# How to fuse quantum mechanics and general relativity?

I am very new to this topic but I have started reading Kevin Wray's lecture notes about string theory (PDF) and in the introduction he says:

"Sometimes it is said that we don’t understand how to fuse quantum mechanics and GR. This statement is really incorrect, though for “NY times purposes”, it’s fine. In fact we understand perfectly well how to include quantum mechanical effects into gravity, as long we we dont ask questions about whats going on at distances, less than the Planck length."

My question now is how is this done, i.e. how can we incorporate quantum mechanics into gravity or where can I read more about this?

I know that there are similar questions here on StackExchange (e.g. here: A list of inconveniences between quantum mechanics and (general) relativity?) but they mainly revolve around the question what the problems are not what we actually can do even without string theory, i.e. above the Planck length.

• "The assumption that gravity is classical necessarily modifies the dynamical laws of quantum mechanics -- the theory must be fundamentally stochastic involving finite sized and probabilistic jumps in space-time and in the quantum field." From the abstract of a paper on arxiv that my phone stubbornly refuses to copy the link to, apologies. The paper is by J. Oppenheim. If you do a search it should turn up easily.
– user214814
Dec 5 '18 at 8:52

"In fact we understand perfectly well how to include quantum mechanical effects into gravity, as long we we dont ask questions about whats going on at distances, less than the Planck length."

This excerpt is probably alluding to the approach described in this review:

In that approach, gravity is treated perturbatively and with the understanding that the results are only valid at sufficiently low resolution compared to the Planck length.

Here's how it works: We know how to formulate quantum field theory (QFT) in flat spacetime. "Flat spacetime" refers to a special metric field. In general relativity, the metric field $$g_{ab}$$ is a dynamic entity, one that both influences and is influenced by all of the other dynamic entities (matter, the electromagnetic field, and so on). For applications involving only weak gravitational fields, we can write the spacetime metric $$g_{ab}$$ in the form $$g_{ab}=\eta_{ab}+\kappa h_{ab}$$, where $$\eta_{ab}$$ represents flat spacetime and $$\kappa h_{ab}$$ is a remainder with some small coefficient $$\kappa$$. Just like most calculations in quantum electrodynamics (QED) are done using an expansion in powers of the fine-structure constant (which defines the strength of electromagnetic interactions), we can treat the remainder $$h_{ab}$$ as a quantum field by using an expansion in powers of $$\kappa$$.

As in QED, the terms in this expansion involve some integrals that would be ill-defined ("infinite") if we naively assumed that the theory was valid at arbitrarily fine scales; but we know that it isn't, and we can account for our ignorance of finer scales by using a cutoff $$\epsilon$$, which amounts to deleting any part of any integral that involves distances less than $$\epsilon$$. The cutoff is obviously artificial, and that's okay, because this isn't supposed to be a theory of everything. It's only supposed to be a theory of things at sufficiently low resolution. This is the idea behind Effective Field Theory, which is the modern view of almost every application of QFT, whether or not it tries to include gravity. In particular, this is the foundation for the modern understanding of renormalization in QFT, as reviewed in

The definition of any particular model involves several parameters $$\lambda$$, including coupling constants, masses of the elementary particles, and so on. The precise value of $$\epsilon$$ shouldn't matter, because it's an artificial cutoff whose only role is to account for our ignorance. In fact, we can compensate for small changes in the value of $$\epsilon$$ by making corresponding small changes in all of the other parameters $$\lambda$$ that define the model, in such a way that the model's predictions remain essentially unchanged at scales much coarser than $$\epsilon$$. This is what "renormalization" means. In other words, when we take the model's parameters to be appropriate functions $$\lambda(\epsilon)$$ of $$\epsilon$$, the model's low-resolution predictions become practically insensitive to the precise value of the cutoff $$\epsilon$$.

Saying that a model is renormalizable means that the model has a short list of special parameters such that all of the model's other parameters may be expressed as functions of these special ones, with no additional $$\epsilon$$-dependence. This was clarified in the technical paper

QED (without gravity) is an example of a renormalizable model. Conversely, saying that a model is non-renormalizable means roughly that each of the model's parameters must depend on $$\epsilon$$ in its own way, independently of the others, in order to keep the model's low-resolution predictions fixed. Despite the negative-sounding name, non-renormalizable models are also useful. We use them all the time, with great empirical success. The non-relativistic approximation to QED is one example of a non-renormalizable model. The approach described above for incorporating gravity is another example.

By the way, we can also formulate ordinary QFT in any given prescribed spacetime background, such as the one that describes the gravitational field of the earth, without trying to treat gravity as one of the quantum fields at all. This is much simpler (though still challenging, and still involving some subtleties). This is the approximation Hawking used to do his famous calculation that led to the once-surprising conclusion that black holes radiate. This calculation did not require treating gravity as a quantum field at all.

Although small-parameter expansions like those described above are very useful (almost all calculations in particle physics use them), they do have limitations. In particular, they are almost certainly only asymptotic expansions, which means that even though the first several terms give an excellent approximation, the series eventually goes astray at sufficiently large powers of the expansion parameter, ultimately diverging. Asymptotic expansions are not peculiar to QFT; they have been well-studied by mathematicians in much simpler contexts. One simple example is the function of $$\lambda$$ defined by $$f(\lambda)\equiv \int_{-\infty}^\infty dx\ \exp(-x^2-\lambda x^4).$$ By inspection, we can see that this integral is well-defined for all $$\lambda > 0$$, but it is undefined ("infinite") for all $$\lambda\leq 0$$. Remarkably, we can expand it in powers of $$\lambda$$, evaluate each of the resulting integrals exactly, and the first several terms in this expansion gives an excellent approximation to the exact function if $$\lambda$$ is sufficiently small (and positive). However, the radius of convergence of this series is zero (because the exact integral becomes undefined as soon as $$\lambda\leq 0$$), and we reap the consequences of this when we try to carry the expansion to higher orders: every individual term is well-defined, but the series does not converge.

QED is presumably like this, and the weak-gravity approach to quantum gravity described above is also presumably like this. And, on top of that, the individual terms in that series are only well-defined if we keep the cutoff $$\epsilon$$ non-zero. So even though that approach is presumably adequate for some weak-gravity applications, it's utility is limited to low resolution and to low orders in the expansion. That's why people say that it doesn't define a proper theory of quantum gravity.

• Thanks for much for providing the link to the Lepage paper, thats just about my level
– user214814
Dec 6 '18 at 13:22