Foundations of Quantum Mechanics: The Connection Between QM and the Central Limit Theorem

In this paper we unravel the connection between the quantum mechanical formalism and the Central limit theorem (CLT). We proceed to connect the results coming from this theorem with the derivations of the Schrödinger equation from the Liouville equation, presented by ourselves in other papers. In those papers we had used the concept of an infinitesimal parameter x that raised some controversy. The status of this infinitesimal parameter is then elucidated in the framework of the CLT. Finally, we use the formal apparatus developed in our previous papers and the results of the present one to advance an alternative objective interpretation of quantum mechanics in which its relations with the classical framework are made explicit. The relations between our approach and those using the Wigner–Moyal transformation are also addressed.     

On the predictivity of the non‐renormalizable quantum field theories

Following a Four Dimensional Renormalization approach to ultraviolet divergences (FDR), we extend the concept of predictivity to non‐renormalizable quantum field theories at arbitrarily large perturbative orders. The idea of topological renormalization is introduced, which keeps a finite value for the parameters of the theory by trading the usual order‐by‐order renormalization procedure for an order‐by‐order redefinition of the perturbative vacuum. One additional measurement is then sufficient to systematically compute quantum corrections at any loop order, with no need of absorbing ultraviolet infinities in the Lagrangian.     

创立量子力学的睿智才思(续2)——纪念矩阵力学和波动力学诞生80～81周年

1922年前后玻尔-索末菲的旧量子论在原子结构理论上取得辉煌成就的同时，它的致命弱点也开始暴露出来．反常塞曼效应是其一，原子光谱的多重线结构是其二，更不要说氢分子离子H2^＋问题了．一时多少杰出的物理学家为这些问题绞尽脑汁，设计了各种物理模型，提出一些“代用（德文Er-satz）”理论，诸如海森伯和泡利的Zwang之类，到头来还只得愁眉以对，一筹莫展．无怪乎海森伯把反常塞曼效应叫做“光谱项的动物学和塞曼植物学（Term Zoology and Zeeman Botany）”，意即在这个领域里物理学家只能像生物学家那样记录现象和描绘事实，提不出像样的理论．到1925年下半年，人们手里有了矩阵力学，问题是否可以解决了呢？万事俱备，只欠“东风”，那就是“自旋”．     

Quantum equilibrium and the origin of absolute uncertainty

The quantum formalism is a measurement formalism-a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from Bohmian mechanics, which is what emerges from Schrödinger's equation for a system of particles when we merely insist that particles means particles. While distinctly non-Newtonian, Bohmian mechanics is a fully deterministic theory of particles in motion, a motion choreographed by the wave function. We find that a Bohmian universe, though deterministic, evolves in such a manner that anappearance of randomness emerges, precisely as described by the quantum formalism and given, for example, by = ¦¦ ². A crucial ingredient in our analysis of the origin of this randomness is the notion of the effective wave function of a subsystem, a notion of interest in its own right and of relevance to any discussion of quantum theory. When the quantum formalism is regarded as arising in this way, the paradoxes and perplexities so often associated with (nonrelativistic) quantum theory simply evaporate.This paper is dedicated to the memory of J. S. Bell.     

On the variational formulation of systems with non‐holonomic constraints

In a previous article by the author, it was shown that one could effectively give a variational formulation to non‐conservative mechanical systems by starting with the first variation functional instead of an action functional. In this article, it is shown that this same approach will also allow one to give a variational formulation to systems with non‐holonomic constraints. The key is to use an adapted anholonomic local frame field in the formulation, which then implies the replacement of ordinary derivatives with covariant ones. The method is then applied to the case of a vertical disc rolling without slipping or friction on a plane.     

On the weak‐field approximation of nonlinear vacuum electrodynamics of the Plebański class

Plebański's class of nonlinear vacuum electrodynamics is considered, which is for several reasons of interest at the present time. In particular, the question is answered under which circumstances Maxwell's original field equations are recovered approximately and which ‘post‐Maxwellian’ effects could arise. To this end, a weak field approximation method is developed, allowing to calculate ‘post‐Maxwellian’ corrections up to Nth order. In some respect, this is analogue of determining ‘post‐Newtonian’ corrections from relativistic mechanics by a low velocity approximation. As a result, we got a series of linear field equations that can be solved order by order. In this context, the solutions of the lower orders occur as source terms inside the higher order field equations and represent a ‘post‐Maxwellian’ self‐interaction of the electromagnetic field, which increases order by order. It becomes apparent that one has to distinguish between problems with and without external source terms because without sources also high frequency solutions can be approximately described by Maxwell's original equations. The higher order approximations, which describe ‘post‐Maxwellian’ effects, can give rise to experimental tests of Plebańksi's class. Finally, two boundary value problems are discussed to have examples at hand.     

Exact solution of the ion‐laser interaction in all regimes

We show that in the trapped ion‐laser interaction all the regimes may be considered analytically. We may solve not only for different laser intensities, but also away from resonance and from the Lamb‐Dicke regime. It is found a dispersive Hamiltonian for the high intensity regime, that, being diagonal, its evolution operator may be easily calculated.     

Raman spectroscopy of collective modes in charge‐density‐wave systems: the mean‐field microscopic theory

The electron‐mediated coupling of external electromagnetic fields and Raman‐active oscillations is derived for a general electronic model with multiple bands using the adiabatic approach and the explicit diagrammatic approach. The theory is illustrated on the quasi‐one‐dimensional (Q1D) quarter‐filled charge‐density‐wave (CDW) model. It is shown how the long‐range Coulomb forces and the single‐electron relaxation processes affect the Raman spectroscopy of the amplitude‐oscillation mode in clean CDW systems. It is also argued that the adiabatic treatment of the photon‐phonon coupling functions can be safely used in this case. Copyright © 2008 John Wiley & Sons, Ltd.     

A solution of the non‐uniqueness problem of the Dirac Hamiltonian and energy operators

In a general spacetime, the possible choices for the field of orthonormal tetrads lead (in standard conditions) to equivalent Dirac equations. However, the Hamiltonian operator is got from rewriting the Dirac equation in a form adapted to a particular reference frame, or class of coordinate systems. That rewriting does not commute with changing the tetrad field (u_α ). The data of a reference frame F fixes a four‐velocity field v, and also fixes a rotation‐rate field Ω . It is natural to impose that u₀ = v. We show that then the spatial triad (u_p) can only be rotating w.r.t. F, and that the title problem is solved if one imposes that the corresponding rotation rate Ξ be equal to Ω – or also, if one imposes that Ξ = 0 . We also analyze other proposals which were aimed at solving the problem of the non‐uniqueness of the Dirac Hamiltonian.     

X‐ray fluorescence induced by standing waves in the grazing‐incidence and grazing‐exit modes: study of the Mg–Co–Zr system

The characterization of Mg–Co–Zr tri‐layer stacks using X‐ray fluorescence induced by X‐ray standing waves, in both the grazing‐incidence (GI) and the grazing‐exit (GE) modes, is presented. The introduction of a slit in the direction of the detector improves the angular resolution by a factor of two and significantly improves the sensitivity of the technique for the chemical characterization of the buried interfaces. By observing the intensity variations of the Mg Kα and Co Lα characteristic emissions as a function of the incident (GI mode) or detection (GE mode) angle, it is shown that the interfaces of the Si/[Mg/Co/Zr]_×30 multilayer are abrupt, whereas in the Si/[Mg/Zr/Co]_×30 multilayer a strong intermixing occurs at the Co‐on‐Zr interfaces. An explanation of this opposite behavior of the Co‐on‐Zr and Zr‐on‐Co interfaces is given by the calculation of the mixing enthalpies of the Co–Mg, Co–Zr and Mg–Zr systems, which shows that the Co–Zr system presents a negative value and the other two systems present positive values. Together with the difference of the surface free energies of Zr and Co, this leads to the Mg/Zr/Co system being considered as a Mg/Co_xZr_y bi‐layer stack, with x/y estimated around 3.5.     

A non‐uniqueness problem of the Dirac theory in a curved spacetime

The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under which a change of the coefficient fields leads to an equivalent Hamiltonian operator H, or to an equivalent energy operator E. We do that for the standard version of the gravitational Dirac equation, and for two alternative equations based on the tensor representation of the Dirac fields. The latter equations may be defined when the spacetime is four‐dimensional, noncompact, and admits a spinor structure. We find that, for each among the three versions of the equation, the vast majority of the possible coefficient changes do not lead to an equivalent operator H, nor to an equivalent operator E, whence a lack of uniqueness. In particular, we prove that the Dirac energy spectrum is not unique. This non‐uniqueness of the energy spectrum comes from an effect of the choice of coefficients, and applies in any given coordinates.     

Integrability and the variational formulation of non‐conservative mechanical systems

It is shown that one can obtain canonically‐defined dynamical equations for non‐conservative mechanical systems by starting with a first variation functional, instead of an action functional, and finding their zeroes. The kernel of the first variation functional, as an integral functional, is a 1‐form on the manifold of kinematical states, which then represents the dynamical state of the system. If the 1‐form is exact then the first variation functional is associated with the first variation of an action functional in the usual manner. The dynamical equations then follow from the vanishing of the dual of the Spencer operator that acts on the dynamical state. This operator, in turn, relates to the integrability of the kinematical states. The method is applied to the modeling of damped oscillators.     

Representation of the Universe as a Dendrogramic Hologram Endowed with Relational Interpretation

A proposal for a fundamental theory is described in which classical and quantum physics as a representation of the universe as a gigantic dendrogram are unified. The latter is the explicate order structure corresponding to the purely number-theoretical implicate order structure given by p-adic numbers. This number field was zero-dimensional, totally disconnected, and disordered. Physical systems (such as electrons, photons) are sub-dendrograms of the universal dendrogram. Measurement process is described as interactions among dendrograms; in particular, quantum measurement problems can be resolved using this process. The theory is realistic, but realism is expressed via the the Leibniz principle of the Identity of Indiscernibles. The classical-quantum interplay is based on the degree of indistinguishability between dendrograms (in which the ergodicity assumption is removed). Depending on this degree, some physical quantities behave more or less in a quantum manner (versus classic manner). Conceptually, our theory is very close to Smolin’s dynamics of difference and Rovelli’s relational quantum mechanics. The presence of classical behavior in nature implies a finiteness of the Universe-dendrogram. (Infinite Universe is considered to be purely quantum.) Reconstruction of events in a four-dimensional space type is based on the holographic principle. Our model reproduces Bell-type correlations in the dendrogramic framework. By adjusting dendrogram complexity, violation of the Bell inequality can be made larger or smaller.