Nonrenormalizable but quantizable theory: gravity? In p.8 of Michio Kaku book Introduction to Superstrings and M-Theory-Springer (1998), he said

The gravitational force. Gravity research was totally uncoupled from research in the other interactions. Classical relativists continued to find more and more classical solutions in isolation from particle research. Attempts to canonically quantize the theory were frustrated by the presence of the tremendous redundancy of the theory. There was also the discouraging realization that even if the theory could be successfully quantized, it WOULD ... still be nonrenormalizable.

My question is that

*

*what does Kaku mean for nonrenormalizable but quantizable theory?

what are the criteria to be nonrenormalizable?
what are the criteria to be quantizable?


*What are examples of nonrenormalizable but quantizable theories?

Fermi weak interaction theory? Gravity? and how come?
 A: The most basic definition of the terms (and I think the one Kaku is using) is
Quantizable theory - A classical theory that can be quantized in a way where the UV divergences are all cancelled by introducing counterterms.
Renormalizable theory - A quantizable theory which needs a finite number of counterterms.
Non-renormalizable theory - A quantizable theory which needs an infinite number of counterterms.
A theory is renormalizable as long as the classical coupling constants have non-negative mass dimension. For instance,

*

*QED has coupling constant $e$ (electric charge) which has mass dimension 0 so it is renormalizable.


*Fermi theory has coupling constant $G_F$ (Fermi coupling constant) which has mass dimension $-2$ so it is non-renormalizable.


*Gravity has coupling $G$ (Newton's constant) which has dimension $-2$ so it is non-renormalizable.
Mass dimension of a quantity is the total power of mass dimension in a quantity when working in natural units where $[M]=[L]^{-1}=[T]^{-1}$. For instance, the dimension of Newton's constant are $[G] = [L]^3 [M]^{-1} [T]^{-2}$ (because $G = 6.674 \times 10^{-11} m^3 kg^{-1} s^{-2}$). In natural units, we have $[G] = [M]^{-2}$. This implies that $G$ has mass dimension $-2$.
