en>Gareth Jones: rewrite lead to include 'facility location problem' in bold typeface and removed 'etc's

2014-01-30T14:43:30Z

rewrite lead to include 'facility location problem' in bold typeface and removed 'etc's

New page

The '''Bethe–Salpeter equation''',<ref>
{{cite journal
|author=H. Bethe, E. Salpeter
|year=1951
|title=A Relativistic Equation for Bound-State Problems
|journal=[[Physical Review]]
|volume=84 |issue= 6 |page=1232
|doi=10.1103/PhysRev.84.1232
|bibcode = 1951PhRv...84.1232S }}</ref> named after [[Hans Bethe]] and [[Edwin Ernest Salpeter|Edwin Salpeter]], describes the [[bound state]]s of a two-body (particles) [[quantum field theory|quantum field theoretical]] system in a relativistically covariant formalism. The equation was actually first published in 1950 at the end of a paper by [[Yoichiro Nambu]], but without derivation.<ref name=nambu>
{{cite journal
|author=Y. Nambu
|year=1950
|title=Force Potentials in Quantum Field Theory
|journal=[[Progress of Theoretical Physics]]
|volume=5 |issue= 4 |page=614
|doi=10.1143/PTP.5.614
}}</ref>

[[Image:JxBSE.pdf|right|thumb|A graphical representation of the Bethe–Salpeter equation]]

Due to its generality and its application in many branches of theoretical physics, the Bethe–Salpeter equation appears in many different forms. One form, that is quite often used in [[high energy physics]] is
:<math> \Gamma(P,p) =\int\!\frac{d^4k}{(2\pi)^4} \; K(P,p,k)\, S(k-\tfrac{P}{2}) \,\Gamma(P,k)\, S(k+\tfrac{P}{2}) </math>
where ''Γ'' is the Bethe–Salpeter amplitude, ''K'' the interaction and ''S'' the [[propagator]]s of the two participating particles.

In quantum theory, [[bound state]]s are objects that live for an infinite time (otherwise they are called [[Resonance (particle)|resonances]]), thus the constituents interact infinitely many times. By summing up all possible interactions, that can occur between the two constituents, infinitely many times, the Bethe–Salpeter equation is a tool to calculate properties of bound states and its solution, the Bethe–Salpeter amplitude, is a description of the bound state under consideration.

As it can be derived via identifying bound-states with poles in the [[S-matrix]], it can be connected to the quantum theoretical description of scattering processes and [[Green's functions]].

The Bethe–Salpeter equation is a general quantum field theoretical tool, thus applications for it can be found in any quantum field theory. Some examples are [[positronium]], bound state of an [[electron]]–[[positron]] pair, [[exciton]]s, [[bound state]] of an electron–[[electron hole|hole]] pair<ref>
{{cite journal
|author=M. S. Dresselhaus et al.
|year=2007
|title=Exciton Photophysics of Carbon Nanotubes
|journal=[[Annual Review of Physical Chemistry]]
|volume=58 |page=719
|doi=10.1146/annurev.physchem.58.032806.104628
|bibcode = 2007ARPC...58..719D }}</ref> and [[meson]] as [[quark]]-antiquark bound-state.<ref name="maristandy2006">
{{cite journal
|author=P. Maris and P. Tandy
|year=2006
|title=QCD modeling of hadron physics
|journal=[[Nuclear Physics B]]
|volume=161 |page=136
|doi=10.1016/j.nuclphysbps.2006.08.012
|arxiv = nucl-th/0511017 |bibcode = 2006NuPhS.161..136M }}</ref>

Even for simple systems such as the positronium, the equation cannot be solved exactly although the equation can in principle be formulated exactly. Fortunately, a classification of the states can be achieved without the need for an exact solution. If one of the particles is significantly more [[mass]]ive than the other, the problem is considerably simplified as one solves the [[Dirac equation]] for the lighter particle under the external [[potential]] of the heavier particle.

== Derivation ==

The starting point for the derivation of the Bethe–Salpeter equation is the two-particle (or four point) [[Dyson equation]]

:<math> G = S_1\,S_2 + S_1\,S_2\, K_{12}\, G </math>

in momentum space, where "G" is the two-particle [[Green's function (many-body theory)|Green function]] <math> \langle\Omega| \phi_1 \,\phi_2\, \phi_3\, \phi_4 |\Omega\rangle </math>, "S" are the free [[propagator]]s and "K" is an interaction kernel, which contains all possible interaction between the two particles. The crucial step is now, to assume that bound states appear as poles in the Green function. One assumes, that two particles come together and form a bound state with mass "M", this bound state propagates freely, and then the bound state splits in its two constituents again. Therefore one introduces the Bethe–Salpeter wave function <math> \Psi = \langle\Omega| \phi_1 \,\phi_2|\psi\rangle </math>, which is a transition amplitude of two constituents <math>\phi_i</math> into a bound state <math>\psi</math>, and then makes an ansatz for the Green function in the vicinity of the pole as

:<math> G \approx \frac{\Psi\;\bar\Psi}{P^2-M^2},</math>

where ''P'' is the total momentum of the system. One sees, that if for this momentum the equation <math> P^2 = M^2</math> holds, what is exactly the Einstein [[Energy-momentum relation|Einstein energy-momentum relation]] (with the [[Four-momentum]] <math> P_\mu = \left(E/c,\vec p \right)</math> and <math> P^2 = P_\mu\,P^\mu </math> ) the four-point Green function contains a pole.
If one plugs that ansatz into the Dyson equation above, and sets the total momentum "P" such the energy-momentum relation holds, on both sides of the term a pole appears.

:<math> \frac{\Psi\;\bar\Psi}{P^2-M^2} = S_1\,S_2 +S_1\,S_2\, K_{12}\frac{\Psi\;\bar\Psi}{P^2-M^2} </math>

Comparing the residues yields

:<math> \Psi=S_1\,S_2\, K_{12}\Psi, \, </math>

This is already the Bethe–Salpeter equation, written in terms of the Bethe–Salpeter wave functions. To obtain the above form one introduces the Bethe–Salpeter amplitudes "Γ"

:<math> \Psi = S_1\,S_2\,\Gamma </math>
and gets finally

:<math> \Gamma= K_{12}\,S_1\,S_2\,\Gamma </math>

which is written down above, with the explicit momentum dependence.

==Ladder approximation==

[[Image:JxBSEtr.gif|right|thumb|A graphical representation of the Bethe–Salpeter equation in Ladder-approximation]]

In principle the interaction kernel K contains all possible two-particle-irreducible interactions that can occur between the two constituents. Thus, in practical calculations one has to model it and only choose a subset of the interactions. As in [[quantum field theory|quantum field theories]], interaction is described via the exchange of particles (e.g. [[photon]]s in [[quantum electrodynamics]], or [[gluon]]s in [[quantum chromodynamics]]), the most simple interaction is the exchange of only one of these force-particles.

As the Bethe–Salpeter equation sums up the interaction infinitely many times, the resulting [[Feynman graph]] has the form of a ladder.

While in [[Quantum electrodynamics]] the simplicity of the ladder approximation caused a lot of problems and thus crossed ladder terms had to be included, in [[Quantum chromodynamics]] this approximation is used quite a lot to calculate [[hadron]] masses,<ref name="maristandy2006"/> since it respects [[Chiral symmetry breaking]] and therefore an important part of the generation these masses.

== Normalization ==

As for any homogeneous equation, the solution of the Bethe–Salpeter equation is determined only up to a numerical factor. This factor has to be specified by a certain normalization condition. For the Bethe–Salpeter amplitudes this is usually done by demanding probability conservation (similar to the normalization of the quantum mechanical [[Wave function]]), which corresponds to the equation <ref>
{{cite journal
|author = N. Nakanishi
|year = 1969
|title = A general survey of the theory of the Bethe–Salpeter equation
|journal = [[Progress of Theoretical Physics|Progress of Theoretical Physics Supplement]]
|volume = 43 |pages = 1–81
|doi=10.1143/PTPS.43.1
|bibcode = 1969PThPS..43....1N }}</ref>

:<math>2 P_\mu = \bar\Gamma \left( \frac{\partial}{\partial P_\mu} \left( S_1 \otimes S_2 \right) - S_1\,S_2\, \left(\frac{\partial}{\partial P_\mu}\,K\right)\, S_1\,S_2\right) \Gamma </math>

In ladder approximation the Interaction kernel does not depend on the total momentum of the Bethe–Salpeter amplitude, thus, for this case, the second term of the normalization condition vanishes.

== See also ==
*[[Lippmann–Schwinger equation]]
*[[Schwinger–Dyson equation]]
*[[Breit equation]]
*[[two-body Dirac equations]]

== References ==
{{reflist}}

== Software supporting the Bethe–Salpeter equation ==
*[http://www.berkeleygw.org BerkeleyGW] – plane-wave pseudopotential method
*[[YAMBO code]] – plane wave
*[http://www.bethe-salpeter.org/ ExC] - plane wave
*[[ABINIT]] – plane wave

== Bibliography ==
Many modern quantum field theory textbooks and a few articles provide pedagogical accounts for the Bethe–Salpeter equation's context and uses. See:
*{{cite book
|author=W. Greiner, J. Reinhardt
|year=2003
|title=Quantum Electrodynamics
|publisher=[[Springer (publisher)|Springer]]
|edition=3rd
|isbn=978-3-540-44029-1
}}
*{{cite arxiv
|author=Z.K. Silagadze
|year=1998
|title=Wick–Cutkosky model: An introduction
|class=hep-ph
|eprint=hep-ph/9803307
}}
Still a good introduction is given by the review article of Nakanishi
*{{cite journal
|author = N. Nakanishi
|year = 1969
|title = A general survey of the theory of the Bethe–Salpeter equation
|journal = [[Progress of Theoretical Physics|Progress of Theoretical Physics Supplement]]
|volume = 43 |pages = 1–81
|doi=10.1143/PTPS.43.1
|bibcode = 1969PThPS..43....1N }}

For historical aspects, see
*{{cite journal
|author=E.E. Salpeter
|year=2008
|url=http://www.scholarpedia.org/article/Bethe-Salpeter_equation_(origins)
|title=Bethe–Salpeter equation (origins)
|journal=[[Scholarpedia]]
|volume=3 |issue=11 |pages=7483
|doi=10.4249/scholarpedia.7483
}}

{{DEFAULTSORT:Bethe-Salpeter Equation}}
[[Category:Quantum field theory]]
[[Category:Equations of physics]]
[[Category:Quantum mechanics]]

Facility location problem - Revision history

en>Gareth Jones: rewrite lead to include 'facility location problem' in bold typeface and removed 'etc's