Time machines and the Principle of Self-Consistency as a consequence of the Principle of Stationary Action (II): the Cauchy problem for a self-interacting relativistic particle

We consider the action principle to derive the classical, relativistic motion of a self-interacting particle in a 4-D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. In particular, we study the case of a pointlike particle subject to a `hard-sphere' self-interaction potential and which can traverse the wormhole an arbitrary number of times, and show that the only possible trajectories for which the classical action is stationary are those which are globally self-consistent. Generically, the multiplicity of these trajectories (defined as the number of self-consistent solutions to the equations of motion beginning with given Cauchy data) is finite, and it becomes infinite if certain constraints on the same initial data are satisfied. This confirms the previous conclusions (for a non-relativistic model) by Echeverria, Klinkhammer and Thorne that the Cauchy initial value problem in the presence of a wormhole `time machine' is classically `ill-posed' (far too many solutions). Our results further extend the recent claim by Novikov et al. that the `Principle of self-consistency' is a natural consequence of the `Principle of minimal action.'Comment: 39 pages, latex fil

Testing the Standard Model by precision measurement of the weak charges of quarks

In a global analysis of the latest parity-violating electron scattering measurements on nuclear targets, we demonstrate a significant improvement in the experimental knowledge of the weak neutral-current lepton-quark interactions at low energy. The precision of this new result, combined with earlier atomic parity-violation measurements, places tight constraints on the size of possible contributions from physics beyond the Standard Model. Consequently, this result improves the lower-bound on the scale of relevant new physics to ~1 TeV.Comment: 4 pages, 3 figures; v2: further details on extraction of electroweak parameters, new figur

Time machines: the Principle of Self-Consistency as a consequence of the Principle of Minimal Action

We consider the action principle to derive the classical, non-relativistic motion of a self-interacting particle in a 4-D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. For the case of a `hard-sphere' self-interaction potential we show that the only possible trajectories (for a particle with fixed initial and final positions and which traverses the wormhole once) minimizing the classical action are those which are globally self-consistent, and that the `Principle of self-consistency' (originally introduced by Novikov) is thus a natural consequence of the `Principle of minimal action.'Comment: 26 pages, plain latex; modified version includes extra constraint for collinear collision case and other minor misprints correction

Perturbation of matrices and non-negative rank with a view toward statistical models

In this paper we study how perturbing a matrix changes its non-negative rank. We prove that the non-negative rank is upper-semicontinuos and we describe some special families of perturbations. We show how our results relate to Statistics in terms of the study of Maximum Likelihood Estimation for mixture models.Comment: 13 pages, 3 figures. A theorem has been rewritten, and some improvements in the presentations have been implemente

Extracting nucleon strange and anapole form factors from world data

The complete world set of parity violating electron scattering data up to Q^2~0.3 GeV^2 is analysed. We extract the current experimental determination of the strange electric and magnetic form factors of the proton, as well as the weak axial form factors of the proton and neutron, at Q^2 = 0.1 GeV^2. Within experimental uncertainties, we find that the strange form factors are consistent with zero, as are the anapole contributions to the axial form factors. Nevertheless, the correlation between the strange and anapole contributions suggest that there is only a small probability that these form factors all vanish simultaneously.Comment: 4 pages, 3 figs; v2: version to appear in PR

A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations

We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the dependence of the coefficients is nonlinear and nonlocal in time with respect to the unknowns. We extend the numerical scheme proposed and studied recently by the authors for a single FPK equation of this type. We analyse the convergence of the scheme and we study its applicability in two examples. The first one concerns a population model involving two interacting species and the second one concerns two populations Mean Field Games

Process reconstruction from incomplete and/or inconsistent data

We analyze how an action of a qubit channel (map) can be estimated from the measured data that are incomplete or even inconsistent. That is, we consider situations when measurement statistics is insufficient to determine consistent probability distributions. As a consequence either the estimation (reconstruction) of the channel completely fails or it results in an unphysical channel (i.e., the corresponding map is not completely positive). We present a regularization procedure that allows us to derive physically reasonable estimates (approximations) of quantum channels. We illustrate our procedure on specific examples and we show that the procedure can be also used for a derivation of optimal approximations of operations that are forbidden by the laws of quantum mechanics (e.g., the universal NOT gate).Comment: 9pages, 5 figure

Time-optimal CNOT between indirectly coupled qubits in a linear Ising chain

We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with equal coupling $J$ plus a local magnetic field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the $\mathrm{CNOT}(1, 3)$ between the indirectly coupled qubits 1 and 3 is $T=\sqrt{3/2} J^{-1}$, i.e. faster than a previous estimate based on a similar Hamiltonian and the assumption of local unitaries with zero time cost. Furthermore, performing a simple Walsh-Hadamard rotation in the Hlibert space of qubit 3 shows that the time-optimal synthesis of the $\mathrm{CNOT}^{\pm}(1, 3)$ (which acts as the identity when the control qubit 1 is in the state $\ket{0}$, while if the control qubit is in the state $\ket{1}$ the target qubit 3 is flipped as $\ket{\pm}\rightarrow \ket{\mp}$) also requires the same time $T$.Comment: 9 pages; minor modification