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  Particle/Pole correspondence in QFT Green's functions

+ 7 like - 0 dislike
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The standard lore in relativistic QFT is that poles appearing on the real-axis in momentum-space Green's functions correspond to particles, with the position of the pole yielding the invariant mass of that particle. (Here, I disregard complications tied to gauge-fixing and unphysical ghosts)

Schematically, I take this to mean:

$$\text{one particle state} \Leftrightarrow \text{pole on real axis}$$

It is easy to show the correspondence going $\text{particle} \Rightarrow \text{pole} $. This is done in many textbooks, e.g. Peskin and Schroeder's text, where a complete set of states diagonalizing the field theoretic Hamiltonian is inserted into a $n$-point correlator, and a pole is shown to emerge.

However, how does one make the argument going the other way $\text{particle} \Leftarrow \text{pole} $? That is, the appearance of a pole in a Green's function indicates an eigenstate of the Hamiltonian, with the eigenvalue corresponding to the particle energy?


This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user QuantumDot

asked Jul 2, 2014 in Theoretical Physics by QuantumDot (195 points) [ revision history ]
edited Jan 9, 2017 by Dilaton
Just to add extra information, it is possible to show within non-relativistic potential scattering theory (for well-behaved potentials) the particle/pole correspondence both ways: essentially, existence of zeros of the Jost function establishes that wavefunctions will give a normalizable bound state, and also would give a pole on the real axis of the corresponding partial wave $s$-matrix eigenvalue. What is the generalization of this to relativistic field theory?

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user QuantumDot

2 Answers

+ 6 like - 0 dislike

For simplicity, take $\hbar=1$ and consider a Hermitian scalar, renormalized field $\phi(x)$; other fields are treated analogously. Then $$G(E)=i\int_0^\infty dt e^{itE}\langle \phi(0)\phi(t,0)\rangle =i\int_0^\infty dt e^{itE}\langle \phi(0)e^{-itH}\phi(0)\rangle\\ =\Big\langle\phi(0)\int_0^\infty dt ie^{it(E-H)}\phi(0)\Big\rangle =\langle \phi(0)(E-H)^{-1}\phi(0)\rangle =\psi^*(E-H)^{-1}\psi,$$ where $\psi=\phi(0)|vac\rangle$, since the vacuum absorbs the other exponential factors. The spectral theorem for the self-adjoint operator $H$ guarantees a spectral decomposition $\psi=\int d\mu(E')\psi(E')$ into proper or improper eigenvectors $\psi(E')$ of $H$ with eigenvalus $E'$, where $d\mu$ is the spectral measure of $H$. Orthogonality of the eigenvectors gives $$G(E)=\psi^*(E-H)^{-1}\psi =\int d\mu(E')\psi(E')^*(E-E')^{-1}\psi(E') =\int d\mu(E')\frac{|\psi(E')|^2}{E-E'}.$$ Therefore $$(1)~~~~~~~~~~~~~~~~~~~~~~~~~~G(E)=\int\frac{d\rho(E')}{E-E'},$$ with the measure $d\rho(E')=d\mu(E')|\psi(E')|^2$. For negative $E$, the Greens function is finite; since the measure $\rho$ is positive, this implies that the integral is well-behaved and has no other singularities apart from those explicitly visible in the right hand side of (1). Note that (1) holds for every self-adjoint Hamiltonian, no matter what its spectrum is. It thus describes the most general singularity structure an arbitrary Greens function can have.

In general, the spectral measure consists of a discrete part corresponding to the bound states and a continuous part corresponding to scattering states. (There might also be a singular spectrum, which is usually absent and does not affect the main conclusion.) Formula (1) therefore says that (in the absence of a singular spectrum) the only possible singularities of the Greens function are poles and branch cuts. Moreover, (1) implies that $G(E)$ has a pole precisely at those discrete eigenvalues $E'$ of $H$ for which $\psi(E')$ is nonzero, and branch cuts precisely at the part of the continuous spectrum in the support of the measure $d\rho$.

In particular, any pole of $G(E)$ must be a discrete eigenvalue of $H$, whereas the converse only holds under the qualification that $\psi=\phi(0)|vac\rangle$ has a nonzero projection to the corresponding eigenspace of $H$.

The above argument applies to quantum field theories after elimination of the center of mass degrees of freedom. Thus we consider the subspace of states in which the spatial momentum vanishes. On this subspace, the bound states have a finite multiplicity only, as each mass shell is intersected with the line in momentum space defined by vanishing spatial momentum.

In terms of the original question, the direction ''pole $\to$ particle'' holds without qualification, whereas the direction ''particle $\to$ pole'' (claimed by the OP to be the obvious part) is not universally correct but only holds under a nondegeneracy condition.

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user Arnold Neumaier
answered Jan 3, 2017 by Arnold Neumaier (14,069 points) [ no revision ]
In my understanding (which might be wrong), a theory obeying the Wightman axioms is "already renormalized", i.e. you have no notion of "bare propagators" in it, so how are you defining the self-energy in a Wightman theory? The self-energy is a "perturbative object".

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user ACuriousMind
@ACuriousMind there are non-perturbative definitions of irreducible (aka, fully connected) correlation functions (obtained by taking functional derivatives of the Legendre transform of the partition function). In practice, the irreducible two-point function is just the inverse of the full two point function: $\Pi(p)=G_2(p)^{-1}$, where $G_2(x)=\langle\phi(x)\phi(0)\rangle$. In other words, and to be clear: I am asking about the behaviour of the two-point function, in momentum space, at $p\to\infty$.

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user AccidentalFourierTransform
This may be helpful.physics.stackexchange.com/questions/274265/…

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user ved
@ved thanks, but that post is discussing the propagator (ie, the free correlation function). What I'd like to know is the behaviour of the interacting correlation function.

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user AccidentalFourierTransform
Interaction terms will basically modify the mass term of propagator and the propagator will involve a physical mass for on-shell regularization scheme (or at some scale for others), so If you consider $p\to\infty$ limit then behaviour of propagator would stay same.

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user ved
If I am not mistaken, this is still $\text{evalue}\Rightarrow\text{pole}$. To go the other way, you'd have to start with something that already has a pole $$G(E) \sim \frac{-g}{E-\lambda} + \text{regular}$$ and then show that this implies $\hat{H}|\psi\rangle = \lambda |\psi\rangle$.

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user QuantumDot
Should not the $e^{-itE}$ inside the bracket instead read $e^{-it\hat{H}}$?

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user QuantumDot
@QuantumDot: I corrected the typo. The argument goes both ways. From your assumption and ny equation one can see that my right hand side, expanded in a spectral basis, must have a pole at $\lambda$, which is not the case if there is no discrete eigenvalue in $H$, since the rhs has poles only there.

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user Arnold Neumaier
Are you sure? To rephrase the question in a couple different ways, (1) how do we know that $\mathcal{FT} \langle\hat\phi(x)\hat\phi(y)\rangle$ wouldn't "accidentally" run to infinity? (2) Why couldn't there be more poles in $\mathcal{FT} \langle\hat\phi(x)\hat\phi(y)\rangle$ than as warranted by the spectral resolution of $(E-\hat{H})$?

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user QuantumDot
@QuantumDot: yes, I am sure. The rhs of my expression, when expanded into a Stieltjes integral over the spectrum of $H$, has poles exactly at the discrete eigenvalues and a branch cut at the continuous spectrum. This is a standard result from complex analysis. You can also see it by integrating over a tiny circle containing your $\lambda$ and using Cauchy's integral theorem.

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user Arnold Neumaier
Hi @ArnoldNeumaier, I think that QuantumDot and I could use more details in your answer. For example, it is not clear how one would "expand ... into a sum or integral of eigenvectors and evaluate the matrix element"; for example, it seems to me that you are already assuming the existence of a discrete eigenvalue in this reasoning. In any case, I have the feeling that $G(E)$ is a very complex function of the complex variable $E$; it is clear that all the particles lead to poles, but it is not clear that all the poles come from particles (and this is precisely the question).

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user AccidentalFourierTransform
In other words, the analytic structure of $G(E)$ in the complex plane is not obvious to me. Id like to see a formal proof that this function cannot have a pole at some E_0 if there is no eigenstate of $H$ with energy E_0. That is, all the pole-singularities of this function are in one-to-one correspondence with the eigenvalues of $H$. I have the feeling that your reasoning only proves that if there is a particle, then there is a pole. But I just cannot seem to understand why the converse is true from what youve written in the post (and thats why I need you to add more details to it). Thanks

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user AccidentalFourierTransform
For the bounty: I'm looking for the asymptotics of two-point functions in a non-perturbative setting. Thanks!

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user AccidentalFourierTransform
Not sure if this is what you had in mind, but there is a non-perturbative bound which follows from unitarity that the propagator cannot fall off faster than $1/p^2$ as $p\rightarrow \infty$ (for example see equations 84 and 85 of the notes my Matt Schwartz isites.harvard.edu/fs/docs/icb.topic1146665.files/…). He shows it for spin 0 but I believe this result holds for general spins.

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user Andrew
@AccidentalFourierTransform: I added all details, with full mathemtical rigor.

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user Arnold Neumaier
Thank you for expanding your answer. It is especially helpful since I can now more sharply state my objections/confusions. (1) minor but probably important: In relativistic QFT with continuously infinitely many DOF, only the vacuum is discreet; single particle states are part of the continuum on account of Lorentz transformations. (2) What forbids the numerator in your final equation $(*)$ from blowing up (exhibiting a pole outside the spectrum of $H$)? After all, the $|\psi(E')|^2$ are not strictly normalizable in relativistic QFT.

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user QuantumDot
Hi @ArnoldNeumaier I still have to go through the details, but now it looks way better, thank you for taking the time to fill in the details of the argument!

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user AccidentalFourierTransform
@QuantumDot: (2) is guaranteed by the fact that for negative $E$, the Greens function is finite, and the measure $\rho$ is positive. I added the response to your (1) to my answer.

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user Arnold Neumaier
@Andrew thanks! I have edited the question to discuss your comment.

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user AccidentalFourierTransform
@Andrew The unitarity fall off $1/p^2$ that you quote actually applies only for correlation functions that are assumed to vanish at infinity. There are perfectly healthy CFT where $\phi$ is a primary operator with dimension $\Delta>2$ where this is not the case. One has to perform the so-called subtractions in order to do the Fourier transforms, and this implies the presence of a finite polynomial in the propagator, on top of the decreasing contribution when $p\rightarrow\infty$. Just try to Fourier transform the 2pt-function of a field $\phi$ with dimension $\Delta>2$, and you see the point.

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user TwoBs
beside unitarity, see my comment above to Andrew though, one `requires' polynomially boundedness that comes from the tempered distribution nature of the Wightman functions. It is believe that string theory and other very peculiar theories (such as the Galileon) seem to violate this condition as they have some degree of non-locality built-in. As for your last comment about the Froissart bound, there exist plenty of interesting and well defined theories (e.g. all gapless ones, CFTs, gravity,...) where it is violated.

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user TwoBs
@TwoBs huh, that's really interesting and I hadn't thought of that. Maybe the question is more complicated than I thought, and I should narrow it down a bit? Anyway, you gave me a couple of topics to think about, thanks!

This post imported from StackExchange Physics at 2017-11-26 12:36 (UTC), posted by SE-user AccidentalFourierTransform
Oh, I see; the point is that the 2-point Green function has the property that it can be given so explicitly by the spectral decomposition, that together with constraints on $\rho$, there is no room for any additional singularities to develop. I am now happy with the answer. I will now, on my own, try to understand how this would work for higher-point functions. Thanks for your patience and time.

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user QuantumDot
+ 3 like - 0 dislike

Particles are very rarely eigenstates of the Hamiltonian.

In particle physics, only (dressed) electrons (and protons?) might be eigenstates. All the other particles have a finite life-time, implying that they decay, which means that they are obviously not eigenstates.

What we want to call a particle is something which has a clear signature in a correlation function and a long life-time compare to its energy. If the theory is perturbative (e.g. QED), the free theory already gives an intuition of what we are looking for: a pole of a Green's function very close to the real axis. Once again, these appear as poles.

In strongly coupled theory (e.g. low-energy QCD), it is much less clear what particles will be if we start from the bare action (quarks+gluons). Nevertheless, there are still particles (stable or unstable, protons and neutrons for instance). There are also resonances (when the life-time is too short).

Finally, there are also strongly-coupled theories where no (quasi)-particles exist. They appear frequently in condensed-matter, and are very difficult to study since one does not have a nice picture in term of scattering, propagation, etc. to figure out what's going on.

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user Adam
answered Jul 2, 2014 by Adam (105 points) [ no revision ]
Most voted comments show all comments
I am perfectly aware of this technicality, and for that reason I refer, in the question, only to poles on the real axis, which correspond to strictly stable particles. Also, I am looking for a non-perturbative argument (possibly involving axiomatic field theory) for poles corresponding to particles.

This post imported from StackExchange Physics at 2017-01-09 20:54 (UTC), posted by SE-user QuantumDot

'' free (quasi)-particles in such theories are those without interaction (coupling). They make sense since it is from them the interacting theory is built, isn't it? '' No. Quasiparticles are collective modes constructed from the interacting theory in a way that they are effectively noninteracting at low accuracy, thus weakly interacting at higher accuracy, so that perturbation theory can be applied. 

@ArnoldNeumaier: I completely agree with you and I don't understand your "No", neither somebody's downvote to my comment since in my papers I speak precisely about these possibilities.

Well, your comment is misleading since you claimed that it is from the quasiparticles that the interaction is built, which is just the opposite of what I asserted.

You can see how what you write is read by others. Maybe you should learn to express yourself more clearly. Note that you can edit all your comments at any time.

Most recent comments show all comments

If it is obsolete, would you agree if I hide it completely?

No, I disagree. I my comment I was speaking of the second answer where I found a strange statement. This statement has not been clarified yet.

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