# Structure of the Phonon Free Propagator

• dejo-ro
In summary: In this way, the operators for the simplest bosons are determined by the fact that they are not observables in an individual experiment, but rather their combination produces observable amplitudes.

#### dejo-ro

Hey Everyone,

I've a quick a question regarding the make-up of bosonic Green's functions, taking the free propagator for phonons as example. According to Mahan, 3rd ed. it is given by:

D(q, $\lambda$, t-t')=-i$\langle$0|TA$_{q}$(t)A$_{-q}$(t')|0$\rangle$

with A$_{q}$=a$_{q}$+a$^{+}_{q}$

[ Eqs. 2.66 & 2.67]

The operators showing up in the propagator ( i.e. A ) are not the simple creation & annihilation operators ( i.e. a, as would be the case for fermions ) but linear combinations thereof. What's the reason for this? Does this imply that phonons are only produced in pairs? Is this the same for other bosonic propagators ( I'm only familiar with phonons ). Hints, Corrections & Solutions greatly appreciated!

I think this is done like it is because A_q is the observable amplitude of the phonon. In contrast to this the fermionic field operators aren't observables. The coupling to the phonons occurs either via the fields A_q or via the momenta $\partial A_q /partial t$. So it makes sense to work with the linear combinations of the creation/ anihilation operators. Note that these are not bilinear operators in anihilation/creation but linear combinations. Hence there is no pair production involved. Rather the operator can produce coherent superpositions of states with different quanta.

OK, that sounds like a physical reason, I can appreciate that. Thanks for your answer! Do you know if this is done analogously for other bosons?

Most notably photons in QED.

At least in relativistic field theories, the combination of creation and anihilation operators in the field operators for the simplest bosons is due to them not carrying charge, in contrast to e.g. electrons. Obviously there are also charge carrying bosons, but it turns out that they can be described in terms of two "chargeless" bosonic fields, so this does not change the mechanism. I am not quite sure how a non-relativistic argument in many body theory would look like.

I think the basic reason is the following:
The time dependent Dirac equation for the electron is a first order differential equation while the Klein Gordon equation is second order. Hence a solution of the Dirac equation requires only one field amplitude a while the KG equation requires two field amplitudes a and b.
For a chargeless field, a and b must be complex conjugate.
The time independent Schroedinger equation can be reduced to a first order differential equation in a similar way
http://www.itp.uni-bremen.de/~noack/PauliDirac.pdf [Broken]
(sorry, in German, but I think the formulas should be clear)
and the argument can be taken over.

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## 1. What is the structure of the Phonon Free Propagator?

The Phonon Free Propagator is a mathematical representation of the dynamics of phonons, which are quantized lattice vibrations in a crystal. It is used in the study of condensed matter physics, particularly in the field of solid state physics.

## 2. What factors determine the structure of the Phonon Free Propagator?

The structure of the Phonon Free Propagator is determined by the crystal lattice structure, the material properties of the crystal, and the temperature at which the phonons are being studied. These factors affect the energy and momentum of the phonons, which in turn determine their behavior.

## 3. How is the Phonon Free Propagator used in research?

The Phonon Free Propagator is used in theoretical studies of phonon dynamics, as it allows researchers to calculate the behavior of phonons in different materials and conditions. It is also used in experimental studies, where it can be compared to experimental data to validate theoretical models.

## 4. What is the importance of studying the Phonon Free Propagator?

Studying the Phonon Free Propagator allows us to better understand the behavior of phonons in different materials, and how they contribute to the overall properties of a material. This knowledge can be applied in the development of new materials for various applications, such as in electronics and energy storage.

## 5. Are there any limitations to the Phonon Free Propagator?

Like any mathematical model, the Phonon Free Propagator has its limitations. It assumes an ideal crystal lattice and does not account for any defects or impurities in the material. It also does not take into account interactions between phonons, which can play a significant role in certain materials. Therefore, the results obtained from the Phonon Free Propagator should be interpreted with caution and validated with experimental data.