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arXiv:quant-ph/9611048

The Quantum Theory of Ur-Objects as a Theory of Information

Holger Lyre

Published Nov 26, 1996• 5 citations
arXivHTMLPDF

Abstract

The quantum theory of ur-objects proposed by C. F. von Weizsaecker has to be interpreted as a quantum theory of information. Ur-objects, or urs, are thought to be the simplest objects in quantum theory. Thus an ur is represented by a two-dimensional Hilbert space with the universal symmetry group SU(2), and can only be characterized as ''one bit of potential information''. In this sense it is not a spatial but an ''information atom''. The physical structure of the ur theory is reviewed, and the philosophical consequences of its interpretation as an information theory are demonstrated by means of some important concepts of physics such as time, space, entropy, energy, and matter, which in ur theory appear to be directly connected with information as ''the'' fundamental substance. This hopefully will help to provide a new understanding of the concept of information.

DOI
10.48550/arxiv.quant-ph/9611048

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The quantum theory of ur-objects proposed by C. F. von Weizsaecker has to be interpreted as a quantum theory of information. Ur-objects, or urs, are thought to be the simplest objects in quantum theory. Thus an ur is represented by a two-dimensional Hilbert space with the universal symmetry group SU(2), and can only be characterized as ''one bit of potential information''. In this sense it is not a spatial but an ''information atom''. The physical structure of the ur theory is reviewed, and the philosophical consequences of its interpretation as an information theory are demonstrated by means of some important concepts of physics such as time, space, entropy, energy, and matter, which in ur theory appear to be directly connected with information as ''the'' fundamental substance. This hopefully will help to provide a new understanding of the concept of information.
The quantum theory of ur objects postulates that all existing physical objects and their properties are constructed from fundamental objects called ur objects being described by an element of a two dimensional complex Hilbert space. This approach is based on the assumption that quantum theory represents a theory being constitutive for human knowledge. Physical objects are characterized by the information one can gain from them being contained in the quantum state they are described by. Since every Hilbert space can be represented as a tensor product of two dimensional Hilbert spaces, one is led to the ur objects. According to this approach relativistic quantum fields and thus the existence of a Minkowski space-time are the consequence of an iteration of a quantization of binary alternatives. In the original formulation there was only obtained a description of quantum fields on a flat Minkowski space-time. In this work there is made the attempt to incorporate general relativity and to describe the gravitational field within this approach. Thus the existence of a (3+1)-dimensional space-time in the sense of general relativity is assumed to be a consequence of quantum theory interpreted in an abstract sense.
The quantum theory of Ur-alternatives of Carl Friedrich von Weizsaecker tries to constitute general quantum theory based on the concept of logical alternatives in time. Based on this interpretation of quantum theory the existence of free objects in space, their symmetry properties and their interactions shall be inferred. The alternatives are represented by binary alternatives, which are called Ur-alternatives because of their logically fundamental character. Through Ur-alternatives as elementary quantum theoretical units of information the Copernican revolution with respect to the question of space is realized in a consequent way. This means that not the objects of nature are in a given space, but the existence of space arises as a kind of indirect representation of relations between abstract quantum theoretical objects. The Ur-alternatives do not exist in a given physical reality, but the existence of space is constituted by Ur-alternatives at all. Such a concept of reality is implicitly contained within the uncertainty relation and can be seen especially in the EPR-paradoxon. It is shown in this thesis in a mathematical consistent way that a state in the tensor space of many Ur-alternatives can directly be mapped into a real three dimensional space which means that together with the dynamics a representation in a (3+1)-dimensional space-time becomes possible. By considering the G2G_2G2​ an approach for the incorporation of the internal symmetries can be suggested. Furthermore the Ur-alternatives enable the constitution of a concept of interaction, which is based on quantum theoretical entanglement. By using this concept it is tried to obtain a purely quantum theoretical description of electromagnetism and gravity. This corresponds to a much more principle and in a radical sense background independent way of quantization.
The concept of quantum information is introduced as both normed superposition of two orthogonal subspaces of the separable complex Hilbert space and invariance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen.The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing a state of a quantum system) as its value as the bound variable.A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is representable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above.Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and invariance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper.Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries 𝑈𝑈(1), 𝑆𝑆𝑆 (2), and 𝑆𝑆𝑆 (3) of the Standard model.
Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time is what generates choices by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latter. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of “Hume’s principle” to be interpreted in terms of quantum mechanics as equivalence of “many” and “much” underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis.
Introduction Many philosophers and physicists have expressed great hope that quantum information theory will help us understand the nature of the quantum world. The general problem is that there is no widespread agreement on what quantum information is. Hence, such pronouncements regarding quantum information theory as the savior of the philosophy of physics are hard to evaluate. Much work has been done producing and evaluating concepts of information. In I have defended and articulated the Schumacher concept of quantum information. Roughly speaking, quantum information is construed as the statistical behavior associated with the measurement of a quantum system. Hence it is a coarse-grained operational description of quantum systems, with no recourse to the fundamental ontological features of quantum systems responsible for such behavior. From this perspective, construing quantum mechanics as a theory of quantum information departs from the traditional interpretive endeavors of philosophers and physicists. The question is whether there is any motivation for taking such a view. The theorem of Clifton, Bub, and Halvorson (CBH) provides just such a motivation. The theorem guarantees that, if a theory T satisfies certain conditions, there will exist an empirically equivalent C *-algebraic theory that has a concrete representation in Hilbert space, which it is notoriously difficult to interpret as a constructive or mechanical theory. In such a case, any underlying ontologies philosophers develop that are compatible with T will be undermined by the C*-algebraic equivalent. Bub suggests in light of this in-principle uncertainty regarding ontology that we re-conceive of quantum mechanics as a theory about quantum information.
A scientific explanation of consciousness and its evolution has become possible by a generalized quantum information concept. Crucial to this is a new understanding of matter and energy. Our consciousness, being an information structure, processes the information that we absorb from our bodies and our environment. This information reaches us as properties of material objects and the massless quanta of light, the photons. One of the properties of matter is motion – physically, kinetic energy. Einstein's E = mc 2 shows that motion can be converted into matter. Matter is thus equivalent to one of its properties. The mathematical-physical structure of quantum theory has opened up a comprehensive view of the construction of complex systems from simple structures. Such a construction can and should begin with the simplest quantum structure , a quantum bit . Matter and also photons can thereby ultimately be understood as manifestations of abstract quantum information. Abstract, i.e. still meaning-free, and absolute, i.e. cosmically based, bits of quantum information, AQIs, provide the common basis for consciousness, energy and matter. This explains the mutual influences between matter, energy and the information that is meaningful for a living being.