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Localizability in the Hamii Tonlan Formalism and in Quantum Mechanics

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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 * <x> x^ is an integral curve for X. we say that CMlrfl-^) and O^tCh ^) are h-compatible. If they are h-compatible for any h, we say that *• 5 - 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- • <m • wi •'•• I'» ">'<« ^^w^^F^i^"^-^»^— » responds to the intuitive concept of localizability. 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.
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