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Rigorous Theoretical Arguments for Suppression of the Lamb Shift saqarTvelos mecnierebaTa erovnuli akademiis moambe, t. 7, #2, 2013 BULLETIN OF THE GEORGIAN NATIONAL ACADEMY OF SCIENCES, vol. 7, no. 2 , 2013 Physics Rigorous Theoretical Arguments for Suppression of the Lamb Shift Anzor Khelashvili* and Tamar Khachidze** * Academy Member, Institute of High Energy Physics, I. Javakhishvili Tbilisi State University; St. Andrew the First-Called Georgian University of the Patriarchy of Georgia, Tbilisi ** Akaki Tsereteli State University, Kutaisi ABSTRACT. The main purpose is to elucidate the role of the hidden symmetry of the Dirac-Coulomb problem and to show algebraic possibilities for derivation of spectra. It is shown that the requirement of invariance of the Dirac Hamiltonian under some kind of Witten’s superalgebra picks out the Coulomb potential only. It follows that the traditional view on the Coulomb potential is to be changed in the context of N=2 supersymmetry. © 2013 Bull. Georg. Natl. Acad. Sci. Keywords: Dirac Hamiltonian, hidden symmetry, Lamb shift, Coulomb potential, Witten superalgebra, Laplace-Runge-Lentz vector. 1. Relativistic Quantum Mechanics solved only 30-40 years later [3]. The fact is that the (Dirac Equation) hydrogen atom spectrum is given by Sommerfeld for- mula. This formula removes all the degeneracy , which In 1916 A.Sommerfeld [1] derived the formula for the took place in NR quantum mechanics, but one de- energy spectrum of hydrogen atom in relativistic generacy still remains – the spectrum depends only mechanics using his quasiclassical quantization on the eigenvalues of total momentum (or method. This formula looks like: JLS on its quantum number j). Therefore there remains a 1/ 2 twofold degeneracy for (j 1/ 2) , according to 2 Z E m 1 . (1) which levels ES()1/ 2 and EP()1/ 2 must be degener- 2 2 n Z ate, but an experimentally small shift was found by Lamb and Retherford [4]. This level shift is named as Here j 1/ 2 . After several years P.Dirac [2] the Lamb shift. It was explained in QED [5]. It seems introduced the correct relativistic equation, in which that one of the motivations of creation of the quantum the spin of electron arises automatically. Dirac solved electrodynamics was the aspiration to explain the Lamb this equation himself for hydrogen atom with the shift [6]. Coulomb potential. Surprisingly enough his result The natural question arises – Is there some sym- coincides with Sommerfeld’s one. This paradox was metry in the Dirac equation behind the prohibition of © 2013 Bull. Georg. Natl. Acad. Sci. 46 Anzor Khelashvili and Tamar Khachidze the Lamb shift? One can now construct new operators Below we shall see that the answer to this ques- Q Q iQ . tion is affirmative. 1 2 Let us consider the general Dirac Hamiltonian They are nilpotent, H p m V r , (2) 2 QH, 0 Q 0 and QQH, 2 , i . (6) where V() r is an arbitrary central potential, which is These algebraic relations (6) define the structure, the 4th component of the Lorentz vector, in accord- which is called Witten’s algebra or N=2 superalgebra ance with the minimal gauge switching. This Hamil- [7]. (There are anticommutators together with com- tonian commutes with the total momentum operator mutators in superalgebras. Such structures in math- ematics are known as graded Lie algebras). 1 JL , where diag(,) is the spin ma- 2 What happens if we require invariance of Dirac trix of fermions. It is easy to confirm, that the follow- Hamiltonian with respect to this algebra? Or if we require ing operator KL 1 commutes with H. Qi , H 0, i 1,2 . (7) It is named as Dirac operator. The eigenvalue of this operator is exactly k, men- Thus, we are faced with the following problem: tioned above and the degeneracy with respect to Find (construct) the operator(s), which anticommutes two signs of k remains in the solution of Coulomb with the Dirac K operator and commutes with the problem. Dirac Hamiltonian, H. Let us remark that: Let us first construct the anticommuting opera- j 1/ 2 , when j l 1/ 2 , leading to levels tor. One of such operators is Dirac’s 5 matrix. What (S , P , etc .) and j 1/ 2 , when else? There is a simple Theorem [8]: 1/ 2 3/ 2 If V is a vector with respect to angular momen- j l 1/ 2 ( P1/ 2 , D 3/ 2 , etc .) . tum operator Therefore the forbidden of the Lamb shift results from symmetry which at the same time L , i.e. if Li, V j i ijk V k means the reflection of the spin direction with re- and simultaneously it is perpendicular to it, spect to the angular momentum direction. Let us find the operator which reflects this sign. (LVVL ) ( ) 0, then the following operator It is evident that such an operator, say Q1 , if it exists, ()V , which is scalar with respect to total momen- must be anticommuting with K, i.e. 1 tum JL , anticommutes with K: 2 Q1, K Q 1 K KQ 1 0 . (3) Clearly the following operator KV, 0 . QK In general, Q i 1 (4) 2 2 K KOV,ˆ 0, (8) would be anticommuting both with K and Q1 . Moreo- where Oˆ is commuting with K. ver, the introduced operators have equal squares Armed with this theorem, one can choose physi- 2 2 cal vectors at hand, which obey the conditions of QQQQH1, 2 0, 1 2 . (5) Bull. Georg. Natl. Acad. Sci., vol. 7, no. 2, 2013 Rigorous Theoretical Arguments for Suppression of the Lamb Shift 47 this Theorem. They are:V rˆ - Unit radius-vector,, 2. The Physical Meaning of the Constructed Conserved Operator and V p - linear momentum vector.. There is also the Laplace-Runge-Lentz (LRL) vec- To elucidate the physical meaning of the derived operator let us make use of the obtained relations tor for the Coulomb potential V r , namely and rewrite it to more transparent form by r application of Dirac’s algebra: 1 i i ˆ , (9) ˆ 5 (12) A p L L p m r Q1 r (. p L L p K 2 2ma mr but its inclusion is not needed, because Lippmann and Johnson [9] in 1950 published this i form in a brief abstract, where it was said only that ( A ) rˆ K ( p ). ma this operator commutes with the Dirac Hamiltonian in Coulomb potential and replaces the LRL vector, and the relevant operator is expressible through known in NR quantum mechanics. But for an un- known operators, owing to the relation known reason they never published the derivation of this operator (a very curious fact in the history of 1 K V i V L L V . (10) th 2 20 century physics). For our aim it is useful to perform non-relativistic Therefore we take the most general K-odd opera- limits 1, 5 0 . Then it follows tor in the following form QA , (13) ˆ 5 1 Q1 x 1 r ixK 2 p ixKfr 3 . (11) i.e. this new operator turns into the spin projection of Now requiring the commutativity with the Hamil- the LRL vector. tonian, The Lamb shift is explained in QED by taking into ˆ account the radiative corrections in the photon propa- Q, H r x2 V r x 3 f r gator and photon-electron vertex, which gives the following additional piece in Hamiltonian [5]: 5 x1 2i K mf r x3 0 , r 2 4 m 1 3 VLamb 2 ln r it follows the relations: 3m 5 (14) 2 x2 V r x 3 f r, 2 3 L. 2 m r x1 x3 mf r . r This expression does not commute with our ob- tained Johnson-Lippmann (JL) operator. Therefore Then we find when only Coulomb potential is considered in the x 1 Dirac equation as in the one-electron theory, Lamb V r 1 . x2 mr shift would be always forbidden. We see that the hidden symmetry of the Coulomb So, we can conclude that the only central poten- potential governs the physical phenomena in a suffi- tial for which the Dirac Hamiltonian is supersymmetric ciently wide range – from planetary motion to the in the above sense, is the Coulomb one. fine and hyperfine structure of atomic spectra. Bull. Georg. Natl. Acad. Sci., vol. 7, no. 2, 2013 48 Anzor Khelashvili and Tamar Khachidze 3. Calculation of the Hydrogen Atom Then the Sommerfeld formula follows. Spectrum Thus, in the case of Dirac equation the spectrum To this end let us calculate the square of obtained of the hydrogen atom is obtainable algebraically from conserved operator by analogy to classical mechan- the symmetry considerations alone. ics. This gives [8] We see that the supersymmetry requirement ap- pears to be a very strong constraint in the framework KH2 2 2 . (15) of the Dirac Hamiltonian. While the supercharge op- Q1 1 2 1 a m erator, commuting with the Dirac Hamiltonian in the All the operators entering here commute with each case of pure vector component only is intimately re- other. Therefore, one can replace them by correspond- lated to the LRL vector, but unlike the latter one rela- ing eigenvalues and then solve from it for energy. tivistic supercharge participates in transformations 2 of spin degrees of freedom. In passing to non-rela- Because of positive definiteness of Q1 as the square of Hermitian operator, its minimal value is zero. This tivistic physics, information concerning spin–degrees gives for the ground state energy of freedom disappears and hence LRL vector as a generator of algebra does not transform anything 2 1/ 2 Z and symmetry becomes hidden as a relic of relativis- E m1 .
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