量子力學與相對論

The precise statement is (1) the quantization of the Einstein-Hilbert action is non-renormalizable, and (2) there is no known quantum non-abelian gauge theory that in low energy reduces to gravity through spontaneous symmetry breaking process.

Elaboration on (1):
In general relativity, the field is represented as the space-time metric g, carrying spin-2. The Christoffel symbol (think of potential) is something like Gamma = dg/g, and the curvature (think of force) is like R = d Gamma + (Gamma)^2. So if you expend R in g, you would get terms like g^(-1)ddg and dg^(-1)dg, these are the kinetic terms; and other terms like (g^(-1)dg)^2, which are the interaction terms and are proportional to g^4.

The Einstein-Hilbert action (without matter fields or cosmological constant) is the space-time integral of the Ricci scalar (something like R). Think of this as a functional of g. Now try to quantize the theory using functional integral (take the functional integral of exponential of the action over the configuration space of g).

Again like many field theories (with interaction) in d=4, there are many infinities. One gets rid of the divergences by adding counter terms to the original action. Unlike a renormalizable theory (such as phi^4 theory, QED, non-abelian gauge theory), where a finite number of counter terms is needed, here in quantum gravity, an infinite number of counter terms are required to regularize the theory.

This means we need an infinite number of parameters to have a predictable quantum theory. But in experiments, we are only allowed to measure a finite set of parameters. This means the quantum theory is vacuous.


Short comment on (2):
This is the center idea of a Grand Unified Theory, which is a non-abelian gauge theory, that in low energy breaks into the standard model of strong, weak and electromagnetic interaction, as well as gravity. Up to now, there is no satisfactory theory for that.

不是互不相容，而是兩個理論的不完全性。

量子力學是用來描述粒子類型的東西，廣義相對論是用來描述星體怎麼大的

東西。但是量子力學不能用來描述星體，相對論也不能夠用來描述粒子。

現在物理學家正努力把這2個理論統一。