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
\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.
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.
Edward Witten and the \(i\varepsilon\) prescription
![Edward Witten and the \(i\varepsilon\) prescription]()
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July 21, 2013
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