# From noncommutative –Minkowski to Minkowski space-time

###### Abstract

We show that free -Minkowski space field theory is equivalent to a relativistically invariant, non local, free field theory on Minkowski space-time. The field theory we obtain has in spectrum a relativistic mode of arbitrary mass and a Planck mass tachyon. We show that while the energy momentum for the relativistic mode is essentially the standard one, it diverges for the tachyon, so that there are no asymptotic tachyonic states in the theory. It also follows that the dispersion relation is not modified, so that, in particular, in this theory the speed of light is energy-independent.

The -Minkowski space kappaM1 , kappaM2 is a noncommutative space-time, in which time and space coordinates satisfy the following Lie-type commutation relation

(1) |

As shown in kappaM1 this spacetime arose quite naturally from -Poincaré algebra qP . More recently it played a central role in Doubly Special Relativity research program Amelino-Camelia:2000ge , Amelino-Camelia:2000mn , rbgacjkg (for recent review see Kowalski-Glikman:2004qa .)

To investigate properties of -Minkowski spacetime it seemed quite natural to construct field theories on it and then discuss their physical properties. This endeavor was undertaken by many authors Kosinski:1999ix , Kosinski:1998kw , Amelino-Camelia:2001fd , Kosinski:2003xx , Daszkiewicz:2004xy (see also Dimitrijevic:2004nv ). Unfortunately, in spite of this effort our present understanding of physics (on the background) of -Minkowski spacetime is still rather incomplete.

The aim of this note is to fill this gap. Using star product for -Minkowski space whose construction we will briefly describe below, we will be able to recast free scalar field theory on this non-commutative spacetime to the form of surprisingly simple, equivalent scalar field theory living on ordinary Minkowski space. The action of this theory contains infinite powers of the wave operator, reflecting non-locality of the original theory, and reminding somehow the situation discussed in context of p-adic string theory (see Moeller:2002vx and references therein).

In this letter we will concentrate on deriving the relevant field theory on Minkowski space skipping many technical points, especially the relation to the equivalent construction on -Minkowski space, leaving them to a longer paper LJStoappera .

Let us start with a simple, but crucial observation. The algebra (1), in the context of DSR understood as an (sub)algebra of tangent vectors to de Sitter space of momenta Kowalski-Glikman:2003we can be regarded a Lie algebra of a Lie group that we will call below Borel algebra and group, respectively. This group arises naturally in Iwasawa decomposition of the into “Lorentz” and “translational” parts, and can be therefore identified with (a portion of) de Sitter space of momenta. Thus the ordered plane waves on -Minkowski spacetime Amelino-Camelia:1999pm

(2) |

can be equivalently regarded as Borel group elements. Accordingly, the composition of plane waves, usually understood in terms of the coproduct of -Poincaré algebra, can be regarded as a simple group elements product. Explicitely

(3) |

Note that Borel group can be coordinatized by labels in the plane waves. These coordinates correspond to the “cosmological coordinates” on (the portion of) de Sitter space of momenta Kowalski-Glikman:2003we .

We now analyze the property of Fourier transformation on -Minkowski space. All the results obtained below are natural extension of the ones obtained in Freidel:2005ec , Freidel:2005bb in the context of 3-dimensional -Minkowski space.

First in order to do physics it is convenient to map, under a Weyl map, the algebra of fields on non commutative -Minkowski space into the space of usual fields on Minkowski spacetime equipped with a star-product. The key point is that among all possible choices of Weyl maps one is preferred, as explained in more details in LJStoappera . This is the one which preserves the Lorentz invariant bicovariant calculus or equivalently the one under which the action of the Poincaré group becomes linear. This Weyl map is fully characterized by its action on plane waves: it maps the -Minkowski plane waves which form a non abelian group to usual plane waves as follows

(4) |

where

(5) |

are naturally understood as labelling points in a de Sitter space of radius (with

This choice of map is dictated by the bicovariant differential calculus 5dcalc1 , which is the starting point of construction of Poincaré invariant field theory on -Minkowski space. Note that not all of is in the image of since only the points for which are reached. Moreover this mapping covers only half of , the one for which . In order to define a notion of Fourier transformation which is covariant under Lorentz transformation we need to have a space of momenta which is without boundaries. This is possible if one interpret the image of to be the quotient of de Sitter space by the identification . This space is usually called elliptic de Sitter space. Since the identification is an isometry which possess no fixed point in de Sitter the quotient is well defined and is a symmetric space under the action of . The invariant measure on is given by

(6) |

A point in is labeled either by or by
with the restriction^{1}^{1}1This parameterizations
does not cover all of since we have to exclude the
hyperplane . this hyperplane is exactly the place where the
star product is not well defined. However this hyperplane is of
measure and this problem should not bother us in the definition
of the Fourier transform. . When written in terms of the group variables the
measure reads^{2}^{2}2From now, in order to simplify notations, on we choose the Planck units,
in which the Planck mass scale as well as the Planck length
scale are equal 1.

(7) |

which is a left invariant measure on the group

(8) |

In order to prove (7) let us integrate a function on

(9) | |||||

where and is the Heaviside distribution.

Given a field we define its Fourier components

(10) |

where we introduce a conjugation

(11) |

This formula involves the star product, which is defined to satisfy

(12) |

where on the right hand side is defined by , i.e. is a label of the group element obtained as a product of group elements labeled by and . Similarly we will use the abbreviation to denote the label of the element . Explicitly, has components , and equals , from which it follows that , which is relevant to the calculation below, is equal .

In order to reconstruct from its Fourier mode we need to relate this Fourier transform to the usual one where the inversion formula is known. This can be achieved by using the following key integration identity which relates the integration of star product of fields to the integration of usual fields product

(13) |

where denotes the complex conjugation and

(14) |

is the differential operator arising in the bicovariant differential calculus 5dcalc1 .

Using this identity the Fourier modes can be written in term of a usual Fourier transform

(15) |

We can then reconstruct using the usual Fourier transformation

(16) |

where is the Lebesgue measure.

This can be written in terms of the right invariant group measure using the fact that

(17) |

hence

(18) |

In order to prove (13) it is sufficient to establish it for plane waves .

(19) |

To prove it let us first notice that

(20) |

Note that the last term in drops out since it is proportional to . Then from the relation we compute

(21) |

where again only the first two terms are relevant. It now follows that

(22) | |||||

(23) |

where we have use the restriction . And since

we can conclude

(24) |

which proves the key identity (13).

The action for a massive field is given by

(25) | |||||

(26) |

This action is manifestly invariant under Poincaré transformations

(27) |

where is a Lorentz transformation.

The formula (26) is the main result of our paper. It shows that with the right choice of star product, that is the one compatible with the bicovariant differential calculus on -Minkowski space, and careful implementation of Lorentz symmetry (related to the choice of conjugation (11)), the resulting Minkowski space action is simple and covariant. It should be stressed that the action (26) is fully equivalent to the -Minkowski space action investigated in Kosinski:1999ix ,Kosinski:2001ii, Kosinski:2003xx , Daszkiewicz:2004xy , (while our method can be readily applied in the case of a bit less natural choice taken in Amelino-Camelia:2001fd , Dimitrijevic:2004nv who use slightly different kinetic term for field theory on -Minkowski space) and therefore it carries equivalent physical information. Further details of the construction will be discussed in the forthcoming paper LJStoappera .

Let us investigate the main properties of the action (26), in the case of real field . First of all the field equations read

(28) |

and describe a mode of mass along with what seems to be a Planck scale tachyon.

In order to get more insight into physical properties of the theory, let us compute the energy-momentum tensor. Since the theory has infinitely many derivatives it is convenient to use the standard formula for symmetric energy momentum tensor (see Moeller:2002vx and references therein for explanation how to handle energy momentum tensor in the case of theories with infinite number of derivatives)

(29) |

where is the covariant wave operator. As it is well know the energy momentum tensor (29) is conserved by construction. Let us consider generic term that arises in expansion of the right hand side of (29)

(30) |

This expression can be calculated by using the identities

Using these expressions and integrating out delta functions we get the following formula

While calculating energy-momentum tensor, in the final formula we can use equations of motion, i.e., replace with , where equals for particle, and for tachyon modes, respectively. Using this observation we can simplify the formula above to give

(31) |

With this formula we are able to calculate energy momentum tensor for the action of the form

with analytic , to wit

(32) |

One can easily check that this formula leads to the right answer in the standard case . In our case, (26), we have to do with two expressions: for the mass term. for the kinetic and

In the particle of mass case the energy-momentum tensor acquires just the multiplicative term and reads

(33) |

note that since the factor in front of expression above is negligible. Thus the energy and momentum of the free particle modes is essentially just classical.

In the case of the tachyonic mode, as a result of the presence of the denominator with , the components of energy-momentum tensor are divergent. This is quite a desirable feature of the model, since it means that one needs infinite energy to create a tachyon mode in the free theory. Moreover it is in agreement with the fact that we initially restricted the momentum space to be such that so that the tachyon is not in the spectra.

In conclusion, we managed to formulate a theory of free scalar field on Minkowski spacetime, which is equivalent to a free theory formulated on -Minkowski spacetime. The spectrum of this theory contains a standard particle of arbitrary mass along with the Planck mass tachyon, which however requires infinite energy to be produced.

We don’t expect expect that this latter feature change in the interacting theory since the ‘tachyonic’ factor comes from the conversion from star product to usual product. This means that this term will also factorise in front of the interacting equation of motion and do not propagate. In other terms this means that we expect this term to appear as a modification of the integration measure over loop momenta but not as a modification of the propagator in agreement with the analysis performed in Freidel:2005bb .

Of course it is a main challenge to investigate in detail such interacting theory at the quantum level. In principle the construction of polynomial interactions does not pose great problem, since we know from the construction presented here how to use star product and conjugation to produce Lorentz invariant terms of any order. For a real field , and since , local non derivative interactions are given by star product powers of the field like Amelino-Camelia:2001fd , Daszkiewicz:2004xy .

It should be also stressed that the construction of the energy momentum tensor (33) leads to a standard dispersion relation, so that, at least in free theory there is no room for energy dependent speed of light. This conclusion contrast with the result of Agostini:2006nc and can be trace back to their use of a non covariant differential calculus but is in agreement with the discussion of Kowalski-Glikman:2004qa on DSR theory based on -Minkowski space. Although it is not excluded that this will change in quantum interactive theory the effect, if any, is expected to be extremely small.

acknowledgment

For JK-G and SN this work is partially supported by the grant KBN 1 P03B01828.

## References

- (1) S. Majid and H. Ruegg, “Bicrossproduct structure of kappa Poincare group and noncommutative geometry,” Phys. Lett. B 334 (1994) 348 [arXiv:hep-th/9405107].
- (2) J. Lukierski, H. Ruegg and W. J. Zakrzewski, “Classical quantum mechanics of free kappa relativistic systems,” Annals Phys. 243 (1995) 90 [arXiv:hep-th/9312153].
- (3) J. Lukierski, H. Ruegg, A. Nowicki and V. N. Tolstoi, “Q deformation of Poincaré algebra,” Phys. Lett. B 264 (1991) 331; J. Lukierski, A. Nowicki and H. Ruegg, “New quantum Poincare algebra and k deformed field theory,” Phys. Lett. B 293 (1992) 344.
- (4) G. Amelino-Camelia, “Testable scenario for relativity with minimum-length,” Phys. Lett. B 510, 255 (2001) [arXiv:hep-th/0012238].
- (5) G. Amelino-Camelia, “Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale,” Int. J. Mod. Phys. D 11, 35 (2002) [arXiv:gr-qc/0012051].
- (6) N. R. Bruno, G. Amelino-Camelia and J. Kowalski-Glikman, “Deformed boost transformations that saturate at the Planck scale,” Phys. Lett. B 522 (2001) 133 [arXiv:hep-th/0107039].
- (7) J. Kowalski-Glikman, “Introduction to doubly special relativity,” Lect. Notes Phys. 669 (2005) 131 [arXiv:hep-th/0405273].
- (8) P. Kosinski, J. Lukierski and P. Maslanka, “Local D = 4 field theory on kappa-deformed Minkowski space,” Phys. Rev. D 62 (2000) 025004 [arXiv:hep-th/9902037].
- (9) P. Kosinski, P. Maslanka, J. Lukierski and A. Sitarz, “Towards kappa-deformed D = 4 relativistic field theory,” Czech. J. Phys. 48 (1998) 1407.
- (10) P. Kosinski, P. Maslanka, J. Lukierski and A. Sitarz, “Generalized kappa-deformations and deformed relativistic scalar fields on noncommutative Minkowski space,” arXiv:hep-th/0307038.
- (11) G. Amelino-Camelia and M. Arzano, “Coproduct and star product in field theories on Lie-algebra non-commutative space-times,” Phys. Rev. D 65 (2002) 084044 [arXiv:hep-th/0105120].
- (12) M. Dimitrijevic, L. Jonke, L. Moller, E. Tsouchnika, J. Wess and M. Wohlgenannt, “Field theory on kappa-spacetime,” arXiv:hep-th/0407187; M. Dimitrijevic, F. Meyer, L. Moller and J. Wess, “Gauge theories on the kappa-Minkowski spacetime,” Eur. Phys. J. C 36 (2004) 117 [arXiv:hep-th/0310116]; M. Dimitrijevic, L. Jonke, L. Moller, E. Tsouchnika, J. Wess and M. Wohlgenannt, “Deformed field theory on kappa-spacetime,” Eur. Phys. J. C 31 (2003) 129 [arXiv:hep-th/0307149].
- (13) M. Daszkiewicz, K. Imilkowska, J. Kowalski-Glikman and S. Nowak, “Scalar field theory on kappa-Minkowski space-time and doubly special relativity,” Int. J. Mod. Phys. A 20, 4925 (2005) [arXiv:hep-th/0410058].
- (14) N. Moeller and B. Zwiebach, “Dynamics with infinitely many time derivatives and rolling tachyons,” JHEP 0210, 034 (2002) [arXiv:hep-th/0207107].
- (15) L. Freidel, J. Kowalski-Glikman and S. Nowak, “Star product for –Minkowski space-time and field theory” to appear.
- (16) J. Kowalski-Glikman and S. Nowak, “Doubly special relativity and de Sitter space,” Class. Quant. Grav. 20 (2003) 4799 [arXiv:hep-th/0304101].
- (17) G. Amelino-Camelia and S. Majid, “Waves on noncommutative spacetime and gamma-ray bursts,” Int. J. Mod. Phys. A 15, 4301 (2000) [arXiv:hep-th/9907110].
- (18) L. Freidel and S. Majid, “Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity,” arXiv:hep-th/0601004.
- (19) L. Freidel and E. R. Livine, “Ponzano-Regge model revisited. III: Feynman diagrams and effective field theory,” Class. Quant. Grav. 23 (2006) 2021 [arXiv:hep-th/0502106]. L. Freidel and E. R. Livine, “Effective 3d quantum gravity and non-commutative quantum field theory,” Phys. Rev. Lett. 96, 221301 (2006) [arXiv:hep-th/0512113].
- (20) A. Sitarz, “Noncommutative differential calculus on the kappa Minkowski space,” Phys. Lett. B 349 (1995) 42 [arXiv:hep-th/9409014].
- (21) A. Agostini, G. Amelino-Camelia, M. Arzano, A. Marciano and R. A. Tacchi, “Generalizing the Noether theorem for Hopf-algebra spacetime symmetries,” arXiv:hep-th/0607221.