Path Integral Quantization

Question

I was reading through my notes on the path integral quantization of bosonic string theory when a general question about path integral quantization arised to me. 

The widely used intuitive explanation of a path integral is that you sum over all paths from spacetime point $x$ to spacetime point $y$. The classical path has weight one (is this correct?), whereas the quantum paths are weighed by $\exp(iS)$, where $S$ is the action of the theory you are considering. In my current situation we have the Polyakov path integral:
$$
Z = \int\mathcal{D}X\mathcal{D}g_{ab}e^{iS_p[X,g]},
$$
where $S_p$ is the Polyakov action.
I have seen the derivation of the path integral by Matrix kernels in my introductory QFT course. A problem which occured to me is that if the quantum paths are really "weighted" by the $\exp(iS_p)$, it only makes sense if $\mathrm{Re}(S_p) = 0$ and $\mathrm{Im}(S_p)\neq0$. If this were not the case, the integral seems to be ill-defined (not convergent) and in the case of an oscillating exponential we cannot really talk about a weight factor right? Is this reasoning correct? Is the value of the Polyakov action purely imaginary for every field $X^{\mu}$ and $g_{ab}$?

Secondly, when one pushes through the calculation of the Polyakov path integral one obtains the partition function
$$
\hat{Z} = \int \mathcal{D}X\mathcal{D}b\mathcal{D}ce^{i(S_p[X,g] + S_g[b,c])},
$$
where we have a ghost action and the Polyakov action. My professor now states that this a third quantized version of string theory (we have seen covariant and lightcone quantization). I am wondering where the quantization takes place. Does it happen at the beginning, when one postulates the path integral on the grounds of similarity to QFT? I am looking for a well-defined point, like the promotion to operators with commutation relations in the lightcone quantization.

Finally, in a calculation of the Weyl anomaly and the critical dimension, the professor quantizes the ghost fields. This does not make sense to me. If the path integral is a quantization of string theory, why do we have to quantize the ghost fields afterwards again?

This post imported from StackExchange Physics at 2014-04-07 15:52 (UCT), posted by SE-user Funzies

Comments

Dear Erik, this is a really bizarre question. Why are you asking about string theory? It's very self-evident that you are confused about the path integrals in any quantum mechanical theory, including non-relativistic quantum mechanics for one particle. The action S is always real in the physical space which is why the integral of exp(iS) is never naively convergent.

 

It may still be defined and calculated. Also, path integrals interpreted as complex probability amplitudes always mean that one is doing quantum mechanics - the theory is quantized.

 

Not a single among these facts and your confusions depends on string theory in any way whatsoever.

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Dear Erik, yes, you should remove the tag "string theory" but you should also reformulate the question so that it asks either about a simple quantum mechanical theory or any quantum mechanical theory.

 

In the way it's phrased here, it's counterproductive because it is hiding extremely simple things behind "string theory" which has nothing to do with them and it seems like you want to boast that you are learning string theory even though you have clearly not understood the pre-requisites such as quantum mechanics (especially in the path integral language) yet.

 

If you're in a course that is teaching string theory in this way but you haven't encountered path integrals before, you will either have to learn path integrals first - and almost everyone doing so starts with simpler physical systems than string theory when he or she is learning path integrals - or you will have to drop the course because you're not ready for it.

 

I think it is an extremely important thing for me to tell you and I am flabbergasted if you're not grateful to me for the answers I am generously giving you.

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Ron, the idea of going around and delete stuff just like that really scares me.

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@Ron I am slightly getting the impression that you still not approve that the Q&A section is for graduat level questions too, and I really think for not obviously bad things like bad crap, spam, etc we should make use of the community moderation threads. This is not more bureaucratic than review queues, and is somehow needed if the site is meant to be community driven. Also, I personally think about heavy edits we should be careful, as there is still no "roll-back" feature at present.
 

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Ok, ok, my bad. I agree with your last statements. But why import a question you intend to close? Also, if you ask the three questions properly, with the proper scope, perhaps it can be useful to someone.

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By the way, I've reverted your edit. Cutting out content is not a valid solution for splitting a question! You have not split the question, just cut it's content. The rest of the content gets lost.  

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Three people acting? That is clearly incorrect. That's less than the number of admins!   

Answer 1

1. In Minkowski spacetime, the action has to be real. Btw, that's necessary for the classical limit to give principle of least action. Yes, such sums are ill-defined, so some might say that the theory is mathematically defined by analytically continuing (Wick rotating) to Euclidean time, where you have nice exponentially decaying weights. You'll get saddle points given by the extrema of the action and you can expand around those solutions and deal with the theory perturbatively.

2. Think of the path integral in QM. Going from paths of least action (classical mechanics) to a weighted sum over all paths gives us quantum mechanics ("first quantization"). Going from quantum mechanics to QFT involves a sum over all field configurations ("second quantization"). Similarly, the moment you write out the path integral summing over all possible string configurations, you're studying a quantum mechanical system of strings.

3. Even in QFT when you do the Fadeev-Popov method and introduce ghosts, you have to quantize them so that diagrams with ghosts consistently cancel amplitudes of (some) diagrams with longitudinal gluons. I'm at a loss for a more insightful answer.

This post imported from StackExchange Physics at 2014-04-07 15:52 (UCT), posted by SE-user Siva

Comments

Thanks for the clear answer! Just a small point concerning my third question: indeed it makes sense that if we consider an action of several fields, all of them have to be quantized. My confusion is that after using the path integral to calculate the partition function, only the $X$ and $g$ fields seem to be quantized, whereas the ghost fields have to be quantized separately. Why is this the case?

This post imported from StackExchange Physics at 2014-04-07 15:52 (UCT), posted by SE-user Funzies

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@Erik: I'm not satisfied with my answer for the third part. I've made an edit. There are examples where we couple a quantized field to classical background sources to get a sensible effective theory, but I guess you can't do anything like that in the QCD example since you want the contributions from the ghosts to cancel contributions from longitudinal gluons. I don't recall the details of string quantization now. I'll respond to your comment if I get time to look through my notes and jog my memory.

This post imported from StackExchange Physics at 2014-04-07 15:52 (UCT), posted by SE-user Siva

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You could leave the question open for a better answer, or maybe post your exact doubts (as expressed in the comment) as a separate string theory question. My answer was pretty generic, hardly referring to the details of string theory.

This post imported from StackExchange Physics at 2014-04-07 15:52 (UCT), posted by SE-user Siva

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1. No, the action can have an imaginary part, this is the case when you change variables for example, or in certain other cases where you have a determinant.

2. String quantization is not a "third quantization", this part of your answer is bullshit.

3. This is  a non-sequitor.

This answer is nonsense, it should be deleted.

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3. Like I said, I'm at a loss for anything better, since I'm limited by my lack of understanding of what the OP is trying to ask.

2. It's not the terminology I would choose, but I'm simply trying to motivate a choice of words for OP since his prof seems to have used them.

1. I would appreciate an example or a reference. Thanks.

This Q/A is not a particularly insightful or useful exchange. I was just trying to help out someone who was confused. This question probably didn't need to be imported. I'm okay with it being deleted.

Answer 2

I am afraid, I am just going to provide the standard lore here. I will do so nonetheless as not many have attempted answering this question yet. Allow me look at the questions from simplistic QM(not QFT) picture. I'll take the question in reverse order: When did we quantize? Well, we never have to, that is the beauty(?) of path integrals! You just pretend your p's and q's are real numbers locating the particle in phase space. In absence of "momentum dependent potential", after the path integral program(PIP) all you are left with is sum (superposition) over histories giving you some amplitude. In QFT the extension of this naive picture only works for bosons and you have to introduce complications (grassmann nos for fermions; additional ghosts for gauge fields etc) so that PIP works (i.e. you have some classical field with given properties which is summed over all possible configuration). All this lets you bypass quantisation, deriving appropriate Wick's theorem and Feynman diagrams etc. Now, what if action S is real? Indeed it is real in usual QM, the lore is that only neighborhood of classical path contributes to the total amplitude which becomes superposition of "amplitude" contribution from individual paths. Consider a set of chosen path which is not close to the classical trajectory. Since for them phase factor is not stationary they could in principle have widely different values from one another. One expects that when you sum these phase factors there will be "wild" phase cancellations associated with it.

This post imported from StackExchange Physics at 2014-04-07 15:52 (UCT), posted by SE-user Noob Rev B

Comments

Not a grad level answer.

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@Ron Maimon I agree, have you posted it in the request for delete votes?  

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It's an IMPORT, it should just be unilaterally deleted. If the author wants to, the author can repost it.