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Theory of Dirac particles with oriented spins and parity non-conservation
OF
PARTICLES WITH ORIENTED SPINS A I NON-CONSERVATION A. A. SOKOLOV Moscow State University Received 23 August 1953
Abstract : The Lilders-Pauli theorem is investigated in connection with a type of theory chosen to describe Dirac particles with oriented spins.
1.
troductlo
The discovery of parity non-conservation by Lee and Yang 1) plays an important role in the theory of weak interactions. It has been shown in a preceding note 2) that Lee and Yang's results can be derived from the theory of Dirac particles with oriented spins. Besides the Dirac equation, the free particle wave function V now satisfies the auxiliary condition 3 . 4 )
sv
y z-%1(-®2) since the operator a commutes with the Hamiltonian . The quantity s = ± 1 characterizes the double spin projection on the direction of motion . It has been remarked 2,5) that in the case of ß-decay or meson decay with neutrino emission summation over spin states should be performed not by the Casimir formula in which averaging is carried out over spin states, but by the following formula in which the spin states enter explicitly:
= s try'
k' ° k' Ko 1 +P1E° s È° +p3e , r a ,k, ( k ko X l +s Y 1 +Ples +P38 k K K
1-i-s
o' , k k
Here p., a. are the well known Dirac matrices and P4 = 04 = I; the matrices y' and y specify the type of interaction : scalar (ys = p3), pseudoscalar (Y p = P2), vector (yv = I), axial vector (y' = a) or tensor (),T = P3Q) . The last three interactions have been written down for the non-relativistic case . The quantity s = ± 1 specifies the sign of the energy ; the latter is related to the particle momentum and rest mass ko by the equation K = 1/(ko2+k2) . 420
THEORY OF DIRAC PARTICLES
42 1
Formula (2) can conveniently be used to study phenomena connected with longitudinally polarized electrons 6, 7, 8). It may also be employed to describe polarization properties in weak interactions involving neutrinos. For this purpose the value of s should be fixed (the neutrino will be specified by unprimed letters K, k, k ® = 0, whereas the primed quantities will refer to particles produced along with the the neutrino, e.g., an electron). If for ß--decay one puts e' = 1 for an electron and e = -1 for an antineutrino, then, depending on the type of interaction, the matrix elements characterizing the spin states will be T Rs, .%
- P33
-
P44
1±ss'ß
where cß is the velocity of the emitted electrons and the values for P33 and P44 are given in ref.4) (see formulas (21.17) and (21.18)) . The definition of the neutrino and antineutrino requires some clarification and we shall therefore consider this point in the present paper. It seems most proper to use the term antineutrino for a particle which is created simultaneously with an electron in ß--decay. If - the experimental results of Frauenfelder et al.9) are taken into account (according to these data electrons are emitted in ß -- decay predominantly with s' = -1), then, to be consistent with (3) one must put, in the case of S, T coupling, s = -1 (left screw, according to Lee and Yang's definition) for an antineutrino and s = 1 (right screw, Feynman and Gell-Mann 1°)) in the case of V, A coupling . For a combination of these types of interaction it may be necessary to introduce two different types of antineutrinos. 2. Lüders
ault Theorem in the New Theory of the Neutrino
Tkle spinor amplitude for the free Dirac equation has the form
11)
f (eK) cos 0$ bSS (k)
where
08
1 _ ~/2
= 120 -14-n (1-S),
f (eK)
sin ®Se =e
sef (-eK) cos 0S
~
(~)
sef (-eK) sin 08 efl P f(eK) =
V 1 + (koleK) ,
and 0, 99 are spherical angles of vector For a neutrino (ko = 0) the wave functions will be related by the equation 2) (V3
`Ws~ . R °Ya)
42 2
A. A. SOKOLOV
where
â = ess. , (6) It follows from here that A can assume four values which specify the spin direction for positive (s = 1) and negative (s -= -1) energy values. If s+ and s_ are respectively the values of s for s =1 and s = -1, we may have (a) A -- 1 : then s+ = -s_ = 1, (k)) A = -1 : then s+ = -s-=-I, (c) A = s : then s+ = s_ = 1, (d) A = -s : then s+ = s_ = -1 . S .nce the I9irac equation satisfies all relativistic invariance requirementE it will be sufficient for us to confine ourselves to an investigation of the . invariance of the auxiliary condition (5) . Under space inversion (transformation from right to left handed coordinate system) the wave function transforms according to V' = psy V1 1) == (VI) 13 (V3 '2 2 W~4 4 Moreover, in this case s' = s, k' = - (polar vector), s' = s (axial vector) where s is a unit spin vector connected with s by the relation _ s ~k s K Hence s' = -s, A' = -â and therefore in the primed coordinate system the auxiliary condition (5) retains its invariant form (I = const) t. For time reversal we have yr' = p2iV :
V V rl ^ r i 3 V,3 = 2 1 (9 . V2 V2 (V4) V '4 In this case s' = -s, k'T == -k± , s't = -ST-, s'± = s:F , where k+ and kare the particle momenta when s is equal respectively to + 1 and -1 Closely related to time reversal is the charge conjugation transformatioi considered in detail by Pauli 12) Y' = -ia2P3eT or i~ '3 - -iQ . 'a = iQ' 2 y3+ 2 ('2 2+ (Y4+) (Y 4 where Q'2 = (°-ô) is a two-row Pauli matrix and V+T is a transposed matrix It can easily be shown in this case that s' = --s, k'± = kT- , s'± -:~_- sT , s'± = ST . t Under space inversion the spin conserves its helicity relative to the momentum .
THEORY OF DIRAC PARTICLES
423
Thus under time reversal as well as charge conjugation we have t Let us first consider cases (c) and (d) when s+ = s_ and therefore Hence it follows, after taking into account the transformation laws for the wave functions (9) and (10), that condition (5) conserves its invariant form under time reversal as welt as under charge conjugation : (12) T = const, C = const. These cases are equivalent to the new formulation proposed by Case 13) (also see ref.l4)) in the 1Viajorana theory of the neutrino if in the latter theory one assumes ko = 0. Developing auli's work 12), Case showed (see his eq. (3)) that the generalized ajorana equation can be obtained if one imposes on the Dirac wave fl motions the auxiliary condition (13) -ia2P3 V+ where V+T is the transposed matrix and 114) w± - 2(1~P1) ~ . It can easily be shown that conditions (13) and (14) can be derived from (1) if for any value of the energy (E _ 1) we retain the sign befoi e s and rote that the wave function y L?so satisfies the Dirac equation (15) and that the neutrino charge is zero:
- Zx2P3 +T e=1
From the physical standpoint the possibility of deducing some new res alts in the Majorana theory from conditions (c) or (d) is due to the fact that only one state is assigned to the neutrino (,- = 1) and one to the antineutrino (s = -1), the spin directions of these states being identical (s+ = s_). Therefore, if the neutrino charge is zero these particles will be indistinguishable and the equation itself (as was just remarked) should be invariant under charge conjugation . In cases (a) and (b), however, we have s+ = -s_ and hence A'=A=g'Se=E'S'e. .
It is easy to show with the hAp of the wave function transformation formulas (9) and (10), that the sign in the right hand side of (5) changes under time const) . reversal (T -- const) as well as under charge conjugation (C
t Under time reversal the positon annihilation operator with a = -1 becomes a negaton creation operator with e' --- I ; under charge conjugation it becomes the positon annihilation operator with e' = 1 .
424
A. A. SOKOLOV
This auxiliary condition will be invariant under combined inversion of the type TC -== const, I = const . On the other hand, in cases (a) and (b) condition (1) conserves the value of s only for particles possessing identical energy signs since s+ = --s_. If the separation constant A is to be the same for any of the values of s = 1, we should write instead of (1) ( H a " p11V==O. (16) `~
IHI
Iii
Hence for particles with a non-vanishing rest mass we obtain k ko 0 p3 a ~+ Pl W -k ` In particular for a neutrino (ko .-- 0) we obtain the condition (A-PI)v = 0
(17) (18)
which yields either the results of Lee and Yang (A, = 1, case (a)) or the results of Gell-Mann and Feynman (A = -1, case (b) ) . The experimental data indicate Is) that in #+-decay (p -> n+e+ +v) the violation of symmetry is opposite to that observed in the case of ß--decay and hence there is much reason to believe that cases (a) and (b) are those that are actually realized, i.e. the neutrino (v) and antineutrino (v") spins are oppositely directed in these cases. In order to give a final formulation of the theory and to verify the ICT-theorem it seems to be important to have data relating to asymmetry of antineutron decay and absorption of neutrinos or antineutrinos by neutrons . In conclusion we may note that in our analysis parity non-conservation was related to the orientation of the neutrino spin characterized by the quantity s. Under space inversion the sign of s is to be changed, even though s in both cases refers to the same helicity. If one attempts to relate parity non-conservation in some way to the properties of space in which 154- const, it will be necessaxy, in order to retain the Lüders-Pauli theorem, to introduce combined inversion of the type IC = const, T = const (Landau ie) ) although the possibility of IT = const, C = const is also theoretically permissible. eferences 1) T. D. 2) A. A. A. A. 3) A. A. 4) A. A.
Lee and C. N. Slang, Phys . Rev. 104 (1956) 254; 105 (1957) 1671 Sokolov, Nuovo Cimento 7 (1958) 240; Sokolov and B. K. Kerirnov, Ann . d. Phys . 2 (1958) 46 Sokolov, Journ. of Phys . 9 (1945) 363 Sokolov, Quantenelektrodynamik (Akad. Verl . Berlin, 1957)
THEORY OF DIRAC PARTICLES
5) 0) 7) 8) 9) 10) 11) 12) 13) 14) 15) 10)
425
A. A. Sokolov. Atomnaya Energiya 4 (1958) 385 A. A. Sokolov and Yu. M. Loskutov, JETP 34 (1958) 1022 A. A. Sokolov and V. A. Lysov, JETP 34 (1958) 1351 E. K. Kerimov and 1. :4i. Nadzhafov, Scientific Reports of Higher School ser . Fiz . Matem. 1 (1958) 79 H. Frauenfelder et al., Phys. Rev. 106 (1957) 380 R . Feynman and M . Gell-Mann, Phys. Rev. 109 (1958) 193 A. A. Sokolov and B. K. Kerimov, Nuovo Cimento 5 (1957) 921 W. Pauli, Revs. Mod . Phys. 13 (1941) 203 K. Case, Phys. Rev. 107 (1957) 307 C. Enz, Nuovo Cimento 6 (1957) 250; W. Pauli, Nuovo Cimento 6 (1957) 204 H. Postma, W . Huiskamp, A. Miedema, M. Steenland, H. Tolhoek and C. Gorter, Physica 23 (1957) 259 L. D. Landau, Nuclear Physics 3 (1957) 127
PARTICLES WITH ORIENTED SPINS A I NON-CONSERVATION A. A. SOKOLOV Moscow State University Received 23 August 1953
Abstract : The Lilders-Pauli theorem is investigated in connection with a type of theory chosen to describe Dirac particles with oriented spins.
1.
troductlo
The discovery of parity non-conservation by Lee and Yang 1) plays an important role in the theory of weak interactions. It has been shown in a preceding note 2) that Lee and Yang's results can be derived from the theory of Dirac particles with oriented spins. Besides the Dirac equation, the free particle wave function V now satisfies the auxiliary condition 3 . 4 )
sv
y z-%1(-®2) since the operator a commutes with the Hamiltonian . The quantity s = ± 1 characterizes the double spin projection on the direction of motion . It has been remarked 2,5) that in the case of ß-decay or meson decay with neutrino emission summation over spin states should be performed not by the Casimir formula in which averaging is carried out over spin states, but by the following formula in which the spin states enter explicitly:
= s try'
k' ° k' Ko 1 +P1E° s È° +p3e , r a ,k, ( k ko X l +s Y 1 +Ples +P38 k K K
1-i-s
o' , k k
Here p., a. are the well known Dirac matrices and P4 = 04 = I; the matrices y' and y specify the type of interaction : scalar (ys = p3), pseudoscalar (Y p = P2), vector (yv = I), axial vector (y' = a) or tensor (),T = P3Q) . The last three interactions have been written down for the non-relativistic case . The quantity s = ± 1 specifies the sign of the energy ; the latter is related to the particle momentum and rest mass ko by the equation K = 1/(ko2+k2) . 420
THEORY OF DIRAC PARTICLES
42 1
Formula (2) can conveniently be used to study phenomena connected with longitudinally polarized electrons 6, 7, 8). It may also be employed to describe polarization properties in weak interactions involving neutrinos. For this purpose the value of s should be fixed (the neutrino will be specified by unprimed letters K, k, k ® = 0, whereas the primed quantities will refer to particles produced along with the the neutrino, e.g., an electron). If for ß--decay one puts e' = 1 for an electron and e = -1 for an antineutrino, then, depending on the type of interaction, the matrix elements characterizing the spin states will be T Rs, .%
- P33
-
P44
1±ss'ß
where cß is the velocity of the emitted electrons and the values for P33 and P44 are given in ref.4) (see formulas (21.17) and (21.18)) . The definition of the neutrino and antineutrino requires some clarification and we shall therefore consider this point in the present paper. It seems most proper to use the term antineutrino for a particle which is created simultaneously with an electron in ß--decay. If - the experimental results of Frauenfelder et al.9) are taken into account (according to these data electrons are emitted in ß -- decay predominantly with s' = -1), then, to be consistent with (3) one must put, in the case of S, T coupling, s = -1 (left screw, according to Lee and Yang's definition) for an antineutrino and s = 1 (right screw, Feynman and Gell-Mann 1°)) in the case of V, A coupling . For a combination of these types of interaction it may be necessary to introduce two different types of antineutrinos. 2. Lüders
ault Theorem in the New Theory of the Neutrino
Tkle spinor amplitude for the free Dirac equation has the form
11)
f (eK) cos 0$ bSS (k)
where
08
1 _ ~/2
= 120 -14-n (1-S),
f (eK)
sin ®Se =e
sef (-eK) cos 0S
~
(~)
sef (-eK) sin 08 efl P f(eK) =
V 1 + (koleK) ,
and 0, 99 are spherical angles of vector For a neutrino (ko = 0) the wave functions will be related by the equation 2) (V3
`Ws~ . R °Ya)
42 2
A. A. SOKOLOV
where
â = ess. , (6) It follows from here that A can assume four values which specify the spin direction for positive (s = 1) and negative (s -= -1) energy values. If s+ and s_ are respectively the values of s for s =1 and s = -1, we may have (a) A -- 1 : then s+ = -s_ = 1, (k)) A = -1 : then s+ = -s-=-I, (c) A = s : then s+ = s_ = 1, (d) A = -s : then s+ = s_ = -1 . S .nce the I9irac equation satisfies all relativistic invariance requirementE it will be sufficient for us to confine ourselves to an investigation of the . invariance of the auxiliary condition (5) . Under space inversion (transformation from right to left handed coordinate system) the wave function transforms according to V' = psy V1 1) == (VI) 13 (V3 '2 2 W~4 4 Moreover, in this case s' = s, k' = - (polar vector), s' = s (axial vector) where s is a unit spin vector connected with s by the relation _ s ~k s K Hence s' = -s, A' = -â and therefore in the primed coordinate system the auxiliary condition (5) retains its invariant form (I = const) t. For time reversal we have yr' = p2iV :
V V rl ^ r i 3 V,3 = 2 1 (9 . V2 V2 (V4) V '4 In this case s' = -s, k'T == -k± , s't = -ST-, s'± = s:F , where k+ and kare the particle momenta when s is equal respectively to + 1 and -1 Closely related to time reversal is the charge conjugation transformatioi considered in detail by Pauli 12) Y' = -ia2P3eT or i~ '3 - -iQ . 'a = iQ' 2 y3+ 2 ('2 2+ (Y4+) (Y 4 where Q'2 = (°-ô) is a two-row Pauli matrix and V+T is a transposed matrix It can easily be shown in this case that s' = --s, k'± = kT- , s'± -:~_- sT , s'± = ST . t Under space inversion the spin conserves its helicity relative to the momentum .
THEORY OF DIRAC PARTICLES
423
Thus under time reversal as well as charge conjugation we have t Let us first consider cases (c) and (d) when s+ = s_ and therefore Hence it follows, after taking into account the transformation laws for the wave functions (9) and (10), that condition (5) conserves its invariant form under time reversal as welt as under charge conjugation : (12) T = const, C = const. These cases are equivalent to the new formulation proposed by Case 13) (also see ref.l4)) in the 1Viajorana theory of the neutrino if in the latter theory one assumes ko = 0. Developing auli's work 12), Case showed (see his eq. (3)) that the generalized ajorana equation can be obtained if one imposes on the Dirac wave fl motions the auxiliary condition (13) -ia2P3 V+ where V+T is the transposed matrix and 114) w± - 2(1~P1) ~ . It can easily be shown that conditions (13) and (14) can be derived from (1) if for any value of the energy (E _ 1) we retain the sign befoi e s and rote that the wave function y L?so satisfies the Dirac equation (15) and that the neutrino charge is zero:
- Zx2P3 +T e=1
From the physical standpoint the possibility of deducing some new res alts in the Majorana theory from conditions (c) or (d) is due to the fact that only one state is assigned to the neutrino (,- = 1) and one to the antineutrino (s = -1), the spin directions of these states being identical (s+ = s_). Therefore, if the neutrino charge is zero these particles will be indistinguishable and the equation itself (as was just remarked) should be invariant under charge conjugation . In cases (a) and (b), however, we have s+ = -s_ and hence A'=A=g'Se=E'S'e. .
It is easy to show with the hAp of the wave function transformation formulas (9) and (10), that the sign in the right hand side of (5) changes under time const) . reversal (T -- const) as well as under charge conjugation (C
t Under time reversal the positon annihilation operator with a = -1 becomes a negaton creation operator with e' --- I ; under charge conjugation it becomes the positon annihilation operator with e' = 1 .
424
A. A. SOKOLOV
This auxiliary condition will be invariant under combined inversion of the type TC -== const, I = const . On the other hand, in cases (a) and (b) condition (1) conserves the value of s only for particles possessing identical energy signs since s+ = --s_. If the separation constant A is to be the same for any of the values of s = 1, we should write instead of (1) ( H a " p11V==O. (16) `~
IHI
Iii
Hence for particles with a non-vanishing rest mass we obtain k ko 0 p3 a ~+ Pl W -k ` In particular for a neutrino (ko .-- 0) we obtain the condition (A-PI)v = 0
(17) (18)
which yields either the results of Lee and Yang (A, = 1, case (a)) or the results of Gell-Mann and Feynman (A = -1, case (b) ) . The experimental data indicate Is) that in #+-decay (p -> n+e+ +v) the violation of symmetry is opposite to that observed in the case of ß--decay and hence there is much reason to believe that cases (a) and (b) are those that are actually realized, i.e. the neutrino (v) and antineutrino (v") spins are oppositely directed in these cases. In order to give a final formulation of the theory and to verify the ICT-theorem it seems to be important to have data relating to asymmetry of antineutron decay and absorption of neutrinos or antineutrinos by neutrons . In conclusion we may note that in our analysis parity non-conservation was related to the orientation of the neutrino spin characterized by the quantity s. Under space inversion the sign of s is to be changed, even though s in both cases refers to the same helicity. If one attempts to relate parity non-conservation in some way to the properties of space in which 154- const, it will be necessaxy, in order to retain the Lüders-Pauli theorem, to introduce combined inversion of the type IC = const, T = const (Landau ie) ) although the possibility of IT = const, C = const is also theoretically permissible. eferences 1) T. D. 2) A. A. A. A. 3) A. A. 4) A. A.
Lee and C. N. Slang, Phys . Rev. 104 (1956) 254; 105 (1957) 1671 Sokolov, Nuovo Cimento 7 (1958) 240; Sokolov and B. K. Kerirnov, Ann . d. Phys . 2 (1958) 46 Sokolov, Journ. of Phys . 9 (1945) 363 Sokolov, Quantenelektrodynamik (Akad. Verl . Berlin, 1957)
THEORY OF DIRAC PARTICLES
5) 0) 7) 8) 9) 10) 11) 12) 13) 14) 15) 10)
425
A. A. Sokolov. Atomnaya Energiya 4 (1958) 385 A. A. Sokolov and Yu. M. Loskutov, JETP 34 (1958) 1022 A. A. Sokolov and V. A. Lysov, JETP 34 (1958) 1351 E. K. Kerimov and 1. :4i. Nadzhafov, Scientific Reports of Higher School ser . Fiz . Matem. 1 (1958) 79 H. Frauenfelder et al., Phys. Rev. 106 (1957) 380 R . Feynman and M . Gell-Mann, Phys. Rev. 109 (1958) 193 A. A. Sokolov and B. K. Kerimov, Nuovo Cimento 5 (1957) 921 W. Pauli, Revs. Mod . Phys. 13 (1941) 203 K. Case, Phys. Rev. 107 (1957) 307 C. Enz, Nuovo Cimento 6 (1957) 250; W. Pauli, Nuovo Cimento 6 (1957) 204 H. Postma, W . Huiskamp, A. Miedema, M. Steenland, H. Tolhoek and C. Gorter, Physica 23 (1957) 259 L. D. Landau, Nuclear Physics 3 (1957) 127