Here is a survey of some answers to the OP, a topic upon which lots of both scientific journal article consideration and educated layman's level treatments have been devoted.
An Overview From Wikipedia
Wikipedia has a brief general treatment of the issue
here. It notes that:
If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to
solid-state physics, would be the other. However, how to reconcile quantum theory with general relativity is still an open question.
Ordinary
quantum field theories, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth. In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime. Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as
Hawking radiation leading to the possibility that they
evaporate over time. As briefly mentioned
above, this radiation plays an important role for the thermodynamics of black holes.
The same article goes onto explain why it has been challenging to develop a quantum gravity theory despite numerous attempts using quite different approaches.
The demand for consistency between a quantum description of matter and a geometric description of spacetime, as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics. Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exist.
Projection of a
Calabi–Yau manifold, one of the ways of
compactifying the extra dimensions posited by string theory
Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems. Some have argued that at low energies, this approach proves successful, in that it results in an acceptable
effective (quantum) field theory of gravity. At very high energies, however, the perturbative results are badly divergent and lead to models devoid of predictive power ("perturbative
non-renormalizability").
Simple
spin network of the type used in loop quantum gravity
One attempt to overcome these limitations is
string theory, a quantum theory not of
point particles, but of minute one-dimensional extended objects. The theory promises to be a
unified description of all particles and interactions, including gravity; the price to pay is unusual features such as six
extra dimensions of space in addition to the usual three. In what is called the
second superstring revolution, it was conjectured that both string theory and a unification of general relativity and
supersymmetry known as
supergravity form part of a hypothesized eleven-dimensional model known as
M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity.
Another approach starts with the
canonical quantization procedures of quantum theory. Using the initial-value-formulation of general relativity (cf.
evolution equations above), the result is the
Wheeler–deWitt equation (an analogue of the
Schrödinger equation) which, regrettably, turns out to be ill-defined without a proper ultraviolet (lattice) cutoff. However, with the introduction of what are now known as
Ashtekar variables, this leads to a promising model known as
loop quantum gravity. Space is represented by a web-like structure called a
spin network, evolving over time in discrete steps.
Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced, there are numerous other attempts to arrive at a viable theory of quantum gravity, some examples being the lattice theory of gravity based on the Feynman
Path Integral approach and
Regge calculus,
dynamical triangulations,
causal sets, twistor models or the path integral based models of
quantum cosmology.
All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests (and thus to decide between the candidates where their predictions vary), although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available.
Gravity Is Not A Gauge Theory
A Q&A about Gravity Probe B notes that:
Quantum mechanics is incompatible with general relativity because in quantum field theory, forces act locally through the exchange of well-defined quanta.
In other words, the forces in the Standard Model are governed by gauge theories, while Einstein's Field Equations are not. This is closely related to another problem with integrating GR and quantum mechanics discussed in the following 2018 scientific journal article which explains in its abstract that:
Some of the strategies which have been put forward in order to deal with the inconsistency between quantum mechanics and special relativity are examined. The EPR correlations are discussed as a simple example of quantum mechanical macroscopic effects with spacelike separation from their causes. It is shown that they can be used to convey information, whose reliability can be estimated by means of Bayes' theorem. Some of the current reasons advanced to deny that quantum mechanics contradicts special relativity are refuted, and an historical perspective is provided on the issue.
Marco Mamone-Capria, "
On the Incompatibility of Special Relativity and Quantum Mechanics", 8(2) Journal for Foundations and Applications of Physics 163-89 (2018).
The position that quantum entanglement can be used to convey information superluminally, however, it should be noted, is a hotly disputed proposition and many scientists would disagree with that conclusion.
A Good Educated Layman's Introduction
The Guardian, a British newspaper, has a surprisingly good and balanced for a newspaper article take on the issue from
November 4, 2015. A highlight in the longish, educated layman oriented article explains:
Basically you can think of the division between the relativity and quantum systems as “smooth” versus “chunky”. In general relativity, events are continuous and deterministic, meaning that every cause matches up to a specific, local effect. In quantum mechanics, events produced by the interaction of subatomic particles happen in jumps (yes, quantum leaps), with probabilistic rather than definite outcomes.
Quantum rules allow connections forbidden by classical physics. This was demonstrated in a
much-discussed recent experiment in which Dutch researchers defied the local effect. They showed that two particles – in this case, electrons – could influence each other instantly, even though they were a mile apart. When you try to interpret smooth relativistic laws in a chunky quantum style, or vice versa, things go dreadfully wrong.
Relativity gives nonsensical answers when you try to scale it down to quantum size, eventually descending to infinite values in its description of gravity.
Likewise, quantum mechanics runs into serious trouble when you blow it up to cosmic dimensions.
Quantum fields carry a certain amount of energy, even in seemingly empty space, and the amount of energy gets bigger as the fields get bigger. According to Einstein, energy and mass are equivalent (that’s the message of E=mc2), so piling up energy is exactly like piling up mass. Go big enough, and the amount of energy in the quantum fields becomes so great that it creates a black hole that causes the universe to fold in on itself. Oops.
The first proposition in bold is primarily addressing quantum mechanical concept of entanglement, but there are also issues like tunneling and virtual particles that classical GR says should be possible.
One of the issues that is being flagged by the language in the second bolded section above is that the Standard Model assumes point-like particles, but a particle with finite mass and zero volume is problematic in classical General Relativity. A naive merge of the Standard Model and GR turns every fundamental particle into a singularity.
The third bolded section is addressing the disconnect between the large non-zero value of vacuum energy in quantum mechanics compared to the very tiny value of the cosmological constant (which nonetheless accounts for roughly three-quarters of the mass-energy of the universe in the LambdaCDM model).
Later in the Guardian article, another subtle issue is discussed. The Standard Model is basically a scientific theory describing sub-systems and components. General Relativity is basically a scientific theory describing the entire mass-energy content of a closed universe in a complete system as a whole.
For example, this poses the tricky issue of localizing or quantifying a gravitational field that is not present in ordinary QFTs.
The Nature of Time In the SM v. GR
A 2008 scientific journal article focuses in its abstract on stating that:
The aim of this work is to review the concepts of time in quantum field theory and general relativity to show their incompatibility. We prove that the absolute character of Newtonian time is present in quantum mechanics and also partially in quantum field theories which consider the Minkowski metric as the background spacetime. We discuss the problems which this non-dynamical time causes in general relativity, a theory characterized by a local dynamical spacetime.
Alfredo Macías, Abel Camacho, "
On the incompatibility between quantum theory and general relativity" 663(1-2) Physics Letters B 99-102 (May 25, 2008).
Classical v. Quantum: An Issue But Not Inherently Incompatible
It bears mention, however, that GR is not incompatible with the Standard Model simply because one is classical and the other is a quantum mechanical theory. This plays into the analysis, but it is not inherently impossible for classical theories, in general, and quantum mechanical theories to play nice with each other. The problems of mixing GR and the Standard Model are particular to GR and not general to classical theories in general.
Indeed, for the most part, the theoretical incompatibility of GR and the Standard Model is not much of a problem, because in the domains of applicability of the theories where they are applied in practice, they don't overlap in troublesome ways.
There are even important
ad hoc instances where elements of GR and the Standard Model are combined in a common sense sort of way under circumstances where the incompatibility of the theories isn't manifest, such as the theory of
Hawking Radiation from black holes, in efforts to understand the equation of state (EOS) of neutron stars, and in the use of quantum electrodynamics and Standard Model neutrino physics to analyze of astronomy observations.
Renormalization
The renormalization question raised in the OP is discussed, for example, here:
We present a short and intuitive argument explaining why gravity is non-renormalizable. The argument is based on black-hole domination of the high energy spectrum of gravity and not on the standard perturbative irrelevance of the gravitational coupling. This is a pedagogical note, containing textbook material that is widely appreciated by experts and is by no means original.
Assaf Shomer, "
A pedagogical explanation for the non-renormalizability of gravity" (December 3, 2007).
Note also that perturbatively non-renormalizable doesn't inherently mean internally inconsistent or internally flawed. For example, infrared quantum chromodynamics (the SM theory of the strong force) is also not capable of being addressed with perturbative renormalization, and instead used non-perturbative methods like lattice QCD. But, no one doubts that QCD is a mathematically rigorous and internally mathematically consistent theory. It's just that the mathematical methods perturbative methods that work so well for the electro-weak interactions in the SM doesn't work so well in low energy QCD.
String Theory and Quantum Gravity
I've heard statements that string theory combines quantum mechanics and gravity successfully. So is it an algebra which does contain QFT and GR and can predict what happens? Or why is this not the accepted theory of quantum gravity?
String theory addresses the renormalization problem and more generally avoids issues that QFT creates for GR by assuming point particles.
But, string theory is embedded in an overall mathematical structure whose unity and connectedness is part of its attractiveness. The generalization of string theory called M-Theory is conceived in a 10 or 11 dimensional space-time when we only observe 4 space-time dimensions,
Also string theory is really not just one theory but a whole legion of different possible "vacua" and no one has found one that corresponds to the Standard Model plus GR in its low energy approximation.
Indeed, there is basically no positive observational evidence that prefers string theory to the Standard Model.
It remains to be seen if the insights of string theory relevant to quantum gravity can be successfully extracted from the overall edifice of string theory as a "theory of everything" candidate.
Do We Have The Right Classical Target To Quantize?
Now, of course, we know that gravity actually works seamlessly and without glitches, so the problem of how to mathematically describe gravity has to be with our ability to use the right math to describe it and not with the fundamental unsoundness of the theory.
Also, the theoretical and phenomenological study of gravity includes many hypothetical very subtle tweaks to Einstein's Field Equations, such as, for example, "Conformal Gravity" or theories that consider torsion in a way not found in basic GR. Other examples are
Whitehead's theory,
Brans–Dicke theory,
teleparallelism,
f(R) gravity and
Einstein–Cartan theory.
It could be that one of the problems with our efforts to develop a theory of quantum gravity is that the classical theory we are trying to find a quantum gravity counterpart for, is actually the wrong one in some seemingly inconsequential way that throws off the rigorous mathematical enterprise that is crafting a quantum theory of gravity. As an analogy, if you tried to create a quantum theory of Newtonian gravity, you could probably do it, but the end product would be wrong and might even have pathological issues in a rigorous mathematical analysis or a comparison to astronomy observables, because the classical analogy isn't quite right.
More generally, gravity seems to behave in a manner that is significantly more constrained than an unthoughtful quantization of Einstein Field Equations on their face would suggest.
Thus, for example, one fruitful path of inquiry into quantum gravity has been the QCD squared approach that finds that in many cases, setting up an analogous QCD problem to a gravity problem, and then squaring it, produces correct results.
