2 2 1 Fe b 20 06 Self-consistency in Theories with a Minimal Length

The aim of this paper is to clarify the relation between three different approaches of theories with a minimal length scale: A modification of the Lorentzgroup in the ’Deformed Special Relativity’, theories with a ’Generalized Uncertainty Principle’ and those with ’Modified Dispersion Relations’. It is shown that the first two are equivalent, how they can be translated into each other, and how the third can be obtained from them. An adequate theory with a minimal length scale requires all three features to be present. PACS numbers: 11.10. Gh, 11.30. Cp, 12.90. +b 1. The Role of the Planck Scale Gravity itself is inconsistent with physics at very short scales. The introduction of gravity into quantum field theory appears to spoil their renormalizability and leads to incurable divergences. It has therefore been suggested that gravity should lead to an effective cutoff in the ultraviolet, i.e. to a minimal observable length. It is amazing enough that all attempts towards a fundamental theory imply the existence of such a minimal length scale. It is expected that the minimal length, Lm is close by, or identical to the Planck length. Motivations for the occurrence of a minimal length are manifold. A minimal length can be found in String Theory [1, 2, 3, 4], Quantum Loop Gravity [5, 6, 7, 8], and Non-Commutative Geometries [9, 10]. It can be derived from various studies of thoughtexperiments [11, 12, 13, 14], phenomenological examinations of precision measurements [15, 16, 17, 18], from black hole physics [19, 20], the holographic principle [21], a Tduality of the path-integral [22, 23, 24] and probably further more. For reviews, the interested reader is referred to [25, 26, 27]. The listed points are cross-related in many ways. The examination of these similarities in the presently availably approaches is a promising way to increase our knowledge about the quantum nature of gravity. Besides the various attempts to pin down the emergence of a finite resolution of spacetime, the inclusion of the minimal length into the theoretical framework of the Standard Model (SM) has been examined from different sides: The Generalized Uncertainty Principle (GUP), Deformed Special Relativity (DSR) and Modified Self-consistency in Theories with a Minimal Length 2 Dispersion Relations (MDR). These theories are an effective description of the expected effects of quantum gravity and provide us with a useful framework to describe the phenomenology of physics beyond the SM. The aim of this work is to clarify the interrelationships of the different approaches to ensure the self-consistency of theories with a fundamental minimal length scale. This paper is organized as follows. In the next section, we will briefly introduce the basic formalism of the three approaches. In section three we will examine the relations between them. We conclude in section 4. Throughout this paper we use the convention h̄ = c = 1. 2. The Three Faces of the Minimal Length The phenomenology that arises from a finite resolution of space-time, and the mathematical structure associated with it, have been investigated closely [28, 29, 30, 31, 32, 33, 34]. In the scenario without extra dimensions, the derived modifications are important mainly for structure formation and the early universe [35, 36, 37, 38, 39, 40, 41, 42]. The importance to deal with the minimal length is sensibly enhanced if we consider a spacetime with large extra dimensions [43, 44, 45].