The principle of least action, which has so successfully been applied to diverse fields of physics looks back at three centuries of philosophical and mathematical discussions and controversies. They could not explain why nature is applying the principle and why scalar energy quantities succeed in describing dynamic motion. When the least action integral is subdivided into infinitesimal small sections each one has to maintain the ability to minimize. This however has the mathematical consequence that the Lagrange function at a given point of the trajectory, the dynamic, available energy generating motion, must itself have a fundamental property to minimize. Since a scalar quantity, a pure number, cannot do that, energy must fundamentally be dynamic and time oriented for a consistent understanding. It must have vectorial properties in aiming at a decrease of free energy per state (which would also allow derivation of the second law of thermodynamics). Present physics is ignoring that and applying variation calculus as a formal mathematical tool to impose a minimization of scalar assumed energy quantities for obtaining dynamic motion. When, however, the dynamic property of energy is taken seriously it is fundamental and has also to be applied to quantum processes. A consequence is that particle and wave are not equivalent, but the wave (distributed energy) follows from the first (concentrated energy). Information, provided from the beginning, an information self-image of matter, is additionally needed to recreate the particle from the wave, shaping a “dynamic” particle-wave duality. It is shown that this new concept of a “dynamic” quantum state rationally explains quantization, the double slit experiment and quantum correlation, which has not been possible before. Some more general considerations on the link between quantum processes, gravitation and cosmological phenomena are also advanced.
The principle of least action, which has so successfully been applied to diverse fields of physics looks back at three centuries of philosophical and mathematical discussions and controversies. They could not explain why nature is applying the principle and why scalar energy quantities succeed in describing dynamic motion. When the least action integral is subdivided into infinitesimal small sections each one has to maintain the ability to minimize. This however has the mathematical consequence that the Lagrange function at a given point of the trajectory, the dynamic, available energy generating motion, must itself have a fundamental property to minimize. Since a scalar quantity, a pure number, cannot do that, energy must fundamentally be dynamic and time oriented for a consistent understanding. It must have vectorial properties in aiming at a decrease of free energy per state (which would also allow derivation of the second law of thermodynamics). Present physics is ignoring that and applying variation calculus as a formal mathematical tool to impose a minimization of scalar assumed energy quantities for obtaining dynamic motion. When, however, the dynamic property of energy is taken seriously it is fundamental and has also to be applied to quantum processes. A consequence is that particle and wave are not equivalent, but the wave (distributed energy) follows from the first (concentrated energy). Information, provided from the beginning, an information self-image of matter, is additionally needed to recreate the particle from the wave, shaping a “dynamic” particle-wave duality. It is shown that this new concept of a “dynamic” quantum state rationally explains quantization, the double slit experiment and quantum correlation, which has not been possible before. Some more general considerations on the link between quantum processes, gravitation and cosmological phenomena are also advanced.
Abstract The uncertainty principle is a fundamental principle of quantum mechanics, but its exact mathematical expression cannot obtain correct results when used to solve theoretical problems such as the energy levels of hydrogen atoms, one-dimensional deep potential wells, one-dimensional harmonic oscillators, and double-slit experiments. Even after approximate treatment, the results obtained are not completely consistent with those obtained by solving Schrödinger’s equation. This indicates that further research on the uncertainty principle is necessary. Therefore, using the de Broglie matter wave hypothesis, we quantize the action of an elementary particle in natural coordinates and obtain the quantization condition and a new deterministic relation. Using this quantization condition, we obtain the energy level formulas of an elementary particle in different conditions in a classical way that is completely consistent with the results obtained by solving Schrödinger’s equation. A new physical interpretation is given for the particle eigenfunction independence of probability for an elementary particle: an elementary particle is in a particle state at the space-time point where the action is quantized, and in a wave state in the rest of the space-time region. The space-time points of particle nature and the wave regions of particle motion constitute the continuous trajectory of particle motion. When an elementary particle is in a particle state, it is localized, whereas in the wave state region, it is nonlocalized.
Abstract We show that the formulations of non-relativistic quantum mechanics can be derived from an extended least action principle. The principle can be considered as an extension of the least action principle from classical mechanics by factoring in two assumptions. First, the Planck constant defines the minimal amount of action a physical system needs to exhibit during its dynamics in order to be observable. Second, there is constant vacuum fluctuation along a classical trajectory. A novel method is introduced to define the information metrics to measure additional observability due to vacuum fluctuations, which is then converted to an additional action through the first assumption. Applying the variational principle to minimize the total actions allows us to recover the basic quantum formulations including the uncertainty relation and the Schrödinger equation in the position representation. In the momentum representation, the same method can be applied to obtain the Schrödinger equation for a free particle while further investigation is still needed for a particle with an external potential. Furthermore, the principle brings in new results on two fronts. At the conceptual level, we find that the information metrics for vacuum fluctuations are responsible for the origin of the Bohm quantum potential. Even though the Bohm potential for a bipartite system is inseparable, the underlying vacuum fluctuations are local. Thus, inseparability of the Bohm potential does not justify a non-local causal relation between the two subsystems. At the mathematical level, quantifying the information metrics for vacuum fluctuations using more general definitions of relative entropy results in a generalized Schrödinger equation that depends on the order of relative entropy. The extended least action principle is a new mathematical tool. It can be applied to derive other quantum formalisms such as quantum scalar field theory.
The principle of least action is known to apply to many domains of Physics. Yet its justifications and derivations usually come from archaic classical principles like the Fermat or Maupertuis principles, or mathematical analytical frameworks used for classical mechanics like the Lagrange, Hamilton and D’Alembert formalisms. As is well known, it is widely applicable to other domains, without necessarily a clear justification for it, especially when applied to Physics where Lagrangians or Hamiltonians may not be defined or do not exist. In addition, the Feynman Path Integral formulation of Quantum Mechanics gave it a second life, and another way to justify its validity not only in classical, but also in quantum Physics. However, the conventional logic to reach these conclusions is at time circular, and the principle is applied without proof that it is applicable: it just works and everybody does it. It is this applicability that led us in the past to treat the Action path integral as God’s equation.The paper justifies the principle of least action and Feynman’s path integrals through different physical reasoning instead of the traditional mathematical formalism, and on that basis we confirm its validity in most situations, even if Hamiltonians or Lagrangians may be shaky to define, or the system may be dissipative. We also explain when, and why, sometimes, it may not conventionally apply. The derivation uncovers different links between Entropy and Action, which also opens a different way to look at the origin of the probability distribution of even just a single lone particle; especially when complemented with the W-type multi-fold hypothesis, and makes some considerations on Feynman diagrams, instantons, renormalons and resurgence. Also discontinuities in the paths of the path integrals imply that spacetime is discrete, and non-commutative, and supersymmetry in flat spacetime is unphysical. Surfaceology, as optimized computation of the Feynman diagram is shown to trivially predict that all gauge theories double copy dual to gravity share the same forbidden scatterings. Relying on a recently encountered paper, we show how the results of the multi-fold theory, in particular the multi-fold space time matter induction and scattering, built explicitly on the path integral formalism, its underlying principles, and the principle of least action, allows recovering almost everything in Today’s physics. It leaves out essentially only some of the coupling constants and mixing angles/parameters. This goes a long way towards the ambition of a Theory Of Everything (TOE), even if not presented as usually expected. It maybe the best that can be achieved considering Gödel's incompleteness theorems. Our justification of the principle of least action, and path integrals, and our work on multi-fold theory encounters relationships between Entropy and Action. We review these encounters, including how our early prediction that entanglement might be irreversible match recent published results that argue against the existence of a (second) Law of entanglement analogous to the second Law of Thermodynamics. Also, from our approach, we obtain a formulation of the evolution of entropy, including for irreversible systems, which matches Onsager and Prigogine’s famous models and principles.
The principle of least action is known to apply to many domains of Physics. Yet its justifications and derivations usually come from archaic classical principles like the Fermat or Maupertuis principles, or mathematical analytical frameworks used for classical mechanics like the Lagrange, Hamilton and D’Alembert formalisms. As is well known, it is widely applicable to other domains, without necessarily a clear justification for it, especially when applied to Physics where Lagrangians or Hamiltonians may not be defined or do not exist. In addition, the Feynman Path Integral formulation of Quantum Mechanics gave it a second life, and another way to justify its validity not only in classical, but also in quantum Physics. However, the conventional logic to reach these conclusions is at time circular, and the principle is applied without proof that it is applicable: it just works and everybody does it. It is this applicability that led us in the past to treat the Action path integral as God’s equation.The paper justifies the principle of least action and Feynman’s path integrals through different physical reasoning instead of the traditional mathematical formalism, and on that basis we confirm its validity in most situations, even if Hamiltonians or Lagrangians may be shaky to define, or the system may be dissipative. We also explain when, and why, sometimes, it may not conventionally apply. The derivation uncovers different links between Entropy and Action, which also opens a different way to look at the origin of the probability distribution of even just a single lone particle; especially when complemented with the W-type multi-fold hypothesis, and makes some considerations on Feynman diagrams, instantons, renormalons and resurgence. Also discontinuities in the paths of the path integrals imply that spacetime is discrete, and non-commutative, and supersymmetry in flat spacetime is unphysical. Surfaceology, as optimized computation of the Feynman diagram is shown to trivially predict that all gauge theories double copy dual to gravity share the same forbidden scatterings. Relying on a recently encountered paper, we show how the results of the multi-fold theory, in particular the multi-fold space time matter induction and scattering, built explicitly on the path integral formalism, its underlying principles, and the principle of least action, allows recovering almost everything in Today’s physics. It leaves out essentially only some of the coupling constants and mixing angles/parameters. This goes a long way towards the ambition of a Theory Of Everything (TOE), even if not presented as usually expected. It maybe the best that can be achieved considering Gödel's incompleteness theorems. Our justification of the principle of least action, and path integrals, and our work on multi-fold theory encounters relationships between Entropy and Action. We review these encounters, including how our early prediction that entanglement might be irreversible match recent published results that argue against the existence of a (second) Law of entanglement analogous to the second Law of Thermodynamics. Also, from our approach, we obtain a formulation of the evolution of entropy, including for irreversible systems, which matches Onsager and Prigogine’s famous models and principles.
Abstract Recently it is shown that the non-relativistic quantum formulations can be derived from an extended least action principle Yang (2023). In this paper, we apply the principle to massive scalar fields, and derive the Schrödinger equation of the wave functional for the scalar fields. The principle extends the least action principle in classical field theory by factoring in two assumptions. First, the Planck constant defines the minimal amount of action a field needs to exhibit in order to be observable. Second, there are constant random field fluctuations. A novel method is introduced to define the information metrics to measure additional observable information due to the field fluctuations, which is then converted to the additional action through the first assumption. Applying the variation principle to minimize the total actions allows us to elegantly derive the transition probability of field fluctuations, the uncertainty relation, and the Schrödinger equation of the wave functional. Furthermore, by defining the information metrics for field fluctuations using general definitions of relative entropy, we obtain a generalized Schrödinger equation of the wave functional that depends on the order of relative entropy. Our results demonstrate that the extended least action principle can be applied to derive both non-relativistic quantum mechanics and relativistic quantum scalar field theory. We expect it can be further used to obtain quantum theory for non-scalar fields.
Investigations into the nature of the principle of least action have shown that there is an intrinsic relationship between geometrical and topological methods and the variational principle in classical mechanics. In this work, we follow and extend this kind of mathematical analysis into the domain of quantum mechanics. First, we show that the identification of the momentum of a quantum particle with the de Broglie wavelength in 2-dimensional space would lead to an interesting feature; namely the action principle S=0 would be satisfied not only by the stationary path, corresponding to the classical motion, but also by any path. Thereupon the Bohr quantum condition possesses a topological character in the sense that the principal quantum number is identified with the winding number, which is used to represent the fundamental group of paths. We extend our discussions into 3-dimensional space and show that the charge of a particle also possesses a topological character and is quantised and classified by the homotopy group of closed surfaces. We then discuss the possibility to extend our discussions into spaces with higher dimensions and show that there exist physical quantities that can be quantised by the higher homotopy groups. Finally we note that if Einstein’s field equations of general relativity are derived from Hilbert’s action through the principle of least action then for the case of n=2 the field equations are satisfied by any metric if the energy-momentum tensor is identified with the metric tensor, similar to the case when the momentum of a particle is identified with the curvature of the particle’s path.