We argue about the following concepts:(i)introduction of endo and exo-observer of a physical system (ii) possible relation between endo, exo-observer and continuum/discrete nature of the same system (iii)the distinction about two categories of observers do not help to solve the basic nature(continuum/discrete)of the system considered. We argue that discrete math tool utilized to describe behavior of a physical system is to ascribe to the exo-observer description. In other words, discreteness nature of a system is due at the presence of an external observer description, the observation act. Vice versa, we retain, that endo-observer description give us a continuum nature of the system but only in a Platonic realm. Finally, we retain that a real theory of the state of matter could to build a real independence from the observer. These brief considerations open a discussion on the foundations of discrete math tools and their relation with the observation act, included the possible extension to foundations of the quantum mechanics.
We argue about the following concepts:(i)introduction of endo and exo-observer of a physical system (ii) possible relation between endo, exo-observer and continuum/discrete nature of the same system (iii)the distinction about two categories of observers do not help to solve the basic nature(continuum/discrete)of the system considered. We argue that discrete math tool utilized to describe behavior of a physical system is to ascribe to the exo-observer description. In other words, discreteness nature of a system is due at the presence of an external observer description, the observation act. Vice versa, we retain, that endo-observer description give us a continuum nature of the system but only in a Platonic realm. Finally, we retain that a real theory of the state of matter could to build a real independence from the observer. These brief considerations open a discussion on the foundations of discrete math tools and their relation with the observation act, included the possible extension to foundations of the quantum mechanics.
We argue about the following concepts:(i)introduction of endo and exo-observer of a physical system (ii) possible relation between endo, exo-observer and continuum/discrete nature of the same system (iii)the distinction about two categories of observers do not help to solve the basic nature (continuum/discrete) of the system considered. We argue that discrete math tool utilized to describe behavior of a physical system is to ascribe to the exo-observer description. In other words, discreteness nature of a system is due at the presence of an external observer description, the observation act. Vice versa, we retain, that endo-observer description give us a continuum nature of the system but only in a Platonic realm. Finally, we retain that a real theory of the state of matter could to build a real independence from the observer. These brief considerations open a discussion on the foundations of discrete math tools and their relation with the observation act, included the possible extension to foundations of the quantum mechanics.
The distinction of external (exo) and internal (endo) perspectives is addressed with respect to the issues of observation and measurement in quantum theory. It is demonstrated how the exo/endo-terminology can be fruitfully applied to two equally important concepts of reality in the quantum world: local (common sense) reality and holistic (nonlocal) reality. Basic elements of the exo/endo-distinction are then used to interpret certain features concerning the issues of time, histories, and presence.
Any scientific system has a unified basic theory. But physics has no unified basic theory in the modern sense. Classical mechanics, relativity and quantum mechanics have their own basic concepts, categories and principles, so none of them can be regarded as true basic theories of physics. Cosmic Continuum Theory holds that the continuity and discreteness of the universe are fundamental issues related to the unification of physics. Because the contradiction between quantum non-locality and local reality is the fundamental obstacle to the unification of physics, while locality and non-locality correspond to the continuity and discreteness of physical reality respectively. The cosmic continuum theory introduces mathematical continuum and axiomatic ideas to reconstruct the basic theory of physics, and by the correspondence of existence and its dimensions to achieve the unification of the essence of physical reality, by introducing the cosmic continuum hypothesis to achieve the unification of the continuity and discreteness of physical reality, by introducing axiomatic methods to achieve formal unification of the foundations on physics. From the perspective of Cosmic Continuum, classical mechanics, relativity and quantum mechanics are no longer the basic theories of physics, but three branch theories of physics that are respectively applicable to macroscopic, cosmoscopic and microcosmic systems.
``Quantum'' stands for for the concepts (both operational and formal) which had to be added to classical physics in order to understand otherwise unexplainable observed phenomena such as the structure of the spectral lines in atomic spectra. While the basic part of classical mechanics deals with the (essentially) reversible dynamics, quantum required adding the notions of ``measurement'' and (possibly non-local) ``correlations'' to the discussion. Crucially, all this comes with a ``probabilistic calculus''. The corresponding mathematical formalism was considered to have reached maturity in [von Neumann 1932], but there are some manifest problems with that formalism: (i) While measurements are applied to physical systems, application of their formal counterpart (i.e. a self-adjoint linear operator) to the vector representing that state of the system in no way reflects how the state changes during the act of measurement. Analogously, the composite of two self-adjoint operators has no physical significance while in practice measurements can be effectuated sequentially. More generally, the formal types in von Neumann's formalism do not reflect the nature of the corresponding underlying concept at all! (ii) Part of the problem regarding the measurements discussed above is that in the von Neumann formalism there is no place for storage, manipulation and exchange of the classical data obtained from measurements. Protocols such as quantum teleportation involving these cannot be given a full formal description. (iii) The behavioral properties of quantum entanglement which for example enable continuous data exchange using only finitary communication are hidden in the formalism. In [Abramsky and Coecke 2004] we addressed all these problems, and in addition provided a purely categorical axiomatization of quantum mechanics. The concepts of the abstract quantum mechanics are formulated relative to a strongly compact closed category with biproducts (of which the category FdHilb of finite dimensional Hilbert spaces and linear maps is an example). Preparations, measurements, either destructive or not, classical data exchange are all morphisms in that category, and their types fully reflect their kinds. Correctness properties of standard quantum protocols can be abstractly proven. Surprisingly, in this seemingly purely qualitative setting even the quantitative Born rule arises, that is the rule which tells you how to calculate the probabilities. Indeed, each such category has as endomorphism Hom of the tensor unit an abelian semiring of `scalars', and a special subset of these scalars will play the role of weights: each scalar induces a natural transformation which propagates through physical processes, and when a `state' undergoes a `measurement', the composition of the corresponding morphisms gives rise to the weight. Hence the probabilistic weights live within the category of processes. J. von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-Verlag (1932). English translation in Mathematical Foundations of Quantum Mechanics. Princeton University Press (1955). S. Abramsky and B. Coecke. A categorical semantics of quantum protocols. In the proceedings of LiCS'04 (2004). An extended version is available at arXiv:quant-ph/0402130 A more reader friendly version entitled `Quantum information flow, concretely, abstractly' is at http://www.vub.ac.be/CLEA/Bob/Papers/QPL.pdf
## Abstract Physics uses continuous mathematics - fields, manifolds, wave functions, differential equations - to make predictions. Physical observations are recorded as discrete events: detector clicks, particle counts, energy transitions. This paper argues that this is not an accident to be explained away, but a structural fact that should be treated as a working principle. The claim: continuous mathematical objects describe distributions over possible outcomes. They are not physical entities occupying space. Discrete events are what physically happens. Confusing these two produces recurring conceptual difficulties—most visibly in the quantum measurement problem and in spacetime singularities. This distinction reframes, but does not solve, several open problems. It does not derive the Born rule, does not specify a concrete discrete causal structure, and does not yield new experimental predictions. Its value is diagnostic: it identifies where existing puzzles are category errors versus where they are genuine open technical problems, and it identifies a specific reframing of how discrete causal models (such as causal set theory) might recover Lorentz-invariant physics. Keywords: continuity, discreteness, possibility, measurement, causality, causal set theory
We study deterministic and quantum dynamics from a constructive “finite” point of view, since the introduction of the continuum or other actual infinities in physics poses severe conceptual and technical difficulties, and while all of these concepts are not really needed in physics, which is in fact an empirical science. Particular attention is paid to the symmetry properties of discrete systems. For a consistent description of the symmetries of dynamical systems at different time instants and the symmetries of various parts of such systems, we introduce discrete analogs of gauge connections. These gauge structures are particularly important to describe the quantum behavior. The symmetries govern the fundamental properties of the behavior of dynamical systems. In particular, we can show that the moving soliton-like structures are inevitable in a deterministic (classical) dynamical system, whose symmetry group breaks the set of states into a finite number of orbits of the group. We demonstrate that the quantum behavior is a natural consequence of symmetries of dynamical systems. This behavior is a result of the fundamental inability to trace the identity of indistinguish-able objects during their evolution. Information is only available on invariant statements and values related with such objects. Using general mathematical arguments, any quantum dynamics can be shown to reduce to a sequence of permutations. The quantum interferences occur in the invariant subspaces of permutation representations of the symmetry groups of dynamical systems. The observables can be expressed in terms of permutation invariants. We also show that in order to describe quantum phenomena it is sufficient to use cyclotomic fields—the minimal extensions of natural numbers suitable for quantum mechanics, instead of a non-constructive number system—the field of complex numbers. The finite groups of symmetries play the central role in this review. In physics there is an additional reason for such groups to be of interest. Numerous experiments and observations in particle physics point to the importance of finite groups of relatively low orders in a number of fundamental processes. The origin of these groups has no explanation within presently recognized theories, such as the Standard Model.