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

[Andrewmiller5432]

TL;DR Summary

The authors were talking about path integral peaking around a single semiclassical history in specific bottom up cosmologies and then went on to say it doesn't work because path integral is appropriate for quantum cosmologies

Hi, I am new here so apologies if i am not using the right subforum. I don't have a physics background so i am not very technical but i do have a little bit of understanding. I was reading this paper by hawking/hertog and came across something that ended up confusing me.

Here is it:

"Pre-big bang cosmologies [10] are examples of models that are based on a bottomup 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 withan 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 beenfound 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 nearthe 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. 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?

 

[PeterDonis]

Isn't path integral always supposed to be about summing over all possibilities.

Yes. But in many cases, most of the possibilities make no significant contribution to the integral. See below.

What do they mean by "path integral peaking around a single semiclassical history"

They mean that for the particular path integral they are describing in this quote, the only significant contributions to the integral, i.e., the only possibilities whose amplitudes do not end up canceling each other out, are the possibilities that are very close to the single semiclassical history. In such cases, one can use that single semi-classical history itself as a good enough approximation.

describing cosmologies by a path integral in quantum cosmology rather than in terms of a single semi-classical trajectory.

They are saying that, for the cosmologies they are describing in that quote, the approximation described above does not work: there is no single semi-classical history that all of the possibilities that make significant contributions to the path integral are close to. So the only option is to just use the path integral itself as the description of what is going on.

What is a path integral in terms of a single semi-classical trajectory?

That's not what is being done: see above.

 

[Andrewmiller5432]

Yes. But in many cases, most of the possibilities make no significant contribution to the integral. See below.They mean that for the particular path integral they are describing in this quote, the only significant contributions to the integral, i.e., the only possibilities whose amplitudes do not end up canceling each other out, are the possibilities that are very close to the single semiclassical history. In such cases, one can use that single semi-classical history itself as a good enough approximation.They are saying that, for the cosmologies they are describing in that quote, the approximation described above does not work: there is no single semi-classical history that all of the possibilities that make significant contributions to the path integral are close to. So the only option is to just use the path integral itself as the description of what is going on.That's not what is being done: see above.

Thanks for the response. I think i understood what they were saying. It just didn’t quite make sense to me, because as I understand, only the points where the action doesn’t change contribute to the propagator in the limit of h-bar to zero. So contributions from other paths cancel out. They seem to be implying that many different paths (probably not all) contribute. It didn’t make sense to me because what i’ve said is expected when things decohere. How can it be invalid for certain cosmologies that imply it, as they seem to suggest. So I thought they probably really were talking about single classical trajectory vs a path integral treatment, not path-integral peaked semiclassical trajectory vs path integral quantum cosmology. Quite frankly, it still doesn’t make sense. Apologies if i am saying anything dumb, i don’t have a background in the field.

 

[PeterDonis]

as I understand, only the points where the action doesn’t change contribute to the propagator in the limit of h-bar to zero

More precisely, only the single path where the action is a minimum contributes to the propagator in the limit of $\hbar \to 0$. That is the usual classical limit. But it only works in cases where the magnitude of the action is always much, much larger than $\hbar$.

However, the paper you reference is not talking about the classical limit. In other words, they are dealing with cases in which you cannot assume that the magnitude of the action is always much, much larger than $\hbar$.

They seem to be implying that many different paths (probably not all) contribute.

Yes.

It didn’t make sense to me because what i’ve said is expected when things decohere.

No, what you've said is what happens in the limit $\hbar \to 0$. That limit, in itself, has nothing to do with decoherence. Indeed, it can be applied to cases, such as light traveling through vacuum, in which decoherence does not occur. (Feynman's discussion of how to derive the fact that light travels in straight lines, in his book QED: The Strange Theory of Light and Matter, gives a good layman's discussion of this case.)

However, as noted above, the paper you reference is not talking about this limit.

How can it be invalid for certain cosmologies that imply it, as they seem to suggest.

Because, as above, they are talking about cases where you cannot take the limit $\hbar \to 0$ because you cannot assume that the magnitude of the action is always much larger than $\hbar$.

 

[Andrewmiller5432]

More precisely, only the single path where the action is a minimum contributes to the propagator in the limit of $\hbar \to 0$. That is the usual classical limit. But it only works in cases where the magnitude of the action is always much, much larger than $\hbar$.

However, the paper you reference is not talking about the classical limit. In other words, they are dealing with cases in which you cannot assume that the magnitude of the action is always much, much larger than $\hbar$.Yes.No, what you've said is what happens in the limit $\hbar \to 0$. That limit, in itself, has nothing to do with decoherence. Indeed, it can be applied to cases, such as light traveling through vacuum, in which decoherence does not occur. (Feynman's discussion of how to derive the fact that light travels in straight lines, in his book QED: The Strange Theory of Light and Matter, gives a good layman's discussion of this case.)

However, as noted above, the paper you reference is not talking about this limit.Because, as above, they are talking about cases where you cannot take the limit $\hbar \to 0$ because you cannot assume that the magnitude of the action is always much larger than $\hbar$.

I think my confusion is cleared up. I lazily overlooked the “semiclassical” bit before posting here. I take it what they mean by semiclassical in this context is the system otherwise being quantum mechanical but ignoring quantum gravitational effects?

 

[PeterDonis]

I take it what they mean by semiclassical in this context is the system otherwise being quantum mechanical but ignoring quantum gravitational effects?

No. What they mean by "semiclassical" is cases where taking the limit $\hbar \to 0$ is useful, i.e., quantum effects end up canceling out and the system can be adequately described by a single semiclassical history. The paper explicitly says that they are considering cases where this is not a useful approximation.

In the context of the paper, there is no such thing as a case where the system is treated as quantum mechanical but ignoring quantum gravitational effects; quantum gravitational effects are the primary quantum effects being considered.

 

[Andrewmiller5432]

No. What they mean by "semiclassical" is cases where taking the limit $\hbar \to 0$ is useful, i.e., quantum effects end up canceling out and the system can be adequately described by a single semiclassical history. The paper explicitly says that they are considering cases where this is not a useful approximation.

In the context of the paper, there is no such thing as a case where the system is treated as quantum mechanical but ignoring quantum gravitational effects; quantum gravitational effects are the primary quantum effects being considered.

Ah, I see. This clears it up. Thanks Peter!