Non-commutative Geometry & Physics

This article is an introduction to the ideas of non-commutative geometry and star products. We will discuss consequences for physics in two different settings: quantum field theories and astrophysics. In case of quantum field theory, we will discuss two recently introduced models in detail. Astrophy...

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Date:	2010
Main Author:	Wohlgenannt, M.
Format:	Article
Language:	English
Published:	Відділення фізики і астрономії НАН України 2010
Subjects:	Поля та елементарні частинки
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Cite this:	Non-commutative Geometry & Physics / M. Wohlgenannt // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 5-14. — Бібліогр.: 34 назв. — англ.

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spelling	Wohlgenannt, M. 2010-11-04T09:31:08Z 2010-11-04T09:31:08Z 2010 Non-commutative Geometry & Physics / M. Wohlgenannt // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 5-14. — Бібліогр.: 34 назв. — англ. 2071-0194 PACS 11.10.Nx, 02.40.Gh, 95.30.Cq https://nasplib.isofts.kiev.ua/handle/123456789/13280 This article is an introduction to the ideas of non-commutative geometry and star products. We will discuss consequences for physics in two different settings: quantum field theories and astrophysics. In case of quantum field theory, we will discuss two recently introduced models in detail. Astrophysical aspects will be discussed, by considering modified dispersion relations. Робота мiстить деякi основнi iдеї некомутативної геометрiї. Її застосування в фiзицi розглянуто у двох напрямках: у квантовiй теорiї поля та астрофiзицi. Детально описано деякi сучаснi моделi в квантовiй теорiї поля. В контекстi астрофiзичних аспектiв некомутативної геометрiї отримано модифiкованi дисперсiйнi спiввiдношення. en Відділення фізики і астрономії НАН України Поля та елементарні частинки Non-commutative Geometry & Physics Некомутативна геометрія і фізика Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Non-commutative Geometry & Physics
spellingShingle	Non-commutative Geometry & Physics Wohlgenannt, M. Поля та елементарні частинки
title_short	Non-commutative Geometry & Physics
title_full	Non-commutative Geometry & Physics
title_fullStr	Non-commutative Geometry & Physics
title_full_unstemmed	Non-commutative Geometry & Physics
title_sort	non-commutative geometry & physics
author	Wohlgenannt, M.
author_facet	Wohlgenannt, M.
topic	Поля та елементарні частинки
topic_facet	Поля та елементарні частинки
publishDate	2010
language	English
publisher	Відділення фізики і астрономії НАН України
format	Article
title_alt	Некомутативна геометрія і фізика
description	This article is an introduction to the ideas of non-commutative geometry and star products. We will discuss consequences for physics in two different settings: quantum field theories and astrophysics. In case of quantum field theory, we will discuss two recently introduced models in detail. Astrophysical aspects will be discussed, by considering modified dispersion relations. Робота мiстить деякi основнi iдеї некомутативної геометрiї. Її застосування в фiзицi розглянуто у двох напрямках: у квантовiй теорiї поля та астрофiзицi. Детально описано деякi сучаснi моделi в квантовiй теорiї поля. В контекстi астрофiзичних аспектiв некомутативної геометрiї отримано модифiкованi дисперсiйнi спiввiдношення.
issn	2071-0194
url	https://nasplib.isofts.kiev.ua/handle/123456789/13280
citation_txt	Non-commutative Geometry & Physics / M. Wohlgenannt // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 5-14. — Бібліогр.: 34 назв. — англ.
work_keys_str_mv	AT wohlgenanntm noncommutativegeometryphysics AT wohlgenanntm nekomutativnageometríâífízika
first_indexed	2025-11-26T13:17:10Z
last_indexed	2025-11-26T13:17:10Z
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fulltext	FIELDS AND ELEMENTARY PARTICLES ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 5 NON-COMMUTATIVE GEOMETRY & PHYSICS1 M. WOHLGENANNT Institute for Theoretical Physics, Vienna University of Technology (8-10, Wiedner Hauptstrasse, Vienna A-1040, Austria; e-mail: michael. wohlgenannt@ univie. ac. at ) PACS 11.10.Nx, 02.40.Gh, 95.30.Cq c©2010 This article is an introduction to the ideas of non-commutative ge- ometry and star products. We will discuss consequences for physics in two different settings: quantum field theories and astrophysics. In case of quantum field theory, we will discuss two recently intro- duced models in detail. Astrophysical aspects will be discussed, by considering modified dispersion relations. 1. Why Non-Commutativity? Non-commutative spaces have a long history. Even in the early days of quantum mechanics and quantum field theory, the continuous space-time and the Lorentz symmetry were 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 was sug- gested to cure the ultraviolet divergences occurring in quantum field theory. H. Snyder was the first who for- mulated these ideas mathematically [4]. He introduced non-commutative coordinates. Therefore, a position un- certainty arises naturally. The success of the renormal- ization program made people forget about these ideas for some time. But when the quantization of gravity was considered thoroughly, it became clear that the usual concepts of space-time are inadequate, and the space- time has to be quantized or non-commutative, in some way. There is a deep conceptual difference between quan- tum field theory and gravity: The space and the time are 1 This submission is a part of the Project of Scientific Coopera- tion between the Austrian Academy of Sciences (ÖAW) and the National Academy of Sciences of Ukraine (NASU) No. 01/04, Quantum Gravity, Cosmology, and Categorification. considered as parameters in the former and as dynamical entities in the latter. In order to combine quantum the- ory and gravitation (geometry), one has to describe both in the same language, this is the language of algebras [5]. Geometry can be formulated algebraically in terms of Abelian C∗ algebras and can be generalized to non- Abelian C∗ algebras (non-commutative geometry). The quantized gravity can even act as a regulator of quan- tum field theories. This is encouraged by the fact that a non-commutative geometry introduces a lower limit for the precision of position measurements. There is also a very nice argument showing that, on the classical level, the self-energy of a point particle is regularized by the presence of gravity [6]. Let us consider an electron and a shell of radius ε around the electron. The self-energy of the electron is the self-energy of the shell m(ε), in the limit ε→ 0. The quantity m(ε) is given by m(ε) = m0 + e2 ε , where m0 and e are, respectively, the rest mass and the charge of an electron. In the limit ε → 0, m(ε) will diverge. Including the Newtonian gravity, we have to modify this equation, m(ε) = m0 + e2 ε − Gm2 0 ε , where G is Newton’s gravitational constant. The self- energy m(ε) will still diverge for ε→ 0, unless the mass and the charge are finely tuned. Considering general rel- ativity, we know that the energy, particularly the energy of electron’s electric field, is a source of the gravitational field. Again, we have to modify the above equation, m(ε) = m0 + e2 ε − Gm(ε)2 ε . M. WOHLGENANNT The solution of this quadratic equation is straightfor- ward: m(ε) = − ε 2G ± ε 2G √ 1 + 4G ε (m0 + e2 ε ). We are interested in the positive root. Miraculously, the limit ε→ 0 is finite, m(ε→ 0) = e√ G . This is a non-perturbative result, since m(ε→ 0) cannot be expanded around G = 0. The quantity m(ε → 0) does not depend on m0; therefore, there is no fine tun- ing. Classical gravity regularizes the self-energy of an electron on a classical level. However, this does not make the quantization of space-time unnecessary, since quantum corrections to the above picture will again in- troduce divergences. But it provides an example for the regularization of physical quantities by introducing grav- ity. So the hope is raised that the introduction of grav- ity formulated in terms of a non-commutative geometry will regularize physical quantities even on the quantum level. On the other hand, there is the old simple argument that a smooth space-time manifold contradicts quan- tum physics. If one localizes an event within a region of extension l, an energy of the order of hc/l is trans- ferred. This energy generates a gravitational field. A strong gravity field prevents, on the other hand, signals to reach an observer. Inserting the energy density into Einstein’s equations gives a corresponding Schwarzschild radius r(l). This provides a limit on the smallest possible l, since it is certainly operationally impossible to local- ize an event beyond this resulting Planck length. To the best of our knowledge, the first time this argument was cast into precise mathematics was in the work by Do- plicher, Fredenhagen, and Roberts [7]. They obtained what is now called the canonical deformation but aver- aged over 2-spheres. At which energies this transition to discrete structures might take place, or at which ener- gies the non-commutative effects occur is a point much debated on. From various theories generalized to non-commutative coordinates, limits on the non-commutative scale have been derived. These generalizations have mainly consid- ered the so-called canonical non-commutativity,[ x̂i, x̂j ] = iθij , θij = −θji ∈ C. Let us name a few estimates of the non-commutativity scale. A very weak limit on the non- commutative scale ΛNC is obtained from an additional energy loss in stars due to the coupling of neutral neu- trinos to photons, γ → νν̄ [8]. They get ΛNC > 81 GeV. The estimate is based on the argument that, within any new mechanism, the energy losses must not exceed the standard neutrino losses from the Standard Model by much. A similar limit is obtained in [9] from the calcu- lation of the energy levels of a hydrogen atom and the Lamb shift within non-commutative quantum electrody- namics, ΛNC & 104 GeV. If ΛNC = O(TeV), measurable effects may occur for the anomalous magnetic moment of a muon which may ac- count for the reported discrepancy between the Stan- dard Model prediction and the measured value [10]. In cosmology and astrophysics, non-commutative ef- fects might be observable. One suggestion is that the modification of a dispersion relation due to the (κ−)non-commutativity can explain the time delay of high-energy γ rays, e.g., from the active galaxy Makar- ian 142 [11, 12]. We will discuss this point in Sec- tion 4. A brief introduction to non-commutative geom- etry will be provided in Section 2. In Section 3, we consider the quantum field theory on non-commutative spaces. We will put emphasis on scalar field theories and will only briefly discuss the case of gauge theo- ries. 2. Some Basic Notions of a Non-Commutative Geometry At the present time, there are three major approaches tackling the problem of quantizing gravity: String The- ory, Quantum Loop Gravity, and Non-Commutative Ge- ometry. Before we discuss some basic concepts of non- commutative geometry, let us state some advantages and disadvantages of the other theories, cf. [13]. Back- ground independence will be a major issue. General Relativity can be described in a coordinate-free way. In some cases, theories for gravity are expanded around the Minkowski metric. They explicitly depend on the back- ground Minkowski metric, i.e., the background indepen- dence is violated. In String Theory, the basic constituents are 1- dimensional objects, strings. The interaction between strings can be symbolized by two-dimensional Riemann manifolds with boundary, e.g., a vertex: 6 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 NON-COMMUTATIVE GEOMETRY & PHYSICS 2 1 The interaction region is not a point anymore. Hence, there is also the hope for that the divergences in the per- turbation theory of quantum field theory are not present. Advantages Disadvantages graviton contained in higher dimensions needed: the particle spectrum superstrings D = 10, 11 bosonic string D = 26 black hole entropy dependence on background space-time geometry mathematical beauty many free parameters (dualities, ...) and string-vacua almost no predictions Quantum Loop Theory studies the canonical quanti- zation of General Relativity in 3+1 dimensions. Advantages Disadvantages background independence very few predictions quantized area operator no matter included 3 + 1 dimensional space-time technical difficulties We are going to discuss the third approach in more details in the next subsection. The three approaches are connected to one another. In [14], the connection be- tween κ-deformation and quantum loop gravity is stud- ied. The authors conclude that the low-energy limit of quantum loop gravity is a κ-invariant field theory. This is a far reaching result which deserves a lot of at- tention. Also String Theory is related to certain non- commutative field theories in the limit of the vanishing string coupling [15]. A better understanding of the in- terrelations will provide clues how a proper theory of quantum gravity should look like. 2.1. Non-commutative geometry In our approach, we consider a non-commutative geome- try as a generalization of quantum mechanics. Thereby, we generalize the canonical commutation relations of the phase space operators x̂i and p̂j . Most commonly, the commutation relations are chosen to be either constant or linear, or quadratic in the generators. In the canonical case, the relations are constant, [x̂i, x̂j ] = iθij , (1) where θij ∈ C is an antisymmetric matrix, θij = −θji. The linear or Lie algebra case [x̂i, x̂j ] = iλijk x̂ k, (2) where λijk ∈ C are the structure constants, basically has been discussed in two different approaches, fuzzy spheres [16] and the κ-deformation [17–19]. Last but not least, we have the quadratic commutation relations [x̂i, x̂j ] = ( 1 q R̂ijkl − δ i lδ j k)x̂ kx̂l, (3) where R̂ijkl ∈ C is the so-called R̂-matrix. For a ref- erence, see, e.g., [20, 21]. The relations between coor- dinates and momenta (derivatives) can be constructed from the above relations in a consistent way [19, 22]. Most importantly, the usual commutative coordinates are recovered in a certain limit, θij → 0, λijk → 0 or Rijkl → 0, respectively. In quantum mechanics, the com- mutation relations lead to the Heisenberg uncertainty, Δxi Δpj & δij ~ 2 . Similarily, we obtain an uncertainty relation for the co- ordinates in the non-commutative case, e.g., Δxi Δxj & \|θij \| 2 . (4) In a next step, we need to know which functions of the non-commutative coordinates are. Classically, the smooth functions can be approximated by power series. So, a function f(x) can be written as f(x) = ∑ I aI (x1)i1(x2)i2(x3)i3(x4)i4 , ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 7 M. WOHLGENANNT where I = (i1, .., i4) is a multiindex, in a four- dimensional space. The commutative algebra of func- tions generated by the coordinates x1, x2, x3, and x4 is denoted by A = C〈〈x1, ..., x4〉〉 [xi, xj ] ≡ C[[x1, ..., x4]], (5) i.e., [xi, xj ] = 0. The generalization to the non-commutative algebra of functions Â on a non- commutative space Â = C〈〈x̂1, ..., x̂n〉〉 I , (6) where I is the ideal generated by the commutation re- lations of coordinate functions, can be found in (1-3). Again, an element f̂ of Â is defined by a power series in the non-commutative coordinates. There is one com- plication: Since the coordinates do not commute, the monomials x̂ix̂j and x̂j x̂i, e.g., are different operators. Therefore, we have to specify basis monomials with some care. This means that we have to give an ordering pre- scription. Let us discuss two different orderings briefly which will be denoted by : :. The normal ordering means the following: : x̂i : = x̂i, i = 1, 2, 3, 4 : x̂2x̂4x̂2x̂1 : = x̂1(x̂2)2x̂4. (7) Powers of x̂1 come first, then powers of x̂2, and so on. A non-commutative function is given by the formal ex- pansion f̂(x̂) = ∑ I bI : (x̂1)i1(x̂2)i2(x̂3)i3(x̂4)i4 := = ∑ I bI (x̂1)i1(x̂2)i2(x̂3)i3(x̂4)i4 . (8) A second choice is the symmetric ordering. There, we define : x̂i : = x̂i, : x̂ix̂j : = 1 2 (x̂ix̂j + x̂j x̂i), : x̂ix̂j x̂k : = 1 6 ( x̂ix̂j x̂k + x̂ix̂kx̂j + x̂j x̂ix̂k + (9) +x̂j x̂kx̂i + x̂kx̂ix̂j + x̂kx̂j x̂i ) , ... A non-commutative function is given by the formal ex- pansion f̂(x̂) = ∑ I cI : (x̂1)i1(x̂2)i2(x̂3)i3(x̂4)i4 : . symmetric ordering can also be achieved by exponentials, eikix̂ i = 1 + ikix̂ i − 1 2 kix̂ ikj x̂ j + · · · = 1 + ikix̂ i− −1 2 (k1x̂ 1 + · · ·+ k4x̂ 4)(k1x̂ 1 + · · ·+ k4x̂ 4) + . . . , (10) and, therefore, f̂(x̂) = ∫ d4k c(k)eikix̂ i , (11) with a coefficient function c(k). This formula will be of vital importance in the next subsection. The normal and symmetric orderings define different choices of a basis in the same non-commutative algebra Â. Most importantly, many concepts of differential ge- ometry can be formulated using the non-commutative function algebra Â such as differential structures. In the following sections, we will concentrate on the first two cases of non-commutative coordinates, namely canonical (1) and κ-deformed (2) space-time structures. 2.2. Star product Star products are a way to return to the familiar con- cept of commutative functions f(x) within the non- commutative realm. In addition, we have to include a non-commutative product denoted by ∗. Earlier, we have introduced the algebras A and Â and have dis- cussed the choice of a basis or an ordering in the latter. We need to establish an isomorphism between the non- commutative algebra Â and the commutative function algebra A. Let us choose symmetrically ordered monomials as a basis in Â. We now map the basis monomials in A onto the according symmetrically ordered basis elements of Â W : A → Â, xi 7→ x̂i, (12) xixj 7→ 1 2 (x̂ix̂j + x̂j x̂i) ≡ : x̂ix̂j : . The ordering is indicated by : :; W is an isomorphism of vector spaces. In order to extend W to an algebra iso- morphism, we have to introduce a new non-commutative multiplication ∗ in A. This ∗-product is defined by W (f ∗ g) := W (f) ·W (g) = f̂ · ĝ, (13) 8 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 NON-COMMUTATIVE GEOMETRY & PHYSICS where f, g ∈ A, f̂ , ĝ ∈ Â. Explicitly, we have f(x) = ∑ I aI(x1)i1(x2)i2(x3)i3(x4)i4 ‖ ‖ Quantization map W ⇓ f̂(x̂) = ∑ I aI : (x̂1)i1(x̂2)i2(x̂3)i3(x̂4)i4 : . The star product is constructed in the following way: f̂ · ĝ = ∑ I,J aIbJ : (x̂1)i1(x̂2)i2(x̂3)i3(x̂4)i4 : : (x̂1)j1(x̂2)j2(x̂3)j3(x̂4)j4 := (14) = ∑ K cK : (x̂1)k1(x̂2)k2(x̂3)k3(x̂4)k4 : , (15) where ĝ = ∑ J bJ : (x̂1)j1(x̂2)j2(x̂3)j3(x̂4)j4 :. Conse- quently, we obtain f ∗ g (x) = ∑ J bJ(x̂1)j1(x̂2)j2(x̂3)j3(x̂4)j4 . (16) The information on the non-commutativity of Â is en- coded in the ∗-product. The Weyl quantization proce- dure uses the exponential representation of the symmet- rically ordered basis. The above procedure yields f̂ = W (f) = 1 (2π)n/2 ∫ dnk eikj x̂ j f̃(k), (17) f̃(k) = 1 (2π)n/2 ∫ dnx e−ikjx j f(x), (18) where we have replaced the commutative coordinates by non-commutative ones (x̂i) in the inverse Fourier trans- formation (17). Hence, we obtain (A, ∗) ∼= (Â, ·), (19) i.e., W is an algebra isomorphism. Using Eq. (13), we are able to compute the star product explicitly, W (f ∗ g) = 1 (2π)n ∫ dnk dnp eikix̂ i eipj x̂ j f̃(k)g̃(p). (20) Because of the non-commutativity of the coordinates x̂i, we need the Campbell–Baker–Hausdorff (CBH) formula eAeB = eA+B+ 1 2 [A,B]+ 1 12 [[A,B],B]− 1 12 [[A,B],A]+.... (21) Clearly, we need to specify the commutation relations of the coordinates in order to evaluate the CBH formula. We will consider the canonical and linear cases as exam- ples. 2.2.1. Canonical case Due to the constant commutation relations [x̂i, x̂j ] = iθij , the CBH formula will terminate, terms with more than one commutator will vanish, exp(ikix̂i) exp(ipj x̂j) = = exp ( i(ki + pi)x̂i − i 2 kiθ ijpj ) . (22) Relation (20) now reads f ∗ g (x) = 1 (2π)n ∫ dnkdnp ei(ki+pi)x i− i 2kiθ ijpj f̃(k)g̃(p), (23) and we get, for the ∗-product the Moyal–Weyl product [23], f ∗ g (x) = exp( i 2 ∂ ∂xi θij ∂ ∂yj ) f(x)g(y) ∣∣∣ y→x . (24) The same reasoning can be applied to the case of normal ordering. In this basis, a non-commutative function f is given by f̂(x̂) = 1 (2π)n/2 ∫ dnk f̃(k)eik1x̂ 1 eik2x̂ 2 eik3x̂ 3 eik4x̂ 4 . (25) Relation (20) has to be replaced by f̂ · ĝ = 1 (2π)n ∫ dnkdnpf̃(k)g̃(p)eik1x̂ 1 . . . . . . eik4x̂ 4 eip1x̂ 1 . . . eip4x̂ 4 . (26) Using the CBH formula, eiax̂ i eibx̂ j = eibx̂ j eiax̂ i e−iabθ ij , we obtain, for the star product for normal ordering, f ∗N g (x) = exp( ∑ i>j i ∂ ∂xi θij ∂ ∂yj ) f(x)g(y) ∣∣∣ y→x . (27) In both cases, we can now explicitly show that Eq. (1) is satisfied. The star product enjoys a very important property,∫ d4x f ? g ? h = ∫ d4xh ? f ? g, ∫ d4xf ? g = ∫ d4x f · g. (28) This is called the trace property. ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 9 M. WOHLGENANNT 2.2.2. κ-Deformed case The following choice of a linear commutation relation is called κ-defomation: [x̂1, x̂p] = iax̂p, [x̂q, x̂p] = 0, (29) where p, q = 2, 3, 4. Because the structure is more in- volved, the computation of a star product is not as easy as in the canonical case. Therefore, we will just state the result. The symmetrically ordered star product is given by [22] f ∗ g (x) = ∫ d4k d4p f̃(k)g̃(p) ei(ωk+ωp)x 1 × ×eix(keaωpA(ωk,ωp)+pA(ωp,ωk)), (30) where k = (ωk,k), and x = (x2, x3, x4). We have used the definition A(ωk, ωp) ≡ a(ωk + ωp) ea(ωk+ωp) − 1 eaωk − 1 aωk . (31) The normal ordered star product has the form [22] f ∗N g (x) = lim y → x z → x e xj ∂ ∂yj (e −ia ∂ ∂z1 −1) f(y)g(z) = = ∫ d4p d4k (2π)4 eix 1(ωk+ωp)eix(keaωp+p)f̃(k)g̃(p). (32) In the κ-deformed case, the trace property is modified. We have to introduce an integration measure µ(x):∫ d4xµ(x) (f ? g) (x) = ∫ d4xµ(x) (g ? f) (x). (33) The above relation also determines the function µ(x), see, e.g., [19]. 3. Non-Commutative Quantum Field Theory Many models of non-commutative quantum field theory have been studied in recent years, and a coherent picture is beginning to emerge. One of the surprising features is the so-called ultraviolet (UV)/infrared (IR) mixing, where the usual divergences of field theory in the UV are reflected by new singularities in the IR. This is es- sentially a reflection of the uncertainty relation: deter- mining some coordinates to a very high precision (UV) implies a large uncertainty (IR) for others. This leads to a serious problem for the usual renormalization pro- cedure of quantum field theories which has only recently been overcome for a scalar-field theoretical model on the canonical deformed Euclidean space [24,25]. This model will be discussed in subsection 3.1. Most models con- structed so far use the canonical space-time, the simplest deformation. Therefore, we will also describe a quantum field theory on a more complicated structure, namely a κ-deformed space, here. Nevertheless, the problem of UV/IR mixing could not be solved by this deformation. 3.1. Scalar field theory In this subsection, we want to sketch two different mod- els of scalar fields on a non-commutative space-time. The first model is formulated on a κ-deformed Euclidean space [26], the second model given in [24,25] on a canon- ically deformed Euclidean space. The commutation relations of coordinates for the κ- deformed case are given by Eq. (29). For simplicity, we concentrate on the Euclidean version and use the symmetrically ordered star product given in Eq. (30). The κ-deformed spaces allow for a generalized coordinate symmetry, the so-called κ-Poincaré symmetry [17, 19]. Therefore, also the action should be invariant under this symmetry. In [19], the κ-Poincaré algebra and the action of its generators on commutative functions are explicitly calculated starting from the commutation relations (29). In order to describe scalar fields on a κ-deformed space, we need to write down an action. Therefore, we have to know the κ-deformed version of the Klein–Gordon oper- ator and an integral invariant under κ-Poincaré transfor- mations. The Klein–Gordon operator is a Casimir one in the translation generators (momenta) [19]. Acting on commutative functions, it is given by the expression �∗ = 4∑ i=1 ∂i∂i 2(1− cos a∂1) a2∂2 1 . (34) A κ-Poincaré invariant integral is given in [27] and has the form (φ, ψ) = ∫ d4xφ(Kψ̄), (35) where K is a suitable differential operator, K = ( −ia∂1 e−ia∂1 − 1 )3 . (36) In the momentum space, this amounts to (φ, ψ) = ∫ d4q ( −aωq e−aωq − 1 )3 φ̃(q) ¯̃ ψ(q). (37) 10 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 NON-COMMUTATIVE GEOMETRY & PHYSICS Therefore, the action for a scalar field with the φ4 inter- action is given by S[φ] = −(φ, (�∗ −m2)ψ)+ + g 4! (b(φ ∗ φ, φ ∗ φ) + d(φ ∗ φ ∗ φ ∗ φ, 1)) . (38) In the above action, we have not included all possible interaction terms. A term proportional to (φ ? φ ? φ, φ) is missing. This term would lead, however, to a peculiar behavior; therefore, it is ignored. For more details, see [26]. In the momentum space, the action has the form S[φ] = ∫ d4q ( −aωq e−aωq − 1 )3 φ̃(q) ( q2 2(cosh aωq − 1) a2ω2 q + + m2 ) ¯̃ φ(q) + b g 4! ∫ d4z 4∏ i=1 d4ki× × ( a(ωk3 + ωk4) ea(ωk3+ωk4 ) − 1 )3 φ̃(k1)φ̃(k2)φ̃(k3)φ̃(k4)× ×eiz 1∑ωki exp ( iz [k1e aωk2A(ωk1 , ωk2)+ + k2A(ωk2 , ωk1) + k3e −aωk4A(−ωk3 ,−ωk4)+ + k4A(−ωk4 ,−ωk3)] ) + d g 4! ∫ d4z 4∏ i=1 d4ki× ×eiz 1∑ωki φ̃(k1)φ̃(k2)φ̃(k3)φ̃(k4)× × exp [iz(k1e aωk2A(ωk1 , ωk2) + k2A(ωk2 , ωk1)) × × ea(ωk3+ωk4 )A(ωk1 + ωk2 , ωk3 + ωk4) ] × × exp [iz(k3e aωk4A(ωk3 , ωk4) + k4A(ωk4 , ωk3)) × × A(ωk3 + ωk4 , ωk1 + ωk2)] . (39) Note that ¯̃ φ(k) = φ̃(−k) for real fields φ(x). The x- dependent phase factors are a direct result of the star product (30), and b and d are real parameters. In the case of a canonical deformation, the phase factor is in- dependent of x. Like the commutative case, we want to extract the amplitudes for Feynman diagrams from a generating functional by differentiation. The generating functional can be defined as Zκ[J ] = ∫ Dφe−S[φ]+ 1 2 (J,φ)+ 1 2 (φ,J). (40) The n-point functions G̃n(p1, . . . , pn) are given by func- tional differentiation: G̃n(p1, . . . , pn) = δn δJ̃(−p1) . . . δJ̃(−pn) Zκ[J ] ∣∣∣ J=0 . (41) Let us first consider the free case. For the free generating functional Z0,κ, Eq. (40) yields Z0,κ[J ] = ∫ Dφ exp [ −1 2 ∫ d4k ( −aωk e−aωk − 1 )3 φ̃(k) × ×(Mk +m2)φ̃(−k)+ + 1 2 ∫ d4k (( −aωk e−aωk − 1 )3 + ( aωk eaωk − 1 )3 ) × × J̃(k)φ̃(−k) ] , (42) where we have defined Mk := 2k2(cosh aωk − 1) a2ω2 k . (43) The same manipulations, as in the classical case, yield Z0,κ[J ] = Z0,κ[0]e 1 2 ∫ d4k ( −aωk e−aωk−1 )3 J̃(k)J̃(−k) Mk+m2 . (44) We will always consider the normalized functional ob- tained by dividing by Z0,κ[0]. Now, the free propagator is given by G̃(k, p) = δ2 δJ̃(−k)δJ̃(−p) Z0,κ[J ] ∣∣∣ J=0 = = L(ωk) δ(4)(k + p) Mk +m2 ≡ δ(4)(k + p)Qk. (45) For the sake of brevity, we have introduced L(ωk) := 1 2 (( −aωk e−aωk − 1 )3 + ( aωk eaωk − 1 )3 ) . (46) ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 11 M. WOHLGENANNT We can rewrite the full generating functional in the form Zκ[J ] = e −SI [1/L(ωk) δ δJ̃(−k) ] Z0,κ[J ]. (47) The full propagator to the first order in the coupling pa- rameter is given by the connected part of the expression G̃(2)(p, q) = δ2 δJ̃(−p)δJ̃(−q) Zκ[J ] ∣∣∣ J=0 . (48) The aim is to compute tadpole diagram contributions. In order to do so, we expand the generating functional (47) in powers of the coupling constant g. Using Eq. (50), we obtain Zκ[J ] = Z0,κ[J ] + Z1 κ[J ] +O(g2). (49) The details of the calculation are given in [26]. Let us just state the results. As in the canonically deformed case, we can distinguish between planar and non-planar diagrams. The planar diagrams display a linear UV di- vergence. The non-planar diagrams are finite for generic external momenta, p and p, respectively. However, in the exceptional case ωp = ωk = 0, the amplitudes also diverge linearly in the UV cut-off. Let us switch the second model. Remarkably, the problem of UV/IR divergences is solved in this case, and the model turns out to be renormalizable. We will briefly sketch the model and its peculiarities. Again, it is a scalar field theory. It is defined on the 4- dimensional quantum plane R4 θ with the commutation relations [xµ, xν ] = iθµν . The UV/IR mixing was taken into account through a modification of the free La- grangian, by adding an oscillator term which modifies the spectrum of the free action: S = ∫ d4x (1 2 ∂µφ ? ∂ µφ+ Ω2 2 (x̃µφ) ? (x̃µφ)+ + µ2 2 φ ? φ+ λ 4! φ ? φ ? φ ? φ ) (x) . (50) Here, ? is the Moyal star product (24). The harmonic oscillator term in Eq. (50) was found as a result of the renormalization proof. The model is covariant under the Langmann–Szabo duality relating the short-distance and long-distance behaviors. The renormalization proof proceeds by using a matrix base bnm. The remarkable feature of this base is that the star product is reduced to a matrix product, bkl ? bmn = δlmbkn. (51) We can expand the fields in terms of this base: φ = ∑ m,n φnmbnm(x). (52) This leads to a dynamical matrix model of the type S = (2πθ)2× × ∑ m,n,k,l∈N2 (1 2 φmnΔmn;klφkl+ λ 4! φmnφnkφklφlm ) , (53) where Δm1 m2 n1 n2; k1 k2 l1 l2 = ( µ2+ 2+2Ω2 θ (m1+n1+m2+n2+2) ) × ×δn1k1δm1l1δn2k2δm2l2 − 2−2Ω2 θ × × (√ k1l1δn1+1,k1δm1+1,l1 + √ m1n1× ×δn1−1,k1δm1−1,l1 ) δn2k2δm2l2− −2−2Ω2 θ (√ k2l2 δn2+1,k2δm2+1,l2 + + √ m2n2 δn2−1,k2δm2−1,l2 ) δn1k1δm1l1 . (54) The interaction part becomes the trace of a product of matrices, and no oscillations occur in this basis. In the κ-deformed case we have discussed before, x−dependent phases occurred. Here, the interaction terms have a very simple structure, but the propagator obtained from the free part is quite complicated. For the details, see [25]. These propagators show asymmetric decay properties: they decay exponentially on particular directions, but have power law decay in others. These decay prop- erties are crucial for the perturbative renormalizabil- ity (respectively, the nonrenormalizability) of models. The renormalization proof follows the ideas of Polchin- ski [28]. The integration of the Polchinski equation from some initial scale down to the renormalization scale leads to divergences after removing the cutoff. For rel- evant/marginal operators, one therefore has to fix cer- tain initial conditions. The goal is then to find a pro- cedure involving only a finite number of such operators. Through the invention of a mixed integration procedure 12 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 NON-COMMUTATIVE GEOMETRY & PHYSICS and by proving a certain power counting theorem, they were able to reduce the divergences to only four rele- vant/marginal operators. A somewhat long sequence of estimates and arguments leads then to the proof of renormalization. Afterwards, they could also derive β- functions for the coupling constant flow, which shows that the ratio of the coupling constants λ/Ω2 remains bounded along the renormalization group flow up to the first order. The renormalizability of this model is a very important result and, so far, the only example of a renor- malizable non-commutative model. 3.2. Gauge theories At present, particles and their interactions are described by gauge theories. The most prominent gauge theory is the Standard Model which incudes the electromagnetic force and the strong and weak nuclear forces. There- fore, it is of vital interest to extend the ideas of non- commutative geometry and the renormalization method described above to gauge field theories. Let us sketch two approaches: 1. Non-commutative gauge theories can be formulated by introducing the so-called Seiberg–Witten maps [15,29]. There, the non-commutative gauge fields are given as a power series in non-commutativity param- eters. They depend on the commutative gauge field and the gauge parameter and are solutions of gauge equivalence conditions. Therefore, no additional de- grees of freedom are introduced. A major advantage of this approach is that there are no limitations to the gauge group. For an introduction, see, e.g., [30]. The Standard Model of elementary particle physics is discussed in [31–33] using this approach. How- ever, these theories seemingly have to be considered as effective theories, since the non-renormalizability of non-commutative QED has explicitly been shown in [34]. 2. The second approach starts from covariant coordi- nates Bµ = θ−1 µν x ν+Aµ [29]. These objects are trans- formed covariantly under the gauge transformations Bµ → U∗ ? Bµ ? U, with U∗ ? U = U ? U∗ = 1. This is analogous to the introduction of covariant derivatives. Covariant coordinates only exist on non-commutative spaces. We can write down a gauge-invariant version of ac- tion (50): S = ∫ d4 ( 1 2 φ ? [Bν ?, [Bν ?, φ]] + + Ω2 2 φ ? Bν ?, {{Bν ?, φ}} ) , (55) where we have used [xµ ?, f ] = iθµν∂νf . 4. Astrophysical Considerations In this section, we want to discuss a modification of the dispersion relations in a κ-deformed space-time. This modification leads to a bound on the non-commutativity parameter. We will follow the presentation given in [11]. In Section 3, we have discussed a scalar model on a κ-deformed Euclidean space. Here, we consider a κ- Minkowski space-time with the relations [x̂i, t̂] = iλx̂i, [x̂i, x̂j ] = 0, (56) i, j = 1, 2, 3. The modification of the dispersion rela- tion by a modified d’Alembert operator has been briefly discussed in Section 3: λ−2 ( eλω + e−λω − 2 ) − k2e−λω = m2. (57) In the commutative limit, λ → 0, we obtain, of course, the usual relation ω2 − k2 = m2 . The velocity for a massless particle is given by v = dω dk = λk λ2k2 + λω \|λω\| √ λ2k2 . (58) One obtains v = e−λω ≈ 1− λω. (59) This means that the velocity of a particle depends on its energy. Particles with different energies will take differ- ent amounts of time for the same distance. Let us consider γ-ray bursts from active galaxies such as Makarian 142. The time difference δt in the arrival times for photons with different energies can be esti- mated as \|δt\| ≈ λL c δω, (60) where L is the distance of the galaxy, δω is the energy range of a burst, and λ is the non-commutativity pa- rameter. A usual γ-ray burst spreads over a range of 0.1 − 100 MeV. Data already available seem to imply that λ < 10−33 m. ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 13 M. WOHLGENANNT This work has been supported by FWF (Austrian Science Fund), project P16779-N02. Among the other forms of support, the author is especially grateful to the Austrian Academy of Sciences which, in the framework of the collaboration with the National Academy of Sci- ences of Ukraine, co-financed the multilateral research project “Quantum Gravity, Cosmology and Categorifi- cation” and which also supported his travel expenses to Ukraine. The author also could have not succeeded in pursuing this program for many years without the col- laborations with Prof. W. Kummer and Prof. J. Wess. 1. E. Schrödinger, Naturwiss. 31, 518 (1934). 2. A. Mach, Z. Phys. 104, 93 (1937). 3. W. Heisenberg, Ann. Phys. 32, 20 (1938). 4. H.S. Snyder, Phys. Rev. 71, 38 (1947). 5. S. Majid, J. Math. Phys. 41, 3892 (2000); {\tthep-th/ 0006167}. 6. A. Ashtekar, Lectures on Nonperturbative Canonical Gravity (World Scientific, Singapore, 1991), Chapter 1. 7. S. Doplicher, K. Fredenhagen, and J.E. Roberts, Com- mun. Math. Phys. 172, 187 (1995); {\tthep-th/ 0303037}. 8. P. Schupp, J. Trampetič, J. Wess, and G. Raffelt, Eur. Phys. J. C 36, 405 (2004); {\tthep-ph/0212292}. 9. M. Chaichian, M. M. Sheikh-Jabbari, and A. Tureanu, Phys. Rev. Lett. 86, 2716 (2001); {\tthep-th/0010175}. 10. N. Kersting, Phys. Lett. B 527, 115 (2002); {\tthep-ph/ 0109224}. 11. G. Amelino-Camelia and S. Majid, Int. J. Mod. Phys. A 15, 4301 (2000); {\tthep-th/9907110}. 12. G. Amelino-Camelia, J.R. Ellis, N.E. Mavromatos, D.V. Nanopoulos, and S. Sarkar, {\ttastro-ph/9810483}. 13. H. Nicolai, K. Peeters, and M. Zamaklar, Class. Quant. Grav. 22, R193 (2005); {\tthep-th/0501114}. 14. G. Amelino-Camelia, L. Smolin, and A. Starodubt- sev, Class. Quant. Grav. 21, 3095 (2004); {\tthep-th/ 0306134}. 15. N. Seiberg and E. Witten, JHEP 09, 032 (1999); {\tthep-th/9908142}. 16. J. Madore, Class. Quant. Grav. 9, 69 (1992). 17. J. Lukierski, H. Ruegg, A. Nowicki, and V.N. Tolstoi, Phys. Lett. B 264, 331 (1991) 18. S. Majid and H. Ruegg, Phys. Lett. B 334, 348 (1994); {\tthep-th/9405107}. 19. M. Dimitrijević, L. Jonke, L. Möller, E. Tsouchnika, J. Wess, and M. Wohlgenannt, Eur. Phys. J. C 31, 129 (2003); {\tthep-th/0307149}. 20. N. Reshetikhin, L. Takhtadzhyan, and L. Faddeev, Leningrad Math. J. 1, 193 (1990). 21. A. Lorek, W. Weich, and J. Wess, Z. Phys. C 76, 375 (1997); {\ttq-alg/9702025}. 22. M. Dimitrijević, L. Möller, and E. Tsouchnika, J. Phys. A 37, 9749 (2004); {\tthep-th/0404224}. 23. J.E. Moyal, Proc. Cambridge Phil. Soc. 45, 99 (1949). 24. H. Grosse and R. Wulkenhaar, JHEP 12, 019 (2003); {\tthep-th/0307017}. 25. H. Grosse and R. Wulkenhaar, Commun. Math. Phys. 256, 305 (2005); {\tthep-th/0401128}. 26. H. Grosse and M. Wohlgenannt, Nucl. Phys. B 748, 473 (2006); {\tthep-th/0507030}. 27. L. Möller, JHEP 0512, 029 (2005); {\tthep-th/ 0409128}. 28. J. Polchinski, Nucl. Phys. B 231, 269 (1984). 29. J. Madore, S. Schraml, P. Schupp, and J. Wess, Eur. Phys. J. C 16, 161 (2000); {\tthep-th/0001203}. 30. M. Wohlgenannt, {\tthep-th/0302070}. 31. X. Calmet, B. Jurčo, P. Schupp, J. Wess, and M. Wohlge- nannt, Eur. Phys. J. C 23, 363 (2002); {\tthep-ph/ 0111115}. 32. B. Melic, K. Passek-Kumericki, J. Trampetic, P. Schupp, and M. Wohlgenannt, Eur. Phys. J. C 42, 499 (2005); {\tthep-ph/0503064}. 33. B. Melic, K. Passek-Kumericki, J. Trampetic, P. Schupp, and M. Wohlgenannt, Eur. Phys. J. C 42, 483 (2005); {\tthep-ph/0502249}. 34. R. Wulkenhaar, JHEP 03, 024 (2002); {\tthep-th/ 0112248}. Received 28.05.09 НЕКОМУТАТИВНА ГЕОМЕТРIЯ I ФIЗИКА М. Волґенант Р е з ю м е Робота мiстить деякi основнi iдеї некомутативної геометрiї. Її застосування в фiзицi розглянуто у двох напрямках: у кванто- вiй теорiї поля та астрофiзицi. Детально описано деякi суча- снi моделi в квантовiй теорiї поля. В контекстi астрофiзичних аспектiв некомутативної геометрiї отримано модифiкованi дис- персiйнi спiввiдношення. 14 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1

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