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HAMILTON RELATIVITY GROUP FOR NONINERTIAL STATES IN QUANTUM MECHANICS 8 0 STEPHENG.LOW 0 2 n a Abstra t. Physi alstatesinquantumme hani sareraysinaHilbertspa e. J Proje tiverepresentationsofarelativitygrouptransformbetweenthequantum 8 physi al states that are in the admissible lass. The physi al observables of 2 position, time, energy and momentum are the Hermitian representation of the generators of the algebra of the Weyl-Heisenberg group. We show that ] h there is a onsisten y ondition that requires the relativity group to be a p subgroupofthegroupofautomorphismsoftheWeyl-Heisenbergalgebra. This, - together with the requirement of the invarian e of lassi al time, results in h the inhomogeneous Hamilton group. The Hamilton group is the relativity t groupfornoninertialframesin lassi alHamilton'sme hani s. Theproje tive a representationofagroupisequivalenttounitaryrepresentationsofthe entral m extensionofthegroup. The entralextensionoftheinhomogeneousHamilton [ group and its orresponding Casimir invariants are omputed. One of the Casimirinvariantsisageneralizedspinthatisinvariantfornoninertialstates. 2 It is the familiar inertial Galilean spin with additional terms that may be v omparedtononinertialexperimentalresults. 9 9 5 1. Introdu tion 3 A relativity group de(cid:28)nes a universal transformation between physi al states . 0 in an admissible lass. For spe ial relativity, the inhomogeneous Lorentz group 1 transforms inertial states that have relative rotation, velo ity and translations in 7 position and time into one another. It spe i(cid:28)es how the measuring rods, lo ks, 0 : momentum meters and energy meters of these states are related. For small ve- v lo ities, the Lorentz group is approximated by the Eu lidean group of Galilean i X relativity. In the quantum formulation, the physi al states are realized as rays in r a Hilbert spa e and the group a ts through a proje tive representation. This is a equivalent to the unitary representation of the entral extension, both algebrai and topologi al, of the relativity group. The Poin aré group, that is the entral extension of the inhomogeneous Lorentz group, is simply its over as it does not have an algebrai extension [1℄. The Galilei group is the entral extension of the inhomogeneous Eu lidean group that has a entral mass generator as an algebrai extension. Wereviewproje tiverepresentationsonphysi alstatesthatareraysinaHilbert spa e and then show that this property of quantum me hani s leads dire tly to a onsisten y ondition that relativity groupsmust be subgroupsof the group of au- tomorphismsofthe Weyl-Heisenbergalgebra. TheHamilton groupisthesubgroup of this group of automorphisms that leaves time invariant. In a previous paper it Date:February 2,2008. Key words and phrases. noninertial, Weyl Heisenberg group, Hamilton group, quantum me- hani s,proje tiverepresentation, symple ti ,relativity,Bornre ipro ity. 1 2 STEPHENG.LOW wasshownthattheHamiltongroupistherelativitygroupfornoninertialframesin lassi alHamilton'sme hani s[2℄. Theproje tiverepresentationsoftheinhomoge- neous Hamilton group must be determined for its a tion on a quantum noninertial state. We will show that the entral extension of the Hamilton group admits three I entral generators; for the position-momentum and time-energy quantum om- M A mutation relations, as the mass generator and a new entral element with dimensions that are the re ipro al of tension. The Galilei group is the subgroup of this group for inertial states and the mass generator is the same. The Casimir invariants are al ulated and this leads to a generalized de(cid:28)nition of Galilei spin invariant for noninertial physi al states. 2. Proje tive representations of groups in quantum me hani s Ψ PhyHsi al states are represented in quantum m|ψe ih∈anHi s by rays in a Hilbert spa e Rays are|ψeqi,u|iψ˜vialen e lasses of states thaΨt di(cid:27)er only in phase [3℄,[4℄. Two states are in the same equivalen e lass if |ψ˜i=eiω|ψ ,ω ∈R. E (1) G A symmetry of the physi al system, des ribed by a Lie group with elements g π , a ts on the raysthroughaproje tiverepresentation to transformonephysi al state into another. Ψ˜ =π(g)Ψ. (2) π(g)π(g˜)=eiω(g,g˜)π(gg˜) ω(g,g˜)∈ R Theproje tiverepresentationhastheΨp≃ropeieωr(tgy,g˜)Ψ ω(g,g˜)∈R , . Astheraysareequivalen e lasses, forany andtherefore π(g)π(g˜)Ψ=π(gg˜)Ψ. (3) A lass ofXobservables is de(cid:28)nead(Gb)y lifting πth′(eXp)ro=je( Ttivπe)(rXep)resentation to a t e oHn elements of theXˆ˜Lie algebra sothΨãt is an opeXˆratoron . An observable a ting on the state is related to the observable a ting Ψ g on the state by the proje tive representation of π(g)XˆΨ=π(g)Xˆπ(g)−1π(g)Ψ=π(g)Xˆπ(g)−1Ψ˜ =π′(X˜)Ψ˜ =Xˆ˜Ψ˜. (4) Therefore π′(X˜)=π(g)π′(X)π(g)−1 =π′(gXg−1) (5) 1 for whi h a su(cid:30) ient ondition is X˜ =gXg−1. (6) X Xˆ˜ =Xˆ Finally, we note that if an operator is an invariant su h that , then X =gXg−1. (7) π A theorem in representation theory states that the proje tive representations G ̺ 2 of a Lie group are equivalent to the unitary or anti-unitary representation of 1If the representation is an isomorphism,then it is also ne essary. Otherwise, equivalen e is de(cid:28)neduptotheequivalen e lassde(cid:28)nedbytakingthequotientofthegroupwiththekernel of therepresentation. 2 Inwhatfollowswewilluseonlyunitary withanti-unitaryimpli itlyin luded HAMILTON RELATIVITY IN QUANTUM MECHANICS 3 Gˇ 3 the entral extension . (See Se tion 2.7 and Appendix B of [1℄.) Therefore, the physi al̺system may beGˇstudied by hara terizingHth̺e unitaryirredu ible representations of thegroup a tingonaHilbertspa e . TheHilbertspa eislabeled bytheunitaryirredu iblerepresentationbe auseitisnotgivena priori, butrather is determined by the unitary representation. |ψi H̺ The unitary representationsa t on the states in the Hilbert spa e |ψ˜i= ̺(g)|ψi, g ∈Gˇ. (8) ̺ The unitary representatioXˆn may be lifted to the tangXent spa e to de(cid:28)ne the Hermitian representation of the element of the algebra Xˆ =̺′(X)=T ̺′(X). e (9) Physi al observables are hara terized by the eigenvalues of the Hermitian representation of the generators Xˆ|ψ =x|ψ , x∈R, E E (10) and these generatorstransform as ̺(g)Xˆ|ψi=̺(g)Xˆ̺(g)−1|ψ˜i=Xˆ˜|ψ˜i. (11) 3. Quantum me hani s onsisten y ondition with a relativity group Measurements of the basi physi al observables su h as position, time, energy and momentum depend on the relative physi al state in whi h they are measured. For ertain lasses of physi al states, there is a relativity prin ipal that de(cid:28)nes a universal group relating the states. Examples were given in the introdu tion: the inhomogeneousLorentzgroup,andits entralextensionthe Poin arégroupforthe lass of inertial quantum states in spe ial relativisti quantum me hani s [1℄ and the inhomogeneousEu lidean group and its entral extension the Galilei group for 4 the lass of inertial states in `nonrelativisti ' quantum me hani s. In quantum me hani s,the obervablesof position, time, energy and momentum are the Hermitian representationsof the algebra orresponding to the unitary rep- H(n+1) 5 resentation of the Weyl-Heisenberg group [5℄ It is the real matrix Lie group H(n+1)≃T(n)⊗ T(n+1). s (12) T(n) ≃ (Rn,+) Rn where . That is, onsidered to be a Lie group under addition. {Z } = {P ,Q ,E,T} ∈ a(H(n+1)) i,j,.. = 1,..n α i i The algebra has a basis , , and α,β,..=1,..2n+2 . The algebra has the familiar ommutation relations. [P ,Q ]=δ I, [E,T]=−I, i j i,j (13) The a tπion of the group on the quantum sta̺tes is given by the proje tivGˇe representation that is the unitary representation of the entral extension . The Weyl-Heisenberg isaspe ial aseofthealgebrai extensionofthetranslationgroup G≃Gˇ 3 If ,thegroupdoes nothaveanyintrinsi proje tiverepresentations and theproje tive representations are justtheunitary representations. c 4`Nonrelativisti ' means the approximation for velo ities small relative to . The `nonrelativisti 'hasitsownrelativitygroupthat istheGalileigroupintheinertial ase. 5 This is also often alled the Heisenberg group. It was Weyl who is generally redited with re ognizing the ommutationrelationswasaLiealgebraanddeterminingthegroup. 4 STEPHENG.LOW T(2n+2) X α . If arethegeneratorsoftheabelianalgebraofthetranslationgroup, then the entral extension is [X ,X ]=M , [X ,M ]=0, M =−M α β α,β γ α,β α,β β,α (14) M = ζ I ζ α,β α,β α,β The Weyl-Heisenberg group is the spe ial ase where is the skewsymmetri symple ti metri (0). Nowasdis ussedinAppendixA,the entral extension of a group may be onstrained if it is the subgroup of a larger group. We know that position, time, energy and momentum are orre tly realized in the quantumme hani sbytheunitaryrepresentationoftheWeyl-Heisenberggroupand the asso iated Hermitian representation of its algebra and so must be onstrained in this manner. Before turning to what this larger group is, we brie(cid:29)y review this familiar Hermitian representation. 6 The Ma keytheoremsforsemidire tprodu ts (orthe Stone von Neumann theorem)maybeusedto omputetheunitaryirredu iblerepresentations. Theresults are wellIkno̺w′(nI.)|ψTih=e rce|ψpriesentations are labelledc=by~the eigenvalue of the single CaHsim≃irL2,(Rn+1,C) . The physi al a|qse,stiis forwhi hthQêHilbeTrˆtspa e i is . If we hoose a basis that diagonalizes and , hq,t|Qˆ |ψ =qiψ(q,t), hq,t|Tˆ|ψ =−tψ(q,t), i E E hq,t|Pˆ |ψ =i~ ∂ ψ(q,t), hq,t|Eˆ|ψ =i~∂ ψ(q,t), i E ∂qi E ∂t (15) hq,t|Iˆ|ψ =~ψ(q,t). E 7 The representations satisfy the quantum ommutation relations . Pˆ,Qˆ =i~δ , Eˆ,Tˆ =−i~, i j i,j h i h i (16) O|p,ft iourse, one ould ePqûallyTwêll hoose the momentum rep|rpe,seeintation with basis i Pˆ,Eˆthat|dqi,aegionalizes and [6Qˆ℄,o,rEˆforthatmatter,abasis thatdiagonalizes i i or that diagonalizes . We now return to the question of the larger group onstraining the algebrai entral extension. A basi onsisten y ondition exists between the physi al ob- servablesbelonging to the algebraof the Weyl-Heisenberg group and the relativity group that a ts on them. We onsider the lass of linear relativity groups that transform one state into another so that in both states, position, time, energy and momentumarerepresentedbytheHermitianrepresentationoftheWeyl-Heisenberg algebra. Ea h observers' spe i(cid:28) measurements of position, time, energy and momentum may di(cid:27)er, due to ontra tions and dilations of the relativity transformation. However, in any physi al state thatgt∈heGˇy are measured, they de(cid:28)ne a Weyl-Heisenberg algebra. Then, from (0), for , we have ̺(g)Zˆ |ψi=̺(g)Zˆ ̺(g)−1̺(g)|ψi=Zˆ′ |ψ′i, α α α (17) Therefore, as in (0-0), it follows that for a faithful representation Zˆ′ =̺′(Z′ )=̺(g)Zˆ ̺(g)−1 =̺′(gZg−1). α α α (18) 6TheStonevonNeumanntheoremisspe i(cid:28) totheWeyl-HeisenberggroupwhereastheMa key theoremsapplytoageneral lassofsemidire tprodu t groups i 7 The is inserted in the exponential relating group and algebra so that the algebra is represented byHermitianrather thananti-Hermitianoperators andthisresultsinitsappearan e in the ommutationrelations. See[12℄. HAMILTON RELATIVITY IN QUANTUM MECHANICS 5 and Z′ =gZg−1, α (19) Z′ ,Z ∈ a(H(n + 1)) g α α where we require that both . This means that is an AutH elementofthe automorphismgroupoftheHeisenbergalgebra, andtherefore G G ⊂AutH 8 the relativity group must be a subgroup, . The automorphism group is given in [7℄ and in Appendix A (0-0), AutH ≃A⊗sZ2⊗sHSp(2n+2), (20) where HSp(2n+2)≃Sp(2n+2)⊗ H(n+1). s (21) Therefore we have the result that, to be onsistent with quantum me hani s in whi h position, time, energy, and momentum are the Hermitian representation of the Weyl-Heisenberg group, the relativity group must be a subgroup of the automorphism group de(cid:28)ned in (0-0). This onsisten ymayalsobearguedintheotherdire tion. Supposetheposition, time, energy and momentum degrees of freedom are des ribed by the generators {Z } = {P ,Q ,E,T} T(2n+2) α i i that span the algebra of R2.n+T2his is the expe ted lassi al des ription where the extended phase spa e is . The generators [Z ,Z ] = 0 α β satisfy the ommutation relations π′(Z )The quZantu∈ma(oTp(e2rnat+or2s)a)re the α α proje tiverepresentationsof the generators, , with . The proje tive representatio̺n′(isZeq)uivaleXnt to∈tah(eTˇu(2nnita+ry2)r)epresentation of the entral α α extensionof the group, with . Considered as a group by itself, the algebrai extension of the translation group is given in (0). G Suppose that a relativity group relates the position, time, energy and momentum Hermitian operators in di(cid:27)erent physi al states. The translation gen- IG = erators may be onsidered to be a subgroup of the inhomogeneous group G⊗ T(2n+2) s . The group a ting on the quantuImGˇ states is the HGerm≃iStipa(n2nre+pr2e)- sentationofthe algebraofthe entralextension . Ifweassume then IG ≃ISp(2n+2)≃Sp(2n+2)⊗ T(2n+2). s (22) The entralextIeSnˇspi(o2nno+ft2h)ei=nhHomSpo(g2enne+ou2s)symple ti groupisshowninAppendix B (0-0) to be de(cid:28)ned in (0). The entral elements for a entral extension of the translation group (0) have been onstrained by its M =ζ I α,β α,β embedding in the inhomogeneous symple ti group to be . Therefore,inthis ase,therelativitygrouprestri tstheadmissible entralexten- sionofthetranslationgroupsothatitispre iselytheWeyl-Heisenberggroup. This Sp(2n+2) turns out to be also true for ertain subgroups of the symple ti group that we shall turn to next. The system is quantizedsimply throughthe proje tive representation that is required be ause physi al states are represented by rays in the Hilbert spa e. To summarize, onsisten y of the relativity group and quantum me hani s re- quiresthatthe relativitygroupisasubgroupofthegroupofautomorphismsofthe 8 If the observables are the Hermitianrepresentation of the element of the algebra ofanother group, then the same arguments hold and the relativity group must be a subgroup of the auto- morphismgroupofthatalgebra. 6 STEPHENG.LOW Weyl-Heisenbergalgebraofthe basi physi alobservablesofposition, time, energy and momentum. On the other hand, we an start by assuming a relativity group that is the symple ti group,or ertainsubgroups,andthatphysi alstatesarerepresentedby rays in the Hilbert spa e with the orresponding proje tive representations of the group. Thenitdire tlyfollowsthatthephysi alquantitiesoftime,position,energy andmomentum,represented lassi allybytranslationgenerators,aretheHermitian representations of the Weyl-Heisenberg algebra in the quantum formulation. The quantization is dire tly a result of the states being rays in a Hilbert spa e. 4. Hamilton Relativity Group: Invarian e of time Nonrelativisti quantum me hani s has the fundamental assumption that time isaninvariantforinertialandnoninertialphysi alstates. Allobservers' lo ksti k at the sameZrat∈e.a(HT(inme+, 1p)o)sition, e{nZerg}y=an{dPm,Qom,eEn,tuTm} are represented by the α α i i generators where . The requirement that G T g ∈G a relativity group leave invariant is, for all , gTg−1 =T. (23) This ondition, together with the requirements that the group is a subgroup of AutH g , requires that the be elements of the group HSp(n)≃Z ⊗ Sp(2n)⊗ H(n). 2 s s (24) HSp(n) This is established in Appendix A, (0,0) with the matrix realization of given in (0). This identity is also established in [2℄ and iPn≃tRha2tn+re2feren e it is shown that the di(cid:27)eomorphismsofthe extendedphasespa e into itself whoseJa obians HSp(n) areelementsofthegroup areHamilton'sequations. The Weyl-Heisenbevrig subgroup off(i0) is paramerterized by the relative rate of hange of position , momentum and energy with time. That is velo ity, for eand power. Poweris G F R i i 9 the entral element. The generators of velo ity are , for e and power . A H(n) general element of this algebra is viG +fiF +rR. i i (25) The full lassi al relativity group in luding the time, position, energy and momentum generatorsis the group IHSp(n)≃HSp(n)⊗ T(2n+2). s (26) For the a tion on the quantum physi al states, we must determine the unitary representation of the entral extension. The most general relativity group that has time as an invariant Ia HˇtiSnpg(no)n a quantum theory is therefore given by the unitaryrepresentationsof . The methodfor omputing entralextensions is reviewed in Appendix A. It an be shown that the algebrai entral extension I requires the addition of the entral element that turns the translation group T(2n+2) H(n+1) in (0) into . Another of the entral elements is mass and the third has the dimensions of re ipro al tension as dis ussed in what follows. A further simpli(cid:28) ation is posδsiib,jlQe tQhat makes the physi al meaning learer by i j requiringinvarian eofthelength intheinertialrestframe. Thiseliminates substantial mathemati al omplexity asso iated with non-orthonormal frames and 9 Notethat theunitsoftheseare 1/velo ity,1/for e and1/power respe tively HAMILTON RELATIVITY IN QUANTUM MECHANICS 7 enables the physi HalSmp(ena)ning to be more transparenvt.i =Tfhie=inrer=tia0l rest frame a tion of the group in (0) has the parameters (see 0) and h∈Sp(2n) so we need only onsider the symple ti subgroup. Thus for δi,jQ Q =hδi,jQ Q h−1. i j i j (27) O(n) ⊂ Sp(2n) The required subgroup is{Q } (see [2℄ and{QÃp}pendix A) and it i i maps an orthonormal basis into an orthonormalbasis This de(cid:28)nes the Hamilton group Ha(n)=Z ⊗ O(n)⊗ H(n)⊂Sp(2n)⊗ H(n), 2 s s s (28) that is the relativity group for transformations between frames in Hamilton's me- hani s where the inertial rest position frames are orthonormal. Again, the quantum theory requires us to onsider the entral extension of IHa(n)≃Ha(n)⊗ T(2n+2). s (29) 10 As we will show in the next se tion, this entral extension is of the form QHa(n) =IHˇa(n)≃Ha(n)⊗ (T(2)⊗H(n+1)) s ≃(Z2⊗Z2)⊗sSO(n)⊗sH(n)⊗s(T(2)⊗H(n+1)) (30) Again, the entral extension of the translation group in (0) is restri ted by the inhomogeneous Hamilton relativity group pre isely so that it de(cid:28)nes the Weyl- Heisenbergsubgrouprequiredforthequantumrealizationofposition, time, energy andmomentumastheHermitianrealizationofHeisenberggenerators. Wewillshow in the following se tion that it has 3 algebrai entral generators. Like the Galilei M group,it has a nontrivial algebrai entral extension that is the mass generator A and furthermore has a se ond entral element that has the physi al dimensions that are the re ipro al of tension (length/for e) in addition to the entral element I H(n+1) M A that appears in the subgroup . and are generatorsof the algebra T(2) of the translation subgroup that appears in (0). 5. Central Extension of the inhomogeneous Hamilton algebra The homogeneous and inhomogeneous Hamilton and Eu lidean groups as well as the Weyl-Heisenberg group are real matrix Lie groups. The matrix realization ofthese groupsisgiven in Appendix A. The Lie algebras an thereforebe dire tly omputed from these matrix realizations. The resulting nonzero ommutation re- Ha(n) lations of the algebra of are [J ,J ]=J δ +J δ −J δ −J δ , i,j k,l j,k i,l i,l j,k i,k j,l j,l i,k [J ,G ]=G δ −G δ , i,j k j i,k i j,k [Ji,j,Fk]=Fiδj,k−Fjδi,k, (31) [G ,F ]=Rδ . i k i,k IHa(n) The inhomogeneous Hamilton group requires the additional nonzero ommutation relations: [J ,Q ]=−Q δ +Q δ , [J ,P ]=−P δ +P δ , i,j k j i,k i j,k i,j k j i,k i j,k [G ,Q ]=δ T, [F ,P ]=δ T, i k i,k i k i,k [E,Gi]=−Pi, [E,Fi]=Qi, (32) [E,R]=2T. 10Z2⊗Z2 isthe4elementparity,timereversaldis rete group 8 STEPHENG.LOW T That is a entral element as expe ted is lear from the stru ture of the ommutation relations. All physi alstatesrelatedby grouptransformationsgenerated T by this algebraleave invariant. All these states havethe samede(cid:28)nitionof time. A general element of the algebra is Z =αi,jJ +viG +fiF +rR+qiP +tE+piQ +eT. i,j i i i i (33) αi,j n(n−1) vi fi r qi The are the 2 rotation angles, velo ity, for e, power, position, t pi e time, momentum and energy. Correspondingly the generators have the Z dimensions su h that is dimensionless. 5.1. Galilei group. Before ontinuing with the Hamilton group, we brie(cid:29)y review the inertial spe ial ase of the Hamilton group that is the familiar Eu lidean group of Galilean relativity [2℄. The Galilei group is the entral extension of the IE(n) inhomogeneousEu lidean group IE(n)≃E(n)⊗ T(n+1)=SO(n)⊗ T(n)⊗ T(n+1). s s s (34) IE(n)⊂IHa(n) {J ,G ,P ,E} i,j i i Thealgebraof isspannedbythegenerators that are a subset of the full set of generatorsin (0,0) [J ,J ]=J δ +J δ −J δ −J δ , i,j k,l j,k i,l i,l j,k i,k j,l j,l i,k [J ,G ]=G δ −G δ , i,j k j i,k i j,k [Ji,j,Pk]=−Pjδi,k+Piδj,k, (35) [E,G ]=−P . i i Ga(n) ≃ IEˇ(n) The entral extension may be dire tly omputed using the method in Appendix A. It is well known the algebrai entral extension is the M single generator with the asso iated additional nonzero ommutation relation [G ,P ]=δ M. i k i,k (36) M Notethat hasthedimensionsofmassandhasa onsistentphysi alinterpretation as mass. The Galilei group may be written as Ga(n)= T(1)⊗SO(n) ⊗ H(n). s (37) (cid:0) (cid:1) E T(1) J i,j where isthegeneratorofthealgebraofthe group, arethegeneratorsof SO(n) {G ,P ,M} H(n) i i the algebra of and generate the Weyl-Heisenberg group M n=3 SO(3)=SU(2) with the entralelement. Of ourse,inthephysi al ase , . 5.2. Central extensionIHoˇaf(tnh)e inhomogeneous Hamilton group. Returning tothe entralextension oftheHamiltongroup,dire tsymboli Liealgebra omputation using Mathemati a with the method des ribed in Appendix A results in the addition of the entral elements [P ,Q ]=δ I, [E,T]=−I, [G ,P ]=δ M, [F ,Q ]=δ A, i k i,k i k i,k i k i,k (38) {M,A,I} totheHamiltonalgebrade(cid:28)nedin(0,0). Thethreenew entralelements , N = 3 e ( ), result in the following terms being added to a general element of the algebra given in (0) Z =αi,jJ +viG +fiF +rR+qiP +tE+piQ +eT+aA+mM+ιI. i,j i i i i (39) M The entral extension ondition for in (0) is identi al to the mass entral Ga(n) extensionin the Galileigroup(0)and ispre iselythe onditionfor to bethe QHa(n) I inertialsubgroupof . The entralextension ,withdimensionsofa tion,is pre iselythe ondition forthe unitaryrepresentationto yield the usualHeisenberg HAMILTON RELATIVITY IN QUANTUM MECHANICS 9 A ommutation relations. The (cid:28)nal extension is new and has the dimensions a that is the re ipro al of tension, length/for e. Therefore the parameter has m ι dimensions of tension, of re ipro al mass and of re ipro al a tion. 6. Casimir invariants C α = 1,..N α c Casimir invariant operators , are polynomials in the enveloping a[Clge,bZra]th=at0 omZmu∈teaw(Git)haAll=the1,g.e..nNeratorsNofthealgebraofthegroGupαin=qu1e,s.t.iNon α A A g g c . , and is the dimension of . N ν c α where isthenumberofCasimirinvarianCtˆs.|ψTihe=eiνge|nψvialuνes ∈ RoftheHermitian α α α representations of the Casimir invariants , are invariants G for all physi al states related by the relativity group . These invariants typi ally labelirredu ibleunitaryrepresentations(butnotalways ompletely)andrepresent fundamentalphysi alquantities. Forexample,inboththeGalilei andthePoin aré relativity group, mass and spin are the eigenvalues of the representations of the orresponding Casimir invariants. N c The number ofNCa=simNiri−nvNarianNtsmaybe omputedNdire× tNly fromathzeAocrCem that states that it is c g r. r is the rank of the g g matrix A,B [Z ,Z ]=cC Z that is the adjoint representationof the algebra A B A,B C [8℄. A general Z =zAZ A elementofthe alpgle(bZra)is . The Casimirs anbe found by onstru tinga A general element of the enveloping algebra u[ppl(tZo a)g,Zive]o=rd0er of polynomial A A inthegeneratorswithgeneral oe(cid:30) ients. Setting reatesalinear set of equations in the oe(cid:30) ients that an be solved to determine the Casimir invariant operators. A entral element is a polynomial or order 1 and so the (cid:28)rst N e Casimir invariants are the entral elements. This is a on eptually a simple al ulation but is best arried out using a symboli omputation pa kage written inMathemati a [9℄asthenumberofbra ket omputationsandthelinearequations is large for the algebras in question. The results for the dimensions are given in the followingtable. Of ourse,while N n N n N g e c may be determined for general and is independent of , the must be omputed from the rank of the symboli matrix on a ase by ase basis. For the n≤3 remainder of this se tion we will restri t our attention to . N N N (n=1) N (n=2) N (n=3) g e c c c Ga(n): 1 n2+3n+4 1 2 3 3 2 IHˇa(n): 1 (cid:0)n2+7n+12(cid:1) 3 4 5 5 2 (cid:0) (cid:1) (40) QHa(n)=IHˇa(n) Before onsideringthegroup Ga(n) ≃ IEˇ(n) webnrie≤(cid:29)y3reviewthewellknown results from the Galilei group, . For , there are 3 Casimir M invariants, one of whi h is the entral element , C =M, C =2ME−P P , 1 2 i i C3 =M2Si,jSi,j, Si,j =Ji,j − M1 (GjPi−GiPj). (41) C n = 1 3 Note that is identi ally zero for . From (0), it follows that these are Ga(n) n ≤ 3 the omplete set of Casimir invariants for for . For learer physi al C C 2 3 insight, note that the Casimir invariants and an be written as the energy and spin of an inertial (free) parti le 1 1 E−E◦ = P P ,S2 =S S where E◦ = C . i i i,j i,j 2 2M 2M (42) 10 STEPHENG.LOW QHa(n) The Casimir invariants of may be dire tly omputed using the same n ≤ 3 method. For , there are 5 Casimir invariants, 3 of whi h are the entral I,M,A elements , C =I,C =M,C =A, 1 2 3 C =TT −IR, 4 (43) C =C2B B , 5 i,j i,j where C =C C +C =−AM +T2−IR, 2 3 4 Bi,j =Ji,j + C1Di,j. (44) C is a Casimir invariant asany polynomial ombinations of a Casimir is a Casimir D i,j and the are given by D =AD1 +MD2 +RD3 +ID4 +T D5 +D6 , i,j i,j i,j i,j i,j i,j i,j (45) (cid:0) (cid:1) where D1 =G P −G P , D2 =F Q −F Q , i,j j i i j i,j j i i j D3 =P Q −P Q , D4 =F G −F G , i,j i j j i i,j i j j i (46) D5 =F P −F P , D6 =G Q −G Q . i,j i j j i i,j i j j i B n = 1 i,j vanishes for . It is a straightforward omputation using the Lie algebra relations(0,0,0) to verifythat these areinvariant. From (0), it follows that these QHa(n) n ≤ 3 B i,j are the omplete set ofBCasim=irSinv+ari1aDn˜ts for S for . Note that may also be written as i,j i,j C i,j where i,j is the Galilean spin de(cid:28)ned 1 + A = C4 in (0) and as M C MC C D˜ = 4D1 +MD2 +RD3 +ID4 +T D5 +D6 i,j M i,j i,j i,j i,j i,j i,j (47) (cid:0) (cid:1) The eigenvalues of the Casimirs in the unitary representation usually label ir- C redu ible representations. In a representation where the eigenvalue of goes to D i,j zero, the term will be negligible and the spin redu es to the usual Galilean lim B = S . C = A → 0 C → 0 i,j i,j 3 4 spin C→0 A su(cid:30) ient ondition for this is and . A=0 1/A As means that the tension is in(cid:28)nite. 7. Dis ussion Physi alstatesinquantumme hani sarerepresentedbyraysinaHilbertspa e. Quantum me hani s realizes position, tZiˆme, momentum and energy as the Her- α mitian representation of the generators Zoˆf|tψhie algebra of the Weyl-Heisenberg α group. These generators a ting on a state, , are transformed by the unitary represeZˆñta|ψ˜tiionsofarelativitygroupto de(cid:28)negeneratorsa tingon thetransformed α state, . In order for the transformed generators to also be generators of the Weyl-Heisenberggroup,therelativitygroupmustbeasubgroupofthegroupofau- tomorphisms of the Weyl-Heisenbergalgebra. If the relativity group does not have this property, then the position, time, momentum and energy degrees of freedom would not satisfy the Heisenberg ommutation relations in the transformed state, as given in (0). This provides a basi onsisten y ondition between the relativity group and the Weyl-Heisenberg group of quantum me hani s. FollandhasproventhattheautomorphismgroupoftheWeyl-Heisenbergalgebra HSp(2n+2) A (and group) is the group together with a onformal s aling group Z 2 and a two element dis rete group that reverses the sign of time and energy.