Editing Loop quantum gravity (section)

== The semiclassical limit and loop quantum gravity ==
The [[Classical limit]] is the ability of a physical theory to approximate classical mechanics. It is used with physical theories that predict non-classical behavior.{{Citation needed|date=July 2021}} Any candidate theory of quantum gravity must be able to reproduce Einstein's theory of [[general relativity]] as a classical limit of a [[quantum]] theory. This is not guaranteed because of a feature of quantum field theories which is that they have different sectors, these are analogous to the different phases that come about in the thermodynamical limit of statistical systems. Just as different phases are physically different, so are different sectors of a quantum field theory. It may turn out that LQG belongs to an unphysical sector – one in which one does not recover general relativity in the semiclassical limit or there might not be any physical sector.

Moreover, the physical Hilbert space <math>H_{phys}</math> must contain enough semiclassical states to guarantee that the quantum theory obtained can return to the classical theory when <math>\hbar \to 0</math> avoiding [[quantum anomalies]]; otherwise there will be restrictions on the physical Hilbert space that have no counterpart in the classical theory, implying that the quantum theory has fewer degrees of freedom than the classical theory.

Theorems establishing the uniqueness of the loop representation as defined by Ashtekar et al. (i.e. a certain concrete realization of a Hilbert space and associated operators reproducing the correct loop algebra) have been given by two groups (Lewandowski, Okołów, Sahlmann and Thiemann;{{sfn|Lewandowski|Okołów|Sahlmann|Thiemann|2006|pp=703–733}} and Christian Fleischhack{{sfn|Fleischhack|2006|p=061302}}). Before this result was established it was not known whether there could be other examples of Hilbert spaces with operators invoking the same loop algebra – other realizations not equivalent to the one that had been used. These uniqueness theorems imply no others exist, so if LQG does not have the correct semiclassical limit then the theorems would mean the end of the loop representation of quantum gravity.

=== Difficulties and progress checking the semiclassical limit ===
There are a number of difficulties in trying to establish LQG gives Einstein's theory of general relativity in the semiclassical limit:
# There is no operator corresponding to infinitesimal spatial diffeomorphisms (it is not surprising that the theory has no generator of infinitesimal spatial 'translations' as it predicts spatial geometry has a discrete nature, compare to the situation in condensed matter). Instead it must be approximated by finite spatial diffeomorphisms and so the Poisson bracket structure of the classical theory is not exactly reproduced. This problem can be circumvented with the introduction of the so-called master constraint (see below).{{sfn|Thiemann|2008|loc=Section 10.6}}
# There is the problem of reconciling the discrete combinatorial nature of the quantum states with the continuous nature of the fields of the classical theory.
# There are serious difficulties arising from the structure of the Poisson brackets involving the spatial diffeomorphism and Hamiltonian constraints. In particular, the algebra of (smeared) Hamiltonian constraints does not close: It is proportional to a sum over infinitesimal spatial diffeomorphisms (which, as noted above, does not exist in the quantum theory) where the coefficients of proportionality are not constants but have non-trivial phase space dependence – as such it does not form a [[Lie algebra]]. However, the situation is improved by the introduction of the master constraint.{{sfn|Thiemann|2008|loc=Section 10.6}}
# The semiclassical machinery developed so far is only appropriate to non-graph-changing operators, however, Thiemann's Hamiltonian constraint is a graph-changing operator – the new graph it generates has degrees of freedom upon which the coherent state does not depend and so their quantum fluctuations are not suppressed. There is also the restriction, so far, that these coherent states are only defined at the Kinematic level, and now one has to lift them to the level of <math>\mathcal{H}_{Diff}</math> and <math>\mathcal{H}_{Phys}</math>. It can be shown that Thiemann's Hamiltonian constraint is required to be graph-changing in order to resolve problem 3 in some sense. The master constraint algebra however is trivial and so the requirement that it be graph-changing can be lifted and indeed non-graph-changing master constraint operators have been defined. As far as is currently known, this problem is still out of reach.
# Formulating observables for classical general relativity is a formidable problem because of its non-linear nature and spacetime diffeomorphism invariance. A systematic approximation scheme to calculate observables has been recently developed.{{sfn|Dittrich|2007|pp=1891–1927}}{{sfn|Dittrich|2006|pp=6155–6184}}

Difficulties in trying to examine the semiclassical limit of the theory should not be confused with it having the wrong semiclassical limit.

Concerning issue number 2 above, consider so-called [[weave states]]. Ordinary measurements of geometric quantities are macroscopic, and Planckian discreteness is smoothed out. The fabric of a T-shirt is analogous: at a distance it is a smooth curved two-dimensional surface, but on closer inspection we see that it is actually composed of thousands of one-dimensional linked threads. The image of space given in LQG is similar. Consider a large spin network formed by a large number of nodes and links, each of [[Planck scale]]. Probed at a macroscopic scale, it appears as a three-dimensional continuous metric geometry.

To make contact with low energy physics it is mandatory to develop approximation schemes both for the physical inner product and for Dirac observables; the spin foam models that have been intensively studied can be viewed as avenues toward approximation schemes for said physical inner product.

Markopoulou, et al. adopted the idea of [[noiseless subsystems]] in an attempt to solve the problem of the low energy limit in background independent quantum gravity theories.{{sfn|Dreyer|Markopoulou|Smolin|2006|pp=1–13}}{{sfn|Kribs|Markopoulou|2005}} The idea has led to the possibility of matter of the [[standard model]] being identified with emergent degrees of freedom from some versions of LQG (see section below: ''LQG and related research programs'').

As Wightman emphasized in the 1950s, in Minkowski QFTs the <math>n-</math> point functions
<math display="block">W (x_1, \dots , x_n) = \langle 0 | \phi (x_n) \dots \phi (x_1) |0 \rangle , </math>
completely determine the theory. In particular, one can calculate the scattering amplitudes from these quantities. As explained below in the section on the ''Background independent scattering amplitudes'', in the background-independent context, the <math>n-</math> point functions refer to a state and in gravity that state can naturally encode information about a specific geometry which can then appear in the expressions of these quantities. To leading order, LQG calculations have been shown to agree in an appropriate sense with the <math>n-</math>point functions calculated in the effective low energy quantum general relativity.