Does quantum chaos exist?If it exists, what is it?If it does not, what was it supposed to have been?Why should we associate chaos with a "third revolution in physics" if it fails to be expressible in terms of quantum mechanics, our most fundamental theory of physical reality?Is there something wrong with quantum mechanics?Or is chaos merely a mathematical construct relevant only for models in classical mechanics?Is the human mind doomed to interpret and understand quantum mechanics in classical terms?The list of unanswered questions in quantum chaos research itself shows symptoms of the phenomenon it attempts to grasp. 1 How do we recognize classical Hamiltonian chaos?We have dealt with this question in our two previous columns.2,3 In Part I we discussed the implications of integrability and non-integrability for the phase-space trajectories of classical Hamiltonian systems with two degrees of freedom.We described the method of Poincaré surfaces of section as a convenient and striking discriminant between the two possibilities.In Part II we searched for an alternate mode of representation for classical Hamiltonian systems, a mode that is an equally powerful indicator of chaos, but one that can be more directly translated into quantum mechanics than the Poincaré map.We proposed a representation based on the construction of classical invariants via time averages of dynamical variables, a representation that can be employed under very general circumstances including integrable and nonintegrable models.In this column we shall construct quantum invariants, demonstrate the impact of non-integrability on these quantities, and discuss their properties in relation to their classical counterparts.In a future column we intend to introduce quantum Poincaré surfaces of section for stationary states and bring full circle our survey of quantum and classical Hamiltonian chaos.Consider a quantized integrable model.Its eigenstates are labeled by quantum numbers.There are as many quantum numbers as there are degrees of freedom.Each quantum number represents a quantized action.In the still evolving language of quantum chaos, quantum numbers are said to have been "lost" in the transition from a quantum system whose classical counterpart is integrable to one whose counterpart is chaotic.The broken, or lost, tori of classical chaos become lost quantum numbers of quantum chaos.As we shall see, the loss of quantum numbers is vividly mirrored in the behavior of the quantum invariants.We begin by describing the construction of the quantum invariants for a
Does quantum chaos exist?If it exists, what is it?If it does not, what was it supposed to have been?Why should we associate chaos with a "third revolution in physics" if it fails to be expressible in terms of quantum mechanics, our most fundamental theory of physical reality?Is there something wrong with quantum mechanics?Or is chaos merely a mathematical construct relevant only for models in classical mechanics?Is the human mind doomed to interpret and understand quantum mechanics in classical terms?The list of unanswered questions in quantum chaos research itself shows symptoms of the phenomenon it attempts to grasp. 1 How do we recognize classical Hamiltonian chaos?We have dealt with this question in our two previous columns.2,3 In Part I we discussed the implications of integrability and non-integrability for the phase-space trajectories of classical Hamiltonian systems with two degrees of freedom.We described the method of Poincaré surfaces of section as a convenient and striking discriminant between the two possibilities.In Part II we searched for an alternate mode of representation for classical Hamiltonian systems, a mode that is an equally powerful indicator of chaos, but one that can be more directly translated into quantum mechanics than the Poincaré map.We proposed a representation based on the construction of classical invariants via time averages of dynamical variables, a representation that can be employed under very general circumstances including integrable and nonintegrable models.In this column we shall construct quantum invariants, demonstrate the impact of non-integrability on these quantities, and discuss their properties in relation to their classical counterparts.In a future column we intend to introduce quantum Poincaré surfaces of section for stationary states and bring full circle our survey of quantum and classical Hamiltonian chaos.Consider a quantized integrable model.Its eigenstates are labeled by quantum numbers.There are as many quantum numbers as there are degrees of freedom.Each quantum number represents a quantized action.In the still evolving language of quantum chaos, quantum numbers are said to have been "lost" in the transition from a quantum system whose classical counterpart is integrable to one whose counterpart is chaotic.The broken, or lost, tori of classical chaos become lost quantum numbers of quantum chaos.As we shall see, the loss of quantum numbers is vividly mirrored in the behavior of the quantum invariants.We begin by describing the construction of the quantum invariants for a
This paper discusses the problem of finding and defining chaos in quantum mechanics. While chaotic time evolution appears to be ubiquitous in classical mechanics, it is apparently absent in quantum mechanics in part because for a bound, isolated quantum system, the evolution of its state is multiply periodic. This has led a number of investigators to search for semiclassical signatures of chaos. Here I am concerned with the status of semiclassical mechanics as a distinct third theory of the asymptotic domain between classical and quantum mechanics. I discuss in some detail the meaning of such crucial locutions as the "classical counterpart to a quantum system" and a quantum system's "underlying classical motion". A proper elucidation of these concepts requires a semiclassical association between phase space surfaces and wave-functions. This significance of this association is discussed in some detail.
Through semiclassical methods the subject of quantum chaos motivates and depends on Hamiltonian chaos research. Presented here is a selection of some of the most important Hamiltonian chaos topics that in this way get directly related to any of a variety of quantum chaos research problems. The overall emphasis is on intuitive explanations and illustrations of various ideas with the references containing more mathematically rigorous expositions. The chapter begins with a description of various useful theoretical and computational tools of chaos research, e.g.~surfaces of section, paradigms of chaos, stability analysis, and symbolic dynamics... This is followed by discussions regarding the geometry of chaos, how chaotic systems respond to perturbations, and the complexification of Hamiltonian dynamics. Most of the contents concern topics that are long established and well known by now. The exceptions involve complications that arise from the complexification of phase space variables and time. There are relatively recent results summarized regarding the appropriate branches of determinant phases and how to distinguish complex trajectories that are associated with classically allowed processes versus those associated with non-allowed ones.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.
The chaos exhibited by a quantum mechanical system is revealed in various aspects such as level spacing, avoiding of level crossing on changing a parameter value, irregular nodal lines of wave function, random matrix and so on. On the other hand since the wave function, the propagator or any physical quantity represented by the average of a phase function is usually quasi-periodic in time development, no exponential divergence of its value takes place between the neighboring two initial values. Furthermore the energy of a kicked oscillator ceases to increase after a certain period of time. These observations lead one to suspect the existence of chaos in quantal systems. It is shown that the nature of quasi-periodicity in quantum system is shared with the distribution function of classical hamiltonian system. The action integral which appears in the formulation of the propagator is the key quantity which can connect the classical and quantal systems, and is considered as serving to search for the origin of quantum chaos. The roles of the action integral in the determination of the energy levels and the behavior of the wave function are discussed.
The intrinsic multivaluedness of interaction process, revealed in Part I of this series of papers, is interpreted as the origin of the true dynamical (in particular, quantum) chaos. The latter is causally deduced as unceasing series of transitions, dynamically probabilistic by their origin, between the equally real, but incompatible 'realisations' (modes of interaction) of a system. The obtained set of realisations form the causally derived, intrinsically complete "space of events" providing the crucial extension of the notion of probability and the method of its first-principle calculation. The fundamental dynamic uncertainty thus revealed is specified for Hamiltonian quantum systems and applied to quantum chaos description in periodically perturbed systems. The ordinary semiclassical transition in our quantum-mechanical results leads to exact reproduction of the main features of chaotic behaviour of the system known from classical mechanics, which permits one to "re-establish" the correspondence principle for chaotic systems (inevitably lost in any their conventional, single-valued description). The causal dynamical randomness in the extended quantum mechanics is not restricted, however, to semiclassical conditions and generically occurs also in essentially quantum regimes, even though partial "quantum suppression of chaos" does exist and is specified in our description, as well as other particular types of the quantum (truly) chaotic behaviour.
Book of Physics — On Difference Tractatus Principio, Volume II What if the fundamental entities of physics are not the foundations of reality, but the results of the relations through which reality becomes measurable? Book of Physics, the second volume of Tractatus Principio — On Difference by C.H.F. Warnke, extends the relational foundation developed in Volume I into the domain of physical reality. The work challenges a fundamental assumption underlying many physical descriptions: that mathematical abstractions such as points, particles, fields, and measurable quantities must correspond to independently existing constituents of the world. Its central distinction is between the mathematical representation of a physical phenomenon and the ontological interpretation of that representation. Beginning with a critical examination of classical mechanics, Cartesian geometry, Newtonian point-mass models, relativity, and quantum physics, the book investigates how relational descriptions can become interpreted as properties of self-contained physical objects. The work proposes a reversal of this perspective: physical objects are not the original bearers of relations. Rather, their determinacy emerges through relational structures, interactions, and the conditions under which they are observed and measured. Building upon the mathematical language of difference, represented by |, and its self-referential unfolding through Δ, the volume develops alternative interpretations of space, time, energy, mass, gravitation, and light. It examines dimensional measurement, relational invariance, the proposed significance of 39.19 within its own formal framework, and the organization of physical appearances through recursive structures and degrees of freedom. Gravitation is reconsidered through a relational shell model. Motion and time are investigated through the concept of complete relational states, or Frozen Frames, whose differences become readable as temporal succession. A central part of the work concerns measurement itself. Rather than treating measurement as the passive discovery of an already fully determined property, the book examines the relationship between observer, apparatus, registration, and interpretation. A measuring apparatus is understood as a materially realized question: its construction determines which differences can become detectable and how their results can be interpreted. From this perspective, particles and other physical entities are investigated as stable forms of relational determination rather than necessarily self-standing objects independent of every measurement context. The later chapters explore the uniformity of relational processes across different scales, including a reinterpretation of Bose–Einstein condensation, before returning to the distinction between scientific explanation and the conceptual images through which reality is understood. These proposals constitute an alternative theoretical interpretation within the framework of Tractatus Principio. Their physical validity is not established merely by the mathematical or conceptual arguments presented. The investigation ultimately reaches the observer himself. If every measurement is relational, the question arises of how the observer becomes capable of recognizing his own position within those relations. This question forms the transition to Volume III, the Book of Self-Knowledge. The central question of Volume II is not merely what physics measures, but what makes a physical reality measurable as something determinate in the first place.