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Comment on spin one theory
Volume 36B, number 5
PHYSICS LETTERS
COMMENT
ON S P I N
ONE
4 October 1971
THEORY*
T. GOLDMAN** and W. TSAI
Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts 02138, uSA Received 15 June 1971
A way to explain the inconsistency of the spin one theory with anomalous magnetic moment couplings observed by Tsai and Yildiz is proposed. By suiably adding infinitely many non-minimal coupling terms which are tensor functions of the field strength Fbt v, the square of the energy eigenvalues can become positive definite. A p~articular model of replacing Kby g (/_/2) = K'(1 +¼K'2 e2H 2 m - 4 ) - l , where no restriction is imposed on K', is discussed. T h i r t y - o n e y e a r s ago, Schiff et al. [1] found that for the spin zero p a r t i c l e moving in a sufficiently deep e l e c t r o s t a t i c potential well, the eigenvalue equation can have complex eigenvalues. Recently, it was d i s c o v e r e d by T s a i and Yildiz [2] that the eigenvalues of the spin one s y s t e m with the a n o m a l o u s m a g n e t i c m o m e n t coupling of the form ieqK F~tv~ v, moving in a homogeneous magnetic field, can b e come pure i m a g i n a r y at high magnetic f i e l d s . - I t has been suggested that this i n c o n s i s t e n c y can be r e moved by taking into account c o r r e c t i o n s due to p a i r c r e a t i o n and radiative c o r r e c t i o n s . Other a r g u m e n t s have also been proposed, for example, that spin one p a r t i c l e s a r e n e c e s s a r i l y composite and s i m p l y decompose into lower spin p a r t i c l e s at sufficiently high magnetic field. However, these a r e just hand waving a r g u m e n t s and it is in i n t e r e s t i n g to find a convincing way to explain this i n c o n s i s t e n cy. The p u r p o s e of this paper is to propose a way to explain the i n c o n s i s t e n c y . We o b s e r v e that the most g e n e r a l form of the eigenvalue equation is *** (m2 +Tr vlr ~ ~b~t - rr• (~Vdpv) + ieq[l +K(H2)]Fl~vq~ v + e2×(H2)m2 Fp?~ F~V v = 0 ,
(1)
w h e r e ~(H2) and x(H 2) a r e functions of H 2 and we have used the fact that
F~k f k j -- Hill ) - 6 i j n 2 ,
H i Fi) = O.
In a s t r o n g m a g n e t i c field, this takes into account the m a g n e t i c p o l a r i z a b i l i t y of the spin one p a r t i c l e which is r e p r e s e n t e d by t e r m s of o r d e r H 2 or higher. By going through the s a m e calculation outlined in ref. [2], it is easy to obtain the eigenvalues of eq. (1) which a r e (for lr3 = 0) p °2 = m 2 { l + ( 2 n + l ) ~ }
for S 3 = 0 ;
(2)
for S3 = + 1,
w h e r e 7 7 = ( 2 n + l - 2 q S 3 ) ~ , ~=eHIm 2, n = 0 , 1 , 2 , 3 , . . . . Note that eq. (2) i s p o s i t i v e d e f i n i t e and that (3) is r e a l s i n c e
X~22 - (n2-~)(I-(i_~)~) ×2~2 = {[_(~+n~-y:-~j ~ , X~.2+ ~4( 772-~9)(I+n)} >- 0 (2 + ~-IZ~K' The r e s u l t s of ref. [2] can be easily reproduced by letting × = 0 and K(H2) = K = constant in the above
* This work is supported in part by the U.S. Air Force Office of Scientific Research under contract F44620-70-c0030. ** National Research Council of Canada Postgraduate Fellow. *** In the following, the meaning of all the unexplained notations can be found in ref. [2]. 467
Volume 36B, number 5
PHYSICS
LETTERS
4 October 1971
e q u a t i o n s ; i.e., eq. ( 3 ) b e c o m e s pO2=m2~I+v+(1'-K)~2+q'S3(1-K)(2+~)(I~
~
~ 2 _ } 2 )1/2 ~+~2 "
~
(3')
F r o m eq. (3') f o r the g r o u n d s t a t e with n = 0 a n d qS 3 = + 1, we have p o2 = m 2 - KeH, w h i c h c a n b e c o m e n e g a t i v e w h e n K > 0 a n d K e H / m 2 > 1. F o r the s t a t e with n = 0 a n d qS 3 = - 1, we have p °2 ~ 3 KeH
a s e H / m 2 ~ ~o ,
w h i c h b e c o m e s n e g a t i v e w h e n K < 0. T h e r e f o r e only w h e n K = 0 c a n p o 2 b e c o m e p o s i t i v e d e f i n i t e . O u r o b j e c t is to s e e w h e t h e r t h e r e is a c h o i c e of K (H 2) ¢ 0 a n d X (//2) s u c h that po2 is p o s i t i v e d e f i nite. We find it is s u f f i c i e n t to d i s c u s s the c a s e w h e n X (H 2) = 0. In this c a s e , the c o n d i t i o n s for eq. (3) to be p o s i t i v e d e f i n i t e a r e the following: (i) F o r the g r o u n d s t a t e with n = 0 a n d qS 3 = + 1, we have p o2 = rn2(1 - K ~ ) . T h i s i m p l i e s the c o n dition K(~ 2) --< 1/4 ,
(4)
be s a t i s f i e d . (ii) F o r the o t h e r s t a t e s , we have the c o n d i t i o n s I+V
+ ½ ( 1 - K ) 4 2 >/ 0,
and
(1+77+½(1-K)~2) 2~
¼(1-K)2(4+477+~2),
(5)
w h i c h c a n be s i m p l i e d to the f o r m (1 +~) >/ K(K- 1)4 2 .
(6)
A n o t h e r c o n d i t i o n f o r a p a r t i c l e with a n o m a l o u s m a g n e t i c m o m e n t is K (4 2) ~ K' = c o n s t a n t ~,
(7)
a s ~ t e n d s to z e r o . By v i r t u e of eq. (4), eq. (5) is a l w a y s s a t i s f i e d . T h e r e f o r e a n y c h o i c e of K(~2) w h i c h s a t i s f i e s c o n d i t i o n s (4), (6) and (7) c a n give a c o n s i s t e n t r e s u l t , i . e . , po 2 is p o s i t i v e d e f i n i t e . F r o m eqs. (4) a n d (6) it is e a s y to s e e that
i K (~2) I --<1/4
a s ~ -~ ~o.
T h i s i m p l i e s that if the n o n - m i n i m a l c o u p l i n g t e r m s a r e added a s a p o w e r s e r i e s i n ~, they m u s t f o r m a n i n f i n i t e s e r i e s that s u m s to a n o n - p o l y n o m i a l function. T h e p a r t i c u l a r c h o i c e [4] K(~ 2) = K'(1 +¼K'2~2) -1, w h e r e X' i s a n a r b i t r a r y n u m b e r , is a s i m p l e m o d e l of K(~ 2) w h i c h s a t i s f i e s c o n d i t i o n s (4) -(7). With t h i s c h o i c e , the g r o u n d s t a t e e n e r g y b e c o m e s p o2 = m2(1 + ¼K'2~2) -1 (1 - ½K'~) 2, w h i c h is s i m i l a r to the s p i n ½ c a s e . T h u s the i n c o n s i s t e n c y of the (vector) s p i n one t h e o r y a s d i s c u s s e d by T s a i a n d Y i l d i z a p p e a r s to be due to the e x c l u s i o n of m a g n e t i c p o l a r i z a b i l i t y e f f e c t s a n d e s p e c i a l l y due to the fact that the a n o m a l o u s m a g n e t i c m o m e n t did not v a n i s h in v e r y s t r o n g m a g n e t i c f i e l d s ~f with t h e i r m o d e l for K. T h e s a m e r e a s o n i n g c a n b e c a r r i e d out f o r the 6 - c o m p o n e n t t h e o r y a n d the m u l t i s p i n o r t h e o r y . W e a l s o e x p e c t that a s i m i l a r a r g u m e n t c a n be u s e d to e x p l a i n the i n c o n s i s t e n c y of the e l e c t r o s t a t i c p o t e n t i a l w e l l c a s e of ref. [1] a n d the i n c o n s i s t e n c y of the s p i n ~ t h e o r y d i s c o v e r e d b y J o h n s o n a n d S u d e r s h a n t e n y e a r s a g o [3]. We t h a n k P r o f . S i d n e y C o l e m a n for h e l p f u l c o n v e r s a t i o n s . J The physical meaning of K' can be found by going to the weak field case. In this limit, the total magnetic moment of the spin one particle is ~ = (eq/2rn)(l +K')S. J'J" This bears out the contention of Tsai and Yildiz that "the disease is in the formalism itself ~ [2]. 468
Volume 36B, number 5
PHYSICS
LETTERS
4October1971
R efF~'encFs [1] L. I. Shfff, H. Snyder and J. Weinberg, Phys. Rev. 57 (1940) 315. [2] W. Tsai and A. Yildiz, Phys. Rev. D, to be published. T. Goldman and W. Tsai, Phys. Rev. D, to he published; P. Lu and R.H. Good J r . , Phys. Rev. D3 (1971) 1275; W. Tsai, to be published. [3] K.Johnson and E. C. G. Sudarshan, Ann. Phys. 13 (1961) 126.
469
PHYSICS LETTERS
COMMENT
ON S P I N
ONE
4 October 1971
THEORY*
T. GOLDMAN** and W. TSAI
Jefferson Physical Laboratory, Harvard University, Cambridge, Massachusetts 02138, uSA Received 15 June 1971
A way to explain the inconsistency of the spin one theory with anomalous magnetic moment couplings observed by Tsai and Yildiz is proposed. By suiably adding infinitely many non-minimal coupling terms which are tensor functions of the field strength Fbt v, the square of the energy eigenvalues can become positive definite. A p~articular model of replacing Kby g (/_/2) = K'(1 +¼K'2 e2H 2 m - 4 ) - l , where no restriction is imposed on K', is discussed. T h i r t y - o n e y e a r s ago, Schiff et al. [1] found that for the spin zero p a r t i c l e moving in a sufficiently deep e l e c t r o s t a t i c potential well, the eigenvalue equation can have complex eigenvalues. Recently, it was d i s c o v e r e d by T s a i and Yildiz [2] that the eigenvalues of the spin one s y s t e m with the a n o m a l o u s m a g n e t i c m o m e n t coupling of the form ieqK F~tv~ v, moving in a homogeneous magnetic field, can b e come pure i m a g i n a r y at high magnetic f i e l d s . - I t has been suggested that this i n c o n s i s t e n c y can be r e moved by taking into account c o r r e c t i o n s due to p a i r c r e a t i o n and radiative c o r r e c t i o n s . Other a r g u m e n t s have also been proposed, for example, that spin one p a r t i c l e s a r e n e c e s s a r i l y composite and s i m p l y decompose into lower spin p a r t i c l e s at sufficiently high magnetic field. However, these a r e just hand waving a r g u m e n t s and it is in i n t e r e s t i n g to find a convincing way to explain this i n c o n s i s t e n cy. The p u r p o s e of this paper is to propose a way to explain the i n c o n s i s t e n c y . We o b s e r v e that the most g e n e r a l form of the eigenvalue equation is *** (m2 +Tr vlr ~ ~b~t - rr• (~Vdpv) + ieq[l +K(H2)]Fl~vq~ v + e2×(H2)m2 Fp?~ F~V v = 0 ,
(1)
w h e r e ~(H2) and x(H 2) a r e functions of H 2 and we have used the fact that
F~k f k j -- Hill ) - 6 i j n 2 ,
H i Fi) = O.
In a s t r o n g m a g n e t i c field, this takes into account the m a g n e t i c p o l a r i z a b i l i t y of the spin one p a r t i c l e which is r e p r e s e n t e d by t e r m s of o r d e r H 2 or higher. By going through the s a m e calculation outlined in ref. [2], it is easy to obtain the eigenvalues of eq. (1) which a r e (for lr3 = 0) p °2 = m 2 { l + ( 2 n + l ) ~ }
for S 3 = 0 ;
(2)
for S3 = + 1,
w h e r e 7 7 = ( 2 n + l - 2 q S 3 ) ~ , ~=eHIm 2, n = 0 , 1 , 2 , 3 , . . . . Note that eq. (2) i s p o s i t i v e d e f i n i t e and that (3) is r e a l s i n c e
X~22 - (n2-~)(I-(i_~)~) ×2~2 = {[_(~+n~-y:-~j ~ , X~.2+ ~4( 772-~9)(I+n)} >- 0 (2 + ~-IZ~K' The r e s u l t s of ref. [2] can be easily reproduced by letting × = 0 and K(H2) = K = constant in the above
* This work is supported in part by the U.S. Air Force Office of Scientific Research under contract F44620-70-c0030. ** National Research Council of Canada Postgraduate Fellow. *** In the following, the meaning of all the unexplained notations can be found in ref. [2]. 467
Volume 36B, number 5
PHYSICS
LETTERS
4 October 1971
e q u a t i o n s ; i.e., eq. ( 3 ) b e c o m e s pO2=m2~I+v+(1'-K)~2+q'S3(1-K)(2+~)(I~
~
~ 2 _ } 2 )1/2 ~+~2 "
~
(3')
F r o m eq. (3') f o r the g r o u n d s t a t e with n = 0 a n d qS 3 = + 1, we have p o2 = m 2 - KeH, w h i c h c a n b e c o m e n e g a t i v e w h e n K > 0 a n d K e H / m 2 > 1. F o r the s t a t e with n = 0 a n d qS 3 = - 1, we have p °2 ~ 3 KeH
a s e H / m 2 ~ ~o ,
w h i c h b e c o m e s n e g a t i v e w h e n K < 0. T h e r e f o r e only w h e n K = 0 c a n p o 2 b e c o m e p o s i t i v e d e f i n i t e . O u r o b j e c t is to s e e w h e t h e r t h e r e is a c h o i c e of K (H 2) ¢ 0 a n d X (//2) s u c h that po2 is p o s i t i v e d e f i nite. We find it is s u f f i c i e n t to d i s c u s s the c a s e w h e n X (H 2) = 0. In this c a s e , the c o n d i t i o n s for eq. (3) to be p o s i t i v e d e f i n i t e a r e the following: (i) F o r the g r o u n d s t a t e with n = 0 a n d qS 3 = + 1, we have p o2 = rn2(1 - K ~ ) . T h i s i m p l i e s the c o n dition K(~ 2) --< 1/4 ,
(4)
be s a t i s f i e d . (ii) F o r the o t h e r s t a t e s , we have the c o n d i t i o n s I+V
+ ½ ( 1 - K ) 4 2 >/ 0,
and
(1+77+½(1-K)~2) 2~
¼(1-K)2(4+477+~2),
(5)
w h i c h c a n be s i m p l i e d to the f o r m (1 +~) >/ K(K- 1)4 2 .
(6)
A n o t h e r c o n d i t i o n f o r a p a r t i c l e with a n o m a l o u s m a g n e t i c m o m e n t is K (4 2) ~ K' = c o n s t a n t ~,
(7)
a s ~ t e n d s to z e r o . By v i r t u e of eq. (4), eq. (5) is a l w a y s s a t i s f i e d . T h e r e f o r e a n y c h o i c e of K(~2) w h i c h s a t i s f i e s c o n d i t i o n s (4), (6) and (7) c a n give a c o n s i s t e n t r e s u l t , i . e . , po 2 is p o s i t i v e d e f i n i t e . F r o m eqs. (4) a n d (6) it is e a s y to s e e that
i K (~2) I --<1/4
a s ~ -~ ~o.
T h i s i m p l i e s that if the n o n - m i n i m a l c o u p l i n g t e r m s a r e added a s a p o w e r s e r i e s i n ~, they m u s t f o r m a n i n f i n i t e s e r i e s that s u m s to a n o n - p o l y n o m i a l function. T h e p a r t i c u l a r c h o i c e [4] K(~ 2) = K'(1 +¼K'2~2) -1, w h e r e X' i s a n a r b i t r a r y n u m b e r , is a s i m p l e m o d e l of K(~ 2) w h i c h s a t i s f i e s c o n d i t i o n s (4) -(7). With t h i s c h o i c e , the g r o u n d s t a t e e n e r g y b e c o m e s p o2 = m2(1 + ¼K'2~2) -1 (1 - ½K'~) 2, w h i c h is s i m i l a r to the s p i n ½ c a s e . T h u s the i n c o n s i s t e n c y of the (vector) s p i n one t h e o r y a s d i s c u s s e d by T s a i a n d Y i l d i z a p p e a r s to be due to the e x c l u s i o n of m a g n e t i c p o l a r i z a b i l i t y e f f e c t s a n d e s p e c i a l l y due to the fact that the a n o m a l o u s m a g n e t i c m o m e n t did not v a n i s h in v e r y s t r o n g m a g n e t i c f i e l d s ~f with t h e i r m o d e l for K. T h e s a m e r e a s o n i n g c a n b e c a r r i e d out f o r the 6 - c o m p o n e n t t h e o r y a n d the m u l t i s p i n o r t h e o r y . W e a l s o e x p e c t that a s i m i l a r a r g u m e n t c a n be u s e d to e x p l a i n the i n c o n s i s t e n c y of the e l e c t r o s t a t i c p o t e n t i a l w e l l c a s e of ref. [1] a n d the i n c o n s i s t e n c y of the s p i n ~ t h e o r y d i s c o v e r e d b y J o h n s o n a n d S u d e r s h a n t e n y e a r s a g o [3]. We t h a n k P r o f . S i d n e y C o l e m a n for h e l p f u l c o n v e r s a t i o n s . J The physical meaning of K' can be found by going to the weak field case. In this limit, the total magnetic moment of the spin one particle is ~ = (eq/2rn)(l +K')S. J'J" This bears out the contention of Tsai and Yildiz that "the disease is in the formalism itself ~ [2]. 468
Volume 36B, number 5
PHYSICS
LETTERS
4October1971
R efF~'encFs [1] L. I. Shfff, H. Snyder and J. Weinberg, Phys. Rev. 57 (1940) 315. [2] W. Tsai and A. Yildiz, Phys. Rev. D, to be published. T. Goldman and W. Tsai, Phys. Rev. D, to he published; P. Lu and R.H. Good J r . , Phys. Rev. D3 (1971) 1275; W. Tsai, to be published. [3] K.Johnson and E. C. G. Sudarshan, Ann. Phys. 13 (1961) 126.
469