The interaction between matter and fields is a classical problem. Still there are difficulties. Time symmetry is broken. So the radiation damping belongs to a class of phenomena, which includes transport properties. We propose a radically different approach based on the extension of the dynamics of integrable systems. We consider a simple model, a harmonic oscillator interacting with a field. For integrable systems, it is well known that there exists a unitary transformation U. However, in the radiation damping we have resonances between the action of the particle and the actions of the field. This makes the system an example of Poincar\'e nonintegrable systems. We extend the unitary operator to a new star-unitary operator $\ensuremath{\Lambda}.$ This changes the dynamical description of radiation damping. Once we know the Hamiltonian, we can of course write the Hamilton equations. But we have the possibility to go to new descriptions. The invertible $\ensuremath{\Lambda}$ transformation gives many new aspects, which are hidden in the initial description. For example, we show that there are fluctuations, and that there exists an irreducible probability description. The transformation to $\ensuremath{\Lambda}$ representation corresponds to a transformation to Markovian probability equations. We can always come back to the initial representation by the inverse transformation. We have verified this remarkable prediction by detailed numerical calculations. We need the $\ensuremath{\Lambda}$ transformation to obtain the definition of a dressed unstable mode, which has a well-defined lifetime. In the initial representation there are various time scales and there is no strictly exponential lifetime. The situation is the same as the one we have studied in the quantum case in recent papers.
The interaction between matter and fields is a classical problem. Still there are difficulties. Time symmetry is broken. So the radiation damping belongs to a class of phenomena, which includes transport properties. We propose a radically different approach based on the extension of the dynamics of integrable systems. We consider a simple model, a harmonic oscillator interacting with a field. For integrable systems, it is well known that there exists a unitary transformation U. However, in the radiation damping we have resonances between the action of the particle and the actions of the field. This makes the system an example of Poincar'e nonintegrable systems. We extend the unitary operator to a new star-unitary operator \ensuremathΛ. This changes the dynamical description of radiation damping. Once we know the Hamiltonian, we can of course write the Hamilton equations. But we have the possibility to go to new descriptions. The invertible \ensuremathΛ transformation gives many new aspects, which are hidden in the initial description. For example, we show that there are fluctuations, and that there exists an irreducible probability description. The transformation to \ensuremathΛ representation corresponds to a transformation to Markovian probability equations. We can always come back to the initial representation by the inverse transformation. We have verified this remarkable prediction by detailed numerical calculations. We need the \ensuremathΛ transformation to obtain the definition of a dressed unstable mode, which has a well-defined lifetime. In the initial representation there are various time scales and there is no strictly exponential lifetime. The situation is the same as the one we have studied in the quantum case in recent papers.
Recent development of spectral analysis of the Liouville-von Neumann equation has revealed the fact that irreversibility is a rigorous dynamical property of Poincaré non-integrable systems with an infinite degrees of freedom interacting among each other through resonance coupling. In the present work we discuss this role of resonance in some examples of matter-field coupling systems for both classical and quantum mechanics: the one is a classical motion of a charged particle in electromagnetic waveguide, and the other is the decoherence problem of quantum matter-field interacting systems. In the first part of this dissertation, we study an accelerated motion of a charged classical dipole molecule with frequency ω1 inside the rectangular waveguide. If the particle is in free space, it is well known that its accelerated motion will eventually stop by radiating the field through the resonance interaction. This result is the so-called radiation damping. For the case in the waveguide, there are two possible situations, due to the existence of the cut-off frequency ωc of the waveguide. Under the cut-off frequency electromagnetic wave cannot propagate inside the waveguide. The stability of the dipole depends on the relation between ω1 and ωc. For ω1 < ωc, the dipole cannot resonate with the field. This corresponds to the Poincaré integrable system. For this case the dipole keeps its accelerated motion without emitting the radiating field. Therefore the radiation damping of the dipole molecules is suppressed inside the waveguide under the absence of resonance interaction. The motion of this steady state somewhat resembles a quantum ground state. We show that this steady state is dressed by electromagnetic field. The overlap of the dressing field leads to a force analogous to van der Waals force in quantum mechanics. The critical frequency determined by ω1 = ωc gives a critical size of the waveguide. For heavy molecules, such as HCl, this is of order 10−5m. We show that the size of the dressing field is the same order of the size of the waveguide. Hence we have a macroscopic size of the dressing in the waveguide. For ω1 > ωc, the dipole can resonate with the field, and the system becomes non-integrable in the sense of Poincaré. As a result, the accelerated motion eventually stop by emitting the resonance field. This corresponds to the problem of classical radiation damping. We show that there is non-negligible deviation of exponential decay in a short time scale of the order t ∼ 1/ω1. This corresponds to the quantum Zeno effect, well known in quantum unstable systems. After this period, the dipole decays exponentially in time by emitting the resonance field. We found by choosing ωc very close to ω1, we can increase the decay rate 105 times faster than the case where the dipole is in the free space, at the same time the emitted field travel 10−4 times slower than the speed of light. This is again a consequence of the existence of the cut-off frequency in the waveguide. Indeed, the cut-off frequency leads to a non-linear dispersion relation for the electromagnetic field. To some extent, the electromagnetic field is sticky inside the waveguide. Due to the large decay rate and slower speed of light, the size of the wavepacket emitted by the dipole is significantly small (about 10cm for HCl). This is even smaller than the quantum case in free space, where the wavepacket of the field emitted by the decay of electron in hydrogen atom is about 1m. In the second part of this dissertation, we study a quantum matter-field coupled system. We focus our attention on the problem of quantum decoherence in a system of a particle coupled with a field, the Hamiltonian of which has a similar structure to the problem of classical radiation damping mentioned above. We apply the complex spectral representation of the Liouville-von Neumann operator that gives a rigorous approach to the irreversible processes. We focus our attention on the time evolution of the field, which is commonly neglected in phenomenological approaches to the decoherence problem. We found a signature of decoherence in the field which has a characteristic time dependence proportional to t that comes from the secular effect between the particle and the field through the resonance interaction that breaks time-symmetry.
In recent papers we have discussed the meaning of dressed excited quantum states. We have shown that these states can be formulated in terms of distribution functions outside the Hilbert space. Our approach applies to ``nonintegrable'' systems in the sense of Poincar\'e. It involves the analytic continuation of the unitary operator U describing the transformation from bare to dressed stable states. This leads to a ``star-unitary'' operator $\ensuremath{\Lambda}.$ Interacting fields are nonintegrable systems obtained as a result of resonances. It is therefore natural to expect that our previous results remain valid for interacting fields. We consider a simple example, which corresponds to an extension of the usual Friedrichs model. This involves a local field in interaction with a bilocal scalar field. This model has been previously studied by one of the co-authors (I.P.). It is a simplified version of the model describing $\stackrel{\ensuremath{\rightarrow}}{A}B+C$ transition with quadratic interaction. The usual Bogoliubov transformation eliminates the field corresponding to A, while our method leads to strictly exponential decay of unstable field. The dressed field A corresponds to a singular distribution function outside the Liouville-Hilbert space. As in the Friedrichs model there exists two time scales---one for the preparation of a dressed state and depending on initial conditions called ``Zeno period'' and the other universal for the decay.
Nonintegrable Poincaré systems with continuous spectrum (so-called Large Poincaré Systems, LPS) lead to the appearance of diffusive terms in the framework of dynamics. These terms break time symmetry. They lead, therefore, to limitations to classical trajectory dynamics and of wave functions. These diffusive terms correspond to well-defined classes of dynamical processes (i.e., so-called “vacuum-vacuum” transitions). The diffusive effects are amplified in situations corresponding to persistent interactions. As a result, we have to include already in the fundamental dynamical description the two aspects, probability and irreversibility, which are so conspicuous on the macroscopic level. We have to formulate both classical and quantum mechanics on the Liouville level of probability distributions (or density matrices). For integrable systems, we recover the usual formulations of classical or quantum mechanics. Instead of being irreducible concepts, which cannot be further analyzed, trajectories and wave functions appear as special solutions of the Liouville-von Neumann equations. This extension of classical and quantum dynamics permits us to unify the two concepts of nature we inherited from the 19th century, based on the one hand on dynamical time-reversible laws and on the other on an evolutionary view associated to entropy. It leads also to a unified formulation of quantum theory avoiding the conventional dual structure based on Schrödinger’s equation on the one hand, and on the “collapse” of the wave function on the other. A dynamical interpretation is given to processes such as decoherence or approach to equilibrium without any appeal to extra dynamic considerations (such as the many-world theory, coarse graining or averaging over the environment). There is a striking parallelism between classical and quantum theory. For LPS we have, in general, both a “collapse” of trajectories and of wave functions for LPS. In both cases, we need a generalized formulation of dynamics in terms of probability distributions or density matrices. Since the beginning of this century, we know that classical mechanics had to be generalized to take into account the existence of universal constants. We now see that classical as well as quantum mechanics also have to be extended to include unstable dynamical systems such as LPS. As a result, we achieve a new formulation of "laws of physics" dealing no more with certitudes but with probabilities. The formulation is appropriate to describe an open, evolving universe.
A previously developed novel theory for the formal canonical quantization of classically dissipating systems will be the starting point for a detailed discussion of the quantum statistical aspects of the simple linearly damped harmonic oscillator. The formalism essentially involves complex classical canonical coordinates and momenta, and thus quite naturally leads to the possibility of creation and annihilation operators. Furthermore, the occurrence of quantal noise operators appears to be of principal importance for the conservation of the fundamental commutator in the course of time, as will be expressed in a simple fluctuation-dissipation relation. Making a canonical transformation back to the real, Cartesian Hermitian position and momentum an "effective" Hermitian Hamiltonian will be derived, with which a transformation is made from the Heisenberg frame to the Schr\"odinger frame where the density operator equation will be computed. This will make it obvious that no proper Schr\"odinger equation exists for the dissipative subsystem on its own, thus reflecting an incomplete knowledge. The master equation will then be translated into its Wigner representation. The intimate connection between the diffusion coefficients in the resulting Fokker-Planck equation and the uncertainty relation will be demonstrated in a clear fashion.