arXiv subjects

Joseph F. Johnson

Publications and source records attributed to Joseph F. Johnson.

9 recordsLinked to original sources

arXiv2014

Hilbert's Sixth Problem: Descriptive Statistics as New Foundations for Probability

Hay esbozos según los cuales las probabilidades se cuentan como la fundación de la teorí a matemática de las estadísticas. Mas la significación física de las probabilidades matemáticas son oscuros, muy poco entendidos. Parecí era mejor que las probabilidades físicas se fundaran en las estadísticas descriptivas de datos fisicales. Se trata una teorí a que así responde a una cuestiona de Hilbert propuesta en su Problema Número Seis, la axiomatización de la Física. Esta está basada en las auto-correlaciónes de los series temporales. Casi todas las funciones de auto-correlación de las trayectorí as de un sistema dinámico lineal (con un número de grados de libertad bastante grande) son todas aproximadamente iguales, no importan las condiciones iniciales, aún si el sistema no sea ergódico, como conjeturó Khintchine en 1943. Usually, the theory of probability has been made the foundation for the theory of statistics. But the physical significance of the concept of probability is problematic, with no consensus. It would seem better to make the descriptive statistics of physical data the foundations of physical probability. This will answer a question posed by Hilbert in his Sixth Problem, the axiomatization of Physics. It is based on the auto-correlation function of time series. Almost all trajectories of a linear dynamical system (with sufficiently many degrees of freedom) are approximately equal, no matter their initial conditions, even when the system is not ergodic, as conjectured by Khintchine in 1943.

cond-mat.stat-mech↗

arXiv2011

Some Special Cases of Khintchine's Conjectures in Statistical Mechanics: Approximate Ergodicity of the Auto-Correlation Functions of an Assembly of Linearly Coupled Oscillators

We give Sir James Jeans's notion of 'normal state' a mathematically precise definition. We prove that normal cells of trajectories exist in the Hamiltonian heat-bath model of an assembly of linearly coupled oscillators that generates the Ornstein--Uhlenbeck process in the limit of an infinite number of degrees of freedom. This, in some special cases, verifies some far-reaching conjectures of Khintchine on the weak ergodicity of a dynamical system with a large number of degrees of freedom. In order to estimate the theoretical auto-correlation function of a time series from the sample auto-correlation function of one of its realisations, it is usually assumed without justification that the time series is ergodic. Khintchine's conjectures about dynamical systems with large numbers of degrees of freedom justifies, even in the absence of ergodicity, approximately the same conclusions. Para emplear el correlograma de los valores muestrales de un proceso estocástico para estimar su función teórica de autocorrelación, por regla general se asume, sin justificación, que el proceso es ergódico. Pero en 1943, Khintchine conjeturó proposiciones de gran importancia en este asunto, que justificarí an una aproximación a las mismas estimaciones aún sin la ergodicidad del sistema. Mostraremos casos particulares de las conjeturas de Khintchine para asambleas de osciladores lineales.

math-ph↗

arXiv2007

Problems of Quantum Measurement

We derive the probabilities of measurement results from Schroedinger's equation plus a definition of macroscopic as a particular kind of thermodynamic limit. Bohr's insight that a measurement apparatus must be classical in nature and classically describable is made precise in a mathematical sense analogous to the procedures of classical statistical mechanics and the study of Hamiltonian heat baths.

quant-ph↗

arXiv2007

The Axiomatisation of Physics

Analysing Quantum Measurement requires analysing the physics of amplification since amplification of phenomena from one scale to another scale is essential to measurement. There still remains the task of working this into an axiomatic logical structure, what should be the foundational status of the concepts of measurement and probability. We argue that the concept of physical probability is a multi-scale phenomenon and as such, can be explicitly defined in terms of more fundamental physical concepts. Thus Quantum Mechanics can be given a logically unexceptionable axiomatisation. We introduce a new definition of macroscopic observable which implements Bohr's insight that the observables of a measurement apparatus are classical in nature. In particular, we obtain the usual non-abelian observables as limits of abelian, classical, observables. This is the essential step in Hilbert's Sixth Problem.

quant-ph↗

arXiv2007

La Derivada del Coseno

The simplest rigourous, non-circular proof that d(cos x) = -sen x. Some details omitted if the gap is intuitive, nevertheles, each gap is easily filled, rigourously. As Ehrenpreis and others have pointed out, the usual text book `proof' is circular, since it assumes the area of circular sectors is already known. Various Cours d'Analyse tomes (Hermite, Jordan, etc.) have correct proofs which, however, are more difficult.

math.GM↗

arXiv2006

Remarks on an attempted axiomatisation of Quantum Mechanics, due to Lucien Hardy, and Ten Theses on Hilbert's Sixth Problem and Quantum Measurement

From the standpoint of Hilbert's Sixth Problem, which is the axiomatisation of Physics, the famous paper of Lucien Hardy's, Quantum Theory from Five Reasonable Axioms, is not relevant. The present paper argues that Hardy does not give a physical definition of `limit', and if we assume the usual mathematical definition of limit of a sequence, he fails to define a sequence in physical terms to which the usual definition is applicable. We argue that one should not, in fact, try to define probability in terms of the usual notion of limit of a sequence of results of a measurement because of seemingly insurmountable difficulties in axiomatising the notion of function or sequence in this context. Von Plato's and the authour's work (see http:arxiv.org/abs/quant-ph/0502124 and euclid.unh.edu/~jjohnson/axiomatics.html for larger context and further references) on the definition of physical probability needs to be used in this context. We conclude with ten theses on quantum measurement, from the standpoint of the Hilbert problem.

quant-ph↗

arXiv2006

Re-formulation of combined system wave-function formalism

We introduce a formulation of combined systems in orthodox non-relativistic quantum mechanics, mathematically equivalent to the usual one. For context and larger issues, see http://euclid.unh.edu/~jjohnson/axiomatics.html and http://arxiv.org/quant-ph/0502124

math-ph↗

arXiv2005

Logical Structure of Physical Probability Assertions

A modification and generalisation of von Plato's fix of the frequency theory of probability is presented. It is thermodynamic in nature. Von Plato already fixed the logical circle in the frequency theory, we generalise his results to not necessarily ergodic systems of classical and quantum mechanics. This turns out to be precisely what is needed for the problem of Quantum Measurement and the problem of induction.

quant-ph↗

arXiv2005

Thermodynamic Limits, Non-commutative Probability, and Quantum Entanglement

We construct a rigourous model of quantum measurement. A two-state model of a negative temperature amplifier, such as a laser, is taken to a classical thermodynamic limit. In the limit, it becomes a classical measurement apparatus obeying the stochastic axioms of quantum mechanics. Thus we derive the probabilities from a deterministic Schroedinger's equation by procedures analogous to those of classical statistical mechanics. This requires making precise the notion of `macroscopic.'

quant-ph↗