3 removed spurious "quantum"

General relativity is nonrenormalizable.

What this actually means is that there is not a semigroup parametrized by some scale (length or wavenumber) that allows the equations of gravity at one scale to be rewritten as identical-looking equations with different parameters at another scale. The existence of such a semigroup is what renormalizability means. The semigroup is called the renormalization group.

The best way to understand renormalization intuitively is to consider the real-space renormalization of the 1D Ising model (or the 2D Ising model on a triangular lattice), or even simpler examples: for instance, an charged particle in an electrolyte attracts a shell of counterions, which in turn attract another shell of counter-counter-ions, etc. The net effect is to transform the normal scale-invariant potential into a scale-dependent potential. This particular example is called Debye screening.

Although many nonabelian gauge theories (such as pure $SU(N)$ Yang-Mills or the Standard Model) are renormalizable (although this has not been proven at a strictly mathematical degree of rigor, but see, e.g., BRST) and general relativity is also a (classical) gauge theory, the space of field configurations or gauge equivalence classes $A/G$ in GR is not well understood, principally because the gauge group (diffeomorphisms) is infinite-dimensional. This also complicates attempts to take the approach of lattice gauge theory for GR.

Regarding the lattice approach: for a "nice" nonabelian quantum gauge theory (such as the Standard Model) the gauge equivalence classes are better understood, but as I mentioned above, still not perfectly. Indeed, showing that discretized SU(2) quantum gauge theory has a well-defined limit (in particular, one that depends only on the size of the discretization and not its detailed structure) is half of a Millennium Problem. This has only been done at the level of rigor of mathematical physics, not mathematics.

Returning to the original focus, the easiest way to see that gravity is nonrenormalizable is the appearance of higher-order (< 4) terms in the action (this is called "power counting").

What this actually means is that there is not a semigroup parametrized by some scale (length or wavenumber) that allows the equations of gravity at one scale to be rewritten as identical-looking equations with different parameters at another scale. The existence of such a semigroup is what renormalizability means. The semigroup is called the renormalization group.

The best way to understand renormalization intuitively is to consider the real-space renormalization of the 1D Ising model (or the 2D Ising model on a triangular lattice), or even simpler examples: for instance, an charged particle in an electrolyte attracts a shell of counterions, which in turn attract another shell of counter-counter-ions, etc. The net effect is to transform the normal scale-invariant potential into a scale-dependent potential. This particular example is called Debye screening.

Although many nonabelian gauge theories (such as pure $SU(N)$ Yang-Mills or the Standard Model) are renormalizable (although this has not been proven at a strictly mathematical degree of rigor, but see, e.g., BRST) and general relativity is also a (classical) gauge theory, the space of field configurations or gauge equivalence classes $A/G$ in GR is not well understood, principally because the gauge group (diffeomorphisms) is infinite-dimensional. This also complicates attempts to take the approach of lattice gauge theory for GR.

Regarding the lattice approach: for a "nice" nonabelian quantum gauge theory (such as the Standard Model) the gauge equivalence classes are better understood, but as I mentioned above, still not perfectly. Indeed, showing that discretized SU(2) quantum gauge theory has a well-defined limit (in particular, one that depends only on the size of the discretization and not its detailed structure) is half of a Millennium Problem. This has only been done at the level of rigor of mathematical physics, not mathematics.

Returning to the original focus, the easiest way to see that gravity is nonrenormalizable is the appearance of higher-order (< 4) terms in the action (this is called "power counting").

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General relativity is nonrenormalizable.