Localizability in the Hamii Tonlan Formalism and in Quantum Mechanics
CENTRAL INSTITUTE OF PHYSICS INSTITUTE FOR PHYSICS AND NUCLEAR ENGINEERING Bucharest, ROMANIA
ţ"FifJ - FT-346-3-9&T January
Localizability in the hamii tonlan formalism and in quantum mechanics
D.R. GRIGORE
ABSTRACT. The problem of localizability is studied in the framework of Hamiltonian formalism. Only the sys tems of nonzero mass and any spin, and of zero mass and zero helicity are localizable for the Poincare group. These results suggests a possible so.lution for the pro blem of localizability of the photon in quantum mecha - nics. It is proved that the photon is localizable in the senrje of Wightman, on the manifold of bidimensional 3 planes in R . So the photon must be considered as a bi dimensional object : the plane wave. §1.INTRODUCTION
It is reasonable to suppose that for a physical system there exists a notion of localizubility. This problem was proposed first in the framework of quantum theory by Newton
and Wigner JIJ, and rigorously stated . and analysed by Wight- man /2/ Cs«e also J3J). The idea is the following. Suppose the configuration space of the system is the diffe rent lab le manifold Q and let PCQ) the natural Borel structure of Q. Then a position observable is a projector valued mea - sure based on Q and taking values in the set if OO of orthogonal projectors from the Hllbert space **~ of the system ;
the physical interpretation is : if • « is an one dimensional projector describing a state of the system, and Range P CRange PJJ, then the system is localized in ECQ.
Suppose now that the group Q is a group of symmetries
for the system, i.e. one has in ?£• a projective unitary re«- presentation of G :
G>gl—*• U C "IL CK\ LB the set of unitary operators in Vi}, and G acts naturally on Q. Then a natural compatibili ty condition is Csee /3/ for the physical argument) :
CHere g.£ is the image of E under the action of g£GX.
The couple CU,P) is called a system of imprimitivity . Of1 course CU,P) and CU*P*) must be considered equivalent from the physical point of view if one has an unitary C or anti - 2 - unitary ) operator V :Jt.-^^{. such tha«t :
On the other hand, one knows that the elementary relativ vistic systems are described by irreducible projective unitary representations of the Poincare group U± . Given such a re presentation, one can obtain the restriction a projective uni tary representation U of the euclidian group SE(.?>. This 3 group has a natural action on fR . So one can defjne a system 3 to be localizable on IR if there exists a projector valued 3 measure P based on /R such that CU,P) is a system of im- primitivity.
As it is well known, this question has a negative answer for the unitary representations of T7* which are believed to descrihe the photon. There are a number of attempts to circum vent this difficulty in the literature Cwe list only the at tempts in the spirit of 72/):
IX One can abandon the requirement ti at the map Ei->PE is a projector valued measure as in /4/, H^re one suppose es sentially that the position operators o. do not commute in general,
2) One can replace the group SEC3) by another subgroup 3 of the Poincare group which also acts in IR J5J. 3) One can suppose that for the photon, the relevant aotion is that of degree of localizability. This amounts to suppose that the map Ef-*PE takes values into the set of positive operators of norm between 0 and 1 76/.
In this taper, we propose another solution to this problem. Namely, following a suggestion from J7J, we consia^r 3 instead of the configuration space 'R , all SEC2X*hcnogeneo«s manifolds. These homogeneous 3 manifolds are completely classified, up to covering, in J8J, This quantum problem seems to be rather difficult to '•% solve, so we attack a simpler problem. Namely, we try to find the proper equivalent for the notion of localizability in the framework for analytical mechanics (more precisely in the canonical formalism). This is done in. $2. Then we clasr- sify all the localizable systems covariant with respect to the Poincare group. This problem can be analysed completely in the framework of analytical mechanics and leads to a very niee rosults. The localizable systems corresponds to ; 1) particles of nonzero mass and any spin ; 2) zero mass and zero helicity. In the first case the only configuration 3 space is DR , In the second case, the configuration space 3 3 can be (R , or the manifold of bidimensional planes in IR . 3 What is interesting is that for the configuration space .1 we get the same elementary systems found by Wightman in the quantum case, and the new configuration space Ci.e. the manifolds of bidimensional planes in IR ) corresponds from the physical point of view to the photon. Indeed , a photon is characterized by the plane wave, so it can be imagined as a bidimensional object, CThis idea appears in J5J but is exploited differently). This analysis is contained in §8. Then,' in §4 we return to the quantum case. As remarked before we are not able to solve it in general, as for the classical case, but guided by the result obtained in §3, we •fltahllsh that the photon is localizable, in the sense 3 of Wightman on the manifold of bidimensional planes in |R .
Some final comments are made In §5. *• 4 -
§ 2. LOCALIZABILITX IN ANALYTICAL MECHANICS
1. As it is «ell known, if the configuration of a system is the manifold Q, then the prescription of the canonical formalism is to take as phase space the cotangent bundle T*CQ) with the canonical symplectic form. In this case
the canonical projection fr : T*CQX—>Q can be conside red as a Q-valued observable - the configuration C or position) of the system. If SEC3) acts on Q, then we have a lifted action of this group on T*CQ) and Î* becomes an SEC3)-morphysm.
2. Generalizing the preceeding scheme, we sa/ that the system
described by the symplectic manifold CM,A) is localizable
on the manifold Q if SEC3) acts symplectically on M, Q
is SEC3) - homogeneous and connected, and there exists an
SEC3)-morphism CD • M->T*CQX.
Remarks ,* 1) If ^ and Q2 are SE(J3)-dif feomprphic, they are considered identical
2) If we want to include spatial inversion as well,
then we must replace SEC3) by E(3).
3. In the canonical formalism, the evolutions of anhamilto-
nian HC «7 CM) are integral curves of the vector field XH, defined by; 'ixV n = M Let (M* ,0..) i • 1,2, two symplectic manifolds and y : Mj^ M2 a smooth map. Let h€ TCM2) and x1 : tR-»»M1 an integral curve for Xhr^ .If x2 *
is natural. If the system OiXll is localizable on Q and the map ^ : M —^ T*CQ) is natural, we say that the system is strictly localisable on Q. It is clear that this notion coi- •
4. Let now G a Lie group such that SEC3)C G . By definition a elementary relativisfic system for G is a triplet CM ,&,<)), where (JipO.) is a homogeneous symplectic manifold for G, and CH,n.) considered as a SE(J3) * symplectic manifold, is strictly localizable on Q.
Remark. The cases of interest are G m Poincare or Galilei group.
5. The problem of lacalizahility can be decided by applying the following elementary.
Lemma. Let lit and BLj to G-spaces and M* • U G,^ >/*• m *» 2 their decomposition in G*orbits. CAi are index sets). Suppose £p : M^-^ V^ is a G -nor- phi8m, and let 0^~ ^HiH ^i*^0 are clo8ed 8Ub *
groups). Then ^ oi i A1 , B fi ^ ^2 SUch that Hj^ is included, up to a conjugacy in Hgd , i.e. 3 g^G such
that g Kx< g~*C Hgg .
6. We intend to apply lemma 5 for Mj « M, M2 m T*tQ) and G-SE13). So we must find first all the SEC3) - orbits from T*CQ) where Q is an arbitrary SEC3).-homogeneous manifold. We have first: frepomition 1. Let Q£ SEtfl/K (XCSEC3) is a closed subgroup). Then, the SEC3)-orbits from T*CQ) are of the
type SEtnyG^ Xy , where A 6 (.Lie SEC3)yLie I)* and Lie 0^ ^ -^J* Lie K/* tfiţ^J) - O.fy Lie K} - 6 -
Proof. We identify T*C3EC3),/K)K c iLie SEC3),/Lie K)* C>
C, CLie SEC?))*. Then : .
and we get the formula from the statement. M
New, we identify as in J9J,CLie SEC3))* <*A2IR3 * IR3 and we have :
Proposition 2. Let W, x) £ Lie SE(J3). Then :
7. We want to apply proposition 2 from 6 to al SEC3) - homo
geneous and connected manifolds. In JBJ,there are classified, up to conjugation, all Lie subalgebras of Lie 8EC3). If we
construct the corresponding connected Lie subgroups K, we fi
a set of SEC3)-manifolds SEC3X/K. These manifolds,are
maximal in the sense that any SEC3) «-homogeneous connected
manifold is covered by one in ""G .
The list "£? corresponds to the following subgroups : en {Ot^ljefc}
cav U KMU^1?* .>,**), I'*'** en t(*M*«°i
C8) Iff,*)} - 7 -
^Rotations ,* An element of SEC3) is a couple (it, a_i with R£S0C3), a_^ (R ; the composition law is U*l. Sl>-^«2. *2> " CR!^.^ + Rag). RCi.cP ) is the rotation of angle <2> around the versor Î. ^ S , **- i» ^ 2»^ 3 is tne canonical basis in IR3 ,) . Ye remark now that if Q covers Q , the covering map Y-:Q-t Q, which is a SEC3)-morphisro, lifts to an SE(J3)-map ţp:T*C^)-^ T*CQ) which is also an SEC3)-morphism. So if a system CM ,Q. ) is localizable on Q is also localizable on Q » if <^ : M -*»T*CQ) is an SE(3)-morphism, then V c ^f ; M-=»T*CQ) is also an SEC3)-morphism. So we will study the localizability on the maximal manifolds from ~\j, , In case of affirmative answer we must find all the SEC3) manifolds covered by Q
8. We want to characterize more conveniently the condition of strict localizability. We have the following slight gene ralization of Jacobi theorem Csee J10J, p.194):
Theorem. Let (# ,fi^) i • \t2 two symplectic manifolds and
Cp: M^-* M2 smooth. Then <£ is natural iff :
Here ^ ; 3^ are the Poisson brackets on M^. In
general, {i, g^M * -Xf g.
Remark. Suppose that IM,A) is strictly locaUzable on Q and q covers Q . Then Ot,Q) is strictly localizable on Q also, because U) is a local condition. 8
So, the strategy at the end of 7 applies also for strict localizability.
9. To verify CD we need convenient realization for uome of the SE(J3)-homogeneous manifolds from -t. and for the cor responding cotangent boundles.
with the action:
Indeed, the fact that we have a transitive action is easy to establish, and G(t , ' ^t^i,V, Q}\ • We identify now
by: (ţi, «.) | * £ ftt"* t^J; J, A*)|^ o V^t^lR
CHere (. , ) _ is the usual scalar product in IR X •
Then, by a routine calculation we find the lifted action of SEC3) to TCS2 x fR3):
Finally, we identify T* OS2 x iR3) «i T. CS2 x )R3) CM) y±,ţ) with the nondegenerated biliniar form:
Then the action of SE(J3) on T*CS2 x fR3) is: _ 9 with the action
Here *y is the orthogonal projector on the linear sub- space generated by £ and P1 • 1 - P .We identify then r by
Then we identify • T* iq) ^ Tx (J^) as at (2 ) and we get for the lifted action of SEC3) ;
where :
and P^ v lis the orthogonal projection on the plane gene rated by J*, and W .
(B, S£C3;/^CR,4>i ^ ^ with the action <
We identify T CIR3)* IR3
3 3 Then, we identify T*.(jR W TaC»R ) with C , ) , and l * a IR* we get for the lifted action of SEC3) ; 10 _
Sxi (60) s£(3j>/^(*cveia) l«/-°> -• ^ with the action ;
2 2 Then we identify T* , CS x JR) * T vi x JR> with the nondegenerated hiliniar form;
and we get fcr the lifted action of SE(J3) :
(7) rec?; /{C*cvr>,a>i *5* with, the action :
6 R We identify "^ U*J * ^ ' Qh *)(RS * °> by
Then we identify T* CS2) *- T CS2) with, the nondegenerated biliniar form:
and get for the lifted action of SEU): - 11 -
10. We give nov the Polsnon brackets for the natural «ympiecţlc structure» os T*(Q) where Q la auong these analysed at ». Is indicate oaly the elementary Poisson brackets.
C5) 0») uj-rr*> (3) I /«i. *jţ - £ ij l»> C7) f {?fVj} - 0 a j - 573) C5> | *»i.rd -J«-Vj i«) Vi-rji -fi'j-r/t t7)
§ S. ELEMENTARI RELATr'ÎSTIC SXSTEMS FOR TEE POIUCARE Q90UP
1. By definition the Poincare group is ;
with, the composition law:
Hore (*»1M«»Y ^0Qk"~(&>iL)g£ • *e consider the proper ortho- chronous Poincare group :
A Here tf Q, «-^, *2l<-3 is the canonical basis in (R verifying
2. One knows J9J, JllJ that the homogeneous symplectic aani- folds for \f+ are coadjoin orbits. As in JQJ we iden tify (JLlel* )*d^2(R4 • (R4 ; the coadjoint action is :
Then, the nontrivial coadjoint orbit:* r.re :
\ - 13 -
(d) M •*.! [r f) 1 UFA*-""*j lir^fll^^f J
2 Here //P/J - CP, PLd . Also C , ) A induces a natural scalar product in^fR4, and * is the Hodge operator.
3. We study first the question of localizability\ It is conve nient to use three dimensional notation. We can write P uniquely in the1form :
P ««o* K + *(A,A2)
with £ and K three dimensional vectors. Then (LieCfJ. )*af»R x IR x (R x IR and the coadjoint action of QŢ restricted to SEC3) is ;
In the cases U), Cc),- Cd) and Ce) it is preferably to work with the Pauli-Liuban8ki quadrivector : or, in three dimensional notations ; 14
Tuen we have u.> M View tXM1* -"«i-^-r«-T/H*f "-'^
V le«)-M| V /w»,^
Here ^ £ (P, Wl/H . Then, the action of SEU) is in these new coordinates: %. a*, a,Mo-(^ **+#3, *r,«; *;2. 4. Now we have the following
Proposition. The possible configuration «paces QC"£ for the Polncare* invariant systems are among the following
ones: 1) for M^ IB3 J* .*-* S 2) for ul Ut*+), Ml «iwfflj ' R**, **j
3) for II ^ and II £ • s* and O,
4) for M. ^ ;S?
Proof. *ir«t, we apply §2.6 for K (1) to CIO) from §2.7 and we get that the SEC3) - orbits from T*(£V(3)/K) are
of the form 3EC3JVN where N ca* be; 15
- for U), (2„ ) and U) : I*, 0]f and K.. r and K. - for 13) K\oti)W/Jtj)\j*(R'] - for C6) and K - for C6^ ) ^fe^lWi^^tt'Stf aDd K r and K i(§jfi)i - - for C7) - for 18) - UO) : K.
Secondly, we determine easily the SE13) - orbits from CLie Ji )* and then we establish the SE(_3)-orbit content of the manifolds from 2. Finally , we apply §2.5 and get the desired result.! This proposition enables us now to elucidate the question of localizability, by a case by case analysis.
Theorem. The only localizable Poinca e invariant systems are:
3 1) Mm70 : on |R
! M 3 2 2 > w «ts«|R*>, ul and M* : on R , S , S x R, i _2 _33 . - x R and QQ
2 3) Mm . and ÎT a : on S m>)
Proof r- We treat separately the five possibilities permitted by the preceedinpr< g proposition. i) Q*«R.*3
il) M Ml m,o Let CJ, *, P, H) 4. 1M *, then one proves easily that there exists CR, a) € SE(3) such that : - 16 -
C£, K, P, H) - CR, a). Co, o,*^, H) with k « l\P H „ IR
J = k Re3 x a K - H a
P - k R e3
It follows that 11 - K/'H and R can be chosen as a function only of P. The morphism CP ; M^ -^ T*GR3) «^ |R3 x tR3 , if — j m,o exists verifies :
^ (J. £, I, H ) -
Evidently, <|> (j), 0, k^ ,H) is of the form Cq»U> , £&)) with £, £ : iR^U^oJ^JR smooth. Then, the preceeding relation and the fornula for Because R depends only on P and & • K/H, we get: with g, £ :ţR -»lft3 smooth. Now, the condition of SEC3)-fflorphl0a Is equivalent with ; q(RP) • B fiC P) £0») - R £(£) i.e. £ and £ are of the form; a(P) - P f t| PMj> £(P) - P fUPjiJ) 17 with f, g : IR+ U <{o) -+ R smooth. So, the most general SEC3)- morphism By the same type of argument we establish that the most general form of Op is: f (¥,15, t,H>- U/4I + lCt.W), -£ C£,W)J where <ţ and £ are of the form: with A, B, C smooth. i3) II m. MJ 14) II »mv\ T is of the same type as at 12). 11) Q - S2 x pi ill) M • MJ The most general form of ^ is : with q, p ; |R^-»fl smooth, and 2 equal to 1 or -1. 113) M - Mf The most general form of iii) Q.J- S2 #*/ iiil) II - II. . CP is of the general form : where — ,£- are smooth and of the form C*),n£M* 1 and C2,{L ) 3 • 0. We observe that for ^1 - G,^ " 0 the con dition fIJW , = 1 is contradicted for J - 0 and K • 0. a? - - - - iii2kM - 1L - with i. and P. as above. iil3) M a it m,s ^P is of the same form as at 1112). 1114) M - MJ8 Ue«+) i MI !*• 9 is of the same form as at 1112) 1115) M » *2 0? is of the form : with £ *qual to 1 or «1. lv) Q » QQ lvn *»*.., for *> 0 wo find that vhlch cannot be extended smoothly .for E • 0. iv2) M.HI m, s The same argument as at ivl). iv3) II » lA s with £_ equal to 1 or -1. iv4X It - Mi a with 1 and £ of the form C*), Dili » 1 and C£ ,* X • 0 v) Q = S2 x fR3' vl) H - M*t Cse«+) with«\^f* ,£, £ of the form (*), • £» - - 1, t£ ,£) 3 - 0. v2) U - ll!t with x, p ; fR^o^R smooth and £ equal to 1 or -1. v3) M " M ţ ? is of the same form as at vl). • * 2D Remark. We see that the notion of localizability has "too many" solution. In particular one cannot eliminate on this ground tahions (Mm - and Mm'fS ) or particles of infinite spin (J<£ ) which are not found in nature. 5. To verify the condition of strict localizability in the form (J.) form §2 we need the Poisson bracket on the coadjoint orbits. One knows J9J that the symplectic structure on co- adjoint orbit is given by : Here î^g Lie Q, and ^yt? <$\, 0 C CLie G)* is a co- adjoint orbit, is given by : iy(f) » We denote as usual 3± = f,^^ y\- ««^«^ »^ 5 f ^ U - 177),tt - f . . Then, the elementary Poisson brackets are : (V ^Wj^ijk^^k {K±, V^m 0 i, J «1, 3 K' Pji "HSij (Wlf H^- 0 (KJ, a}- PA i - ITS (Pi( HJ- 0 6. We have now all the elements to prove the Hain Theovam i The only elementary relativistic systems for the group - 21 - 2 where the action of Zg on S x R is : Proof. We verify CD from §2 for all cases from 4. We indicate only a limited number of details. i> Q = IR3 il) M = M t m,o We verify (J2) - C4) from §2.10. (4) is equivalent with g = 1, and (Ji) and (J3) are verified identically. So the most general solution is : 12) M = M I IseiRj. m ,s * As before, we verify C2)-C4) from §2.10. C3) is equivalent with ; p_C£,W) = A(>,H)P. Then C4) imposes A 5 1. So Finally a tedious computation shows that C2) is equi valent with the relations : These equations admit solutions. For instance we can take B = 0. Then it follows that C * '«("H+IM*) and we can take A * 0. This solution already appears in the literature /12/. But there are other solutions. For instance we can take C * 0 and ve get : - 22 ~ with £= ^ 1. This solution seems to be new. i3) M = M^ As at iu, C4) gives g = 1, 13) is identically verified, but C2) is true iff s =0. So we have strict localizability iff s = 0 and the most general solution is the save as at il) i4) M = M L A tedious computation shows that C2) canno be satisfied (see 713/ for some details). ii) Q = S2 x ]R iil) We verify C5) - U) and (12) - (16 , from §2.10. The relations (5), (13) and C16) are verified automatically; (G) gives p(JI) = £ K and then (J.2) is also verified. Finally (7) and (.14) are verified iff s = 0. So, the r/stem is strictly localizable iff s = 0, and the most ger*, il solution is : In all remaining cases, one proves by tedious computa tions that (1) from §2 cannot be fulfilled. Finally it must 3 2 be decided if R or S x |R can cover other SE(.3)-homogeneous connected manifolds. For the first case the answer is negative and for the second we have only one possibility namely (S x »R)/Z2JI 3 7. The physical interpretation for Q = |R is clear : o the system is pointlike. 0 • CS x IR)/ Z2 is the manifold 3 of bidimensional planes from R . 23 ~ 3 Indeed a plane in \R is givpn analytically by an equation of the type : where C-i ,q) «*nd C-Ij , -q) must be identified. In the same o spirit Q » S x |R is the manifold of oriented planes from 3 |R . So it seems that a zero mass particle as the photon must be imagined as a bidimensional object : the wave plane. *) 3 8. We remark that for the system CM ' ,CL , »R ) if we per form the canonical transform : then f becomes : so the solution is unique, up to a canonical transform . The same remark applies also for the system OQ XL, S x |R) namely we preform the canonical transform on M& U,fe,£,w;i-*u, J< -*£ jo», ^ t*nd in this way get rid" of qX; after that we make eventual* If the canonical transform in T*(£ x ft).; and arrange such that £ - 1. Summing up, we can arrange such, that : We have not succeded to elucidate the similar problem for Finally, we UmentioH n that the last formula can be inverted to: f t 24 * LOCALIZABILITI OP THE PROTON IN QUANTUM MECHANICS The elementary relativistic systems of zero mass for j + are described in the notations of ]3J, by the projective unitary irreducible reprezentations Ut,nC7"x^ { nfl). If we consider now the orthochronous Poincare group 3r, then the representations U?'° and u7'n Q_ u7' ~n Ca^lN*) of J^. have unique extensions to irreducible represent tations of y . The photon is associated with U ' ® ur--2. We study the localizability of this system on the confir- 2 " guration manifold Q • (S x IRX/ Z» . * 2 * -2 We describe the re^rssentation U -* (JLU * using the Hilbert space bundle formalism Csee 73/, J14JX, let : Here X£~*ţfelfiIt» "^'«e*} is the upper li«llt cone; -C ',' ^ is the Minkowski biliniar form in (R given by •fp» P'j " P0P0 r Cj>iP'l3 , Q is the LorentZrHermite biliniar form in € i and XQ~ ifl the Lorentz Invariant measure on X -, In U we identify functions which are different only on zero measure sets. Then is complete but is only a semi norm. Let presentation of (more precisely of the covering group in SLC2, C% given by : Here C • the proper orthochronous Lorentz group) is the covering homoicorphism. Then U fnotorizes to an unitary representation ofwSUC2, €) which is equivalent with U*'2 S> U+»"2. Restricting to in SUC2) Cwhich is the covering group of SEC3X)we have from (l) U i(S f ( AltLV(t^' ' ^"((^"t) (2) where A£SU(2). It is convenient to identify naturally # 3 functions defined on X ~ vith functions defined on JR . Also we remark that if Hw « 0 then x 4*Cp) • pACp) where A s 0"-*C. We can eliminate this "gauge degree of freedom" by Imposing the "gauge fixing conditions» f Qip) - 0. rinally we rescale <£ffc)-->v£||A|l_5 and (2) becomes ; C3) where here we denote also by b the covering homomor- phism SUC2)-^ SOC3). - 26 - 4. We pass now to raâiusr-angle variables ; A function of J_ becomes a function of Jtf and E defined 2 on S x R+ . We extend this functions to a function on S2 x |R*• by imposing the parity condition : as the Hilbert space : 2 Here d (5) 5. Finally, we perform the "partial Fourier transform" if: $( -*!% given by By transport 4 5) becomes ; o 6, We identify now naturally functions defined on S xfR and verifying 4) with functions defined on Q • (5" x 2. Here the measure (^ is obtained from d 7. If we define now the projector valued measure based on Q and taking values in J\ by fe )(r iî, V) = Tetei, î^ &*> 1J) r C8> Chen a simple computation establish that : tyft^fl "%,«>£ • (9) where the action of in SUC2) on Q is obtained from the one in §2.9 (jB) : This proves that the photon is localizable on Q in the sense of Wightman. Hmavk. : The Fourier transform j from 5 was suggested by the fact that in §3.8 f H. is canonically con m iugated with, a -. (K,P ) \ /R2. - 28 §5. FINAL COMMENTS In this paper we have analyzed a reasonable version of the notion of localizability in analytical mechanics. This analysis has suggested a solution for the localizability of the photon in quantum mechanics. It would be interesting to study completely the pro<- blem of localizability in quantum mechanics in the same spi rit as in §3., i.e. by taking into account all the possible configuration spaces. This will be done elsewhere. This could clarify the connection between the quantum and classical pic ture of physical systems. • 29 r REFKMKHCRS 1. T.D,Newton, E.Wigner, Rev. Mod. Phys. 21 (,1949) 400. 2. A.S.Wightman, Rev. Hod. Phys. 34 C1962) 845. 3. T.S.Varadarajan, "Geometry ol Quantum Theory", vol. II, von Nostrand Reinholi Co., 1970. 4. W.Amrein, Helv. Phys. Acta, 42 C1969) 149. 5. E.Angelopoulos, F.Bayen, H. Flato, Phys. Scr. 9 C1974) 173. 6. K.Krauss, "Position Observable of the Photon", in "Uncertainty Principle and Foundation of Quantum Mechanics", W.C.Price, S.S.Chissick, eds., Wiley, London, 1977. 7. A.O.Barut, R.Racza, "Theory of Group Representations and Applications", PWN, Warszawa, 1980. 8. H.Bacry, Ph.Combe, P.Sorba, Rep. Math. Phys. 5 C1974) 145. 9. V.Guillemin, S.Sternberg "Symplectic techniques in Physics" Cambridge University Press, 1984. 10. R.Abraham, J.E.Marsden, "Foundation of Mechanics", Benjamin Inc., 1982. 11. J.M.Souriau, "Structure des syst'emes dynamiques", Dunod, 1970. 12. M.Pauri, G.M.Prosperi, Journ. Math. Phys. 16 (.1975) 1503. 13. D.RvGrigore, "Localizability and covariance in analytical mechanics", preprint FT<-320-1987. 14. A.S.Wightman. " L'Invariance dans la mlcanique quantique relativiste", in "Relations des dispersions et particules lletaentaires", R . Omnes ed.