Relativistic quantum dynamics requires a unitary representation of the
Poincare group on the Hilbert space of states. The dynamics of many-body
systems must satisfy cluster separability requirements. In this paper we
formulate an abstract framework of four dimensional Euclidean Green functions
that can be used to construct relativistic quantum dynamics of N-particle
systems consistent with these requirements. This approach should be useful in
bridging the gap between few-body dynamics based on phenomenological mass
operators and on quantum field theory.Comment: Latex, 9 Pages, Submitted to World Scientific - 50 Years of Quantum
  Many-Body Theory - A Conference in Honor of the 65-th Birthdays of John W.
  Clark, Alpo J. Kallio, Manfred L. Ristig, and Sergio Rosat

Coester, F.

Polyzou, W. N.

English

arXiv

UNT (University of North Texas) Digital Library

?0-,“.fhe submitfad manuscript has been createdby the University of Chicago as OWrafor ofArgonne National LaboratoW ~Argonne-)under Contract No. W-31-109-ENG-38 wllhthe U.S. Department of Energy. The U.S.Government retains foritself,and others act-ing on its behaff, a @d-up, nonexclusive,irrevocable worldwfde license in said @“deto repmdum, prepare derivative work, ds.tribute copies to the pubtic, and perform pub.Iicly and display publicly. by oron behalf ofthe Government.RELATIVISTIC QUANTUM DYNAMICS OF MANY-BODYSYSTEMSF. COESTERPhysics Division, Argonne National Laboratory, Argonne. IL 604.39. US.4E-mail: coester@anl.govW. N. POLYZOUDepartment of Physics and Astronomy, University o} Iowa. Iowa City IA .5.2242.l’S.-lE-mail: polyzou(i%iowa.eduRelativistic quantum dynamics requires a unitary representation of the Poincar6group on the Hilbert space of states. The Dynamics of many-body systems mustsatisfy cluster separability requirements. In this paper we formulate an abstractframework of four-dimensional Euclidean Green functions that can be used to con-struct relativistic quantum dynamics of N-particle systems consistent with theserequirements. This approach should be useful in bridging the gap between few-body dynamics based on phenomenological mass operators and on quantum fieldtheory.1 IntroductionThe superposition principle and the space-time symmetry are realized in relativis-tic quantum mechanics by a Hllbert space of states with a unitary representationof the inhomogeneous Lorentz group (Poincar6 group).1 Various representations ofsingle particle states are well kno-ivn.2Since the components of the four-momentump are constrained by the mass, P* = –rnz, there is a choice of convenient indep-endent momentum variables in the wave function: For instance the components 17orthogonal to some fixed time-like vector, or the components p+. pl orthogonal tosome null-vector. For particles with spin the functions representing state vectorsare functions of spin variables undergoing lVigner rotations. All these representa-tions are equivalent. States of noninteracting particles are represented by tensorproducts of single particle states.In quantum mechanics the Hilbert space of state vectors is the same for freeand interacting particles. The interactions are implemented by modifications ofthe Poincar6 generators. Following Bakamjian and Thomas3 this has been clonemodifyhg the mass operator, leaving the spin operator independent of interac-tions. The Poincar6 generators obtain as functions of kinematic components ofthe four momentum operator and canonically conjugate positions, the mass oper-ator, the spin operator. The choice of these kinematic components determines the“form of dynamics”.4 The principal difficulty in thk approach is the realization ofcluster separability.s The properties of any isolated cluster of particles should notdep&nd on the presence or absence of other clusters. The solution involves the re-cursive construction of appropriate many-body interactions in the many-body massoperators.6 There are no theorems defining minimal many-body interactions.Alternatively, single-particle states can be represented by equivalence classesof covariant functions of the four-momentum with the positive semi-definite innereqmws5: submitted to World Scientific on. October 2, 2000 1,mz_ ~- ~-. . .. ..$? .. . . . . . . . . . . . ...-=-. . . . . . .. --— ---. . .DISCLAIMERThis report was.prepared as an account of work sponsoredbyan agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.....,.—Y4, , ,,. . . ... ,,. ,. 7.. .,.., .,, ,? ,. ,., ,-7. ... ., ,, .7, ,.. r ,.. ., ..17-,.. .,., , , -. . . ., . . ). ,. .——. . . . . .DISCLAIMERPortions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.,,,’.,product measure alp(p) := d~p~(pa+ rn2)8(po). For particles with spin the functionsrepresenting state vectors are Lorentz covariant functions of spinor indices with asemi-definite inner product measure. Starting from kinemat ically covariant func-tions representing multi-particle states, interactions can be introduced modifyingthe semi-definite inner product measure.7For free fields (free particles) Poincar6 generators obtain by integration ofthe energy-momentum tensor over a three-dimensional manifold in the L1inkowskispace.8 Interactions in the energy-momentum tensor require local commutationrelations and infinitely many degrees of freedom. The action of these Poincar6generators on Fock-space vectors produces linear functional over the Fock space.not vectors in Fock-space. The Hilbert spaces of free and interacting fields arenecessarily inequivalent.Minkowski-Green functions are defined by vacuum expectation values of time-ordered products of local renormalized Heisenberg fields. Using assumed spectralproperties of the intermediate states and the asymptotic properties of the fielcloperators there are simple relations to observable bound-state masses, scatteringamplitudes and form factors. The principal problem is to establish a quantitativerelation of “approximations” to a theory local operators.A central feature in the formulation of relativistic quantum theory is the ab-sence of finite-dimensional unitary representations of the Lorentz group, 0(1. 3).This is the reason the representation of states by Lorentz covariant functions re-quires a semidefinite inner product measure. However, the Lorentz group is relatecl.by complexification, to the orthogonal group in four dimensions, O(4) which doeshave finite dimensional unitary representations. The unitary representations ofthe real Euclidean group E(4) together with invariant Green operators are USe-ful in the formulation of Poincar6 invariant dynamics. Thk. connection has beenexploited extensively in quantum field theory. The equivalence of the Wightmanaxioms with the Osterwalder-Schrader axiomsg establishes that Euclidean Greenfunctions (Schwinger functions) satisfying the Osterwalder-Schrader axioms implythe existence of field operators. In the context of Lagrangean field theory theSchwinger functions obtain as moments of the functional measure defined by theLagrangean. In the Euclidean formulation the locality axiom is independent of theaxioms which establish unitary Poincar6 representations with the appropriate spec-tral properties. This feature is essential for the formulation of Poincar6 covariantdynamics dynamics finite many-body systems.The purpose of this paper is to explore the formulation of relativistic quan-tum dynamics based on Euclidean invariant Green functions. Ordinary quantummechanics provides some heuristic indications. The Hilbert space: ‘H of states ofN particles is independent of the dynamics. It is the same for free and interact-ing particle% The dynamics is specified by the invariant Casimir Hamiltonian.h := H – P 2/2M or the resolvent operator G(E) := l/(h - iE). Interactionsare introduced by invariant additions to the Casimir Hamiltonian h = ho + V. orG(E)’1 = GO(E)-l + V. While the approach explored here should be applicable tomany qualitatively different physical systems and illuminate relations to quantumfield theory the focus of this exploratory study is limited systems with a &xed num-ber of particles, for instance nucleons. The abstract framework of the dynamicseqmws5: submitted to World Scientific on C)ctober 2, 2000 2,. —,. ,,, . y-~ ,. .,:..,:.: .. ...,,,.~- .. .. .,.. ?.-. !. u ,/. .. . . . .> ;- -,-. >---- . —,-.,.to be explored is formulated in Section 2. Section 3 provides a realization for sin-gle part icles. Two-body interactions are formulated in Section 4. The many-bm Iydynamics with the realization of cluster separability is discussed in Sec. 5.2 The Abstract Framework2.1 The Aw”liaqj Hilbert Space.. Since there are no finite dimensional unitary representations of the Lorentz groupSO(1, 3) we assume the representatives of physical states to be a subset of a largerspace, subject to the following assumptions.H1. Physical states are a linear subset of vectors V in an auxiliary Hilbert space‘H= with the norm IIVII: = (V, T).H2. There is a unitary representation U(R, a) of the Euclidean group E(4) with7? c 0(4) on the Hilbert space ?f.. The self-adjoint generators of E(4) aredenoted by P~ and J~” = –J”~.H3. There is a self-adjoint, unitary operatorthe 3-dimensional Euclidean subgroupp, Pk] = 0,. .~, on ‘H=, which is invariant uncler[~, J’k] = O (1)and satisfiesThe two Casimir operators & of 0(4) are functions of the generators J~”,10(3)The spectra of the Casimir operators are a(~~) = S+(SA + 1) with non-negativeinteger or half-odd integer values ofs+.The operatorsPO := ipo , pk := pk , ~ok ;= ~Jrk, Jik := Jik (4)satisfy the Poincar6 Lie algebra, and the spectrum of P2 is the real line.–00 < C(P2) < co. The operatorsexp(iPOt) s exp(–POt) and exp(iJOkX) - exp(–JOkX) (.5)are self-adjoint. Together with the unitary representations of the 3-dimensionalEuclidean group they define a non-unitary representation of the Poincar6 group.The inner product (V, !l?) := (~!l, V) s (V, (3V) defines a pseud~Hilbert spacellof the vectors in H.. The inner product (V, W) of the pseudo-Hilbert space isPoincar6 invariant, (U(A, a)~, U(A, a)~) = (V, 14).eqmws5: submitted to World Scientific on October 2, 2000 31- —-. ., .,.,.,,..,*. ,,,4-—7 . -.4 . , . .,,VF7.7.-7 . . . . . . . . . . . . ..●✎✌✌✌2.2 Green Operators and the Physical Hilbert Space.The representation of physical states and the dynamical properties of the systenlare specified by a Green operator, G, with the following properties.G1. The Green operator G is a bounded normal operator on Ha with an inversedefined on a dense set.G2. The Green operator’ G commutes with PP and JP” and hence with Pp. .J~”.G3. The operator @G is Hermitean, 61G = Gt~.G4. There is a Poincar6 invariant linear manifold S of vectors W c W. that allV G S satisfy the inequalitiesBy assumption G4 the inner product (W, GV) of vectors in this manifold is semi-definite. Physical states are represented by equivalence classes of vectors. Twovectors Ila and ~b are equivalent, Va ~ ~b iff]]~g - ~b[l’ := ([~. - Wb];G[!J= - ~b)]) = O (7)The physical Hilbert space H is equipped with a unitary representation of thePoincar6 group. Single-particle states iJ!M of mass M, elementary or composite.satisfy (P2 + M2)TM w O.2.3 Perturbations of Green OperatorsA perturbati~n Go d G maybe defined byG-1 := G;l +U (8)where U is an E(4) invariant, pseudo-Hermiteafi operator with domainD(U) > D(G;l) BY assumption U is bounded rekitive to G;l,IIUVII=s allG;’1411a + 611VII=, (9)with O s a < 1 and O s b. The operators UGO, GOU and are bounded with abound less than 1. It follows that G-l Go, and GoG-l are bounded operators withbounded inverses.3 Realization for Single Particles3.1 Single Particles with Spin OThe auxiliary Hilbert space % of a spin-zero single particle is realized by squareintegrable functions V(x) with x:= {x”, X1,X2,X3} with the inner product(w, v)= Jd’xp(x)y (lo)eqmws5: submitted to World Scientific on October 2, 2000 41--.-----—- . ---- . —- ., --.-TzT7----c ------ ,....Schwartz functions f(x) are dense in this Hilbert space. The involution operator Ois defined byWI!(x) := W(–XO,Z) (11)The self-adjoint generators of the real Euclidean group E(4) arelap/J:= ._~~xv ‘ ( aJW := ! #- “ai &v–x ~ )The associated Poincar6 generators are then defined by eq. (4). The Green operatoris represented by the Green function G(x – x’)401G(x – X’) := ; ~4P exp[ip(x – x’)]P2+ m20113exp (i~(~ – 5’) – W(p)lx’” – X“l)‘%~3p2w(p)(13)where w(p) := /--- Schwartz functions f(x) with support restricted topositive values of X“ represent a linear manifold of vectors in 7-L=-which satisfies thethe requirement G4. With the support restriction it follows that(f>Gf) = (f>.f):=/ ‘lf(~l’ =/~4xlf(x)12 (14)where/f(x) ,= (Zn)-; ~3peiF=-u(P)x”@ (xo)$(~ (15)The Hilbert space ‘H is constructed by the usual procedure of moding out zero-norm vectors and adding Cauchy sequences. The equivalence classes of Schwartzfunctions are dense in 7-/. Two functions fl and ~z are equivalent, jl - ~z iffIlfl -f’11 =0 o (16)It f~llow~ that two functions ~1 and ~2 are in the same equivalence class, ~1 * f...iff ~1 = ~2.Since3()/O(x’O)~G(x’ – X)8(X0) = 6(X’O)e(XO) ;e—w(p)(x’”+x qeijw-qd3p2w(p). (17)it follows that the inner product (~a, G .fb) of functions satisfying the support con-dition is manifestly Lorentz invariant,J(fb, Gfa) = -@w(p) fb(~”fa(d = /&W2 + m2) 6(P0) fb(p)”fa(~) (18)withf(P):= /d4x=p(-+” ~- POXO)f(x) - (19)I eomws5: submitted to World Scientific on October 2, 2000 51... *The time evolution evolution ~=(t) := e- ‘p’)t~= is given explicitly by(fb, @-iP”tfa) = (.fb? ~a(t))(~~)with ~c(t, p) := e–iWtPlt~a(p7).3.2 Single Particles with Spin ~For a single spin 1/2 particle the Green function isG(x – y) := —(2:)4 I “’eip”(x-’)(pP~:z)owhere the spinor matrices ~~ := ifl, @ with 6:= -[w? satisfy*{7w 7.U} = -J.v , (w)The involution operator @ must also act on the spinor indices,(Elf)(x) := pf(-xo,q. (23)As in the case of spin O it is easy to verify positivity of the inner product (j. G~)for Schwartz functions with support restricted to positive values of X“.(~, G~) := ~d4xd4y~t(x)@G(x - y)j(y) = ~d’xd4y~t(x)OG(-x0 - y“.~ - ~~(y)where4 Two-Body DynamicsThe auxiliary Hilbert space ‘?-f=is the tensor product of the single-particle awciliaryHHbert spaces. The involution operator Cl is the tensor product of single particleinvolution operators. Schwartz functions ~(xl, X2) s ~(X, x) with X := ~(xl + Xz)and x := x1 – X2 are dense in this Hilbert space. For spin 1/2 particles thesefunctions depend on spinor variables as well. The E(4) generators are additive.In general Green operators are realized by -E(4) invariant tempered d~tributionsG(xl, x2; x;, xi). For free particles the Green function is the product of single particleGreen functions, Interactions are added according to eq. (8). A simple example ofa nucleon-nucleon interaction is of the formp,veqmws5: submitted to World Scientific on October 2, 2000 61, 9eScattering wave functions obtain by the weak time limits(il~, Gfla*) = ,~~=(~~, GeipOt~.(t)) (2s)- (1)where V=(t) := 14a (t) x @!2)(t) is the tensor product of single particle states. *ZThe S-matrix obtains in the limit(& = l~hlm &llIl ~b(t’), Ge iP”(&t’)&@ (29)5 Many-Body DynamicsThe auxiliary Hilbert space ?& is the N-fold tensor product of the single-particleauxiliary Hilbert spaces. The involution operator (3 is the outer product of singleparticle involution operators. Schwartz functions, j(xl,..., x~), of Npoints and Xspinor indices are dense in this Hilbert space. The -E(4) generators are additive.In general Green operators are realized by E(4) invariant tempered distributionsGN(x1,.. .XN; yN,. ... yl ). The cluster properties can be conveniently realized usingformal annihilation operators a(x), b(x) and creation operators at(x), bt(x) whichsatisfi the commutation relations13{a(x), at(y)} = 6(4)(x -y), {b(x): t$(y)} = fw(x - y){a(x), b(y)} = {a(x), bt(y)} = 0. (31)The cluster structure of the general N-particle Green functions is realized by theexpression14GN(x1, . . . ,xN; yN, . . . ,y~) = (I)la(xl) . . .U(X1~)b(yjv) ()-.. b(yl)exp ~Sn 10) .n(qwhere 10) is the cyclic vector that is annihilated by a(x), b(x), and “‘n’=51d4xlld4ylx at(xn). --at(xl)bt(yl) .. .bt(yn)Sn(xl, . . . 9xn; Yn9 . . . !Yl) (33)The functions Sn vanish for separation of the points into widely separated clusters.Representations of the inverse Green operators have the same cluster structure.G-l = exp[1 Id4x d4yat(x)bt(y)S-l(x – y) + ~ U. 1 (34)n~zeqmws5: submitted to World Scientific on October 2, 2000 71-— .- —..,.. .,. ..(, . .. . . ——,. 4 .,..... ~ <., ,... -,>Aqqr --:--<“ . J . -. . . . . ,. . ..,..7 .,,,, ,. .-, ... .$, ,- -, ..-,.. . -.,. ..’r,-A many-body Green operator can be completely determined by two-bocly interac-tions, U* c O for n >2,‘2=+’’/’4’’/’4”/d~y~at(xl)at(xz) bt(yz)bt (yl)U(xl. xz: yz. yl) (35)6 ConclusionsKinematic Poincar6 covariance of state vectors of many-body systems requires a dy-namically determined semi-definite inner product measure. An effective realizationis based on an awxiliary Hilbert space endowed with a unitary representation of thefour-dimensional Euclidean group. A self-adjoint unitary involution operator pr-ovidesa Poincar6 invariant indefinite inner product. The Euclidean invariant Greeuoperator specifies the Poincar6 invariant semi-definite inner product of the sub-space of physical states. In this framework two-body Green operators are sufficientto determine many-body Green functions satiswlng cluster separability.AcknowledgmentsThis work was supported in part by the Department of Enerbg, Nuclear PhysicsDivision, under contracts W-31-109-ENG-38 and DEFG02-86ER40286.References1.2.3.4.5.6.7.8.9.10.11.12.13.14.E. P. Wgner, Ann Math. 40141 (1939)R. Haag, Local Quantum Physics (Springer Verlag (1992) Chap. 3:S. Weinberg The Quantum Theory of Fields (Cambr. Univ. Press 1995) Chap.2B. Bakamjian and L. H. Thomas, Phys. Rev. 921300 (1953)P.A.M . Dirac, Rev. Mod. Phys. 21392 (1949)L. L. Foldy, Phys. Rev. 122275 (1961)F. Coester, Helv. Phys. Acts 387 (1965)S. N. Sokolov, Dold. Akad. Nauk. USSR 233575 (1977) andTeor. Mat Fiz. 36193 (1978)F. Coester and W. N Polyzou, Phys. Rev. D 261348 (1982)W.N. Polyzou, Phys. Rev. D 32,995 (1985)J. Schwinger, Phys. Rev. 127324 (1964)K. Osterwalder and R. Schrader, Comm. Math. Phys. 3183 (1973 ) ancl42281 (1975).S.S. Schweber Relativistic Quantum Field Theory (Harper and Row, 1961) p.33N. N. Bogolubov, A.A. Logunov, A.I. Oksak, L.T. Todorov, General Principlesof Quantum Field Theory (Kluver Acad. Pub. 1987) Chapter 10W. Brenig and R. Haag, Fortschritte der Physik: 7183 (1959) [translation inQuantum Scattering Theory M. Ross ed. (Indiana Univ. Press 1963) p. 13F. Coester, Phys. Rev. 951318 (1954)J.G. Valatin, Proc. Roy. Sot. (London) 229221 (1955)F. Coester, Nut. Phys. 8421 (1958)eqmws5: submitted to World Scientific on October 2, 2000 81

Relativistic quantum dynamics of many-body systems.

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RELATIVISTIC QUANTUM DYNAMICS OF MANY-BODY SYSTEMS

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Relativistic Quantum Dynamics of Many-Body Systems

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