Band 30 a Zeitschrift für Naturforschung Heft 4
Angular Momentum and Discrete Symmetries in the Spinor Bethe-Salpeter Equation
W. Bauhoff
Institut für Theoretische Physik, Universität Tübingen
Z. Naturforsch. 30 a, 395-401 [1975] ; received January 8, 1975)
The group-theoretical basis for the angular momentum reduction of the spinor Bethe-Salpeter equation is discussed. We construct amplitude of definite angular momentum by analogy to the nonrelativistic problem and discuss the differences. The additional decomposition because of discrete symmetries is explained. Finally, we relate the states to those obtained in the non- relativistic quark model.
I. Introduction and Fundamentals example the meaning of the degeneracy parameters in the angular momentum decomposition or the It is well known that the Bethe-Salpeter (BS) relation to the state assignments of the nonrela- equation describing bound states is invariant under tivistic quark model. It is the purpose of this paper the little group of the total momentum pfl of the to clarify these points and to give a useful review. bound state, that is under three-dimensional rota- The fermion-antifermion BS-amplitude is defined tions for time-like pß . Hence the solutions can be classified according to their angular momentum j by and the equation may be decomposed into a set of r(x1,x2) = (0\TyJ(x1)y)(x2\\a) . (1.1) equations corresponding to definite values of the For a bound state a) the amplitude satisfies the 2 angular momentum operators J and /3. In the homogeneous BS-equation which reads in coordi' scalar case this is achieved simply by expanding the nate space wave-functions in terms of spherical harmonics. In v the spinor case which we will consider in the present ( - i 7'" 3/3x/ + m) X [xt, :r2) ( - i y d/dx2v + m) paper, the problem is much more intricate since we = f®(x1,x2; x3,x4) r(x3,x4) dx3 dar4 . (1.2) must combine the space dependent part of the am- plitudes with their Dirac structure. Besides the an- ;x3,x4) is the interaction kernel which is gular momentum, we have as additional symmetries assumed to be invariant under Lorentz transforma- the parity P, and the charge conjugation ^ in the tions including reflections. It should be noted that case of the equal mass fermion-antifermion problem. we do not restrict the discussion to ladder-like inter- So the set of coupled equations obtained by the actions where angular momentum decomposition is split up into ^(x^rro; x3,x4) ~ (x1-x2) d^-^) d(x2~x4) four distinct ones, according to the eigenvalues of The field operator yj(x) transforms under Lorentz the states with respect to P and transformations x = Ax according to Recently, some additional symmetries of the spinor U(A)yj(x)U~1(A) =S~1(A)y(Ax) (1.3) BS-equations have been discussed, for example the 1 reflection of the total momentum pu — pu but where S(yl) is the (§, 0)©(0, |) representation we will not discuss it here. matrix of the homogeneous Lorentz group. Hence The importance of the spinor BS-equation must we have for an infinitesimal Lorentz rotation not be stressed here. It will be sufficient to say that x'f = xu + a/ xv: (1.4) in all bound-state models of elementary particles S(A) = 1 - (i/2) or , 2„ = (£/4) 7r] . such as the quark model 2 and the nonlinear spinor theory 3, the fundamental fields are assumed to be So we get from (1.4) the commutation relation of spinors. So it is not astonishing that the kinematical y'(x) with the generator M/(I, defined by analysis of the spinor BS-equation has been con- uv sidered by several authors4-6. But nevertheless U (A) = 1 + (i/2) a M/lv , (1.5) there remain some problems to be discussed, for [MßV,y (*)] = (ixfldv-ixvdß +2ßV) rp(x) . (1.6)
Dieses Werk wurde im Jahr 2013 vom Verlag Zeitschrift für Naturforschung This work has been digitalized and published in 2013 by Verlag Zeitschrift in Zusammenarbeit mit der Max-Planck-Gesellschaft zur Förderung der für Naturforschung in cooperation with the Max Planck Society for the Wissenschaften e.V. digitalisiert und unter folgender Lizenz veröffentlicht: Advancement of Science under a Creative Commons Attribution Creative Commons Namensnennung 4.0 Lizenz. 4.0 International License. 396 W. Bauhoff • Symmetries in the Spinor Bethe-Salpeter Equation
Similarly, we have the commutation relation for the generator Pu of translations x'" = x" + a":
(1-7) The corresponding formulas for y){x) are obtained trivially from (1.6), (1.7). Next we want to construct the operators Pu and acting on t(xx,X2) which are equivalent to Pft and Mftv acting on the state a). That means we require:
P.z (xltxg) = (OlTMxJvM} Pu | a) , (1,8)
r(xi,x2) = (0 | T {y>{Xl) r(x2)} Mllv \ a) . (1.9) From the commutation relations (1.6), (1.7) we find:
( 3 3 3 3 \ Xu ~dx 7 ~ Xlv dx^ + X'2" Sx7" ~ X'2v 3x7/ T^'^ + 'T^'' *1 11 ^
Now we require a) to be a state of definite momen- Additionally we have a definite spin projection ;3: tum pß and definite angular momentum j. So we L3
(we are dealing with massive particles only). U(is)y(x)U-His)=y0y(x) (1.22) Translating (1.12) into an equation for r(x1,x2) by means of (1.10) we have as solutions: where we have arbitrarily chosen a phase factor r]p'= +1. The corresponding operator P acting on t(xx,x2) =99(2) exp { -ip- +x2)/2} (1.15) the amplitude r(x1,x2) is found from (1.22) to 7 with the relative coordinate 2 = xx — x2 . be :
The transcription of (1.13) into an equation for Pr(x1,z2) = y0 T (Xl ,x2) J'O . (1.23) cp(z) and its solution is greatly simplified if we If the bound state a) has a definite parity ?]p we pass to the rest frame p/4 = (m, 0, 0, 0). Then we have get from (1.23) an eigenvalue equation which we will write down immediately for the relative coordi- /O = 0, Ji = heiklMkl. (1.16) nate wave function (p(z) : Furthermore we define P
U(ic)v(x)U-i(ic)=C^(x) (1.25) I (Li + Oi ® 1-1 ®o{T)2cp{z) =j(j+l)cp(z) . i = 1 (C is the well-known charge conjugation matrix), (1.19) where we have again chosen a phase factor y}q = + 1 • (1.19) may look somewhat strange, but it should In the by now familiar manner we deduce from be noted that (1.25) the operator acting on the amplitude:
T t _1 IW, We will now start with the construction of am- aj,o2,o2 f°r 5 = 1. (2.6) plitudes with definite angular momentum and its o0 is also a solution of (2.2). For s = 1 we have to projection, i.e. with the solution of (1.19), (1.20). pass to the spherical basis9. This is achieved by In doing so we will be guided by the corresponding defining nonrelativistic problem of a bound state of two spin 0lsj= Us si Oi (2.7) 1/2-particles: There one couples first the two in- dividual spins, and then the resulting spin s is with the unitary transformation matrix coupled to the orbital angular momentum I. So we begin by looking for the solutions of (2.8) S2 The solutions are the spherical harmonics multiplied by a function not depending on the angles. Com- bining this with (2.9) we have SSs 11, For (pss"n> the Clebsch-Gordan expansion holds: The inversion of (2.14) is easily obtained with the help of the orthogonality relations for the Clebsch- Gordan coefficients and the spherical harmonics with dSS} = ( — 1)söSSj: Alternatively, the construction of the amplitude can be done with help of the vector spherical har- monics Y(jijt)i(Qz). This method avoids the introduction of the spherical basis and is of some computa- tional advantage. The vector spherical harmonics are defined by9: Y(mi = ~ y2 <11 z >3 — 11 ih) Yih-i + ^{1-llh + l\ih)Yiit + i (IUh-l\ih) Ylh.{l-llh + l\ih)Yijt + i (2.16) Y aim = (101 j31 jj3) Y,h. These three equations can be summarized with help of the transformation matrix U: Y(jih)i = 2 U& (1 s3lj3-s3\jj3) YUi_Ss. (2.17) Sj Inverting (2.7) we get another expression for the amplitude (fjj,{z) : Vih=Yih (oSSi) aß = V2S+T(! ßss3\la) (2.19) where the matrix indices run through — g, + |. Inserting this in (2.10) and changing the normalization a little, we get 1 = 1 01, }'5=1 00,, J'o = 1 ® (2-21) °jk = Oi 0 1 , O0i = o-t 0 Qt, yi y5 = Oi 0 03 , v. = „. (g)g301. (2.22) 399 W. Bauhoff • Symmetries in the Spinor Bethe-Salpeter Equation Summarizing, the amplitude (p{z) is decomposed with respect to the Dirac algebra 3 S(z)=Yjjt(Qz)sjh(z0,r) . (2.24) For the "vector" amplitudes we have an expansion in vector spherical harmonics which we get from (2.18) with a slight change of the notation, for example: Vi{z) = y<;,(Ö,) i>fc> («0, r) + Y(njl)i{Qz)vf, (z0, r) + Y(jj + lh)i(Qz) > (z0, r) . (2.25) Substituting this into the BS-equation (1.2) we get a set of equations for the amplitudes Sjjt, ..., v^jfj,... From now on we will omit the subscript j j3 . Table 1. The different states of the system and the amplitudes belonging to them. Parity (-1 )J Parity (-1); + ! Triplet j = l— 1 Triplet j = l + \ Singlet ;' = / Triplet j = l States Particles jtPic States Particles j'ipic States Particles j'iric States Particles jir'lc ++ + 3Po e, .tin 0 % r/, Ji 0~ 3 3 3 ++ D1 co, Q r~ s, CO, o 'Pt ?, B V Pt ?, Al 1 3 ++ 3 2" + 3D, 2" F2 f, A 2 2 P, I, A 2 »D., + ++ 3 3 3 " 3F; 3 G3 3" D; 'Fl Amplitudes Amplitudes Amplitudes n n (4) 11 (0) ll (0) n n (4) 1) (0) ll C0) 5e. 'o , öp • t pe, ae , v0 , u0 Po» ao ' vc i "e + M (+) „ (+) a_ (-) , (+) , (-) ve< >, ve , ae , n , 'e > ^e The two triplets j=l±l are degenerate. We have included the corresponding isoscalar and isovector particles, taken from16. 3. Parity In the space dependent part only the angles are changed. The parities of the scalar and vector The set of sixteen coupled equations obtained spherical harmonics are well-known: after angular momentum reduction is decoupled into smaller sets because of the discrete symmetries. Y}h(-Q) = {-l)>YJh{Q), (3.5) As the first one we investigate parity. We have to yw,)i(-ß) = (-DzFaeis)i(ß). (3.6) look for solutions of (1.24) and consider as a first From (3.3) — (3.6) we can easily deduce the parity step the algebraic part. Since y0 is the unit matrix of the sixteen amplitudes. They decouple into two in 2, it is now sufficient to deal with o-matrices. distinct sets of eight amplitudes in each set. For the This has to be expected since under space inversion parity f/p = ( — 1); (natural parity) of the bound the two representations (|, 0) and (0, |) are inter- state a) we have the amplitudes: changed. From the Pauli algebra we have: 5, 1/4), a<0\ t<0\ v( + \v(~\ u( + \ u(~),r)v =(-!)> QiQ3Q3 = Q3' (3.1) (3.7) 3 2 9s 9I93 = - 9i > 93 9i 93 93 = ~ 9i 93 • ( - ) and for rjv=( — 1)' + 1 (unnatural parity) we have Hence the elements of the Dirac algebra may be p,a{*\ v(0\u(0\ au\ a(-\ + \ t(~\ classified as even or odd under the operation of 1)/ + 1. (3.8) 7o ® 7o: So the set of sixteen equations decouples into two re={l,y0,yiy5,oik} , y0 Te y0 = Te, (3.3) sets each consisting of eight equations. The solutions of (1.24) are now easily constructed. {75>7o75>7iöo/} > 7oT0 7O= -r0. (3.4) 400 W. Bauhoff • Symmetries in the Spinor Bethe-Salpeter Equation 4. CP-Invariance note that we have never used the reality of the time coordinate z0 . As the next step we consider now CP-invariance, After Wick rotation, the amplitudes can be ex- i.e. we are looking for solutions of (1.28). In this panded in terms of four-dimensional scalar and vec- case the algebraic part of the problem is not simply tor spherical harmonics, defined by 6: discussed in terms of 2- or P-algebra alone. So we have to deal with the complete i^-algebra. Again it Y nll,m=Gn®(0)YihW, Se, »o(4\ «e(0), *o(0), + *.<->, "e( + ), «o^ of the bound state is unequal zero, we have no (4.3) invariance of the equation under four-dimensional and the same amplitudes for t]\> = (— 1);, r/p 7]q = — 1 rotations. Hence the equations will not decouple with "e" and "o" interchanged. with respect to n: = N — j. The considerations on For rji<=( — 1)' + 1 and rj? rj^ = —1 we have the angular momentum and parity are not altered by amplitudes: the Gourdin expansion. But CP-invariance requires the reversal of the relative time, so we have to dis- Pe, He«, ^o(0), "of0), «o( + ), «„("J, + cuss this in terms of four-dimensional spherical co- (4.4) ordinates. The symmetry properties of the spherical and for t]i>={ — 1); + 1, ~ 1 the same with harmonics (5.1), (5.2) with respect to time reversal again "e" and "o" interchanged. (0-^-0) are: Hence CP-invariance does not decouple the set of . YNjh (* - 0,tp) = ( - 1) •" / YNjh (0,cp), equations into smaller ones, but allows the restric- (5.5) tion of the range of the variable z0 to positive values. Y(Nmi(n- G>, 5. Gourdin Expansion So the even functions for z4 —> — z4 have even val- ues of n = N — j in the expansion (5.3), (5.4), and For the efficient numerical solution of the BS- the odd functions odd values. So to (4.3) corre- equation, one has to perform the Wick rotation and sponds the coupling scheme go over to Euclidean coordinates. Whether this ro- c „(4) (0) ,( + ) ,,(-) <-) „( + ) ..(-) (c n\ tation is allowed, is an entirely analytical problem s2ni v2n + li "2n •> l1 n 5 v2n 1 V»n ? u1n 5 u2n of the behaviour of the amplitudes at infinity etc. and from (4.4) we have the coupling scheme We will not discuss it here. The angular momentum „(4) „(0) „(0) f + ) <-) ,(+) ,(-) decomposition and the considerations on the dis- P2ni "2n ? v2n + ii "2n + l> «2/1 + 1? «2/1 +1 , »2/1 + 1 » '2/t crete symmetries are not affected by this transfor- (5.8) mation, as is assured by the Hall-Wightman theo- Hence after Gourdin expansion the equation de- rem n. A more pedestrian way of seeing this is to couples into four distinct sets. 401 W. Bauhoff • Symmetries in the Spinor Bethe-Salpeter Equation 6. Nonrelativistic Quark Model So we have for the unnatural parity rjp = — ( — 1); a singlet state with r]pi]c= j = l and a triplet For a given value j of the angular momentum, state with +1? } = I- we have four distinct solutions corresponding to For the natural parity rjP = ( —1); the situation >?P=±(- 1); and r]p y)q = i 1. We will now dis- is more complicated. It is impossible nonrelativisti- cuss the relation of these different states to those cally to build a singlet state with parity ( — 1)*. In obtained from a nonrelativistic quark model. the quark model these states are called exotics of the second kind 14, and the corresponding particles have Of basic importance for this is the equivalence of not been found so far. In the context of the rela- the CP-transformation with the spin exchange of the tivistic BS-equation these states cannot be aban- particle-antiparticle system 13. States which are odd doned by group-theoretical reasons. But it seems under spin exchange are singlet states, and the even possible to exclude them by the normalization con- ones are triplet states. Hence we have dition because they have negative norm at least for zero bound state mass15. We will neglect these states : VP VC= + 1 triplet states , in the following. So we have only triplet states with 11PVC= — 1: singlet states. ; = I ± 1 which are degenerate. The possible states are summarized in the follow- Furthermore the parity is given by — ( —l)z non- ing table which is similar to a table in 5. It should relativistically where I is the orbital angular mo- be noted that for j = 0 some states are missing be- mentum. The additional minus sign stems from the cause in this case all vector amplitudes with "o" or fact that the two particles have different intrinsic " —" subscripts vanish. This is easily seen from parity. (2.14). 1 R. F. Keam, Progr. Theor.. Phys. 50, 957 [1973], 9 A. R. Edmonds, Angular Momentum in Quantum Me- 2 M. Böhm, H. Joos, and M. Krammer, Nucl. Phys. B 51, chanics, Princeton University Press, Princeton 1957. 397 [1973]. 10 H. P. Dürr and F. Wagner, Nuovo Cim. 53 A, 255 3 W. Heisenberg, An Introduction to the Unified Theory [1968], of Elementary Particles, Wiley, London 1967. 11 D. Hall and A. S. Wightman, Dan. Mat. Fys. Medd. 31, 4 R. F. Keam, J. Math. Phys. 9, 1462 [1968]. No. 5 [1957]. 5 M. K. Sundaresan and P. J. S. Watson, Ann. Phys. 59, 12 M. Gourdin, Nuovo Cim. 7, 338 [1958]. 375 [1970]. 13 J. M. Jauch and F. Rohrlich, Theory of Photons and 6 K. Ladänyi, Ann. Phys. 77, 471 [1973]. Electrons, Addison-Wesley Publ., Cambridge 1955. 7 R. F. Keam, J. Math. Phys. 10, 594 [1969]. 14 H. J. Lipkin, Phys. Rep. 8, 175 [1973]. 8 W. Bauhoff and K. Scheerer, Z. Naturforsch. 27 a, 1539 15 M. Krammer, DESY T-73/1 [1973]. [1972]. 16 Particle Data Group, Rev. Mod. Phys. 45 [1973].