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%%% Primary Title: space-time quantization in quantum gravity theories
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\begin{document}
\subsection{Physical and Mathematical Problems with Quantizing Spacetime in Quantum Gravity Theories}
Beginning with Riemann there has been a prevailing tradition among mathematicians to leave \htmladdnormallink{space-time}{http://planetphysics.us/encyclopedia/SR.html} structure
problems for theoretical physicists to solve; this tradition has been however punctuated by
mathematical contributions made to the space-time structure problem by major contributors such as
Minkowski, Poincar\'e, Weyl, viewed in conjunction with, or separate from, those of \htmladdnormallink{Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html} and Lorentz.
This tradition is currently changing very rapidly with the major contributors being dedicated mathematicians.
As there are so many excellent contributing mathematicians it is very hard to provide a short list, and thus
only two will suffice here: \'E. Cartan (\htmladdnormallink{spinor}{http://planetphysics.us/encyclopedia/ECartan.html} theory) and A. Connes (\htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} in
quantum gravity theories and \htmladdnormallink{SUSY}{http://planetphysics.us/encyclopedia/SUSY2.html}). Arguably, such a trend will continue in favor of mathematicians contributing heavily to the foundation of unified physical theories of space and time as well as developing new mathematical \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} for their `own sake', or in their own right. Examples abound to the point that there is talk not only of mathematical, or \htmladdnormallink{theoretical physics}{http://planetphysics.us/encyclopedia/PhysicalMathematics2.html}, but also of physical mathematics (that is, just for the sake of Abstract mathematics); such is the reaction from many in the \htmladdnormallink{high-energy physics}{http://planetphysics.us/encyclopedia/LargeHadronCollider.html}, `elementary \htmladdnormallink{particle' physics}{http://planetphysics.us/encyclopedia/SUSY4.html} community. Negative attitudes aside, this is a relatively new, fertile and rather exciting borderline \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} at the very junction of mathematics and physics-- where the `intertwinner \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html}' act on both physical and mathematical spacetimes, or topoi, etc., in order to create new science that is neither `pure' mathematics nor `pure' physics (which has never existed anyway, as it is always expressed in a mathematical `language', theory, model, \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} or formalism). \\
The new trend means however much more: it means to employ deeper mathematics and mathematical tools in order to either develop or derive deeper physics. On both sides of the fence many would argue against the other side to maintain a `status quo of pure' mathematics- that has never existed; this would be, of course, as counter-productive as it can be to sciences, in general, and especially to mathematics, \htmladdnormallink{computer}{http://planetphysics.us/encyclopedia/SupercomputerArchitercture.html} science and informatics, for example.
\subsection \htmladdnormallink{{quantum fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}, symmetry, space-time and connections to general relativity}
As the experimental findings in high-energy physics--coupled with
theoretical studies-- have revealed the presence of new fields and symmetries,
there appeared the need in modern physics to develop systematic procedures for
generalizing/generating space-times and \htmladdnormallink{quantum state space}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} (\htmladdnormallink{QSS}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html})
representations that reflect the existence of such new fields and symmetries.
In the general relativity (\htmladdnormallink{GR}{http://planetphysics.us/encyclopedia/SR.html}) formulation, the local structure of space-time -- which is characterized by its \htmladdnormallink{tensors}{http://planetphysics.us/encyclopedia/Tensor.html} and curvature -- incorporates \htmladdnormallink{The Gravitational Fields}{http://planetphysics.us/encyclopedia/GravitationalField.html} surrounding various \htmladdnormallink{masses}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}. In Einstein's own representation, the `physical space-time of GR' has the structure of a Riemann $R^4$ space over large distances,
although the detailed local structure of space-time -- as Einstein suggested -- is likely to
be significantly different.
On the other hand, there is a growing consensus in theoretical physics that
a valid theory of \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html} requires a much deeper understanding of the
small (est)--scale structure of \htmladdnormallink{quantum space-time}{http://planetphysics.us/encyclopedia/SUSY2.html} (\htmladdnormallink{QST}{http://planetphysics.us/encyclopedia/SUSY2.html}) than currently
developed. In Einstein's GR theory and his subsequent attempts at developing an
\htmladdnormallink{unified field theory}{http://planetphysics.us/encyclopedia/QAT.html} (as in the space concept advocated by Leibnitz), space-time
does \emph{not} have an \emph{independent existence} from \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, matter or
fields, but is instead an entity generated by the \emph{continuous}
transformations of fields \cite{Einstein} (Einstein, 1950, 1954). Hence, the
continuous nature of space-time adopted in GR and Einstein's subsequent field theoretical
developments. Furthermore, the quantum, or `quantized', versions of space-time, \emph{QST},
are operationally defined through local \htmladdnormallink{quantum measurements}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} in general \htmladdnormallink{reference frames}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} that
are prescribed by GR theory. Such a definition is therefore subject to the postulates of both
GR theory and the axioms of \htmladdnormallink{local quantum physics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} (that are briefly summarized in Subsection
3.3). We must empasize, however, that this is \emph{not} the usual definition of \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} and
time \htmladdnormallink{observables}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} in `standard' \htmladdnormallink{QM}{http://planetphysics.us/encyclopedia/FTNIR.html}. Therefore, the general reference frame positioning in QST
is itself subject to the \htmladdnormallink{Heisenberg uncertainty principle}{http://planetphysics.us/encyclopedia/QuantumParticle.html}, and therefore it acquires through
quantum measurements a certain `fuzziness' at the Planck scale which is intrinsic to all
microphysical quantum \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}, as further explained in this \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}. Whereas Newton, Riemann,
Einstein, Weyl, Hawking, Weinberg and many other exceptional theoreticians regarded the
physical space as being represented by a \emph{continuum}, there is an increasing number of
proponents for a \emph{discrete, `quantized'} structure of space-time. The latter view is not
without its problems and advantages. The biggest problem for any discrete, `point-set' (or
discrete topology), view of physical spacetime is not only its immediate conflict with
Einstein's General Relativity representation of spacetime as a \emph{continuous Riemann} space, but also the
impossibility of carrying out quantum measurements to localize precisely either quantum events
or masses at singular (in the sense of disconnected, or isolated), sharply defined, geometric
points in space-time. One of the proposed resolutions of this problem is \htmladdnormallink{non-commutative geometry}{http://planetphysics.us/encyclopedia/NAQAT2.html} (\htmladdnormallink{NCG}{http://planetphysics.us/encyclopedia/QAT.html}), or `Quantum Geometry', where QST has `no points' (or perhaps no point!), in
the sense of visualization of such a geometrical space as some kind of a distributive and commutative
lattice of space-time `points'. The quantum `\htmladdnormallink{metric}{http://planetphysics.us/encyclopedia/MetricTensor.html}' of QST in NCG would be related to a
certain, fundamental quantum field \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html}, or `fundamental triplet (or quintet)'
construction (Connes, 2004). Although \htmladdnormallink{quantization}{http://planetphysics.us/encyclopedia/MoyalDeformation.html} is standard in \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html} (QM) for
most of the \htmladdnormallink{quantum observables}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}, it does run into major difficulties when applied to position
and time. In standard QM, there are at least two implemented approaches to solve the problem,
one of them designed 72 years ago by von Neumann (1933).
\bigbreak
Another potential concern is the inadequacy of the long-standing model of
space-time as a $4$--dimensional \htmladdnormallink{manifold}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} \emph{with a Lorentz metric}. The hope of some of
the earlier approaches to quantum gravity (\htmladdnormallink{QG}{http://planetphysics.us/encyclopedia/SUSY2.html}) was to cope with extremely small length scales
where a manifold structure may be justifiably foresaken (for instance, at the Planck length
$L_p = (\frac{G\hslash}{c^3})^{\frac{1}{2}} \approx 10^{-35}m$). On the other hand, one needs to
reconcile the discreteness versus continuum approach in view of
space--time diffeomorphisms and that space--time may be suitably modeled as some \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} of
`combinatorial space' (such as a \htmladdnormallink{simplicial complex}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html}, a poset, or a \htmladdnormallink{spin network}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html}) The
monumental difficulty is that to the present day, apart from a distinct lack of experimental
evidence, there is no specific agreement on the kind of data, plus no agreement on the actual
conceptual background to obtaining the data in the first place(!) This difficulty equates with
how one can relate the approaches to QG to run the gauntlet of conceptual problems in \htmladdnormallink{QFT}{http://planetphysics.us/encyclopedia/HotFusion.html} and
(General Relativity) GR. To quote an example, the space--time \htmladdnormallink{metric tensor}{http://planetphysics.us/encyclopedia/Curved4DimensionalSpace.html}:
$\gamma = (\gamma_{ab})$ is less a fundamental field than perhaps once thought since it leads
to describing an essentially classical gravitational field. A case study in ref. \cite
{BIsham1} involves quantizing one side of Einstein's field equations by a quantum expectation value, so that a coupling of $\gamma$ to quantized matter is given by an expression such as:
$$ G_{\mu \nu} (\gamma) = \langle~ \psi ~\vert T_{\mu \nu} (g, \hat{\phi} \vert~ \psi
~\rangle~, $$
\bigbreak
where $ \vert \psi \rangle$ denotes a state in the \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} of quantized matter variables $\hat{\phi}$, and the subsequent source of the gravitational field is given by the expectation %%@
of the corresponding energy--momentum tensor $T_{\mu \nu}$~. Unfortunately, this expression is
not without its ontological and `physical' problems sufficiently serious to prevent the
development of a complete QG theory that includes this expression. Three possible approaches
were suggested by Butterfield and Isham in ref. \cite {BIsham1} (cf. also an extensive survey
article by Rovelli, 1997):
\begin{itemize}
\item[(1)] to develop and test a quantized form of classical relativity theory;
\item[(2)]to recover GR as the low \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} limit of a QFT approach which is not a quantization
of a classical theory (e.g., \emph{via \htmladdnormallink{quantum algebras/}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html} and their representations});
\item[(3)]to develop a new theory, such as a `quantization of topology' or
`causal' structures where, for instance, microphysical states provide amplitudes to the values
of quantities whose \htmladdnormallink{norms}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} squared define probabilities of occurrence for \emph{physical},
quantum events.
\end{itemize}
We turn now to another facet of quantum measurement. Note first that QFT
\htmladdnormallink{pure states}{http://planetphysics.us/encyclopedia/PureState.html} resist description in terms of field configurations since the former are not
always physically either observable or interpretable. \htmladdnormallink{Algebraic Quantum Field Theory}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} (\htmladdnormallink{AQFT}{http://planetphysics.us/encyclopedia/MetaTheorems.html}) as
expounded by Roberts (2004) points to various questions raised by considering theories of
(unbounded) operator--valued distributions and \emph{quantum field nets} of \htmladdnormallink{von Neumann algebras}{http://planetphysics.us/encyclopedia/CoordinateSpace.html}. Using in part a gauge theoretic approach, the idea is to regard two field theories
as equivalent when their associated nets of observables are isomorphic. More specifically,
AQFT considers taking \emph{additive nets of quantum field algebras} over subsets of Minkowski
space, which among other properties, enjoy Bose--Fermi commutation \htmladdnormallink{relations}{http://planetphysics.us/encyclopedia/Bijective.html}. There may be
analogs with \htmladdnormallink{sheaf theory}{http://planetphysics.us/encyclopedia/PAdicMeasure.html} in this approach, even though these analogs appear to be limited.
The typical AQFT net does not seem to give rise to a \htmladdnormallink{presheaf}{http://planetphysics.us/encyclopedia/CommutativeRingWithUnit.html} because the relevant \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} orientations are in reverse. Closer then is to regard a net as a precosheaf, but the
additivity does not allow proceeding to a cosheaf structure. This may be a \htmladdnormallink{reflection}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html} of some
deeper incompatibility of AQFT with those aspects of quantum gravity (QG) where the sheaf--
theoretic/topos approaches are advocated (as, for example, in \cite {BIsham1} (Butterfield and
Isham, 1999-2004).
\begin{thebibliography}{9}
\bibitem{BIsham1}
Butterfield, J. and C. J. Isham: 2001, Space-time and the
philosophical challenges of quantum gravity., in C. Callender and
N. Hugget (eds. ) \emph{Physics Meets Philosophy at the Planck
scale.}, Cambridge University Press,pp.33--89.
\bibitem{BIsham2}
Butterfield, J. and C. J. Isham: 1998, 1999, 2000--2002, A topos
perspective on the Kochen--Specker theorem I - IV, \emph{Int. J.
Theor. Phys}, \textbf{37} No 11., 2669--2733 \textbf{38} No 3.,
827--859, \textbf{39} No 6., 1413--1436, \textbf{41} No 4.,
613--639.
\end{thebibliography}
\end{document}
