Quantum Field Theory and Gravity on Non-Commutative Spaces

Transcription

1 Projektbeschreibung Quantum Field Theory and Gravity on Non-Commutative Spaces Grant Application to the Austrian Academy of Sciences Vienna, Austria October

2 1 Background Non-commutative spaces have a long history. Even in the early days of quantum mechanics and quantum field theory, continuous space-time and Lorentz symmetry was considered inappropriate to describe the small scale structure of the universe [1]. It was also argued that one should introduce a fundamental length scale limiting the precision of position measurements. In [2,3], the introduction of a fundamental length is suggested to cure the ultraviolet divergencies occuring in quantum field theory. H. Snyder was the first to formulate these ideas mathematically [4]. He introduced non-commutative coordinates. Therefore, a position uncertainty arises naturally. The success of the renormalisation programme made people forget about these ideas for some time. But when the quantisation of gravity was considered thoroughly, it became clear that the usual concepts of space-time are inadequate and that space-time has to be quantised or non-commutative, in some way. In order to combine quantum theory and gravitation (geometry), one has to describe both in the same language, this is the language of algebras. Geometry can be formulated algebraically in terms of abelian C algebras and can be generalised to non-abelian C algebras (non-commutative geometry). Quantised gravity may even act as a regulator of quantum field theories. This is encouraged by the fact that non-commutative geometry introduces a lower limit for the precision of position measurements. There is also a very nice argument showing that, on a classical level, the selfenergy of a point particle is regularised by the presence of gravity [5]. The approach to non-commutative geometry considered in the proposed project is based on commutator relations of the coordinates [ˆx µ, ˆx ν ] = iθ µν (ˆx), (1) where θ µν (ˆx) is an antisymmetric tensor depending on the coordinates which will be specified in the latter. Functions and physical fields are given as formal power series in the non-commutative coordinates [6]. In some of the discussed cases non-commutative space-time allows for a deformed symmetry structure (quantum group), e.g. κ-deformed Poincaré symmetry. In those cases, the space-time algebra is a module algebra of the symmetry. A main question concerns renormalisability of quantum field theories defined on non-commutative spaces. A novel feature, namely UV/IR mixing [7] seems to threaten it. Ultraviolet degrees of freedom result in infrared divergencies. Qualitatively, UR/IR mixing can be easily understood, from a quantum mechanics point of view [8]. The commutation relation (1) supplies us with an uncertainty x µ x ν θ µν. Together with Heisenberg s uncertainty 11

3 relation x µ p µ 1 - no summation implied -, we obtain δx ν θ µν p µ. Above formula connects the ultraviolet regime in the x µ -direction with infrared effects in the x ν -direction and vice versa. Therefore, it is of vital importance to study renormalisability and UV/IR mixing, in the context of both, constant and space-time dependent non-commutativity structures. A way to circumvent these infrared problems is to consider an expansion in the non-commutativity parameter which we will also use in the proposed project and discuss in greater detail in Section 3. Let us briefly summarise the recent progress in the subject, concentrating on three different types of non-commutative spaces. There are of course many other interesting aspects, ranging from studies of non-commutative geometries to applications to the quantum Hall effect. The selection below is based on the relevance to the proposed project. 1.1 The Quantum Plane R n θ The quantum plane R n θ is obtained by the simplest non-trivial choice for the antisymmetric tensor θ µν (ˆx), namely θ µν = const. This kind of space-time structure has widely been studied being one of its advantages that explicit calculations are possible. Only recently, the symmetry group was identified as a twisted Poincarè group [9, 10]. This twisted group has the same representation theory as the classical Poincaré group. Therefore, a deformed theory contains the same representations for the particles as the undeformed one. A main hope was that by introducing non-locality via non-commuting coordinates divergencies of quantum field theories would go away. However, divergencies persist. Loop integrals in scalar theory differ only by phase factors e iθµν k µp ν form ordinary ones [11]. For non-planar diagrams the ultraviolet divergencies disappear due to these phases, but for planar diagrams the divergencies are unaltered. Although, the non-planar graphs do not contain ultraviolet (uv) divergencies, they explode for vanishing external momentum. That means they are infrared (ir) divergent even for non-zero mass. This property, obtaining ir divergencies when integrating out uv degrees of freedom, is called UV/IR mixing. Also, problems with unitarity occur extracting the Feynman rules directly from the Lagrangian [12]. The solution is provided by a consequent analysis of perturbation theory in a Hamiltonian approach, cf. [13 15]. Many studies are devoted to avoiding of UV/IR mixing and to finding a renormalisable quantum field theory. A proper adjustment of parameters for 12

4 complex φ 4 theory has been discussed in [16]. Using a superfield formulation, it has been proved that this mixing is absent in the Wess-Zumino model [17]. A different approach to real scalar field theory has been presented in [18,19]. An oscillator term - breaking translational invariance explicitly - is introduced in order to modify both, the uv and the ir regime. This theory is renormalisable to all orders and does not show UV/IR mixing. Its generalisation to gauge theory is not at all obvious. A.A. Slavnov [20] advocates an approach to U(1) gauge theory where additional constraints are introduced which are even present in the commutative limit. In the Lagrangian, the additional term λ(x)σ µν F µν is considered, where F µν = µ A ν ν A µ i[a µ, A ν ] is the non-commutative field strength, and λ(x) is a multiplier field. Note that σ µν merely resembles the index structure of the non-commutativity. We have θ µν = ξ σ µν, and the commutative limit is performed by ξ 0. This model is free of non-integrable singularities, at least at 1-loop order. However, we want to emphasise that the analysis presented in [20] lacks the presence of ghosts and a BRS formalism. Still, for higher loop-order there are the problems of overlapping singularities and of the socalled commutant divergencies [21]. In the same line, Yang-Mills theory was studied on the quantum plane in much detail. N. Seiberg and E. Witten [22] argued that there is a map relating non-commutative gauge fields and parameters to their counterparts on commutative space-time. This map is a formal power series in the noncommutativity parameter θ µν. The approach widely used in the proposed project is to expand Seiberg-Witten maps and star products up to some fixed order in θ. On the one hand it avoids the problem of UV/IR mixing, on the other hand these theories seem to be non-renormalisable [23]. Therefore, they might have to be interpreted as effective theories. However, there might be additional symmetries or structures evolving from more elaborate space-time structures than canonical (such as κ-deformed) which may prove useful for renormalisation. Thus, a study of these aspects may provide deep insights still. Using Seiberg-Witten maps arbitrary gauge groups can be considered, and therefore we can describe realistic gauge models [6,24]. One of the most interesting ones is the Standard Model of elementary particle physics [25,26]. Remarkably, there is an arbitrariness in choosing the representation of the gauge group in the gauge kinetic sector. Therefore, we distinguish between the simplest choice - minimal model - and other choices - extended models [25]. The most striking difference is the absence of a triple photon vertex in the minimal model. The phenomenology of both models has been initiated. Emphasis has been put on processes which are forbidden in the usual Standard Model 13

5 [26 29], such as neutral particles coupling to photons [30]. There have been various attempts to incorporate canonical non-commutative space-time into gravity, ranging from pure insertion of star products to string theoretical approaches. Here, I just want to mention the connection between non-commutative field theory and ordinary field theory with a gauge field dependent background field provided in [31]. 1.2 κ-deformed Euclidean and Minkowski Space-Time In [32], κ-deformed Minkowski space was first introduced as a module algebra of the κ-deformed Poincaré algebra constructed in [33]. The space algebra is generated by coordinates ˆx 0,..., ˆx 3, which satisfy linear commutation relations [ˆx 0, ˆx i ] = iaˆx i, where i = 1, 2, 3. All other commutators vanish. The κ-poincaré invariant field equations and scalar field theory on κ-minkowski space have been studied intensively. A Klein-Gordon operator has been introduced, e.g. in [32,33], and a Dirac operator for κ-deformed Poincaré algebra has been constructed in [34 36]. The field equations could be deduced from a variational principle [36] using an integral enjoying the trace property. Scalar field theories have been studied in [37 40]. Nevertheless, a thorough analysis of scalar models in the spirit of [36] is still missing. The study of gauge theories based on Seiberg-Witten maps has also been initiated by [36, 41]. The most remarkable feature of Seiberg-Witten maps is that they are derivative valued, i.e., κ-poincaré algebra valued since the derivatives generate translations. It is due to the non-trivial Leibnitz rule for the derivatives and prepares the ground for novel structures. 1.3 More General Quantum Spaces Yet, different space-time structures have been considered in [42]. The authors discuss two different Lie algebra commutation relations for space-time, namely a space-dependent case and a space-time dependent one [ˆx 1, ˆx 2 ] = iαˆx 3, [ˆx 1, ˆx 3 ] = [ˆx 2, ˆx 3 ] = 0, [ˆx, ẑ] = i Rˆx +, [ˆx, ẑ] = i Rˆx. Both choices can be motivated from String Theory. In both cases, the nonplanar tadpole contribution is evaluated for real scalar φ 4 theory. The uv 14

6 divergence is softened from quadratic in canonical non-commutative field theory to linear and logarithmic, respectively. Also, ir divergencies are softened with respect to the canonical case. Note that θ-expansion has not been employed here in order to study the UV/IR mixing property. It seems reasonable to study further generalisations of the non-commutativity structure to a generic function θ µν (x) satisfying the Jacobi identity. In [43], we have constructed Seiberg-Witten maps for gauge fields using covariant coordinates which had been calculated for a general Poisson structure in [6]. The results have been obtained as a power series expansion in the noncommutativity parameters. Also, some first phenomenological considerations are provided. For the lack of symmetry, the theory is considered as an effective theory. Seiberg-Witten maps for this kind of space-time are also studied in [44]. Remarkably, they also give an example of a non-commutative gauge theory which reduces to scalar electrodynamics on a curved background, in the commutative limit. 2 Specific Aims There are several problems in non-commutative field theory we want to address. We place emphasis on the quantisation of non-commutative theories and on phenomenology. 2.1 Non-Commutative Standard Model In the usual Standard Model, renormalisability and gauge invariance fix the representation for the gauge fields in the kinetic sector. In the noncommutative case, criteria fixing the representations, especially renormalisability, still have to be studied. So far, different models have been discussed which employ different representations. Also, experimental evidence is not yet available. We want to exploit both versions, the minimal and the nonminimal ones in order to study processes which might lead to experimental evidence for non-commutativity. A further aim of this project is to remove the arbitrariness in choosing the representation for the gauge field studying renormalisation, discrete symmetries of the action and phenomenological implications. In a next step, we want to study the expansion of the Non- Commutative Standard Model (NCSM) [25] up to second order in θ. This is necessary in order to obtain reliable bounds on the non-commutativity scale using scattering-type of processes if these processes are forbidden in the usual Standard Model. First steps in this direction have been done in [45]. 15

7 2.2 Field Theory on κ-deformed Minkowski Space For the construction of Lagrangian models, we need a measure invariant under κ-poincaré transformations [46, 47]. The trace property is also a desireable feature for the integral [36]. This property ensures that variational methods can be applied. We will be concerned to ensure both properties. Once we are able to examine actions, we can study physical models, their predictions and the structure of the newly found interactions. Still we are missing a consequent analysis of φ 4 theory on κ-deformed space-time in the spirit of [36]. So far for the construction of actions, integrals have been considered [40, 48] which do enjoy neither κ-poincaré invariance nor trace property. Therefore, it is of vital importance to discuss models using an integral which satisfies both diserable features (or at least the trace property). A main issue will be its renormalisability. As mentioned above, a lot of new features and structures arise considering the Seiberg-Witten map on κ-deformed spaces [36, 41]. In a first step, we will compute the covariant coordinates [6] and compare their relation to covariant derivatives with results obtained in [43]. The aim is to study and to understand these structures better and to construct realistic gauge theoretical models. These models will then be both, covariant under κ-deformed symmetry and under arbitrary gauge symmetry. Above all, we will study the deformation of the Standard Model of elementary particle physics. One of the main tasks will be the study of its renormalisability and its phenomenological implications. In [49], the connection between κ-deformation and quantum loop gravity is studied. The authors conclude that the low energy limit is a κ-invariant theory. This is a far reaching result which surely deserves a lot of attention. The far goal the proposed project is aiming at is the construction of a κ- deformed toy model of gravity. 2.3 Consistent Models and General Space-Time Structures We plan to study a U(1) pure gauge model on quantum space-time with a general non-commutativity structure θ(x) satisfying the Jacobi identity. The study of phenomenological implications is the main aim. Maybe it is also possible to some study non-perturbative effects, such as UV/IR mixing, if we restrict the non-commutativity parameters a bit further and do not use Seiberg-Witten maps. It is also very interesting and instructive to work out the transition between these models and models that enjoy a deformed space-time symmetry. 16

8 Another aim of the project is the study of UV/IR mixing and renormalisability of another model proposed by Slavnov [20]. As we mentioned before, novel constraints have been introduced which seem to fix UV/IR mixing problems. Foremost, we want to have a close look at the classical model in a Hamiltonian approach. In this formulation, the discussion of constraints is the most central point. Relations between fields and conjugated momenta imply the socalled primary constraints. The total Hamiltonian is obtained from the naive Hamiltonian by adding the primary constraints with unknown coefficient functions. Discussing the time evolution of the primary constraints, the unknown coefficient functions are determined and socalled secondary constraints might appear. For the model to be consistent the series of constraints must stop at some point. The pure gauge sector of the Slavnov model is characterised by three primary constraints and three additional secondary constraints. The first step will be done at the classical level. In a second step, we want to study the quantised theory. Furthermore, we have to study the properties of the model in two-loop order in order to discuss the mentioned problems of overlapping singularities and commutant divergencies. A further task is the discussion of the meaning of the constraint σ µν F µν = 0 in a Hamiltonian approach. This part of the project will be conducted in coorperation with M. Schweda and his group at the Vienna Technical University. 3 Methods The main methods we want to employ in the proposed project are presented in [6, 25, 36, 41]. First of all, this is the star product approach to non-commutativity. In the very beginning, the coordinates are generators of the non-commutative (function) algebra. Symmetry algebras can be introduced as acting consistently on the space-time algebra, cf. [36]. Even field equations and field theory can be done. However, at some point we change ends and swap on to usual commutative function spaces endowed with the non-commutative star product. In this way we can take advantage of both structures. Secondly, we use Seiberg-Witten maps. These are formal power series in the non-commutativity parameter resembling a map form commutative fields to their non-commutative counterparts. This does not only work for θ µν = const, but also for general x-dependent structures [41, 43, 44]. In most cases, there will be no closed form of the Seiberg-Witten map. Therefore, an expansion in the non-commutativity parameter seems to be obligatory. When considering non-perturbative effects, such as UV/IR mixing, we have to drop Seiberg-Witten maps or use closed expressions in θ, if they are 17

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