these statistical field theories to include non-trivial gauge symmetries and show that these theories have all the features of a Quantum Yang-Mills theory in four-dimensional space-time. We generalize the Gaussian measure to allow for the definition of Hamiltonians which are non-polynomial in the fields, such as in Classical Statistical Field Theory and Quantum Gravity. Finally, we test the consistency of our formalism with the quantization of the free Electromagnetic field.
One common way to define spontaneous symmetry breaking involves explicit symmetry breaking. This definition can be used in any approach to Effective Field Theory, from perturbation theory to lattice simulations. It allows us to study the spontaneous breakdown of global symmetries without assuming that the local gauge symmetry is spontaneously broken. This is important since perturbation theory is insufficient to study extended Higgs sectors: it is insufficient to predict the physical spectrum of the SU(5) Grand Unified Theory (Georgi-Glashow) or to predict the spontaneous breakdown of global symmetries. We also study background symmetries: these are symmetries that despite they are already explicitly broken, can be still spontaneously broken. We analyse examples where a background CP (charge-parity) symmetry is not spontaneously broken: in the Standard Model, in rephasing symmetries and in geometrical CP-violation. We show that all fields are real representations of the group of symmetries, since CP is a unitary transformation. There are consequences: to study accidental symmetries (e.g. custodial symmetry, pseudo-golstone bosons) we must consider real representations; CP is a symmetry of order 4 if the neutrinos are Majorana particles and the notion of CP-violating phases is inconsistent in some Lagrangians; a recent claim that a toy model exhibits physical CP-violation while the CP symmetry is conserved by the Lagrangian and the vacuum is false.
Extending the scalar sector helps in studying the Higgs mechanism and some Standard Model problems. We implement the correspondence between the gauge-dependent elementary states and the non-perturbative non-abelian gauge-invariant asymptotic states, necessary to study the non-perturbative phenomenology of two-Higgs-doublet models. The Flavour and CP violation in experimental data follows a hierarchical pattern, accounted by the Standard Model. We define the Minimal Flavour Violation condition with six spurions in effective field theories, implying Flavour and CP violation entirely dependent on the fermion mixing matrices but independent of the fermion masses hierarchy; it is renormalization-group invariant. We study the phenomenology of renormalizable two-Higgs-doublet models which verify the defined condition as consequence of a symmetry; new light physical scalars, mediating Flavour Changing Neutral Currents, are allowed by flavour data without flavour coefficients beyond the Standard Model; we tested the models with C++ libraries linked by the symbolic skills of the GiNaC library and we propose more libraries supporting a systematic search for Flavour Changing Neutral Currents. We also map the complex to the real Poincare group representations, derive the free Dirac equation requiring covariant localizability of the representations and study Localization and gauge symmetries in Quantum Field Theory.
Observable states are gauge-invariant. In a non-Abelian gauge theory, these are necessarily composite operators. We investigate the spectrum of these operators in the two-Higgs-doublet model. For this purpose, we are working along the lines of the Fröhlich-Morchio-Strocchi mechanism to relate the physical spectrum to the spectrum of the elementary particles. We also investigate the consequences of spontaneous breaking of the global (custodial) symmetry group. Finally, we briefly comment on how to test the results using lattice methods.
There are Poincare group representations on complex Hilbert spaces, like the Dirac spinor field, or real Hilbert spaces, like the electromagnetic field tensor. The Majorana spinor is an element of a 4 dimensional real vector space. The Majorana spinor field is a space-time dependent Majorana spinor, solution of the free Dirac equation. The Majorana-Fourier and Majorana-Hankel transforms of Majorana spinor fields are defined and related to the linear and angular momenta of a spin one-half representation of the Poincare group. We show that the Majorana spinor field with finite mass is an unitary irreducible projective representation of the Poincare group on a real Hilbert space. Since the Bargmann-Wigner equations are valid for all spins and are based on the free Dirac equation, these results open the possibility to study Poincare group representations with arbitrary spins on real Hilbert spaces.
The formulation of quantum mechanics with a complex Hilbert space is equivalent to a formulation with a real Hilbert space and particular density matrix and observables. We study the real representations of the Poincare group, motivated by the fact that the localization of complex unitary representations of the Poincare group is incompatible with causality, Poincare covariance and energy positivity. We review the map from the complex to the real irreducible representations---finite-dimensional or unitary---of a Lie group on a Hilbert space. Then we show that all the finite-dimensional real representations of the identity component of the Lorentz group are also representations of the parity, in contrast with many complex representations. We show that any localizable unitary representation of the Poincare group, compatible with Poincare covariance, verifies: 1) it is a direct sum of irreducible representations which are massive or massless with discrete helicity. 2) it respects causality; 3) if it is complex it contains necessarily both positive and negative energy subrepresentations 4) it is an irreducible representation of the Poincare group (including parity) if and only if it is: a)real and b)massive with spin 1/2 or massless with helicity 1/2. Finally, the energy positivity problem is discussed in a many-particles context.
We analyse the constraints and some of the phenomenological implications of a class of two Higgs doublet models where there are flavour-changing neutral currents (FCNC) at tree level but the potentially dangerous FCNC couplings are suppressed by small entries of the CKM matrix $V$. This class of models have the remarkable feature that, as a result of a discrete symmetry of the Lagrangian, the FCNC couplings are entirely fixed in the quark sector by $V$ and the ratio $v_2 /v_1$ of the vevs of the neutral Higgs. The discrete symmetry is extended to the leptonic sector, so that there are FCNC in the leptonic sector with their flavour structure fixed by the leptonic mixing matrix. We analyse a large number of processes, including decays mediated by charged Higgs at tree level, processes involving FCNC at tree level, as well as loop induced processes. We show that in this class of models one has new physical scalars beyond the standard Higgs boson, with masses reachable at the next round of experiments.