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arXiv:0910.3303

About the Origin of the Division between Internal and External Symmetries in Quantum Field Theory

Martin Kober

Published Oct 17, 2009• 1 citation
arXivHTMLPDF

Abstract

It is made the attempt to explain why there exists a division between internal symmetries referring to quantum numbers and external symmetries referring to space-time within the description of relativistic quantum field theories. It is hold the attitude that the symmetries of quantum theory are the origin of both sorts of symmetries in nature. Since all quantum states can be represented as a tensor product of two dimensional quantum objects, called ur objects, which can be interpreted as quantum bits of information, described by spinors reflecting already the symmetry properties of space-time, it seems to be possible to justify such an attitude. According to this, space-time symmetries can be considered as a consequence of a representation of quantum states by quantum bits. Internal symmetries are assumed to refer to relations of such fundamental objects, which are contained within the state of one single particle, with respect to each other. In this sense the existence of space-time symmetries, the existence of internal symmetries and their division could obtain a derivation from quantum theory interpreted as a theory of information.

DOI
10.48550/arxiv.0910.3303

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It is made the attempt to explain why there exists a division between internal symmetries referring to quantum numbers and external symmetries referring to space-time within the description of relativistic quantum field theories. It is hold the attitude that the symmetries of quantum theory are the origin of both sorts of symmetries in nature. Since all quantum states can be represented as a tensor product of two dimensional quantum objects, called ur objects, which can be interpreted as quantum bits of information, described by spinors reflecting already the symmetry properties of space-time, it seems to be possible to justify such an attitude. According to this, space-time symmetries can be considered as a consequence of a representation of quantum states by quantum bits. Internal symmetries are assumed to refer to relations of such fundamental objects, which are contained within the state of one single particle, with respect to each other. In this sense the existence of space-time symmetries, the existence of internal symmetries and their division could obtain a derivation from quantum theory interpreted as a theory 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.
A foundational result in relativistic quantum information theory due to Peres, Scudo, and Terno, is that von Neumann entropy is not Lorentz invariant. Motivated by the "It from Qubit" paradigm, here we show that Lorentzian symmetries of quantum information emerge naturally in a pre-spacetime setting, without any reference to external variables such as position or momentum. In particular, we derive the natural action of the restricted Lorentz group SO+(1,3)\text{SO}^+(1,3)SO+(1,3) on the internal degrees of freedom of a single qubit from a simple, information-theoretic principle we refer to as preservation of linear entropy. It is then shown that the Lorentz invariance of the linear entropy of a relativistic qubit is a special case of a much more general phenomenon, namely, that any spectral invariant of an operator we term the 'WWW-matrix' is an SL(2,C)⊗n\text{SL}(2,\mathbb C)^{\otimes n}SL(2,C)⊗n invariant scalar. Consequently, the linear nnn-partite quantum mutual information is shown to be an SL(2,C)⊗n\text{SL}(2,\mathbb C)^{\otimes n}SL(2,C)⊗n invariant for all nnn-qubit states. Finally, we show that the correlation function associated with a pair of qubits in the singlet state yields the Minkowski metric on the space of qubit observables, whose symmetry group is the full Lorentz group SO(1,3)\text{SO}(1,3)SO(1,3). In accordance with the "It from Qubit" paradigm, our results thus establish the natural emergence of relativistic spacetime structure from intrinsic properties of quantum information.
An interpretation of the quantum theory is proposed, allowing an explanation of the origins of the theory principles and its mathematical formalism. In contrast to the hitherto interpretations it is based on the ontology assuming existence of omniscience. The interpretation proposed explains ‐ in consistence with the relativity theory ‐ the results of the correlation experiments implying that an unobserved elementary particle does not exist objectively (beyond our consciousness) or that a quantum system makes o non-divisive unity bound through non-spacetime interactions. According to this interpretation, both these possibilities occur simultaneously.
Abstract A new interpretation of the quantum theory is proposed, allowing an explanation of the origins of the theory principles and its mathematical formalism. In contrast to the hitherto interpretations it is based on the objective idealistic ontology in the version proposed by George Berkeley. The interpretation proposed explains-in consistence with the relativity theory - the results of the correlation experiments implying that an unobserved elementary particle does not exist objectively (beyond our consciousness) or that a quantum system makes a non-divisive unity bound through non-spacetime interactions. According to this interpretation, both these possibilities occur simultaneously.
The paper proposes the theory of temporal extension for fundamental particles as a key to intuitively interpret and understand quantum symmetries and related quantum and relativistic phenomena. While space-time as a principal world view is accepted and utilized in fundamental theories, certain traditional concepts of time have not been challenged before. Such a conception is the view that physical objects can only have a point like feature on the time line. When introducing a more space like quality of time, which allows the temporal extension for physical objects and particles, a fundamentally new world view arises. In this view quantum symmetries and certain relativistic phenomena become more intuitive to describe and deal with and realism gets a new definition. As a result the temporal length theory also offers to eliminate the tension between the nonlocality of quantum physics and the locality of relativistic phenomena. The paper provides a detailed description of the theory.
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.