Bethe–Salpeter equation

Due to its common application in several branches of theoretical physics, the Bethe–Salpeter equation appears in many forms. One form often used in high energy physics is

${\displaystyle \mathrm{\Gamma }(P,p)=\int \frac{d^{4}k}{(2\pi {)}^4}K(P,p,k)S(k-\frac{P}{2})\mathrm{\Gamma }(P,k)S(k+\frac{P}{2})}$

where ${\displaystyle \mathrm{\Gamma }}$ is the Bethe–Salpeter amplitude (BSA), ${\displaystyle K}$ the Green's function representing the interaction and ${\displaystyle S}$ the dressed propagators of the two constituent particles.

In quantum theory, bound states are composite physical systems with lifetime significantly longer than the time scale of the interaction breaking their structure (otherwise the physical systems under consideration are called resonances), thus allowing ample time for constituents to interact. By accounting all possible interactions that can occur between the two constituents, the BSE is a tool to calculate properties of deep-bound states. Its solution ${\displaystyle \mathrm{\Gamma }}$, the BSA, then encodes the structure of the bound state under consideration.

As it can be derived via identifying bound-states with poles in the S-matrix of the 4-point function involving the constituent particles, the equation is related to the quantum-field description of scattering processes applying Green's functions.

As a general-purpose tool the applications of the BSE can be found in most quantum field theories. Examples include positronium (bound state of an electron–positron pair), excitons (bound states of an electron–hole pairs), and mesons (as quark-antiquark bound states).

Even for simple systems such as the positronium, the equation cannot be solved exactly under quantum electrodynamics (QED), despite its exact formulation. A reduction of the equation can be achieved without the exact solution. In the case where particle-pair production can be ignored, if one of the two fermion constituent is significantly more massive than the other, the system is simplified into the Dirac equation for the light particle under the external potential of the heavy one.

Derivation

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

${\displaystyle G=S_1{S}_2+S_1{S}_2{K}_{12}G}$

in momentum space, where "G" is the two-particle Green function ${\displaystyle ⟨\mathrm{\Omega }|{\varphi }_1{\varphi }_2{\varphi }_3{\varphi }_4|\mathrm{\Omega }⟩}$, "S" are the free propagators and "K" is an interaction kernel, which contains all possible interactions 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 ${\displaystyle \mathrm{\Psi }=⟨\mathrm{\Omega }|{\varphi }_1{\varphi }_2|\psi ⟩}$, which is a transition amplitude of two constituents ${\displaystyle {\varphi }_i}$ into a bound state ${\displaystyle \psi }$, and then makes an Ansatz for the Green function in the vicinity of the pole as

${\displaystyle G\approx \frac{\mathrm{\Psi }\overline{\mathrm{\Psi }}}{P^2-M^2},}$

where P is the total momentum of the system. One sees, that if for this momentum the equation ${\displaystyle P^2=M^2}$ holds, which is exactly the Einstein energy-momentum relation (with the Four-momentum ${\displaystyle P_{\mu }=\left(E/c,\overrightarrow{p}\right)}$ and ${\displaystyle P^2=P_{\mu }{P}^{\mu }}$ ), 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 that the energy-momentum relation holds, on both sides of the term a pole appears.

${\displaystyle \frac{\mathrm{\Psi }\overline{\mathrm{\Psi }}}{P^2-M^2}=S_1{S}_2+S_1{S}_2{K}_{12}\frac{\mathrm{\Psi }\overline{\mathrm{\Psi }}}{P^2-M^2}}$

Comparing the residues yields

${\displaystyle \mathrm{\Psi }=S_1{S}_2{K}_{12}\mathrm{\Psi },}$

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"

${\displaystyle \mathrm{\Psi }=S_1{S}_2\mathrm{\Gamma }}$

and gets finally

${\displaystyle \mathrm{\Gamma }=K_{12}{S}_1{S}_2\mathrm{\Gamma }}$

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