Regge theory

In quantum physics, Regge theory (/ˈrɛdʒeɪ/) is the study of the analytic properties of scattering as a function of angular momentum, where the angular momentum is not restricted to be an integer multiple of ħ but is allowed to take any complex value. The nonrelativistic theory was developed by Tullio Regge in 1959. In quantum physics, Regge theory (/ˈrɛdʒeɪ/) is the study of the analytic properties of scattering as a function of angular momentum, where the angular momentum is not restricted to be an integer multiple of ħ but is allowed to take any complex value. The nonrelativistic theory was developed by Tullio Regge in 1959. The simplest example of Regge poles is provided by the quantum mechanical treatment of the Coulomb potential V ( r ) = − e 2 / ( 4 π ϵ 0 r ) {displaystyle V(r)=-e^{2}/(4pi epsilon _{0}r)} or, phrased differently, by the quantum mechanical treatment of the binding or scattering of an electron of mass m {displaystyle m} and electric charge − e {displaystyle -e} off a proton of mass M {displaystyle M} and charge + e {displaystyle +e} . The energy E {displaystyle E} of the binding of the electron to the proton is negative whereas for scattering the energy is positive. The formula for the binding energy is the well-known expression where N = 1 , 2 , 3 , . . . {displaystyle N=1,2,3,...} , h {displaystyle h} is the Planck constant, and ϵ 0 {displaystyle epsilon _{0}} is the permittivity of the vacuum. The principal quantum number N {displaystyle N} is in quantum mechanics (by solution of the radial Schrödinger equation) found to be given by N = n + l + 1 {displaystyle N=n+l+1} , where n = 0 , 1 , 2 , . . . {displaystyle n=0,1,2,...} is the radial quantum number and l = 0 , 1 , 2 , 3 , . . . {displaystyle l=0,1,2,3,...} the quantum number of the orbital angular momentum. Solving the above equation for l {displaystyle l} , one obtains the equation Considered as a complex function of E {displaystyle E} this expression describes in the complex l {displaystyle l} -plane a path which is called a Regge trajectory. Thus in this consideration the orbitalmomentum can assume complex values. Regge trajectories can be obtained for many other potentials, in particular also for the Yukawa potential. Regge trajectories appear as poles of the scattering amplitude or in the related S {displaystyle S} -matrix. In the case of the Coulomb potential considered above this S {displaystyle S} -matrix is given by the following expression as can be checked by reference to any textbook on quantum mechanics: where Γ ( x ) {displaystyle Gamma (x)} is the gamma function, a generalization of factorial ( x − 1 ) ! {displaystyle (x-1)!} . This gamma function is a meromorphic function of its argument with simple poles at x = − n , n = 0 , 1 , 2 , . . . {displaystyle x=-n,n=0,1,2,...} . Thus the expression for S {displaystyle S} (the gamma function in the numerator) possesses poles at precisely those points which are given by the above expression for the Regge trajectories; hence the name Regge poles. The main result of the theory is that the scattering amplitude for potential scattering grows as a function of the cosine z {displaystyle z} of the scattering angle as a power that changes as the scattering energy changes: where l ( E 2 ) {displaystyle l(E^{2})} is the noninteger value of the angular momentum of a would-be bound state with energy E {displaystyle E} . It is determined by solving the radial Schrödinger equation and it smoothly interpolates the energy of wavefunctions with different angular momentum but with the same radial excitation number. The trajectory function is a function of s = E 2 {displaystyle s=E^{2}} for relativistic generalization. The expression l ( s ) {displaystyle l(s)} is known as the Regge trajectory function, and when it is an integer, the particles form an actual bound state with this angular momentum. The asymptotic form applies when z {displaystyle z} is much greater than one, which is not a physical limit in nonrelativistic scattering.