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Journal of Physics Conference Series

Quantum Theory From a Non-Conventional Perspective

Dieter Schuch

Published May 26, 2017• 0 citations
Publisher

Abstract

Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex quantities are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics as well as in in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. It turns out that this dissipative effect can either affect the amplitude or the phase of the complex quantity describing the open quantum system. This suggests a relation between non-unitary transformations and gauge-transformations for these systems. A change of the amplitude also seems to be connected with a second quantum of action for "radial" changes, compared to ħ for "angular" changes, leading to an interpretation of Sommerfeld's constant.

DOI
10.1088/1742-6596/839/1/012008

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Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex quantities are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics as well as in in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. It turns out that this dissipative effect can either affect the amplitude or the phase of the complex quantity describing the open quantum system. This suggests a relation between non-unitary transformations and gauge-transformations for these systems. A change of the amplitude also seems to be connected with a second quantum of action for "radial" changes, compared to ħ for "angular" changes, leading to an interpretation of Sommerfeld's constant.
Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex wave function are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics and has a counterpart in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of nonlinear real Ermakov equations or equivalent complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. In this context, different approaches for describing open quantum systems, particularly effective ones, are discussed and compared. Certain kinds of nonlinear modifications of the Schrödinger equation are discussed as well as their interrelations and their relations to linear approaches via non-unitary transformations. The modifications of the aforementioned Ermakov and Riccati equations when environmental effects are included can be determined in the time-dependent case. From formal similarities conclusions can be drawn how the equations of time-independent quantum mechanics can be modified to also incluce the enviromental aspects.
In contrast with classical physics, complex quantities have a fundamental physical meaning in quantum physics and action, being essentially the quantized entity, should be given more attention instead of focusing mainly on Hamiltonians or Lagrangians that have the dimension of energy. Phase and amplitude of the complex quantities in (time-dependent and time-independent) quantum mechanics are not independent of each other but coupled via some conservation law. This coupling can be understood if the systems are described in terms of complex nonlinear Riccati equations. These equations not only enable a connection to the Pythagorean triples, probably the oldest and most abstract 'quantization' problem, but also lead to dynamical invariants with the dimension of action. Factorization of the corresponding operator provides generalized creation and annihilation operators, which is also possible for dissipative systems where no conventional Hamiltonian formalism exists. Formal similarities with other fields, particularly with nonlinear dynamics, are shown.
Phase and amplitude of the complex wave function are not independent of each other, but coupled, which becomes obvious looking at Madelung's hydrodynamic formulation of quantum mechanics. In the time-independent case, this leads to a kind of conservation law that allows for the reformulation of the linear Schrodinger equation in terms of a nonlinear Ermakov equation which is equivalent to a complex Riccati equation where the quadratic term in this equation explains the origin of the phase-amplitude coupling. A similar conservation law and corresponding nonlinear equations can also be found in the time-dependent case. The gain from the nonlinear formulations emerges when open systems with dissipation and irreversibility are considered. Describing this kind of systems by an effective nonlinear Schrodinger equation leads to a modification of the above-mentioned equations with new qualitative effects like Hopf bifurcations.
An aim of this paper is to show that the major difference between classic and quantum physics is not only that in the latter case complex quantities are a conceptional element, but also that real and imaginary parts, or phase and amplitude, respectively, are not independent of each other but coupled via some conservation laws. This also holds for systems where the Hamiltonian is no longer a constant of motion but a so-called Ermakov invariant with the dimension of an action still exists. In particular, this is valid for certain dissipative system where a transition between formal Hamiltonian level and physical nonlinear level can be achieved via a non-unitary transformation.
Theoretical physics seems to be in a kind of schizophrenic state. Many phenomena in the observable macroscopic world obey nonlinear evolution equations, whereas the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. I claim that linearity in quantum mechanics is not as essential as it apparently seems since quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown where complex Riccati equations appear in time-dependent quantum mechanics and how they can be treated and compared with similar space-dependent Riccati equations in supersymmetric quantum mechanics. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation. Finally, it will be shown that (real and complex) Riccati equations also appear in many other fields of physics, like statistical thermodynamics and cosmology.
Searching for the first well-documented attempts of introducing some kind of "quantization" into the description of nature inevitably leads to the ancient Greeks, in particular Plato and Pythagoras. The question of finding the so-called Pythagorean triples, i.e., right-angled triangles with integer length of all three sides, is, surprisingly, connected with complex nonlinear Riccati equations that occur in time-dependent quantum mechanics. The complex Riccati equation together with the usual Newtonian equation of the system, leads to a dynamical invariant with the dimension of an action. The relation between this invariant and a conserved "angular momentum" for the motion in the complex plane will be determined. The "Pythagorean quantization" shows similarities with the quantum Hall effect and leads to an interpretation of Sommerfeld's fine structure constant that involves another quantum of action, the "least Coulombic action" e 2 / c . Since natural evolution is characterized by irreversibility and dissipation, the question of how these aspects can be incorporated into a quantum mechanical description arises. Two effective approaches that also both possess a dynamical invariant (like the one mentioned above) will be discussed. One uses an explicitly time-dependent (linear) Hamiltonian, whereas the other leads to a nonlinear Schrödinger equation with complex logarithmic nonlinearity. Both approaches can be transformed into each other via a non-unitary transformation that involves Schrödinger's original definition of a (complex) action via the wave function.