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EUREKA Social and Humanities
Top-3 citing journals
E3S Web of Conferences
(5 citations)
Sustainability
(4 citations)
Lecture Notes in Networks and Systems
(2 citations)
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Vytautas Magnus University
(2 publications)
Bauman Moscow State Technical University
(1 publication)
Surgut State University
(1 publication)
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Publications found: 699
Quantum Field Theory
Wetterich C.
Quantum field theory emerges in a very natural and straightforward way from our general probabilistic setting. We have already seen that very simple unique jump chains or probabilistic cellular automata, like the transport automata, describe twodimensional quantum field theories for free massless fermions. Actually, it is the quantum field theory which arises directly at the most basic level. Single-particle quantum mechanics is obtained as a special case or approximation, rather than being the starting point, as in the historical development of quantum field theory. The generic situation for interesting probabilistic systems describes states with many particles, with a continuum limit admitting an infinite number.
Fundamental Probabilities
Wetterich C.
The advent of quantum mechanics has opened a probabilistic view on fundamental physics. It has come, however, with new concepts and rules like wave functions, non-commuting operators, and the rules for associating these quantities with observations. The unusual probabilistic features have opened many debates on their interpretation, as well as on suggestions for extensions of quantum mechanics.
Qubit Automaton
Wetterich C.
In the present section, we concentrate on quantum mechanics for a single qubit. A simple local chain with three classical Ising spins already realizes many characteristic features of quantum mechanics, such as non-commuting operators for observables, the quantum rule for computing expectation values, discrete measurement values corresponding to the spectrum of operators, the uncertainty principle, unitary evolution, and complex structure.
Quantum Mechanics
Wetterich C.
In this chapter, we discuss probabilistic automata that realize all features of quantum mechanics.We start with quantum mechanics for a two-state system or a single qubit. The quantum spin in an arbitrary direction is associated with a corresponding classical Ising spin. For the quantum subsystem, the deterministic evolution for the automaton results in the unitary evolution according to the von Neumann equation for the quantum density matrix. Suitable updating rules can realize an arbitrary Hamiltonian for a single qubit. Quantum mechanics for a single qubit is the extension of the quantum clock system to rotations in three-dimensional space or on the two-dimensional sphere.
Discussion and Conclusions
Wetterich C.
This book proposes a probabilistic formulation of the fundamental laws of nature. On a fundamental level, the description of the universe is entirely based on the concepts of classical statistics: observables that take definite values in the states of the system, probabilities for these states, and expectation values of observables computed from these probabilities. The overall probability distribution covers everything in the world, including all times, all locations, and all possible events.
Fermions
Wetterich C.
In this section we present a first simple classical statistical model that is equivalent to a quantum field theory. It is a generalized Ising model which describes free massless fermions in one time and one space dimension. Even though this two-dimensional model remains extremely simple, all aspects of the formalism of quantum mechanics are already visible. This concerns both the Schr¨odinger or von Neumann equation for a unitary evolution and the appearance of non-commuting operators for observables like position or momentum.
Fundamental Probabilism
Wetterich C.
The starting point of the present work is the assumption that the fundamental description of our world is probabilistic [25,52–54]. The basic objects for this description are probability distributions and observables. Deterministic physics arises as an approximation for particular cases. Our description of probabilities remains within the standard setting of classical statistics. No separate laws for quantum mechanics will be introduced. They follow from the classical statistical setting for particular classes of subsystems.
Probabilistic and Deterministic Evolution
Wetterich C.
Different types of evolution are distinguished by the way the probabilistic information propagates. This is reflected by the question of how boundary properties influence behavior in the bulk.
Entanglement in Classical and Quantum Statistics
Wetterich C.
Entanglement describes situations where two parts of a system are connected and cannot be separated. The properties in one part depend on the properties of the other part. The quantitative description of such situations is given by correlation functions. There is no conceptual difference between entanglement in classical statistics and entanglement in quantum mechanics [185]. In this chapter, we will explicitly construct probabilistic automata that realize the maximally entangled state of a twoqubit quantum system.
Embedding Quantum Mechanics in Classical Statistics
Wetterich C.
It has often been asserted that an embedding of quantum mechanics in classical statistics is not possible. The present work demonstrates by explicit examples that this claim is not justified.
Continuous Classical Variables
Wetterich C.
The use of continuous classical variables brings us closer to the quantum particle, for which we have argued that it involves an infinite number of degrees of freedom. If we allow for an arbitrary orthogonal evolution of the classical wave function, we can find a bit-quantum map which maps this to the evolution of a quantum particle in a potential according to the usual Schrödinger equation.
Subsystems
Wetterich C.
Subsystems are a central ingredient for a probabilistic description of the world. For most practical purposes, one does not want to deal with the overall probability distribution for all events in the universe from the past to the future. One rather concentrates on subsystems.
Probabilistic Time
Wetterich C.
Time is a fundamental concept in physics. It is the first structure among observables that we will discuss. Rather than being postulated as an “a priori concept” with physics formulated in a pregiven time and space, probabilistic time is a powerful concept to order and organize observables. There is no time outside the correlations for the observables of the statistical system.
Correlated Computing
Wetterich C.
As already emphasized, the crucial feature of quantum computing is the large number of correlations between the associated classical bits. These correlations result from the quantum constraint of positivity of the density matrix for the quantum subsystem.
Introduction
Wetterich C.
Since the discovery of quantum mechanics, probabilities have play a central role in fundamental physics. The mathematical formalism of quantum mechanics makes statements about expectation values of observables. The expectation value describes the mean value of an observable for an ensemble of many identical measurements. It is a genuinely probabilistic concept since every possible measurement value of the observable is weighted by the probability of finding it. The quantum laws for the computation of these probabilities are often felt to be mysterious.
