Using a single index i =1; 2; 3; 4 instead of the double index (; ) =

HEP-TH-9408111

LMU-TPW94-4August,1994

Hilb ert Space Represen tation of an Algebra of Observ ables for q-Deformed Relativistic Quan tum Mec hanics

W. Zipp old Sektion Ph ysik, Univ ersit at M unc hen Theresienstr. 37, D-80333 Munic h, German y

AbstractUsingarepresentationoftheq-deformedLorentzalgebraasdierentialoperatorsonquantumMinkowskispace,wedeneanalgebraofobserv-ablesforaq-deformedrelativisticquantummechanicswithspinzero.WeconstructaHilbertspacerepresentationofthisalgebrainwhichthesquareofthemassp2isdiagonal.

1 Intro duction

TheconceptofLiegroupsandLiealgebrashasfoundanaturalgeneralizationintheframeworkofnon-commutativeHopfalgebrasandquantumgroups.Thishasposedthequestion,whetherthesecanappearassymmetriesofphysicaltheories.Inthispaperwewanttoaddresstheproblemofquantumsymmetricrelativisticquantummechanicswithspinzero.Inordinaryrelativisticquantummechanics,wehaveaHilbertspaceofstateswhichisarepresentationofthePoincarealgebra.Actingonthisspace,wehavethealgebraofobservablesgeneratedbypositionandmomentumcoordinates,whichareessentiallyself-adjointoperatorsdenedonacommondomain,whichisdenseintheHilbertspace.TheysatisfytheHeisenbergalgebra[

x

i

;p

j]=

ig

ij.PositionandmomentumspacearebothisomorphictoMinkowskispaceandcarryarepresentationoftheLorentzgroup.

1

In this paper, we dene a deformation of the algebra of observables. The po- sition and momentum coordinates generate algebras, which are isomorphic as

-comodule algebras of the q-deformed Lorentz group. Introducing an algebra of angular momentum operators, which is a representation

U

q(

SL

(2

;C

), we con- struct a Hilbert space representation of this algebra of observables. Coordinates and momenta are represented on this Hilbert space by unbounded operators.

2 Preliminaries

2.1

^SLq

(2

^;C

) and Quantum Lorentz Group

The complex quantum group

SL

q(2

;C

) ([4]) is a ^-Hopf algebra constructed by taking two copies of the quantum group

SL

q(2) (cf. [5]) with generators (

t

);⁼¹;²and (^

t

);⁼¹;²which are connected by the mixed relations ^

R

 ^

t



t

 =

t

^

t



R

^, where ^

R

denotes the ^

R

-matrix of

SL

q(2). The involution is (

t

)^ :=

S

(^

t

),

S

being the antipode of

SL

q(2).

The quantum Lorentz group

SO

q(3

;

1) (cf. [1]) is the ^-sub-Hopf algebra ^

SL

q(2

;C

) generated by the elements

M

^()( ⁾ := ^

t



t

 with relations and Hopf structure induced by

SL

q(2

;C

). Its ^

R

-matrix is a product of

SL

q(2)- ^

R

- matrices. Using a single index

i

= 1

;

2

;

3

;

4 instead of the double index (

;

) = (1

;

1)

;

(1

;

2)

;

(2

;

1)

;

(2

;

2), the reality condition for the generators is given by (

M

ij)^ =



jb

S

(

M

ba)



ai, where the matrix



ij is essentially given by the ^

R

- matrix of

SL

q(2).The ^

R

-matrix has the projector decomposition characteristic of orthogonal quantum groups:

R

^ijkl=

q

^PSijkl^,

q

^,1PAijkl+

q

^,3P1ijkl

;

(1)

PS, ^PA and ^P1 being the symmetric, antisymmetric and trace projector, re- spectively. ^P1ijkl can be expressed in terms of the q-Minkowski metric

C

ij as

P

1ijkl =

Q

^,1

C

^ij

C

kl with

Q

:=

C

^ij

C

ij = [2]²_q. Here we have introduced the q-numbers [

x

]_q := (

q

^x,

q

^,x)

=

(

q

^,

q

^,1).

2.2 Quantum Minkowski Space and Dierential Calculus

A comodule algebra for

SO

q(3

;

1) can be dened as the unital C-algebra gener- ated by the coordinates

x

ⁱ with relations^PAijkl

x

^k

x

^l= 0. This is the algebra of functions on quantum Minkowski space and is denoted

A

x(^Mq). The coaction



of

SO

q(3

;

1) is given by



(

x

ⁱ) :=

M

ij

x

^j. For



to become a homomorphism of^-algebras, we have to dene the involution on

A

x(^Mq) by (

x

ⁱ)^:=

x

^k

C

kl



li. The element

r

² :=

C

ij

x

ⁱ

x

^j is central in

A

x(^Mq). Another convenient set of generators

t;x;y;z

for

A

x(^Mq) is given by

t

:=

q

^,1

=

[2]_q(

q

^,1

x

^2,

x

³)

;

y

:=

x

¹

; z

:=

q

^,1

=

[2]_q(^,

qx

^2,

x

³)(2)

;

x

:= ^,

x

⁴

:

The generator

t

becomes central in

A

x(^Mq) and factorization with respect to the relation

t

= 0 yields an

SO

q(3)-comodule algebra (cf. [1]).

We can now introduce partial derivatives acting on

A

x(^Mq) as linear operators.

We dene the partial derivative

@

^{i 2}EndC(

A

x(^Mq))

;i

= 1

;:::;

4

;

by its action

@

ⁱ(1) := 0 on the unit element 1²

A

x(^Mq) and by the Leibniz rule

@

ⁱ(

x

^j

f

) :=

C

^ij

f

+

q R

^{^,1}ijkl

x

^k

@

^l(

f

) ⁸

f

²

A

x(^Mq)

;

(3) which determines

@

ⁱ on the whole algebra

A

x(^Mq). The partial derivatives satisfy^PAijkl

@

^k

@

^l= 0 and



(

@

ⁱ(

f

)) = (

M

ij

@

^j)^



(

f

)

;

⁸

f

²

A

x(^Mq).

Therefore the algebra

A

@ generated by (

@

ⁱ) is isomorphic to

A

x(^Mq) as an

SO

q(3

;

1)-comodule algebra. The coordinates

x

ⁱ act on

A

x(^Mq) as linear oper- ators by left multiplication. So we can consider the algebra

A

x;@ :=

<

(

x

ⁱ)i⁼¹;:::;⁴

;

(

@

ⁱ)i⁼¹;:::;⁴

>

^EndC(

A

x(^Mq)) with relations

P

Aijkl

x

^k

x

^l = ^PAijkl

@

^k

@

^l= 0

;

@

ⁱ

x

^j =

C

^ij+

q R

^{^,1}ijkl

x

^k

@

^l

:

(4) Given this algebra, we can recover the Leibniz rule and therefore the action of

@

ⁱ as a dierential operator on

A

x(^Mq). Next, we want to dene a ^- structure on

A

x;@. To that end, we rst note, that there exists an operator  ²

A

x;@

(cf. [7]) satisfying (1) = 1, 

x

ⁱ =

q

²

x

ⁱ and 

@

ⁱ =

q

^,2

@

ⁱ. Consequently

²EndC(

A

x(^Mq)) is a positive and invertible operator, so we can dene an invertible operator



:=

q

²¹⁼². We can now introduce the algebra

D

(^Mq) generated by the coordinates (

x

ⁱ), the derivatives (

@

ⁱ) and the operators



and



^,1, and we call it algebra of dierential operators on quantum Minkowski space. In this extended algebra we introduce the elements 

@

ⁱ by

@

ⁱ:= ^,1(

@

ⁱ+

q

³

x

ⁱ
)

;

(5) and it was shown in [7] that these dierential operators satisfy the relations of the second possible covariant dierential calculus on

A

x(^Mq), i.e. we have

P

Aijkl

@

^k

@

^l = 0 and 

@

ⁱ

x

^j =

C

^ij +

q

^,1

R

^{^}ijkl

x

^k

@

^l. The involution on the partial derivatives can now be dened by (

@

ⁱ)^ := ^,

q

^,4

@

^k

C

kl



li, such that

D

(^Mq) nally becomes a^-algebra in which



^ =



^,1.

2.3 Regular Functionals and Vector Fields

In [4] it was shown that the dual algebra

SL

q(2

;C

)^ of

SL

q(2

;C

) contains a

-sub-Hopf algebra

U

^R, called algebra of regular functionals.

U

^R is generated by functionals (

L

^)_;⁼¹_;² and (^

L

^)_;⁼¹_;². The action of these functionals is dened using the ^

R

-matrix of

SL

q(2):

L

^(1) :=





L

^(

t

) := ^

R

^1

L

^(^

t

) := ^

R

^1 (6) and the same relations for ^

L

^ with

t

and ^

t

exchanged. These denition is extended to products by

L

^(

ab

) :=

L

^(

a

)

L

^(

b

),⁸

a;b

²

SL

q(2

;C

), and

the same for ^

L

^. The commutation relations in

U

^R and the Hopf algebra structure are a direct consequence of these denitions (cf. [4]). The involution on the generators of

U

^R is (

L

^)^y =

S

(^

L

^). Using the properties of the

SL

q(2)- ^

R

-matrix, some of these generators can be eliminated, and it turns out that

U

^R is generated by

L

⁺¹¹ ,

L

⁺²² ,;

L

⁺¹² , ^

L

⁺²¹ ,

L

^,11,

L

^,12,

L

^,21 ,

L

^,22 with

L

⁺²² = (

L

⁺¹¹)^,1. Moreover, in a certain minimal extension also

L

^,11 is invertible, so we are essentially left with six generators (cf. [4]). We can now also dene the algebra of vector elds as the unital

C

-algebra generated by the elements

Y

 :=

L

⁺

S

(

L

^, ) ²

U

^R and ^

Y

 := ^

L

⁺

S

(^

L

^{,  2}

U

^R. This algebra was introduced in [2, 3], and it was shown that for

SL

q(2

;C

) it is a sub-Hopf algebra of

U

^R and that a certain natural extension of the algebra of vector elds is isomorphic to

U

^R. The left invariant vector elds are then given by

X

:= 1

=

(1^,

Y

). Let us nally mention that there exist two Casimir operators

C

¹ and

C

² in

U

^R.

3 Angular Momentum Representation of

^UR

3.1 Angular Momentum Algebra

In the sequel we will always assume

q

^1.

In the undeformed case we have a representation of the Lie algebra of

SL

(2

;C

) in terms of antisymmetric generalized angular momentum operators acting on functions over Minkowski space. In this section, we will dene the analogue of this in the q-deformed framework, i. e. a representation of the quantum universal enveloping algebra of

SL

q(2

;C

) (which is essentially given by

U

^R) by dierential operators on quantum Minkowski space

A

x(^Mq). In [6] such a representation was found for the closely related case of

SO

q(

N

) and, apart from the star structure, these results can be applied to the case of

SL

q(2

;C

).

Therefore we rst introduce the elements

u

^{ij 2}

D

(^Mq) as

u

^ij := (1 +

q

^,4)

C

^ij+

q

^,4(

q

^,1

R

^{^}ijkl

x

^k

@

^l,

q R

^{^,1}ijkl

x

^k

@

^l) (7) and dene

V

^ij :=



^PAijkl

u

^kl =

q

^,4



[2]_q^PAijkl

x

^k

@

^l

;

(8)

U

:=

C

ij

u

^ij = (1 +

q

^,4)



((

q

+

q

^,1)²1 + (

q

^,4,1)

C

ij

x

ⁱ

@

^j)

:

(9) These dierential operators commute with

r

² and we are led to the following

Denition 3.1

The^-subalgebra ^^Uq :=

<

(

V

^ij)i;j⁼¹;:::;⁴

;U >

^

D

(^Mq) with in- volution^ given by

(

V

^ij)^ =

V

^kl

C

km

C

ln



ni



mj

; U

^ =

U

(10) is calledangular momentum algebra on quantum Minkowski space.

As the rank of the antisymmetric projector in four dimensions is six, only six of the generators

V

^ij are linearly independent. To determine the action of the elements of ^^Uq as dierential operators on

A

x(^Mq), we consider the algebra

A

x( ^^Uq

;

^Mq) :=

<

1

;

(

V

^ij)

;U;

(

x

^j)

>

i;j⁼¹;:::;^4

D

(^Mq)

:

(11) Using the relations in

D

(^Mq), we can derive the relations in

A

x(^^Uq

;

^Mq), which determines the action of the generators of ^^Uq as dierential operators. The result is (cf. [6]):

V

^ij

x

^k = ^,[2]_q^PAijmn

x

^m

V

^nk+

q

(1 +

q

⁴)[2]_q^PAijmn

C

^nk

x

^m

U;

(12)

Ux

^k = (

q

^4,

q

^,4) 1



[2]_q²

x

^k

U

^,

q

2

C

mn

x

^m

V

^nk

!

:

(13)

Furthermore the relations in

A

x(^Mq) imply the following identities:

0 =

x

³

V

¹²+

x

²

V

³¹+

x

²

V

^12,



[2]_q

x

¹

V

^14,2

q

^,1 [2]_q

x

¹

V

²³

;

0 =

x

⁴

V

¹²+

q

^,2

x

¹

V

^24,

q

^,1

[2]_q(

q

²+

q

^,2)

x

²

V

^14,

q

^,2



[2]_q

x

²

V

²³

;

0 =

x

⁴

V

³¹+

x

¹

V

⁴³+

x

¹

V

^24,

q

²



²

[2]_q

x

²

V

¹⁴ (14)

,

2

q

[2]_q

x

²

V

²³+ 2

q

[2]_q

x

³

V

^14,



[2]_q

x

³

V

²³

;

0 =

x

³

V

²⁴+

q

^,2

x

²

V

^43,

q

²



[2]_q

x

⁴

V

^{14, 2}

q

[2]_q

x

⁴

V

²³

:

For explicit calculation a more convenient choice for the generators is:

V

¹⁺ := ^q_²

V

¹²

; V

^1, := ^q_²

V

⁴³

; M

¹ := ^[2]q2

q

(^,

V

¹⁴+

V

²³+

q

^,2

C

)

; V

²⁺ := ^q_²

V

³¹

;

V

^2, := ^q_²

V

²⁴

; M

² := ^[2]q2

q

(^,

q

²

V

^14,

V

²³+

q

^,2

C

)

;

(15)

where

C

is dened by

C

:= 1

=

((1 +

q

^,4)[2]_q)

U

. For the commutators of the generators of ^^Uq with each other we have the identities (cf. [6])

PAijkl

C

mn

V

^km

V

^nl=

q

^,1



(

q

²+

q

^,2)[2]_q²

V

^ij

U; V

^ij

U

=

UV

^ij (16) and

1 = 1

(1 +

q

^,4)²[2]_q⁴

U

^2,

q

⁶

2[2]_q²

C

ij

C

kl

V

^ik

V

^lj

:

(17)

The second equation yields two identities which read with

i

= 1

;

2 1 +

M

_i^2, _[2]²

q

M

i

C

+

q

²

[2]_q(

V

_i^,

V

_i⁺+

q

^,2

V

_i⁺

V

_i^,) = 0

;

(18) and (16) gives another six relations (i=1,2):

V

_i^

M

i =

q

^,2

M

i

V

_i^

;

q

^,1

V

_i^,

V

_i^,,

qV

_i^,

V

_i⁺ = 1



^(1,

M

_i²)

:

(19) Note that the algebras generated by^f

V

¹⁺

;V

^1,

;M

^1g or^f

V

²⁺

;V

^2,

;M

^2galone are isomorphic to the algebra of left invariant vector elds on the quantum group

SU

q(2) as dened in [9]. This is exactly analogous to the undeformed case, but unlike the classical case, these two subalgebras do not commute:

V

¹⁺

V

^2, =

q

²

V

^2,

V

¹⁺

; V

¹⁺

V

²⁺ =

q

^,2

V

²⁺

V

¹⁺

; V

^1,

V

^2, =

q

^,2

V

^2,

V

^1,

; V

^2,

M

¹ =

M

¹

V

^2,

;

V

¹⁺

M

² =

M

²

V

¹⁺

;

(20)

M

²

M

^1,

M

¹

M

² =



³

V

¹⁺

V

^2,

;

M

¹

V

^2+,

V

²⁺

M

¹ =

qV

¹⁺([2]_q

M

^2,

C

)

; M

²

V

^1,,

V

^1,

M

² =

qV

^2,(

C

^,[2]_q

M

¹)

; V

²⁺

V

^1,,

q

^,2

V

^1,

V

²⁺ = [2]



^q(

qM

²

M

¹+

q

^,1

M

¹

M

²

,(

M

¹+

M

²)

C

+ 1[2]_q

C

²)

:

(16) means that

C

is central ^^Uq. Finally, the commutation relations of the scaling operator



with the coordinates imply that it commutes with all the generators of ^^Uq. The extension of

A

x( ^^Uq

;

^Mq) by



and



^,1 will be denoted by

A

x;( ^^Uq

;

^Mq). From (10) we obtain the involution on the generators as (

V

¹⁺)^ =

,

qV

^2,, (

V

^1,)^=^,

q

^,1

V

²⁺, (

M

¹)^ =

M

²and

C

^=

C

. Actually there is a natural extension of the Algebra ^^Uq. We dene the element

H

^{2U^}q by

H

:=

M

¹

M

^2,

q

^,1



²

V

¹⁺

V

^2,

:

(21)

H

satises

H

(

x

¹

;x

²

;x

³

;x

⁴) = (

q

^,2

x

¹

;x

²

;x

³

;q

²

x

⁴)

H;

(22) which together with

H

(1) = 1 implies that

H

is a positive and invertible dif- ferential operator on

A

x(^Mq). Therefore the operator

H

¹⁼² and its inverse are well dened by their commutation relations with the coordinates following from (22). The extension of ^^Uq by

H

^1=2 will be denoted by ^Uq.

3.2 Dierential Representation of

^UR

In the undeformed case the algebra of angular momentum operators on Minkowski space is a representation of the Lie algebra of

SL

(2

;C

) , i.e. a representation of the left invariant vector elds on the Lie group. It turns out that a similar statement is true in the deformed case. To see this, we consider an algebra representation



:

U

^{R !} EndC(

A

x(^Mq))

; a

^7!



(

a

) =:

a



;

of

U

^R by linear operators on

A

x(^Mq).

a

 is dened for an arbitrary

f

²

A

x(^Mq) by

a

(

f

) = (

S

^,1(

a

)

id

)^



(

f

)

;

(23)

where



is the

SO

q(3

;

1)-coaction dened in section 1.

U

 :=



(

U

^R) is a subal- gebra of EndC(

A

x(^Mq)). It turns out that

U

 =^Uq and we obtain the

Proposition 3.2

The algebra ^Uq ^ EndC(

A

x(^Mq)) is a ^- representation of the algebra

U

^R. In this representation the two Casimir operators of

U

^R coincide, i.e.



(

C

¹) =



(

C

²) =

C

.

In particular, it turns out that



maps the algebra of vector elds on

SL

q(2

;C

) to ^^Uq. However,

A

x(^Mq) being a comodule algebra,

U

and therefore^Uq is also a bialgebra and even a Hopf algebra. Dening the mappings
:

U

^!

U



U

,

"

:

U

^!

C

and

S

:

U

^!

U

 by

(
(

a

))(

f

g

) :=

a

(

fg

) ⁸

f;g

²

A

x(^Mq)

;

"

(

a

) :=

i

(

a

(1))

;

(24)

S

(

a

) :=



(

S

^,1(

a

))

;

where

i

:

A

x(^Mq) ^!

C

projects the subalgebra of

A

x(^Mq) generated by 1 ²

A

x(^Mq) onto the complex numbers, we deduce⁸

a

²

U

^R :
(

a

) = (





)(



^

(

a

)),

"

(

a

) =

a

(1)

;

where



denotes the transposition of tensor components and
the coproduct in

U

^R, and get the following

Proposition 3.3 U

 together with the coproduct
, the counit

"

, the an- tipode

S

 and the involution

a

^:=



(

a

^y) is a ^-Hopf algebra.

Uq contains a ^-subalgebra corresponding to vector elds on

SU

q^,1(2). It is generated by

H

^1=2

;X

^ with

X

⁺ :=

q

[2]_q

!

1=²

(

qV

¹⁺

C

^,

q

²

V

¹⁺

M

^2,

q

^,1

V

²⁺

M

¹)

;

(25)

X

^, :=

q

^,1

[2]_q

!

1=²

(^,

V

^2,

C

+

q

^,1

V

^2,

M

^1,

q

^,1

V

^1,

M

¹) (26) and involution

H

^ =

H

and (

X

⁺)^ =

qX

^,. It will be denoted^U_q³, its extension by the generators of

A

x(^Mq) by

A

x(^U_q³

;

^Mq). The relations in ^U_q³ are those already encountered in (19). The element

W

^2U_q³, related to the central element

S

^{2 2}

U

q^,1(

SU

(2)) (cf. [9]) by

W

:=

q

(



²

S

²+ [2]_q1) is the central Casimir operator of^U_q³. It satises

HW

=

q

²1 +

H

²+

q

²



²[2]_q

X

^,

X

⁺

; W

^=

W:

(27) Finally, we give the relations determining

H;X

^and

W

as operators on

A

x(^Mq).

They read

H

(

t;y;z;x

) = (

t;q

^,2

y;z;q

²

x

)

H;

X

⁺(

t;y;z;x

) = (

t;y;z;x

)

X

⁺+ (0

;

0

;q

^,1

y;

^,

z

)

H

X

^,(

t;y;z;x

) = (

t;y;z;x

)

X

^,+ (0

;z;

^,

qx;

0)

H

(28)

W

(

t;y;x;z

) = (

t;q

²

y;z;q

^,2

x

)

W

+



[2]_q(0

;

^,

y;qz;qx

)

H

+



²[2]_q[(0

;q

²

z;

^,

qx;

0)

X

⁺+ (0

;

0

;qy;

^,

z

)

X

^,]

:

(29) As

H

and

X

^ are a representation of vector elds of the q-deformed rotation group, they commute with the generator

t

²

A

x(^Mq) corresponding to the time coordinate in the limit

q

^!1.

4 Algebra of Observables

We want to dene momentum coordinates

p

ⁱ, such that the algebra generated by the momenta is isomorhpic to

A

x(^Mq) as a ^-comodule algebra of

SO

q(3

;

1).

Unlike the undeformed case, the partial derivatives

@

ⁱ are not closed under involution.

Lemma 4.1

The momentum operators(

p

^j)j⁼¹;:::;⁴²

D

(^Mq) dened by

p

^j :=^,

i

(

@

^j+

q

^,4

@

^j) (30)

generate the momentum quantum space

A

p(^Mq) of functions over q-Minkowski space, i.e. they satisfy

PAijkl

p

^k

p

^l= 0

;

(

p

^j)^=

p

^k

C

kl



lj

:

(31)

The complete symmetry between coordinates and momenta as ^-comodule al- gebras of

SO

q(3

;

1) is established by the following

Lemma 4.2

The algebra

A

p(^Uq

;

^Mq) generated by (

p

ⁱ) and^Uq is isomorphic to

A

x(^Uq

;

^Mq) with the isomorpism

i

given on the generators by

i

(

p

ⁱ) =

x

ⁱ,

i

(

V

^ij) =

V

^ij and

i

(

U

) =

U

.

Now we are in the position to introduce the algebra of observables (cf. [6]) by the following

Denition 4.3

The algebra of observables^O:=

<

1

;

(

x

ⁱ)i⁼¹;:::;⁴

;

(

p

^j)j⁼¹;:::;⁴

>

^

D

(^Mq) is the algebra generated by coordinates and momenta.

Proposition 4.4

In^O we have relations

p

ⁱ

x

^j,

q R

^{^,1}ijkl

x

^k

p

^l=^,

iu

^ij (32) where

u

^{ij 2}

D

(^Mq) are the elements dened in (7).

The next aim is to construct a Hilbert space representation of this algebra of observables, which can be interpreted as the state space for q-deformed relativis- tic quantum mechanics. To make the connection with the angular momentum algebra, we rewrite (32) as

p

ⁱ

x

^j,

q R

^{^,1}ijkl

x

^k

p

^l=^,

i

^,1(

V

^ij +

U

)

:

(33) It was shown in [6] that we can reconstruct the partial derivatives given the coordinates and the angular momentum algebra with the help of the following identities in

A

x;(^Uq

;

^Mq):

r

²

@

ⁱ = 1 +

q

^,2

1^,

q

²

x

ⁱ+ 1

q

^8,1

x

ⁱ

U

+ 1 +

q

^,2

2(

q

^2,1)

C

kl

x

^k

V

^li

;

(34)

r

²

@

^i = ^, 1 +

q

²

q

^,2,1

x

ⁱ+ 1

q

^,8,1

x

ⁱ

U

^,1+

q

^{2 1 +}

q

²

2(1^,

q

^,2)

C

kl

x

^k

V

^li



^,1

:

(35) Due to the complete symmetry between coordinates and momenta, the same relations hold with coordinates and momenta exchanged. This means that we can nd Hilbert space representations ^O by constructing Hilbert space repre- sentations of

A

x;(^Uq

;

^Mq) with invertible

r

². This will be done in the next section.

5 Hilbert Space Representation of

^O

5.1 Representation of

^Ap

(

^Uq3;Mq

)

In analogy to the undeformed case we expect a relativistic one particle state to be an element of an irreducible representation of the Poincare algebra. As a maximal set of commuting observables we take the square of the momentum

p

²:=

C

ij

p

ⁱ

p

^jwhich is the square of the particle mass, the energy

p

⁰(correspond- ing to the generator t in the coordinate algebra), the angular momentum

W

and the 3-component of angular momentum

H

. In the limit

q

^!1 this is a com- plete set of commuting observables for spin 0. We will construct an irreducible

-representation of

A

p(^Uq

;

^Mq) as a direct sum of irreducible representations of

A

p(^U_q³

;

^Mq).

The hermitean elements

p

²

; p

⁰are central in

A

p(^U_q³

;

^Mq). In an irreducible rep- resentation we can therefore choose them as multiples of the identity operator.

We dene for abitrary

M

²

; E

²

IR

a linear space

H

^M;E := Lin^f

a

^j

M;E;

0

;

0ⁱ:

a

²

A

p(^U_q³

;

^Mq)^g

;

(36)

where the cyclic vector^j

M;E;

0

;

0ⁱhas the properties

X

^j

M;E;

0

;

0ⁱ:= 0

; H

^j

M;E;

0

;

0ⁱ:=^j

M;E;

0

;

0ⁱ

:

(37)

p

^2j

M;E;

0

;

0ⁱ:=

M

^2j

M;E;

0

;

0ⁱ

; p

^0j

M;E;

0

;

0ⁱ:=

E

^j

M;E;

0

;

0ⁱ

;

(38) i.e. it carries a scalar representation of^U_q³.

The commutators (28) imply

H

^M;E =

Lin

^f

a

^j

M;E;

0

;

0ⁱ:

a

²

A

p(^Mq)^g. Con- sidering the vector

p

_ly^j

M;E;

0

;

0^{i 2}

H

^M;E with

l

²

IN

⁰, we nd it to be an eigenvector of

H

and

W

with eigenvalue

l

for both operators which is annihi- lated by

X

⁺. Therefore it is a highest weight for the irreducible representation of

U

q(

SU

(2)) characterized by the eigenvalue l of the Casimir

W

.

If we dene

E;l^j

M;E;l;l

ⁱ:=

p

_ly^j

M;E;

0

;

0ⁱ, then a basis of this representation is given by the states^j

M;E;l;m

ⁱwith

m

=^,

l;:::;l

dened by

j

M;E;l;m

ⁱ:=

q

^m+1 _[

l

+

m

+ 1]^[2]q_q[

l

^,

m

]_q

!

,1=²

X

^,j

M;E;l;m

+ 1ⁱ

:

(39) Being eigenvectors to dierent eigenvalues of

H

, these states are linearly inde- pendent.

Lemma 5.1

The set

B

:= ^fj

M;E;l;m

ⁱ mit

l

²

IN

⁰

; m

²

Z; Z

^j

m

^{j }

l

^g is a linear basis of

H

^M;E

:

For^h

M;E;l

⁰

;m

^0j

M;E;l;m

ⁱ:=



l⁰l



m⁰m to be a consistent denition of a scalar product on

H

^M;E, the constant

E;l = ^h

M;E;

0

;

0^j(

y

^)^l

y

^lj

M;E;

0

;

0ⁱ must be positive. Using the denition of the central element

p

² and the relations (28), we get a recursion formula for

E;l:

E;l⁺¹ =

q

²[

l

+ 1]_q [2

l

+ 3]_q ( 1

f

l

+ 1^g_q

t

^2,f

l

+ 1^g_q

M

²)

E;l (40) with ^f

x

^g_q := ^q_q^x++_q^q,1,x. As

E;⁰ = 1 this xes

E;l for all

l

²

IN

⁰. We have to distinguish two cases:

i

)

M

^20 :

E;l

>

0 ⁸

l

²

IN

⁰

:

(41)

ii

)

M

²

>

0 :

E;ll^!1

! ,1

:

(42)

In the rst case there is no restriction on the values of

E

, but for the physically interesting case of real mass the requirement of positivity forces the recursion to terminate, i.e. for each xed value of

M

²

>

0 there must be an

N

²

IN

⁰, such that

E;l = 0 ⁸

l > N

. This means that in this case we get only the discrete energy eigenvalues

E

=^f

N

+ 1^g_q

M:

(43)

In the sequel we take

M

²

>

0. We can then use the positive integer

N

to label the states by^j

M;N;l;m

ⁱ. The sign of the energy eigenvalues is an additional

invariant of the representation, because all the generators of

A

p(^U_q³

;

^Mq) com- mute with

p

⁰. So we have two orthonormal bases which span the linear spaces

H

^M;N := Lin^fj

;M;N;l;m

ⁱ:

N;l

²

IN

⁰

;l

^

N ;m

²

z

^j

m

^j

l

^g

:

(44) Completing

H

^M;N to Hilbert spaces ^HM;N with respect to the scalar product dened above, we get

Lemma 5.2

For every

M

²

>

0 and

N

²

IN

⁰ the Hilbert spaces ^HM;N are irreducible^- representations of the algebra

A

p(^U_q³

;

^Mq).

5.2 Representation of

^Ap

(

^Uq;Mq

)

We consider now the linear space

H

^M := ^M1

N⁼⁰

H

^M;N (45)

with scalar product ^h

;M;N

⁰

;l

⁰

;m

^0j

;M;N;l;m

ⁱ :=



N⁰N



l⁰l



m⁰m. To show that these spaces are representations of

A

p(^Uq

;

^Mq), it is sucient to determine the action of the generators of ^Uq on the ^U_q³-invariant vectors ^j

;M;N;

0

;

0ⁱ, because the action on an arbitrary vector^j

;M;N;l;m

ⁱcan then be deduced using the relations in

A

p(^Uq

;

^Mq). Making use of the commutators of the gen- erators of^Uq and^U_q³and of the discreteness of the spectrum of

p

⁰, we make the following ansatz:

V

_k^+j

;M;N;

0

;

0ⁱ = ^X1

i⁼⁰

c

^+_;N;k;i^j

;M;i;

1

;

1ⁱ

; V

_k^,j

;M;N;

0

;

0ⁱ = ^X1

i⁼⁰

c

^,_;N;k;i^j

;M;i;

1

;

^,1ⁱ

;

(46)

M

k^j

;M;N;

0

;

0ⁱ = ^X1

i⁼⁰

c

^;N;k;i^j

;M;N;

1

;

0ⁱ+

c

^0_;N;k;i^j

;M;N;

0

;

0ⁱ

;

with

k

= 1

;

2. This also determines the action of the Casimir

C

, because one can prove

C

^j

;M;N;

0

;

0ⁱ= (

qM

¹+

q

^,1

M

²)^j

;M;N;

0

;

0ⁱ. The commutation relations of the generators of^Uq with

X

⁺ and

X

^, can be used to eliminate all but two coecients:

c

^0_;N;¹_;i =

c

^0_;N;²_;i

;

c

^+_;N;¹_;i = (

q

[2]_q)¹⁼²_¹

c

^;N;¹;i

; c

^,_;N;¹_;i = ^,(

q

^,1[2]_q)¹⁼²_¹

c

^;N;¹;i

; c

^;N;²;i = ^,

q

²

c

^;N;¹;i

;

c

^+_;N;²_;i =

q

³(

q

[2]_q)¹⁼²¹_

c

^;N;¹;i

; c

^,_;N;²_;i = ^,(

q

[2]_q)¹⁼²¹_

c

^;N;¹;i

:

(47)