Table Of ContentSpin 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
oftheL´evy-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, rota-
tions 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 be-
ing 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 K˜a 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 ˜a a5 = d3xdx5Ψ¯(x)γ4Σ˜aΨ(x), (50)
S ≡S √2Z
1
= d3xdx5 Ψ¯(x)γ4ΣΨ(x),
S √2Z where Σ˜a Σaγ5.
The non≡ze−ro commutation relations including K˜a 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). [K˜a,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
˜a = d3xdx5Ξ†σaΞ
[Q˜,Ka] = iKa, [Q˜,K˜a] = iK˜a 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 Schr¨odinger
+ + 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 L´evy-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 Schr¨odinger 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).
