If general relativity (GR) describes gravity as a fictitious force, then shouldn't it be easy to quantize gravity in the sense that the spacetime distortion of a body changes in discrete values as a particle (e.g. an electron) is emitted or absorbed by the same body? Analog quantum fields give rise to discrete elementary particles and the particles and bodies made of particles distort the analog specetime discretely. What is theoretically or experimentally inconsistent in that picture? Or if the existence of gravitons is discovered would that mean GR is wrong in describing gravity and that gravity is an actual force?

Can we say that some of the questions that a unification of general relativity and quantum mechanics may answer are if the gravitational interaction between spacetime and matter and energy happens somehow directly or through a quantum gravity field or another undiscovered field(s) by means of real and/or virtual particles of that field? Also, can we say whether spacetime itself is also somehow quantized or not?


To explain the problem properly we need to think not in terms of "quantum mechanics" (by which you probably mean a quantization of Newtonian mechanics), but rather quantum field theory (the quantization of a Lorentz-invariant field theory). The key difference is that QM considers the state of a small fixed number of particles, whereas in QFT each particle species is associated with a "field" analogous to a QM wavefunction, so that the field's state may leave even the number of particles fuzzy just as QM creates uncertainty regarding a particle's position. For example, the Higgs field $\hat{\phi}(x)$, which quantizes a hypothetical "classical" scalar field $\phi (x)$, is associated with the Higgs boson, and in principle you could consider a quantum-mechanical model in which one Higgs boson's wavefunction is $\phi (x)$.

Similarly, the electromagnetic field $A_\mu$ is promoted to $\hat{A}_\mu$, while general relativity's metric tensor $g_{\mu\nu}$ becomes $\hat{g}_{\mu\nu}$. It's not too hard to write down a QFT in a curved spacetime for a non-quantized $g_{\mu\nu}$ even if it's dissimilar to Minkowski space (although depending on the spacetime considered all manner of things can go wrong, such as a vacuum state being undefined or lacking the expected spacetime symmetries). But when we quantise the metric tensor we face additional difficulties. I'll give just a few details with some reading material:

  1. General relativity's high energy entropy-energy relation is a power law, as expected of a quantum field theory, but the exponent is wrong. This paper gives a fuller explanation, including the finding that, in anti de Sitter space, there is a fortunate change in the power law that has motivated a lot of string-theoretic research.
  2. The fact that $G$ has a negative mass dimension accounts for the non-renormalizability of quantum gravity, so at high energies we need more and more free parameters to predict anything (though some theories, such as string theory and loop quantum gravity, have some very complicated ways around that). However, as this paper explains, the same techniques that consider the quantization of other interactions in nature meet some degree of success for gravity too.
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    $\begingroup$ Nice answer J.G. Though, can you explain in a few words, what non-renormalizabilty of a theory means? Thank you :) $\endgroup$ – Robert Poenaru Jul 26 '17 at 13:24
  • $\begingroup$ @J.G. That is exactly what I’m wandering. Why would we want to quantize the metric tensor? That would imply the essence of the fabric of spacetime is itself quantized. Are there any experimental or well established enough theoretical hypotheses for that idea. Can’t we just express the curvature tensor as some function of the metric and the stress-energy tensor? In that way can’t we present the gravitational spacetime distortion in terms of a quantized stress-energy tensor and a constant metric? $\endgroup$ – Georgi Pavlov Jul 29 '17 at 13:12
  • $\begingroup$ This is a great answer $\endgroup$ – Señor O Jul 29 '17 at 14:58

Why is there a need to unify Quantum mechanics and General Relativity and what is meant by such a unification?

One could have asked a similar question of Maxwell: was there a need to unify electricity and magnetism? The need was intellectual, to get a mathematical model that described existing observations of electric and magnetic effects and be predictive of new situations/systems. And look where satisfying this need got us.

Physicists, who appeared as a different discipline from mathematics and philosophy from the appearance of Newton on the scene, have been actively pursuing theoretical models of unification.

Physics has developed different mathematical models for different ranges of the variables of space and time. These different models meld and emerge at the boundaries where both frameworks apply: classical mechanics emerges from quantum mechanics. Classical electromagnetism emerges from quantum electrodynamics. This happens seamlessly.

General relativity reduces to Newtonian gravitation in flat spaces . Special relativity reduces to classical mechanics for non relativistic phase spaces.

With the establishment of the standard model of particle physics , the path to unifying three of the known fundamental forces through GUT theories is well trodden. It really is impressive how weak and electromagnetic theories have been unified at the quantum level and there is real evidence for the unification of the strong force to the other two.

At present physics assumes that the underlying level of nature is quantum mechanical, and this assumption works. The difficulty arises with General Relativity, which works with the coordinates themselves on which the quantum mechanical effective field theories are built. The need for quantizing gravity rigorously arises when cosmological models are built and the variables are stretched to their limits.

At present the usual way of building quantum field theories gives only effective quantization of gravity, used in cosmologial models with good resuls. The mathematics though has infinities that cannot be brushed away or renormalized. The search is on for a quantization that is rigorous. String theories offer the possibility of embedding the standard model of particle physics and quantizing gravity, but they are still at a research stage.

Can we say then that some of the questions that a unification of General relativity and Quantum mechanics may answer are if the gravitational interaction between spacetime and matter and energy happens somehow directly or through a quantum gravity field or another undiscovered field(s) by means of real and/or virtual particles of that field. Also whether spacetime itself is also somehow quantized or not?

In effective field theories the standard way of building a model is used, having a graviton exchanged, the graviton building up the classical gravitational waves of general relativity. These cannot be unified with the existing SU(3)xSU(2)xU(1) current particle theory in an elegant manner.

As pointed out above, research in string theories offers a path for embedding the standard model , its elementary particles being vibrations of a basic unite , a string. The different particles are vibrations on this string and have the group symmetries of the standard model, and in addition a spin two graviton exists as a vibrational level to model the quantization of gravity. The future research will show if this is a good path for unifying all four known forces in a quantum mechanical underlying framework.

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I would try to provide an answer in a simplified manner. We need to unify Quantum Mechanics and General Relativity in order to formulate a Theory of Everything. Such a theory would combine all the existing four fundamental forces - 1.) Weak Nuclear, 2.) Strong Nuclear, 3.) Electromagnetism, 4.) Gravity into a single force. General Relativity and Quantum Mechanics are the two opposite branches of Modern Physics. Quantum Mechanics deals with small scale objects like electrons, protons, atoms etc. whereas General Relativity deals with large scale objects like galaxies, clusters, stars etc. The combination of these two theories will yield us a Relativistic Quantum Theory of Gravity, which may or may not be related to the existence of Gravitons. If gravitons are proven to exist it has nothing to do with falsifying General Relativity. Since, we would then have formulated a theory of gravity for quantum mechanics, which will explain gravity for small scale objects but will fail when applied on large scale objects, just like General Relativity fails when applied on small scale objects.

Therefore the combination of General Relativity and Quantum Physics will yield a Relativistic Quantum Theory of Gravity which will include Quantum as well as Relativistic effects to explain the nature of Gravity on medium scale objects. We need such a theory in order to get the Theory of Everything (T.O.E) as stated earlier but such a theory would also help us study Singularities. Since in a Singularity, the density of a body is so tremendous that Quantum Effects cannot be ignored whereas the gravitational force is so strong that General Relativity cannot be ignored too. Therefore a combined Relativistic Quantum Theory of Gravity would help us study the heart of Black Holes or the Big Bang singularity.

There have been many successful attempts to quantize all the three fundamental forces and then to combine them into a Grand Unified Theory, the only force that is left out is Gravity.

Reasons for the difficulty to Quantize Gravity :-

1.) Quantizing gravity is partially related to the background independence of General Relativity. If we try to quantize General Relativity we always end up getting it background dependent. There are some theories in which background independent quantization procedures follow eg. Loop Quantum Gravity but it is impossible to prove the existence of such procedures experimentally.

2.) It is hard to establish a natural unification of gravity and Standard Model. It is mostly an issue of obtaining a complex gravity + gauge Lagrangian from simple geometrical model.


We therefore conclude that we will get closer and closer in obtaining a precise Relativistic Quantum Theory of Gravity through time. It is hard to obtain a Quantum Theory of Gravity because it is hard to formulate a background independent i.e. independent of any background field (like Electromagnetism is dependent on the background of Minkowski's space-time but General Relativity is consisting of the space-time itself) theory and even if such theories are formulated they are only mathematically true but never verified experimentally.

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Since the physical world that presents itself to us is a unity we expect a single theory to explain it, hence a situation where we have two theories, general relativity and quantum mechanics, is not ideal, though admittedly they cover very different scales.

Secondly, we already have partial successes in unifying both theories, for example QFT is the result of unifying special relativity with quantum mechanics and such theories are the basic ingedients that go in constructing the standard model.

Further, there is also the semi-classical result of the Bekenstein-Hawking radiation which is used as guide to unification, for example both Loop Quantum Gravity and String Theory have proposals to explain this through microstates of some kind.

Actually, LQG quantises spacetime via area and volume operators which are discrete valued, and this notion is taken as the starting point for causal set theory that dispenses with the whole quantisation procedure and attempts to build a theory by taking the discrete structure as a given.

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