# Scalar Field theory in -Minkowski spacetime from twist

###### Abstract

Using the twist deformation of , the linear part of the diffeomorphism, we define a scalar function and construct a free scalar field theory in four-dimensional -Minkowski spacetime. The action in momentum space turns out to differ only in integration measure from the commutative theory.

###### pacs:

02.20.Uw, 02.40.Gh## I Introduction

Doplicher, Fredenhagen, and Roberts Doplicher showed that, in the presence of gravity, the Heisenberg’s uncertainty relation has to be generalized to include the uncertainty between coordinates, which may be reproduced from the noncommutativity of coordinates such as the canonical noncommutativity Saxell or the -Minkowski spacetime Majid94 ; Lukierski95 . Especially in the -Minkowski spacetime, the position satisfies an algebra-like commutational relation,

(1) |

with all other commutators vanishing.

The -Minkowski spacetime may arise as an effective low energy description of quantum gravity Freidel ; Smolin ; amelino . Such space first appeared in the investigation of -Poincaré algebra. Later, it was related to Doubly Special Relativity (See DSR and references therein) which might have a quantum gravitational origin kappa:QG .

The differential structure of the -Minkowski spacetime has been constructed in kappa-diff and based on this differential structure, the scalar field theory has been formulated Kosinski ; KRY ; int-scalar ; Freidel ; Noether ; arzano1 ; Meljanac . It was shown that the differential structure requires that the momentum space corresponding to the -Minkowski spacetime becomes a de-Sitter section in five-dimensional flat space. Various physical aspects of -Minkowski spacetime have been investigated in Refs. KRY and was extended to -Robertson-Walker spacetime kim . The Fock space and its symmetries amelino ; arzano , -deformed statistics of particles das ; other2 , and interpretation of the -Minkowski spacetime in terms of exotic oscillator Ghosh were also studied. In addition, the properties of scalar field theory on this spacetime has started being analyzed in depth Freidel ; Kosinski ; Noether ; int-scalar .

Recently, the -Minkowski spacetime is realized in terms of twisting procedure lightlike ; Bu ; other ; twist:other2 ; Ballest . This twist approach can be seen as an alternative to the -like deformation of the quantum Weyl and conformal algebra Ballest , which is obtained by using the Jordanian twist jordantwist ; kulish ; Borowiec . The light-cone -deformation of Poincaré algebra can be given by standard twist (see eg. lightlike ). The realization for the time-like -deformation was constructed in Ref. Bu ; other by embedding an abelian twist in whose symmetry is bigger than the Poincaré, and their differential structure was studied in twist:diff . One can also find other approach to the differential structure and twist realization of -Minkowski spacetime by using the Weyl algebra twist:other2 .

In this paper, we construct a free scalar field theory by using the twist approach Bu ; twist:diff . In Sec. II, we review the -Minkowski spacetime from twist and then define the -product between vectors. We also provide an interesting relation between the generators when acting on coordinates space. In Sec. III, we introduce a new action of the generators on function and define a -product between functions. In Sec. IV, we find a transformation rule for a scalar function and then construct an action for a real free scalar field in Sec. V. We summarize the results and discuss the physical applications in Sec. VI.

## Ii Review on the -Minkowski spacetime from twist

Twisting the Hopf algebra of the universal enveloping algebra of inhomogeneous general linear group is considered in Bu ; other . The group of inhomogeneous linear coordinate transformations is composed of the product of the general linear transformations and the spacetime translations. The inhomogeneous general linear algebra in (3+1)-dimensional flat spacetime is composed of generators where represents the spacetime translation and represents the boost, rotations and dilations. The generators satisfy the commutation relations,

(2) |

The universal enveloping Hopf algebra with the counit and antipode can be constructed starting from the base elements and coproduct with .

The infinitesimal transformation by the general linear group is given in terms of generators:

(3) |

where denotes an abstract action of on vectors, scalar fields, or vector fields. Consider the action of on an algebra where satisfying

(4) | |||||

In the previous paper twist:diff , we considered the vector space or equivalently . To define a scalar function in this paper we extend the module algebra of to include a dual space to . Therefore, we generalize the relations in Eqs. (11) and (12) in Ref. twist:diff to both of the covariant and contravariant vectors. The action on the product of vectors is given by the coproduct,

(5) |

where . If we choose and , this equation provides the well-known Leibnitz rule.

### ii.0.1 Abelian twist

A new Hopf algebra is obtained by twisting a given Hopf algebra. The new Hopf algebra has the algebra part in common with the original, however, the coproduct is changed by the twist. A twist is a counital 2-cocycle satisfying and . The new Hopf algebra, is given by the original counit and antipode (, ), but with a twisted coproduct:

(6) |

An abelian twist can be constructed by exponentiating two commuting generators such as the momentum operators and , which gives the canonical noncommutativity Chaichian . Other choice of a twist by using two commuting operators and Bu

(7) |

generates the -Minkowski spacetime with the twisted Hopf algebra . is a constant representing different ordering of the exponential kernel function in the conventional -Minkowski spacetime formulation. In this paper, , which corresponds to the time-symmetric ordering.

For convenience, we explicitly write down the twisted coproduct, ()

(8) | |||||

The spatial (rotational) parts are undeformed and keeps the rotational symmetry. On the other hand, the boost parts are deformed nontrivially due to the presence of the spatial dilatation term. It is noted that the twisted Hopf algebra is different from that of the conventional -Poincaré in two aspects. First, the algebraic part is nothing but those of the un-deformed inhomogeneous general linear group (II) rather than that of the deformed Poincaré. Second, the co-algebra structure is enlarged due to the bigger symmetry and its co-product is deformed as (8).

### ii.0.2 -product between vectors

In this subsection, we study the non-commutative -product by using the twisted Hopf algebra. The twist with new product, , given by

(9) |

defines a new associated algebra as a module algebra of in the sense that

(10) |

Explicitly, the -product between and gives the usual -Minkowski relation (1) and the -product between and leads to the nontrivial commutation relation twist:diff . The -product with gives

(11) | |||||

which results in commutation relations

(12) |

Note that any vector having non-vanishing spatial index does not commute with the time coordinate. It was also shown in Ref. twist:diff that the relation (12) is related to the -dimensional differential structure of the -Minkowski spacetime from twist.

### ii.0.3 Relation between the actions of and

In this subsection, we provide a nontrivial relation satisfied by the two actions of and on the vector space of coordinate vector :

(13) |

We prove this by the method of induction. We use the notation for simplicity. In the case of , it is clear from the definition of in Eq. (4). Let us assume that satisfies Eq. (13). Then, we can show that also satisfies the relation. As an illustration we post the proof for the case of :

where we use the deformed coproduct (8) and the definition (10) of the action on the -product. The first term in the right-hand-side of Eq. (II.0.3) becomes

where we use the definition of to replace the -product with a normal one , by using in the first equality and use the property in the second equality. Similarly, the second term in the right-hand-side of Eq. (II.0.3) becomes

where we replace the normal product with a -product in the first equality, exchange the order of product in the second equality, and then replace the -product with a normal product in the last equality. Adding the above two equations we have

The two terms exactly cancels the last two terms in Eq. (II.0.3). In the case of the action of , one may similarly show by using . One may similarly demonstrate that the relation (13) is satisfied for other cases too. This implies that satisfies Eq. (13).

We emphasize that the action of is not equivalent to the action of if the target is a tensor composed of whole module space since

(15) |

Eq. (13) holds only when the target space of is the coordinates vector space which is a subset of .

## Iii Action on functions and -product between functions

We now consider the action of generators on the algebra of functions , where denotes the space of functions and denotes the ordinary product given by , where . For convenience, we introduce a new notation denoting the action of the generators on a function with the form:

(16) | |||||

Explicitly, the two actions of on a simple function give different results,

(17) |

In the first action , acts on both and so that is invariant under and in the second action , it acts only on . On the other hand the two actions of momentum are equivalent:

since the action of momentum to the vector vanishes.

Given the action of the algebras on , we may define an associated algebra as module algebra of by using the twist (7). The -product between two functions is given by twisting the ordinary product by using the action as

If the functions and are ordinary functions without -product, Eq. (III) leads to the conventional -Minkowski star-product,

One may also try to get a deformed associated algebra by acting the action on rather than on its functional form , as in Eq. (III). However in this case, the resulting module algebra will be the same as the original one, , since any action of generators on a scalar makes it vanish. Thus the star-product defines a scalar function ,

due to and Eq. (17). Consider two scalar functions and so defined. Explicit calculation shows that the two functions commutes for the -product,

This implies that the -product between functions reduces to the ordinary product. However, the -product does not commute,

(20) |

From Eqs. (11) and (20), one notices that the -product satisfies

(21) |

This is the reason why we construct the noncommutative star product between scalar functions using .

Finally, the partial derivative is identified with and its coproduct is defined by ,

(22) | |||||

We have shown in Ref. twist:diff that the differential structure is consistent with the Jacobi identity and the relation .

## Iv -product and vectors

Up to this point, we have defined the -product between the scalar functions. However, it is not yet defined the -product between a vector and a function. To find one, we calculate the action of on the product of two scalar functions in two different ways. First, we calculate it by using the definition of coproduct (8),

Second, we calculate the -product before acting the momentum operator,

Since the two results should be the same we have the -product between ’s and a function:

(25) |

This shows that commutes with the -product so that

The same result holds for the contravariant vector .

We can also do the same calculation for as in Eqs. (IV) and (IV). For example, we calculate

which should be the same as

Equating the two equations for all , we get

(26) | |||||

We now provide an independent check of Eqs. (25) and (26). One may conjecture that . However, one immediately realizes that this conjecture is not valid since Eq. (IV) is different from (IV) with this conjecture. Noting and Eq. (21), one has to try a weaker form:

(27) |

Then, by assuming in Eq. (27), Eq. (20) results in Eq. (26). In addition, one may have as in Eq. (25) if one requires

Given the relations (25) and (26), we may show that the actions and satisfy

Similarly, we also get

(28) |

which is consistent with Eq. (13).

In general, one may construct any function as a series of . Therefore, Eq. (28) must be satisfied for all scalar functions. In this sense, we propose the transformation law of a scalar field under a general linear group as

(29) |

Especially, the time-symmetric exponential function

(30) |

and their product satisfy the property (29). It is remarked in passing that other ordered exponential functions also satisfy the same property (29). Note that we have used the -product to define the ordered exponential function in Eq. (30) rather than the -product. This is a crucial difference from the previous work twist:diff in which the ordered product was implemented by using the -product and was absent in the module algebra .

## V Action of a free scalar field

In this section, we construct an action of a free scalar field which is invariant under the action of general linear algebras in the sense of Eqs. (II) and (4).

To find a metric we choose and in Eq. (20) as and , the tetrad of the coordinates (Note that the tetrad is independent of the coordinate vector because of Eq. (4)). Multiplying the flat Minkowski metric to Eq. (20) we arrive at

(31) |

with signature transforms under the actions of generators as

Since the RHS of Eq. (31) is invariant under the general linear transformation, the LHS satisfies the transformation rule,

(32) |

To construct a free scalar field theory in -Minkowski spacetime from twist, we need to know some useful identities for the time-symmetric exponential function (30). From Eq. (25) we have

(33) |

(This simple relation is also satisfied by the exponential function of different ordering if one uses different twist parameter ). Eq. (33) confirms that the exponential function acts as a scalar function following Eq. (29). In addition, the multiplication of two exponential functions is given by a new scalar exponential function:

(34) |

where . . Note that

In the presence of a non-commutativity we introduce the integration of a scalar function using the property,

In the first equality, we use Eq. (25) and in the second equality, we use the fact that the metric tensor is independent of coordinate vector . (If the metric were dependent on coordinate vector, the -product might be relevant in this calculation.)

We calculate the -product of two exponential functions to get a -function,

Since the left-hand-side is a scalar quantity, the right-hand-side is also a scalar under the transforms in . Note that the -function appears with the normalization factor .

We define the Fourier transform of a scalar field with the form:

(35) |

where is an appropriate measure to be determined below. The inverse Fourier transform must be given by the -product. We have the consistency condition,

which determines the measure

(36) |

The Hermitian conjugate of becomes where implies the complex conjugate of . If is a real scalar field, the mode function satisfies .

The integration of the product of two scalar fields is

One may calculate any number of products of scalar fields in a similar way.

The action for a free scalar field in commutative spacetime with metric is

(38) |

This action is invariant under the general linear transformation in -dimensions in the sense of linear diffeomorphism: and . The corresponding action in noncommutative spacetime is obtained by the following modifications: 1) The partial derivative is replaced by the nontrivial action on ordered functions. 2) The normal products between functions and vectors are replaced by the -products. Since the metric is independent of coordinate vector, the -product between the metric and a function is irrelevant. Thus, the action of a free real scalar field in -Minkowski spacetime from twist is written as

(39) |

Note the way how transforms under the action :