EEQT, Quantum Jumps and Quantum Fractals

Note: These two applets crash on certain browsers. Internet Explorer seems to be OK and Mozilla seems to be OK. The applets crash with Opera (for reasons that are not understood) and with older Netscape. Therefore the best thing is to download the .jar files from SourceForge, unpack them, download Java SDK appropriate to the operating system, install it, and then run the .jar files with "java -jar qf.jar" or "java -jar wave.jar"

EEQT or Event-Enhanced-Quantum-Theory

1. Isn't it so that EEQT is a step backward toward classical mechanics
that we all know are inadequate?

EEQT is based on a simple thesis: not all is "quantum" and there are things in this universe that are NOT described by a quantum wave function. Example, going to an extreme: one such case is the wave function itself. Physicists talk about first and second quantization. Sometimes, with considerable embarrassment, a third quantization is considered. But that is usually the end of that. Even the most orthodox quantum physicist controls at some point his "quantizeeverything" urge - otherwise he would have to "quantize his quantizations" ad infinitum, never being able to communicate his results to his colleagues. The part of our reality that is not and must not be "quantized" deserves a separate name. In EEQT we are using the term "classical." This term, as we use it, must be understood in a special, more-general-than-usually-assumed way. "Classical" is not the same as "mechanical." Neither is it the same as "mechanically deterministic." When we say "classical" - it means "outside of the restricted mathematical formalism of Hilbert spaces, linear operators and linear evolutions." It also means: the value of the "Planck constant" does not govern classical parameters. Instead, in a future theory, the value of the Planck constant will be explained in terms of a "non-quantum" paradigm.

(...)

12. The name "Event Enhanced Quantum Theory" is misleading.

As we have stated: "EEQT is the minimal extension of orthodox quantum theory that allows for events." It DOES enhance quantum theory by adding the new terms to the Liouville equation. When the coupling constant is small, events are rare and EEQT reduces to orthodox quantum theory. Thus it IS an enhancement. (...)

According to EEQT quantum jumps are not directly observable. What we see are the accompanying "events". This part is somewhat tricky, and I will try to explain the trickiness here, in few paragraphs, but without any hope that there will be even one person who will understand what I mean. Well, perhaps mathematicians will do, but are they going to read this page? I doubt. Physicists certainly will think that it is too weird. And they have better things to do than following someone's weird ideas - as every physicist with guts has weird ideas of his/her own! But I would feel guilty if I did not give it a try. So here it is.

Physicists do consider quantum jumps. In particular those who deal with theoretical quantum optics and/or quantum computing and information. But these quantum jumps are not being taken as "real". If not for any other reason, than because there are infinitely many jump processes that can be associated with a given Liouville equation, and there is no good reason to choose one rather than other. Thus discontinuous quantum jumps in theoretical quantum optics are considered mainly as a convenient numerical methods for solving the continuous Liouville equation. It is not so in EEQT. But EEQT splits the world into a quantum and a classical part, and quantum physicists deny that the classical part exist. They think all is quantum - the same way Ptolemeian physicists thought that all is perfectly round. Can we propose a clever idea that will show that not all is quantum? Indeed, according to quantum physics the only thing that exists is the quantum wave function. Now, let us us ask this: is the wave function itself a classical or a quantum object? That is, we ask, is location of the wave function in the Hilbert space governed by classical or by quantum laws? Most quantum physicists would pretend they do not understand the question. Some will understand, and will answer: "sure, there is an uncertainty in the state vector, but that is altogether different story. They will point me to Braginsky or Vaidman or some other, more recent, paper - but they will not answer my question: is the quantum wave function a classical or a quantum object? Is it an object at all? And if it is an object, then what kind of animal it is, and where it fits? Philosophers perhaps will point me to Eccles and Popper, but this is not an answer either.
What is my answer to my own question? I do not know the answer, but I can speculate. So, here it is: we are talking about models. Models of "Reality". Perhaps nothing but models "exists", but that is not our problem now. If all is about models, then we can think of a model in which wave function is both classical and quantum. In which Wave Function "observes" itself - as John Archibald Wheeler has imagined:

"The universe viewed as a self-excited circuit. Starting small (thin U at upper right), it grows (loop of U) to observer participancy - which in turn imparts 'tangible reality' (cf. the delayed-choice experiment of Fig. 22.9) to even the earliest days of the universe"
"If the views that we are exploring here are correct, one principle, observer-participancy, suffices to build everything. The picture of the participatory universe will flounder, and have to be rejected, if it cannot account for the building of the law; and space-time as part of the law; and out of law substance. It has no other than a higgledy-piggledy way to build law: out of statistics of billions upon billions of observer participancy each of which by itself partakes of utter randomness." (J.A. Wheeler, "Beyond the Black Hole", in "Some Strangeness in the Proportion", Ed. Harry Woolf, Addison-Wesley, London 1980)

To observe itself "It" must split into two "personalities", a quantum one and a classical one. So, here comes the model: consider a pair of wave functions, the function trying to determine its own shape. One element of the pair is considered to be "quantum" - as it determines probabilities and quantum jumps, while the second element of the pair is interpreted as a classical one - its shape is the classical variable.

They dance together and they jump together. More details can be found in "Topics in Quantum Dynamics". And here we come to the mathematical description of quantum jumps in EEQT. Of course the simplest situation is when we separate jumps from the continuous evolution. To analyze this particular situation let us think of the simplest possible "toy model". Physicists like toy models, as they usually provide us with explicit solutions whose properties we can study in order to try to understand more complex, real world situations, where the problems get so complicated that there is no hope even for an approximate solution. Physicists usually replace real-world problems with other problems, build out of their toy models, which are still simple enough to be solvable, even if only approximately, and yet mirror some essential features of the "true problems."

So, what would be the simplest toy model to play with, that teaches us something about quantum jumps? The quantum system, to be nontrivial, must live in a Hilbert space of at least two complex dimensions. The classical system must have at least two states. Such a toy model was indeed studied in connection to the Quantum Zeno effect., where it was demonstrated that a flip-flop detector strongly coupled (that is "under intensive observation" - watched pot never boils... ) to a two-state quantum system effectively stops the continuous quantum evolution. This model is not interesting though if we want to study pure quantum jumps. Here we need a more complicated model and that is how "tetrahedral model" was developed. It was found that it leads to chaotic dynamics and to fractals of a new type: fractals drawn by a quantum brush on the quantum canvas - a complex projective space. And that is how we come to quantum fractals.

The details and the bibliography are given in "Quantum Jumps, EEQT and the Five Platonic Fractals." Here let us describe the algorithm and the Java applet. (The applet is a part of an OpenSource project, so additions and enhancements will probably follow its release. )

The canvas, is a surface of the unit sphere In coordinates its points are represented by vectors n = (n¹,n²,n³) of unit length, thus (n1)²+(n2)²+(n3)² = 1. There are five Platonic solids: tetrahedron (N=4), octahedron (N=6), cube (N=8), icosahedron (N=12), dodecahedron (N=20), where N is the number of vertices. They have equal faces, bounded by equilateral polygons. It was Euclid who proved that only five such can exist in a three dimensional world. In his Mysterium Cosmographicum (1595) Johannes Kepler attempted to account for the orbits of the six then known planets by radii of concentric spheres circumscribing or inscribing the solids.