Quantization of (2+1)-spinning particles and bifermionic constraint problem

0.32 MB·English

by  R. Fresneda

+
Quantization of (2 1)-spinning particles and
bifermionic constraint problem
R. Fresneda∗, S.P. Gavrilov†, D.M. Gitman‡, and P.Yu. Moshin§
Instituto de F´ısica, Universidade de S˜ao Paulo,
Caixa Postal 66318-CEP, 05315-970 S˜ao Paulo, S.P., Brazil
1st February 2008
4
0
0
2
Abstract
n
a Thisworkisanaturalcontinuationofourrecentstudyinquantizingrelativisticparticles.
J Thereitwasdemonstratedthat,byapplyingaconsistentquantizationschemetotheclassical

7 modelofaspinlessrelativisticparticleaswellastotheBerezin-Marinovmodelof3+1Dirac
particle,itispossibletoobtainaconsistentrelativisticquantummechanicsofsuchparticles.

2 Inthepresentarticleweapplyasimilar approach totheproblemofquantizingthemassive
v
2+1Diracparticle. However,westressthatsuchaproblemdiffersinanontrivialwayfrom
8
the one in 3+1 dimensions. The point is that in 2+1 dimensions each spin polarization
0
describesdifferentfermionspecies. Technicallythisfactmanifestsitselfthroughthepresence
2
of a bifermionic constant and of a bifermionic first-class constraint. In particular, this
1
1 constraintdoesnotadmitaconjugategaugeconditionattheclassicallevel. Thequantization
3 problemin2+1dimensionsisalsointerestingfromthephysicalviewpoint(e.g. anyons). In
0 ordertoquantizethemodel,wefirstderiveaclassicalformulationinaneffectivephasespace,
/ restricted by constraints and gauges. Then the condition of preservation of the classical
h
symmetries allows usto realize theoperator algebra in an unambiguous way and construct
t
- an appropriate Hilbert space. The physical sector of the constructed quantum mechanics
p
contains spin-1/2 particles and antiparticles without an infinite number of negative-energy
e
h levels,andexactlyreproducestheone-particlesectorofthe2+1quantumtheoryofaspinor
: field.
v
i
X
1 Introduction
r
a
This work is a natural continuation of our recent study [1, 2, 3] which was devoted to the
consistentquantizationofclassicalandpseudoclassicalmodels ofrelativistic particles. We recall
thatinthefirstarticle[1]itisdemonstratedthat,byapplyingtheconsistentquantizationscheme
(canonicalquantization, combinedwith the analysisof constraints and symmetries,as wellas of
thephysicalsector)totheclassicalmodelofaspinlessrelativisticparticle,itispossibletoobtain
aconsistentrelativisticquantummechanics(QM)ofsuchaparticle. Remarkably,theproblemof
∗E-mail: [email protected]
†Dept. F´ısicaeQu´ımica,UNESP,CampusdeGuaratingueta´, Brazil;onleavefromTomskState Pedagogical
University,634041 Tomsk,Russia;email: [email protected]
‡E-mail: [email protected]
§OnleavefromTomskStatePedagogical University,634041 Tomsk,Russia;e-mail: [email protected]
1
the infinite number of energy levels and of the indefinite metric is solvedin the same manner as
inthecorrespondingquantumfieldtheory(QFT),i.e.,byproperlydefiningthephysicalsectorof
the Hilbertspace. Inexternalbackgroundsthatdo notviolatevacuumstability,the constructed
QM turns out to be completely equivalent to the one-particle sector of the QFT. We stress that
the Schr¨odingerequationof the QMis equivalentto a pair ofrelativistic waveequations (in this
particularcasetothepairofKlein-Gordonequations),namely,toanequationforaparticlewith
charge q and to an equation for an antiparticle with charge q.
−
As a logical extension of the approach [1], we consider the quantization of pseudoclassical
modelsofspinningparticles. Inthisrelationoneoughttorecallthattheredoesnotexistaunique
frameworkfor the descriptionofrelativistic spinning particleswhich embracesallpossible cases:
particles with integer and half-integer spin, massive particles and massless particles, particles
in even and in odd dimensions. In all these cases, the action’s structure differs in an essential
manner, and therefore each case requires a completely different treatment in the course of its
quantization. In our works [2, 3], we started the quantization program of spinning particles
considering the pseudoclassical Berezin-Marinov [4] action of the massive 3+1 Dirac particle,
thestructureofwhichistypicalforallmassiverelativisticparticleswithhalf-integerspinineven
dimensions. The constraintstructureofthismodelintheHamiltonianformulationallowsoneto
fixcompletelythegaugefreedomattheclassicallevel. Inspiteoftheessentialtechnicaldifficulties
involvingtherealizationofthecommutationrelationsandtheHamiltonianconstruction,onecan
carry out the canonical quantization scheme leading to the consistent (as in the spinless case)
QM of the 3+1 Dirac particle.
In the present article we consider the problem of quantizing the massive 2+1 Dirac particle
using the pseudoclassical model first proposed by Gitman, Gonc¸alves, and Tyutin (GGT) in
[5]. From this particular model, one can devise the general quantization scheme for half-integer
spinning particles in odd dimensions [6], since the action for the general case has the same
essential structure as that of the GGT model. More remarks are in order regarding the choice
of the model for the massive 2+1 Dirac particle. Namely, we note that a number of alternative
models have been proposed for the description of a spinning particle in 2+1 dimensions (see,
e.g.,[5,7]). Wealsonotethatin2+1dimensions,adirectdimensionalreductionoftheBerezin-
Marinovactiondoesnotreproducetheminimalquantumtheoryofspinningparticles,whichmust
provideonlyonespinprojectionvalue(1/2or 1/2). Inthepapers[7,8],twomodificationswere
−
proposed. OneofthemisnotminimalandisP-andT-invariant,sothatananomalyispresent.
The other does not possess the desirable gauge supersymmetries. The GGT action is gauge
supersymmetric and reparametrization invariant. Furthermore, it is P- and T-noninvariant, in
accordance with the expected properties of the minimal theory in 2+1 dimensions.
We stress that the quantization of the GGT model differs in a nontrivial way from the one
presentedfortheBerezin-Marinovmodel. Thepointisthatin2+1dimensions(aswellasinany
odd-dimensional case) to each spin projection corresponds a different particle, because distinct
spinprojectionsofa2+1spinningparticlebelongtodistinctirreduciblerepresentations,andthus
describe differentfermionspecies. Technicallythis factmanifests itselfin the modelthroughthe
presenceofabifermionicconstantandofabifermionicfirst-classconstraint. Thisconstraintdoes
not admit a conjugate gauge condition at the classical level. However, since the corresponding
operatorhasacompactspectrum,itcanbeconsistentlyusedtofixtheremaininggaugefreedom
at the quantum level according to Dirac. Such problems do not appear in the Berezin-Marinov
case. Our interest in the GGT model does not reside entirely upon these mentioned departures
from the Berezin–Marinov model. The quantization problem of a particle in 2+1 dimensions
is a very interesting one from the physical viewpoint. There is a direct relation to field theory
in 2 + 1 dimensions [9, 10], which has recently attracted much attention, due to non-trivial
topological properties, and especially due to the possible existence of particles with fractional
2
spinandexoticstatistics(anyons). Thereisalsoastrongrelationofthe2+1quantumtheoryto
the fractional Hall effect, high-T superconductivity, etc. [9]. Thus, we hope to have motivated
c
the construction of a consistent relativistic QM of a spinning particle in 2+1 dimensions.
The paper is organized as follows. In Section 2, we study the classical properties of the
given pseudoclassical model and present its detailed Hamiltonian analysis. We focus on the
selection of the physical degrees of freedom and on an adequate gauge-fixing. We obtain a
Hamiltonian formulation of the model in an effective phase space, restricted by constraints and
gauges. We gauge-fix two of the initial gauge symmetries, and retain an effective bifermionic
first-class constraint, which does not admit gauge-fixing. In Section 3, we apply a quantization
approach,being a combinationof the canonicaland the Diracschemes,in which the bifermionic
first-class constraint is imposed at the quantum level to select admissible state-vectors. We
present a detailed construction of the Hilbert space. Then we reformulate the time-evolution in
terms of the physical time, and verify that the constructed theory has the necessary symmetry
properties. WeselectaphysicalsectorwhichdescribestheconsistentrelativisticQMofparticles
in 2+1 dimensions without an infinite number of negative-energy levels. In Section 4, we make
a comparison of the constructed QM with the one-particle sector of the 2+1 QFT. In Section
5, we summarize all the results obtained in our paper. In the Appendix, we justify the selected
Hamiltonian realization, considering the semiclassical limit of the QM constructed.
2 Pseudoclassical model and its constraint structure
2.1 Lagrangian and Hamiltonian formulations
In order to describe classically (that is, pseudoclassically) massive relativistic spin-1/2 charged
particles in 2+1 dimensions, we take the action first proposed in [5]. It has the form
1 z2 m2 1
S = Ldτ, L= e qx˙µA +ieqF ξµξν imξ3χ θmκ iξ ξ˙n,
µ µν n
−2e − 2 − − − 2 −
Z0
zµ =x˙µ iξµχ+iεµνλξ ξ κ. (1)
ν λ
−
Here e, κ, and xµ, µ = 0,1,2, are even variables, while χ and ξn, n = (µ,3), are odd variables;
theMinkowskimetricin2+1dimensionsreadsη =diag(1, 1, 1),andin3+1dimensionsis
µν
− −
η =diag(1, 1, 1, 1);θisanevenconstant;ελµv istheLevi-Civitatensorin2+1dimensions
mn
− − −
normalized as ε012 =1, and summation over repeated indices is assumed. The particle interacts
with an arbitrary external gauge field A (x), which can be of Maxwell and/or Chern-Simons
µ
nature, F = ∂ A ∂ A is the strength tensor of this field, and q is the U(1)-charge of the
µν µ ν ν µ
−
spinning particle. We assume that the coordinates xµ and ξµ are 2+1 Lorentz vectors;e, κ, ξ3,
andχ areLorentzscalars. Allthe variablesdependonthe parameterτ [0,1],whichplayshere
∈
the role of time. Dots above the variables denote their derivatives with respect to τ. The action
(1) is invariant under the restricted Lorentz transformations, but is P- and T-noninvariant, in
accordance with the expected properties of the minimal theory in 2+1 dimensions.
We recall that the action is invariant under the reparametrizations
d d d
δxµ =x˙µε, δe= (eε) , δξn =ξ˙nε, δχ= (χε) , δκ= (κε) ,
dτ dτ dτ
where ε(τ) is an even gauge parameter, and under two types of gauge supertransformations
zµ m
δxµ =iξµǫ, δe=iχǫ, δξµ = ǫ, δξ3 = ǫ, δχ=ǫ˙, δκ=0,
2e 2
1
δxµ = iεµνλξ ξ ϑ, δξµ = εµνλz ξ ϑ, δκ=ϑ˙, δe=δξ3 =δχ=0,
ν λ ν λ
− e
3
where ǫ(τ) is an odd gauge parameter and ϑ(τ) is an even gauge parameter.
We note that e, χ, and κ are degenerate coordinates, since their time derivatives are not
presentintheaction. Inwhatfollows,weconsiderareducedhamiltonizationschemefortheories
withdegeneratecoordinates[11], inwhichmomentaconjugateto thedegeneratecoordinatesare
notintroduced. Toproceedwiththe hamiltonization,we introducethe velocitiesυµ andαµ and
write the action (1) in the first-order formalism as
1 1
Sυ = Lυ+p (x˙µ υµ)+π ξ˙n αn dτ = p x˙µ+π ξ˙n Hυ dτ,
µ n µ n
− − −
Z0 h (cid:16) (cid:17)i Z0 (cid:16) (cid:17)
where
z¯2 m2 1
Lυ = e qυµA +iqeF ξµξν imξ3χ θmκ iξ αn,
µ µν n
−2e − 2 − − − 2 −
z¯µ =υµ iξµχ+iεµνλξ ξ κ, Hυ =p υµ+π αn Lυ.
ν λ µ n
− −
The variablesp andπ shouldbe treatedasconjugate momentato the coordinatesxµ andξn ,
µ n
respectively. TheorderingofvariablesintheHamiltonianHυ complieswiththeusualconvention
forthechoiceofderivativeswithrespecttocoordinatesasright-handonesandthosewithrespect
to momenta as left-hand ones.
The equations of motion with respect to the velocities and the degenerate coordinates read
δSυ 1 δSυ z¯2 m2
= p z¯ qA =0, = +iqF ξµξν =0,
δυµ − µ− e µ− µ δe 2e2 − 2 µν
δSυ i 1 δSυ
= z¯µε ξνξλ θm=0, =π +iξ =0,
δκ −e µνλ − 2 δαa n n
δSυ 1
= υµξ +iεµνλξ ξ ξ κ imξ3 =0.
µ µ ν λ
δχ e −
(cid:0) (cid:1)
The equations δSυ/δαn =0 lead to the primary constraints
ϕ =π +iξ , (2)
n n n
and the equations δSυ/δυµ =0 can be used to express the velocities υµ, viz.,
υµ = e(pµ+qAµ)+iξµχ iεµνλξ ξ κ. (3)
ν λ
− −
Substituting this relation into the other equations, we obtain more primary constraints
φ =(p+qA)µξ +mξ3,
1 µ
φ =(p+qA)2 m2+2iqF ξµξν,
2 µν
−
i
φ =ε (p+qA)µξνξλ+ θm. (4)
3 µνλ
2
Upon substituting (3) into the Hamiltonian Hυ, we get the total Hamiltonian
H(1) =Λ φ +Λ φ +Λ φ +λnϕ , (5)
1 1 2 2 3 3 n
where Λ = iχ, Λ = e/2, Λ = iκ, and λn = αn are henceforth Lagrange multipli-
1 2 3
− − − −
ers. One can see that the total Hamiltonian is proportional to the constraints. The resulting
dynamically equivalent action is
1
S(1) = p x˙µ+π ξ˙n H(1) dτ.
µ n
−
Z0 (cid:16) (cid:17)
4
Inthefollowing,weshallmakeuseoftheDiracbracketswithrespecttoasetϕofsecond-class
constraints,
A,B = A,B A,ϕ Cab ϕ ,B .
{ }D(ϕ) { }−{ a} { b }
HereCab ϕ ,ϕ =δa andthePoissonbracketsofthefunctionsF andGofdefiniteGrassmann
{ b c} c
parities ε(F) and ε(G) are given by
∂F ∂G ∂F ∂G
F,G = + ( 1)ε(F)ε(G)(F G). (6)
{ } ∂xµ∂p ∂ξn∂π − − ↔
µ n
One oughtto say that in the classicaltheory the quantity θ is a bifermionic constant. In the
quantum theory, though, it turns out to be a real number (see Sect. 5 Discussion).
2.2 Constraint reorganization and gauge-fixing
In analogy to [2, 3], we first reorganize the constraints (2) and (4) into an equivalent set (T,ϕ)
such that T=(T ,T ,T′) is a set of first-class constraints, and ϕ is a set of second-class con-
1 2
straints,
T,T = T,ϕ =0.
{ }|T,ϕ=0 { }|T,ϕ=0
The new constraints T have the form
T =(p+qA) (πµ iξµ)+m π3 iξ3 ,
1 µ − −
T2 =p0+qA0+ζr, ζ = sgn(cid:0)[p0+qA0(cid:1)] ,
−
1
T′ =ε (p+qA)µξνπλ+ θm, (7)
µνλ
2
where ζ = 1, and r = m2+(p +qA )2+2qF ξµπν is the principal value of the square
k k µν
±
root1. One can see that Tqand φ are related as
2 2
i
φ = 2ζrT ϕ ηnn˜ ϕ ,φ +(T )2 .
2 2 n n˜ 2 2
− − 2 { }
In terms of the T-constraints, the Hamiltonian (5) becomes
H(1) =Λ T +Λ T +Λ′T′, (8)
1 1 2 2
with redefined Lagrange multipliers.
Our goalis to quantize this theory,so supplementary gaugeconditions will be imposedupon
the first-class constraints T and T , except the constraint T′, which is of bifermionic nature.
1 2
The problem related to its gauge fixing is still open (see discussion in [6, 12, 13, 14]). In the
end, there will remain a first-class constraint T′ reduced on the constraint surface, while the
second-classsetof allthe other constraintsand gaugeconditions will be used to constructDirac
brackets. The surviving first-class constraint will be enforced on state vectors of the quantized
theory in order to fix the remaining gauge freedom according to Dirac.
We impose the following gauge-fixing conditions
φG =π0 iξ0+ζ π3 iξ3 , (9)
1 − −
φG2 =x0−ζτ, ζ =(cid:0)±1. (cid:1) (10)
1Wedefine theprincipalvalueofthesquarerootofanexpressioncontaining Grassmannvariablesas theone
whichispositivewhenthegeneratingelementsoftheGrassmannalgebraaresettozero.
5
Thegauge(9),chosentofixthegaugefreedomrelatedtoT ,reducesthesetofindependentspin
1
variables. The gauge (10), chosen to fix the gauge freedom related to T , is the chronological
2
gauge [15, 1, 2]. The resulting set of constraints (T ,T ,φG,φG,ϕ) is second-class.
1 2 1 2
In order to simplify the Poisson brackets between constraints, and to write the dynamics
in terms of independent variables, we pass to a set Φ,T˜ which is equivalent to the set of
constraints (T,φG,φG,ϕ). Here Φ are the second-class(cid:16)const(cid:17)raints
1 2
Φ =p +qA +ζω˜, Φ =φG, Φ =ϕ , Φ =ϕ ,
1 0 0 2 2 3 1 4 2
i
Φ = T +bT +cφG, Φ =φG, Φ =ϕ , Φ =ϕ , (11)
5 −2 1 2 2 6 1 7 0 8 3
where
2ζqF
ω˜ = ω˜2+ k0 (p+qA) (ξkπl+πkξl),
0 ω˜ +m l
r 0
ω˜ = m2+(p +qA )2+2qF ξiπk,
0 k k ik
iqφG,T iT +bT ,Φ
b= 2 1 , c= −2 1 2 1 ,
2 φG,T − φG,Φ
(cid:8) 2 2(cid:9) (cid:8) 2 1 (cid:9)
and T˜ reads (cid:8) (cid:9) (cid:8) (cid:9)
T,Φ
T˜ =T { 5} Φ , T = T′ .
− Φ ,Φ 6 |Φ=0
6 5
{ }
TheconstraintT˜isfirst-class,soitisorthogonaltoallthesecond-classconstraints, T˜,Φ =
Φ=0
0. n o(cid:12)
(cid:12)
However,forthefurtherconsideration,itisconvenienttousetheconstraintT,whichco(cid:12)incides
with T˜ and T′ on the constraint surface,
1
T = mζ ξ1π2 ξ2π1 + θm. (12)
− − 2
(cid:0) (cid:1)
The constraint T, which is responsible for parity violation with respect to reflection of one
of the coordinate axes, does not depend on the space-time coordinates and momenta. The
corresponding operator can be realized as a finite matrix. Therefore its spectrum is compact,
and we thus do not expect standard difficulties with the Dirac quantization in such a case.
The nonzero Poisson brackets (taken on the constraint surface Φ = 0) between second-class
constraints are
Φ ,Φ = Φ ,Φ =1, Φ ,Φ = Φ ,Φ = 2i,
2 1 1 2 3 3 4 4
{ } −{ } { } { } −
Φ ,Φ = Φ ,Φ =ζ(ω˜ +m), Φ ,Φ = Φ ,Φ =2i.
5 6 6 5 0 7 7 8 8
{ } { } { } −{ }
In terms of the new constraints, the Hamiltonian (8) becomes
H(1) =Λ Φ +ΛT˜, a=1,2,5,6.
a a
with redefined Lagrange multipliers.
6
2.3 Effective dynamics in the reduced space
Evidently, not all variables are independent, due to the presence of constraints. In fact, it is
possible to reduce the number of the variables using some of the second-class constraints. In
doing so, we retain the following set of variables
η = xk,p ,ξk,π ,ζ , k=1,2. (13)
k k
We hereafter refer to these variables(cid:0)as the basic va(cid:1)riables. All the initial phase-space variables
can be expressed in terms of these basic variables. We remark that the basic variables are not
independent, since there still are constraints among them.
Therefore, we seek to write the dynamics in the reduced space determined by the evolution
of the basic variables (13) alone. Despite the fact that the corresponding constraints are time-
dependent, it is still possible to write the evolution equation for the basic variables by means
of Dirac brackets if we introduce a momentum ǫ canonically conjugate to the time-evolution
parameter τ as was done in [16]. This equation reads
η˙ = η,ΛT˜+ǫ , Φ=0, T˜ =0, (14)
D(Φ)
n o
where the Dirac brackets , are constructed with respect to the constraints (11).
D(Φ)
{ }
The new set of constraints Φ,T˜ allows a number of simplifications in equation (14). In
this connection, we divide the s(cid:16)et Φ i(cid:17)nto second-class subsets U and V given by
U = Φ ,Φ ,Φ ,Φ , V = Φ ,Φ ,Φ ,Φ ,
1 2 3 4 5 6 7 8
{ } { }
so that it is possible to apply the rule
A,B = A,B A,V Cab V ,B , Cac V ,V =δa, (15)
{ }D(Φ) { }D(U)−{ a}D(U) { b }D(U) { c b}D(U) b
for any dynamical variables A and B. As a result, thanks to the vanishing of V ,ǫ on the
{ b }D(U)
constraint surface, equation (14) is simplified to
η˙ = η,ΛT +ǫ , U =0, T =0. (16)
{ }D(U)
Inthesamespirit,wefurtherdividethesubsetU intotwosetsu=(Φ ,Φ )andv =(Φ ,Φ ).
3 4 1 2
This time, application of the rule (15) gives rise to a new simplification due to the vanishing of
η,ǫ . Thus,
{ }D(u)
η,ǫ = η,v cab v ,ǫ =ζ η,Φ . (17)
{ }D(U) −{ a}D(u) { b }D(u) { 1}D(u)
Sinceneitherthebasicvariablesnortheconstraintsuinvolvethecoordinatex0,andp isalonein
0
the constraintΦ as anadditive factor,we caneliminate p altogether from(17). Consequently,
1 0
we are now able to take Φ 0 identically, and thus substitute x0 =ζτ into the same brackets.
2
≡
Finally, we express π in terms of ξk in ω˜, using the constraints u, so that the equations of
k
motion in the reduced phase space space (13) become
η˙ = η, +ΛT , u =π +iξ =0, T =0, (18)
{ Heff }D(u) k k k
where is an effective Hamiltonian given by
eff
H
=[ζqA +ω] , ω = ω˜ = ω2+ρ,
Heff 0 x0=ζτ |πk=−iξk 0
q 4iζqF
ω = m2+(p +qA )2 2iqF ξkξl, ρ= − k0 (p +qA )ξkξl, (19)
0 k k kl l l
− ω +m
0
q
7
and T (12) is an effective first-class constraint.
The nonzero Dirac brackets between the basic variables η are given by
i
xk,p =δk, ξ ,ξ = π ,π =i ξ ,π = η . (20)
l D(u) l { k l}D(u) −{ k l}D(u) { k l}D(u) 2 kl
(cid:8) (cid:9)
3 Quantization
3.1 Operators of basic variables
Theequal-timecommutationrelationsforthe operatorsXˆk,Pˆ ,Ξˆk,andζˆ,correspondingtothe
k
basicvariablesxk,p ,ξk,andζ, aredefinedaccordingtotheir Diracbrackets(20). Thenonzero
k
comutators [, ] (anticomutators [, ] ) are
+
~
Xˆk,Pˆ =i~δk, [Ξˆk,Ξˆl] = ηkl. (21)
l l + −2
h i
Weassumeζˆ2 =1andselectapreliminarystatespace ofx-dependent16-componentcolumns
R
Ψ(x),
Ψ (x)
Ψ(x)= +1 , (22)
Ψ (x)
−1
(cid:18) (cid:19)
where Ψ (x), ζ = 1,are 8-componentcolumns. The inner product in is defined as follows2,
ζ
± R
ΨΨ′ = Ψ ,Ψ′ + Ψ′ ,Ψ , (Ψ,Ψ′)= Ψ†(x)Ψ′(x)dx. (23)
| +1 +1 −1 −1
Z
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Later on, we shall see this construction of the inner product is Lorentz-invariant.
We realize all the operators in the following block-diagonal form3,
Xˆk =xkI , Pˆ =pˆ I , pˆ = i~∂ ,
16 k k 16 k k
−
Ξˆk =bdiag ξˆk,ξˆk , ζˆ=bdiag(I , I ) . (24)
8 8
−
(cid:16) (cid:17)
Here, I and I are the 16 16 and 8 8 unit matrices, respectively, whereas ξˆk are 8 8
16 8
× × ×
matrices which obey the equal-time commutation relations
~
ξˆk,ξˆl = ηkl.
+ −2
h i
3.2 Hamiltonian, first-class constraint and spin variables
Let us constructthe operatorwhich is a quantum versionof the classicalfunction (19). We
eff
H
select it as follows
Hˆ =qζˆAˆ +Ωˆ =bdiag Hˆ ,Hˆ , (25)
eff 0 +1 −1
H →
2Inwhatfollows,wedefinethebilinearform(ψ,ϕ)as (cid:16) (cid:17)
(ψ,ϕ)= ψ†ϕdx
Z
forvectors ofanyfinitenumberofcomponents.
3Hereandinwhatfollowsweusethenotationbdiag(A,B)= A 0 ,whereAandB arematrices.
0 B
(cid:18) (cid:19)
8
where Aˆ =bdiag A I , A I , Ωˆ =bdiag Ωˆ ,Ωˆ , Ωˆ = ωˆ , and
0 0|x0=τ 8 0|x0=−τ 8 +1 −1 ζ 0|x0=ζτ
(cid:0) (cid:1) (cid:16) (cid:17)
Hˆ =qζ A I +Ωˆ = (ζqA I +ωˆ ) . (26)
ζ 0|x0=ζτ 8 ζ 0 8 0 |x0=ζτ
WerealizetheoperatorTˆcorrespondingtothefirst-classconstraintT (12)bytheprescription
Tˆ = T and the classical bifermionic constant θ as a constant matrix Θˆ = bdiag θˆ,θˆ ,
|ζ=ζˆ,ξ=Ξˆ
where θˆare 8 8 matrices. Thus, Tˆ has a block-diagonalstructure, (cid:16) (cid:17)
×
1 1
Tˆ =m 2iζˆΞˆ1Ξˆ2+ Θˆ =bdiag tˆ,tˆ , tˆ =m 2iζξˆ1ξˆ2+ θˆ . (27)
1 −1 ζ
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
We are going to use this operator to fix the gauge at the quantum level according to Dirac,
Ψ , TˆΨ=0. One can see this condition implies
∈R
tˆ Ψ =0. (28)
ζ ζ
The conservation of (28) in time is guaranteed by Tˆ,Hˆ = 0, which follows from the corre-
sponding relation of the classical theory, {T,Heff}D(hu) =0i. It is equivalent to
tˆ ,Ωˆ =0. (29)
ζ ζ
h i
Now we postulate a manifest form for the operator Ωˆ , subject to the relation (29). This
ζ
form ensures the hermiticity of the operator Hˆ with respect to the inner product (23), provides
the gauge invariance under U(1) transformations, and Lorentz invariance of the inner product
(23). Here, we note that ωˆ cannot be realizedin analogywith the 3+1 dimensional case [2, 3],
0
that is, in the form of the familiar one-particle Dirac Hamiltonian, and with ξˆk proportional to
gamma-matrices. This is because tˆ contains the term ξˆ1ξˆ2, which would not commute with ωˆ
ζ 0
in such a realization. In order to fulfill the condition (29), and simultaneously ensure that ωˆ2
0
corresponds to the classical ω2, we select
0
0 m γk(pˆ +qA )
ωˆ0 = m+γk(pˆ +qA ) − 0k k , (30)
k k
(cid:18) (cid:19)
where γk, k = 1,2, are any 4 4 matrices that obey the relation γk,γl = 2δ . In fact,
× + − kl
we can consider them as two matrices of a specific 4 4 realization of gamma-matrices in 2+1
× (cid:2) (cid:3)
dimensions,[γµ,γν] =2ηµν. Moreover,wecanconsiderthesematricesasapartofthecomplete
+
set of gamma-matrices in 3+1 dimensions, for which it is convenient to select the following
representation [17]
σ3 0 iσ2 0 iσ1 0 0 I
γ0 = , γ1 = , γ2 = − , γ3 = 2 ,
0 σ3 0 iσ2 0 iσ1 I 0
2
(cid:18) − (cid:19) (cid:18) − (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
(31)
where σk are the Pauli matrices, and I is the 2 2 unit matrix.
2
Then, the expression Ωˆ2 = bdiag ωˆ2,ωˆ2 × , where
0 0 x0=ζτ
m2+(pˆ +qA )2+(cid:0)i~qF (cid:1)γ(cid:12)(cid:12)kγl 0
ωˆ2 = k k 2 kl , (32)
0 0 m2+(pˆ +qA )2+ i~qF γkγl
(cid:18) k k 2 kl (cid:19)
is consistent with the semiclassical limit (see Appendix).
9
The realization of the operators ξˆk , k = 1,2, and of the matrix θˆ, corresponding to the
classical quantity θ, is constrained by the relation (29). The latter relation is a constraint on
tˆ , and implies that ξˆ1ξˆ2 and θˆmust commute with ωˆ . Additionally, we require the condition
ζ 0
ξˆk,θˆ =0 in accordance with the classical theory.
h Thie above restraints are not sufficient to single out a representation, so we impose further
restrictions to the form of ξˆ1ξˆ2 and θˆ. The matrix of ξˆ1ξˆ2 is chosen to be composed of blocks
whichareproducts oftwo4 4gamma-matrices,andthe matrixθˆis chosento be diagonalwith
eigenvalues ~. Thelastres×trictionis consistentwiththe relationθˆ2 =~2,validinthe subspace
± 2
ofstatessatisfyingthecondition(28),whereθˆ2 canbeidentifiedwith 4iξˆ1ξˆ2 =~2. Moreover,
it is clear that ξˆ1ξˆ2 cannot be the unit matrix, since this would lead t(cid:16)o a con(cid:17)tradiction with the
commutation relations for ξˆk. There is only one realization in the space of 8 8 matrices which
×
fulfills all the aforementioned demands, viz.,
γ0Σ3 0 i~ 0 Σ3
θˆ=~ , ξˆ1ξˆ2 = , (33)
0 γ0Σ3 4 Σ3 0
(cid:18) (cid:19) (cid:18) (cid:19)
where Σ3 =iγ1γ2. Then,
i 0 γ1 i γ2 0
ξˆ1 = ~1/2 , ξˆ2 = ~1/2 . (34)
2 γ1 0 2 0 γ2
(cid:18) (cid:19) (cid:18) (cid:19)
Taking into account the concrete realization of the operator tˆ , we can see that states Ψ
ζ ζ
that obey the condition (28) have the following form
1 ψ (τ,x)
Ψ (τ,x)= ζ , (35)
ζ √2(cid:18) ζγ0ψζ(τ,x) (cid:19)
where the factor 1/√2 has been inserted for convenience.
3.3 Schr¨odinger equation
The Schr¨odinger equation
i~∂ Ψ= Hˆ +ΛTˆ Ψ, (36)
τ
(cid:16) (cid:17)
with Hˆ given by (25), for vectors Ψ subject to TˆΨ=0, has the form
i~∂ Ψ=Hˆ Ψ. (37)
τ
SolutionsoftheaboveequationcanbechosenaseigenstatesofthematrixΘˆ =bdiag θˆ,θˆ . Let
us denote eigenstates of θˆ by Ψ , which are subject to θˆΨ = θ~Ψ , with the e(cid:16)igen(cid:17)values
ζ,θ ζ,θ ζ,θ
θ = 1. The latter implies that these solutions have the specific structure
±
1 ψ (τ,x)
Ψ (τ,x)= ζ,θ ,
ζ,θ √2(cid:18) ζγ0ψζ,θ(τ,x) (cid:19)
ψ(+1)(τ,x) 0
ψ (τ,x)= ζ , ψ (τ,x)= , (38)
ζ,+1 0 ! ζ,−1 σ1ψζ(−1)(τ,x) !
where ψ(θ)(τ,x) are 2-component columns. We can see that, due to the constraint (28), these
ζ
states obey the eigenvalue equation 2iξˆ1ξˆ2Ψ = ~θζΨ . Consequently, the eigenvalues θ
− ζ,θ 2 ζ,θ
10