Correspondingly , f r o m Eqs. (20), (16), and (22), the kinetic ene rgy of the pa r t i c l e is

T m ] u I ~- _ rn%~ ( 1 ~, ~t) {R 2 + 2 R r sin (~ -- ,3 - "':0) T' r ~} e ~xz. (24) 2 4r~

Cer ta in commen t s on these r e su l t s a r e app rop r i a t e . Suppose that ini t ia l ly the pa r t i c l e is at r e s t (R = 0) at a d is tance r f r o m the solenoid ax i s . Af ter the va r i ab l e field in Eq. (1) h a s been in opera t ion fo r a t ime of t t s e c , the pa r t i c l e ene rgy is

T{eV) 1,I 10-SH, 2 (erg 2 = r e;,, / 8~ ; (25) :: �9 ),~,~x (c~) ; ,~ .... V 1 . + ~t-----~ -~,--~

H, = H 0 ( l + ~t,).

It follows f r o m Eq. (25) that the pa r t i c l e energy r i s e s in accordance with the law

T ~ ( ! + ~t) e ~0+'I '~,

and in the ca se of adiabat ic va r ia t ion in the field the pa r t i c l e energy i n c r e a s e s s imply in accordance with T ~ (1 + vt). In addit ion, i t follows f r o m the f o r m u l a desc r ib ing the m a x i m u m dis tance of the pa r t i c l e f r o m the solenoid axis that for a pure ly per iodic field (v = O) Pmax i n c r e a s e s exponentially; tMs leads to i n c r e a s e in the g e o m e t r y of the s y s t e m . If an adiabat ic component of the field is p r e sen t (the ca se when v ~ 0), a pa r t i c l e of l a r g e r energy may be obtained with constant g e o m e t r y of the s y s t e m in the t i m e in te rva l for which the expo- nent , which includes t e r m s depending on w0, v, h, and t , does not r i s e too rapidly in compar i son with the f a c - t o r ( r / ~ ) 4 - 8 J T ' ~ 0 .

Thus , by choosing the a p p r o p r i a t e values of the p a r a m e t e r s on which the pa r t i c l e energy and the m a x i - m u m dis tance f r o m the solenoid axis depend, i n c r e a s e in pa r t i c l e energy may be obtained without changing the d imens ions of the s y s t e m .

In conclusion, thanks a r e due to the par t ic ipan ts in Prof . Sokolov 's s e m i n a r for d i scuss ion of the work.

1. 2. 3. 4.

L I T E R A T U R E C I T E D

V. P. Kr ive t s and B. P. Pe regud , Zh. Tekh. F i z . , 41, 1174 (1971). K. B. A b r a m o v , V. B. Boshnyak, and B. P. Pe regud , Zh. Tekh. F l z . , 46, 1042 (1976). A. G. Kul 'k in , Yu. G. Pavlenko, and A. A. Sokolov, At. E n e r g . , 31, 292 (1971). P. L. Kapi t sa , Zh. Eksp . T e o r . F l z . , 21, 588 (1951); Usp. F iz . Nauk, 44, 7 (1951).

M O D E L O F T H E P R I M I T I V E S P I N O R

D . D . I v a n e n k o a n d G . A . S a r d a n a s h v i l i UDC 539.12.01

In an in te rp re ta t ion of the p r i m i t i v e sp inor model , the authors d i scuss s y s t e m s whose s y m m e t r i e s a r e infinite Coxeter g roups d e s c r i b e d , as degene ra t e s y s t e m s , by f lbrat ions over the genera t ing space of t hese g roups as a b a s e s .

The model of the p r i m i t i v e sp inor [1-4] is an a t t empt to r e a l i z e the idea of the nonl inear Ivanenko--Hei - senberg theo ry f r o m the posi t ions of the method of fo rma l i za t ion of se l f -ac t ing s y s t e m s with f ibra t ions [5-8], v i z . , s y s t e m s of objects with a s i m p l e a lgebra ic s t r u c t u r e .

The a lgeb ra i c s t r u c t u r e of a s y s t e m is taken to mean the ca tegory E = ({A}, S = horn(A, A ' ) , A, A ' e{A]) ([9], 1.7), which can be de t e rmined , to within i s o m o r p h i s m , by a c l a s s of m o r p h l s m s of a s y s t e m (with p a r - t ia l ly de t e rmined multipl ication) fo rming an a b s t r a c t ca tegory S ([9], 1.7, E x e r c i s e 3). It was re la ted to other s t r u c t u r e s of the s y s t e m by the r e p r e s e n t a t i o n E [byass ign tng a functor ([9], 1.8)] of a ca tegory with the c o r - responding spaces as i ts objects whereas S is r ea l i zed by the i r mappings .

M. V. Lomonosov Moscow State Univers i ty . T rans l a t ed f r o m Izves t iya Vysshikh Uchebnykh Zavedeni l , F iz ika , No. 10, pp .78-81 , October , 1978. Original a r t i c l e submit ted November 17, 1977.

1314 0038-5697/78/2110-1314507.50 �9 1979 Plenum Publishing Corpora t ion

The s imp le s t a b s t r a c t ca tegory of S is the group Z 2 = (s, s 2 = 1) and the s y s t e m of p r imi t ive spin0rs is c h a r a c t e r i z e d by the ex is tence in S of a subc lass S' = {s ES, such that s z is de te rmined and is unity in S}. Let us cons ider in S' a re la t ion R such that s R s ' , s , s ' ~ S , if the product s s ' is defined in S. It can be shown that R is the equivalence re la t ion ([10], II. 5) in S ' , m o r p h i s m s of one equivalence c lass const i tute a se t Si = {Sy, y~Y'~i~I and in the ca tegory E fo rm ed by S, S i C h o m ( A , A), of s o m e object A f r o m E. Let W i Chom(A, A) be the se t of al l poss ib le finite products of e lements f r o m S i and let Wi be a Coxeter g roup with ~ the g e n e r - ating se t S i ([11], IV, 1.3) . Now, set t ing ourse lves the t a s k of studying complexes of connected s t r u c t u r e s poss ib l e in one and the s a m e se t of p r imi t i ve sp ino r s , we shal l a s s u m e that al l se ts Yi, i e I a r e equipotent and that g roups Wi, i ~ I can be quas io rde red ([10], II. 9) with r e spec t to embedding, and fo r any i , j ~ I t he r e exis ts a EEI, such that W i and Wj a r e embedded in Wk. Groups W i can be t r a n s f o r m e d into topological g roups by introducing into t h e m t o p o l o g i c a l and un i form ([12], II. 1) s t r u c t u r e s defined by ce r ta in fami l i e s of subgroups in W i ([13], III. 1 .1 , 2; Note 3 .1) , and as such s t r u c t u r e s in each W i we shal l take images of all poss ib le embeddings of fi. in W. of al l groups W., such that i < j , j ~ I The bas is of the topology so genera ted

j i ] - - �9

cons is t s of open-c losed se t s and the topology is comple te ly d isconnected, i . e . , the connected component of any point w~W i is ~ ([13], III , 2 .1 , Co r . ; [12], 1.11.5) and which sa t i s f i es axiom O,II I ([12], 1 . 8 . 4 , 1 1 .1 .2 , Cot . 3). The topology in W i will be s e p a r a b l e if the in te r sec t ion of all fJi (Wi) and se t s adjoint to it r educes to 1 in W i ([13], IH, 1 .2 , Cor . ) and will be d i s c r e t e if t h e r e exis ts a finite in te rsec t ion o f t M s kind. In the topologies desc r ibed in the fami ly {Wi, i~I}, al l mappings fij will be open--c losed ([12], 1 . 5 . 1 , Note 1). If we now extend the s y s t e m so that any open subgroup of the base of the topology in Wi, i e I t h e r e always exis ts a g roup Wj e{Wi, iEI}, whose image under embedding in Wi was this group , then for a l l Wi, Wj, j-<i: the topology in Wj will be a p r e i m a g e of the topology in Wi, the inject ions fJi a r e continuous, and a l l Wi, i61 a r e local ly i somorph i c ([12], 1 .2 ,3 , Note 1; [13], I I I . 1 . 3 , Def. 2).

Conclusion. By v i r tue of i ts s y m m e t r i e s , a s y s t e m of p r imi t i ve spinors has a family of locally i s o m o r - phic Coxeter g roups .

Since for any Coxeter g roup W i t he r e exis ts a h o m o m o r p h i s m in the group Z z x (group of permuta t ions Yi ~ Si) {[11], IV. l , L e m m a 1), it is na tura l to take as objects of ca tegory E the se t Pi of sec t ions of local ly t r i v i a l f ibra t ions k i = (Mi, V, Yi, Z2) ([14], 1.3.2a) with a typ ica l f ibe r V, a two-e l emen t se t in the d i s c r e t e topology [object of ca tegory (V, Z2) of the p r i m i t i v e spinor] of the s t ruc tu r a l group Z 2 (in the d i s c r e t e topology) and base Yi with topological and uni form s t r u c t u r e s , induced in it f r o m Wi (Yi ~ Si C Wi). J u s t as in models with spontaneous s y m m e t r y violat ion [8], such f ibra t ions c h a r a c t e r i z e the s y s t e m as degenera te with r e sp ec t to the object of the p r imi t i ve spinor V and group Z 2 with a se t of degenerac ies Y (Y ~ Yi, i~ I as a set) . The i s o m o r p h i s m s of the f ibra t ions ki a r e defined by a bundle of loca l ly-cons tan t functions f r o m Yi in Z z and the c l a s se s of i s o m o r p h i s m s bi jec t ive to the e lements of the cohomological se t H1yiZ 2 ([14], I , T h e o r e m 3 .2 .1 ) .

Conclusion: A s y s t e m of p r i m i t i v e sp inors can be cons idered as degenerac ies over a set of p r imi t ive sp inors and its objects a r e rea l i zed by sect ions of the cor responding f ibra t ions .

The in te r sec t ions Pi a r e wri t ten in t e r m s of the (discontinuous) functions ~, which a r e local ized in V, in the r e f e r e n c e f r a m e { ~ : 7r-l(Uc~) ~ U ~ • V, U~ ( Y'~ of a t las ~ of f ibra t ion ki. The in te rsec t ions Pi a r e bi ject ive to the se t 2Yi and on this se t we can introduce the topology F ~ , t h e w e a k e s t o f t h e topologies in 2Y~so that 2U is open in 2Y fo r any nonempty set U C Y i ([12], 1 .2 , Exe rc i s e 7a), which topology, genera l ly speaking, will no longer be comple te ly disconnected. The effect o f Wi on Pi is defined by W i ~ SiSsy: ~ ~ ~ = {~(~,) = s6Y, Y'~0 (y')} where the effect of W i on Yi, Wi"gw:Y -* ~ = {s~ = wsvw -l} is a h o m e o m o r p h i s m ([11], IV. l , L-emma 1; III. i . 1 , 2 .4 , L e m m a 1). It can then be shown that the e'tfect o-f Wi o:. Pi in the topology F~ is a l so a homeo- m o r p h i s m . The embeddings of the groups Wj ~ Wi, i ~ j , induce continuous open injections f ' : y j ~ Yi and the topology in Yj will be the p r e i m a g e of the topology in Yi and all Yi, i~ I , will be local ly homeomorph ic ([12], 1.11, E x e r c i s e 25). Then in the f ibra t ions k: and {kj, j -< f} the re can a lways be such congruent a t l a ses ~i and {~j, j <_ i} that t he r e will be na tura l continuous open embeddings Pj ~ Pi ([12], I , 2, E x e r c i s e 7e).

Conclusion. Objects of a degenera te s y s t e m of p r imi t ive spinors admit a nontr ivia l a lgebra ic and topo- logical s t r u c t u r e which in p r inc ip le makes it poss ib le to r e a l i z e intuit ive ideas of "nonlinear p r o m a t t e r . "

The effect of Wi in Yi in t roduces into Yi the equivalence re la t ion yRy ' if t he r e exis ts a w~Wi such that y = wy 'w -1 and defines the space of orbi ts X i =Yi /Wi as a f a c t o r - s p a c e of the space Yi by this re la t ion ([13], I I I . 2 . 4 ) w h i c h w i l l b e s e p a r a b l e ([13], I I I . 4 . 1 , Notes 1, 2. P rop . 3). The canonical pro jec t ion P : Y i ~ X i will be continuous and open ([13], I I I . 2 . 4 . L e m m a 2; [12], L5.2, Def. 2) and the p r e i m a g e of every open set in X i is an open Wi - s t ab le se t in Yi. Then Pi can be r e p r e s e n t e d as a bundle above the space Xi, co r re la t ing with eve ry open se t U C X i a subse t to in te r sec t ions f r o m Pi, defined on p-l(U) E Yi ([15].II.1) and, the re fo re , as a

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se t of continuous in te rsec t ions of the cor responding covering space (f ibered space with base Xl and f iber Px, x E Xi, se t in the d i s c r e t e topology of the ge rms of the functions f ro m Pi with r e spec t to the f i l t e r of the neighborhoods of the point x) ([15].H.2, Def. 2.1, Theorem 2.2; [14].L2.1, 3). The effect of W i is induced f rom Pi on Px, x ~ Xi, and Px, x E X i is s table with r e s p e c t to W I. The effect of W i on X i is t r ivial , Wi = ldxi. Then on Pi we can define a continuous effect of local t rans format ions W i, defined by the bundle Wi(X i) of con- tinuous functions f rom X i in W i ~ W l C Wi(Xi). Commuting with W i on Pl a r e t ransformat ions of the a t lases of the f ibrat ions X i whose cover ings a r e s table with r e spec t to W i (they always exist); the t ransformat ions a re induced on Pi- These t r ans fo rmat ions can be r ep resen ted by a bundle Z2(X i) of local ly constant functions f rom X i in Z 2 and the i r action on Px, x E X, as the space of representa t ions W i is an equivalence.

Conclusion. The Coxeter groups W of symmet r i e s of the pr imi t ive spinors can be considered as i n t e r - nal s y m m e t r i e s , the i r orbi ts on the pr imi t ive spinor set Y can be considered as Internal spaces , and the space of these orbi t s , as ex terna l spaces . In this in terpre ta t ion the t ransformat ions Z20D will play the ro le of ex terna l local s y m m e t r i e s .

The bundles Wi(Xi) , Z2(Xi) f o rm a se t of morph isms hom(Ai, A i) of the object Ai = Pi, i ~ I, of the ca te - go ry Eo The morph isms horn (Al, Aj), i ~ j, can be reproduced as induced morphisms f rom hom(Ak, Ak) of A k such that there exist embeddings Pi "* Pk, Pj "* Pk, taking the images Pi, Pj in Pk into each other . The ca tegory E of the sys tem of p r imi t ive spinors has thus been const ructed, although in ex t r eme ly general fo rm.

Conclusion. The Coxeter groups hi ther to dropped out of the discussion as groups of symmet r i e s of rea l s y s t e m s , although any spinor and informat ion sys tem can in pr inciple be classi f ied as such. Thus, finite Coxeter g roups , groups of symmet r i e s of root d iagrams of s imple Lie a lgebras , can be defined on the spec t ra of par t ic les ; infinite Abelian groups can desc r ibe s ta t i s t ica l fe rmton sys t ems ; and infinite groups with finite nonabelian subgroups hold out p romise for descr ibing pa r t i c l e - l ike s t ruc tu re s . In this case , a sys t em is de- f ined, e . g . , as a quantum sys t em, by a C*-a lgebra of continuous functions f ro m Pi in a s imple r a lgebra (a +, a-) of canonical anticommuting re la t ions , and as an informat ion sys t em, by a Boolean a lgebra of functions on Pi with values i n the Boolean algebra~ (p, -~, A , 1), etc. The foregoing discussion, t h e r e f o r e , is of signifi- cance not only for models of p r imi t ive spinors and for the development of sys tems with s y m m e t r i e s , Coxeter groups , seems to be ex t remely promis ing .

L I T E R A T U R E C I T E D

1. D . D . Ivanenko, Int roductory Ar t i c le in: Quantum Gravitat ion and Topology [in Russian], Mir , Moscow (1973).

2. D . D . Ivanenko and G. A. Sardanashvi l t , Izv. Vyssh. Uchebn. Zaved . , F i z . , No. 5 (1976). 3. D . D . Ivanenko and G. A. Sardanashvi l l , in: Current Problems of Theore t ica l Physics [In Russian],

Moscow State Univ. , Moscow (1976), p. 91. 4. D . D . Ivanenko and G. A. Sardanashvi l i , Paper read at All-Union Meeting on Contemporary Problems

of the Theory of Gravi ty and E lementa ry P a r t i c l e s , Mendeleevo (1977). 5. G . A . Sardanashvlll , Digests of Papers of Eighth International Conference on the Theory of Relativity

and Gravi tat ion, Canada (1977), p. 311. G. A. Sardanashvil l , Izv. Vyssh. Uchebn. Zaved . , F i z . , No. 5 (1977). 6p

7. G . A . Sardanashvi l l , Izv. Vyssh. Uchebn. Zaved . , F i z . , No. 5 (1977). 8. G . A . Sardanashvi l i , Izv. Vyssh. Uchebn. Zav ed . , F i z . , No. 5 (1977). 9. S. MacLane, Homology, Spr inger -Ver lag (1975).

10. K. Kuratowski and A. Mostowski, Set Th eo ry , North-Holland, Ams te rdam (1967). 11. N. Bourbakt , Lie Groups and Algebra , Addison-Wesley. 12. N. Bourbaki , Genera l Topology. Fundamental S t ruc tu re s , Addison-Wesley. 13. N. Bourbaki , Genera l Topology. Topological Groups, Addison-Wesley. 14. F. Hi rzebruch , Topological Methods in Algebraic Geomet ry , Spr inger -Ver lag (1966). 15. R . O . w e l l s , J r . , Different ia l Analysis on Complex Manifolds, P ren t i ce -Hal l (1973).

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