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. Consequently, one can distinguish local physical interactions describable by a single Hilbert space (or by any factorizable tensor product of such ones) and non-local physical interactions describable only by means by 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. 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.

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
• vasildinev@gmail.com
 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!