Space, time and probabilities not only form the arena of many physical theories, but also the arena of our everyday life. Therefore, the problem of their ontological status becomes even more interesting. It seems that this problem cannot be solved before we have the final physical theory (theory of everything). However, various methods of looking for such a theory lead to converging results, and are able to elucidate the problem in question. We analyze noncommutative algebraic methods, and show that, when applied to model the fundamental level, they suggest a nonlocal regime in which there is no space and no time, in their usual sense, and which is essentially probabilistic (in a generalized sense). In the noncommutative approach there are many probabilistic measures (not just one as in macroscopic physics), and none of them seems to be distinguished in any way. The tentative conclusion is that space, time and probability are not absolute categories, but rather aspects of the world structure emergent from the fundamental level. One should expect that the ultimate theory will be well off our traditional philosophical categories. This shows that our mind has a peculiar property allowing it to reach far beyond its usual limitations.
Space, time and probabilities not only form the arena of many physical theories, but also the arena of our everyday life. Therefore, the problem of their ontological status becomes even more interesting. It seems that this problem cannot be solved before we have the final physical theory (theory of everything). However, various methods of looking for such a theory lead to converging results, and are able to elucidate the problem in question. We analyze noncommutative algebraic methods, and show that, when applied to model the fundamental level, they suggest a nonlocal regime in which there is no space and no time, in their usual sense, and which is essentially probabilistic (in a generalized sense). In the noncommutative approach there are many probabilistic measures (not just one as in macroscopic physics), and none of them seems to be distinguished in any way. The tentative conclusion is that space, time and probability are not absolute categories, but rather aspects of the world structure emergent from the fundamental level. One should expect that the ultimate theory will be well off our traditional philosophical categories. This shows that our mind has a peculiar property allowing it to reach far beyond its usual limitations.
After briefly reviewing classical and quantum aspects of probability, basic concepts of the noncommutative calculus of probability (called also free calculus of probability) and its possible application to model the fundamental level of physics are presented. It is shown that the pair (M, *), where M is a (noncommutative) von Neumann algebra, and a state on it, is both a dynamical object and a probabilistic object. In this way, dynamics and probability can be unified in noncommutative geometry. Some philosophical consequences of such an approach are indicated.
Since noncommutative measure theory plays a central role in the quantum theory, one might say that quantum mechanics is a new generalized theory of probability. That assertion, however, raises many problems with interpreting quantum probabilities and quantum events. Although one can take a radically instrumentalist or pragmatist approach toward this question, it is much more interesting, both theoretically and heuristically, to adopt a realistic or ontological strategy that treats the quantum measure as an effective parameter of an underlying time-symmetric regime of physical dynamics, which only generates the classical events and probabilities at its limit. It is argued that time-symmetric relativistic reformulations of the quantum theory with time-like nonseparability have important advantages over space-like nonlocal hidden-variable Bohmian-type models, which have Aristotelian spacetime as their natural environment.
The paper can be regarded as a short and informal introduction to noncommutative calculi of probability. The standard theory of probability is reformulated in the algebraic language. In this form it is readily generalized to that its version which is virtually present in quantum mechanics, and then generalized to the so-called free theory of probability. Noncommutative theory of probability is a pair (M, φ) where M is a von Neumann algebra, and φ a normal state on M which plays the role of a noncommutative probability measure. In the standard (commutative) theory of probability, there is, in principle, one mathematically interesting probability measure, namely the Lebesgue measure, whereas in the noncommutative theories there are many nonequivalent probability measures. Philosophical implications of this fact are briefly discussed.
After briefly reviewing classical and quantum aspects of probability, basic concepts of the noncommutative calculus of probability (called also free calculus of probability) and its possible application to model the fundamental level of physics are presented. It is shown that the pair (M, *), where M is a (noncommutative) von Neumann algebra, and a state on it, is both a dynamical object and a probabilistic object. In this way, dynamics and probability can be unified in noncommutative geometry. Some philosophical consequences of such an approach are indicated.
The main objective of the present article is the methodological analysis of the structures of physical theories that could apply to the theory of quantum gravitation or the unifi ed theory of all interactions (other terms are the “Final Theory” or “the Theory of Everything”, TOE). In the fi rst step of this discussion, it is shown that, unlike widely believed, quantum theory, in principle, allows representation by local classical hidden variables. The possibility of such representation is proved by the possibility of an exhaustive simulation of quantum systems by means of a local and classical device — the computer equipment plays a role of the local classical hidden variables. Details regarding the realization of such a representation are discussed using the example of an actual computer program simulating a correlation experiment using Eeinstein-Podolsky-Rozen pairs. This example explicitly violates the theorems of impossibility for hidden classical variables in quantum theory. The fact that these theorems ignore the possibility for a situation, which, in this article, is characterized as the splitting of the layers of reality, is the reason that a possible violation of the theorem of impossibility for hidden variables exists. There are two such layers in the computer example — a reality layer where computer equipment exists, and a layer for simulated quantum reality. The analysis of this example results in a general idea about the layers of reality, which are the main subject of the subsequent discussion. The computer example plays a role of the existence theorem. It follows that, in principle, a fundamental local and classical structure may exist behind physical quantum reality. However, such a “local realism” leads to the idea that an immense “space container” exists for classical objects of such a layer of reality. The problem is overcome if the fundamental ontology is classical, but nonlocal. Then, a space container for it is not required. It is shown that such a classical, but nonlocal, structure is very similar to a formal mathematical system. It leads to a thought that the ideal mathematical system can be a fundamental ontology of the TOE, or it is reminiscent of something beyond mathematics — a nonreducible pseudomathematical structure. In this regard, the analysis of the nature of mathematics is given. It is shown that mathematics are not only a result of the imagination of people, but that mathematical forms and all mathematics holistically exist objectively. Moreover, the statement about the objective existence of mathematical forms has an empirical status based on Popper’s criterion of falsifi ability. It transfers a question of objectivity for mathematics from the fi eld of philosophy to the fi eld of empirical science. Then, a connection on the bases of mathematics and physics is established — namely, with the existence of the classical sector of quantum theory and with a causal structure of space-time. In this sense, the existing mathematics are not only objective, but they also possess physics. That is, mathematics, in a sense, is a thing but it is not a thought, and this quality can be considered as an objective layer of reality that can be a substrate of the physical world. In the fi nal part of this paper, several modern directions in the quantum gravity theory or TOE (string theory, loop quantum gravity, and causal sets) are considered regarding, as far as these theories are concerned, the formation of an abstract mathematical substratum. It is shown that the tendency to develop a of structure like an abstract mathematical substratum defi nitely exists. This means that a TOE can be not just a physical theory, but rather an abstract mathematical structure
The question of what ontological message (if any) is encoded in the formalism of contemporary physics is, to say the least, controversial. The reasons for this state of affairs are psychological and neurobiological. The processes by which the visual world is constructed by our minds, predispose us towards concepts of space, time, and substance that are inconsistent with the spatiotemporal and substantial aspects of the quantum world. In the first part of this chapter, the latter are extracted from the quantum formalism. The nature of a world that is fundamentally and irreducibly described by a probability algorithm is determined. The neurobiological processes responsible for the mismatch between our "natural" concepts of space, time, and substance and the corresponding aspects of the quantum world are discussed in the second part. These natural concepts give rise to pseudoproblems that foil our attempts to make ontological sense of quantum mechanics. If certain psychologically motivated but physically unwarranted assumptions are discarded (in particular our dogged insistence on obtruding upon the quantum world the intrinsically and completely differentiated spatiotemporal background of classical physics), we are in a position to see why our fundamental physical theory is a probability algorithm, and to solve the remaining interpretational problems.