# -Minkowski spacetime as the result of Jordanian twist deformation

###### Abstract

Two one-parameter families of twists providing Minkowski -product deformed spacetime are considered: Abelian and Jordanian. We compare the derivation of quantum Minkowski space from two perspectives. The first one is the Hopf module algebra point of view, which is strictly related with Drinfeld’s twisting tensor technique. The other one relies on an appropriate extension of ”deformed realizations” of nondeformed Lorentz algebra by the quantum Minkowski algebra. This extension turns out to be de Sitter Lie algebra. We show the way both approaches are related. The second path allows us to calculate deformed dispersion relations for toy models ensuing from different twist parameters. In the Abelian case one recovers Poincaré dispersion relations having numerous applications in doubly special relativity. Jordanian twists provide a new type of dispersion relations which in the minimal case (related to Weyl-Poincaré algebra) takes an energy-dependent linear mass deformation form.

###### pacs:

11.10.Nx, 11.30.Cp, 02.40.Gh## I Introduction

Noncommutative geometry has found many applications in physical theories in recent years. It has been suggested as a description of spacetime at the Planck scale and proposed as a background for the unification of gravity and quantum field theory. The first idea of non commuting coordinates was suggested as long ago as the 1940s by Snyder Snyder . More recently, deformed coordinate spaces based on algebraic relations (with constant) have been introduced in Ref. Dop1 as a consequence to quantum gravity Dop1 -Dop2 . These presently known as ”canonical” spacetime commutation relations have been the subject of many investigations (see, e.g., Dop1 -ABDMSW ). Alternatively, this type of noncommutative coordinates was introduced in string theory as coordinates on the spacetime manifold attached to the ends of an open string in a particular gauge field background SW . The same authors have also introduced the so-called Seiberg-Witten map relating gauge theories on commutative and noncommutative (NC) spaces. This research was a beginning of extensive studies on quantum field theories (QFT) defined over NC spaces DN -Zupnik , particularly with the twisted (deformed) Poincaré invariance of NC QFT in Refs. Chai1 ; Chai2 . A growing number of investigations concern gauge theories MSSW -ADMSW and, moreover, NC gravity ADMW -Chaichian .

The Lie-algebraic type of noncommutativity has also been widely investigated. Inspired by the -deformed Poincaré algebra Luk1 ; Luk2 , the -Minkowski spacetime has been introduced in Refs. Z ; MR , together with dimensionfull masslike deformation parameter (usually connected with Planck mass ). It has been furt her used by many authors Luk1 -BACKG as a starting point to construct quantum field theories and then to discuss Planck scale physics. The past few years have experienced an ever-growing interest in -deformed spacetime motivated by so-called doubly special relativity (DSR) AC ; BACKG which includes, besides the velocity of light, a second invariant mass parameter (). In the framework of DSR, the kinematic consequences of deformed spacetime have been examined; see, e.g., Smolin -Gosh .

Lie-algebraic quantum deformations (19) are the most physically appealing from a bigger set of quantum deformations (14) in the Hopf-algebraic framework of quantum groups Drinfeld -Fadeev . It appears that quantum deformations of Lie algebras are controlled by classical -matrices satisfying the classical Yang-Baxter (YB) equation: homogeneous or inhomogeneous. Particularly, an effective tool is provided by the so-called twisted deformations Drinfeld which classical matrices satisfy the homogeneous YB equation and can be applied to both Hopf algebra (coproduct) as well as related Hopf module algebra ( product) Oeckl . Two types of explicit examples of twisting two tensors are the best known and investigated in the literature: Abelian Reshetikhin and Jordanian Ogievetsky as well as the extended Jordanian one Kulish (see also Bonneau ; VNT1 ). The -deformation of Poincaré algebra is characterized by the inhomogeneous classical YB equation, which implies that one should not expect to get Minkowski space from a Poincaré twist. However, twists belonging to extensions of the Poincaré algebra are not excluded Bu ; MSSKG . An interesting result has been found in Ref. Ballesteros , where starting from nonstandard (Jordanian) deformed conformal algebra the Minkowski space has been obtained in two ways: by applying the Fadeev-Reshetikin-Takhtajan technique Fadeev or exploiting a bialgebra structure generated by a classical matrix.

As a final remark, let us remind the reader that the classification of quantum deformations strongly relies on classification of the classical matrices. In a case of relativistic Lorentz and Poincaré symmetries, such a classification has been performed some time ago in Ref. Zakrzewski . A passage from the classical -matrix to twisting two-tensor and the corresponding Hopf-algebraic deformation is a nontrivial task. Explicit twists for Zakrzewski’s list have been provided in Ref. Varna as well as their superization VNT1 ; VNT2 . Systematic quantizations of the corresponding Lorentz and Poincaré Hopf algebras are carried on in Ref. BLT2 . In particular, the noncommutative spacetimes described by three types of Abelian Poincaré twists have been calculated in Ref. LW .

In this article we shall work with a flat spacetime of arbitrary dimension . The paper is organized as follows: in Sec. II, we establish formalism and notation. We also show how to associate to a given -product the corresponding left and right operator realizations of noncommutative coordinates by means of generalized differential operators. Section III describes two families of twists providing -Minkowski deformed spacetime with deformed coproducts and antipodes. These are Abelian and Jordanian families. The Abelian family has been previously investigated Bu . One finds out, in an explicit form, the operator realization of the -Minkowski coordinates for each value of twist parameters. In this section we also point out the smallest possible subalgebras, containing Poincaré algebra, to which one can reduce the deformation procedure. Lie algebra comprising the -deformed Minkowski algebra and deformed realizations of the standard Lorentz algebra is introduced in Sec. IV. This turns out to be algebra. Similarly as in Ref. MS one introduces deformed generators , d’Alambert operator , and Dirac derivatives in the Lorentzian case. It should be noted that in Ref. MS it was Euclidean algebra. The generalized d’Alambert operator allows one to calculate, in explicit form, dispersion relations for plane wave solutions. A number of toy models based on different values of twist parameters are considered. Appendix A collects improved ordinary differential equations (ODE) for functions describing deformed generators. Appendix B refers to Weyl-Poincaré algebra as to minimal algebra containing both the Poincaré subalgebra as well as the twist element.

## Ii Preliminaries and notation

It is well -known that representations of Lie algebras (Lie groups) bring
these objects into the broader context of operator algebras which are essential
for quantum theories. In a case when representation space is, like in any
field theory, a space of functions (fields), this context is provided by an
algebra of (partial) differential operators on an underlying spacetime
manifold. In this case the algebra of functions itself becomes automatically
a Hopf module algebra over Hopf algebra of differential operators. Twisted
deformations in this broader context are a step forward in the geometrization of the
traditional Drinfeld scheme Drinfeld .
It has been argued in a seminal paper ADMW (see also ABDMSW ; Oeckl ; Aschieri ) that such a framework is very
useful in noncommutative geometry and deformed field theories and gives a hope
to describe gravity at the Planck scale ^{3}^{3}3See, e.g., Chaichian for a different framework..

As an example, let us consider Lie algebra of the inhomogeneous general linear group as a semidirect product of with translations . We choose a basis in with the following standard set of commutation relations:

(1) |

, and -denotes a dimension of spacetime which is not yet provided with any metric structure. Nevertheless, for the sake of future applications, we shall use ”relativistic” notation with spacetime indices and running and space indices .

This Lie algebra contains several interesting subalgebras, e.g., of the inhomogeneous special linear transformations. In this case instead of diagonal generators we shall use its traceless counterparts:

(2) |

denotes a central element in ,^{4}^{4}4However, it is not central in . with . Therefore, we have a basis of elements in .

For any (constant of arbitrary signature) metric tensor
:
on , one can associate a subalgebra of the
inhomogeneous orthogonal transformations , ^{5}^{5}5We shall write whenever the signature will become
important. which is defined by the following set of commutation relations:

(3) |

(4) |

where is defined by the embedding

(5) |

The algebra , as well as its classical subalgebras, acts on the algebra of smooth (complex-valued) functions on the spacetime manifold via first-order differential operators (derivationsvectors fields): this is defined by natural representation - the so-called Schwinger realization:

(6) |

of into infinite dimensional Lie algebra of complex-valued vector fields . For the purpose of deformations, one needs to work over a (algebraically closed) field of complex numbers . In what follows, we shall use complex Lie algebras , etc., instead their real counterparts. A real Lie algebra structure can be eventually encoded in a corresponding reality structure (involutive antiautomorphism) but we shall not focus on this point here.

In order to simplify the notation, we shall use the same letter to denote an abstract element in and its (first-order) differential operator realization in . This induces an embedding of the corresponding enveloping algebras

(7) |

as an embedding of Hopf algebras with primitive coproducts

(8) |

for . Nondeformed counit and antipode maps read as , and . Let us emphasize that the enveloping algebra is simultaneously an algebra of linear (complex-valued, partial) differential operators over . It can be equipped with a natural Hermitian involution defined on generators by , . The embedding (7) provides a real (anti-Hermitian) realization (, for ) only for the subalgebra and its subalgebras like . In these cases Hermitian conjugation is compatible with the corresponding reality structures. The action via derivations of on extends to a Hopf module algebra action of on the algebra (for details concerning Hopf module algebras, see, e.g., ABDMSW ; ADMW ).

Since is a Hopf module algebra over , it becomes automatically a Hopf module algebra over its sub-Hopf algebras, too, particularly over as well as its subsequent Hopf subalgebras. This can be further deformed, by a suitable twisting element , to achieve deformed Hopf module algebra , where the algebra is equipped with a twisted star -product

(9) |

Hereafter the twisting element is symbolically written in the following form:

(10) |

Quantized Hopf algebra has nondeformed algebraic sector (commutators), while coproducts and antipodes are subject to deformation:

(11) |

where . Let us recall that twisting two-tensor is an invertible element in which fulfills the 2-cocycle and normalization conditions Drinfeld ; Chiari :

(12) |

Its relation with a corresponding classical -matrix satisfying the classical Yang-Baxter equation is via a universal (quantum) - matrix :

(13) |

where denotes the deformation parameter. As it is well -known from the general framework of quantum deformations Drinfeld , a twisted deformation requires a topological extension of the enveloping algebra of some Lie algebra into an algebra of formal power series in the formal parameter (see, e.g., Bonneau ; Chiari ). For the purpose of the present paper, we shall call an algebra of formal (or generalized) differential operators. Accordingly, the Hopf module algebra has to be extended to as well. Particularly, deformed algebra can be represented by deformed commutation relations

(14) |

replacing the nondeformed (commutative) one

(15) |

where the coordinate functions play a role of generators for the corresponding algebras: deformed and nondeformed. The action of differential operators on functions induced by derivations (vector fields) remains the same in deformed and nondeformed cases. Moreover, the twisted star product enables us to introduce two operator realizations of the algebra in terms of (formal) differential operators on . The so-called left-handed and right-handed realizations are naturally defined by

(16) |

with , satisfying the operator commutation relations

(17) |

correspondingly. In other words, the above formulas describe embedding of into . These operator realizations are particular cases of the so-called Weyl map (see, e.g., Weyl and references therein for more details). They allow us to calculate the commutator:

(18) |

It has been argued in Ref. JSSW (see also BMS ) that any Lie-algebraic star product (when generators satisfy the Lie algebra structure)

(19) |

can be obtained by twisting an element in the form

(20) |

The star product

(21) |

would imply

(22) |

i.e., a vector field action on . The last formula is particularly important for obtaining a Seiberg-Witten map for noncommutative gauge theories (see JSSW ; BMS ). We are going to show using explicit examples that this formula is not satisfied for an arbitrary twist in the form of (20). However, we shall find an explicit twist for deformed Minkowski spacetime which belongs to the class described by (22).

Before proceeding further, let us comment on some important differences between the canonical and Lie-algebraic cases. In the former, related to the Moyal product case

(23) |

with a constant antisymmetric matrix ; cf. (14), one finds that

(24) |

is a total derivative, e.g.,

(25) |

This further implies the following tracial property of the integral:

(26) |

which is rather crucial for a variational derivation of Yang-Mills field equations (see ADMSW ). In contrast to (25), Eq. (22) rewritten under the form

(27) |

indicate obstructions to the tracial property (26) provided
that . ^{6}^{6}6For the
-deformation [see (28) and (42) below], one has
.

## Iii Minkowski spacetime from twist: Hopf module algebra point of view

Our first task is to find explicit twists in order to achieve a twisted star-product realization of the well-known deformed Minkowski spacetime algebra Z ; MR :

(28) |

for with remaining elements commuting. Here is the above-mentioned formal parameter and has the mass dimension. Strictly speaking, formulas (28) mean that the corresponding algebra of functions has been provided with a twisted star product (9) which leads to the commutation relations (28). Of course, as we shall see later on, different twisted star products may lead to the same commutation relations (28). In what follows, we shall present explicit results for two one-parameter families of twists providing quantum Minkowski spacetime.

### Jordanian family

To this aim we shall consider a one-parameter family of two-dimensional Borel subalgebras :

(29) |

with a numerical factor .
In terms introduced before the basis, the element
has the form ^{7}^{7}7The Borel subalgebra commutation relation leads to the validity of the
cocycle condition (12) which in turn guarantees associativity of the
corresponding star product. can be expressed as (see Ogievetsky –VNT1 for Jordanian twists): . The
corresponding one-parameter family of Jordanian twists

(30) |

where with formal parameter . Direct calculations show that, regardless of the value of , twisted commutation relations (14) take the form of that for Minkowski spacetime (28).

The twists (30) can be used to deform the entire Hopf
algebra. For generic , the smallest subalgebra containing
simultaneously the Borel subalgebra (29) and one of the orthogonal
subalgebras [e.g., Poincaré subalgebra ] is . However, there are three exceptions.

(A) For in dimensional spacetime, the smallest subalgebra is .

(B) For () in an arbitrary dimension, the smallest subalgebra
is Weyl-orthogonal algebra . It contains a central
extension of any orthogonal algebra . ^{8}^{8}8The signature of the metric is irrelevant from an algebraic point of view. In
this case, the commutation relation (3-4) should be
supplemented by

(31) |

Of course, for physical applications we will choose the Weyl-Poincaré
algebra. This minimal one-generator extension of the Poincaré algebra has
been used in Ref. Ballesteros (cf. Appendix B).

(C) in dimensions, is a boost generator for
nondiagonal metric , with the Loerentzian signature. This corresponds
to the so-called light-cone deformation of the Poincaré algebra LLM .

Since in the generic case we are dealing with Lie algebra, we shall write deformed coproducts and antipodes in terms of its generators . The deformed coproducts read as follows:

(32) |

(33) |

(34) |

(35) |

Here

and denotes the so-called falling factorial. The antipodes are:

(36) |

(37) |

(38) |

The Jordanian one-parameter family of twists (30) generates, of course,
left- and right-hand representations, respectively, which give a realization of
(16) and (17) in the following form:

I. Left-handed representations :

(39) |

Hereafter one introduces for convenience a Hermitian operator . ^{9}^{9}9This notation will be particularly convenient and utilized in the subsequent
section.

II. Right-handed representations (Hermitian for ):

(40) |

Particularly, using (18), we obtain

(41) |

which is different from (19). However for one obtains the desired commutator

(42) |

providing the deformed Minkowski spacetime, i.e., .

### Abelian family

Minkowski spacetime can be also implemented by the one-parameter family of Abelian twists Bu ; MSSKG (with being a numerical parameter):

(43) |

All are twists in a sense explained before. A
special case has been treated in Ref. Bu . Thus one gets the following:

I. Left-handed representation (Hermitian for ) :

(44) |

II. Right-handed representation (Hermitian for ):

(45) |

This implies

(46) |

which is different from (19) for any value of the parameter . The deformed coproducts read as follows (cf. Bu ):

(47) |

(48) |

(49) |

(50) |

Here . The antipodes are

(51) |

(52) |

(53) |

(54) |

The above relations are particularly simple for .

Before closing this section, let us point out that all twists considered here correspond to the same classical -matrix (Poisson bi-vector on ):

(55) |

## Iv Lie algebra of deformed Lorentzian spacetime.

The noncommutative Minkowski space can be realized in quantized relativistic phase space LRZ or in a Schrödinger representation () in terms of generalized differential operators.

In Refs. MSSKG ; MS ; MS2 , Meljanac et al., following an earlier development DJMTWW ; Dimitrijevic , have found an interesting realization of the noncommutative coordinates in terms of generalized differential operators. Their approach assumes the following ansatz:

(56) |

for noncommutative coordinates , where as before . Functions , , and are taken to be real analytic; however generalization to complex analytic is straightforward and will not be discussed here. These functions obey initial conditions and , and has to be finite in order to ensure a proper classical limit at . The operators are automatically Hermitian while Hermiticity of requires the additional assumption that

(57) |

where and denotes spacetime dimension. As we will see later on, this condition will be satisfied only in a few exceptional cases. Apparently, the ansatz (56) could be guessed from our twisted realizations (39), (40), (44), (45).

Now, the Minkowski commutation relations [cf. (28)]

(58) |

are equivalent to the property that functions , and do satisfy the ODE MS :

(59) |

hereafter for convenience [cf. (17)]
^{10}^{10}10The sign convention is due to the difference between left- and right-handed realizations; see (17). In fact, it can be eliminated by
rescaling ..

Throughout this paper we shall be interested in solutions of (59) for . In this case, for any one easily gets

(60) |

Specializing further to the linear case , one finds

(61) |

and

(62) |

The Hermiticity restriction (57) is satisfied only for and and in -dimensions for subcases. All of our twisted products realizations turn out to be special cases of the above more general formulas (60)-(62).

I.) One finds that the first case (61), i.e., , corresponds to the Abelian twists with for left-handed realizations [cf. (44)] and for right-handed ones. Two subcases and give rise to Hermitian representations.

II.) The case (62) is related to the Jordanian family under rather restricted values of : for left-handed representation , while for right-handed ones [cf. (40)]. We do not know twist realizations for generic and . Among Jordanian twists only the case with is Hermitian in a spacetime of dimension and can be reduced to the subgroup .

Now, following the general method developed in Ref. MS (see also Dimitrijevic ), one can try to covariantly incorporate the Minkowski algebra (58) into the extension of undeformed orthogonal algebra (3) by assuming

(63) |

(64) |

where and . The main point is the ansatz

(65) |

where are (real) analytic functions to be determined from (63-64). In turn, generators

remain undeformed, i.e., in the Schwinger realization: . Formula (65) describes the deformed realization of the ”boost” generators together with the initial conditions . A difference between our and the original approach MS is that we do not apriori assume the Euclidean signature for the metric . One should notice that the Lorentzian signature is more natural and expected in this context: the commutation relations (58) distinguish one of the variables . In contrast, the Euclidean signature puts all variables on equal footing.

In order to keep under control the difference between the Euclidean and Lorentzian cases, we shall temporarily introduce a coefficient . (Notice that and .)

Inserting ansatz (65) into the algebra (63), we obtain the following equations:

(66) |

(67) |

where (Euclidean case) or (Lorentzian case). Substituting (56) and (65) into

(68) |

(69) |

one obtains some overdetermined system of ODE; see Appendix A for details. Its solutions can be recast into the form (here ) :

(70) |

and

(71) |

The last equation is consistent with (59), (66-70) for both values . Its solution

(72) |

together with (70) determines generators completely. It is
worth noticing that the Hermticity of automatically implies reality
for the boost generators: . Since the Euclidean case has
been already studied in Refs. MS ; MS2 , further on, until the end of the present
paper,we shall use only Lorentzian signature .

Analogously, following Ref. MS we have also obtained (with some sign corrections) a generalized d’Alambert operator :

(73) |

under the form []

(74) |

where above denotes a dimensional, spacelike, Laplace operator.We are now in a position to define Dirac derivatives as

(75) |

This implies

(76) |

where (see Appendix A)

(77) |

Let us remark that only formulas for and are universal in the sense that they do not depend on the parameters and .

Direct calculations performed on solutions with constant give rise to

(78) |

and ()

These relations allow us to enlarge the Lorentz - Minkowski algebra (58), (63), (64) by the following commutation relations:

(79) |

(80) |

(81) |

in order to include the Dirac derivatives. In this way one has obtained a
new (non Lie-algebraic) quantum extension of the Lorentz algebra which
contains generators
in -dimensional spacetime
^{11}^{11}11It is a relativistic version of the Euclidean algebra obtained already in MS ..
Its algebraic structure is
completely described by the commutation relations (58), (63), (64), (79) - (80) and does not depend on a particular differential
operator realization which had been used for its construction.
Particularly, it does not depend on a twisting tensor itself. It
contains undeformed Poincaré algebra :
; cf. (79). Another Lie-algebraic part
splits into Lorentzian subalgebra
generated by combined with
the quantum Minkowski space : (58), (63), (64). It contains one free (formal) parameter and sign convention . From the algebraic point of view this dependence can be removed
by rescaling or equivalently by
setting . It makes this algebra isomorphic by substituting

(82) |

to the (nondeformed) simple Lie algebra for a ”frozen” value of the parameter . This observation can be helpful for studying representations for such systems.

From the physical point of view, however, a dimensionful constant can be related with some fundamental constant of nature,
similarly like in DSR ^{12}^{12}12See JKG1 for interrelations between DSR and de Sitter group . . Assuming for a moment that twist has a physical meaning, we will see how
its parameters enter the so-called dispersion relations obtained from plane
wave solutions of the corresponding Klein-Gordon equations with a deformed
operator . Before doing that, let us emphasize that the presented
formalism has a well-defined classical limit which reconstructs
standard Minkowski spacetime together with the Poincaré group acting on it and
Heisenberg-type relations between position and momenta operators.
Particularly, becomes a standard d’Alambert operator .

### Dispersion relations

To this aim let us consider as a toy model involving d’Alambert operator

in specific realizations (74). One is looking for a plane wave solution of the deformed Klein-Gordon equation

(83) |

where represents the plane wave with the covariant wave vector ; denotes a mass. Straightforward and simple calculations give rise to the general form of deformed ”dispersion relation”

(84) |

where . Further specialization to the case when which corresponds to the Abelian twists give rise to