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In this video Alain Connes made a comment about the ,,quantum corrections'' of the geometry. I would like to understand this notion in some details since I haven't found anything about this in the literature. I'm guessing that this notion is different to the notion of so called ,,inner fluctuations'' of the metric (see for example this discussion) and unlike the inner fluctuations, quantum corrections is less understood. So making the long story short my question is

What are quantum corrections of a geometry?

EDIT: Let me be a little bit more precise. One of the highlights of noncommutative geometry is that it can reproduce the complicated Lagrangian of the Standard Model from the so called spectral action and can be seen as pure gravity on some slightly noncommutative space of the form $M \times F$. Here $F$ is not correctly defined as a set but manifests itself as a noncommutative algebra: the idea is that $F$ represents some noncommutative space. Since this noncommutativity is already present on the electroweak scale (i.e. distances approx. $10^{-16}\text{cm}$) I guess that the quantum corrections of geometry should be:

  • already visible on this (much lower than Planck) energy scale
  • understood within the context of noncommutative geometry i.e. Dirac operators and spectral triples (if we take the lesson from the Standard Model seriously).

So let me state the problem once again, this time more precisely:

How to understand quantum corrections of geometry within the context of noncommutative geometry?

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  • $\begingroup$ This is rather old but might be a nice starting point for diving deeper into quantum gravity: [arxiv.org/pdf/gr-qc/0108040.pdf] $\endgroup$ – Kai May 21 at 3:57
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Physicist chiming in; "quantum corrections of a geometry" represents the vague idea that gravity should be quantum.

Classically, gravity is a geometric theory: "reality" is modelled as a (Lorentzian) manifold, whose metric can be found -- in principle -- by solving a system of PDEs. To be more specific, the metric $g$ is such that the classical action $S[g]$ is stationary, $$ S'[g]\equiv0 $$ where $S$ is, say, the Hilbert action $S=\int_X R$. (This is the canonical choice, which agrees with experiments to a great level of accuracy, but there are other choices that may work better in extreme regimes; a popular alternative is $S=\int_X f(R)$ for some appropriate function $f$, cf. f(R) gravity).

Quantumly, this does not longer hold. Gravity should still be a geometric theory, but "quantum corrected", meaning that the paradigm changes but should reduce to the standard (classical) framework in the classical limit (schematically written as $\hbar\to0$). This is the problem of quantum gravity. The intuitive idea is that the metric is no longer fixed by a system of differential equations, but "fluctuates": there should be a "measure" $\mu[g]$ in the "space of metrics" such that, when we perform a given experiment, a single metric $g$ is measured, drawn at random from this space of metrics, and with probability $\mu[g]$. The "expectation value" for the geometry of spacetime is $$ \langle g\rangle\sim\int \mu[g] $$ where the integral is over "the space of all metrics", whatever this means. A popular choice for the measure is $\mu[g]=e^{-S[g]/\hbar}$, which has the nice property that, in the $\hbar\to0$ limit, is peaked around the stationary point $S'[g]=0$, and so this measure appears to single out the classical configuration as a special point in this space.

Obviously, this is not a well-defined procedure, but it represents what we physicists would like to have. We leave it to the mathematicians to develop a theory of "integration over function spaces" strong enough to accommodate this our intuition.

But anyway, the take home message is that, classically, nature is supposed to be described as a manifold with a fixed (Lorentzian) metric. But if you go to very high energies (or very short distances), the classical picture breaks down and you need to take into account "quantum corrections": the metric is no longer fixed but becomes a random variable, peaked around the classical value but "fluctuating" according to some measure. Making this precise is the open problem in physics.

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I presume this refers to quantum corrections to the metric tensor. These are expected to be important in general relativity when the curvature of space-time approaches the Planck scale $\sqrt{G\hbar/c^3}\simeq 10^{-33}$ cm.

Some pointers to the literature:

Einstein Equation with Quantum Corrections

Quantum Corrections to the Schwarzschild and Kerr Metrics

Quantum Corrections to the Reissner-Nordstrom and Kerr-Newman Metrics

The edit also asks "How to understand quantum corrections of geometry within the context of noncommutative geometry (NCG)?".$^*$ I do not think this understanding exists at present. NCG has given a geometrical description of the Standard Model of elementary particles, and there might be ways to unify this with gravity, but that development is still very much in its infancy.

$^*$ The name noncommutative geometry is misleading, in the sense that it suggests a quantum theory (since we all know that non-commuting operators play such an essential role in the quantum world), but in fact the homological theory that constitutes NCG does not itself contain quantum effects.

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