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Edward Witten and the \(i\varepsilon\) prescription

What it would look like if Wolfgang Amadeus Mozart decided to discuss A440, concert pitch (440 Hz), on 24 pages?

Today, you may see the answer to a very similar question. Edward Witten finally attempted to solve a homework problem given not only to him by his (former) doctoral adviser in 1989 and wrote

The Feynman \(i\varepsilon\) in String Theory.
Almost all particle physicists learn about the \(i\varepsilon\) prescription in their introductory courses. The Feynman propagators have to have the form\[

\frac{-i}{p^2+m^2-i\varepsilon}

\] in the mostly positive \(({-}{+}{+}\cdots{+}{+})\) signature that Witten prefers. The extra infinitesimal term tells us in what direction we should circumvent the singularity when we integrate over the momenta in the loops and that's why it matters. In the position basis, the addition of the infinitesimal imaginary term answers the question whether the propagators are retarded or advanced or something in between. Yes, C) is correct: they are Feynman propagators, stupid.

Note that the extra term adds an imaginary term to something that you could naively try to define by the real principal value because\[

\frac{1}{z-i\varepsilon} = {\rm v.p.} \frac{1}{z}+i\pi\delta(z).

\] I would always say that you may imagine that this \(i\varepsilon\) is an infinitesimal limit of something like \(i\Gamma/2\) coming from a finite width (decay rate) – even if the lifetime is infinite, it has to be there for the stable intermediate particle to behave just like the unstable ones. There can't be any discontinuity if you just send the lifetime to infinity and because the form of the propagator seems obvious for the unstable particles (whose wave functions exponentially decay with time), a "trace" of the exponential decrease with time has to be inserted to the stable particles' propagators, too. This is a moment in which the arrow of time enters the fundamental formulae, by the way.




There exists a more systematic way to derive the \(i\varepsilon\) term in quantum field theories, of course. When inserted as a multiple of the Hamiltonian integrated over an infinitely long period of time, it effectively gives us the suppression \(\exp(-\varepsilon M H)\) which only picks a multiple of the ground state – and that's right because we're evaluating correlators or scattering amplitudes upon the vacuum.

From this need to reduce the path integral to the vicinity of the ground state we learn that all the masses \(m\) should be supplemented with an infinitesimal negative imaginary part so that in the rest frame where \(E=m\), \(\exp(Et/i\hbar)\) contains the factor that exponentially (but slowly) decreases with time, too. Therefore, you may imagine that \(i\varepsilon\) really came along with the mass, not with the squared momentum, and that's another way to look at the prescription.




I won't discuss these issues here. But what happens when we switch to string theory? People know how to calculate loop diagrams and it has almost seemed like we don't even need such a thing.

Well, we implicitly used it in all the calculations but we do need it, anyway. Witten argues that in string field theory, the prescription is straightforward. \(1/L_0\) appears in its propagators and has to be replaced by \(1/(L_0-i\varepsilon)\). However, it's harder to see what the prescription looks like in the normal, non-string-field-theory-based covariant calculations.

Witten's answer is that while the generic world sheets have the Euclidean signature in the conventional Euclideanized calculations, when an intermediate string goes on-shell, one has to change the signature to the Lorentzian one. Where do we change it? What are the variables – counterparts of momenta – that are treated in this way? You will have to read the paper to learn all the answers, assuming that they're right.

At any rate, the complexification of the world sheets is important for a proper definition of string theory (and similarly field theory, especially in the presence of gravity). Here, the complexification commands us to deviate from the expected signature just infinitesimally but finite excursions may be important for other physical applications.

Note that for many years, your humble correspondent has had disputes with various people including Jacques Distler who have various irrational reasons to dislike the analytic continuation and the Wick rotation and changing signatures etc. They believe that those operations make physics shaky and so on. It's reassuring to know that Witten agrees with me – these continuations and a careful incorporation of things calculable in other signatures are not only not ruining the precise consistency of physics but they're actually needed for physics to be precise.

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snail feedback (17) :


reader Dilaton said...

Thanks for these nice additional explanations of the i*epsilon ... in the Feynman propagator ;-)


Can people who do not like the Wick rotation etc really reliable determine if this makes the physics shaky or if they are shaky about the physics ...? I mean because of the actio = reactio for example ... ;-P?


Cheers


reader Michael said...

I believe, and always have, that using the Wick rotation is a perfectly valid, very phyical, reliable, and rock solid calculation tool.
What I am less certain about is that there aren't additional, potentially important, subtleties in the Lorentzian signature calculations that one simply misses when relying solely on the Wick rotation.

After all, in Euclidean space Green's functions are often unique (fixed by a natural boundary condition at infinity) whereas in Minkowski space there is an infite family of such funtions, one differing from another by a solution of the homogenous wave equation.
This seems to be an issue especially in time-dependent and (locally) unstable backgrounds.


reader Vladimir Kalitvianski said...

Nice explanation Lubosh, congratulations! I just did not understand why you call the Feynman propagators "stupid" instead of "clever".


reader Dan said...

Just a typo: Witten works with the mostly positive signature -+...+.
Thanks for the nice post.


reader Ignacio Mosqueira said...

Here is a simple question for those of us who do not follow string progress as closely as we should: why did it take this long to get an answer - any kind of answer - to this question. I am not being flippant here. I am genuinely curious. The Feynman iepsilon was needed to answer physically motivated questions and theory self-consistency of interpretation. Without it the theory fails as far as I can tell. Surely the string theory analogue of this procedure should have been one of the first things one needs to look into.


reader Luboš Motl said...

Dear Ignacio, not really. The calculation in string theory normally proceeds differently than in QFTs - one can calculate the tree-level, one-loop, and multiloop diagrams analytically from scratch, and it's also obvious what the singularities mean and how unitarity imposes the i*epsilon things. But there has never been any explicit i*epsilon inserted anywhere in the usual stringy calculations because it's not needed.


reader Ignacio Mosqueira said...

I see. I get your point that the string theory is not really on field theory approaches. Nevertheless a time tested approach to doing physics is to explicitly demonstrate how prior theories may be viewed as special cases of the more general theory. So it is still surprising to me that string theorists, including you, have not already made a concerted effort to answer the question raised by D. Gross in 89.


reader Luboš Motl said...

Dear Ignacio, I wouldn't find this homework exercise from David deep.


It has been understood for decades why the stringy diagrams reduce to those of QFTs at low energies. In this limit, stringy diagrams also have long QFT-like propagators and the stringy amplitudes of course behave just like those in QFT, including the same i*epsilon prescriptions.


There was really nothing to solve at the level of the QFT limit of string theory. What Witten wrote a paper about is something that goes beyond the limit - extending the simple QFTs' i*epsilon prescription to a structure that exists within the whole string theory,not just in the alpha' = 0 QFT limit, and this structure is kind of there, if his paper is right, but this structure clearly wasn't needed to do the actual calculations - as proven by the simple fact that they have been done without it.


reader Ignacio Mosqueira said...

Lubos

I appreciate you are a string theory advocate and you have sound reasons to do so. But as far as I can tell every theory has in the end physical issues that are in need of clear interpretation before one can be confident that you're not barking up the wrong tree.

You can do math until you turn blue in the face but in the final analysis to do physics you need to make sure you are playing with the right set of lego pieces.

Saying that this is structure is just kind of there doesn't strike me as a reassuring.

BTW from my interested layman's perspective it looks to me like string theory still has a credibility gap, whether justified or not, as evidenced by the question an answer period during Gordon Kane's talk at the Kavli institute this month.



Notice that most of the questions after his talk do in fact refer to the QFT limit of string theory.


reader Luboš Motl said...

Thanks, Dan, fixed, and I added the signs. ;-)


reader Luboš Motl said...

Sorry, you completely misunderstand how physics works and in between the lines, one can see so much arrogance - you are proud to be ignorant - that it makes no sense to try to explain anything to you.


reader Curious George said...

I saw this thread now, and I saw that it didn't end up well, but I will give it a try since a similar question on physics stack exchange didn't get an answer. What about the chiral anomaly, is it obvious how it shows up? (I mean the Pion -> 2 gammas effect)


reader Luboš Motl said...

Dear Curious George,


it did end up well! I banned the Ignacio troll which turned it into a happy end.


My estimates of the probability that Ignacio was just a troll who wasn't really interested in i*epsilon or any other specific science question - and, instead, he was a stupid jerk obsessed with demagogic attacks against modern physics in general - was about 50% at the beginning and 99.99% before I placed him on the black list. Maybe to save some time, I should lower the threshold for the black list to 50%.


Before I try to answer: Are you asking how the chiral anomaly is showing up in string theory or something else?


Cheers
LM


reader Dilaton said...

Ha, interesting relationship between R-symmetry and chiral anomaly !


reader Curious George said...

Dear Lubos,

thank you for your feedback. If I understand correctly you are saying that the chiral anomaly in the low energy limit of string theory is not special, and it will get reproduced in the same way any other QFT low energy phenomena is reproduced.

My original question on Physics Stack Exchange was related to the 4D trace anomaly of massless fields on curved backgrounds. Does this fall in the same category of "Nothing special over there"? Will string theory reproduce the semi-classical results or that is a subtler situation?


reader Luboš Motl said...

Dear CG, the trace anomaly may be seen only if one assumes the theory to be scale-invariant to start with, at least in some way, and the stringy physics at the string/Planck scale inevitably breaks this symmetry, so the symmetry and its related anomalies are artifacts of low-energy approximations. That doesn't mean that Witten can't write an interesting paper on similar issues in the future ;-) but the paper won't be solving any "existential" issues in string theory and it has to find some new structure.


reader Curious George said...

If he doesn't know what to do to kill time on the beach during summer, you could suggest to him this exercise. It seems still better than a bunch of crossword puzzles :)