vF - m~A = J.
(11)
Equations (11), written in tensoriaI form, coincide with the equations studied in [9, 10] (see also [11] and the literature cited therein). They describe a superposition of fields and in [9, 12] are called general-type vector field equations. Thus, our investigated group is the internal symmetry group for the theory of the general-type v e c t o r field. This theory (for J = 0) turns out to be the m o s t a c c e p t a b l e foundation for the c o n s t r u c tion of a theory d e s c r i b i n g e l e c t r o m a g n e t i c interaction of a c h a r g e d v e c t o r field [10]. The author is indebted to S. I. Kruglov, Yu. A. Kurochkin, and E. A. ToLkachev for a c r i t i c a l and s t i m ulating discussion of the p r e s e n t r e s u l t s . LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
CITED
D . P . Zhelobenko, C o m p a c t Lie Groups and T h e i r R e p r e s e n t a t i o n s [in Russian], Nauka, Moscow (1970). Yu. V. Novozhilov, Introduction to the T h e o r y of E l e m e n t a r y P a r t i c l e s [in Russian], Nauka, Moscow (1972). V . I . Strazhev and L. M. TomiL'chik, E l e c t r o d y n a m i c s and Magnetic Charge [in Russian], Nauka i Tekhnika, Minsk (1975). G . A . Z a i t s c v , Izv. Vyssh. Uchebn. Zaved., F i z . , No. 12 (1969). G . A . Z a i t s e v and A. hi. Solunin, Izv. Vyssh. Uchebn. Zaved., F i z . , No. 11 (1969). G . A . Z a i t s e v , A l g e b r a i c P r o b l e m s in Mathematical and T h e o r e t i c a l P h y s i c s [in R u s s i a n ] , Nauka, Moscow 0-974). I. BiaLynieki-Biruia, Bull. Acad. Polon. Sei., 1_1, 135 (1963). S. M a n d e l s t a m , P h y s . R e v . , 175, 1580 (1968). A . A . B o r g a r d t , Zh. l~ksp. T e o r . F i z . , 24, No. 24 (1953). E. Durand, Phys. Rev. D l l , 3405 (1975). A . A . B o r g a r d t , Zh. l~ksp. T e o r . F i z . , 4_55, 116 (1963). A . A . B o r g a r d t , A u t h o r ' s A b s t r a c t of Doctoral D i s s e r t a t i o n , Minsk (1965).
E X I S T E N C E OF A D I S P E R S I O N ZONE IN THE D E F L E C T I O N OF LIGHT RAYS IN AN O P T I C A L L Y LESS DENSE MEDIUM OVER THE I N T E R F A C E BETWEEN TWO O P T I C A L L Y HOMOGENEOUS MEDIA Yu. I. T c r e n t ' e v
UDC 535
According to the F r e s n e i equations, when light p r o p a g a t e s in an optically less dense medium in a d i r e c tion s t r i c t l y p a r a l l e l to the interface between two optically homogeneous m e d i a , t h e r e is no energy flux into the second medium. According to m o d e r n ideas, n e i t h e r does r e f r a c t i o n o c c u r in the c a s e when the Light rays p r o p a g a t e in a homogeneous medium of l o w e r optical density at s m a l l angles with r e s p e c t to the r e f r a c t i n g s u r f a c e , if the continuations of the initial ray t r a j e c t o r i e s a r e r e m o v e d f r o m it at distances that exceed the wavelength [1]. H o w e v e r , as e x p e r i m e n t s have shown, such r a y s can r e f r a c t into an optically d e n s e r m e d i u m , that is, f r o m an a p p r e c i a b l e l a y e r ( c o m p a r e d to the wavelength) of a homogeneous medium of lower optical density. E x p e r i m e n t s w e r e p e r f o r m e d initially using the a p p a r a t u s shown s c h e m a t i c a l l y in Fig. 1. In the Light path f r o m the slit 1, which is 0.1 m m wide and illuminated by a p a r a l l e l b e a m , t h e r e is a ceil 2 with a p l a n e p a r a l l e l polished plate 3 and a fluid 4, which a r e the optically homogeneous media. The cell and the plate a r e mounted p e r p e n d i c u l a r to the b e a m axis so that the latter is s y m m e t r i c with r e s p e c t to the edge of the plate. The deflection angles of the r a y s r e f r a c t e d in the region of the p l a t e - f l u i d interface and at the exit f r o m the ceil a r e r e c o r d e d using a s c a l e 5 and a telescope 6. The ray intensity is m e a s u r e d by using an FI~U-36 Institute of A t m o s p h e r i c Optics, Siberian Branch of the Academy of Sciences of the USSR. T r a n s l a t e d f r o m I z v e s t i y a Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 48-54, August, 1977. Original a r t i c l e s u b mitted N o v e m b e r 5, 1976.
0038-5697/77/2008- 1023 S07.50 9 1978 Plenum Publishing C o r p o r a t i o n
1023
fro
_ _
~o$ a l~'b75"*g
,
2
v
J
pte Fig. 1
Fig. 2
Fig. 1. A r r a n g e m e n t for the experimental investigation of the r e f r a c t i o n of gianctng rays. Fig. 2. A r r a n g e m e n t for a probe of the interface region of adjacent media by a point light source. photomultiplier, which is mounted on the coaxis of rotation of the telescope, which coincides with the v e r t i cal axis of s y m m e t r y of the edge of the plate. A m i c r o m e t e r s c r e w 7 m o v e s the cell p e r p e n d i c u l a r to the beam axis. As the adjacent media, we used different s u b s t a n c e s , p r e f e r a b l y with r e f r a c t i v e indices of negligible difference and dispersions of the highest possible difference. When the plate is fabricated from NS6 optical glass 2.15 mm thick and dimethyl phthalate is the medium of higher optical density, a s p e c t r u m of large angular width is f o r m e d in the direction I. Its c h a r a c t e r i s t i c c u r v e is given in Table 1, which contains data on beam deflection for different adjacent media. The intensity of the s p e c t r u m does not change if the sharp edge of an opaque s c r e e n 8 is placed on the edge of the plate. If opaque aluminum films a r e deposited on the face or edge, the s p e c t r u m is not visible. Consequently, the s p e c t r u m is produced by the p a s s a g e of rays f r o m the plate into the fluid through its edge. If interference filters (6380, 5300, 4520 .~) a r e placed in s u c c e s s i o n along the path of the p a r a l l e l beam, red, g r e e n , and blue lines a r e seen in the spectrum. The rays c o r r e s p o n d i n g to them have divergences of 18', 7', and 36'. When the continuous s o u r c e is replaced by an LG-56 l a s e r , the angular width of the r e d line is 11.4,. F i g u r e 2 shows the a r r a n g e m e n t for a probe of the interface region of adjacent media by a point light s o u r c e . In this a r r a n g e m e n t an objective (lndustar-26m) in the plane of s y m m e t r y of the plate, which is parallel to its face, produces an image S v of the slit, which is 30 p wide. The diaphragm of the objective a d mits rays which form only the c e n t r a l diffraction maximum in its front focal plane. The light beam is seen by the eye as a well-defined spot against a dark background. When the ceil with the plate moves from the position in which the beam of m o n o c h r o m a t i c rays p a s s e s through the plate, at a c e r t a i n time a weak line a p p e a r s in the s p e c t r u m ia the direction in which S' approaches the edge. With a subsequent displacement by Al the intensity of the line f i r s t r e a c h e s a peak and then decays to 0, as can be seen in Fig. 3. TABLE I / Beam deflccU~onangles for different adjacent media, in d. .e_ . [NS6glass- ]ZhS10glass- K8 glass-- K3 glass- ~LK~glass- STKTglassType of ]dimethyl ] b e n z y l d i m e t h y l dimethyl ~imethyl a-monobrorays [phthalate; talcohol ~hthalate phthalate ~hthalate monaphthalene ]nrj [ :oo I 516 l 1,510; l 1.478; t 7n'~. . ,,, o.~.3 oo; ' 1 .515 ....... | ~, 1,o02: ] 1,515 [ 1,515 | 9 1.678S -
1,515
I
.....
Red
10,2--12
5,2-7,2
G t e,e.n
12--14
7,2.-10.2
14--M7
10,2--15,9
_ t
[
--7,05-. ----4,3 --4,3~6
325--6,25
18.5--!9.5
20--17.6
6.25--9,05
19.5--_Ol
1 7 , 6 - - 14.2
6-11,5
9,05~14,09
Blue
Violet
1024
14.2--5.2
44 1,ml.uni~
::I
12 &~. J2 ,,':7 5.0 g,u
40
++ 10 20 30 40 /0g,~
Fig. 3. Curves of the change in intensity of the red lit) and g r e e n (G) lines during displacement of the slit image S' in the region of the edge of an NS6 plate along the n o r mal to it. In r e g a r d to the p r e s e n c e of wings on the c u r v e s , the values of Al a r e read off at points with an intensity of 4.5~} of its peak, and they c o r r e s p o n d to the onset and end of a rapid change in the Line intensity. T h e s e values a r e 42 and 44 p for the red and g r e e n lines, respectively. When rays a r e r e f r a c t e d in dimethyl phthalate directly at the interface, Al should equal the g r e a t e s t width of the beam t within the plate. However, it is l a r g e r than t at a height h. To determine the width of the beam in the plate, it is r e p l a c e d by a plate with a face ab that is tapered at an angle 75+40 ' and that r e f r a c t s the entire beam in the position of the cell until the projections of the f o r w a r d and r e f r a c t e d beams a r e completely separated. As the cell moves so that the p l a n e - p a r a l l e l part of the plate and S' c o n v e r g e , f o r w a r d - t r a v e l i n g rays begin to a p p e a r at a c e r t a i n instant and a r e r e c o r d e d by the photomultiplier. During a subsequent displacement of the ceil by the width of the b e a m , the latter becomes c o m pletely f o r w a r d - t r a v e l i n g . According to the m e a s u r e d data, the widths of the red and g r e e n beams at the entrance to the plate, which w e r e taken under the s a m e conditions as for Al equal 26 and 28 ~, respectively. T h e s e values a r e constant up to 2.8 mm along the beam axis and thus will r e m a i n the same inside the plate. The latter r e s u l t is c o n f i r m e d by m e a s u r e m e n t s of Al with the cell in different positions along the beam since the data indicate that Al r e mains constant to within 1 ~ as the cell with the plate is displaced by +0.4 mm with r e s p e c t to the working p o s i tion. According to the m e a s u r e d values of AI and t, the quantity h equals 16 ~ 2 p for the red and g r e e n beams. In this r e g a r d , it is obvious that the observed s p e c t r u m is the result of r e f r a c t i o n of the rays into dimethyl phthalate from a l a y e r 16 ~ deep in the plate through its edge. The p a s s a g e of light rays from a less dense medium into a denser medium under conditions where the continuations of their initial t r a j e c t o r i e s a r e r e m o v e d by an amount h from the interface leads to the conclusion that in a medium of lower optical density there is a dispersion zone of beam deflection located above the i n t e r face of two optically homogeneous media that deflects the rays towards the medium of higher optical density. The width of the r e f r a c t e d m o n o c h r o m a t i c beams equals the thickness of the plate multiplied by the cosine of the r e f r a c t i o n angle. F o r example, in the c a s e of r e d b e a m s it equals 0.28 m m , instead of 16 p, up to r e fraction. Consequently, the rays deflected in this zone enter the dimethyl phthalate over the entire width of the edge. The given expansion of the r e f r a c t e d b e a m s can be explained as due to a drop in the effectiveness of ray deflection in the zone in the direction of the plate edge. As a result the rays 2 (Fig. 4) that are far r e m o v e d from the edge but still able to reach it a r e deflected over the entire length of the zone, which is equal to the plate thickness. The r a y s 3 and 4, which a r e still f a r t h e r r e m o v e d and not able to r e a c h the edge, enter the dimethyl phthalate through the r e a r of the plate and a r e deflected at different angles. F o r this reason, the cited value of h is less than half the depth of the zone and equals the depth of its active p a r t , within which all the deflected rays reach the edge. The depth of the active part of the zone gradually d e c r e a s e s as the plate thickness d e c r e a s e s , since in this c a s e an even s m a l l e r number of rays reaches the edge. F o r example, with a d e c r e a s e in the plate thickness to 1.35 and 0.75 mm the value of h for the red and g r e e n rays fails to 9 and 5 p, r e s p e c tively. In the latter c a s e this results In an attenuation of the tines roughly by a f a c t o r of 4. With an i n c r e a s e in the plate thickness to 5.7 mm the intensitics of the g r e e n and red lines increases only by a f a c t o r of 1.14 and
1025
2
4
----e I I
3 Fig. 4
Fig. 5
F i g . 4. Motion of the r a y s in the deflection zone. Fig. 5. A r r a n g e m e n t f o r the m e a s u r e m e n t of the depth of the d i s p e r s i o n zone using two NS6 p l a t e s . 1.38, i n d i c a t i n g that with a p l a t e t h i c k n e s s of 2.15 mm the depth of the a c t i v e p a r t a p p r o a c h e s its total value. T h e d e f l e c t e d r a y s , as w e l l as p e n e t r a t i n g into the d e n s e r m e d i u m , a r e p a r t i a l l y r e f l e c t e d by the edge of the p l a t e ; t h e r e f o r e , the ray fluxes of the r e d a n d g r e e n lines c o m p r i s e only 20% a n d 25%, r e s p e c t i v e l y , of t h e i r m a g n i t u d e in the zone. A c c o r d i n g to T a b l e 2, the ray deflection under study is d e s c r i b e d by an equation which a g r e e s with the Snell equation sin 2 .
sin~ . . .
1
w h e r e (~ is the angle b e t w e e n the r a y s which a r e i n i t i a l l y p a r a l l e l to the edge and n o r m a l to it; 3 is the r e f r a c tion angle; n is the r e l a t i v e r e f r a c t i v e index of the i n t e r f a c e . In s p i t e of the deflection in the zone at d i f f e r e n t d i s t a n c e s f r o m the i n t e r f a c e , the r e f r a c t e d m o n o c h r o m a tic r a y s , a c c o r d i n g to the above data, have a s m a l l d i v e r g e n c e . T h e r e a s o n for this is e a s y to u n d e r s t a n d if one c o n s i d e r s the w e a k a t t e n u a t i o n of the r a y s in the m a i n p a r t of the a c t i v e zone, which is the p e r i p h e r y ; n e v e r t h e l e s s , it is s u f f i c i e n t to d i s p l a c e the r a y s by up to 2.15 mm in the p r i m a r y r a y deflection r e g i o n , which is a t the edge. As w e l l as using the m e t h o d d e s c r i b e d a b o v e , the depth of the b e a m deflection zone was m e a s u r e d by using two o p p o s e d p l a t e s a a n d b (Fig. 5), which a r e a d j a c e n t to the edges of the n a r r o w f a c e s and p a r a l l e l to the v e r t i c a l a x i s of the b e a m c r o s s s e c t i o n . A c e n t e r cut a l l o w s the r a y s to p a s s f r o m p l a t e a to p l a t e b. As the c e l l m o v e s in the d i r e c t i o n in which the zone of p l a t e a a p p r o a c h e s S ' , the image of the s l i t a p p e a r s s u c c e s s i v e l y in p o s i t i o n s 1-2 and 3 - 4 , which c o r r e s p o n d to the t i m e s that the b e a m e n t e r s into the p l a t e zones a n d l e a v e s them. T h e r e s u l t i n g change in line intensity is shown in Fig. 6. The shift of c u r v e s a and b r e l a t i v e to each other is a c o n f i r m a t i o n of the e x i s t e n c e of d e f l e c t i o n zones. It is e a s y to s e e that the d i s t a n c e between the o r i g i n s and ends of the c u r v e s equal the depth of the zones in the left and r i g h t p l a t e s . As a r e s u l t , the d e t e r m i n a t i o n of the depth of the zone by this m e t h o d does not r e q u i r e that the width of the b e a m in the p l a t e be known, which i n c r e a s e s its a c c u r a c y . TABLE 2 Refraction angles for rays initially parallel to the interface Type of
optical glass NS6 K8 K3 LK~
1026
ca lculated
ex ~r
6380 A
5300 A
4520 1~
6380 A
530~ A
82~1 ' 8620' 86~13' 77o40,
81o14,
79o35,
82:21' 86c6, 86~10' 77045,
81015'
83:52'
83~50' 76:59'
84o10, 77:8'
4520 79:47' 84~
T A B L E 3. E x p e r i m e n t a l Data f o r D i f f e r e n t A d j a c e n t M e d i a .Type . . . . . .of . . .adjacent ...... i . . . .",, ..
PlatZthick. .... |--/['me:m-----~-v ". . . .d. . . .~. . . .i . . .[. . . red .i . . . .light . . . . . -. . .0.838 . . . . :~'---Light be]in with-S' line inness, /;, # tensity, re 1. mm itensity, tel. angles, '!~el units ~nits deg
media
teen light - 0.53 ,~ ray deft. angles, nte 1 deg
I
Dirnethy I phtha late-] 1,51,5 K8 glass 1,5163 Dimet hy1phtha late - 1,515 NS6 glass 1,502 a . Monob~omonaph- 1.(~88 tha lene- 8F16 1,0709 glass Dimethyl phthalate. ZhSI0 glass
1,515 1,538
C~Monobromonaph-[ 1,6588 thalene-STK7 [ glass | 1,6869 /
Dimethyl phthalate-t LK0 glass { a-Monobromonaph-I ttlalene-TK23 l g~ass [ Dimethy_lphthalate J ZhS4 glass
20 fl
27,2
--2,4
1,00')3
17 pl .
1, OOSS
16 pl
63
~13,8
1,0128
15 pl
13,3
1,0095
20 fl
32
14,1
1,0t3
15 fl
1 ~,5
1,012
20 fl
32
18,55
1,019
I:~ fl
15,25
1,0126
10,8
1,026
18 pl
38, I
1.0,~9
20 fl
?7
2,15
43,3
2,35
28,5
2,07
--l1,5
48,5
23
--18.7
1,0236
10 pl
31
2,32
32
- '9 O
1,05
19 pl
--
2,05
19,5
37,2
1,0~5
20 fl
20
1,515 1,-178! 1,658~ 1,5891 ! ,515 1,632
1,0032
7
2,05
'2
h , b~
Note. fl denotes a fluid and pl a plate. T h e v a l u e s of hi, 2 w h i c h w e r e m e a s u r e d b e t w e e n p o i n t s with i n t e n s i t i e s equal to 4.5% of the t o t a l l i n e i n t e n s i t y , a r e the s a m e in both p l a t e s a n d e q u a l 17 +- 1 p f o r the g r e e n a n d r e d r a y s . W h e n a n NS6 p l a t e is r e p l a c e d by a p l a t e of K8 g l a s s , d i m e t h y l p h t h a l a t e is the d e n s e r m e d i u m only f o r t h e v i o l e t - b l u e a n d g r e e n r a y s . S i n c e the d i s p e r s i o n z o n e is in the m e d i u m of l o w e r d e n s i t y , the r e d r a y s in this c a s e w i l l be d e f l e c t e d in the d i m e t h y l p h t h a l a t e z o n e t o w a r d s the e d g e of the p l a t e , w h i l e the g r e e n a n d s h o r t e r - w a v e l e n g t h r a y s a r e d e f l e c t e d in the p l a t e z o n e into the d i m e t h y l p h t h a l a t e . Due to the p o s i t i o n of the d i f f e r e n t b e a m d e f l e c t i o n z o n e s in the d i f f e r e n t m e d i a , a t f i r s t enly the r e d l i n e a p p e a r s a s the c e l l m o v e s in the d i r e c t i o n in w h i c h S' a p p r o a c h e s the p l a t e . F o r the o t h e r l i n e s to a p p e a r i t is n e c e s s a r y that the c e l l b e m o v e d by a n a d d i t i o n a l 20 ;:, w h i c h ks one of the f a c t s in f a v o r of the e x i s t e n c e of d e f l e c t i o n z o n e s . in r e g a r d to the n o n s i m u l t a n e o u s a p p e a r a n c e of the l i n e s , the c u r v e s in F i g . 7 t h a t show the c h a n g e in the i n t e n s i t i e s of the g r e e n a n d b l u e t i n e s a r e s h i f t e d w i t h r e s p e c t to the r e d l i n e c u r v e . T h e d i s t a n c e b e t w e e n the o r i g i n s of the c u r v e s d e t e r m i n e s the depth of the z o n e hfl in d i m e t h y l p h t h a l a t e , a n d t h a t b e t w e e n t h e i r ends d e t e r m i n e s its v a l u e hpI in a p l a t e . T h e v a l u e s of hfl a n d hpl , w h i c h w e r e m e a s u r e d a t 4.5% of the i n t e n s i t y l e v e l , e q u a l 20 to 17 p , r e s p e c t i v e l y .
40 - I, rel.units
a
b
rel.units m
;
;;I
20
!0 tO 20 53 40
50 50 70 80 l,u
Fig. 6
20 3~
40
5,: 60
70 s
l, u
Fig. 7
Fig. 6. Changes in the intensity of the red lines during displacement of S' in the edge region of two opposed NS6 plates. Fig. 7. Curves of the change in the intensity of the green I, blue III, and red II lines during the displacement of S' in the edge region of a K8 plate.
1(1:27
The basic data p r o v i d e d by the e x p e r i m e n t s a r e shown in T a b l e 3. As can be s e e n , in all the e x p e r i m e n t s the depth of the zone ranged f r o m 16-20 IJ. P o s s i b l e c a u s e s of the phenomenon will be c o n s i d e r e d a f t e r the e x p e r i m e n t s have been c o m p l e t e d . The author is s i n c e r e l y g r a t e f u l to C o r r e s p o n d i n g M e m b e r of the A c a d e m y of Sciences of the USSR V. E. Zuev and S. D. T v o r o g o v f o r t h e i r support of the work, d i s c u s s i o n of the r e s u l t s , and c o m m e n t s during the experiments. LITERATURE 1.
V.A.
CITED
Kizel', Reflection of Light [in Russian] (1973).
RELATIVISTIC SEMIGROUP, THE LORENTZ G R O U P , AND TACHYONS. IV V. V. Yudin
UDC 530.145
The a l g e b r a i c and topological p r o p e r t i e s of the r e l a t i v i s t i c s e m i g r o u p a r e discussed. Its p r o b a b i l i t y - t h e o r e t i c a l f e a t u r e s e s t a b l i s h that the r e l a t i v i s t i c s e m i g r o u p belongs to the type of c o m plex Markov s t r u c t u r e s . F r o m the functional point of view, the r e l a t i v i s t i c s e m i g r o u p is a c o m p a c t Lie s e m i g r o u p which is contracting in p a r t i a l s p a c e s . P r i n c i p l e s of m e a s u r a b i l i t y , obs e r v a b i l i t y , and stochastictty a r e f o r m u l a t e d , and these lead to a s p a c e - t i m e s t r u c t u r e of c o m plex Markov kind. T h u s , a c e r t a i n p r o b a b i l i t y - t h e o r e t i c a l gnosio[ogy is also p o s s i b l e in the theory of r e l a t i v i t y . The c e n t r a l f e a t u r e of the p r e s e n t p a p e r is the illustration of a different way of constructing a s e m i group r e a l i z a t i o n of the h y p e r b o l i c - t a n g e n t s e m i g r o u p (HTS). This method is m o r e s y s t e m a t i c and does not invoke any p a r t i c u l a r s p a c e - t i m e relations. H e r e we have the r a t h e r r a r e opportunity of noting how the concept of s p a c e t i m e is of s e c o n d a r y , derived significance. Our direction is b a s e d on two a x i o m s , which d e t e r m i n e the explicit f o r m of the c e n t e r of the realization. T h e s e a r e the p r i n c i p l e of m e a s u r a b i l i t y and o b s e r v a b i l i t y in the f r a m e w o r k of the theory of relativity and the postulate of s t o c h a s t i c i t y , these a x i o m s being f o r m u l a t e d for the images of the fixed points of the HTS. It should be e m p h a s i z e d that the p r e s u p p o s i t i o n s m a d e in this p a p e r a r e w e a k e r than the preceding one. Our results r e v e a l the p r o b a b i l i t y - t h e o r e t i c a l inferences that provide the b a s i s of the natural t r e a t m e n t of the r e l a t i v i s t i c s e m i g r o u p as a c o m p l e x Markov chain. 1.
Some
Properties
of the
Relativistic
Semigroup
We c o n s i d e r the a l g e b r a i c and topological p r o p e r t i e s of the r e l a t i v i s t i c s e m i g r o u p . 1. The Singular E l e m e n t s Z(fl• kinds of idempotents:
Z(,~ • = or
Z(f~~ = or = Z(~•
Fixed :Points. Quasistochasticity.
The r e l a t i v i s t i c s e m i g r o u p has t h r e e
Z(~=0)=E;
;
Z(~+-=~)=
0) and the r e l a t i v i s t i c idempotent =--
z ( ~ = 1)=
] ---i
~--~
-7- i
;
detZ(}---- 1 ) = 0 , SpZ(~ ~ - = I ) = 1 , z(3• = l).Z (~---'=: 1 ) = Z ( ~ - - = 1).
l--i In a c c o r d a n c e with the p r i n c i p l e of observability and m e a s u r a b i l i t y , we shall be i n t e r e s t e d in F a r E a s t State University. T r a n s l a t e d f r o m I z v e s t i y a Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 55-58, August, 1977. Original a r t i c l e submitted N o v e m b e r 24, 1976.
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