In quantum field theory (QFT), the dimensions of physical quantities, including mass, are determined by the behavior of the corresponding field under scale transformations. The dimensions of a field are given by its scaling behavior under rescaling of space and time coordinates.
In the case of gravity, the gravitational field is described by a field called the graviton. The dimensions of the graviton field, and therefore the associated mass dimension, can be derived by analyzing the interaction terms in the theory.
In conventional QFT, the dimensions of fields are typically positive. However, in the case of gravity, the situation is different. Gravity is described by a theory called general relativity, which is a classical theory of gravity. When attempting to quantize gravity, it is found that the dimensionality of the gravitational coupling constant, which is related to the mass dimension of the graviton, is negative.
The negative mass dimension arises due to the non-renormalizability of general relativity in its naive form as a quantum field theory. When computing loop corrections in perturbation theory, divergences appear that cannot be absorbed by simple renormalization techniques. This indicates that gravity, as described by general relativity, requires a more fundamental and consistent framework for quantization.
The negative mass dimension in quantum gravity implies that the theory is not renormalizable in the conventional sense. It suggests that the quantization of gravity necessitates the inclusion of additional degrees of freedom or a modification of the underlying framework. One prominent approach to addressing these issues is string theory, which aims to provide a consistent quantum theory of gravity by considering strings as the fundamental building blocks of the universe.
It is important to note that our current understanding of quantum gravity and the behavior of gravity at small scales is still an active area of research, and there is ongoing exploration into various theoretical frameworks that may better describe the quantum nature of gravity.