Macroeconomic policy and elections: Theories and challenges

This paper reviews recent developments in the literature of economic policy-making. It focuses in particular on the relation between elections and macroeconomic policy. It should also be noted that in spite of tremendous advances in the area, there are still many important unresolved issues. In particular, both the normative and empirical areas are the ones in most urgent need study.

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

The Equivalence Postulate of Quantum Mechanics: Main Theorems

We consider the two main theorems in the derivation of the Quantum Hamilton--Jacobi Equation from the Equivalence Postulate (EP) of quantum mechanics. The first one concerns a basic cocycle condition, which holds in any dimension with Euclidean or Minkowski metrics and implies a global conformal symmetry underlying the Quantum Hamilton--Jacobi Equation. In one dimension such a condition fixes the Schwarzian equation. The second theorem concerns energy quantization which follows rigorously from consistency of the equivalence postulate.Comment: 30 pages. Standard LaTe

Energy Quantisation and Time Parameterisation

We show that if space is compact, then trajectories cannot be defined in the framework of quantum Hamilton--Jacobi equation. The starting point is the simple observation that when the energy is quantized it is not possible to make variations with respect to the energy, and the time parameterisation t-t_0=\partial_E S_0, implied by Jacobi's theorem and that leads to group velocity, is ill defined. It should be stressed that this follows directly form the quantum HJ equation without any axiomatic assumption concerning the standard formulation of quantum mechanics. This provides a stringent connection between the quantum HJ equation and the Copenhagen interpretation. Together with tunneling and the energy quantization theorem for confining potentials, formulated in the framework of quantum HJ equation, it leads to the main features of the axioms of quantum mechanics from a unique geometrical principle. Similarly to the case of the classical HJ equation, this fixes its quantum analog by requiring that there exist point transformations, rather than canonical ones, leading to the trivial hamiltonian. This is equivalent to a basic cocycle condition on the states. Such a cocycle condition can be implemented on compact spaces, so that continuous energy spectra are allowed only as a limiting case. Remarkably, a compact space would also imply that the Dirac and von Neumann formulations of quantum mechanics essentially coincide. We suggest that there is a definition of time parameterisation leading to trajectories in the context of the quantum HJ equation having the probabilistic interpretation of the Copenhagen School.Comment: 11 pages. The main addition concerns a discussion on the variational principle in the case of discrete energy spectra (Jacobi's Theorem). References adde

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.

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