# Equations of Motion of Glashow-Salam-Weinberg Theory after Spontaneous Symmetry Breaking.

код для вставкиСкачатьAnnalen der Physik. 7. Folge, Band '43, Heft 1/2, 1985, S. 119-133 J. A. Barth, Leipzig Equations of Motion of Glashow-Salam-Weinberg Theory after Spontaneous Symmetry Breaking By DIETERW. EBNER Physics Department, University of Konstanz, Konstanz A b s t r a c t . While for quantum field theoretical calculations it is sufficient t o know the Lagrangian, we give here t h e field equations of the unified gauge-theory of weak and electromagnetic interactions after spontaneous symmetry breaking. With this approach, inhomogeneous Lorentz conditions for the massive vector bosons Z,, W$ are obtained. Bewegungsgleichungen der Glashow-Salam-Weinberg-Theorie nach spontaner Symmetriebrechung I n ha1 t s u b e r s i c h t . Wahrend es fur quantenfeldtheoretische Rechnungen ausreichend ist, die Lagrangefunktion zu kennen, geben wir hier die Feldgleichungen der einheitlichen Eichtheorie der schwachen und elektromagnetischen Wechselwirkung nach spontaner Symmetriebrechung an. Auf diese Weise werden inhomogene Lorentzbedingungen fur die massiven Vektorbosonen Z, W: erhalten. 1. Introduction and Notation Glashow-Salam-Weinberg theory (GSW-theory) [l- 111 is a gauge theory of weak and electromagnet'ic interactions. It is based on the following Lagrangian : 1 YPVY,, f iLfCL"D,L ieiCRaD,eR f (D,@)+(Dp@)- p&@'+@- 3.. (@+@)2 - ge(L+@eR 6 + ef@+L), 1 leading to the following equations of motion (field equations) : OR" = ge@'L, (1.2a) iaL"DuL = ge@eR, D,DQ (1.2b) ii = -p&@ - 2 (@+@) D,YQ"= -gvj;, 3 @ - geeR+L, (1.2c) (1.2dl) 120 Ann. Physik Leipzig 43 (1986) 1/2 1 1 o,P-L-eeRLdReeH-gi(Ue@)+ 2 j,"= -L' 1 @ +-i@ 2 (De@), D o W r = -gwjew,,, (1.2d2) (1.2el) z j"w, = -L+oLQ $ L f i(D@@)+ 2@ - i W . 5 ( D W ) . (1.2e2) 2 2 (Greek indices run from 0 to 3, Latin indices run from 1to 3.) We have used the following abbreviations (1.3a) ~ , e ,= (a, - ig, Y , eR , t + r,) = D,@ = y,, = Wpva I', = gauge WPa= W pa (a, (a, D,L a,y, = 2 - ig, W,, ' a - i 9, Y , 2 - ig,, W,, - a, W v a t a - +, ; .9 Y,)@? f a &Y,) - wfia + glVEabe W f i b W v c . (1.3b) (1.3~) (1.3d) (1.3e) field of the U,(1) (= gauge-symmetxy U(1) of the weak hypercharge Y ) . 5 = gauge field of the SU,(2) 2 (= gauge symmetry s U ( 2 )of weak isospin). cra = z, = Pauli matrices. gv, gu,, go,AH, ,uHare coupling constants. eK = right-handed electron. " = left-handed = (e,) vL = neutrino. eL = left-handed @= (zI) iso-doublet. electron. = iso-doubIet of Higgs-field. We have uaed 2-component spinor calculus ;eR, v,, e, are 2 . 1matriceg. Thq Ccomponent spinor of the electron is v= (2). The advantage of 2-component spinor calculus in constructing solutions of the field equations is that no spurious components of the non-existing right-handed neutrino occur. (1.5a) 01 = 0, 1)2, 3 ) G = ( 1 3 0 1 0 2 , 03); (1.5b) u; = (1, ---GI,- 0 2 , -d3) are the 2-component pieces of Dirac's matrices in chiral representation : (1.5c) We have given the theory in case of the minimal coupling t o a gravitational field g,, . Further formulae for this case will be found in Appendix B. For flat space-time we have: (1.6a) gnu = qPy= diag (1,-1, -1, -l), Jq= 1, r, = 0. (1.6b) (1.6~) D. W. EBNER,GSW Theory after Symmetry Breaking 121 The Lagrangian (1.1)is invariant and the equations (1.2) are covariant with respect to a local (i.e. space-time dependent) element of the group (1.8) U,(1) (8, SUW(2)? assuming the following transformation laws for the fields e x = exp [ --ig,@(z)] eR , (1.9a) (1.9b) (1.9c) Y; = Y, - a,@, (1.9d) (The last two equations are valid for infinitesimal transformations only.) The equations given above are special cases of the equations of a gauge theory. Covariant derivative : D, = a, A, (1.10) - is&, = AaaTa 9 (1.11) g = coupling constant, A,, = gauge fields. ?‘a = generator in the representation of the object y upon which D, acts. Curvature tensor : Pas = P,,,T, = = a,& - - a,& - igpi,, A,], $. gCbcaAabA,9c* (1.12a) (1.12b) Covariant derivative of curvature tensor : D,R, = a,*, - - - -ig[A,, F,,I. (1.13) Transformation law of particle fields: w’ = uy, u = e--igJaTa. (1.14) Transformation law of gauge fields: A”, = U[& + ig-1 a,] u-1. (1.15) The essential difference between particle and gauge - fields is Particles transform homogeneous linear. Gauge-fields transform inhomogeneous linear. (1.16) Ann. Physik Leipzig 43 (1986) 112 122 The particle properties as they appear in (1.3) can be summarized in the following table : I 8 y5 t 6 Y Qle = :1 @ 1 lj2 1/2 1 112 1 1/2 e, (1j2 -I @I 0 112 -1 -1 -1 o o -1 1/2 1/2 -112 1/2 -112 (@JO -2 0 1. 1 1 0 (1.17) ( s = Lorentz-spin, y s = chirality) t = isospin, t3 = third component of isospin, Y = weak hypercharge of U J l ) ) Q = electric charge.) 2. Choosing a Gauge The equations of motion of a gauge theory (e.g. (1.2)) are singular in the sense that the solution of all fields Y,) W,d (2.1) for t < to given as initial data does not determine the solution for t > to uniquely. This non-uniqueness is intuitively clear, because we have a t any time (and space-point) the unpredictable and arbitrary choice of a reference frame. I n order to solve particular problems and to have the concept of particles propagating uniquely in time, it is necessary to choose a gauge (choice of special reference frames). I n the GSW-theory the following gauge conditions are usually assumed : (el27 eL7 V D @, GI = 0, = real. (2.2) As proved in Appendix A l , this gauge-condition is admissible, i.e. every solution yk7A ; of the equations of motion can be transformed by a gauge transformation (comp. (1.9)7 (1.14)) (1.15)) into a solution yh, A , satisfying the gauge condition (2.2). The gauge-condition (2.2) does not yet fix the gauge completely'). The gauge transformations which are still permitted arez): A, =A2 = 0, A3 = -(gy/9uJ (2.3) @(XI. These transforniations do not mix anymore the components of an isodoublet (neutrinos are transformed into neutrinos, electrons are transformed into electrons). They read : (2.4a) e x = exp [-ig,O] e R , V L = VL, (2.4b) (2.4~) e i = exp [-ig,@] e L , @;I = @II) Y', = Y, - if,@, (2.4d) (2.4e) w;,= w,,+ (9,lgw) if,@) (2.4f) W', = W , exp (-ig,@) , (2.4g) 1) We have done this intentionally in order toobtain electrodynamics with its U(1) phaseinvariance as a special case within the GWS-theory. Fixing the gauge completely proceeds on the same lines as in electrodynamics and will not be discussed here. 2) See Appendix A?. D. W. EBNER,GSW Theory after Symmetry Breaking 123 where we have defined w,= w,,+ i W P 2 . ( W p is usually denoted by W ; (2.5) , WE by W t .) For an arbitrary particle with hypercharge Y and third component of isospin t, (conip. (1.17)) we find (see Appendix A3): y' = exp [ig,@( Y P + t3)1 y . (2.6) The residual gauge invariance has the mat heniatical structure of a U (1)-gaugeinvariance. It can be identified with the U (1)-phase transformation of electrodynamics y' = exp [iQ @(@I y , (2.6') where Q is the electric charge of the particle. Since for the electron Q = - e mentary charge), we obtain bhe weak Gell-Mann-Nishijima formula3) Q = (t3 + TY )e . (e = ele- (2.7) I n particular we note that the right and left-handed electron have the same charge, the neutrino and the Higgs field have zero charge, and the W,' now behaves as a particle (not a gauge-field) with electric charge -e. We can write down the equations of motion (1.3) in our particular gauge (2.2), and obtain :4, iY" V%Y- 9,@IIY ioL" V,V, = - 1 + 9% aEvLZn - 1 g,crie,w:, (2.8 b) (2.8 c2) ), See Appendix A4. The factor 1/2 before I' and the factor 1 before T, are conventions (comp. (1.9)) chosen so t h a t (2.7) has the same form as the strong Gell-Mann-Nishijima formula occurring in strong interactions. The essential content of (2.7) is t h a t the electric charge is the sumof hypercharge and third component of isospin. (2.7) has already been anticipated in (1.17). 4, The physical interpretation of the equations (2.8) will be given in the next paragraph. For t h e derivation of (2.8) see Appendix A.5. a * { C.C. means a { } a* . { }*. For further abbreviations used in (2.8) see (2.9)-(2.17). Rf = Rff is the Ricci tensor. (RF = 0 in flat space-time.) + + + Ann. Physik Leipzig 43 (1986) 1/2 124 (2.8d) + 12 + 9%) ZQ= Z'Rf U"U,ZQ - UQUaZo + 1 Q,( Wew*q + i wvde W"* gw cos @,{i 1 2 UGV,AQ - UJeU,A" iw,Qvwe*>+ C.C. + 1 eL+u$,e,(gw cos Ow - gg sin 0,) - g), sin @,,ej;a$en + g, sin 0,) - - (gw cos 0, - 1 2 = - - 9, Y; (2.8e) aivL, cos Ow x {i O,( we WG*) - i w,0.we* + i w, WW,* + C.C. + A"!. (2.8f) + yveYyt) We have used the following abbreviations M = [3 (2.9) tg ow = g,lgw, 8, = cos Ow W$ (2.10)5) + sin Ow Y" A, = cos 0, Ybc- sin 0, F,, = a,A, W$, - avA,, z,,= a,zv - avz,. (2.11 a) (2.11 b) 6, (2.12) (2.13) From these definitions we find the following useful formulas + cos o w = gw(g; 9%)- l j 2 sin 0, = g,(g,: g&)-'/', gw sin 0, = g, cos O,, , Y, = cos @,A" sin @wZp, + 9 + W$ = cos e V r Z @ - sin @,,,A". (3.14a) (2.14b) (2.14 c) (2.11 a') (2.11 b') Under the electroniagnetic gauge transforniat'ions (comp. (2.4)) we find z; = z,, A; = A, - (2.4h) (l/cos 0,) a,@. (2.4i) 5 , 0, is called the Weinberg angle. It gives the relative strength of the coupling constants g, and gw. 6, A , will be identified with the electromagnetic 4-vector potential, F,, with the electromagnetic field sprength-tensor, both in Heaviside-Lorentx units. D. W. EBNER,GSW Theory after Symmetry Breaking 125 Under these transformations only A , has an inhomogeneous transformation law, i.e. behaves as a, gauge field. The other degrees of freedom of the original gauge-fields (W,,, Y,) now behave as particles, with the following charges (comp. (1.17)): 1 Qle (2.15) I n (2.8) we have used the covariant derivative corresponding to the electromagnetic gauge transformations (comp. (2.4)) : V,Y + (2.16) !Pi, ig, cos Ow(Qle)A P Y . Furthermore, in (2.8) we have used as an abbreviation the covariant derivative7) = 0,w,= v, w,+ igw cos OWZ,w,. (2.17) 3. Spontaneous Symmetry Breaking According to the GSW-theory we live in a space-time filled with a large Higgs field which is approximately constant, i.e. in the gauge (2.2): + = (1If2)( ~ 0 X) (3.1) (Q1 = 0, Go, = real, Qo = constant). Experimentally, we are unable t o change this large value in any appreciable way: x < X Go. (3.2) This selects the gauge (2.2) as the most natural one. We have spontaneous symmetry breaking : U,(1) 8 fJU,,(2) W). (3.3) Consider an analogous situation : Suppose we live in a large constant gravitational field produced by a plane mass distribution. When, experimentally, we are unable t o change this mass distribution in any appreciable way, the SO(3) symmetry is hiddens). Then we have a symmetry breaking --f SO(3) + SO(2) (3.3‘) and the “gauge” where the z-axis lies along the gravitational field is the most natural one. A direct experimental test of a symmetry presupposes that we construct experimentally a t t = to a situation where all components of the system (test particles, fields, including gravitational field and Higgs field) are rotated relative t o the original situation. Then we have t o test whether the rotated system evolves for t > to in exactly the ’) I n fact it is the covariant derivative with respect to the gauge-field Z,, (i.e. with respect t o the residual gauge-transformation when instead of (2.2) we choose the gauge GI= 0, GII = complex). I - - As a covariant derivative V, satisfies the product rule and V,W: = (V,W,)*. See also (A5.8). ; p is the covariant derivative of Riemannian geometry. Q is the charge of the particle Y (comp. (1.17 ), (2.15)). For A , we apply (2.16) with Q = 0. 8 , It is remarkable that Newton’s law of gravitation with its O(3) symmetry has been detected only through the observation of astronomical phenomena far outside the earth. 126 Ann. Physik Leipzig 43 (1986) 1/2 same way as the original system-only rotated. When we are unable (depending on the available energy, etc.) to produce the rotated initial dist,ribution, we say that the symmetry is hidden, i.e. cannot be observed directly. Accepting t'hat a physical theory should be formulated in such a way as to reflect the symmetries which can actually be tested a t a particular epoch, we have to use the formulation of § 2 for present-day experiments. x = 0 , g p P = qPv,eR = O,e, = O,Y, = 0, W , = 0, A , = O,Z, = 0 (3.4) is bhe vacuum. By ( 2 . 8 ~ 1we ) find Go = f -6p;lAff For small equation (p$ < 0). (3.5) x (in the absence of other fields), the Higgs field x satisfies a Klein-Gordon (3.6) with C = -m, fi I-. (3.7) Therefore, mH is called the mass of the Higgs boson. I n the same sense, we find V,'VfiWQ + (+vn.) 2 We = 0, (3.8) with C -6m Z C m,, 1 2 1 = - gu" + & @ o , mw = 1 g,@, = (3.10) C m, COB Ow, (3.11) mw are the mass of the Z and W-boson, respectively. (2.8a), (2.8f) for R! = 0, 8, = 0, x = 0) are Dirac's equation and Maxwell's equation V,VvA' - U"V,Av = -he 1 .(Yyfiy+ yy,M), 2 ( W , = 0, (3.12) with (3.13) (3.14) A,, = V4nh A,, (3.15)9) F,, = f 4 n 6 ~F,,. 9) A,, P,, are in cgs units, A,, P,, are in Heaviside-Lorentz units with fi (3.16) = c = 1. D. W. EENER,GSW Theory after Symmetry Breaking 187 4. Discussion The original locally #? @ U,(1)I covariant ,(? field equations ,)(1.2) are 32 equations for 32 fields. (Complex fields and complex equations are counted twice: eR : 4, L : 8, di : 4, Y , : 4, W,, = 12). As explained in § 2, these equations are singular in the sense that initial data for t < to determine the solution only up t o 4 arbitrary functions of space-time. (4 because SU,(2) @ UJ1) is a 4-dimensional Lie group.) I n the formulation of 5 2, because of the gauge-condition (2.2)) 3 real fields drop out, i.e. we are left with 29 fields. The number of equations (32) has remained the same. Therefore, the solution depends on only 1 arbitrary function of space-time (corresponding t o the remaining U(1) gauge freedom). The number of equations in (2.8) is still 32, because 3 additional equations ( ( 2 . 8 ~ 2 ) is 1 real and ( 2 . 8 ~ 3is ) 1 complex equation) occur. These equations are inhomogeneous Lorentz-conditions for the vector bosons Z , and W,. Usually, the gauge condition ( 2 . 2 ) is inserted directly into the Lagrangian (1.1).With this approach the equations (2.8c2), (2.8~3)are lost. Finally we give the physical interpretation of the remaining equations (2.8). (2.8a) is Dirac’s equation. The mass of the electron now depends on the Higgs field. The left and right handed electrons remain united when no vector bosons Z,, W , occur. (2.8b) is the inhomogeneous neutrino equation with the vector bosons as sources. (2.8~1)is a perturbed Klein-Gordon equation for the Higgs field. (2.8d), (2.8e), (2.8f) are inhomogeneous wave equations for Z,, W , and the photon A,. The Z,, W,, has a mass depending on the Higgs-field. The photon and the neutrino has remained massless. Appendix A : Proofs and Auxiliary Calculations A1 Suppose we have in an arbitrary gauge We have t o show the existence of a gauge exp (ig, (312) E U J 1 ) (comp. ( 1 . 9 ~ ) )so ) that fulfills the gauge condition (2.2). Thus we have the equations O L q i-p q z = 0, (-p*@i with + exp (ig,@/2) = real, py + 1. (A 1.1) (A 1.2) (A 1.3) (A 1.2) can obviously be satisfied by a suitable choice of the phase 0. (A 1.1)together with (A 1.3) can be fulfilled with 10112 if = GI$. 0 (for diI = 0 choose p = 0, OL = 1). 128 Ann. Physik Leipzig 43 (1986)1/2 A2 Suppose @ = [ @ I @II and for which (@; = 0, 01 ] fulfills (2.2), we ask for which U = exp (i@g,f2), @' = [z,] = - @iI= real). We find the equations I*:;-[ = u exp (i@g,/2) with /2+ 1 ~ 1 2 = 1 @ again fulfills (2.2) A & - exp (iOg,l2) = 0 , .WII exp (i@g,/2) = real. This leads to ,B = 0. (We never have = 0 as we shall see later on, since the vacuum corresponds to a large Q I I ) . Now we obtain: p@II rc = exp (i@g,/2). Thus to every U,(1) transformation exp (i@g,/2) there is associated a S U w ( 2 ) transformation The residual gauge transformations (leaving (2.2) invariant) are @' = exp (i@ggz312 $- i@g,l2) @. Comparing this with ( 1 . 9 ~we ) find A, = A , = 0 , A, = - (g,lsw) @ * A3 The generalization of (1.9) for an arbitrary representation of U,(1) (with hypercharge Y )and an arbitrary representation of SU,(2) with generators is (comp. (1.17), (1.9)) y' = exp For an isosinglet (t Y + ig, -o - igw A f]y. [ 2 = 0): T , = 0, for an isodoublet an eigenstate of 1T3 : +3y = t3y, i.e. y has the third component of isospin t, (comp. (1.17)). By (2.3) we obtain Y A4 For the electron (& = - e ) (2.6') should coincide with (2.4a) (or ( 2 . 4 ~ ) ) : --ig,@(x) = -iezY(x). Thus (2.6) reads y' = exp [ie6( Y / 2 + t3)]ly, Comparing this with (2.6') we obtain (2.7). D. W. EBNER,GSW Theory after Symmetry Breaking 129 A5 To deduce (2.8) requires straightforward but lengthy calculations which cannot be given in full detail here. By (2.5) we have (A 5.1) By (1.3a), (1.3b), (2.2) the equation of motion (1.2a) and the lower component of (1.2b) read : ~ ic; 8 -,is, Y , 1 (a, + ig, + r,) e R = gedi,,eL w,, 1 . -2 tg, , Y, - 1 r,+e, = yeGIIe, - 1 gwa;vLw,. Interchanging these equations and using (1.5c), (B.ll), (B.12), (2.14c), (2.16), (1.17) we obtain (2.8a). The upper component of (1.2 b) reads 1 ic; 8avI,-Tig,v(W,,vL+ 1 = 0, 2 a which by (2.11a), (2.14b), (2.14c), (2.17), (1.17), (B.12) is equivalent to (2.8b). To proceed ( 1 . 2 ~ )we use (2.2), ( 1 . 3 ~ )and obtain ( W,*eL)--ig,Yav,-I‘+v 4 (A 5.2) Since is a scalar field with zero charge we have (comp. (1.17), (2.16)) ~P@II = VP%+ The covariant derivative Dp@belongs t o the same representation as @. This is t r u e for internal symmetries. Since DP@ has an additional space-time index, we obtain (comp. (1.3c), (B.15)): 1 ap - i g w w p a Z , ( 2 By (2.15), (B.141, (2.16), (1.17) we obtain D ~ D Q= - v,(wP*@II) = aP(wP*QII) + 1 + FzaDQ. ig,Y,) DP@ + r z n W w I I+ ig, cos OivApW’*@II. The upper component of ( 1 . 2 ~ )is then the complex conjugate of ( 2 . 8 ~ 3(VP(W’*@II) ) = EV,( WP@ll)l*). Using V,@II = @ a,VP@,, I I ; ~ ~ =(~FQ~II);~, V,(@IIZI”)= ( @ I I Z ~ ) ; ~ and splitting into real and imaginary part we obtain (2.8cl), (2.8~2).Equations (1.2e) can be summarized in the form (comp. (1.12), (1.13)) 130 Ann. Physik Leipzig 43 (1986) 1/2 with (A 5.4) jk = jgl + ij& = 1 -vi@,eL - -gwWe@&. 2 (A 5.5) We find the following commutat'ion relations (A 5.6) [z+, 2-3 = z,. By (1.12) (g = 9,) we obtain The gravitational covariant derivative can be introduced because of the symmetry of the Christoffel-symbols (conip. (B.15),(B. 21)). (1.13) valid for flat space-time, reads in the general case D&p = - igw[& 3"SI. I n our case we obtain D ~ I + B= 3 2 1 [ ~ $ 3 ; " ; ~ w;;B;~ + Tigw(w"wp* - - w~wJ*);~ + 1 i g ,Wp(wp* ;a - W"* ;B + ig,( W"* Wg wfl*W;)) 1 - ,igwWB*(wp:" - W",s + ig,(W$WB - W g v " ) ) - 1 + -z-l2 [wS;.;p- Wa;S$+ ig,(W;WB - W p " ) $ + igwWp3(Wp;"- W",s + igw(w$Wp W{W")) 1 - igwWS (wg;a- W" + - ig,(W"Wp* - WSW"") 2 11 + 1 [ WB*;"$ - W&*;B$+ ig,(W"*Wt - W@*W");B - 3;L3 TZ+ 3 - igwWS,(w~*;" - W"*:fl+ ig,(W"*Wg - WS*W$)) + 1 W$;a- W$;@ ,igw(W~W~* - WSW"*) 131 D. W. EBNER, GSW Theory after Symmetry Breaking z3 We decompose (A 5.3) into the coefficient,s of ,,z, z-. (Obviously, the coefficient of 2 t+is the complex conjugate of the coefficient oft- and need not be considered separately. Note that W,, is real). For all components we use (comp. [12] I1 § 91.7): (A 5.7) Wfl;B;.- W’;a;B = - WARAX, where R,, is the Ricci-tensor. By (2.11b), (2.14c), (2.15), (2.17), (2.16) we obtain q3, = wv;@ + i9ww,,w,. (A 5.8) By (2.12), (2.13), (2.11b) we obtain W$;fl- W$;Q= - cos OWZefl + sin O w P f l . With these results we can derive (2.8d) For the derivation of (2.8e), (2.8f) we note 1 j%3= 2 ( e k o i e L - v~+c+,v~ 1 i $ = - 3 (vL4vL + fgi + gf ZQ) + 2e$dReR + + 9% ZQ) and have to satisfy Yfl;e;B - YQ;fl$ = - g& and I + ig,(WQ*W{- W@*W$))+ We use v,w,* = w,*;, - igww,,w* , C.C. = -9wjf3. (A 5.8”) VQZa= Za;, and VQAa= Aa;Q to obtain (2.8e), (2.8f). Appendix B We have given all calculations in case of minimal coupling t o a gravitational field. Here, we collect the formulae, pertaining to the gravitational field, which have been used. Spinors are refered t o tetrads : e(+; ( a ) = 0 , 1, 2, 3 , eye(,), = g / l v , e(”)/le(B) = q(”)(B), g g = det (gpp), = - e2. e = det ( e r ) ) , (B. 1) (B-2) (B.3) (B.4) (R.5) 132 Ann. Physik Iieipzig 43 (1986) 112 a,p, etc. are space-time coordinate indices lifted with The covariant derivatives of the right-handed spinors (yR, of left-handed spinors ( y L ) ,of 4-component spinors y (comp. (1.4)),of vectors (d”), A”) are: (B.ll) (B.12) (B.13) (B.14) (B.15) (B.16) (B.17) (B.18) (B.19) (B.20) (B.21) (B.22) (y@),c B ( p ) , c,(p) are the corresponding quantities written in 5 1 without brackets. Now, we have : (B.23) etc.) The following quantities are covariant constant : (B.24a) (B.24 b) (B.24~) (B.24d) (B.24e) References [l] AITCHISON, I.J. R.;HEY,A. J. G. : Gauge Theories in Particle Physics. A practical introduction. Bristol: Hilger 1983. [S] LOPES,J. L. : Gauge Field Theories. An Introduction. Pergamon Press 1981. D. W. EBNER,GSW Theory after Symmetry Breaking 133 [3] BECHER, P.; B o a , M.; JOOS, H. : Eichtheorien der starken und elektroschwachen Wechselwirkung. Stuttgart 1981. [4] JACKIW, R.: Introduction to the Yang-Mills quantum theory. Rev. Mod. Phys. 52 (1980) 661. [5] ABERS,E. ; LEE,B. W. : Phys. Rep. 9 (1973) 1. [6] TAYLOR, J. C. : Gauge Theories of Weak Interactions. Cambridge 1976. [7] FADDEEV, L. D.; SLAVNOV, A. A.: Gauge Fields. Introduction to Quantum Theory. (Reading, Mass., 1980). [8] ITZYKSON, C.; ZUBER,J.-B.: Quantum Field Theory. McGraw-Hilll980. [9] KONOPLEVA, N. P.; POPOV, V. N.: Gauge Fields. (Harwood academic publishers 1981). [lo] OKUN,L. B. : Leptons and Quarks. North-Holland 1982. [li] HUANU, K.: Quarks, Leptons & Gauge Fields. (World Scientific Publishing 1988). [12] LANDAU, L. D.; LIFSCHITZ, E. M.: Lehrbuch der Theoretischen Physik. Akademie-Verlag 1980 Bei der Redaktion eingegangen am 30. Januar 1985. Anschr. d. Verf.: Dr. DIETERW. EBNER Physics Department University of Konstanz D-7750 Konstanz, Box 5560

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