In quantum physics, Regge theory ( REJ-ay, Italian: [ˈrɛddʒe]) 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.
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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}/(4\pi \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 expression
E → E N = − 2 m ′ π 2 e 4 h 2 N 2 ( 4 π ϵ 0 ) 2 = − 13.6 e V N 2 , m ′ = m M M + m , {\displaystyle E\rightarrow E_{N}=-{\frac {2m'\pi ^{2}e^{4}}{h^{2}N^{2}(4\pi \epsilon _{0})^{2}}}=-{\frac {13.6\,\mathrm {eV} }{N^{2}}},\;\;\;m^{'}={\frac {mM}{M+m}},}
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
l → l ( E ) = − n + g ( E ) , g ( E ) = − 1 + i π e 2 4 π ϵ 0 h ( 2 m ′ / E ) 1 / 2 . {\displaystyle l\rightarrow l(E)=-n+g(E),\;\;g(E)=-1+i{\frac {\pi e^{2}}{4\pi \epsilon _{0}h}}(2m'/E)^{1/2}.}
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 orbital momentum 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:
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