What do the authors of the paper mean here exactly by path integral?

Question

First of all, please forgive me if i am asking a dumb question. I don't have a physics background. I was reading this paper by Hawking & Hertog on populating string theory landscape and came across something that ended up confusing me.

Here it is:

"Pre-big bang cosmologies [10] are examples of models that are based on a bottom-up approach. In these models one specifies an initial state on a surface in the infinite past and evolves this forward in time. A natural choice for the initial state would be flat space, but that would obviously remain flat space. Thus one instead starts with an unstable state in the infinite past, tuned carefully in order for the big crunch/bigbang transition to be smooth and the path integral to be peaked around a single semiclassical history. Several explicit solutions of such bouncing cosmologies have been found in various minisuperspace approximations [14]. It has been shown, however, using several different techniques, that solutions of this kind are unstable [15, 16].

In particular, one finds that generic small perturbations at early times (or merely taking in account the remaining degrees of freedom) dramatically change the evolution near the transition. Rather than evolving towards an expanding semi-classical universe at late times, one generically produces a strong curvature singularity. Hence the evolution of pre-big bang cosmologies always includes a genuinely quantum gravitational phase, unless the initial state is extremely fine-tuned. It is therefore more appropriate to describe these cosmologies by a path integral in quantum cosmology, and not in terms of a single semi-classical trajectory. The universe won’t have a single history but every possible history, each with its own probability."

Now my questions is,

Isn't path integral always supposed to be about summing over all possibilities anyway. What do they mean by "path integral peaking around a single semiclassical history" and describing cosmologies by a path integral in quantum cosmology rather than in terms of a single semi-classical trajectory. What is a path integral in terms of a single semi-classical trajectory?

Or could it be a mistake and they were talking about a semiclassical trajectory vs a path integral approach in quantum cosmology rather than path integral peak for a single trajectory. This is what it looks like to me because its not otherwise making sense.

Answer 1

To answer your questions directly, my understanding is as follows:

As you have mentioned, the path integral formulation is a way to compute probabilities of different outcomes by considering all possible paths that a system can take in its evolution. However, "path integral peaking around a single semi-classical history" means that, even though you're summing over all possible paths in the quantum cosmological system, the contributions from most of these paths are negligible (i.e. contribute very little to the overall probability amplitude) compared to one particular path that corresponds to classical/semi-classical behavior.

So in essence, they are suggesting that, according to certain cosmological models, the universe is not well described by a single semi-classical trajectory. More concretely, the dominant path in the path integral formulation doesn't correspond to a classical-like evolution, but instead, the 'quantum nature' of the system does play a significant role, and multiple possibilities contribute to the evolution.