ion i. cotÃescu west university of timiºoara, v. parvan ... · theory of external symmetry for...

Dedicated to Acad. Aureliu Sãndulescu’s 75th Anniversary

DIRAC FERMIONS IN DE SITTERAND ANTI-DE SITTER BACKGROUNDS

ION I. COTÃESCU

West University of Timiºoara, V. Parvan Ave. 4, RO-300223 Timiºoara

Received March 27, 2007

Starting with a new theory of symmetries generated by isometries in fieldtheories with spin, one finds the generators of the spinor representation in back-grounds with a given symmetry. In this manner one obtains a collection of conservedoperators from which one can chose the complete sets of commuting operatorsdefining quantum modes. In this framework, the quantum modes of the free Diracfield on de Sitter or anti-de Sitter spacetimes can be completely derived in static ormoving charts. The discrete quantum modes are presented in the central static charts ofthe anti-de Sitter spacetime, whose eigenspinors can be normalized. The consequenceis that the second quantization can be done in this case in canonical manner. For thefree Dirac on de Sitter manifolds this can not be done in static charts being forced toconsider the moving ones. The quantum modes of the free Dirac field in these chartsare used for writing down the quantum Dirac field and the its one-particle operators .

Pacs: 04.20.Cv, 04.62.+v, 11.30.-j

1. INTRODUCTION

In general relativity [1–3] the development of the quantum field theory incurved spacetimes [4] give rise to many difficult problems related to the physicalinterpretation of the one-particle quantum modes that may indicate how toquantize the fields. This is because the form and the properties of the particularsolutions of the free field equations [5–9] are strongly dependent on theprocedure of separation of variables and, implicitly, on the choice of the localchart (natural frame). Moreover, when the fields have spin the situation is morecomplicated since then the field equations and, therefore, the form of theirparticular solutions depend, in addition, on the tetrad gauge in which one works[10, 1]. In these conditions it would be helpful to use the traditional method ofthe quantum theory in flat spacetime based on the complete sets of commutingoperators that determine the quantum modes as common eigenstates and givephysical meaning to the constants of the separation of variables which are justthe eigenvalues of these operators.

A good step in this direction could be to proceed like in special relativitylooking for the generators of the geometric symmetries similar to the familiar

Rom. Journ. Phys., Vol. 52, Nos. 8–10 , P. 895–940, Bucharest, 2007

896 Ion I. Cotãescu 2

momentum, angular momentum and spin operators of the Poincaré covariantfield theories [11]. However, the relativistic covariance in the sense of generalrelativity is too general to play the same role as the Lorentz or Poincarécovariance in special relativity. In its turn the tetrad gauge invariance of thetheories with spin represents another kind of general symmetry that is not able toproduce itself conserved quantities [1]. Therefore, one must focus only on theisometry transformations that point out the specific spacetime symmetry relatedto the presence of Killing vectors [1, 3, 12].

Another important problem is how to define the generators of theserepresentations for any spin when some spin parts could appear. In the case ofthe Dirac field these spin parts were calculated not only in some particular cases[13] but even in the general case of any generator corresponding to a Killingvector in any chart and arbitrary tetrad gauge fixing [14]. Starting with thisimportant result, we have generalized this theory for any spin, formulating thetheory of external symmetry for any curved manifold [15].

Our approach is a general theory of tetrad gauge invariant fields defined oncurved spacetimes with given external symmetries. This predicts how musttransform these fields under isometries in order to leave invariant the form of thefield equations and to obtain the general form of the generators of thesetransformations. The basic idea is that the isometries transformations mustpreserve the position of the local frames with respect to the natural one. Suchtransformations can be constructed as isometries combined with suitable tetradgauge transformations necessary for keeping unchanged the tetrad fieldcomponents. In this way we obtain the external symmetry group showing that itis locally isomorphic with the isometry group.

Moreover, we define the operator-valued representations of the externalsymmetry group carried by spaces of fields with spin. We point out that these areinduced by the linear finite-dimensional representations of the (2 )SL group.This is the motive why the symmetry transformations which leave invariant thefield equations have generators with a composite structure. These have the usualorbital terms of the scalar representation and, in addition, specific spin termswhich depend on the choice of the tetrad gauge even in the case of the fieldswith integer spin. In general, the spin and orbital terms do not commute to eachother apart from some special gauge fixings where the fields transformmanifestly covariant under external symmetry transformations.

As examples we study the central symmetry and the maximal symmetry ofthe de Sitter (dS) and anti-de Sitter (AdS) spacetimes. In the central charts wedefine a suitable version of Cartesian tetrad gauge which allowed us recently tofind new analytical solutions of the Dirac equation [16, 17]. We show that in thisgauge fixing the central symmetry becomes global and, consequently, the spinparts of its generators are the same as those of special relativity [18, 19]. For thedS and AdS spacetimes we calculate the generators of any representation of theexternal symmetry group in central charts with our Cartesian gauge.

3 Dirac fermions in de Sitter and anti-de Sitter backgrounds 897

Furthermore, we show how can be used these results for finding thequantum modes of the Dirac field on AdS and dS spacetimes. First we consider(static) central charts where the Dirac equation can be analytically solved. OnAdS spacetime we obtain only quantum discrete modes of given energy describedby fundamental solutions that can be correctly normalized. Unfortunately, in thecase of the dS spacetime we find fundamental solutions corresponding to acontinuous energy spectrum that can not be normalized in the generalized sense.In these circumstances we are able to quantize the Dirac field in central chartsonly in the AdS background. However, quantization of the Dirac field on dSspacetime can be solved in moving charts where we can identify the componentsof the momentum operator and normalize the fundamental solutions using themomentum representation [20]. Obviously, to this end our theory of externalsymmetry is crucial.

We start presenting in the second section the basic ideas of the relativisticand gauge covariance which will be embedded in our theory of externalsymmetry in the next section. The fourth section is devoted to the charts withcentral symmetry while the dS and AdS symmetries are discussed in section 5.The main features of the Dirac theory in curved manifolds are reviewed in thenext section. Section 7 is devoted to the quantum modes of the Dirac field incentral charts of AdS and dS spacetime. We show that the second quantization ofthis field can be performed in canonical manner only in AdS case. Forquantizing the Dirac field in dS spacetimes we must choose moving charts wherewe have a suitable momentum representation. The theory of the Dirac field inthese charts and the quantization procedure are given in section 8. Usefulformulas are given in Appendix.

We work in natural units with 1.c

2. RELATIVISTIC COVARIANCE

In the Lagrangian field theory in curved spacetimes the relativisticcovariant equations of scalar, vector or tensor fields arise from actions that areinvariant under general coordinate transformations. Moreover, when the fieldshave spin in the sense of the (2 )SL symmetry then the action must beinvariant under tetrad gauge transformations [10]. The first step to our approachis to embed both these kind of transformations into new ones, called combinedtransformations, that will help us to understand the relativistic covariance in itsmost general terms.

2.1. GAUGE TRANSFORMATIONS

Let us consider the curved spacetime M and a local chart (natural frame) ofcoordinates x , = 0, 1, 2, 3. Given a gauge, we denote by ˆ ( )e x the tetrad


fields that define the local (unholonomic) frames, in each point x, and by ˆ( )ˆ xethose defining the corresponding coframes. These fields have the usualorthonormalization properties

ˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆˆˆ ˆ ˆˆ ˆ ˆ ˆe e e e e e e e (1)

where = diag(1, –1, –1, –1,) is the Minkowski metric. From the line element

ˆˆ2ˆˆ ˆ ˆ ( )ds dx dx g x dx dx (2)

expressed in terms of 1-forms, ˆ ˆˆ ˆ ,dx e dx we get the components of themetric tensor of the natural frame,

ˆ ˆˆ ˆˆ ˆ ˆˆˆ ˆg e g e ee (3)

These raise or lower the natural vector indices, i.e., the Greek ones ranging from0 to 3, while for the local vector indices, denoted by hat Greeks and having thesame range, we must use the Minkowski metric. The local derivatives ˆ ˆ

ˆ e

satisfy the commutation rules

ˆˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ, , ˆ

ˆ ˆ ˆ ˆˆ ˆ[ ] ( )e e e e C (4)

defining the Cartan coefficients which help us to write the connection componentsin local frames as

ˆˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆˆ , ˆ ˆ ˆˆ ˆ ˆ

1ˆ ˆ ˆ( ) ( )2

e e e e C C C (5)

We specify that this connection is often called spin connection (and denoted byˆ

ˆˆ ) in order to do not be confused with the Christoffel symbols, ,

involved in the well-known formulas of the usual covariant derivatives .

The Minkowski metric ˆˆ remains invariant under the transformations of

the gauge group of this metric, G( ) = O(3, 1). This has as subgroup the Lorentzgroup, ,L of the transformations [A( )] corresponding to the transformations

( ) (2 )A SL through the canonical homomorphism [11]. In the standard

covariant parametrization, with the real parameters ˆ ˆˆ ˆ , we have

ˆˆˆˆ2( )

i SA e (6)

where ˆˆS are the covariant basis-generators of the (2 )SL Lie algebra which

satisfy


ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ( )S S i S S S S (7)

For small values of ˆˆ the matrix elements of the transformations can bewritten as

ˆ ˆˆˆ ˆ ˆ[ ( )]A (8)

Now we assume that M is orientable and time-orientable such that L canbe considered as the gauge group of the Minkowski metric [3]. Then the fieldswith spin can be defined as in the case of the flat spacetime, with the help of thefinite-dimensional linear representations, , of the (2 )SL group [11]. Ingeneral, the fields M V are defined over M with values in the vector

spaces V of the representations . In the following we systematically use thebases of V labeled only by spinor or vector local indices defined with respect tothe axes of the local frames given by the tetrad fields. These will not be writtenexplicitly except the cases when this is requested by the concrete calculationneeds.

The relativistic covariant field equations are derived from actions [10, 1],

4ˆ[ ] ( ) det( )e d x g D g g (9)

depending on the matter fields, , and the components of the tetrad fields, e,which represent the gravitational degrees of freedom. Recently it was shown thatthis action can be completed adding a term with the integration measure d4x(instead of 4d x g ) where can be expressed in terms of scalar fieldsindependent on e [21]. This new term allows one to define a new global scalesymmetry which, in our opinion, is compatible with the geometric symmetrieswe study here. Therefore, without to lose generality, we can restrict ourselves toactions of the traditional form (9) in which the canonical variables are thecomponents of the fields and e.

The covariant derivatives,

ˆ ˆˆˆ ˆ ˆ ˆˆ

ˆ ˆˆ ( )2iD e D S (10)

assure the invariance of the whole theory under the tetrad gauge transformations,

ˆˆˆ ˆˆˆ ˆ ˆ( ) ( ) [ ( )] ( )e x e x A x e x

ˆˆ ˆ ˆ ˆ( ) ( ) [ ( )] ( )e x e x A x e x (11)


( ) ( ) [ ( )] ( )x x A x x

determined by the mappings (2 )A M SL the values of which are the local(2 )SL transformations ( ) [ ( )].A x A x These mappings can be organized as

a group, , with respect to the multiplication defined as ( )( )A A x( ) ( ).A x A x The notation Id stands for the mapping identity, ( ) 1Id x

(2 ),SL while A–1 is the inverse of A, 1 1( )( ) [ ( )] .A x A x

2.2. COMBINED TRANSFORMATIONS

The general coordinate transformations are automorphisms of M which, inthe passive mode, can be seen as changes of the local charts corresponding to thesame domain of M [3, 12]. If x and x are the coordinates of a point in twodifferent charts then there is a mapping between these charts giving thecoordinate transformation, ( ).x x x These transformations form a groupwith respect to the composition of mappings, , defined as usual, i.e. ( )( )x

[ ( )].x We denote this group by , its identity map by id and the inversemapping of by –1.

The automorphisms change all the components carrying natural indicesincluding those of the tetrad fields [1] changing thus the positions of the localframes with respect to the natural ones. If we assume that the physicalexperiment makes reference to the axes of the local frame then it could appearsituations when several correction of the positions of these frames should beneeded before (or after) a general coordinate transformation. Obviously, thesehave to be done with the help of suitable gauge transformation associated to theautomorphisms. Thus we arrive to the necessity of introducing the combinedtransformations denoted by (A, ) and defined as gauge transformations, givenby ,A followed by automorphisms, . In this new notation the puregauge transformations will appear as (A, id) while the automorphisms will bedenoted from now by (Id, ).

The effect of a combined transformation (A, ) upon our basic fields, , eand ê is ( )x x x ( ) ( )e x e x ˆ ˆ( ) ( )e x e x and ( ) ( )x x

[ ( )] ( )A x x where e are the transformed tetrads of the components

ˆˆ ˆ ˆ

( )[ ( )] [ ( )] ( )

xe x A x e x

x(12)

while the components of ê have to be calculated according to Eqs. (1). Thus wehave written down the most general transformation laws that leave the action


invariant in the sense that [ ] [ ].e e The field equations derived from

, written in local frames as ( )( ) 0,E x covariantly transform according to

the rule

( )( ) ( )( ) [ ( )]( )( )E x E x A x E x (13)

since the operators E involve covariant derivatives [1].The association among the transformations of the groups and must

lead to a new group with a specific multiplication. In order to find how looks thisnew operation it is convenient to use the composition among the mappings A and (taken only in this order) giving new mappings, ,A defined as ( )( )A x

[ ( )].A x The calculation rules ,Id Id A id A and ( )A A

( ) ( )A A are obvious. With these ingredients we define the newmultiplication

( ) ( ) ( )A A A A (14)

It is clear that now the identity is (Id, id) while the inverse of a pair (A, ) reads

1 1 1 1( ) ( )A A (15)

First of all we observe that the operation * is well-defined and represents thecomposition among the combined transformations since these can be expressed,according to their definition, as ( ) ( ) ( ).A Id A id Furthermore, we canconvince ourselves that if we perform successively two arbitrary combinedtransformations, (A, ) and ( ),A then the resulting transformation is just( ) ( )A A as given by Eq. (14). This means that the combinedtransformations form a group with respect to the multiplication *. It is not

difficult to verify that this group, denoted by , is the semidirect product

where is the invariant subgroup while is an usual one.In the theories involving only vector and tensor fields we do not need to

use the combined transformations defined above since the theory is independenton the positions of the local frames. This can be easily shown even in ourapproach where we use field components with local indices. Indeed, if weperform a combined transformation (A, ) then any tensor field of rank (p, q),

1 2 1 21 1

1 1 21 2 1

ˆˆˆ ˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆ

p p q p

p qq q

e e e e (16)

transforms according to the representation

S


1 21 2 11

11 2 1 2 1

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆˆ ˆ ˆ, , , , ,

( ) ( ) ( )pqp q

A A A (17)

such that the resulting transformation law of the components carrying naturalindices,

1 11 1

1 11 1( ) ( )x xx x

x x(18)

is just the familiar one [1]. In other words, in this case the effect of the combinedtransformations reduces to that of their automorphisms. However, when the halfinteger spin fields are involved this is no more true and we must use the

combined transformations of if we want to keep under control the positions ofthe local frames.

3. EXTERNAL SYMMETRY

In general, the symmetry of any manifold M is given by its isometry groupwhose transformations leave invariant the metric tensor in any chart. The scalarfield transforms under isometries according to the standard scalar representationgenerated by the orbital generators related to the Killing vectors of M [1, 3, 12].In the following we present the generalization of this theory of symmetry tofields with spin, for which we have defined the external symmetry group and itsrepresentations [15].

3.1. ISOMETRIES

There are conjectures when several coordinate transformations,( ),x x x depend on N independent real parameters, a (a, b, c… =

= 1, 2, …, N), such that = 0 corresponds to the identity map, 0 .id The set

of these mappings is a Lie group [22], ,G if they accomplish thecomposition rule

( )f (19)

where the functions f G G G define the group multiplication. These must

satisfy (0 ) ( 0)a a af f and 1 1( ) ( ) 0a af f where –1 are the

parameters of the inverse mapping of , 1 1. Moreover, the structure

constants of G can be calculated as [23]


0

( ) ( )c cabc b aa b

f fc (20)

For small values of the group parameters the infinitesimal transformations,( ) ,a ax x x k x are given by the vectors ka whose components,

0a a

k (21)

satisfy the identities

0a a abc cb bk k k k c k (22)

resulting from Eqs. (19) and (20).In the following we restrict ourselves to consider only the isometry

transformations, ( ),x x which leave invariant the components of the metric

tensor [1, 12], i.e.

( ) ( )x xg x g xx x

(23)

These form the isometry group ( )G I M which is the Lie group giving thesymmetry of the spacetime M. We consider that this has N independentparameters and, therefore, ka, a = 1, 2, … N, are independent Killing vectors(which satisfy 0).a ak k Then their corresponding Lie derivatives form

a basis of the Lie algebra i(M) of the group I(M) [12].However, in practice we are interested to find the operators of the

relativistic quantum theory related to these geometric objects which describe thesymmetry of the background. For this reason we focus upon the operator-valuedrepresentations [24] of the group I(M) and its algebra. The scalar field

: M transforms under isometries as ( ) [ ( )] ( ).x x x This rule

defines the representation T of the group I(M) whose operators have the

action 1.T Hereby it results that the operators of infinitesimal

transformations, 1 ,a aT i L depend on the basis-generators,

1 2a aL ik a N (24)

which are completely determined by the Killing vectors. From Eq. (22) we seethat they obey the commutation rules

[ ]a b abc cL L ic L (25)


given by the structure constants of I(M). In other words they form a basis of theoperator-valued representation of the Lie algebra i(M) in a carrier space of scalarfields. Notice that in the usual quantum mechanics the operators similar to thegenerators La are called often orbital generators.

3.2. THE GROUP OF EXTERNAL SYMMETRY

Now the problem is how may transform under isometries the wholegeometric framework of the theories with spin where we explicitly use the localframes. Since the isometry is a general coordinate transformation it changes therelative positions of the local and natural frames. This fact may be animpediment when one intends to study the symmetries of the theories with spininduced by those of the background. For this reason it is natural to suppose thatthe good symmetry transformations we need are combined transformations inwhich the isometries are preceded by appropriate gauge transformations suchthat not only the form of the metric tensor should be conserved but the form ofthe tetrad field components too.

Thus we arrive at the main point of our theory. We introduce the externalsymmetry transformations, (A , ), as combined transformations involvingisometries and corresponding gauge transformations necessary to preserve thegauge. We assume that in a fixed gauge, given by the tetrad fields e and ˆ,e A isdefined by

ˆ ˆˆ ˆ

( )ˆ[ ( )] [ ( )] ( )

xA x e x e x

x(26)

with the supplementary condition 0 ( ) 1 (2 ).A x SL Since is an isometry

Eq. (23) guarantees that [ ( )]A x L and, implicitly, ( ) (2 ).A x SL Then

the transformation laws of our fields are

( )

( ) ( ) [ ( )]( )

ˆ ˆ ˆ( ) ( ) [ ( )]

( ) ( ) [ ( )] ( )

x x x

e x e x e xA

e x e x e x

x x A x x

(27)

The mean virtue of these transformations is that they leave invariant the form ofthe operators of the field equations, E , in local frames. This is because thecomponents of the tetrad fields and, consequently, the covariant derivatives in

local frames, ˆD , do not change their form.


For small a the covariant (2 )SL parameters of ( ) [ ( )]A x A x can

be written as ˆ ˆˆ ˆ( ) ( )a ax x where, according to Eqs. (6), (8) and (26),

we have

ˆˆˆ ˆ ˆˆ ˆ ˆ

, ˆ0

ˆ ˆa a aa

e k e k e (28)

We must specify that these functions are antisymmetric if and only if ka areKilling vectors. This indicates that the association among isometries and thegauge transformations defined by Eq. (26) is correct.

It remains to show that the transformations (A , ) form a Lie grouprelated to I(M). Starting with Eq. (26) we find that

( )( ) fA A A (29)

and, according to Eqs. (14) and (19), we obtain

( ) ( )( ) ( ) ( )f fA A A (30)

and 0 0( ) ( ).A Id id Thus we have shown that the pairs (A , ) form a

Lie group with respect to the operation *. We say that this is the external

symmetry group of M and we denote it by ( ) .S M From Eq. (30) weunderstand that S(M) is locally isomorphic with I(M) and, therefore, the Liealgebra of S(M), denoted by s(M), is isomorphic with i(M) having the samestructure constants. In our opinion, S(M) must be isomorphic with the universalcovering group of I(M) since it has anyway the topology induced by (2 )SLwhich is simply connected. In general, the number of group parameters of I (M)or S(M) (which is equal to the number of the independent Killing vectors of M)can be 0 N 10.

The form of the external symmetry transformations is strongly dependenton the choice of the local chart as well as that of the tetrad gauge. If we changesimultaneously the gauge and the coordinates with the help of a combinedtransformation (A, ) then each ( ) ( )A S M transforms as

1( ) ( ) ( ) ( ) ( )A A A A A (31)

which means that

1 1A A A A (32)


1 (33)

Obviously, these transformations define automorphisms of S(M).

3.3. REPRESENTATIONS

The last of Eqs. (27) which gives the transformation law of the field

defines the operator-valued representation ( )A T of the group S(M),

( )[ ( )] [ ( )] ( )T x A x x (34)

The mentioned invariance under these transformations of the operators of thefield equations in local frames reads

1( )T E T E (35)

Since ( ) (2 )A x SL we say that this representation is induced by the

representation of (2 )SL [24, 25]. As we have shown in Sec. 2.2, if is avector or tensor representation (having only integer spin components) then theeffect of the transformation (34) upon the components carrying natural indices isdue only to . However, for the representations with half integer spin thepresence of A is crucial since there are no natural indices. In addition, thisallows us to define the generators of the representations (34) for any spin.

The basis-generators of the representations of the Lie algebra s(M) are theoperators

0a a aa

TX i L S (36)

which appear as sums among the orbital generators defined by Eq. (24) and thespin terms which have the action

( )( ) [ ( )] ( )a aS x S x x (37)

This is determined by the form of the local (2 )SL generators,

ˆˆˆˆ

0

( ) 1( ) ( )2a aa

A xS x i x S (38)

that depend on the functions (28). Furthermore, if we derive Eq. (29) withrespect to and then from Eqs. (8), (20) and (28), after a few manipulations,we obtain the identities


ˆ ˆˆ ˆ ˆˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆˆˆ 0a a a a abc cb b b bk k c (39)

Hereby it results that

[ ] [ ] [ ]a a b a abc cb bS S L S L S ic S (40)

and, according to Eq. (25), we find the expected commutation rules

[ ]a abc cbX X ic X (41)

Thus we have derived the basis-generators of the operator-valued representationof s(M) induced by the linear representation of (2 ).SL All the operators ofthis representation commute with the operator E since, according to Eqs. (35)and (36), we have

[ ] 0 1 2aE X a N (42)

Therefore, for defining quantum modes we can use the set of commutingoperators containing the Casimir operators of s(M), the operators of its Cartansubalgebra and E .

Finally, we must specify that the basis-generators (36) of the representationsof the s(M) algebra can be written in covariant form as

ˆˆˆ ˆ

1 ( )2a a a

X ik D k e e S (43)

generalizing thus the important result obtained in Ref. [14] for the Dirac field.

3.4. MANIFEST COVARIANCE

The action of the operators aX depends on the choice of many elements:the natural coordinates, the tetrad gauge, the group parametrization and therepresentation . What is important here is that they are strongly dependent onthe tetrad gauge fixing even in the case of the representations with integer spin.This is because the covariant parametrization of the (2 )SL algebra is definedwith respect to the axes of the local frames. In general, if we consider the

representation ( )A T and we perform the transformation (31) then it

results the equivalent representation, ( , ) .A T Its generators calculated

from Eqs. (32) indicate that in this case the equivalence relations are much morecomplicated than those of the usual theory of linear representations. Without toenter in other technical details we specify that if we change only the gauge with


the help of the transformation (A, id) then the local (2 )SL generators (38)transform as

1

ˆˆ ˆˆ ˆ ˆ

( ) ( ) ( ) ( ) ( )

( ) [ ( )] [ ( )]

a a a

a

S x S x A x S x A x

k x A x A x S(44)

while the orbital parts do not change their form. This means that the gaugetransformations change, in addition, the commutation relations among the spinand orbital parts of the generators .aX

The consequence is that we can find gauge fixings where the local(2 )SL generators Sa(x), a = 1, 2, …, n (n N), corresponding to a subgroup

( ),H S M are independent on x and, therefore, [ ] 0a bS L for all a = 1, 2, …,

n and b = 1, 2, …, N. Then the operators aS a = 1, 2, …, n are just the basis-

generators of an usual linear representation of H and the field behavesmanifestly covariant under the external symmetry transformations of thissubgroup. Of course, when H = S(M) we say simply that the field is manifestcovariant.

The simplest examples are the manifest covariant fields of specialrelativity. Since here the spacetime M is flat, the metric in Cartesian coordinates

is g = and one can use the inertial (local) frames with .ˆe e Then

the isometries are just the transformations [ ( )]x A x a of the Poincaré

group, (4)T L [11]. If we denote by ( ) the (2 )SL

parameters and by ( ) a those of the translation group T(4), then it is asimple exercise to calculate the basis-generators

( )X i (45)

( ) ( ) ( )X i x x S (46)

which show us that transforms manifestly covariant. On the other hand, it is

clear that the group ( ) (4) (2 )S M T SL is just the universal covering

group of ( ) .I MIn general, there are many cases of curved spacetimes for which one can

choose suitable local frames allowing one to introduce manifest covariant fieldswith respect to a subgroup ( )H S M or even the whole group S(M). In our

opinion, this is possible only when H or S(M) are at most subgroups of .

S

S


4. THE CENTRAL SYMMETRY

Let us take as first example the spacetimes M which have sphericallysymmetric static chart that will be called here central charts (or central frames).These manifolds have the isometry group ( ) (1) (3)I M T SO of timetranslations and space rotations.

4.1. CENTRAL CHARTS

In a central chart { },t x with Cartesian coordinates x0 = t and xi

(i, j, k … = 1, 2, 3), the metric tensor is time-independent and transformsmanifestly covariant under the rotations (3)R SO of the space coordinates,

( )i ii j i jj jt t x R x x x (47)

denoted simply by .x x Rx Here the most general form of the line element,

2 2( ) ( ) [ ( ) ( ) ]i j i jijds g x dx dx A r dt B r C r x x dx dx (48)

may involve three functions, A, B and C, depending only on the Euclidean normof ,x .r x In applications it is convenient to replace these functions by newones, u, v and w, defined as

2 2 22

2 2 2 21w w wA w B C

v r u v(49)

Other useful central charts are those with spherical coordinates, {t, r, , },commonly associated with the Cartesian space ones, xi. Here the line elements are

2 22 2 2 2 2 2 2 2

2 2 ( sin )w wds w dt dr r d du v

(50)

and, consequently, we have

42 1 2

2[ ( )]wg B A B r Cuv

(51)

The new functions we introduced here have simple transformation lawsunder the isotropic dilatations which change only the radial coordinate,

( ),r r r without to affect the central symmetry of the line element. Thesetransformations,

( ) ( )( ) ( ) ( ) ( ) ( ) ( )dr r r ru r u r v r v r w r w rdr r

(52)

allow one to choose desired forms for the functions u, v and w.


4.2. THE CARTESIAN GAUGE

The Cartesian gauge in central charts was mentioned long time ago [26] butit is less used in concrete problems since it leads to complicated calculations inspherical coordinates. However, in Cartesian coordinates this gauge has theadvantage of explicitly pointing out the global central symmetry of the manifold.In Refs. [16] we have proposed a version of Cartesian gauge in central chartswith Cartesian coordinates that preserve the manifest covariance under rotations(47) in the sense that the 1-forms ˆ ˆˆ ˆ ( )dx e x dx transform as

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ( ) ( )dx dx e x dx Rdx (53)

If the line element has the form (48) then a good choice of the tetrad fields withthe above property is

0 00 0

ˆˆ ˆ ˆ ˆ ˆ ˆ( ) 0 ( ) ( )i i i ji j ije a r e e e b r c r x x (54)

0 00 0( ) 0 ( ) ( )

i i i ji j ije a r e e e b r c r x x (55)

where, according to Eqs. (3), (48) and (49), we must have

21ˆˆ ˆw w wa w b c

v u vr(56)

21 1v u va b cw w w wr

(57)

When one defines the metric tensor such that 0rg then u(0)2 =

= v(0)2 = w(0)2 = 1. Moreover, it is natural to take (0) = 0. In other respects,from Eqs. (56) and (57) we see that the function w must be positively defined inorder to keep the same sense for the time axes of the natural and local frames. Inaddition, it is convenient to consider that the function u is positively definedtoo. However, the function Pv v has the sign given by the relative parity

P which takes the value P = 1 when the space axes of the local frame at x = 0are parallel with those of the natural frame, and P = –1 if these are anti-parallel.

Now we have all the elements we need to calculate the generators of therepresentations T of the group S(M). If we denote by (0) the parameter of thetime translations and by ( ) 2i jkijk the parameters of the rotations (47), we

find that the local (2 )SL generators of Eq. (38) are just the su(2) ones, i.e.

( ) ( ) 2,i i ijk jkS x S S such that the basis-generators read

( ) ( )(0) ( ) ( )t i i iiX H i X J L S (58)


where ( )j

i ijk kL i x are the usual components of the orbital angular

momentum. Thus we obtain that the group ( ) (1) (2)S M T SU is the universalcovering group of I(M). Its transformations are gauge transformations

(2),A SU independent on x, combined with the isometries of I(M) given by

( )x x R A x and (0) .t t t This means that, in this gauge, the field

transforms manifestly covariant. The generators have the usual physical

significance, namely H is the Hamiltonian operator while ( )iJ are the

components of the total angular momentum operator. Moreover, the totalangular momentum is conserved in the sense that ( )[ ] 0.iH J

Concluding we can say that, in our Cartesian gauge, the local frames playthe same role as the usual Cartesian rest frames of the central sources in flatspacetime since their axes are just those of projections of the angular momenta.

4.3. THE DIAGONAL GAUGE

In other gauge fixings the basis-generators are quite different. A tetradgauge largely used in central charts with spherical coordinates is the diagonalgauge defined by the 1-forms [13]

0 1 2 3ˆ ˆ ˆ ˆ sins s s sw w wdx wdt dx dr dx r d dx r du v v

(59)

In this gauge the angular momentum operators of the canonical basis (where

( ) (1) (2) )J J iJ are [13]

( ) 23 (3)( ) (3)( )sinieJ L S J L (60)

Thus one obtains a representation of SU(2) where the spin terms do not commutewith the orbital ones and, therefore, the field does not transform manifestlycovariant under rotations. In this case we can say that the spin part of the centralsymmetry remains partially hidden because of the diagonal gauge whichdetermines special positions of the local frames with respect to the natural one.However, when this is an impediment one can change at anytime this gauge intothe Cartesian one using a simple local rotation. For the flat spacetimes thesetransformations and their effects upon the Dirac equation are studied inRef. [27]. We note that the form of the spin generators as well as that of thementioned rotation depend on the enumeration of the 1-forms (59).


5. THE dS AND AdS SYMMETRIES

The backgrounds with highest external symmetry are the hyperbolicmanifolds, namely the dS and the AdS spacetimes. These are exact solutions ofthe vacuum Einstein equations with cosmological constant c. We shall brieflydiscuss simultaneously both these manifolds which will be denoted by Mwhere 1 for dS case and 1 for AdS one. Our goal here is to calculate thegenerators of the representations of the group ( )S M induced by those of

(2 ).SLThe dS and AdS spacetimes are hyperboloids in the (4 + 1) or (3 + 2)-

dimensional flat spacetimes, 5 ,M of coordinates ZA, A, B, … = 0, 1, 2, 3, 5, and

the metric 5( ) diag(1 1 1 1 ). The equation of the hyperboloid of

radius ˆ1 3 cR reads

2( ) A BAB Z Z R (61)

From their definitions it results that the dS or AdS spacetimes are homogeneousspaces of the pseudo-orthogonal groups SO(4, 1) or SO(3, 2) which play the roleof gauge groups of the metric ( ) (for 1 and 1 respectively) and

represent just the isometry groups of these manifolds, [ ( )] ( ).G I M Then it

is natural to use the covariant real parameters AB = – BA since in thisparametrization the orbital basis-generators of the representations of [ ( )],G

carried by the spaces of functions over 5 ,M have the usual form

5 ( ) ( )C CAB AC B BC AL i Z Z (62)

They will give us directly the orbital basis-generators of the representations of( )S M in the carrier spaces of the functions defined over dS or AdS spacetimes.

5.1. CENTRAL CHARTS OF AdS AND dS SPACETIME

The hyperboloid equation can be solved in Cartesian dS/AdS coordinates,x0 = t and xi (i = 1, 2, 3), which satisfy

5 1 0 1ˆ ˆcosh if 1 sinh if 1

ˆ ˆ( ) ( )ˆ ˆcos if 1 sin if 1

i i

t tZ r Z r

t t

Z x

(63)

where we have denoted 2 2ˆ( ) 1 .r r The line elements


2

22 2 2 2 2 2

2

( )

( ) ( sin )( )

A BABds dZ dZ

drr dt r d dr

(64)

are defined on the radial domains ˆ[0 1 )rD or [0 )rD for dS or AdSrespectively.

We calculate the Killing vectors and the orbital generators of the externalsymmetry in the Cartesian coordinates defined by Eq. (63) and the mentionedparametrization of ( )I M starting with the identification ( ) .AB AB Then,from Eqs. (24) and (62), after a little calculation, we obtain the orbital basis-generators

(05) ˆ tiL (65)

( 5)

ˆ ˆcosh sinh( )ˆ ( )ˆ ˆcos sin

jj j t

t ti ixL rrt t

(66)

(0 )

ˆ ˆsinh cosh( )ˆ ( )ˆ ˆsin cos

jj j t

t ti ixL rrt t

(67)

( ) ( )i j

ij j iL i x x (68)

Furthermore, we consider the Cartesian tetrad gauge defined by Eqs. (54)–(57)where, according to Eq. (64), we have

2( ) ( ) ( ) ( ) ( )u r r v r w r r (69)

In addition we take = 0. In this gauge we obtain the following local (2 )SLgenerators

(05) ( ) 0S x (70)

( 5) 0 2

0

ˆ ˆsinh cosh1( ) [ ( ) 1] ˆˆ ˆsin cos

ˆsinh( ) ˆsin

kjk

j j

k jk

S xt tS x S r

t tr

tS x xr t

(71)

(0 ) 0 2

0

ˆ ˆcosh sinh1( ) [ ( ) 1] ˆˆ ˆcos sin

ˆcosh( ) ˆcos

kjk

j j

k jk

S xt tS x S r

t tr

tS x xr t

(72)


( ) ( )ij ijS x S (73)

With their help we can write the action of the spin terms (37) and, implicitly, thatof the basis-generators ( )( ) ( )ABAB ABX L S of the representations of ( )S M

induced by the representations of (2 ).SL Hereby it is not difficult to showthat ( )S M is isomorphic with the universal covering group of ( )I M which inboth cases ( 1) is a subgroup of the SU(2, 2) group. As expected, in centralcharts and Cartesian gauge the fields transform manifestly covariant only underthe transformations of the subgroup (2) ( ).SU S M

6. THE DIRAC FIELD IN CURVED BACKGROUNDS

In what follows we study the Dirac particles freely moving on dS or AdSbackgrounds without to affect the geometry. The problems of this type are calledoften Dirac perturbations on a given manifold.

The main purpose is to find the quantum modes of the Dirac fielddetermined by complete sets of commuting operators. The Noether theoremapplied to our theory of external symmetry give us classical conserved quantitiesor conserved operators in quantum theory, corresponding to the generators of thegroup S(M). Thus, the quantum theory is equipped with a large algebra ofconserved operators among them we can select the systems of commutingoperators we need for defining the quantum modes which will help us to performthe second quantization of the Dirac field in canonical manner.

6.1. THE DIRAC FIELD

The form of the Dirac field in curves backgrounds depends on two basicelements: the choice of the tetrad gauge and the representation of the Dirac’s-matrices. These satisfy

ˆ ˆˆ ˆ2 (74)

and give the generators of the reducible spinor representation of the (2 )SL C

group [19] we denote from now directly by ˆˆS instead of ˆˆ( ).S Thesegenerators,

ˆ ˆˆ ˆ4iS (75)

are self-adjoint, ˆ ˆˆ ˆ ,S S (with respect to the Dirac adjoint defined as0 0 )X X and satisfy


ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆ( )S i (76)

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ( )S S i S S S S (77)

In the chiral representation of the spinor field the crucial role is played by thematrix 5 0 1 2 3i which is diagonal in this representation. We note that the

set of matrices ˆˆ ˆ ˆ 5 5{ }S form the basis of a fundamentalrepresentation of the su(2, 2) algebra1.

The theory of the Dirac field of mass m, defined on the space domain D,starts with the gauge invariant action [4],

ˆ ˆ4ˆ ˆ[ ] ( )2

id x g D D m (78)

where 0 is the Dirac adjoint of and

ˆ ˆˆ ˆ ˆ ˆˆ

ˆ ˆ2iD S (79)

are the covariant derivatives of the spinor field given by Eq. (10). From theaction (78) it results the Dirac equation

ˆˆ0D DE m E i D (80)

After a few manipulations the Dirac operator ED can be put in a morecomprehensive form as

ˆ ˆˆ ˆ ˆˆ ˆ ˆ ˆˆ

1 1 ˆ2 4DiE i e ge S

g(81)

On the other hand, starting with the conservation of the electric charge, onecan define the time-independent relativistic scalar product of two spinors is [4].When 0 0,ie i = 1, 2, 3, this takes the simple form

3 0( ) ( ) ( ) ( )D

d x x x x (82)

where00( ) ( ) ( )x g x e x (83)

is the specific weight function of the Dirac field. In the central chart with themetric (50) this weight function reads

1 For extended bibliography see Ref. [28].


3

2 2 21

( )w

b b r c uv(84)

since g is given by Eq. (51).Our theory of external symmetry offers us the framework we need to

calculate the conserved quantities predicted by the Noether theorem. Startingwith the infinitesimal transformations of the one-parameter subgroup of S(M)generated by Xa corresponding to the Killing vector ka, we find that there exists

the conserved current [ ]aX which satisfies [ ] 0.aX Using the action

(78) we obtain the concrete form of these currents,

ˆˆ ˆˆˆ ˆ

1[ ]4a a a

X T k S e (85)

where the functions ˆ ˆa are defined by Eq. (28) while

ˆ ˆˆ ˆ( )2

iT e e (86)

is a notation for a part of the stress-energy tensor of the Dirac field [1, 4].Finally, it is clear that the conserved quantity corresponding to the Killing vectorka is the real number

3 0 1[ ]2a a aD

d x g X X X (87)

We note that it is premature to interpret this formula as an expectation value orto speak about Hermitian conjugation of the operators Xa with respect to thescalar product (82), before specifying the boundary conditions on D.

What is important here is that this result is useful in quantization givingdirectly the one-particle operators of the quantum field theory when becomes afield operator. Indeed, starting with the form (87) of the conserved quantities, wefind that for any self-adjoint generator X of the spinor representation of the groupS(M) there exists a conserved one-particle operator of the quantum field theorywhich can be calculated simply as

XX (88)

respecting the normal ordering ( ) of the operator products [29]. Thequantization rules must be postulated such that the standard algebraic properties

[ ( )] ( ) [ ] [ ]x X x X XX X X (89)

should be accomplished.


6.2. THE REDUCED DIRAC EQUATION IN CENTRAL CHARTS

In what follows we consider manifolds M with central symmetry havingeither charts with Cartesian coordinates { }t x and the line element (48) or charts

with spherical coordinates {t, r, , } where we prefer to work with metrics ofthe form (50). In these charts we chose the Cartesian gauge defined by Eqs. (54),(55), (56) and (57) which bring the rotations in manifest covariant form.

The Dirac equation with central symmetry can be written replacing theconcrete form of the tetrad components in Eq. (81). First, we observe that the lastterm of this equation does not contribute when the metric is spherically

symmetric. The argument is that ˆˆ ˆˆ ˆˆ ˆ 5ˆ{ }S (with 0123 = 1) is

completely antisymmetric, while the Cartan coefficients resulted from (54) and(55) have no such type of components. Furthermore, we bring the remainingequation in a simpler form defining the reduced Dirac field, , as

( ) ( ) ( )x r x (90)where

1 22 3 2( )g b r c b a vw (91)

In this way we obtain the reduced Dirac equation in the central chart ( )t x withCartesian coordinates and Cartesian tetrad gauge,

0( ) ( )( ) ( )( ) 1 ( ) ( ) ( )ti a r b r c r x x x m x (92)

We observe that this equation is invariant under time translations whichmeans that the energy is conserved. For this reason it is convenient to bring it inHamiltonian form,

D tH i (93)using the operators

0 02 2

( ) ( )( ) 1 ( ) ( )i i i i iD iu r v r

H i x x i x K w r mr r

(94)

0 2 1K S L (95)

The operator K which concentrates all the angular terms of DH as well as the

total angular momentum J L S defined by Eq. (58) commute with DH and

satisfy 2 2 1 .4

K J Consequently, all the properties related to the conservation

of the angular momentum, including the separation of variables in spherical


coordinates, will be similar to those of the usual Dirac theory in Minkowskispacetimes. Thus in Eq. (93) one can separate the spherical variables with thehelp of the four-components angular spinors

j jm used in the central problems

of special relativity [19]. These are eigenspinors of the complete set 2 3{ }J J K

corresponding to the eigenvalues { ( 1) }j jj j m where j can take only the

values ( 1 2).j We note that for each set of quantum numbers ( )j jj m

there are two orthogonal spinors, denoted by the superscripts , as defined inRef. [19].

We must specify that in curved manifolds this conjecture can be obtainedonly in our Cartesian tetrad gauge where the representations of the group SU(2)are manifest covariant since [ ] 0.i jL S In other gauge fixings where Li and

Si(x) do not commute among themselves the form of Eqs. (94) and (95) may be

different [13, 7] so that the spinors j jm

become useless. For example, in the

diagonal gauge defined by Eq. (59) one must consider another type of angularspinors, constructed with the help of the so called spin-weighted harmonics [30].

In other respects, we can verify that in our gauge the discretetransformations, P, C and T, have in this gauge the same significance and actionas those of special relativity [18, 19]. This is because the form of theHamiltonian operator (94) is close to that meet in the flat case. We shall see laterthat the charge conjugation transforms each particular solution of positivefrequency of Eq. (92) into the corresponding one of negative frequency.

6.3. THE RADIAL PROBLEM

In the central chart {t, r, , } where the line element can be put in the form(50) we consider the particular solutions of the original Dirac equation (80)having positive frequency and the energy E,

( ) ( )3 2

( ) ( )

( ) ( ) ( ) ( ) ( )( )

j j j j

j jj j j j

E j m E j m

iEtmE m E

U x U t r

v rf r f r e

rw r

(96)

The reduced part of these solutions, ,U U satisfy Eq. (93) from which, afterthe separation of angular variables, we obtain the radial equations

( ) ( )( ) ( ) ( ) [ ( ) ] ( )jdu r v r f r E w r m f rdr r

(97)


( ) ( )( ) ( ) ( ) [ ( ) ] ( )jdu r v r f r E w r m f rdr r

(98)

where we omitted the indices E and j of the radial functions. In practice, theseradial equations can be written directly starting with the line element put in theform Eq. (50) from which we can identify the functions u, v, and w we need.

The angular spinors are normalized such that the angular integral of thescalar product (82) does not contribute and, consequently, this reduces to theradial integral. By using Eqs. (90) and (96) we find that this is

( ) ( ) ( ) ( )1 2 1 21 2 ( ) ( ) ( ) ( )( )rD

dr f r f r f r f ru r

(99)

where Dr is the radial domain corresponding to D. What is remarkable here is

that the radial weight function 2 1 ,u resulted from Eqs. (128) and (91), is

just that we need in order to have ( ) .r ru u This means that the operatorsof the left-hand side of the radial equations are related between themselvesthrough the Hermitian conjugation with respect to the scalar product (99).

Hereby it results that the operator

j

r

j

d vmw udr rH

d vu mwdr r

(100)

is self-adjoint. This is just the radial Hamiltonian, which allows one to write theEqs. (97) and (98) as the eigenvalue problem

rH E (101)

where the two-dimensional eigenvectors ( ) ( )( )Tf f have their own scalarproduct,

1 2 1 2( ) ( )rDdr

u r(102)

as it results from (99). Thus we have obtained an independent radial problemwhich has to be solved in each particular case separately using appropriatemethods.

First of all, we look for possible transformations which should simplify theradial equations. It is known that the transformations of the space coordinates ofa natural frame with static metric do not change the quantum modes. In our casewe can change only the radial coordinate without to affect the central symmetry.A good choice is the chart where the radial coordinate is defined by


( ) .( )sdrr r const

u r(103)

so that (0) 0.sr This chart will be called special central chart (frame). In thefollowing we shall use only this frame starting with metrics with u = 1 while thesubscript s will be omitted.

In radial problems of the Dirac theory in flat spacetime [19] the manifestsupersymmetry play an important role in solving the radial equations. For thisreason we look for transformations of the form ˆ U and

1ˆr r rH H UH U which could simplify the radial problem pointing out a

hidden supersymmetry. This means that the transformed Hamiltonian has thesupersymmetric form

ˆr

d WdrH

d Wdr

(104)

where must be a constant and W is the resulting superpotential [16]. If the radialproblem has this property, then the second order equations for the components

( )f̂ and ( )f̂ of ˆ can be obtained from 2 2ˆ ˆˆ rH E . These equations,

22 2 ( ) 2 ( )

2( ) ˆ ˆ( ) ( ) ( )dW rd W r f r E f r

drdr(105)

represent the starting point for finding analytical solutions.The simplest radial problems are those with manifest supersymmetry, for

which the original radial Hamiltonian Hr has the form (138). These are generatedby the metrics of the central manifolds 3M which have w = 1. In the specialframes (where u = 1) these are determined only by the arbitrary function v whichgives the superpotential W = jv/r. The most popular example is the three-

dimensional sphere with the time trivially added, 3.S A more complicatedsituation is when we need to use a suitable transformation U in order to point outthe supersymmetry. These are problems with hidden supersymmetry which aresimilar with that of the Dirac particle in external Coulomb field, known fromspecial relativity. However, examples of radial problems in which thesupersymmetry is much more hidden will be discussed in the next section.

7. DIRAC FIELD IN CENTAL CHARTS OF THE AdS AND dS SPACETIMES

In our approach the Dirac equation in the central charts (frames) of AdS ordS backgrounds can be analytically solved in terms of Gauss hypergeometric


functions [16,17]. One obtains thus the fundamental solutions giving thequantum modes. Since the central frames are static, these solutions are energyeigenspinors corresponding to discrete or continuous energy spectra. When thesespinors can be normalized (in usual or generalized sense) then the canonicalquantization of the Dirac field in these frames may be done.

7.1. DIRAC QUANTUM MODES IN AdS SPACETIME

We consider first the AdS spacetime, M, and we present how can beselected the quantum modes of the Dirac field and to write down the normalizedenergy eigenspinors of the regular modes. The final objective is to quantize thisfield [16].

The AdS spacetime M of radius R is defined by Eq. (61) for 1.Changing the notation, we denote from now 1 R instead of ˆ . Thismanifold has a special central chart {t, r, , } with the line element [31]

2 2 2 2 2 2 2 22

1sec sin ( sin )ds r dt dr r d d (106)

The radial domain of this chart is Dr = [0, /2 ) because of the event horizon atr = /2 . We specify that here we take t (– , ) which defines in fact theuniversal covering spacetime (CAdS) of AdS [31].

Now, from Eq. (118) we can identify the functions w(r) = sec r and( ) cscv r r r . With their help and using the notation k m we obtain the

radial Hamiltonian,

sec csc

csc sec

j

r

j

dk r rdrH

d r k rdr

(107)

which has to give us the radial functions of the particular solutions (96).Despite of the fact that this is not obvious at all, this Hamiltonian has a

hidden supersymmetry that can be pointed out with the help of the local rotation

( ) ( )ˆ ˆ ˆ,T

R f f produced by

cos sin2 2( )

sin cos2 2

r r

R rr r

(108)

Indeed, after a few manipulations we find that the transformed (i.e. rotated andtranslated) Hamiltonian,


2 2ˆ 1

2T

r rH RH R (109)

has supersymmetry since it has the requested specific form with diagonalconstant terms [19] and superpotential [16] of the Pöschl-Teller type [32] .

In these circumstances the new eigenvalue problem

ˆ ˆˆ2r

H E (110)

involving the transformed radial wave functions ( )ˆ ,f leads to a pair of secondorder equations

22 2 ( ) 2 2 ( )

2 2 2

( 1)( 1) ˆ ˆ( ) ( )cos sin

j jk kd f r f rdr r r

(111)

where we denoted 1 2.E These equations have analytical solutions [16],

( ) 2 2

2

ˆ ( ) sin cos

12 sin2 2 2

s pf r N r r

F s p s p s r(112)

where F are the Gauss hypergeometric functions [33]. Their real parameters aredefined as

2 (2 1) ( 1)j js s (113)

2 (2 1) ( 1)p p k k (114)

while N are normalization factors. The next step is to select the suitable values

of these parameters and to calculate N+/N–such that the functions ( )f̂ should besolutions of the transformed radial problem (137), with a good physical meaning.This can be achieved only when F is a polynomial selected by a suitablequantization condition since otherwise F is strongly divergent for 2sin 1.r

Then the functions ( )f̂ will be square integrable with normalization factorscalculated according to the condition

2 2( ) ( )ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) 1

rDdr f r f r (115)

resulted from the fact that the matrix (135) is orthogonal.The discrete energy spectrum is given by the particle-like CAdS

quantization conditions


2( ) 0n s p (116)

that must be compatible with each other, i.e.

n s p n s p (117)

Hereby we see that there is only one independent radial quantum number,nr = 0, 1, 2, …. In addition, we shall use the orbital quantum number l of the

spinorj jm

[18], as an auxiliary quantum number. On the other hand, if we

express (141) in terms of Jacobi polynomials, we observe that these functionsremain square integrable for 2s > –1/2 and 2p > –1/2. Since l = 0, 1, 2, … weare forced to select only the positive solutions of Eqs. (113). The differentsolutions of Eqs. (114) defines the boundary conditions of the allowed quantummodes, like in the case of scalar modes [5]. We say that for k > –1/2 the values2p+ = k and 2p– = k + 1 define the boundary conditions of regular modes. Theother possible values, 2p+ = –k + 1 and 2p– = –k, define the irregular modeswhen k < 1/2. Obviously, for –1/2 < k


12

( )( 1 2) 1

2

3 1( )2 1 2 21

1ˆ ( )

sin cos (cos2 )

j

r

rj

r

l kl kn

n k lf r N

n l

r rP r

where12

( 1)2

1 12 2

r r

r r

n n k lN

n l n k(120)

is defined up to the phase factor . Notice that from Eq. (118) we understandthat the second equation of Eqs. (119) gives ( )ˆ 0f for nr = 0.

For j = j + 1/2 = l we use the same procedure finding that 2s+ = l + 1,2s– = l, n+ = n– = nr and

2 12 2 1r

N lN n k

(121)

In this case the normalized radial wave functions are

11 22( )

1 2 12

1 1( )1 2 2

ˆ ( )

sin cos (cos2 )

j

r

rj

r

l kl kn

n kf r N

n l

r rP r

(122)

11 22( )

1 2 12

1 1( )1 2 2

ˆ ( )

sin cos (cos2 )

j

r

rj

r

l kl kn

n lf r N

n k

r rP r

The energy levels result from Eq. (116). Bearing in mind that k = m and2,E and defining the principal quantum number n = 2nr + l we obtain

the discrete energy spectrum [16]

3 0 1 22n

E m n n (123)

These levels are degenerated. For a given n our auxiliary quantum number ltakes either all the odd values from 1 to n, if n is odd, or the even values from 0to n, if n is even. In both cases we have j = l 1/2 for each l, which means that


j = 1/2, 3/2, …, n + 1/2. The selection rule for j is more complicated since it isdetermined by both the quantum numbers n and j. If n is even then the even jare positive while the odd j are negative. For odd n we are in the oppositesituation, with odd positive or even negative values of j. Thus it is clear that foreach given pair (n, j) we have only one value of j. With these specifications andby taking into account that for each j we have 2j + 1 different values of mj, wecan conclude that the degree of degeneracy of the level En is (n + 1) (n + 2).

Since the solutions (119) and (122) are completely determined by thevalues of n and j, we denote by ( )n̂ jf the radial wave functions (119) and (122).

With their help we can write the functions ( )n jf (i.e. the components of )

using the inverse of the transformation (135). Then from Eq. (96) we find thedefinitive form of the normalized particle-like energy eigenspinors of the regularmodes,

3 2

( ) ( )

( ) ( )

( ) csc cos

ˆ ˆcos ( ) sin ( ) ( )2 2

ˆ ˆsin ( ) cos ( ) ( )2 2

j

j j

nj j

n j m

n j n j m

iE tmn j n j

U x r r

r rf r f r

r rf r f r e

(124)

The antiparticle-like energy eigenspinors can be derived directly by using thecharge conjugation [29]. These are

2 0( ) ( )j j j

c Tn j m n j m n j mV U C U C i (125)

Furthermore, we can verify that all these normalized energy eigenspinors havegood orthogonality properties obeying

3 0

3 0

( ) ( ) ( )

( ) ( ) ( )

j j

j j j j

n j m n j mD

n j m n j m n n j j m mD

d x x U x U x

d x x V x V x(126)

3 0

3 0

( ) ( ) ( )

( ) ( ) ( ) 0

j j

j j

n j m n j mD

n j m n j mD

d x x U x V x

d x x V x U x(127)

where3

2 32 2 2

( ) 1( ) sin sec( )

w rx r r

v r r(128)


is the specific relativistic weight function [16]. Of course, the factors and 1/ 2can be removed simultaneously from Eqs. (124) and respectively (128).

The final result is that for 2,m when only regular modes are allowed,the quantum Dirac field on CAdS reads

†( ) ( ) ( )j j j j

j

n j m n j m n j m n j mn j m

x U x a V x b (129)

This field can be quantized supposing that the particle (a, † )a and antiparticle

(b, † )b operators satisfy usual anticommutation relations from which the non-vanishing ones are

† †j j j jj j

n j m n j m n n j j m mn j m n j ma a b b 1 (130)

where 1 is the identity operator. Then the one-particle operators obtained viaNoether theorem, using Eq. (88) have correct forms and satisfy the conditions(89). After a little calculation we obtain the Hamiltonian operator

† †j jj j

j

n n j m n j mn j m n j mn j m

E a a b bH (131)

while the charge operator reads

† †j jj j

j

n j m n j mn j m n j mn j m

a a b bQ (132)

The angular operators diagonal in this basis 2 ,J J3 and K, take the same form asH but depending on the eigenvalues j(j + 1), mj and j respectively. Thus, weconclude that the complete set of commuting operators that defines the quantummodes is {H, Q, 2 ,J K, J3}. All these operators have similar structures andproperties like those of the usual quantum field theory in flat spacetime.

7.2. DIRAC QUANTUM MODES IN dS SPACETIME

The first solution of the Dirac equation on dS backgrounds was found inRef. [6] based on a diagonal gauge in central charts. Other interesting solutionsin the null tetrad gauge of these charts were derived recently in Ref. [9].However, here we restrict ourselves to present only the solutions that can beobtained in our Cartesian gauge [17].

Let us consider the problem of the massive Dirac particle on dSbackground M of radius R = 1/ defined by Eq. (61) for 1. There exists aspecial central chart {t, r, , } with the line element


2 2 2 2 2 2 22 2

1 1 sinh ( sin )cosh

ds dt dr r d dr

(133)

from which we can identify the functions v and w we need for writing down theradial Hamiltonian,

cosh sinh

sinh cosh

j

rj

k dr dr rH

d kdr r r

(134)

with the same notation, k = m/ , as in the previous case. This is the basic pieceof the radial problem which must determine the radial functions f ( ) of thesolutions (96).

This radial problem has a hidden supersymmetry like in the AdS case [17].This can be pointed out with the help of the transformation ˆ ( )U r

where ( ) ( )( ) ,Tf f ( ) ( )ˆ ˆ ˆ( )Tf f and

cosh sinh2 2( )

sinh cosh2 2

r riU r

r ri(135)

A little calculation shows us that the transformed Hamiltonian,

12 2

ˆ ( ) ( ) 12r r

H U r H U r i (136)

which gives the new eigenvalue problem

ˆ ˆˆ2r

H E i (137)

has supersymmetry since it has the requested specific form,

ˆr

d WdrH

d Wdr

(138)

Here ( )jk i is a constant and

( ) ( tanh coth )jW r ik r r (139)

is the superpotential of the radial problem [17]. The transformed radial wavefunctions ( )f̂ satisfy the second order equations resulted from the square of Eq.(137). These are


22 2 ( ) 2 2 ( )

2 2 2

( 1)( 1) ˆ ˆ( ) ( )cosh sinh

j jik ikd f r f rdr r r

(140)

where we have denoted 2.E iThe solutions of Eqs. (140) are well-known to be expressed in terms of

Gauss hypergeometric functions [33], ( ) ( ),F y F y depending on

the new variable 2sinh ,y r as

( )ˆ ( ) (1 ) ( )p sf y N y y F y (141)where

122 2 2i is p s p s (142)

N are normalization factors while the parameters p and s are related with kand j through

2 (2 1) ( 1)j js s (143)

2 (2 1) ( 1)p p ik ik (144)

Furthermore, we have to find the suitable values of these parameters suchthat the functions (141) should be solutions of the transformed radial problem(137). If we replace (141) in (137), after a few manipulation, we obtain

1 2 1 2

( )(1 ) ( ) (1 ) ( )

2 2

1 (1 ) ( )2 2

j

s s p pj

dF y iky y y p F y y s F ydy

N M Ei y y F yN

(145)

where = 1. These equations are nothing else than the usual identities ofhypergeometric functions if the values of s , p and N+/N– are correctly matched.First we observe that the differences s+ – s– and p+ – p– must be half-integersince we work with analytic functions of y. This means that the allowed groupsof solutions of (143) are

1 2

1 2

2 , 2 1

2 1 2

j j

j j

s s

s s(146)

while Eq. (144) gives us

1 2

1 2

2 , 2 1

2 1 2

p ik p ik

p ik p ik(147)


On the other hand, it is known that the hypergeometric functions of (141) areanalytical on the domain ( 0],yD corresponding to Dr, only if

( ) ( ) 0 [33]. Moreover, their factors must be regular on this domainincluding y = 0. Obviously, both these conditions are accomplished if we takes > 0. We specify that there are no restrictions upon the values of the parametersp .

We have hence all the possible combinations of parameter values givingthe solutions of the second order equations (140) which satisfy the transformedradial problem. These solutions will be denoted by (a, b), a, b = 1, 2,understanding that the corresponding parameters are ,as ,bp as given by (146)and (147), and

( ) ( ) ( ) 122 2 2

a b a b aa b a b ai is p s p s (148)

The condition s > 0 requires to chose a = 1 when j = –j – 1/2, and a = 2 if = j + 1/2. Thus, for each given set (E, j, j) we have a pair of different radial

solutions, with b = 1, 2. Therefore, it is convenient to denote the transformed

radial wave functions (141) by ( )ˆ ,jE a b

f bearing in mind that the value of a

determines that of j. The last step is to calculate the values of N+/N–. From(145) it results

(11) (1 2)(1 1) (1 2)

(1 1) (1) (1 2) (1)1 12j

N Nj

N N(149)

(2 1) (2 2)(2 1) (2 2)

(2 1) (2) (2 2) (2)1 12j

N Nj

N N(150)

Notice that here (1) (2) 1.jNow we can restore the form of the original radial wave functions of (96)

using the inverse of the transformation (135). In our new notation these wavefunctions are [17]

( ) ( ) ( )ˆ ˆ( ) cosh ( ) sinh ( )2 2j j jE a b E a b E a br rf r f r i f r (151)

For very large r (when y – ) the hypergeometric functions behave as

( ) ~ ( )F y y [33]. Thereby we deduce that ( )ˆ ~ exp( )jE a b

f i r and

( ) ~j

iErE a bf e (152)


This means that the functions (151) represent tempered distributionscorresponding to a continuous energy spectrum. On the other hand, Eq. (152)indicates that this energy spectrum covers the whole real axis, as it seems to benatural since the metric is not asymptotically flat. Anyway, it is clear that theenergy spectrum is continuous, without discrete part, while the energy levels areinfinitely degenerated since there are no restrictions upon the values of j, whichcan be any positive half-integer.

In these conditions, the fundamental solutions can not be normalized in thegeneralized sense and, therefore, the quantization can not be done in this chart.

8. DIRAC FERMIONS IN MOVING FRAMES OF THE dS SPACETIME

The first solutions of the Dirac equation in moving charts with sphericalcoordinates of the dS manifolds were derived in Ref. [8] and normalized inRef. [34]. Other solutions of this equation in moving frames with Cartesiancoordinates have been reported [35] but these solutions are not correctlynormalized. We have constructed the normalized solutions in this gauge usingthe helicity basis in momentum representation [20]. Our theory of externalsymmetry offered us the operators we need for deriving the normalizedfundamental solutions and quantizing the Dirac field in usual manner.

8.1. OBSERVABLES IN dS SPACETIME

We remain in the case of the de Sitter spacetime, of radius R = 1/ , forwhich we keep the definition (61) and the previous notations. This manifold isthe homogeneous space of the pseudo-orthogonal group SO(4, 1). This group isin the same time the gauge group of the metric 5(1) = (1, –1, –1, –1, –1) and theisometry group, I(M), of the dS spacetime. For this reason it is convenient to usethe covariant real parameters AB = – BA since in this case the orbital basis-generators of the representation of SO(4, 1), carried by the space of the scalarfunctions over M5, have the standard form (62). They will give us directly theorbital basis-generators L(AB) of the scalar representations of I(M). Indeed,starting with the functions ZA(x) that solve the equation (61) in the chart {x}, onecan write down the operators (62) in the form (24), finding thus the generatorsL(AB) and implicitly the components ( ) ( )ABk x of the Killing vectors associated to

the parameters AB [15]. Furthermore, one has to calculate the spin parts S(AB),according to Eqs. (38) and (28), arriving to the final form of the basis-generatorsX(AB) = L(AB) + S(AB) of the spinor representation of S(M).

In the dS spacetime there are many static or moving charts of physicalinterest. First we consider the moving local chart {t, r, , } associated to the


Cartesian one { }t x with the line element

2 2 2 2tds dt e dx (153)

The time t (– , ) of this chart is interpreted as the proper time of an observerat 0.x Another moving chart which play here a special role is the chart { }ct xwith the conformal time tc and Cartesian spaces coordinates xi defined by

20 2 22

25 2 22

1 1 ( )2

1 1 ( )2

1

cc

cc

i i

c

tZ rt

tZ rt

Z xt

(154)

with .r x This chart covers only a half of the manifold M, for tc (– , 0)

and 3.x D Nevertheless, it has the advantage of a simple conformal flatline element [4],

2 2 22 21

cc

ds dt d xt

(155)

Moreover, the conformal time tc is related to the proper time t through

tct e (156)

In what follows we study the Dirac field in the chart { }t x using theconformal time as a helpful auxiliary ingredient. The form of the line element(155) allows one to choose the simple Cartesian gauge with the non-vanishingtetrad components [8]

0 00 0

1 1ˆ ˆi i i ic j j c j jc c

e t e t e et t

(157)

In this gauge the Dirac operator reads

0 0

0 0

32

32

c

iD c t i

t it i

iE i t

ii ie(158)

and the weight function of the scalar product (82) is

33( ) tct e (159)


The next step is to calculate the basis-generators X(AB) of the spinorrepresentation of S(M) in this gauge since these are the main operators thatcommute with ED. The group SO(4, 1) includes the subgroup (3)E

(3) (3)T SO which is just the isometry group of the 3-dimensional Euclidean

space of our moving charts, { }ct x and { },t x formed by 3 translations,

,i i ix x a and proper rotations, ii jjx R x with (3)R SO [11]. Therefore,

the basis-generators of its universal covering group, (3) (3) (2)E T SU

( ),S M can be interpreted as the components of the momentum, ,P and total

angular momentum, ,J operators. The problem of the Hamiltonian operatorseems to be more complicated, but we know that in the mentioned static centralcharts with the static time ts this is (05) stH X i [15]. Thus the Hamiltonian

operator and the components of the momentum and total angular momentumoperators (Pi and 2i ijk jkJ J respectively) can be identified as being the

following basis-generators of S(M)

(05) ( )ci

c t iH X i t x (160)

(5 ) (0 )i

i i iP X X i (161)

( ) ( )i j

ij ij j i ijJ X i x x S (162)

after which one remains with the three basis-generators

2 2(5 ) (0 ) 0( ) 2 2 ( )

i i i ji i c i c ijN X X t r P x H S t S x (163)

which do not have an immediate physical significance. The SO(4, 1)transformations corresponding to these basis-generators and the associatedisometries of the chart { }ct x are briefly presented in Appendix.

In the other local chart, { },t x we have the same operators P and

J L S (with )L x P whose components are the (3)E generators, whilethe Hamiltonian operator takes the form

tH i x P (164)

where the second term, due to the external gravitational field, leads to thecommutation rules

[ ]i iH P i P (165)

S

S


We observe that in this chart the operators (0 )i

iK X are the analogous of

the basis-generators of the Lorentz boosts of (2 )SL since in the limit of

0, when ( 153) equals the Minkowski line element, the operators H = P0, Pi,Ji and Ki become the generators of the spinor representation of the group

(4) (2 )T SL (i.e. the universal covering group of the Poincaré group[11, 19]).

In both the charts we used here the generators (160)–(163) are self-adjointwith respect to the scalar product (82) with the weight function (159) if weconsider the usual boundary conditions on 3.D Therefore, for any generatorX we have X X if (and only if) and are solutions of theDirac equation which behave as tempered distributions or square integrablespinors with respect to the scalar product (82). Moreover, all these generatorscommute with the Dirac operator ED.

8.2. POLARIZED PLANE WAVE SOLUTIONS

The plane wave solutions of the Dirac equation with m 0 must beeigenspinors of the momentum operators Pi corresponding to the eigenvalues pi,with the same time modulation as the spherical waves. Therefore, we have to lookfor particular solutions in the chart { }ct x involving either positive or negativefrequency plane waves. Bearing in mind that these must be related amongthemselves through the charge conjugation, we assume that, in the standardrepresentation of the Dirac matrices (with diagonal 0 [19]), they have the form

( )( ) ( )

( ) ( )

cip x

pc

f t pep

g t pp

(166)

( ) ( ) ( )

( ) ( )

c ip xp

c

pg t p

p ef t p

(167)

where ,p p i denotes the Pauli matrices while and are arbitrary Paulispinors depending on .p Replacing these spinors in the Dirac equation given by

(158) and denoting k = m/ and 1 ,2

ik we find equations of the form (202)

whose solutions can be written in terms of Hankel functions as

(1)2 2( ) ( )kc cf f Ct e H pt (168)

S


(1)2 2( ) ( )kc cg g Ct e H pt (169)

The integration constant C will be calculated from the ortonormalizationcondition in the momentum scale.

The plane wave solutions are determined up to the significance of the Paulispinors and . For 0p these can be treated as in the flat case [36, 19] since,in the tetrad gauge (157), the spaces of these spinors carry unitary linearrepresentations of the (3)E group. Indeed, a transformation (27) produced by

( ) (3) ( )A aA E S M where (2)A SU and 3a involves the usual linear

isometry of E(3), ( ) ( )iii i j ijA ax x x A x a with ( ) (3),A SO and

the global transformation ( ) ( ) ( ) ( ).t x t x A t x Consequently, thePauli spinors transform according to the unitary (linear) representation

1( ) [ ( ) ]ia pp e A A p (170)

(and similarly for ) that preserves orthogonality. This means that any pair of

orthogonal spinors ( )p with polarizations 1 2 (obeying )

represents a good basis in the space of Pauli spinors

( ) ( ) ( )p p a p (171)

whose components, ( ),a p are the particle wave functions in momentumrepresentation. According to the standard interpretation of the negativefrequency terms [36, 19], the corresponding basis of the space of spinors isdefined by

2( ) ( )[ ( )] ( ) [ ( )]p p b p p i p (172)

where ( )b p are the antiparticle wave functions. It remains to choose a specificbasis, using supplementary physical assumptions. Since it is not sure that the socalled spin basis [36] can be correctly defined in the dS geometry, we prefer thehelicity basis. This is formed by the orthogonal Pauli spinors of helicity = 1/2which fulfill

( ) 2 ( ) ( ) 2 ( )p p p p p p p p (173)

The desired particular solutions of the Dirac equation with m 0 resultfrom our starting formulas (166) and (167) where we insert the functions (168)and (169) and the spinors (171) and (172) written in the helicity basis (173). It


remains to calculate the normalization constant C with respect to the scalarproduct (82) with the weight function (159). After a few manipulations, in thechart { },t x it turns out that the final form of the fundamental spinor solutions ofpositive and negative frequencies with momentum p and helicity is

(1)22

(1)2

1 ( ) ( )2( )

( ) ( )

k tip x t

pk t

e H qe pU t x iN e

e H qe p(174)

(2)2

2(2)2

( ) ( )( ) 1 ( ) ( )

2

k t

ip x tp

k t

e H qe pV t x iN e

e H qe p(175)

where we introduced the new parameter q p and

3 21

(2 )N q (176)

Using Eqs. (201) and (203), it is not hard to verify that these spinors are charge-conjugated to each other,

2 0( ) ( )c Tp p pV U U i (177)

satisfy the ortonormalization relations

3( )p p p pU U V V p p (178)

0p p p pU V V U (179)

and represent a complete system of solutions in the sense that

3 3 3( ) ( ) ( ) ( ) ( )tp pp pd p U t x U t x V t x V t x e x x (180)

Let us observe that the factor e–3 t is exactly the quantity necessary tocompensate the weight function (159). Other important properties are

i i i ip p p pP U p U P V p V (181)

p p p pW U p U W V p V (182)

whereW J P S P (183)

is the helicity operator which is analogous to the time-like component of thefour-component Pauli-Lubanski operator of the Poincaré algebra [11]. Thus, we


arrive at the conclusion that the fundamental solutions (174) and (175) form acomplete system of common eigenspinors of the operators Pi and W. Since thespin was fixed a priori by choosing the representation , we consider that thecomplete set of commuting operators which determines separately each of thesets of the particle or antiparticle eigenspinors is 2{ }.iDE S P W Finally wenote that these solutions can be redefined at any time with other momentumdependent phase factors as

( ) ( ) ( )i p i pp p p pU e U V e V p (184)

without to affect the above properties.In the case m = 0 (when k = 0) it is convenient to consider the chiral

representation of the Dirac matrices (with diagonal 5 [36]) and the chart { }.ct xWe find that the fundamental solutions in helicity basis of the left-handedmassless Dirac field,

50

0

3 2

1( ) lim ( )2

1( ) ( )2

2 0c

c p cpk

c ipt ip x

U t x U t x

pte

(185)

50

0

3 2

1( ) lim ( )2

1( ) ( )2

2 0c

c p cpk

c ipt ip x

V t x V t x

pte

(186)

are non-vanishing only for positive frequency and = –1/2 or negative frequencyand = 1/2, as in Minkowski spacetime. Obviously, these solutions have similarproperties as (177)–(182).

8.3. QUANTIZATION

The quantization can be done considering that the wave functions inmomentum representation, ( )a p and ( ),b p become field operators (so that

† )b b [36]. Then the quantum field which satisfies the Dirac equation withm 0 in the chart { }t x reads

( ) ( )

3 †

( ) ( ) ( )

( ) ( ) ( ) ( )p p

t x t x t x

d p U x a p V x b p (187)


We assume that the particle (a, † )a and antiparticle (b, † )b operators mustfulfill the standard anticommutation relations in the momentum representation,from which the non-vanishing ones are

† † 3{ ( ) ( )} { ( ) ( )} ( )a p a p b p b p p p (188)

since then the equal-time anticommutator takes the canonical form

3 0 3{ ( ) ( )} ( )tt x t x e x x (189)

as it results from (180). In general, the partial anticommutator functions,

( ) ( ) ( )( ) { ( ) ( )}S t t x x i t x t x (190)

and the total one ( ) ( )S S S are rather complicated since for t t we haveno more identities like (203) which should simplify their time-dependent parts.In any event, these are solutions of the Dirac equation in both their sets ofcoordinates and help one to write the Green functions in usual manner. Forexample, from the standard definition of the Feynman propagator [36],

( ) 0 [ ( ) ( )] 0FS t t x x i T x x (191)

( ) ( )( ) ( ) ( ) ( )t t S t t x x t t S t t x x (192)

we find that3 4[ ( ) ] ( ) ( )tD FE x m S t t x x e x x (193)

Another argument for this quantization procedure is that the one-particleoperators given by the Noether theorem have similar structures and propertieslike those of the quantum theory of the free fields in flat spacetime. Indeed, fromEq. (87) we obtain the one-particle operators of the form (88) which satisfy theEqs. (89). The diagonal one-particle operators result directly using Eqs. (178)–(182). In this way we obtain the momentum components

3 † †( ) ( ) ( ) ( )i i iP d p p a p a p b p b pP (194)

and the helicity (or Pauli-Lubanski) operator

3 † †( ) ( ) ( ) ( )W d p p a p a p b p b pW (195)

The definition (88) holds for the generators of internal symmetries too, includingthe particular case of X = 1 when the bracket

3 † †( ) ( ) ( ) ( )d p a p a p b p b pQ (196)


gives just the charge operator corresponding to the internal U(1) symmetry of theaction (78) [19, 4]. It is obvious that all these operators are self-adjoint andrepresent the generators of the external or internal symmetry transformations ofthe quantum fields [36]. The conclusion is that, for fixed mass and spin, thehelicity state vectors of the Fock space defined as common eigenvectors of theset {Q, Pi, W} form a complete system of orthonormalized vectors ingeneralized sense, i.e. the helicity basis.

The Hamiltonian operator HH is conserved but is not diagonalin this basis since it does not commute with Pi and W as it follows from thecommutation relations (165) and the properties (89). Its form in momentumrepresentation can be calculated using the identity

3( ) ( )2i

ip ppH U t x i p U t x (197)

and the similar one for ,pV leading to

3 † †( ) ( ) ( ) ( )2

i ii p pi d p p a p a p b p b pH (198)

where the derivatives act as ( ) .f h f h f h Hereby it results the expected

behavior of H under the space translations of (3)E which transform theoperators a and b according to (170). Moreover, it is worth pointing out that thechange of the phase factors (184) associated with the transformations

( ) ( )( ) ( ) ( ) ( )i p i pa p e a p b p e b p (199)

leave invariant the operators , Q, Pi and W as well as the equations (188), buttransform the Hamiltonian operator,

3 † †[ ( )] ( ) ( ) ( ) ( )ii pd p p p a p a p b p b pH H (200)

This remarkable property may be interpreted as a new type of gauge transformationdepending on momentum instead of coordinates. Our preliminary calculationsindicate that this gauge may be helpful for analyzing the behavior of the theorynear ~ 0.

In the simpler case of the left-handed massless field with the fundamentalspinor solutions (185) and (186) we obtain similar results and we recover thestandard rule of the neutrino polarization.

The quantum theory presented above opens many new interestingproblems. One of them is if in the de Sitter geometry there exists an orbitalanalysis analogous to the Wigner theory of induced representations of the


Poincaré group [19]. This is necessary if we want to understand the meaning ofthe rest frames (of the massive particles) in the de Sitter spacetime and to findthe “booster” mechanisms changing the value of p or even giving rise to wavesof arbitrary momentum from those with 0p . We believe that this theory maybe done starting with the orbital analysis in M5 since this helps us to findSO(4, 1)-covariant definitions for our basic operators on M 2. More precisely, foreach momentum 5qq M we can write a five-dimensional momentum operator

P(q) of components ( )( )A AC B

BCP q q X while a generalized five-dimensional

Pauli-Lubanski operator in M has to be defined by ( ) ( )1 .8

BC DEA ABCDEW X X

Then it is clear that for the representative momentum ˆ ( 0 0 0 )q of theorbit q2 = 0, associated to the little group E(3) SO(4, 1), we recover ouroperators ˆ( ) ( )P q H P H and ˆ A AW Wq as given by (160), (161) and (183),respectively. In this way one may construct generalized Wigner representationsof the group S(M) in spaces of spinors depending on momentum.

Finally, we would like to hope that our approach based on externalsymmetries could be an argument for a general tetrad gauge covariant theory ofthe quantum fields with spin in which the procedure of second quantizationshould be independent on the frames one uses.

APPENDIX

SOME PROPERTIES OF HANKEL FUNCTIONS

According to the general properties of the Hankel functions [33], we

deduce that those used here, (1 2) ( ),H z with 12

ik and ,z are related

among themselves through

(1 2) (2 1)[ ( )] ( )H z H z (201)

satisfy the equations

(1) (1)( ) ( )kd H z ie H zdz z

(202)

and the identities

(1) (2) (1) (2) 4( ) ( ) ( ) ( )k ke H z H z e H z H zz

(203)

2 For plane wave spinors of a Dirac theory in M 5 see Ref. [37]


REFERENCES

1. S. Weinberg, Gravitation and Cosmology: Principles and Appli

ion i. cotÃescu west university of timiºoara, v. parvan ... · theory of external symmetry for...

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