# Interacting scalar field theory in -Minkowski spacetime

###### Abstract

We construct an complex scalar field theory in -Minkowksi spacetime, which respects -deformed Poincaré symmetry. One-loop calculation shows that the theory is finite and needs finite renormalization to be compatible with the limit. The loop result also has an imaginary valued correction due to the complex poles present in the propagator.

###### pacs:

11.10.Nx, 11.30.Cp^{†}

^{†}preprint: arXiv:0802.3793[hep-th]

^{†}

^{†}preprint: 2008/Feb/26

## I Introduction

Poincaré symmetry has been a primary geometric notion for the Minkowski spacetime (MST) and played the guiding role of construction of quantum field theory. At the short distance of Planck length scale, however, the spacetime itself may change its concept due to the quantum gravity effect. On this purpose, Poincaré algebra in momentum space is deformed kappaP and a new scale parameter is introduced, which will be an order of Planck length. The -deformed Poincaré algebra (KPA) can have many different forms. Based on bicrossproduct basis majid where the four momenta are commuting each other but the boost relation is deformed, the dual picture of the KPA is realized in terms of non-commuting spacetime majid ; zakr . This non-commuting spacetime is called -Minkowski spacetime (KMST), where the rotational symmetry is preserved but time and space coordinates do not commute each other,

(1) |

The Planck scale parameter has the role of a deformation parameter. When approaches infinity, the deformed Poincaré algebra in momentum space reduces to the ordinary Poincaré algebra and therefore, the ordinary Poincaré symmetry is recovered in Minkowski spacetime. The -deformed realization implies that the special relativity is deformed and the energy momentum relation also has a new form. This will result in a change of the group velocity of photon. In this respect, KPA implies the doubly special relativity doubly and the deformation parameter reflect the Planck scale physics.

After the appearance of the KPA, it is soon realized that the differential structure of the KMST of 4 spacetime dimension is not realized in 4 dimensional spacetime but needs to be constructed in 5 dimensional spacetime sitarz ; gonera . This reminds of the Snyder’s approach where non-commutative coordinates are realized in 4 dimensional De Sitter space snyder . The differential calculus is realized in exponential operator with an appropriate ordering of and .

(2) |

where and is the differential element, with and is the new differential element. The momentum realization of the derivatives is given as

(3) |

and behaves as a 4-vector element and as an invariant in KPA KRY-b ,

(4) |

Here and and are the rotation and boost generators of KPA, respectively. It is worth to mention that the corresponding derivative is realized in the dimensional De Sitter space:

(5) |

with and .

The invariant property of leads one to construct 4-dimensional system without invoking the fifth dimensional tangential direction if one requires physical system to respect the -deformed Poincaré symmetry (KPS). Based on this 4-dimensional differential structure, one requires the on-shell condition to be

(6) |

where is the particle mass and constructs the scalar field theory with KPS kosinski ; KRY-a ; KRY-b . Still, an interacting (field) theory needs more elaboration since it is not clear how to construct the many particle states from the right choice of vacuum since the many particle states constructed so far show the non-local nature. (See for example kosinski ; DLW and references there in).

To understand the physical effects of the -deformation, one may study black-body radiation KRY-b and Casimir energy KRY-c using the mass-shell condition only, which uses essentially the free field theory only. It turns out that the thermal energy of the blackbody radiation due to the massless mode of the KMST ( in (6)) reduces to the Stephan-Boltzmann law (proportional to ) when limit is taken if one takes care of ordinary modes (OM) from the mass-shell condition (6) which reduces to the one from Einstein’s special relativity.

In the asymmetric ordering, the ordinary massless mode is explicitly given as

(7) |

In fact, the mass-shell condition (6) also allows high momentum mode (HM) which exist only when its momentum is greater than .

(8) |

It is shown in KRY-b that the Stephan-Boltzmann law would be spoiled if the HM were to be included in the thermal distribution, whose contribution turns out to be proportional to or depending on how one treats the negative energy part of HM. Thus, one needs to eliminate the HM from the on-shell. The same thing applies to the symmetric ordering case.

On the other hand, the study of Casimir energy on a spherical shell shows that in the asymmetric ordering case the vacuum can break particle and anti-particle symmetry at Planck scale: The Casimir energy of the negative mode (anti-particle) in (7) is not the same as the one due to the positive mode (particle) if the HM is not included. Thus, if one requires the vacuum to respect the particle and antiparticle symmetry at Planck scale, one cannot adopt the asymmetric ordering dispersion relation. This reasoning forces us to adopt the symmetric ordering only to have the particle and anti-particle symmetry at the Planck scale.

In this paper we are going to construct an interacting complex scalar field theory imposing the KPS. We present the essential element for the free field theory in Sec. II, and construct its interacting scalar field theory in Sec. III. We evaluate the one-loop correction of propagator in Sec. IV and one-loop correction of vertex in Sec. V. Sec. VI is the summary and discussion.

## Ii Free scalar field theory

To construct the free field theory with KSP one defines a field variable in momentum space,

(9) |

Here both the coordinate variable and momenta are treated as commuting variables. The non-commuting nature of KMST is encoded in - product between field variables: The product of exponential element is required to satisfy the composition rule FGN

(10) |

In this paper, we will adopt the composition law corresponding to the symmetric ordering;

(11) |

The homomorphism of the product of field variables reproduces the KMST effect and this way, one can avoid various conceptual difficulties of non-commuting spacetime geometry.

The KPS is the guiding principle to construct the field theory and is applied to the free scalar action explicitly in KRY-b . The free analogue of massive complex scalar theory is given as

(12) |

is the conjugate of the scalar field and is expressed just as the complex conjugate of the field in this symmetric ordering case:

(13) |

In momentum space, the action in (12) is given as

(14) |

where -Poincaré invariance sets (see below (18)). Explicit form of is given as

is the 4-vector (4) and and are invariants in KPA

(15) | ||||

(16) |

where is the Casimir invariant

(17) |

One notes that the the integration measure given in (14) is invariant under the KPS:

(18) |

Let’s introduce a notation for the propagator function which is explicitly written as

(19) | ||||

The on-shell dispersion relation is given as ;

(20) | ||||

(21) |

The dispersion relation in (20) corresponds to the ordinary mode (OM) since this reduces to the ordinary particle and antiparticle dispersion relation as . The second one (21) corresponds to the tachyon mode since the mode is obtained by putting in (20). The tachyon mode, when its momentum is sufficiently large , becomes a real mode corresponding to the high momentum mode (HM). This HM should not be included in on-shell mode since HM will spoil the blackbody radiation law at limit KRY-b .

The propagator function has the periodic property

(22) |

and thus possesses an infinite number of poles on the complex plane of . It is convenient for later use to separate the OM and TM contribution, each satisfying the periodicity relation (22);

(23) | ||||

## Iii Interacting Scalar field theory

We will assume there is one complex scalar field in this paper. Extension to many fields is straight-forward. To find an interaction which respects KPS, one notices that KPS is preserved in the - product interaction

(24) |

where represents composite two fields. There are two ways to represent ;

(25) |

where

(26) |

This allows two types of interactions.

(27) | ||||

(28) | ||||

where the bosonic permutation symmetry of the scalar field is used in the last identity. It turns out that the B-type interaction, however, spoils the KPS after loop correction. Thus we will consider A-type interaction only (27).

Our action is written as

(29) |

From this action the Feynman rule follows. The propagator is given as (see Fig. 1 for notation)

(30) |

where . Four point vertex is given as where

(31) |

## Iv One loop correction of the propagator

One loop correction of the propagator is given as

(32) |

where is independent of the external momentum

(33) |

Using the explicit form of in (19) one may put this two point function correction as

(34) |

where is used in the integration and

(35) |

Here and and are positive real for the whole range of the 3-momentum integration. Therefore, the integrand has two simple poles at and where we set

(36) |

To avoid this singularity, one employs the small prescription so that the pole lies at . The prescription (, ) amounts to put (, ) in the ordinary field theory when .

After this prescription, one can evaluate the integration using the contour given in Fig. (2). The contribution over the quarter circles at infinity is neglected since it is canceled by the tachyon contribution. This prescription results in the integration

(37) |

where is the pole contribution and is the integrated value along the imaginary axis.

About the tachyon contribution, the real part of changes its sign depending on the 3-momentum range. In addition, becomes complex when . Noting that and poles correspond to the tachyon poles ( from the and pole) one can prescribe to lie on the lower half plane of the complex -plane for the whole range of the 3-momentum and on the upper half plane.

(38) |

where and . The schematic flow is seen in the Fig. 3.

This prescription results in the -integration

(39) |

where is the Heavyside step function so that the pole contribution is not vanishing only when .

After the -integration, one is left with the form,

(40) |

Here, is a function of with and is given in terms of the 3-momentum integration:

(41) | ||||

(42) |

where , , and .

Noting that the one-loop correction is proportional to the measure factor , one can shift the propagator function

(43) |

which will shift the mass or equivalently

(44) |

It turns out that is finite but has imaginary part,

(45) | ||||

(46) |

The quadratic divergence in the particle and anti-particle contribution at large momentum is compensated by the tachyon contribution. Explicit evaluation is given as

(47) |

The price for this finiteness is that is not real. The imaginary contribution arises from the complex poles present in the propagator, which have the role in the off-shell loop correction.

This imaginary contribution can be seen using a different contour integration of as in Fig. 4 where

(48) | |||

which is the same as the one in (IV). The contour in Fig. 4, on the other hand, can be regarded as the Wick rotation at , since . This shows that one cannot do the Wick rotation without including the poles at the complex -plane due to the periodicity of in (22) in this KMST theory.

In addition, as the one loop correction becomes infinite since is quadratic in . This forces one to renormalize away the imaginary mass correction as well as the quadratic term to have the proper theory at limit; with

(49) |

## V one loop correction of the vertex

The one-loop correction of the four point function is given as

(50) |

At one has the one-loop correction, as a function of

(51) | ||||

where and and are defined in (42). is the imaginary axis contribution

(52) |

and is the pole contribution,

(53) |

Let us consider the value at the imaginary axis. One may put in a more convenient form,

(54) |

where and is the complex conjugate of . Introducing , one has and . Integration of over is given as

(55) |

After this, one may interchange the integration of and . Integration over at the massless limit () is given as

(56) |

And the integration over gives

(57) |

which is finite but imaginary, absent in the ordinary field theory at . dependence of the integration is given in powers of or .