Physics Beyond the Limits of Uncertainty Relations
by Micho Durdevich
Table of Contents
1. IntroductionQuantum mechanics is a physics of microworld. Its principal aim is to provide a mathematically coherent picture of physical reality at the deepest possible experimentally accessible level--the quantum level. This means understanding phenomenas involving elementary particles and quantas of interactions, and the study of the internal structure of matter and fields.
Quantum world is very different from the picture given by classical mechanics, which is a physics of macroworld.
One of the principal purely quantum phenomenas is complementarity. If we consider all possible properties/attributes of a given quantum system then it turns out that for every state of the system there exists an infinite collection of properties that are not applicable to the system in this particular state. In other words, in such situations it is impossible to think that a given attribute from the mentioned collection holds or not.
This is very different from classical mechanics, where every possible attribute of a physical system has a definitive (0 or 1) value in every state. In other words, either the attribure holds or its negation holds in a given state of the system.
All the properties of a classical system form a Boolean algebra. All possible attributes of a quantum system form an essentially different structure---a non-distributive lattice.
We can also say that quantum systems are never completely understandable with the help of one single system of classical-type attributes. On the other hand, in every concrete experimental context, the subset consisting precisely of the system attributes actualized in this context necessarily forms a classical Boolean algebra. The situation is somehow similar to the relation between Euclidean geometry--where the space is covered by a single coordinate system, and general Riemannian geometry--where it is only possible to cover the whole space by the atlas of local coordinate systems, each describing a portion of the space.
Another fundamental difference between classical and quantum mechanics is that quantum mechanics is an intrinsically stochastical theory. In other words, probability concepts are incorporated in the very roots of the quantum theory. Even if we know everything about a quantum physical system, it is still not possible to predict with certainty the outcomes of all measurements performed on this system.
Accordingly, physical quantities (observables) do not possess definitive values in quantum states. More precisely, if a quantum system is described by a Hilbert space H then the possible states of the system are described by unit vectors and it is assumed that unit vectors that differ up to a phase factor describe the same physical state. Physical observables are described by selfadjoint operators F acting in H. The mean value of quantum observables in states are then computed by the standard quantum-mechanical formula:
This situation is in a sharp contrast with classical mechanics, which is a completely causal theory. The states of a system are in one-one correspondence with points of the phase space and physical observables are interpreted as the appropriate functions defined on this space. The values of observables in the states are simply the corresponding values of functions in points. In classical mechanics, the probabilities appear as something secondary, whenever we do not know exactly the state of the system. A standard example is given by classical statistical mechanics, which studies very complicated physical systems and so it is practically impossible to determine the state of the system. Accordingly in classical statistical mechanics the system is effectively described by a probability distribution over the phase space (for example--the canonical ensemble distribution).
It is natural to ask the following question: is it possible to explain the stochasticity of quantum mechanics as a simple consequence of an incompleteness of quantum theory. In other words, as a simple consequence of the fact that quantum theory does not include certain deeper parameters (=hidden variables), which if included in the game would re-establish causality as in the classical physics.
In such a way we arive to the idea of subquantum mechanics.
[Next Segment]: Subquantum Mechanics--Basic Ideas