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Spin operator and spin states in Galilean covariant Fermi field theories Fuad M. Saradzhev Centre for Science, Athabasca University, Athabasca, Alberta, Canada∗ Spin degrees of freedom of the Galilean covariant Dirac field in (4+1) dimensions and its non- relativisticcounterpartin (3+1)dimensionsareexamined. Two standardchoicesofspinoperator, the Galilean covariant and Dirac spin operators, are considered. It is shown that the Dirac spin of the Galilean covariant Dirac field in (4+1) dimensions is not conserved, and the role of non- Galilean boosts in its non-conservation is stressed out. After reduction to (3+1) dimensions the Dirac field turnsinto a nonrelativistic Fermi field with a conserved Dirac spin. A generalized form oftheLévy-LeblondequationsfortheFermifieldisgiven. One-particlespinstatesareconstructed. A particle-antiparticle system is discussed. 2 1 PACSnumbers: 03.70.+k,11.10.Kk,03.65.Nk,11.30-j,11.30.Cp 0 2 I. INTRODUCTION nonrelativisticcase. OuraimistodemonstratetheDirac n a spin conservation in (3 + 1) dimensions as a result of J dimensional reduction and to construct the spin states Galileaninvarianceunderliesmanylow-energysystems 6 explicitly. encounteredinnuclearphysics,condensedmatterphysics 1 Theconceptofspinfornonrelativisticparticleswasin- and many-body theory. It is used in the field theoretical ] studyofsuperfluids,superconductors,andBose-Einstein troducedin[12]throughthetheoryofunitaryirreducible representations of the Galilei group [13]-[17]. A nonrel- h condensation [1]-[5]. p ativistic particle described by such representation is lo- A(4+1)-dimensionalcovariantformulationofGalilean - calizable for any value of spin, if it has a nonzero mass. h covariance[6]basedonanextendedspace-timeapproach Although the concept of spin appears here in the same at developed in [7],[8] provides new insights into the prop- way as in the relativistic case, the nonrelativistic spin m ertiesofthesesystems. AformulationofGalileancovari- particles are not involved in spin-orbit interactions and ancewithinarelativisticframeworkinonehigherdimen- [ do not possess electromagnetic multiple momenta [12]. sion makes non-relativistic theories similar to Lorentz Our paper is organized as follows. In Sect. II, we 3 covariant theories. Many procedures and calculations first briefly review a covariant formulation of Galilean v can be carried out in the same way as relativistic ones. covariancein (4+1) dimensions. Then we construct the 9 Some characteristic features of non-relativistic theories 9 Galileancovariantspinoperator. The expressionforthis are likely to appear only after a reduction to (3 + 1) 8 operator was previously given in [15]-[18], and a possi- 4 dimensions. Historically, a dimensional reduction from ble connection between spin and statistics was discussed 9. (4 + 1) to (3 + 1) dimensions was first developed for in [19]. We derive the same expression in a different Lorentz covariant theories [9]. The extended space- 0 waysimilartotheLorentzcovariantprocedurepresented time formulation allows us to treat relativistic and non- 1 in [20]. 1 relativistic theories in (3 + 1) dimensions in a unified We define the five-dimensional Galilei invariant Dirac : approach, depending on how (3+1)-dimensional space- v field model and determine the transformation proper- time is embedded into a (4+ 1)-dimensional manifold. i ties of the Dirac field. Conserved currents and the X A similar approachwas used in the investigationof fluid corresponding generators are constructed. We ob- r dynamics in [10],[11]. serve that the model is invariant with respect to non- a Inthepresentpaper,westudyspindegreesoffreedom Galilean transformations as well and introduce non- of nonrelativistic Fermi fields in both (4+1) and (3+1) Galilean boosts. We discuss similarities and differences dimensions. We start with a Galilean covariant Fermi between the Galileancovariantand Dirac spin operators field theory in (4 + 1) dimensions and then perform a and demonstrate the role of non-Galilean boosts in the reduction to (3+1) dimensions. We want to determine non-conservation of the Dirac spin. Different choices of towhatextentspinanditscharacteristicsareaffectedby spinoperatorinthe relativistic casewerestudied in[21]. this reduction. In Sect. III, we perform a reduction to (3+1) dimen- In relativistic theories, the orbital angular momentum sions by eliminating the x5 coordinate and keeping the and Dirac spin of a moving particle are not separately modelinvariantunderreducedGalileantransformations. conserved. Thecomponentofthe Diracspininanyfixed WeobtaintheeffectiveLagrangiandensityforanonrela- directioncannotbe thereforeusedto enumeratethe par- tivisticFermifieldin(3+1)dimensions. TheFermifield ticle’s spin states. Such enumeration is possible in the isquantized,andone-particlestatesareconstructed. We prove that the one-particle states are spin-1/2 and use the third component of spin to enumerate them. Finally, we study a particle-antiparticle system. The ∗E-mail address: [email protected] possibility to include antiparticles within a Galilean 2 frameworkwasdiscussedin[16],[22]. Weshowthatparti- The non-zero commutation relations of the Galilei Lie clesandantiparticlesbelongtotwodifferentmasssectors algebra are and construct transition operators that connect states with different number of particles and antiparticles. We [Ja,Jb]− =iǫabcJc, [Ja,Kb]− =iǫabcKc, conclude with Discussion in Sect. IV. [Ja,Pb] =iǫabcPc, [Ka,P ] =iPa, (3) − 4 − [Pa,Kb] =igabP , − 5 II. GALILEAN COVARIANT FERMI FIELD THEORIES IN (4+1) DIMENSIONS where a,b,c = 1,2,3, Pa, Ja = (1/2)ǫabcMbc and Ka = Ma4 are generators of space translations, rotations and Galilean boosts, respectively, while P and P The Galilean covariant theories in (4+1) dimensions 4 5 are generators of time and x5-translations. The explicit arebuiltwithLorentz-likeactionfunctionals,exceptthat expressionsforthesegeneratorsinthecaseofFermifields Galilean kinematics is based on the so-called Galilean will be given below. five-vectors [6] Like the Poincare group in (4 + 1) dimensions, the (x,x4,x5)= x,c¯t,s , Galilei group admits three Casimir invariants: c¯ (cid:16) (cid:17) I = P Pµ, 1 µ where c¯is a parameter with the dimensions of velocity, I = P , (4) which will be specified below, and s stands for the addi- 2 5 tional fifth coordinate. By rescaling I3 = W5µW5µ, t x4 =c¯t, s x5 = s, where → → c¯ 1 W ǫ PαMβρ the physicalunits of spaceandtime areadjustedin such µν ≡ 2 µαβρν way that all the components of the Galilean five-vectors have the same dimensions. is the five-dimensionalPauli-Lubanskitensor,ǫµναβρ being the totally antisymmetric tensor in five dimensions. Under homogeneous Galilean transformations, these vectors transform as follows: We assume that ǫ54abc =ǫabc. TheinvariantI is theinertialmassoperator,whichis 2 x′ = Rx βx4, assumedtobepositive(ornegative)definite. Theinertial − x′4 = x4, (1) mass is conserved independently; that is a characteristic feature of Galilean covariant theories. This implies the 1 x′5 = x5 (Rx) β+ β 2x4, so-called Bargmann superselection rule [23], the theory − · 2| | factoring into sectors labeled by the eigenvalue of I . 2 where β = (v/c¯), v is the relative velocity between two The invariantI1 is related to the rest (or internal) en- reference frames, R is a three-by-three orthogonal rota- ergy: tion matrix. P2 These transformationsleave invariantthe scalarprod- I =2P P P2 =2P P =2P P(0), uct, g xµxν, µ,ν = 1,...,5, defined with the Galilean 1 5 4− 5(cid:16) 4− 2P5(cid:17) 5 4 µν metric where 13×3 0 0 P2 gµν =− 0 0 1. (2) P4(0) ≡P4− 2P (5) 0 1 0 5   is the rest energy operator. In Galilean theories the in- The momentum five-vector is ertial and rest masses are not in general the same [24]. ∂ Theyaretakenequalif the Galileantheoryisconsidered p =i∂ = i∇,i t,ic¯∂ =(p,p ,p ). µ µ s 4 5 to be the limit of a Lorentz covariantone. c¯ (cid:16) (cid:17) For a state with momentum p , µ With the usual identification i∂ E, we have t → p = E/c¯. Since p /c¯has the dimension of mass, it is P p =p p , 4 5 µ µ | i | i identifiedwiththeinertialmassmsothatthecoordinate x5 is canonically conjugate to p =mc¯. assuming that 5 TheinhomogeneousGalileigroupalsoincludestransla- tionsinspace,timeandx5-direction,beingasubgroupof pµpµ =k2 >0, (6) thePoincaregroup. While thePoincaregroupin(4+1)- this gives us the dispersion relation dimensions is generatedby 15 elements, five translations Pµ and ten rotations Mµν with µ < ν, its Galilei sub- p2 k2 group has 11 generators. E p = + p . (7) | i (cid:16)2m 2m(cid:17)| i 3 The rest energy k2/2m takes the familiar form m c¯2 if c¯ while for B =0 we get 0 is defined as A=P−1. (17) 5 k c¯= , (8) The first solution is singular in the rest frame limit √2mm 0 P 0, so it is excluded. The second one brings the | | → where m is the rest mass. Both m and c¯are constants spin operator in the form 0 0 with respect to the Galilean transformations. Sa =P−1W a. (18) 5 5 IntermsoftheGalileigroupgenerators,thespinoperator A. Spin operator is written as The axial vector W obeys the following relations S=J P−1[K P]. (19) 5µ − 5 × PµW =0, (9) From the relations (10)-(11), we obtain 5µ [Sa,Pµ] = 0, − [Pσ,W5µ]− =0, (10) [Ja,Sb]− = iǫabcSc, i.e. Sa behaves like a 3-dimensional vector under rota- [W ,W ] =iǫ W λPρ. (11) tions, being invariant under translations. 5µ 5ν − 5µνλρ 5 IncontrastwiththespinoperatorinLorentzcovariant Since W =0, W has four non-vanishing components theories, which transforms in a noncovariant way under 55 5µ Lorentz boosts, the spin operator given by Eq.(19) is in- W =(W ,W ,0), variant under Galilean boosts as well, 5µ 5a 54 where [Ka,Sb] =0. − An alternative way of introducing the Lorentz covari- W = P J ǫ K P , (12) 5a 5 a abc b c − ant spin operator is by performing a transformation to W = P J . (13) 54 a a the rest frame and identifying the spin components as spatial part of the axial vector W . Along the same As a consequence of Bargmann’ssuperselectionrule, the 5µ lines, we perform the transformation Hilbert space is decomposed into a direct sum of sub- spacesrepresentingdifferentinertialmasssectorsinclud- W′ = G λW , (20) 5µ µ 5λ ing a zero-mass one. Let us consider a non-zero inertial P′ = G λP , (21) masssubspace. InsuchsubspacetheoperatorP5iseither µ µ λ positiveornegativedefiniteandthereforeP−1 iswellde- 5 where G λ is a Galilean boost matrix chosen in such a fined. AsinLorentzcovarianttheories[20],[25]wedefine µ the spin operator components Sa as linear combinations way that of the components of W5µ, P′ =0. a Sa =A[W5a−BW54Pa], (14) For Gµλ, we obtain where the coefficients A and B depend only on the five- P G b =δ b, G 5 = G a = a, momentum P . We find A and B from the requirement a a a 4 µ − −P 5 that the usual commutationrelationsfor the spin opera- P2 tor hold: G 5 = , G 4 =G 5 =1, (22) 4 2P2 4 5 5 [Sa,Sb] =iǫabcSc. (15) G 4 =0, G 4 =G a =0. − 5 a 5 Substituting (14) into (15), this gives us a system of two Using (22) in equation (21), this yields equations, P′ =P(0), 4 4 A2(P BP2) = A, 5 − demonstrating that (22) is a transformation to the rest A2BP = AB. 5 − frame. From equation (20), we also have For nonzero B, the system is solved by W′ = 0, 2P 54 A=−P5−1, B = P25, (16) W5′a = W5a, 4 i.e W is invariantunder Galileanboosts. The operator with respect to the inhomogeneous Galilean transforma- 5a tions of coordinates. W5′µ =(W5a,0,0), A transition from one reference frame x to another x′ through a Galilean transformation corresponds to a has only three nonvanishing components, and we use homogeneous linear transformation of the Dirac field, them to introduce the spin operator given again by Eq.(18). Ψ(x) Ψ′(x′)=ΛΨ(x), (25) The invariant I becomes → 3 whereΛis determinedbythe parametersofthe Galilean I =P−2S Sa, 3 5 a transformation. In the manner of Lorentzcovarianttheories, Λ=1 for so that P2I is related to the magnitude of spin. 5 3 space-time translations and for translations in the x5- Multiplying both sides of Eq.(19) by P, we get the direction. For spatial rotations, we have relation PS = PJ which holds in the zero inertial mass subspace as well. It shows that the component of the 1 angular momentum J along the direction of the linear Λ=exp γaγbǫabcφc , (26) (cid:26)2 (cid:27) momentumP,i.ehelicity,canbedefinedinallsubspaces. where (φ1,φ2,φ3) are angles of rotation. Now we compute Λ for Galilean boosts; that is, R = B. Galilean Dirac Lagrangian I in Eq.(1). From Eq.(1), we find the Galilean boost transformations for the derivatives: The five-dimensional Dirac field model is given by the Lagrangiandensity: ∂′ = ∂ β ∂ , a a− a 5 L0(x)=Ψ(x)(iγµ∂µ−k)Ψ(x), (23) ∂4′ = ∂4−βa∂a+ 21|β|2∂5, (27) where k is givenby Eq.(6), andΨ(x) is a free Dirac field ∂′ = ∂ , 5 5 defined on the five-dimensionalmanifold with the (4+1) Galilean metric in Eq.(2). The Dirac matGrices γµ in the where(∂a,∂4,∂5)and(∂a′,∂4′,∂5′)arethederivativeswith extended space-time are four-dimensional and obey the respect to the original and the new coordinates, respec- usual anti-commutation relations: tively. According to Galilean covariance, the Dirac equation γµ,γν =2gµν. must retain its form in terms of the transformed coordi- { } nates and fields; that is, We use the following representation,which is convenient for a particle-antiparticle interpretation [26]: (iγµ∂′ k)Ψ′(x′)=0. (28) µ− γa = 0 iσa , γ4 = 1 1 1 , BymultiplyingEq. (28)ontheleftbyΛ−1,andbyusing (cid:18)iσa 0 (cid:19) √2(cid:18) 1 1(cid:19) Eqs. (25) and (28), we obtain the following relations: − − Λ−1γaΛ = γa βaγ4, 1 1 1 − γ5 = − , Λ−1γ4Λ = γ4, √2(cid:18)1 1(cid:19) − 1 Λ−1γ5Λ = γ5+βaγa+ β 2γ4. where σ1,σ2, and σ3 are the usual Pauli matrices. The 2| | adjoint field is defined as From these relations we find the transformation matrix Ψ(x)=Ψ†(x) γ0, for Galilean boosts in the form 1 1 where Λ=exp γ4γaβ =1+ γ4γaβ , (29) a a (cid:26)2 (cid:27) 2 1 1 0 γ0 = γ4+γ5 = . √2 (cid:18)0 1(cid:19) where we have used (γ4)2 =0. (cid:0) (cid:1) − The Euler-Lagrangeequations of motion for Ψ(x) and its adjoint Ψ(x), respectively, are C. Non-Galilean boosts ← (iγµ∂µ k)Ψ(x)=0, Ψ(x)(iγµ ∂µ +k)=0, (24) The Lagrangiandensity given by Eq.(23) preservesits − form under the replacement ← where ζ ∂ χ = (∂ζ)χ. The action corresponding to the LagrangiandensityinEq.(23)isassumedtobe invariant x4 x5, γ4 γ5. (30) ↔ ↔ 5 This symmetry reflects itself in the invariance of the ac- currentsgroupedtogetherasanenergy-momentum-mass tion with respect to non-Galilean transformations of the tensor Tµν: form ∂ Tµν =0, (36) µ x′ = x τx5, − 1 where x′4 = x4 (x τ)+ τ 2x5, (31) − · 2| | Tµν =Ψ(x)iγµ∂νΨ(x) gµν. x′5 = x5, −L0 By integrating Eq.(36) over a volume limited in x4- the transformation matrix for the Dirac field becoming direction by two four-dimensional hyper-surfaces, x4 = constant, and of infinite extent in other directions, and 1 Λ=1+ γ5γaτ . (32) by assuming that the fields vanish at the infinite bound- a 2 aries,we obtainthe time-independent five-momentumof the Galilean Dirac field: In accordance with (30), the transformation (31) can be obtained from the Galilean one by applying x4 x5, and the transformation matrix given by Eq.(32) r↔esults Pµ = d3xdx5√2T5µ. Z fromEq.(29)byreplacingγ4 withγ5. Theparametersτ a play the same role as the parameters βa in the Galilean This gives the explicit expressions for generators P4 and boosts, P : 5 dxa dxa τa = =c¯ , P = d3xdx5√2 Ψ(x)(iγa∂ k)Ψ(x) dx5 ds 4 Z − a− (cid:0) where the ”velocity” dxa/ds determines the rate of −Ψ(x)iγ5∂5Ψ(x) (37) change of xa in the fifth coordinate s. (cid:1) and Although I is invariant with respect to the transfor- 1 mation(31)aswell,theinvarianceofI andI isviolated. 2 3 P = d3xdx5√2Ψ(x)iγ4∂ Ψ(x). (38) In particular, the energy of the system in two reference 5 5 Z frames connected by (31) is the same, E′ =E, while the inertial mass transforms as follows Whilethefirsttermintheright-handsideofEq.(37)rep- resentsthekineticandmasstermcontributionstotheto- (τp) 1 E m m′ =m + τ 2 . (33) talenergy,whichhavethesameformasin4-dimensional → − c¯ 2| | c¯2 relativistic theories,the x5 dependence introduces a new type of interaction and therefore a new type of energy Assumingthatc¯isthesameinallreferenceframes,either contribution. Galilean or non-Galilean, we conclude from Eq.(8) that We rescale the generators P and P and define the 4 5 m′ m Hamiltonian H and the total mass M of the system as 0 = , (34) follows m m′ 0 1 i.e. the inertial and rest masses are transformed in such c¯P4 =H, P5 =M, (39) c¯ way that the product mm remains unchanged. This 0 givesusthe followingexpressionforthetransformedrest As in the case of Lorentz covariant theories, we can mass m′ in terms of the original m and m : introduce the Belinfante tensor 0 0 1 (τp) 2m 1 p2 m −1 Θµν =Tµν ∂ [Ψ(x)FρµνΨ(x)], (40) m′0 =m0(cid:16)1− k r m0+2|τ|2(cid:16)1+k2(cid:17) m0(cid:17) . (35) − 2 ρ where The transformation (31) can be referred to as a boost in the x-direction with velocity dx/ds or a non-Galilean Fρµν γρ µν γµ ρν γν ρµ boost. A general form of non-Galilean transformations ≡ F − F − F includes rotations in the way similar to Eq.(1). and i µν = [γµ,γν] , − D. Conserved currents and generators F −4 which satisfies the same conservation law as Tµν, As a consequence of the invariance of the action with respect to translations, our model has five conserved ∂ Θµν =0, (41) µ 6 and provides us with the same expression for the time- The Galilean covariant and Dirac spin operators are re- independent five-momentum: lated as S= P−1[K P]+J . (47) PµΘ ≡Z d3xdx5√2Θ5µ =Pµ. (42) S− 5 × 0 Like S, the Dirac spin operator is invariant under ThismeansthatΘµν canalsoberegardedastheenergy- Galilean boosts. In particular, undeSr the transformation momentum-masstensor. TheexplicitexpressionforΘµν (22) Σa transforms as follows is Pc 1 ↔ ↔ Σa Λ−1ΣaΛ=Σa+iγ4ǫabcγb , Θµν = Ψ(x)i(γµ ∂ν +γν ∂µ)Ψ(x) 0gµν, (43) → 2P5 2 −L where Λ is given by Eq.(29) with v = P /P , while ↔ a a 5 S where a ∂b 1[a∂b (∂a)b], Θµν being symmetric, does notchange. In the restframe,the lasttwo terms in Θµν =Θνµ. ≡ 2 − the right-handside of Eq.(47) vanish, so the operatorsS The invariance of the action with respect to spatial and coincide. However,thereisanimportantdifference S rotations and boosts, gives us another set of conserved between the two spin operators. The Dirac spin is not currents: ingeneralconservedintime. Thisisspecific forGalilean covarianttheories in (4+1) dimensions. ∂µMµ,ρσ =0, (44) When we set ρ = a and σ = 4 or vice versa, Eq.(46) gives us the Galilean boost generator where Mµ,ρσ xρΘµσ xσΘµρ. (45) Ka =Ma4 =x4Pa d3xdx5 √2xaT55. (48) ≡ − −Z The conservationof Mµ,ρσ is provided by the symmetry The spin part of Ma4 is ofΘµν. IntegratingEq.(45)inthesamewayasinthecase of translations, we obtain the time-independent tensor 1 Sa4 = d3xdx5 Ψ¯(x)Fa44Ψ(x). √2Z Mρσ = d3xdx5√2M4,ρσ. (46) Z Since Fa44 = 0, it vanishes and does not contribute to Eq.(48). For spatial components of indices ρ and σ, Mab is theangularmomentumtensorrelatedtorotationsinthe xaxb-planes. As in Lorentz covariant models, Mab con- E. Nonconservation of Dirac spin sists of two parts Mab =Mab+ ab, The remaining nonzero components of the tensor (46) 0 S are M45 and Ma5. The conservation of where M45 =x4P d3xdx5√2x5Θ (49) 4 55 −Z Mab d3xdx5√2(xaT4b xbT4a) 0 ≡Z − is another form of Eq.(42) for µ = 4. Using Eq.(41), we is the orbital part of Mab , while can rewrite P4 as d Sab ≡Z d3xdx5 √2Ψ¯(x)γ4FbaΨ(x) P4 = dx4 Z d3xdx5√2x5Θ55. is its spin part. Integratingthisexpressionoverx4,thisgivesusthetime- The tensor Mab has three independent components, independent quantity represented by Eq.(49). that can be represented by the vector Ja in (3), The component Kã Ma5 is a generator of non- ≡ Galileanboosts. IncontrastwiththeGalileanboostgen- Ja ≡J0a+Sa, erator, it has a non-vanishing spin part whereJa =(1/2)ǫabcMbc. Thespinpartofthis vectoris 1 0 0 ã a5 = d3xdx5Ψ¯(x)γ4ΣãΨ(x), (50) S ≡S √2Z 1 = d3xdx5 Ψ¯(x)γ4ΣΨ(x), S √2Z where Σã Σaγ5. The non≡ze−ro commutation relations including Kã are which is often called the Dirac spin operator, where [Ja,K˜b] = iǫabcK˜c, [Pa,K˜b] =igabP , σa 0 i − − 4 Σa =(cid:18) 0 σa (cid:19)= 2γa(γ4γ5−γ5γ4). [Kã,P5]− = iPa 7 (compare to the corresponding commutators in (3)). In and addition, we get 1 a = d3xdx5Ξ†σaΞ . [Ka,K˜b] = iǫabcJc igabQ˜, S 2Z 1 1 − − − However, Ξ contributes to the spin part of the non- i.e. the galilean and non-Galilean boosts generators do 2 notcommute,Q˜ M45playingtheroleofcentralcharge. Galilean boost generator: The nonzero com≡mutation relations for Q˜ are 1 ã = d3xdx5Ξ†σaΞ [Q˜,Ka] = iKa, [Q˜,Kã] = iKã S −√2Z 1 2 − − − 1 [Q˜,P5]− = iP5 [Q˜,P4]− =−iP4. = √2Z d3xdx5Ξ†1σap+i∂5−1Ξ1 The nonconservation in time of the Dirac spin is repre- making its density nonlocal in x5. sented as Equation (51) becomes d 1 = d3xdx5Ψ¯ γ4[Σ˜ ] [Σ˜ ]γ4 Ψ, (51) d 1 dx4S √2Z (cid:16) ×∇ − ×∇ (cid:17) dx4Sa = √2Z d3xdx5∂5(cid:16)∂5−1Ξ†1σap−p+∂5−1Ξ1(cid:17). being related to the spin part of the non-Galilean boost generator. Due to the nonlocality in x5, d a does not vanish un- dx4S The x4 x5 symmetry and the role of x5 coordinate less the x5 coordinate is compactified and/or boundary becomemo↔reobviousifwerepresentthefour-component conditions on ∂−1Ξ are imposed. 5 1 spinorΨ(x)asasetoftwotwo-componentspinorsΨ (x) 1 and Ψ (x) as follows 2 III. REDUCTION TO (3+1) DIMENSIONS Ψ (x) Ψ(x)= 1 . (cid:18)Ψ2(x) (cid:19) The reduction to (3+1) dimensions is performed by factoring the x5 coordinate out of the originalfield Ψ(x) For the linear combinations of these spinors as [6] Ξ (x) Ξ(x) = 1 Ψ(x)=e−imc¯x5ψ (x,t). (54) (cid:18)Ξ2(x)(cid:19) + Ψ (x)+Ψ (x) The field ψ (x,t) is a nonrelativistic Fermi field which 1 2 , (52) + ≡ (cid:18)Ψ1(x) Ψ2(x) (cid:19) represents particles with positive inertial mass m in − (3+ 1) dimensions. The integral over x5, for instance theDiracequationgiveninEq.(24)reducestothesystem in Eqs.(37) and (38), is then interpreted as dx5 of two equations lim (1/l) l/2 dx5,wherel isanarbitraryleRngth,i.→e. l→∞ −l/2 ∂ Ξ = ip Ξ , thex5 coordiRnateisfirstcompactifiedandfactorizedand 4 1 − 2 then integrated out. ∂ Ξ = ip Ξ , (53) 5 2 + 1 − In two reference frames connected by the reduced where Galilean transformation (x,x4) (x′,x′4), the fields ψ′ (x′,t′) and ψ (x,t) are related→as 1 + + p (σa∂ k). ± a ≡ √2 ± ψ′ (x′,t′)=e−i∆(x,t)Λψ (x,t), (55) + + WeseethatthesecondcomponentΞ isnotdynamically 2 where, in addition to the transformation matrix Λ dis- independent. Itstimeevolutioniscompletelydetermined cussedabove,wehaveaspace-time-dependentphasefac- byΞ ,whilethex5-dependenceofthefirstcomponentΞ 1 1 tor with isgovernedbyΞ . Actingatbothsidesoftheseequations 2 by ∂5 and ∂4, respectively, this gives us ∆(x,t) mc¯ (Rx) β 1 β 2x4 . (56) ≡ h · − 2| | i ∂ ∂ Ξ = p p Ξ , 4 5 1 − + 1 Thisphasefactoriscausedbythenontrivialcohomology ∂ ∂ Ξ = p p Ξ , 4 5 2 − + 2 of the Galilei group [16],[27]. The ansatz (54) breaks the invariance of the original i.e. thesecondorderformoftheseequationsisthesame. theorywithrespecttonon-Galileanboosts. Assumingfor The generators of the Galilei Lie algebra can be ex- a moment that the field Ψ(x) in (54) transforms under pressedin terms of Ξ only. For instance, the generators 1 non-Galilean boosts in accordance with that invariance, of translations and the spin operator take the form i.e. like in Eq.(25) with Λ given by Eq.(32), this yields Pµ =Z d3xdx5Ξ†1i∂µΞ1 ψ+′ (x′,t′)=ei(m′−m)c¯x5Λψ+(x,t) 8 or Representing the field ψ (x,t) as + ∂ ψ′ (x′,t′)=i(m′ m)c¯ψ′ (x′,t′), ψ (x,t) 5 + − + ψ+(x,t)=(cid:18)ψ12,,++(x,t) (cid:19), where(x′,t′)and(x,t)arerelatedbythetransformation (31), while m′ = m according to Eq.(33). Therefore, and introducing the independent and dependent fields, 6 evenif we startwith a Galileanreference frame inwhich η (x,t) andη (x,t), respectively,inthe same wayas 1,+ 2,+ ψ+(x,t) is x5 independent, non-Galileanboosts bring us above for the Dirac field in (4+1) dimensions, to a non-Galilean one where the factorization (54) does not hold. η (x,t) = η1,+(x,t) The ansatz (54) is not invariant under translations + (cid:18)η2,+(x,t) (cid:19) in the x5-direction as well. However, the factor ψ (x,t)+ψ (x,t) exp imc¯d ,which appearsin Eq.(54) after the transla- 1,+ 2,+ , (60) tion{−x5 x}5+d, can be removed by performing global ≡ (cid:18)ψ1,+(x,t)−ψ2,+(x,t) (cid:19) → phase transformation: we bring the system given by Eq.(59) to the following form ψ (x,t) eimc¯dψ (x,t). + + → i∂ η +p η = 0, With the factorization given by Eq.(54), the total mass 4 1,+ − 2,+ p η mc¯η = 0, (61) of the system is + 1,+ 2,+ − where M =mN . + 1 The symbol N+, p± ≡ √2(σa∂a±k), N d3x (x,t) i.e the independent field η (x,t) obeys a Schrödinger + + 1,+ ≡ Z N type equation. For k = 0, the rest mass is equal to = d3x√2ψ¯ (x,t)γ4ψ (x,t) (57) zero, and this system coincides with the Lévy-Leblond + + Z equations[12]. Inwhatfollows,wewillassumethatm= m . is the number of particles. We assume that the particles 0 Another way of separating the independent and the have the electric charge +1, so that N stands for both + dependent fields is by introducing the four-component thetotalnumberofparticlesandthetotalelectriccharge. spinors By substituting this factorization into the action, we rewrite the Lagrangiandensity of Eq.(23) as 1 φ = I γ4ψ (62) + − + √2 (x) (x,t) 0 0,+ L → L ↔ and = ψ (x,t)(iγµ¯ ∂ kI )ψ (x,t), (58) + µ¯ − + + 1 χ = I γ4γ5ψ (63) where µ¯ runs from 1 to 4, and the matrix I is defined + − + + 2 by for the independent and dependent fields, respectively. I =I mc¯γ5, The inverse transformationis ± ∓ k 1 1 ψ = γ5γ4I φ + γ4I χ . (64) I being the identity matrix. Below we will also use + 4 − + 2√2 − + the matrix I that differs from I in the sign of m. − + Eq.(58) is the effective Lagrangiandensity in (3+1) di- The equations of motion become mensions. The corresponding action is invariant under 1 global phase transformations of the field ψ+(x,t) and i∂4φ+ = iγaΓ∂a k(γ0+I) χ+, under reduced Galilean transformations of coordinates −2√2(cid:16) − (cid:17) (x,x4) (x′,x′4). k k → Γχ+ = iγa∂a+ Γ φ+, (65) 2 2 (cid:16) (cid:17) where A. Non-relativistic Fermi field in (3+1) dimensions 1 Γ (γ4 γ5)+2i 45, The Euler-Lagrange equation of motion for the field ≡ √2 − F ψ (x,t) is + iγµ¯∂ kI ψ (x,t)=0. (59) Γ2 =0. µ¯ + + − (cid:0) (cid:1) 9 The equations (65) reproduce Eq.(61) if we take φ and By using Eqs.(66)-(67), we find + χ in the form + 1 u(r)(p)u¯(r)(p)= (γµ¯p +kI ). η η 2k µ¯ − φ+ =(cid:18) 10,+ (cid:19), χ+ =(cid:18) 20,+ (cid:19). Xr Quantizing the field ψ (x,t) by assuming that the anni- + With the transformation, Eq.(60), and eliminating η2,+ hilation and creation operators obey the anticommuta- in favor of η1,+, this gives us the Lagrangian density in tion relations: the form a(r)(p),a†(s)(q) =δ δ(p q), i p p rs (x,t)=η† (x,t) ∂ + + − η (x,t) n o − L0,+ 1,+ (cid:16)√2 4 k (cid:17) 1,+ we obtain at equal times x4 =y4 =t the following anti- reflecting once more the Schrödinger nature of the inde- commutator pendent field. 1 ψ (x,t),ψ (y,t) =D (x y) γ4 2δ(x y), n + + o + − −2k2√2 ∇ − B. Quantization where As in Lorentz covariant models, the general solution 1 1 D (x y) (γ0+I)δ(x y)+ iγa∂ δ(x y). to Eq.(59) may be expanded in terms of the plane wave + − ≡ 2 − 2k a − solutions as in Ref. [26]: The appearance of terms that do not vanish unless ψ+(x,t)= (2π1)3/2 Z d3pa(r)(p)u(r)(p)e−ipµ¯xµ¯, kthe→anti∞comismtuhteatcoorn.tFriobrutthioeninodfetpheenddeenptenfideledntwfieehldavteo Xr where r =1,2, the scalar product pµ¯xµ¯ is nη1,+(x,t),η1†,+(y,t)o=δ(x−y), E which is the standard equal-time anticommutation rela- p xµ¯ = px+ x4, µ¯ − c¯ tion for Fermi fields. withEgivenbyEq.(7),anda(r)(p)anda†(r)(p)areanni- hilationandcreationoperatorsofpositivemassparticles. C. One-particle states The positive-energy spinors u(r)(p) = u(r)(p,E,m) obey the equations Theplanewavesolutionexpansionfortheindependent field η (x,t) is (γµ¯p kI )u(r)(p)=0 1,+ µ¯ + − and the orthonormality conditions η (x,t)= 1 d3pa(r)(p)u(r)(p)e−ipµ¯xµ¯, 1,+ (2π)3/2 Z + Xr u¯(r)(p)u(s)(p)=δ , (66) rs where which are solved by u(r)(p)=u(r)(p)+u(r)(p)=d (iσap +2k)ξ(r)(0) + 1 2 u a u(r)(p)=d (γµ¯p +kI )u(r)(0), u µ¯ − and where u(r)(0)=ξ(r)(0). + 1 4mc¯2 1/2 d = . u 2k(cid:18)E+3mc¯2(cid:19) Thespinorsu(r)(p)obeythesameorthonormalitycondi- + tions as u(r)(p) in Eq.(66). The restframe spinorsu(r)(0)=u(r)(p=0) aretaken Usingthe ansatz(54)inthe expressionsforgenerators in the form constructed in the previous section in the same way as in the expression for the total mass, this gives us the ξ(r)(0) u(r)(0)= , (67) correspondinggeneratorsin(3+1)dimensions. Interms (cid:18) 0 (cid:19) oftheannihilationandcreationoperators,thegenerators Pa and H, for instance, become with ξ(1)(0)=(cid:18)10(cid:19), ξ(2)(0)=(cid:18)01 (cid:19). P+a =Xr Z d3ppaa†(r)(p)a(r)(p) 10 and and H = d3pE a†(r)(p)a(r)(p) (S1a,+(q,r)S2a,+(s,q;p) + S2a,+(q,r;p)S1a,+(s,q)) + p Xr Z Xq = 2d2p2δ . (72) with − u rs Let us now introduce one-particle states as p2 E = . p 2m r;p;+ a†(r)(p)0;+ , (73) | i≡ | i TheorbitalangularmomentumandDiracspinarenow where 0;+ is the vacuum state that contains no parti- separately conserved. For the corresponding operators, cles, | i we get a(r)(p)0;+ =0 for all r and p, ∂ | i Ja = d3pa†(r)(p) iδ ǫabcp 0,+ Xr,s Z h − rs b∂pc and belongs to the zero inertial mass subspace. Since − S2a,+(r,s;p)ia(s)(p) M|r;p;+i=m|r;p;+i, P+a|r;p;+i=pa|r;p;+i, and H r;p;+ =E r;p;+ , p | i | i a = d3pa†(r)(p) Sa (r,s)+Sa (r,s;p) a(s)(p), the states r;p;+ are characterized by the mass m, the S+ Xr,s Z h 1,+ 2,+ i momentum| p andithe energy Ep. On one-particlestates, the Galileancovariantspin op- respectively, where erator S1a,+(r,s)≡ 12ξ†(r)(0)σaξ(s)(0), (68) S+a =S+a − m1cεabcK+bP+c +J0a,+, where Ka is the Galilean boost generator in (3+1) di- + Sa (r,s;p) d2ξ†(r)(0) (p p σb p2σa) mensions, reduces to the Dirac one, 2,+ ≡ u { a b − + kǫabc(p σb p σc) ξ(s)(0). (69) Sa r;p;+ = a r;p;+ , c − b } +| i S+| i The numbers Sa (r,s) can be represented as entries of so that we can use any of them. With the conditions 1,+ the following (2 2)-matrices in the (r,s)-space: given by Eqs.(70)-(72), we get × 3 S11,+ = 12(cid:18)01 10 (cid:19), S12,+ = 21(cid:18)0i −0i(cid:19), (S+a)2|r;p;+i= 4|r;p;+i indicating that r;p;+ are spin-1/2 states. | i Foragivenmomentump,therearetwodifferentstates 1 1 0 S3 = , (r = 1,2), corresponding to two possible values of the 1,+ 2(cid:18)0 1(cid:19) third component of spin. The action of S3 on the states − + r;p;+ is as follows while Sa (r,s;p) satisfies | i 2,+ 1 paS2a,+(r,s;p)=0. S+3|1;p;+i = (cid:16)2 −f1(cid:17)|1;p;+i+f2|2;p;+i, 1 TheS2a,+(r,s;p)contributionstotheorbitalangularmo- S+3|2;p;+i = −(cid:16)2 −f1(cid:17)|2;p;+i+f2⋆|1;p;+i, mentum and Dirac spin operators are opposite in sign, so that these contributions cancel each other in the ex- where pression for the total angular momentum. p2+p2 weFgoertbtilhineecaorncdoimtiobninsationsofS1a,+(r,s)andS2a,+(r,s;p), f1 = p21+4k22, (p +ip )(p i2k) 1 2 3 Sa (q,r)Sa (s,q) = 3δ , (70) f2 = p2+4k2− 1,+ 1,+ 4 rs Xq and Sa (q,r;p)Sa (s,q;p) = 2d2p2δ , (71) 2,+ 2,+ u rs Xq f2f2⋆ =f1(1−f1).