Statistical dynamics of multiply-periodic systems

Summary form only given, as follows. Sheet electron beams have the potential to make possible higher power-sources of microwave radiation due to their ability to transport high currents, at reduced current densities, through a single narrow RF interaction structure. Previous investigations have indicated the feasibility for laboratory formation of an elliptical sheet beam using magnetic quadrupoles and a Pierce gun pencil-beam source. The configuration exhibits several unique physical features of phase space evolution not observed in more conventional, paraxial beam transport systems. Previous numerical simulations using MAGIC3-D indicate the r.m.s. phase space volume occupied by the sheetbeam increased longitudinally, which contradicts the principle of phase space conservation for a conservative system (Liouville's theorem). The explanation is that is Liouville's theorem does not strictly apply to r.m.s. emittance calculations and some alternative phase space measure might provide closer agreement with Liouville's theorem. The developed phase space measure is non-statistical and uses principles from mathematical morphology. We discuss and compare the results of numerical simulations using the TRACE3-D and MAGIC3-D codes. In addition, we discuss the morphological algorithm for computing phase space area and compare its results with that of the r.m.s. formalism for calculating phase space area.

Calculation of the VIB-rotational state density of polyatomic molecules by adiabatic switching

A wave dynamics of fields φ(p, q; t) ∈ L2(Γ) over the phase space Γ(p, q) of a classical system 𝒮 is derived from the Liouville theorem. We define the energy contained in a given field φ(p, q; t). We show that for a special class of fields, selected on physical grounds, the energy spectrum is given by a time-independent Schrödinger equation. This allows us to associate with 𝒮 an ordinary quantum system Q such that the values of the quantized energy coincide for the fields in the phase space of 𝒮 and for Q. Then we make use of Wiener's stochastic integral based on the theory of Brownian motion to derive probabilities which are the same as those one would obtain through Born's statistical postulate of quantum theory. From this it follows that we can regard normalized fields φ(p, q; t) as ``probability amplitudes'' leading to a probability density function ρ(p, q; t) = φφ* in the sense of Gibbs' statistical mechanics. Our work therefore appears as a bridge between a statistical theory (in the sense of Gibbs) of a mechanical system 𝒮 and the usual quantum theory of the related quantum system Q.

This Element explores the formal and conceptual foundations of phase space formulations of classical and quantum mechanics. It provides an overview of the core mathematical and physical content of Hamiltonian mechanics, stochastic phase space mechanics, contact Hamiltonian mechanics, and open and closed quantum mechanics on phase space. The formal material is unified via three interpretative themes relating to structured possibility spaces, Liouville's theorem and its failure, and the classical and quantum notions of open and closed systems. This Element book is intended for researchers and graduate students in the philosophy and foundations of physics with an interest in the conceptual foundations of physical theory.

The correspondence principle plays a fundamental role in quantum mechanics, which naturally leads us to inquire whether it is possible to find or determine close classical analogs of quantum states in phase space—a common meeting point to both classical and quantum density statistical descriptors. Here, this issue is tackled by investigating the behavior of classical analogs arising upon the removal of all interference traits displayed by the Wigner distribution functions associated with a given pure quantum state. Accordingly, the dynamical evolution of the linear and von Neumann entropies is numerically computed for a continuous-variable bipartite system, and compared with the corresponding classical counterparts, in the case of two quartic oscillators nonlinearly coupled under regular and chaos conditions. Three quantum states for the full system are considered: a Gaussian state, a cat state, and a Bell-type state. By comparing the quantum and classical entropy values, and particularly their trends, it is shown that, instead of entanglement production, such entropies rather provide us with information on the system (either quantum or classical) delocalization. This gradual loss of information translates into an increase in both the quantum and the classical realms, directly connected to the increase in the correlations between both parties’ degrees of freedom which, in the quantum case, is commonly related to the production of entanglement.

Until quite recently, all phenomena in the world were separated into deterministic and random. The theory of differential equations, mathematical physics, classical mechanics, and electro-dynamics studied the former; probability theory, the theory of random processes, statistical physics, and quantum mechanics, the latter. This sharp borderline seemed unquestionable, and raised no doubts, though certain discrepancies did arise: The probabilistic nature of classical statistical physics laws was not justified accurately, and was accounted for by the very large number of particles and degrees of freedom.

In this paper, I review one of the most important and interesting parts of my new book “Fuzzy Systems to Quantum Mechanics” (see Ref. 1). Several conclusions in this part are worth introducing here. First of all, the motion of a mass point in classic mechanics has also waviness and the wave function of the motion of a mass point is composed of wave functions of countably infinite microscopic particles. Secondly, based on the waviness of the motion of a mass point we surely know the new conclusion described as the wave-mass-point dualism in classic mechanics. And thirdly, by using the closed relation between the wave-mass-point dualism in classic mechanics and the wave-particle dualism in quantum mechanics, unified theory of classic mechanics and quantum mechanics is naturally formed.

A physically relevant unitary irreducible non-projective representation of the Galilei group is possible in the Koopman–von Neumann formulation of classical mechanics. This classical representation is characterized by the vanishing of the central charge of the Galilei algebra. This is in contrast to the quantum case where the mass plays the role of the central charge. Here we show, by direct construction, that classical mechanics also allows for a projective representation of the Galilei group where the mass is the central charge of the algebra. We extend the result to certain kind of quantum–classical hybrid systems.

Classical statistical mechanics, as well as quantum statistical mechanics, are both acknowledgedly statistical theories even if one subscribes to the conventional point of view about the intrinsically disparate natures of classical and quantum mechanics, respectively. Yet, the conventional mathematical frameworks for these two theories are remarkably dissimilar even at the most fundamental level of description of states of ensembles: in the classical case states are described by distribution functions ƒ(q, p) on phase space, whereas in the quantum case they are given by density operators ρ.

A plasma consists of charged particles moving relatively freely, but interacting through the electric and magnetic fields produced by each particle. It is well known that all except a very small (but nevertheless important) fraction of the detectable universe is a plasma. Most plasma physics can be expressed in terms of five fairly straightforward equations, the four Maxwell equations for electromagnetism and the equation of motion for charged particles in electromagnetic fields. Why then is plasma physics such a mathematically difficult subject and why is numerical simulation so important? The most basic reason is that any experimental plasma contains a very large number of particles, 1018–1020 in a typical fusion experiment. Moreover, particles in plasmas have a tendancy to act independently of each other. In conventional hydrodynamics, large numbers of particles can be described by a few simple numbers such as their temperature, density and mean velocity. Sometimes we can do this with a plasma but even then we have a few more numbers to include such as current and charge density, along with the electromagnetic fields. More usually, plasmas are not so simply described. Plasmas are not usually Maxwellian and we have to think in terms of distribution functions in phase space. Whenever we can, we model the non—Maxwellian nature of the plasma by transport models, e.g. for heat flow, but this is a specialised skill in itself. When one considers that conventional hydrodynamics is very complex and involves advanced numerical simulation, it is hardly surprising that simulation plays such an important role in plasma physics.

Classical statistical average values are generally generalized to average values of quantum mechanics. It is discovered that quantum mechanics is a direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, and the general classical statistical uncertain relation is generally generalized to the quantum uncertainty principle; the two lost conditions in classical uncertain relation and quantum uncertainty principle, respectively, are found. We generally expound the relations among the uncertainty principle, singularity and condensed matter stability, discover that the quantum uncertainty principle prevents the appearance of singularity of the electromagnetic potential between nucleus and electrons, and give the failure conditions of the quantum uncertainty principle. Finally, we discover that the classical limit of quantum mechanics is classical statistical mechanics, the classical statistical mechanics may further be degenerated to classical mechanics and we discover that merely stating that the classical limit of quantum mechanics is classical mechanics is a mistake. As application examples, we deduce both the Schrödinger equation and the state superposition principle, and deduce that there exists a decoherent factor from a general mathematical representation of the state superposition principle; the consistent difficulty between statistical interpretation of quantum mechanics and determinant property of classical mechanics is overcome.

The irreversible process of ‘wave packet collapse’ in quantum measurement contradicts the reversible characteristics of the Schrödinger equation and the entire quantum mechanics. This contradiction is actually the origin of measurement problems in quantum mechanics. How to resolve this contradiction is related to the understanding of some philosophical concepts. The Copenhagen School embraces the reversible/irreversible contradictions in quantum measurement through the principle of complementarity. Both von Neumann's measurement hypothesis and Everett's many world explanation adhere to the ‘universal position of the Schrödinger equation’, but due to their detachment from the thermodynamic mechanism analysis of quantum measurement processes, they fall into the dilemma of subjectivism or possible state materialization in ontology. Decoherence theory is a natural extension of dissipative structure theory in quantum mechanics, which helps to solve quantum measurement problems by combining the study of classical conversion mechanisms between reversibility and irreversibility in statistical mechanics. Due to the common statistical roots of the time arrow problem in classical mechanics and quantum mechanics, decoherence theory cannot be seen as a measurement theory that fully implements the ‘universal Schrödinger position’. Many contradictions in quantum mechanics may need to be cleverly resolved through various contradictions in classical physics.

The Weyl representation in classical and quantum mechanics

2 - Traditional Quantum Mechanics

We examine the relationship between the mathematical structures of classical mechanics, quantum mechanics, and special relativity, with a view toward building a consistent framework for all three. The usual idea of ’’canonical quantization,’’ with its emphasis on the transition from functions over classical phase space to operators, appears to be inconsistent with relativistic covariance. On the other hand, the spectral condition of relativistic quantum mechanics, which is merely a covariant statement that the energy is nonnegative, naturally leads to the construction of a covariant extension of classical phase space. By giving up the idea of sharp ’’localizability’’ in space—known to be at odds with covariance— and adopting instead a notion of ’’soft’’ localizability in phase space, a consistent theory of relativistic quantum mechanics is seen to emerge whose structure naturally incorporates the classical sympletic geometry as well. Furthermore, the new theory deals directly and covariantly with extended particles rather than point particles and is free of the various inconsistencies known to plague the usual theory. The new notion of ’’microlocality’’ in phase space leads to a new form of gauge freedom which is similar to the usual one but simpler and more powerful, using methods of complex analysis. The phase-space version of Yang–Mills theory is worked out.