Contribution to the Theory of Bosom and Fermions with Oriented Spins By A. A . S o k o l o v , and Yu.M . L o s k u t o u With 4 Figurcs Abstract A theory is developed for positive and negative energy charged bosons and fermions with spins directed oppositely with respect to the corresponding momenta. Invariance of the equations for a fixed spin direction is investigated. I. General theory of vector particles As is well known, Lee and Y a n g l ) predicted a new phenomenon in the case of weak interactions which has subsequently been cabled parity nonconservation2). For progresses in which the neutrino is involved (fermions with zero rest mass,m, = 0) this phenomenon can be explained either by the twocomponent theory or by the theory of D i r a c particles with oriented ~ p i n s 3 ) ~ ) . I n the latter case the solution corresponding to one of the spin directions (sl = s) should be retained for positive energies ( F = 1 e.g. neutrino) whereas for negative energies ( E = - 1, antineutrino) the solution corresponding to the l-s) should be used. This can be done if we second spin direction (s= require that the wave function y not only ohev the D i r a c equation but also the auxiliary condition5) (.. - or -L + (0;’) y =0 (Lt) + (A - el)y = 0 where 2 = t s, = -1 (or 1) for E = 1 as well as for E = The quantity s, = - s - ~ characterizes the double spin projection (in - 1. units of ti) on the momentum direction. We shall attempt to extend this result to particles possessing spins of 1 (bosons, vector field) and vanishing rest masses. I n the general case when the particles can possess two charge values we obtain Maxwell ian type equations. * 1 aA* + ,H’ = rot c at E+ = _ _ _ + + + 1 aE Ax, rot H - - c at + = 0, divA = cp+ = 0. ( 1 3 2 ) ~ l) T. D. L e e , C. N. Y a n g , Physic. Rev. 104, 254 (1956); 105, 1671 (1957). *) The following will be required of a theory explaining parity nonconservation: a)a n explanation of longitudinal polarization of paricles produced as a result of spontaneous decay and b) a n oxplanation of the asymmetry i n the number of particles emitted at various angles during the decay of longitudinally-polarized particles. A. S o k o l o v , B. K e r i m o v , Ann. Physik 1, 46 (1968). 4) A. Sokolov, n’ucl. Phys. 9, 420 (1959). A. A. S o k o l o v , Jonrn. of Phys. CJSSR 9, 363 (1945); -4.a.S o l ~ o l o v ,ZETF 33, 794 (1957). -4.-1. Sokolov a. 1-u. 34. Loskutov: Theory of Bosons and Fernicoi8s 43 Here the sign (-) refers to the principal function and ( f ) to the complex conjugate one, electromagnetic field notations being used here. + Although the transversality condition for a vector field, divA= = 0, is not invariant under a L o r e n t z transformation. we caii satisfy this requirement in any Loreritz coordinate system by applying the gauge transformations in a manner similar to that in M a x w e l l ' s theory. Taking into account the gauge transformation 1%hen changing to a primed coordinate system which moves along the 2 axis with a velocity c P relative to the unprimed system, we find t h a t the vector potential will transform according to the lan- It is interesting to notice that in changing from one inertial system to another the projection of the spin on the direction of the momentum (quantity s) remains constant for mO= 0 (this has been demonstrated for fermions in ref.6)). I n order to abtain the solutions referring to longitudinally polarized bosons ' : the following auxiliary condition should be imposed on the wave fund,ions + + -t H'=s,l/-V2A'=~i~s,E . (193) It corresponds to condition (1.1)introduced for fermions. Here the eigenvalues of the operator are equal t o x. In this case the solution for the d--f?2 + generalized vectorpotential A+ satisfying equations (1,2) under condition (1,3) can be represented in the form x here + - b+ ( P O , s, and + ,130 is 1 = -(PO v2 -i GO) is, [GO,~]), a unit vector perpendicular to the vector + - (1.5) GO. + I n case of a real field (A+ = A- etc.) we arrive a t the usual theory of circularly polarized photons') Outgoing from. (1,4) i t is not difficult to deter+ +- mine from (1,2) the vectors H + and E - : A. A. Sokolov, I. &I. T e r n o v , Yu. NI. L o s k u t o v , ZETF 36, 930 (1969). D. D. Ivanenko, A. A. Sokolov, Uoklady Acad. Nauk 61, 51 (1948); also see A. 9.S o k o l o v , T.M. Ternov, ZETF 31, 473 (1956). 6) 7) 14 drbnalen der Physik. 7 . Folge. Baud 5. 1959 I n order to find the relation between the amplitudes q ' and dynamic variables directly referring to the particles we determine the integral expressions for the energy * * * H = - J 1d 3 * + x [ ( E + E - )+ ( H f H - ) } = z c f i x q L ( E , s , , x ) ~ -( E . & , ; ) (1,7a) 4n -+ %, & momentum + G = 1 4nc * * j-d3x "E+ H-I * + - [H+E-1) = $-ti E * Z q+ ( F , 8,. 31) 4- Z), (6,S&) (1,7b) %, & and spin + 1 4nc S=-Jld 3 2 I( [ + E +A-1 * - -+.+ + [A+ E-]} = : ti 310 E s, q' * --f (E, S&)x ) q- * ( F , S&, 31). ( ~ 7 4 x, & From (1,7b and (1,7d) i t can be seen that s, characterizes the projection of the spin on the momentum direction for particles with positive as well as negative energy. Relation (1,7d) can easily be deduced from the spin pseudovector whose ~ s[121,4 , , (see space components are defined by the values s [ ~ ~~ r ~g 1 ,] , ~ (1,7d)) and fourth component by 8 4 = 4 yc{ ( *A + Z+)+ (2.i-)) . (138) The momentum, spin and charge of the Bose amplitudes which directly characterize the particles should be found from integral expressions (1,7.b, c, d) (see ref.s)). They obey the following commutation relations a- a+ - a+ u- = 1, ( 3 >9) where a+ and a- are the particle creation and absorption operators respectively. Determining now the commutation relations for q t from the quantum equation of motion and taking into consideration (1,9) we get where =E is the particle momentum. We shall now investigate the dependence of the boson polarization (which we specify by the so-called helicity9) on the magnitude of sE. D. D. Ivanenko, A . A . Sokolov, Sow. Phys. 11, 590 (1937). --t The spin is frequent1 y described by means of the pseudovector s whose direction is a matter of convention. Therefore it seems better to characterize the spin (or polarization) by its helicity. If a screw is moved in the direction of the momentum and the thread is such thet its direction of rotation coincides with that of the polarization vector we obtain in the case of a right-handed screw a right-handed he1 icity and in the case of a left-handed screw a left-handed helicity. It shuolf be noted that for particles with ?no= 0 the helicity is conserved in all Lorentz systems. 9, Consider first the case s, = 1 when (1.11) ?])exp( i c e g t ) . A- - ( p F i & [ g-+o /"0 If the vector Po is directed along the x axis and the momentum g'along the 2 -+ z axis, the vector [go?] will be directed along the y axis. ning only the real part of the functions we get: A ; -sin g c t . A,- N cos g c t , 111 this case, retai(Wa) + It can be seen from here that A* will rotate from the x axis to they axis. i.e. for s, = 1 in a right-handed coordinate system the helicity will be righthanded for .s = 1as well as E = - 1and it will be left-handed in a left-handed system. I n contrast, for s, = - 1 we have a left-handed helicity in a righthanded system and vice versa. 2. Invariance of the vector equations under C, 1' and I transformations We shall now investigate how the vector equations for free particles with oriented spins behave separately under charge conjugation transformations (C), time reversal ( T )and space inversion (I)since the L u d e r s - P a n l i theorem ( 'T I = const refers to the combined triple operation. a ) For charge conjugation transformations (C) we haw + --t 8'' -L AT. Therefore, to retain the iorm of the fundamental equations in the new (primed) system we must perform the following traiisformation in (1,4) ' -+ I x - x , e' = - p . (2.1) Then for A'= we obtain an expression of the form identical to (1,4) and in which the primed quantities are related to the unprimed ones by the equations + * + + b' ( 8 0 , s,,., + X ' O ) = b- ( P O , s . - 2) = ($0 3- i 8;. [Z'"])10), pf -= - t, 1 I v2 p (-A', s-,, + -x ' ) = q'+ (E', s:., I ' x') (W From (2,s) it can be seen t h a t the new amplitudes a'= the arguments vharacterise directly t h e s t a t e o t t h e particlc, are connected w i t h the old ones by the relations a'* I , -+ = 1,sl,Q = - e , g ' ) = oi - + - + s - ~ = s;, Q = Q', g = g '). (2,3) Thus if the initial state is that represented in fig. 1 we obtain as a result of charge conjugation the state shown in fig. 2. The momentum vector (solid arrow) and direction of rotation of the polarization vector (helicity) characterize the partirle state. The direction of the spin pseudovector (dotted arrow) however is only of a conditional nature. Noninvariance (C # const) is exhibited here in the fact that if in the main equation we have a certain helicity for t = 1 then for F' = 1 it will be of opposite sign (s; = s-I = - sI). (F' (E = -F', + We shall not prime the normalization vectors 2, as t h e j are G numbers a n d are --f * completely defined by the arguments p, sE a n d X" (of formula (1,5)). lo) 46 d n n a l e n der Physik. 7. Folge. Band 5. 1959 * -+ + -+ b) Under time reversal (T)we have t’ = - t, F‘ = r , for Ef* = -E*, + - + + HI* = H i , A’= = A * equation (1,4) retains its form in the primed coordinate system if we make the snbstitution +I x = x+, e ’ = e . & I = - & , Then - - f + 6; 4’; (PO, (&I, I + ’ ’ a&,, x o ’ ) = 61 ( P O , s;., Z f ) = qr (- --f S - E , , I t ~ / )= Po]),I I 1 + (80 i .:. o;[f ‘ I &I, s-,,, x ). (2>4) L‘/ &=-I, &)=-I,S-;=s1=-I $,=+I 61 0) Fig. 1.Initial state of longitudinally polarized particles (right-handed system) b, S1=S-, -’ Ji+ig.2. Charge conjugation transfornia- tion (right-handed system) x I=x 9 Fig. 3. Time reversal (right-handed system) 4 Fig. 4. Space inversion (left-handed system) A . A . Sokolou a . Yu.-If. Losliiito~:Theory of Bosofzs and Ferrnions 47 I n this case, as in the case of charge conjugation, the relation (see (2,4)) between the new primed amplitudes and old ones can be easily established a” (E’= l,s;,&’=-e,g -, ) = a ’ ( ~ = - ~ ’ , ~ ~ = s i , & = - - &r , g-’= - g ’*) . (2,s) It can be seen from here t h a t under time reversal the creation operator for negative energy particles transforms into an absorption operator of positive energy particle with an opposite charge. Just as under charge conjugation the helicity of particles of identical = - sl) and in this energy will change its sign under time reversal (8; = sense T # const. The time seversal transformation is represented in fig. 3. However if combined inversion is carried out that is, both the charge conjugation, time reversal are performed we obtain CT (2.6) = COIIS~. c) Finally m e shall consider the space inversion oparation (1): t’ = t + -> + + Then E“ = - E - , H‘I = H and A’ = - ,4*. Changing furA 6 = - r. A + - * == - P O ) thermore the signs of vectors 2 (f! = - 2 ) and o f vector P O (/P’ and of t h e spin projection on the momentum direction (6;. =; - ~ ~ 1 for 1 ) + E‘ = P , we .obtain for A’- equation (1,1) : with unprimed quantities replaced by the primed ones in the right handside. that is, equations ( l , 2 ) and (1,s) are invariant in this case. * For the amplitudes 2’4 and vector h‘ in the primed system w c obtain As in the two preceding transformations the Bose commutation rules (1,9) are conserved under spare inversion, and transition from the second quantised amplitudes a* t o the amplitudes a‘& in this case is defined by the equation It can thus be seen that under space inversion the energy and the charge do not change signs (E‘ = F, &‘ = 8)whereas the momentum and spin projection on the momentum change their signs‘ ;( = Y,S, ’ = - 81). The space inversion transformation is represented in fig. 4. It can be seen that helicities corresponding to the energies of the same sign do not change ( I = const) and in this sense inveriance is preserved, despite the fact that s’ = --s. -- 11) It should be noted t h a t in the right and left-handed coordinate systems the same helicity is described by values of s with opposite signs. Therefore if sschanges sign under space inversion t h e helicitj- should remain the same. 48 Annalen der Physik. 7 . Folge. Band 5. 1959 Thus the L u d e r s - P a u l i theory is valid in the rheory of vector particles with oriented spins as well as in the fermion theor-ylz). TCI = const (229) 3. Comparison with results of the theory of Dirae particles with oriented spins It has been s h o ~ n ~ )t h~a t~the ) ~L~u d) e r s - P a u l i theorem is valid in the two-component theory since combined inversion C I = const, T = const (321) is applicable in this case. It turns out, moreover, that the theory of fermions with oriented spins can also be formulated in such a manner t,hat combined inversion (2,6) is applicable. This has been performed in ref.4). I n the present paper we shall consider the case of fermions with nonvanishing masses (m,, # 0). If auxiliary condition (1,1)15) is imposed on the D i r a c particles the general solution of the free motion and k where c p1 = c - is the velocity of the electron in the intial coordinate system. K The quantity s' 0 1- =_ = ; _ v(i - p pl cos el2 B cos 8 -3) - (1- p < i therefore becomes smaller than unity. This is due t o the fact that the momentum vector is time-like and its fourdimensional components are proportional .+ to kl,, = k , k4 = K whereas the four-dimensional components of the spin = pseudovector are space-like and the components are proportional to slZ3 k. g, sa = k therefore these vectors rotate through different angles under L o r e n t z transformations. For particles with zero rest mass ( K = k ) the 16, = 1. I n this case 6; = 6: and hence in the new coordinate system s' will be also equal unity (8' = s = 1). 12) This is true only for free particles. I n presence of an interaction energy V which involves the product of several wave functions the variation of V as a whole should be taken into account in TCI transformations. la) A. Salam, Nuovo Cimento 6 , 299 (1957), la) L. D. L a n d a u , Nucl. Phys. 3, 127 (1957). 15) I n passing it may be mentioned that for particles with nonvanishing rest masses (m,,# 0) condition (1,l)willnot be Lorentz-invariant. Indeed, if the three dimensional momentum and spin vectors in the initial coordinate system were parallel to each other ( 5 = 1) and formed an angle 0 with the z axis, then in a new L o r e n t z coordinate system moving relative to the initial one with EL velocity c 8 along the z axis the angles would be respectively. A . A . Sokoloc n. 49 l-u, M . Loskulov: Theory of Bosom and Fermions It has permitted us to develop a theory for neutrinos with oriented spins arid zero masses (for details see ref.6)). D i r a c can be expressed as + + ++ y* = L-"* 2 b* (F. s, k ) y* ( F , s, k ) e = i c s K t r i k r (3,2) 4 h,E where -iv /(F b- = K ) cos Bs e * f (K ~ ) sin Bs e 12 [SF~(--F K ) c8:L: 9, s~f(-~K)sin19,e 1 K = 1/#F+ k; (3,s) 0s = 2e - 4x (1-s) , and 0 and y are spherical angles of vector i; s, = & 1 characterizes the double projection of the spin vector (in units of ti) on the momentum direction for particles E = 1 as well as for particles with E = - 1. The matrix b+ is conjugated t o matrix h-. The Hamiltonian for D i r a c particles H = J y+ (c (Z 5)+ e3 m, c2) y d3 2 = 2 c ti e K y+ + (E, s, k ) y- ( e , s, differs from the bosoii Hamiltonian (1,7a) by the sign factor, sum symbol. As a result, the F e r m i commutation rules a+ a- E, Z) (3,4) under the + a- a+ = 1 (33) will apply to the corresponding D i r a e amplitudes u+. This signifies that the amplitude a- is the absorption operator and emplitude a+ - the creation operator. We shall now show that for fermions with s, = 1 the function y will describe particles of right-handed helicity in a right-handed coordinate system for arbitrary values of E (F = 4- 1); in a left-handed system the helicity will be left-handed. Indeed, if the momentum = E k is directed along the z axis we obtain (of 8, = 1) --f + It has been taken into account here t h a t for E = - 1 vector k in accord + -+ with ( l , l O ) , should be replaced by - k (since in 110th cases the vector 2; = E k should be directed along the z axis), i.e. angle 8 should be equal. Ann. Physlk. 5. Folge. Rd. 5 4 50 Annalen der Physik. 7 . Folge. Band 5. 1959 The spinor waves xx=OxwE=l - 8, = 0, yE=-1- - 2. (-TK)) exp (ic K t ) -sin c K t will describe the rotation of the polarization vector from x to y thus forming a right-handed screw in a right-handed coordinate system and a left-handed screw in a left-handed system. I n the case 8, = - 1it is not difficult to show that the helicity will he of opposite sign. Consider now the L u d e r s - P a u l i theorem for P e r m i particles with non zero rest mass. We shall investigate the invariance of directly solution (3,2) which not only obeys the D i r a c equation but also auxiliary condition (1,l). a) As is well known, under the charge conjugation ( C ) the y function varies according to the law yJ* = - f i a 2 p 3 y ’ + T (336) = 1) and the matrix - i a2g, in where f is the normalized coefficient our notations (see ref.16)) has the form 0 0 0 1 (]/I2 --ia,e3= :I. (:-;-; 1 0 0 0 Noting that y’ T is a transposed matrix we can retain y’* in the same form as (3,2) by making the substitution --f -+ e’ = - E , Ic’ = - k , s:. = s,, e’ = - e . Applying then transformation (3,6) we obtain + -+ -++ s, -h’) e i 6 c & ’ E ’ t r i k ’ r y’i = L - ’ / s z b’i ( E ‘ , s,: k’) q’+ -+ k ,E’ 16) A. S o k o l o v , Quantenelektrodynamik, Akademie-Verlag, Berlin 1957. (3,7) 51 A . A . Sokolov a. Yu.111. Loskuloti: Theory of Bosoas utkd Fermions and + -> 6‘- (t’,s;, k’) = - 1.. f 0 1 g, ~ b+ (Thus for the new amplitudes q’ we get E’, s,, - k’),f = i -+ E’. -+ q’+ ( F ’ , sit, k’) = 4’ (- E l , st, - k’). (3,8) It can easily be shown that the F e r m i as well as Bose amplitudes specify the particle properties, transform in accord with relations (2,3) (see also fig. 2), that is, for particles with the same energy sign the particleh elicity changes = - .sl) and in this sense C # const. its sign (si1 = + - + b) Under time reversal (T) : t‘ = - t, r’ = r and in order that the form o f function y‘+ be the same as in (3,2), substitution should be made E’ = -E, s :, = s,, + + k’ = k, e‘ = e. (3,9) The wave functions, on the other hand, are connected as follows $2 For amplitudes q’z we find = -i + E’ 8:. pz y’. + k’) = 4‘ (- F ‘ , s,, k’). q’ ’ ( F ’ , (3JO) It can now be easily demonstrated that the F e r m i amplitudes a* transform in accordance with formiila (2,s) (see fig. 3) derived by us for Bose amplitudes, i.e. under charge conjugation in virtue of the equation s; == - sL1 we get T# const. -+ c) Finally, under space inversion ( I ) when t’ = t, ?‘ = - r the wave functions will transform according to the rule y‘ = - i g, y. I n order that -c -> the new soliition possess the form as (3,2) the substitution E’ = E, k’ = - 1:. s,‘,= - s, should be made. Hence with the aid of (3,7) we find that under space inversion helicity is conserved. I = const (3,13) for particles with positive as well as negative energy (s; = - s1 and s! == - s- see fig. 4). The rrlatiori b e h e e i i the primed Dirac amplitudes (c‘- and the uiiprimetl ones is the same as for bosoms (see 2,s). Although for F e r m i particles C # const and 1’# const one finds that under combined inversion of the C T type (charge conjugation plus time reversal) the field free equations retains its helicity. As it is well known, the experments, which can confirm the reality of F e r m i ’ s coefficients, are interpreted according t o two-component theory as the verification of the law T = const. According to our theory these expcrinents are interpreted as the verification of the law T C = const. I n conclusion we shall consider the transformation of Lagrangians under hpace and time inversion. For vector equations (bosom) the Lagrangian L + + 1 --f* = - ((B’ Ii-) - (H+ H-)) ‘4z ~ 52 Annnlen der Physik. 7 . Folge. Band 5. 1959 is obviously invariant under tome reversal and space inversion inasmuch as -+ + + - + under these transformations E'+ = - E , H'- = H + . On the other hand for D i r a c particles the Lagrangian I where ++ v)+ e3 is invariant under space inversion (v' =- v, y' H = CfL (a ~ + f c2i ~ -+ = - i p3 y ) but changes its sign, as Schwinger has notedl'), under trime reversal (t' = -t, If." = - i F' .:3 &Iz y,) f i a L' = y'f (T- W j y;. at' + Schwinger has inverstigated in the general form the type of the auxiliarjtransformations of functions y that should be performed on order t o preserve the required invariance of the theory. In our case this was attained by substituting E for -&'; from the physical point of view this means transferring particles from positive levels to negative ones and vice versa. This transformation corresponds to a trivial substitution which does not affect the equation itself In this case the Lagrangian retains its invariant form The authors are grateful t o M. M. K o l e s n i k o v a for help in treatment of the results presented in the graphs. 17) J. Schwinger, Physic. Rev. 82, 914 (1951). M o s c o w , State University. Bei der Redaktion eingegangen am ti. April 1969.

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