The M\"obius Symmetry of Quantum Mechanics

The equivalence postulate approach to quantum mechanics aims to formulate quantum mechanics from a fundamental geometrical principle. Underlying the formulation there exists a basic cocycle condition which is invariant under $D$--dimensional M\"obius transformations with respect to the Euclidean or Minkowski metrics. The invariance under global M\"obius transformations implies that spatial space is compact. Furthermore, it implies energy quantisation and undefinability of quantum trajectories without assuming any prior interpretation of the wave function. The approach may be viewed as conventional quantum mechanics with the caveat that spatial space is compact, as dictated by the M\"obius symmetry, with the classical limit corresponding to the decompactification limit. Correspondingly, there exists a finite length scale in the formalism and consequently an intrinsic regularisation scheme. Evidence for the compactness of space may exist in the cosmic microwave background radiation.Comment: 16 pages. Talk presented at the DICE 2014 international conference, Castiglioncello, Tuscany, September 15-19, 201

Equivalence Principle: Tunnelling, Quantized Spectra and Trajectories from the Quantum HJ Equation

A basic aspect of the recently proposed approach to quantum mechanics is that no use of any axiomatic interpretation of the wave function is made. In particular, the quantum potential turns out to be an intrinsic potential energy of the particle, which, similarly to the relativistic rest energy, is never vanishing. This is related to the tunnel effect, a consequence of the fact that the conjugate momentum field is real even in the classically forbidden regions. The quantum stationary Hamilton-Jacobi equation is defined only if the ratio psi^D/psi of two real linearly independent solutions of the Schroedinger equation, and therefore of the trivializing map, is a local homeomorphism of the extended real line into itself, a consequence of the Moebius symmetry of the Schwarzian derivative. In this respect we prove a basic theorem relating the request of continuity at spatial infinity of psi^D/psi, a consequence of the q - 1/q duality of the Schwarzian derivative, to the existence of L^2(R) solutions of the corresponding Schroedinger equation. As a result, while in the conventional approach one needs the Schroedinger equation with the L^2(R) condition, consequence of the axiomatic interpretation of the wave function, the equivalence principle by itself implies a dynamical equation that does not need any assumption and reproduces both the tunnel effect and energy quantization.Comment: 1+10 pages, LaTeX. Typos corrected, to appear in Phys. Lett.

Hamilton-Jacobi meet M\uf6bius

Adaptation of the Hamilton\u2013Jacobi formalism to quantum mechanics leads to a cocycle condition, which is invariant under D\u2013dimensional M\ua8obius transformations with Euclidean or Minkowski metrics. In this paper we aim to provide a pedagogical presentation of the proof of the M\ua8obius symmetry underlying the cocycle condition. The M\ua8obius symmetry implies energy quantization and undefinability of quantum trajectories, without assigning any prior interpretation to the wave function. As such, the Hamilton\u2013Jacobi formalism, augmented with the global M\ua8obius symmetry, provides an alternative starting point, to the axiomatic probability interpretation of the wave function, for the formulation of quantum mechanics and the quantum spacetime. The M\ua8obius symmetry can only be implemented consistently if spatial space is compact, and correspondingly if there exist a finite ultraviolet length scale. Evidence for non\u2013 trivial space topology may exist in the cosmic microwave background radiation

Dendritic spike induction of postsynaptic cerebellar LTP

The architecture of parallel fiber (PF) axons contacting cerebellar Purkinje neurons (PNs) retains spatial information over long distances. PF synapses can trigger local dendritic calcium spikes, but whether and how this calcium signal leads to plastic changes that decode the PF input organization is unknown. By combining voltage and calcium imaging, we show that PF-elicited calcium signals, mediated by voltage-gated calcium channels, increase non-linearly during high-frequency bursts of electrically constant calcium spikes because they locally and transiently saturate the endogenous buffer. We demonstrate that these non-linear calcium signals, independently of NMDA or metabotropic glutamate receptor activation, can induce PF long-term potentiation (LTP). Two-photon imaging in coronal slices revealed that calcium signals inducing LTP can be observed by stimulating either the PF or the ascending fiber pathway. We propose that local dendritic calcium spikes, evoked by synaptic potentials, provide a unique mechanism to spatially decode PF signals into cerebellar circuitry changes

Extra Z′Z^\primes and W′W^\primes in Heterotic--String Derived Models

The ATLAS collaboration recently recorded possible excess in the di--boson production at the di--boson invariant mass at around 2 TeV. Such an excess may be produced if there exist additional $Z^\prime$ and/or $W^\prime$ at that scale. We survey the extra $Z^\prime$s and $W^\prime$s that may arise from semi--realistic heterotic string vacua in the free fermionic formulation in seven distinct cases including: $U(1)_{Z^\prime}\in SO(10)$; family universal $U(1)_{Z^\prime}$ not in $SO(10)$; non--universal $U(1)_{Z^\prime}$; hidden sector $U(1)$ symmetries and kinetic mixing; left--right symmetric models; Pati--Salam models; leptophobic and custodial symmetries. Each case has a distinct signature associated with the extra symmetry breaking scale. In one of the cases we explore the discovery potential at the LHC using resonant leptoproduction. Existence of extra vector boson with the reported properties will significantly constrain the space of allowed string vacua.Comment: 25 pages, 2 figures. Standard LaTeX. References added. Published versio

Likelihood decision functions

In both classical and Bayesian approaches, statistical inference is unified and generalized by the corresponding decision theory. This is not the case for the likelihood approach to statistical inference, in spite of the manifest success of the likelihood methods in statistics. The goal of the present work is to fill this gap, by extending the likelihood approach in order to cover decision making as well. The resulting decision functions, called likelihood decision functions, generalize the usual likelihood methods (such as ML estimators and LR tests), in the sense that these methods appear as the likelihood decision functions in particular decision problems. In general, the likelihood decision functions maintain some key properties of the usual likelihood methods, such as equivariance and asymptotic optimality. By unifying and generalizing the likelihood approach to statistical inference, the present work offers a new perspective on statistical methodology and on the connections among likelihood methods

On maxitive integration

A functional is said to be maxitive if it commutes with the (pointwise) supremum operation. Such functionals find application in particular in decision theory and related fields. In the present paper, maxitive functionals are characterized as integrals with respect to maxitive measures (also known as possibility measures or idempotent measures). These maxitive integrals are then compared with the usual additive and nonadditive integrals on the basis of some important properties, such as convexity, subadditivity, and the law of iterated expectations