Structure of the Phonon Free Propagator

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Discussion Overview

The discussion revolves around the structure of bosonic Green's functions, specifically focusing on the phonon free propagator as described in Mahan's text. Participants explore the implications of using linear combinations of creation and annihilation operators in the context of phonons and other bosons, examining theoretical underpinnings and comparisons with fermionic operators.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions why the phonon propagator involves linear combinations of operators rather than simple creation and annihilation operators, suggesting this might imply phonons are produced in pairs.
  • Another participant argues that the operators represent observable amplitudes for phonons, contrasting them with fermionic field operators, and clarifies that these combinations do not indicate pair production but rather allow for coherent superpositions of states.
  • A participant inquires whether similar structures are used for other bosons, leading to a mention of photons in quantum electrodynamics (QED).
  • Discussion includes a point about the relationship between charge and the structure of bosonic field operators, noting that chargeless bosons can be described using combinations of two fields.
  • Another participant introduces a comparison between the Dirac and Klein-Gordon equations, suggesting that the order of the differential equations influences the number of field amplitudes required.

Areas of Agreement / Disagreement

Participants express varying views on the implications of the operator structure for phonons and other bosons, with no consensus reached on the broader applicability of these concepts or the specifics of non-relativistic arguments in many-body theory.

Contextual Notes

The discussion touches on complex theoretical concepts, including the nature of field operators in different contexts and the implications of charge on their formulation, but does not resolve the nuances of these arguments.

dejo-ro
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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\langle0|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!
 
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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
(sorry, in German, but I think the formulas should be clear)
and the argument can be taken over.
 
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