The qualitatively new concept of dynamic complexity in quantum mechanics is based on a new paradigm appearing within a nonperturbational analysis of the Schroedinger equation for a generic Hamiltonian system. The unreduced analysis explicitly provides the complete, consistent solution as a set of many incompatible components ('realisations') which should permanently and probabilistically replace one another, since each of them is 'complete' in the ordinary sense. This discovery leads to the universally applicable concept of dynamic complexity and self-consistent, realistic resolution of the stagnating problems of quantum chaos, quantum measurement, indeterminacy and wave reduction. The peculiar, 'mysterious' character of quantum behaviour itself is seen now as a result of a dynamically complex, intrinsically multivalued behaviour of interacting fields at the corresponding lowest levels of the (now completely causal) structure of reality. Incorporating the results of the canonical theories as an over-simplified limiting case, this new approach urgently needs support, since its causality and completeness are directly extendible to arbitrary cases of complex behaviour of real systems, in sharp contrast to the dominating inefficient empiricism of 'computer experimentation' with primitive mechanistic (i. e. dynamically single-valued) 'models' of the irreducibly multivalued reality.
The qualitatively new concept of dynamic complexity in quantum mechanics is based on a new paradigm appearing within a nonperturbational analysis of the Schroedinger equation for a generic Hamiltonian system. The unreduced analysis explicitly provides the complete, consistent solution as a set of many incompatible components ('realisations') which should permanently and probabilistically replace one another, since each of them is 'complete' in the ordinary sense. This discovery leads to the universally applicable concept of dynamic complexity and self-consistent, realistic resolution of the stagnating problems of quantum chaos, quantum measurement, indeterminacy and wave reduction. The peculiar, 'mysterious' character of quantum behaviour itself is seen now as a result of a dynamically complex, intrinsically multivalued behaviour of interacting fields at the corresponding lowest levels of the (now completely causal) structure of reality. Incorporating the results of the canonical theories as an over-simplified limiting case, this new approach urgently needs support, since its causality and completeness are directly extendible to arbitrary cases of complex behaviour of real systems, in sharp contrast to the dominating inefficient empiricism of 'computer experimentation' with primitive mechanistic (i. e. dynamically single-valued) 'models' of the irreducibly multivalued reality.
Two major deviations from causality in the existing formulations of quantum mechanics, related respectively to quantum chaos and indeterminate wave reduction, are eliminated within the new, universal concept of dynamic complexity. The analysis involves a new paradigm for description of a system with interaction, the principle of dynamic multivaluedness (redundance), and the ensuing concept of the fundamental dynamic uncertainty. It is shown that both the wave reduction and truly unpredictable (chaotic) behaviour in quantum systems can be completely and causally understood as a higher sublevel of the same dynamic complexity that provides the causally complete picture of the unified wave-particle duality and relativity at its lowest level (quant-ph/9902015,16). The presentation is divided into five parts. The first three parts deal with intrinsic randomness in Hamiltonian (isolated) quantum systems as the basic case of dynamical chaos. In the last two parts a causal solution to the problem of quantum indeterminacy and wave reduction is proposed. Part I introduces the method of effective dynamical functions as a generalisation of the optical potential formalism. The method provides a legal transformation of the Schroedinger equation revealing the hidden multivaluedness of interaction process, i. e. its self-consistent, dynamical splitting into many equally real, but mutually incompatible branches, called 'realisations'. Each realisation incorporates the usual "complete" set of eigenfunctions and eigenvalues for the entire problem. The method is presented in detail for the Hamiltonian system with periodic (not small) perturbation, both in its time-independent and time-dependent versions.
The fundamental impasses and ruptures in various domains of the canonical, unitary science, or the 'end of science', become the more and more evident. The natural unity of being is recovered within a universal nonperturbative method leading to the dynamic redundance paradigm. It is shown that the unreduced behaviour of any nontrivial (isolated) system includes many equally real, but incompatible dynamic regimes, each of them being roughly equivalent to an ordinary 'complete' solution of the unitary science. Therefore the regimes should 'spontaneously' and randomly replace one another, which provides a universal, purely dynamic origin of randomness. The discovered dynamic redundance leads to the universal, reality-based concept of dynamic complexity and its permanently developing hierarchical structure, alias the world. Its lowest levels give the causally complete mechanics of dynamically quantized elementary fields, extending the double solution found by Louis de Broglie. One obtains a physically complete solution for the 'mysteries' of quantum mechanics, unifyng it with the extended, causal versions of relativity (emergent space and time), quantum and classical gravity (dynamical mass), field theory (electric charge and spin), particle physics, and cosmology. The same key features of the unreduced dynamic complexity determine behaviour of any system at higher levels of complexity described by a case of the single, universal equation in its two related versions, the extended Lagrange-Hamilton (trajectorial) and Schroedinger (distributional) equations. The end of the unitary science opens the renaissance of the ultimately complete and universal understanding initiated by Rene Descartes (but then mechanistically falsified), which is confirmed by many sound, practically important results.
The true dynamical randomness is obtained as a natural fundamental property of deterministic quantum systems. It provides quantum chaos passing to the classical dynamical chaos under the ordinary semiclassical transition, which extends the correspondence principle to chaotic systems. In return one should accept the modified form of quantum formalism (presented by the Schroedinger equation) which, however, does not contradict the ordinary form and the main postulates of quantum mechanics. It introduces the principle of the fundamental dynamic multivaluedness (redundance) extending the quantum paradigm to complex dynamical behaviour. Moreover, a causal solution to the well-known problems of the foundation of quantum mechanics, those of quantum indeterminacy and wave reduction, is also found using the same method. The concept of the fundamental dynamic uncertainty thus established is universal in character and provides a unified scheme of the complete description of arbitrary complex system of any origin (physics/9806002). This scheme incorporates, in particular, universal definitions of complexity, (non)integrability, and general solution, as well as the physically complete notion of probability. One obtains thus a self-consistent hierarchic picture of the world characterised by a (high) non-zero complexity and containing the intrinsic causal randomness, where the causally extended quantum mechanics can be consistently interpreted as several lowest levels of complexity (quant-ph/990215,16).
The universal dynamic uncertainty, discovered in Parts I and II of this series of papers for the case of Hamiltonian quantum systems, is further specified to reveal the hierarchical structure of levels of dynamically redundant 'realisations' which takes the form of the intrinsically probabilistic 'fundamental dynamical fractal' of a problem and determines fractal character of the observed quantities. This intrinsic fractality is obtained as a natural, causally derived property of dynamic behaviour of a system with interaction and the corresponding complete solution. Every branch of the fundamental dynamical fractal of a problem, as well as the probability of its emergence, can be obtained within the extended nonperturbative analysis of the main dynamic equation (Schroedinger equation in our case), contrary to basically restricted imitations of fractality within the canonical, single-valued approach. The results of the dynamical chaos analysis in Hamiltonian quantum systems, Parts I-III, are then subjected to discussion and generalisation. The physical origins of the dynamic uncertainty are analysed from various points of view. The basic consequences, involving essential extension of the conventional, unitary (= single-valued) quantum mechanics, are summarised. Finally, we emphasize the universal character of the emerging notions of dynamic multivaluedness (or redundance), causal randomness (or dynamic uncertainty), first-principle probability, (non)integrability, general solution, and physical complexity applicable to real dynamical systems of any kind.
The physical consequences of the analysis performed in Parts I-IV are outlined within a scheme of the complete quantum (wave) mechanics called quantum field mechanics and completing the original ideas of Louis de Broglie by the dynamic complexity concept. The total picture includes the formally complete description at the level of the "average" wave function of Schroedinger type that shows dynamically chaotic behaviour in the form of either quantum chaos (Parts I-III), or quantum measurement (Part IV) with causal indeterminacy and wave reduction. This level is only an approximation, though often sufficient, to a lower (and actually the lowest accessible) level of complexity containing the causally complete version of the unreduced, nonlinear "double solution" proposed by Louis de Broglie. The extended 'double solution with chaos' describes the state of a nonlinear material field and includes the unstable high-intensity "hump" moving chaotically within the embedding smooth wave (quant-ph/9902015,16). The involvement of chaos causally understood within the same concept of dynamic complexity (multivaluedness) provides, at this lower level, de Broglie's "hidden thermodynamics" now, however, without the necessity for any real "hidden thermostat". The chaotic reduction of the "piloting" Schroedinger wave, at the higher sublevel of complexity, conforms with the detailed 'wandering' of the virtual soliton at the lower sublevel. The proposed dynamic multivaluedness (redundance) paradigm serves as the basis for a self-consistent hierarchic picture of the world with a (high) non-zero complexity (and thus irreducible randomness), where the complete extension of quantum mechanics is causally interpreted as several lowest levels of complexity.
Following Max Planck's hypothesis of quanta (quant-ph/0012069) and the matter wave idea of Louis de Broglie (quant-ph/9911107), Erwin Schroedinger proposed, at the beginning of 1926, the concept of wavefunction and wave equation for it. Though endowed with a realistic undular interpretation by its father, the wavefunction could not be considered as a real "matter wave" and has been provided with only abstract, formally probabilistic interpretation. In this paper we show how the resulting "mysteries" of usual theory are solved within the unreduced, dynamically multivalued description of the underlying, essentially nonlinear interaction process (quant-ph/9902015, quant-ph/9902016), without artificial modification of the Schroedinger equation. The latter is rigorously derived instead as universal expression of unreduced interaction complexity. Causal, totally realistic wavefunction is obtained as a dynamically probabilistic intermediate state of a simple system with interaction performing dynamically discrete transitions between its localised, incompatible "realisations" ("corpuscular" states). Causal wavefunction and Schroedinger equation are then extended to arbitrary level of world dynamics. We outline some applications of the obtained causal description, such as genuine quantum chaos (quant-ph/9511034-36) and realistic quantum devices (physics/0211071), and emphasize the basic difference of the proposed dynamically multivalued theory from dynamically single-valued imitations of causality and complexity. The causally complete wavefunction concept, representing the unified essence of unreduced (multivalued) complex dynamics, provides a clear distinctive feature of realistic science, absent in any its unitary imitation.