Quantum Gravity Necessitates the Quantization of SpaceTime: it has a Non-Commutative Algebraic Structure

Quantum Gravity Necessitates the Quantization of SpaceTime: it has a Non-Commutative Algebraic Structure In the search for a complete theory of quantum gravity there has been many proposals
over the last years. One approach that has gained quite some popularity comes from mathematics: non-commutative geometry. The main motivation to study
non-commutative geometry in the realm of physics comes from the “geometrical” measurement problem, [DFR95]. Basically, the problem goes as follows; by combining principles of quantum mechanics and general relativity it turns out that the
measurement of a space-time point with arbitrary precision is not possible and thus
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space-time, around the Planck length, does not have a continuous structure. Hence,
geometry of space-time has to be replaced by a non-commutative version thereof.
One of the most studied and well understood examples of such a non-commutative
geometric structure is known by the Moyal-Weyl plane. In essence, one has a
constant non-commutativity between the space-time coordinates (that are replaced
by operators). This is equivalent to the non-commutativity that is introduced
between the observables, i.e. momentum and coordinate, in quantum mechanics.
Another essential reason to study non-commutative geometry is the search for
quantum gravity. Let us elaborate on this point a bit further. Essentially, the Einstein
Field Equations of general relativity tell us that gravity is a force, experienced
by the curvature of space-time. Hence, ultimately quantization of gravity corresponds
to the quantization of space-time. What is quantization? This question has many
possible answers. However, all answers have something in common. Namely, we
take a classical theory and by introducing, for example, a new product, we extend
the classical framework. An interesting example is to take classical mechanics and
perform the so-called deformation quantization in order to obtain quantum
mechanics but in terms of functions with a non-commuting product. Hence, in the
former example we went from classical mechanics to quantum mechanics. Therefore,
deformation of classical space-time should lead us to a quantized, i.e. quantum
space-time. Another, equally valid approach is to start with a non-commutative
algebra and to obtain by techniques developed in non-commutative geometry the
equivalent of a metric, i.e. a distance function that corresponds to a non-commutative
space-time and therefore a quantum space-time.
Hence, what we thrive to have, in the context of non-commutative geometry,
is a mathematical rigorous deformation quantization of space-time that in addition
displays, at least the first, steps towards the quantization of gravity. Intuitively, there
are two main ideas that motivate this work. The first idea takes into account that
in quantum theory all observables are given by self-adjoint (possibly unbounded)
operators on a Hilbert-space. Hence, in a theory of quantum gravity, a quantity
identified with the metric, has to be given in terms of self-adjoint operators defined
on a dense subset of the Hilbert space as well. The second intuitive idea is the
fact that many proposed quantum gravity theories face the issue of recovering flat
space-time. Therefore, we should start with the Minkowski metric, that is given by
self-adjoint operators, and by a deformation quantization induce a curved space-time
metric. This is the reason why we refer to such an outcome as emergent gravity. In
particular, Gravity, i.e. curved space-time is not preassumed but rather induced by a
well-defined mathematical framework in the context of operator theory.
Hence, the concrete question that we pose is the following: Can we obtain a
curved space-time from a purely flat space-time by deformation quantization of
linear operators that are defined on a Hilbert space? Hence, is there a possibility to
understand the emergence of a gravitational field from a strict deformation of flat
space-time. Moreover, in this context, does such a curved space-time supply us with
strong arguments in favor of a non-commutative space-time?
Primary answers to these questions were given in [And13] and [Muc14], where
the dynamics of a free quantum mechanical particle were deformed and a minimal
substitution was induced. The minimal substitution was understood as a gravitomagnetic
field. The advantage therein was an understanding of gravitational effects
that were obtained by analogous electro-magnetic phenomena. However, the subject
of curved metrics was neither obtained nor investigated in this context.
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Hence, a scheme of obtaining a curved metric from a flat one by a strict deformation
quantization procedure is still missing. This paper intends to resolve this
issue. We present a concrete and moreover (mathematically) strict scheme in which it
is possible to apply a deformation quantization of the flat metric and obtain a curved
space-time. The emergence of curvature by a deformation quantization is achieved
by combining two major mathematical developments in non-commutative geometry.
The first development that we use is the universal differential structure of Connes
[Con95, Chapter 3, Section 1] that associates to any associative unital algebra a
differential structure. While the second framework used is the Rieffel deformation,
[Rie93] (and extensions thereof see [GL07], [GL08], [BS], [BLS11]), that deals with
strict deformation quantizations of C^∗−algebras.