L]~TER]~ AL NUOVO ClM]~N~O
VOL. 36, ~V. 7
12 l~ebbraio 1983
The Spin Kinetic Energy and Intrinsic Mass of Elementary Particles. II.
-
Massless
Particles.
L. S. LEVlT~ T h e Institq~te ]or T h e o r e t i c a l S t u d i e s - B o x 12292 - E l P a s o , T e x a s 79912
(ricev-uto il 22 Giugno 1982)
T h e p h o t o n a n d n e u t r i n o s are the o n l y k n o w n massless particles, t h a t is h a v i n g no (( rest ~)mass. T h e i r a p p a r e n t (~m o v i n g )) mass w i t h velocity c is given b y the E i n s t e i n Compton equation (1)
<~
m
,>
=
hick = p / e ,
w h e r e / 9 is t h e lineal" m o m e n t u m . W e can define a r a d i u s <( r o ~> for such particles, as was done'-in t h e previous paper (~) on massive particles, as the corresponding C o m p t o n wave-lgngth:
(2)
to=
h/cm=
h/c(h/e~)
=
2.
I t is seen from eq. (2), therefore, t h a t the (~r a d i u s )) of a p h o t o n or n e u t r i n o m a y be identified w i t h its wave-length. T h e p h o t o n has spin q u a n t u m n u m b e r equal to u n i t y a n d , therefore, t h e spin angular m o m e n t u m is
(3)
P0(~) = ~ / l ~ i + 1)h = ~/2h = 1.49.10 -2~ e r g . s .
T h e spin r o t a t i o n of a particle m a y also be expressed (2) in t e r m s of its m o m e n t of i n e r t i a I a n d its spin a n g u l a r v e l o c i t y ~o: (4)
~Oo(~) = I c o .
The associated k i n e t i c energy of spin, E~(~), is, therefore, (5)
EK(~) = Ie92/2 = p~(~l/2I,
which w i t h eq. (3) for p h o t o n s becomes (6) (1) (2)
EK(~) = h2/Z. L. S. LEVITT: Lett. Nuovo Cimento, 34, 333 (1982). A. J. SONNESS~: Introduction to Molecular Spectroscopy (New Y o r k , N . Y . , 1966), pp. 34-39.
167
168
L.S. LEWT~
B u t , since I = ~ i r o~, w h e r e m i is t h e <(intrinsic ~>mass of t h e spinning p a r t i c l e and r o its (( r a d i u s ~, one obtains
(7)
Ex(.> =
h~/m,Po.
W e conceive t h a t t h e (( m o v i n g mass )) of a massless e l e m e n t a r y particle, m, is in fact c o m p o s e d of two t e r m s Q): its i n t r i n s i c mass, m i, ~nd t h e mass e q u i v a l e n t of its spin k i n e t i c energy, m~. T h u s , (8)
m :
m i ~- ms 9
T h e r e f o r e , t h e t o t a l m a s s - e n e r g y of t h e p a r t i c l e is (9)
E T = E , + Ex(~) = m c 2 ,
w h e r e EK(~) is t h e kinetic e n e r g y due to i t s spin angular m o m e n t u m , P0. T h e t o t a l e n e r g y is t h e n , as usual, m c 2, b u t its ((intrinsic ~) e n e r g y i s m ~ c 2, w h e r e m ~ < m o. I t is seen t h e n t h a t (10)
E T = me ~ =mic
2 + p~/2mir~o
or
(11) Setting t h e p h o t o n p a r t i c l e or <(spin ~) r a d i u s e q u a l to its C o m p t o n w a v e - l e n g t h (a) ( h / m i c), one obtains
(12)
~ = m~ + h ~ / m i c ~ ( h / m l c ) ~ = cn,i + m~/4zt 2 ,
or
1 )= h/c;~
(13) Therefore,
(14)
(15)
h/c~ qTbi - -
(1 + 1/4~ 2)
-- 0.9753m = 2.152.10-1~/~.
m~ = (1 - - 0 . 9 7 5 3 ) ~ = 0.0247m = 5.450.10-19/Z.
T h e ((intrinsic)) mass of a p h o t o n , therefore, comprises 97.53% of t h e <(o b s e r v e d )) mass, and t h e mass e q u i v a l e n t of its spin k i n e t i c energy is 2.47% of h / c L T h e f o u r n e u t r i n o s , on t h e o t h e r h a n d , h a v e spin 89 and (16)
P0(s) = V 3 h l 2 .
L . S. L E V I T T : Bull. A m . Phys. ~oc., 11-2, 211 ( 1 9 5 6 ) ; Experientia, 1 4 , 223 ( 1 9 5 8 ) ; Lett. JNuovo Qimento{ 12, 537 ( 1 9 7 5 ! ,
(a)
THV. SPI~ ~IN~TIC ~ n G Y
ASD I ~ T ~ S ~ C
~SS
~C.
- H
169
A p p l y i n g t h e s a m e a n a l y s i s t o n e u t r i n o s as h a s a l r e a d y b e e n a p p l i e d (~) t o massive p a r t i c l e s of s p i n 89 o n e f i n d s t h a t
(17)
mi = m
1 -}- ~
= 0.9906m = 2.186. lO-~/)l
and m s = 9.42.1O-3m = 2.074.10-19/A.
(18)
T h u s t h e <(intrinsic >>m a s s of a n e u t r i n o c o n s t i t u t e s 9 9 . 0 6 % of h/cA a n d i t s <~s p i n >> m a s s is 0 . 9 4 2 % of h/cA. T h e s p i n m o m e n t of i n e r t i a of b o t h p h o t o n s a n d n e u t r i n o s c a n b e c a l c u l a t e d as I = mir ~ = (h/cA))~ = h/v
(19)
a n d c a n also b e c M c u l a t e d as (1) I = h~/mic== h=/eA(h/cA) = h/v.
(20)
T h e s p i n a n g u l a r v e l o c i t y co is g i v e n b y (21)
(o :
w h i c h f o r p h o t o n s is s e e n r
po/I,
be
m = V-2h/(h/~) = f / V 2 z
(22)
= 0.225~
a n d for n e u t r i n o s is (23)
m = ~/-3~/2I = ~/3I~/(2h/~) = ~ / 3 ~ / 4 z = 0.138~.
T h e s p i n r o t a t i o n a l f r e q u e n c y ] is r e l a t e d t o ~o b y (3) (24)
/ = ~/2~,
w h i c h w i s h eq. (22) g i v e s f o r p h o t o n s (25)
~I2V~ =
i =
~lVg~
= 0.113~
a n d , w i t h eq. (23), f o r n e u t r i n o s (26)
] = ~ / 3 v / 8 z ~ - - 0.0220v.
Finally, one may calculate a linear (tangential) spin velocity from (27)
v = r o ~ ---- 2 w .
F o r p h o t o n s t h i s is e q u i v a l e n t t o (28)
v :
A v / ~ / 2 = e / n ~ / 2 = 0.225c
170
n.s.
LI~VITT
(where e is t h e s p e e d of l i g h t ) , a n d f o r n e u t r i n o s is e q n i v a l e n t t o (29)
v =
4o
=
~/32v/la
=
~/3c/4~ =
0.138e.
I t s h o u l d b e n o t e d t h a t eq. (29) g i v e s t h e s a m e r e s u l t (1) (fl = 0.138) as f o r m a s s i v e p a r t i c l e s of s p i n 89 T a b l e I g i v e s a s u m m a r y of M1 q u a n t i t i e s c a l c u l a t e d a b o v e .
TABLe, I. -- Calculated spin properties o/ photons and neutrinos. Quantity
yo
vo
p0(B) (erg. s)
1.49.10 -27
0 . 9 1 3 . 1 0 -27
E~(s) (erg)
0.0247hc/~
0.0094he/2
r o (era)
~
I ( g c m 2)
h/~, = h;tle
hl~,
~o (s -1)
~1~ ~/~
~ V~I4~
] (s-l)
~/~ Vg
v V~/8~
v (cm/s)
c/~ V~
~V~/4~
m,
0.0247m
0.0094m
mi
0.9753m
0.9906m
m (g)
h/c~
h/c~
I t is i n t e r e s t i n g t o n o t e t h a t a p h o t o n or n e u t r i n o w i t h 2 = 1.32.10 -13 e m ( t h e n u c l e o n r a d i u s ) h a s t h e s a m e m a s s a n d d e n s i t y as a n u c l e o n . I n f a c t a n e u t r i n o w i t h t h i s w a v e - l e n g t h will h a v e , w i t h t h e e x c e p t i o n of i t s t r a n s l a t i o n a l v e l o c i t y (c) a n d b a r y o n n u m b e r , all t h e p r o p e r t i e s of a n e u t r o n a t r e s t ! T h u s , s, r o, E T, m i, m s, P0(s), I , (o, ] a n d fl will all b e i d e n t i c a l t o t h o s e of n o. B u t , m i g h t n o t t h e b a r y o n number -~ 1 ?