Can anyone shed light on the comparison between these two concepts?

# [Physics] When is quasiparticle same as elementary excitation, and when is not

condensed-matterquantum-field-theoryquasiparticlessolid-state-physics

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Dear Robert, the answer to your question is trivial and your statement holds pretty much by definition.

You know, the Green's functions contain terms such as $$G(\omega) = \frac{K}{\omega-\omega_0+i\epsilon}$$ where $\epsilon$ is an infinitesimal real positive number. The imaginary part of it is $$-2\Im(G) = 2\pi \delta(\omega-\omega_0)$$ So it's the Dirac delta-function located at the same point $\omega$ which determines the frequency or energy of the particle species. At $\omega_0$, that's where the spectrum is localized in my case. If there are many possible objects, the $G$ and its imaginary part will be sums of many terms.

This delta-function was for a particle of a well-defined mass (or frequency - I omitted the momenta). If the particle is unstable, or otherwise quasi-, the sharp delta-function peak will become a smoother bump, but there's still a bump.

Because you didn't describe what you mean by "peak" more accurately, I can't do it, either. It's a qualitative question and I gave you a qualitative answer.

Cheers LM

I would say they are not entirely the same, but it depends on the context. First the definitions:

the Wigner transform of an operator $\hat{A}$ is defined as $$\tilde{W}\left[\hat{A}\right]=\int dz\left[e^{\mathbf{i}pz/\hbar}\left\langle x-z/2\right|\hat{A}\left|x+z/2\right\rangle \right]$$ and this is a strange function. You see that on the left, the operator is projected onto a real-space representation, then Fourier transformed. You may find more details (especially the link with the Weyl transform) on the wonderful review by Hillery, M., O’Connel, R. F., Scully, M. O. & Wigner, E. P.

*Distribution functions in physics: Fundamentals*, Phys. Rep. 106, 121–167 (1984) which is unfortunately beyond a paywall.the Wigner transform of the density operator $\hat{\rho}=\left|\Psi\right\rangle \left\langle \Psi\right|$ is then naturally defined as the Wigner transform $$W\left(p,x\right)=\int dz\left[e^{\mathbf{i}pz/\hbar}\left\langle x-z/2\right|\hat{\rho}\left|x+z/2\right\rangle \right]$$ and it is coined

*Wigner function*in that context.the Green function is not an operator, it is a correlation function, defined as $G\left(x_{1},x_{2}\right)=\left\langle \hat{T}\left[\hat{a}\left(x_{1}\right)\hat{a}^{\dagger}\left(x_{2}\right)\right]\right\rangle $ where $\hat{T}$ is the time-ordering operator, $\hat{a}$ is the (fermionic or bosonic) destruction operator, and $\left\langle \cdots\right\rangle $ represents the averaging process: it could be $\left\langle \cdots\right\rangle =\left\langle N\right|\cdots\left|N\right\rangle $ if you're working with number states $\left|N\right\rangle$, or $\left\langle \cdots\right\rangle =\text{Tr}\left\{ e^{-\beta H}\cdots\right\} /\text{Tr}\left\{ e^{-\beta H}\right\} $ if you're working with thermal averaging ($\beta=\left(k_{B}T\right)^{-1}$ is an inverse temperature in that case), ... Note there are other conventions for the Green functions, but it does not matter here. The

*Fourier*transform of the Green function reads $$G\left(p,x\right)=\int dz\left[e^{\mathbf{i}pz/\hbar}G\left(x-z/2,x+z/2\right)\right]$$ and it looks like a Wigner transform of the Green function, but it should be more appropriate to call it a Fourier transform of the Green function when you choose $x_{1,2}=x\mp z/2$ for the components. In condensed matter theory, $G\left(p,x\right)$ is often called a*mixed-Fourier Green function*(the*full*Fourier transform would have given $G\left(p_{1},p_{2}\right)$ instead) or a*quasi-classical Green function*for the reason to come.

In the limit $\hbar/\tilde{p}\tilde{x}\ll1$ (called quasi-classical limit), with $\tilde{p}\tilde{x}$ the phase-space exploration of the system, the equation of motion of the quasi-classical Green function is the Boltzmann's (transport) equation. The quasi-classical Green functions are not normalised, so they can not be *interpreted* (whatever it means) as quasi-probability distribution.

As far as I remember, the quasi-classical equation of motion for the Wigner function is not the Boltzmann's one, but the Liouville's one: the collision term is absent, since there is no self-energy method associated with the density matrix. One needs to work with the Lindblad equation for the density matrix, whereas the self-energy method is sufficient when you work with the quasi-classical Green function. Other method to deal with open systems when working with the density matrix is the so-called *stochastic method*, see *e.g.* Walls, D. F. & Milburn, G. J. *Quantum optics* (Springer-Verlag, 1994).

To conclude, note I've put the time under the carpet in the above explanation. That's for a good reason: time is always more complicated to deal with in the Wigner-Weyl transform, especially in the quasi-classical limit and with the Green functions method. The use of the Wigner function is not a big problem when time is taken into account. Of course dealing with the Lindblad equation is not a simple issue... but that's an other story :-)

## Best Answer

Dictionary for this answer: Excitation = particle; collective excitation = quasi-particle.

Short answer:Elementary particles are never quasi-particles, by definition of elementary. This does not mean that what now it is thought as an elementary particle could be a quasi-particle of entities to be discovered.

Mathematical answer:Elementaryparticles are those that correspond to irreducible representations of the Poincare group.Physical answer:Quasi-particles require the existence of an external medium or fields, whereas elementary particles do not. For example, phonons require a solid or a fluid to exist (they are collective modes of the atomic lattice vibration), likewise pions require a quark-antiquark sea. These are not fundamental particles, in the sense that they need the existence of other particles. A closed notion is that of composite particle, for example a molecule is

madeof atoms which, in turn, aremadeof a nucleus plus electrons. The difference between quasi and composite particles lies in the fact that quasi-particle are though as collective excitations of many particles (usually of the order on the Avogadro number $\sim 10^{23}$, but there may be far fewer, but not tens), while composite particles are more like building blocks where each constituent may be an elementary particle—such as an electron— or another composite particle—such as an atom— (a molecule is usually made of a few or tens of atoms, an atom usually contains from a few to tens of electrons plus a nucleus). Nevertheless, the difference between both concepts is not sharp; for example, pions are somehow made of quarks and antiquarks, they actually are also collective modes (waves) on the quark-antiquark sea, being quasi-Goldstone bosons of the approximate chiral symmetry.The difference between elementary and composite particles is tied to the human knowledge at the time. At a certain point, it was though that nuclei were elementary, after that people realized that there were in fact more fundamental constituents (protons and neutrons), and later on quarks and gluons were discovered.