Hilbert Space and pseudo-Riemannian Space: The Common Base of Quantum Information
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The document discusses the relationship between Hilbert space in quantum mechanics and pseudo-Riemannian space in general relativity, proposing that both share a foundational basis in quantum information. It argues that local interactions are described by single Hilbert spaces while non-local interactions, such as gravity, require a more complex representation. The author posits that this unified framework allows for a comprehensive understanding of physical processes across time and the nature of entanglement in quantum systems.
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• According to the Standard model, the
electromagnetic, weak, and strong
interaction can be unified as the following
composite symmetry of a single Hilbert
space:
[U(1)]X[SU(2)]X[SU(3)]
Consequently, these three fundamental
physical interactions are local in the sense of
quantum information](https://image.slidesharecdn.com/penchevhilbert-170813092733/85/Hilbert-Space-and-pseudo-Riemannian-Space-The-Common-Base-of-Quantum-Information-18-320.jpg)
Hilbert Space and pseudo-Riemannian Space: The Common Base of Quantum Information
- 1. Hilbert Space & Pseudo-Riemannian Space The Common Base of Quantum Information
- 2. Vasil Penchev • Bulgarian Academy of Sciences: Institute for the Study of Societies and Knowledge: Dept. of Logical Systems and Models • [email protected] ECAP9: 9th Congress of European Society of Analytic Philosophy, Munich, 21-26 August 2017 Venue: Ludwig-Maximilian-University (LMU), Main University Building, Geschwister-Scholl-Platz1
- 3. The thesis
- 4. • Hilbert space underlying quantum mechanics and pseudo-Riemannian space underlying general relativity share a common base of quantum information Hilbert space can be interpreted as the free variable of quantum information, and any point in it, being equivalent to a wave function (and thus, to a state of a quantum system), as a value of that variable of quantum information • In turn, pseudo-Riemannian space can be interpreted as the interaction of two or more quantities of quantum information and thus, as two or more entangled quantum systems
- 5. • Consequently, one can distinguish local and non-local physical interactions The local ones are describable by a single Hilbert space (or by any factorable tensor product of such ones) • The non-local physical interactions are describable only by means of that Hilbert space, which cannot be factorized as any tensor product of the Hilbert spaces, by means of which one can describe the interacting quantum subsystems separately
- 6. • Any interaction, which can be exhaustedly described in a single Hilbert space, such as the weak, strong, and electromagnetic one, is local in terms of quantum information Any interaction, which cannot be described thus, is nonlocal in terms of quantum information • Any interaction, which is exhaustedly describable by pseudo-Riemannian space, such as gravity, is nonlocal in this sense Consequently all known physical interaction can be described by a single geometrical base interpreting it in terms of quantum information
- 7. Arguments “pro” the thesis:
- 8. 1 • Hilbert space is introduced as the fundamental space of the quantum formalism: It is the simplest one, which can contain the solution of any case of the equivalence of a discrete motion (quantum leap) and a smooth motion (any motion according to classical physics)
- 9. 1+ • Consequently, any motion described as a linear automorphism of Hilbert space can be interpreted equally well both as a quantum and as classical motion Any quantity featuring that automorphism (such as any physical quantity definable according to quantum mechanics as a selfadjoint operator in Hilbert space) is referable both to a classic and to a quantum motion
- 10. 2 • However, the probabilistic interpretation of Max Born demonstrates even more: Hilbert space can unify furthermore the description of a possible and an actual state of a quantum system rather than only those of a discrete actual physical motion and of a smooth actual one • Thus it can guarantee the uniform description of a physical process in the future, present, and past, though absolute dissimilarity of these temporal “media”:
- 11. 2+ • The future is unorderable in principle corresponding to a coherent state of a quantum system containing all possible states as a “superposition” On the contrary, the past is always well- ordered being absolutely unchangeable • The present is forced to mediate and agree these two temporal “poles”
- 12. 2++ • Mathematically, this implies the well- ordering theorem equivalent to the axiom of choice • The present is the only temporal “media” which is able to harmonize the “no any ordering” state of the future and the well- ordered state in the past • This can be realized as a relevant series of choices exhaustedly describing any physical process and motion
- 13. 3 • The quantity of information can be described as the quantity of elementary choices necessary for an unordered state to be transformed into an ordered one or for an ordered state to be transformed into another also ordered but otherwise A bit (i.e. a “binary digit”) is the unit of an elementary choice between two equiprobable alternatives (e.g. either “0” or “1”) • A qubit (i.e. a quantum bit) is analogically interpretable as the unit of an elementary choice between infinitely many alternatives if it is defined as usual:
- 14. 3+ • A qubit is defined usually as the normed superposition of two orthogonal subspaces of Hilbert space It is isomorphic to a unit ball with two points chosen in it: the one can be any within the ball, and the other should be only on its surface • Hilbert space can be equivalently represented as an ordered series of qubits, and any point in it (i.e. any wave function or state of any quantum system), as just one value of this series
- 15. 4 • Thus Hilbert space and Minkowski space can be discussed as equivalent or as Fourier “twins” in terms of quantum information • Both represent ordered series of qubits being a discrete series in the case of separable Hilbert space and a continuous but discretizable one in the case of Minkowski space
- 16. 5 • Pseudo-Riemannian space is smooth Thus it possesses a tangent Minkowski space in any point of it • Gravity according the Einstein field equations can be defined only as a relation between two or more points (i.e. tangent Minkowski spaces) of pseudo-Riemannian space, but not in a single one (i.e. in one tangent Minkowski space)
- 17. 5+ • Minkowski space and Hilbert space are equivalent in the sense of quantum information as above Then, any tangent Minkowski space can be substituted by the corresponding Hilbert space and therefore one can demonstrate that gravity is nonlocal in the sense of quantum information
- 18. 6 • According to the Standard model, the electromagnetic, weak, and strong interaction can be unified as the following composite symmetry of a single Hilbert space: [U(1)]X[SU(2)]X[SU(3)] Consequently, these three fundamental physical interactions are local in the sense of quantum information
- 19. 7 • One can discuss that pseudo-Riemannian space, in which the tangent Minkowski spaces are replaced by equivalent Hilbert spaces being even isomorphic in the sense of quantum information, as Banach space Any two or more points of that Banach space possessing one tangential Hilbert space in each of them define an entanglement between those quantum systems, which are describable each in each of those Hilbert spaces
- 20. 7+ • Consequently, entanglement is also nonlocal in terms of quantum information and can be considered as a counterpart of gravity That viewpoint is possible after substituting the pseudo-Riemannian space with Banach space, and the tangent Minkowski spaces with the corresponding tangent Hilbert spaces
- 21. • Arguments “contra” the thesis are not known till now (at least as to me) Thank you for your kind attention! I am waiting for your questions or comments!