1 Spacetime noncommutativity

Transcription

1 APPUNTI2 Giovanni Amelino-Camelia Dipartimento di Fisica, Universitá La Sapienza, P.le Moro 2, I Roma, Italy 1 Spacetime noncommutativity As stressed in the previous lectures, various arguments appear to indicate that the emergence of noncommutative geometry in quantum gravity is plausible, and if spacetime geometry is in general noncommutative then in particular the situations in which we make reference to a classical Minkowski spacetime should admit a more accurate description in terms of noncommutative geometry. There is therefore some interest in the possibility of noncommutative versions of Minkowski spacetime, with [x µ, x ν ] = iθ µν (x α ) (1) where µ, ν, β = 0, 1, 2, 3, x 0 is the time coordinate, and Θ µν is some matrix which in general may depend on the noncommuting coordinates. A familiar example of noncommutative geometry is the phase space (the space mapped by space and spatial-momentum coordinates) of nonrelativistic quantum mechanics. The difference is that, in the noncommutative spacetimes which we shall consider, spacetime itself is noncommutative, whereas in ordinary quantum mechanics the space coordinates do not commute with momentum coordinates. 2 Effective noncommutativity In addition to the idea that noncommutative geometry might be needed for a fundamental description of spacetime structure, it is also possible that noncommutativity would emerge only in some limit of the correct fundamental theory. Let us see the example of a charged particle confined to a plane and subject to a magnetic field orthogonal to the plane Then the momenta are L = mv2 2 + qbxv y (2) p x = mv x (3) p y = mv y + qbx (4) and in the large-b limit the canonical commutator [y, p y ] = i turns into [y, x] = i qb (5) 1

2 The role of the large-b limit in the emergence of noncommutativity means that in the whole Hilbert space of the problem there are states such that x and y can be measured simultaneously with absolute precision, but these states are energetically disfavored when the B field is large. A mechanism analogous to this one has been recently discovered within string theory, leading to an effective-theory description of certain aspects of string theory in terms of a noncommutative geometry. 3 The Weyl map for the phase space of nonrelativistic quantum mechanics In our description and handling of functions of the noncommuting coordinates (fields in the noncommutative geometry) an important role will be played by Weyl maps, which allow to introduce structure for the functions of the noncommuting coordinates in terms of corresponding structures that are meaningful for ordinary commuting functions. Before going to that more modern set of issues, let us just stress that a first example of Weyl map was already introduced in some approaches to the description of nonrelativistic quantum mechanics. One starts by introducing the operators W (α, β) = e i j (α jq j +β j p j ) where q j and p j are the usual operators that represent the position and momentum observables of a particle, whose dynamics is classically described on the phase space R 2n. This form of the operator W can be intuitively seen as a sort of plane wave basis. Given a function f(p, q) on the classical phase space, with Fourier transform f (α, β) d n qd n pf (q, p) e i(α jq j +β j p j ) (7) at the quantum-mechanical level we can associate an operator Ω(f) Ω(f) = 1 (2π) n (6) d n αd n β f (α, β) e i(α j ˆq j +β j ˆp j). (8) [In (7) and (8), for the sake of clarity, I introduced notation such that an operator denoted by  describes in the quantum-mechanical context the observable A of classical mechanics.] The map f Ω(f) was first introduced by Weyl, and is accordingly called Weyl map. Various properties of operators on a Hilbert space and several relationships among these operators can be conveniently characterized through the Weyl map. Just to give an example let us notice that through Ω one can describe hermitian conjugation in terms of complex conjugation: Ω(f ) = Ω(f) (9) 2

3 Already in the context of these studies of the noncommutativity of phase space which is characteristic of nonrelativistic quantum mechanics one finds useful to introduce several structures in terms of the Weyl map, such as a description of the product of operators, but for our purposes it is preferable to postpone a discussion of these structures directly to the level of noncommutative spacetime geometries. 4 Fields in Canonical Spacetimes Most work on noncommutative versions of Minkowski spacetime has explored various parts of the two-matrix parameter space [ˆx µ, ˆx ν ] = iθ µν + iγ β µν ˆx β, (10) where µ, ν, β = 0, 1, 2, 3 and ˆx 0 is the time coordinate. In the description of fields in these spacetimes it is useful to make reference to a suitable Weyl map. Let us see this first in the simple case of the canonical noncommutative spacetimes [ˆx µ, ˆx ν ] = iθ µν (11) obtained from the more general formula (10) by setting γµν β = 0. One can describe a given function of the canonical-noncommutative coordinates as inverse Fourier transform of some commuting functions of energy-momentum-space variables: f (ˆx) = 1 (2π) 2 d 4 k exp (ik µˆx µ ) f (k). (12) On the basis of (13) one can construct a one-to-one correspondence between our noncommutative functions and the ordinary commutative functions of classical-minkowski coordinates. Our function f will be mapped into the commutative function f commut, f commut (x) = 1 (2π) 2 d 4 k exp (ik µ x µ ) f (k), (13) obtained by performing a reverse Fourier transform, according to the standard rules of Fourier transform in classical Minkowski spacetime, of the function of energymomentum variables f (k). In the development of a field theory, with these fields of noncommuting spacetime coordinates, the description of products of fields plays of course a central role. And it is useful to describe the product of two fields F and G of noncommuting spacetime coordinates in terms of a (correspondingly deformed) rule of product for the commuting fields f and g that are associated to F and G through the Weyl map: F = Ω(f) G = Ω(g). (14) Of course, as a result of the noncommutativity, the product F G cannot be described as Ω(fg). Instead one has that F G = Ω(f g), where (f g) = Ω 1 (Ω(f)Ω(g)) (15) 3

4 is the product (often also called Moyal product). We can render this observation more explicit by using the Fourier transform, where the properties of the product are codified in the properties of products of exponentials of the noncommuting variables. And in particular if the functions F and G are exponentials of the noncommuting variables one finds e ipµˆx µ e ikν ˆx ν = e i 2 pµ θ µνk ν e i(p+k)µˆx µ = Ω(e ipµ x µ e i 2 pµ θ µνk ν e ikν x ν ). (16) The Weyl map clearly takes e ipµ x µ into e ipµˆx µ and e ikµ x µ into e ikµˆx µ, and for consistency with the direct calculation of the product e ipµˆx µ e ikν ˆx ν the product [e ipµ x µ ] [e ikµ x µ ] is given by [e ipµ x µ ]e i 2 pµ θ µν k ν [e ikν x ν ]. The Weyl map can also be used to introduce a notion of integration in the noncommutative spacetime. For a given function F of the noncommuting coordinates which can be described through the Weyl map as F = Ω(f) we can set F d 4 xf, (17) i.e. the integral of F is defined in terms of the ordinary notion of integration of the function f obtained from F as f = Ω 1 (F ). It is important to note that the choice of Weyl map is not unique. To make this point explicit let us compare the Weyl map we described so far and the possible alternative in which the Weyl map takes the function f(x), whose Fourier transform is f(p) = 1 (2π) 4 f(x)e ipx d 4 x, into Ω (f) = d 4 k f(k) e ikj ˆx j e ik0ˆx 0. (18) Clearly the difference between Ω and Ω originates in the fact that, if [ˆx j, ˆx 0 ] 0, then also e ikj ˆx j e ik0 ˆx0 e ikµˆx µ. Some alternative ways to write the Fourier exponentials, which are equivalent in the commutative limit, are actually different in the noncommutative case. 5 Fields in κ-minkowski Lie-algebra Spacetime While the application of the Weyl map to the case of canonical noncommutative spacetime is essentially the same thing as applying the Weyl map to the analysis of the phase space of nonrelativistic quantum mechanics, which is also a canonical noncommutative geometry, one must perform a genuine generalization in developing a Weyl map for noncommutative spacetimes which are not canonical. For our scopes it is sufficient to illustrate these difficulties within a specific example of Lie-algebra noncommutative spacetimes (θ = 0), the so-called κ-minkowski spacetime [ˆx m, t] = iλˆx m, [ˆx m, ˆx l ] = 0. (19) 4

5 Also for the case of κ-minkowski it is useful to consider at least two alternative Weyl maps,, which I denote by Ω R and Ω S, so that we get some intuition for the differences which may arise. It is sufficient to specify the Weyl map on the complex exponentials and extend it to the generic function Ω R,S (f(x)), whose Fourier transform is f(p) = 1 (2π) 4 f(x)e ipx d 4 x, by linearity Ω R,S (f) = f(p) ΩR,S (e ipx ) d 4 p. The time-to-the-right Ω R Weyl map is implicitly defined through Ω R (e ipx ) = e i p ˆx e ip 0 ˆx 0 (20) while the alternative Ω S time-symmetrized Weyl map is such that Ω S (e ipx ) = e ip 0ˆx 0 /2 e i p ˆx e ip 0ˆx 0 /2 (21) Notice that it is possible to go from time-to-the-right to time-symmetrized ordering through a transformation of the Fourier parameters Ω R (e ipx ) = Ω S (e i pe λ 2 p0 x ip 0 x 0 ). (22) From Ω R and Ω S of course some corresponding star-products are obtained Ω R (f R g) = Ω R (f) Ω R (g) Ω S (f S g) = Ω S (f) Ω S (g) While in canonical spacetimes the star-product formulas are characterized by a standard addition of the Fourier parameters, in Lie-algebra spacetimes the form of the commutators among coordinates imposes a more complicated formulation of the composition of Fourier parameters. In the case of the Ω R and Ω S Weyl maps for κ-minkowski one finds Ω R (e ipx ) Ω R (e iqx ) = Ω R (e i( p+ qe λp 0) x i(p 0 +q 0 )x 0 ) Ω S (e ipx ) Ω S (e iqx ) = Ω S (e i( pe λ 2 q0 + qe λ 2 p0 ) x i(p 0 +q 0 )x 0 ) Therefore, for example, in time-to-the-right conventions the product is such that e ipx e iqx = e i( p+ qe λp 0) x i(p 0 +q 0 )x 0. HOMEWORK: Verify explicitly that the κ-minkowski commutation rules imply [e ipj ˆx j e ip0ˆx 0 ][e iqj ˆx j e iq0ˆx 0 ] = e i(pj +q j e λp0 )ˆx j i(p 0 +q 0 )ˆx 0. 5

6 5.1 Free scalar fields in classical Minkowski While for canonical noncommutative spacetimes the naive choice of action S(φ) = d 4 x φ( 2 M 2 )φ (for free scalar fields) is fully satisfactory, in the description of free scalar fields in κ-minkowski a nontrivial choice of action emerges very naturally. This originates from the desire to work with a maximally symmetric action, and in the case of κ-minkowski it is possible to introduce an action which is invariant under 10 Poincaré-like symmetries, but this action has nontrivial form. In preparation for this κ-minkowski analysis I find useful to devote this subsection to a description of the simple action S(φ) = d 4 x φ( 2 M 2 )φ for a free scalar field φ in commutative Minkowski spacetime ( 2 = µ µ is the familiar D Alembert operator). Let me start by introducing some notation and conventions for the description of symmetry transformations. The most general infinitesimal transformation is of the form x µ = x µ + ɛa µ (x), with A µ four real functions of the coordinates. A field is scalar if φ (x ) = φ(x), and in leading order in ɛ one finds φ (x) φ(x) = { µ φ(x)}(x µ x µ) = ɛa µ (x) µ φ(x) In terms of the generator T of the transformation, T = ia µ (x) µ, one obtains x = (1 iɛt )x and φ = (1 + iɛt )φ. [For the action of T on x I use the notation T x.] Correspondingly the variation of the action can be written as S(φ ) S(φ) = iɛ d 4 x ( T {φ( 2 M 2 )φ} + φ[ 2, T ]φ ) = iɛ d 4 x ( T L(x) + φ[ 2, T ]φ ) and therefore the action is invariant under T -generated transformations, S(φ ) S(φ) = 0, (23) if and only if d 4 x ( T L(x) + φ[ 2, T ]φ ) = 0. (24) For the action S(φ) = d 4 x φ( 2 M 2 )φ in classical Minkowski spacetime it is well established that the symmetries are described in terms of the classical Poincaré algebra, generated by the elements P µ = i µ M j = ɛ jkl x k P l N j = x j P 0 x 0 P j The operator 2 = P µ P µ is the first Casimir of the algebra, and of course satisfies [ 2, T ] = 0. For this case of a maximally-symmetric theory in commutative Minkowski spacetime it is conventional to describe the symmetries fully in terms of the Poincaré Lie algebra. For κ-minkowski noncommutative spacetime a description of symmetry in terms of a Hopf algebra turns out to be necessary. But I must stress that essentially the difference between symmetries described in terms of a Lie algebra and symmetries described in terms of a Hopf algebra resides in the description of the action of symmetry transformations on products of functions: if for all generators T a one finds that T a (fg) = [T a (f)]g +f[t a (g)], one may say that the coproduct is trivial and a description in terms of a Lie algebra is sufficient, whereas for the case when the coproduct is nontrivial one speaks of a Hopf-algebra symmetry. 6

7 Symmetry analysis and the free-scalar-field action Next I want to discuss the form of the action for a free scalar field in κ-minkowski which most naturally replaces the S(φ) = d 4 x φ( 2 M 2 )φ action of the classical- Minkowski case. A detailed discussion and derivation goes beyond the scopes and the needs of these notes, but I want to at least introduce some elements of the analysis. Of course the first step is the introduction of an action functional, i.e. a map from functions of κ-minkowski coordinates to pure numbers, essentially a suitable rule of integration. Most of the κ-minkowski literature adopts as action functional the following Weyl-map-based formula: d 4 x Ω R (f) = f(x) d 4 x. (25) R While this definition appears to depend explicitly on the choice Ω R of Weyl map, by adopting the Ω S Weyl map, d 4 x Ω S (f) = f(x) d 4 x, (26) S one obtains the same action functional. This is easily verified by expressing (using (22)) the most general element of κ-minkowski both in its Ω R -inspired form and its Ω S -inspired form Φ = d 4 p f(p)ω R (e ipx ) = d 4 p f(p 0, pe λp0/2 )e 3λp0/2 Ω S (e ipx ) (27) and observing that R d 4 x Φ = S d 4 x Φ = (2π) 4 f(0). Because of the equivalence one can omit indices R or S on the integration symbol. Let us assume this rule of integration 1 and look for a suitable Lagrangian density, so that we will have the sought action for a free scalar field in κ-minkowski. A key point is that it is possible to introduce an action for a free scalar field in κ-minkowski which is invariant under translations, space-rotations and boosts, in the Hopf-algebra sense. As stressed earlier, a symmetry transformations T must be such that d 4 x ( T {Φ ( 2 λ M 2) Φ } + Φ[ 2 λ, T ]Φ ) = 0, (28) if the action takes the form S(Φ) = d 4 x Φ( 2 λ M 2 )Φ with 2 λ to be determined. This equation will play the role of guiding ansatz for the analysis. 1 But I should inform the reader of the fact that alternatives have been considered in the literature. 7

8 When several such symmetry generators are available they may or may not combine together to form a Hopf algebra. When they do form a Hopf algebra one has a symmetry (Hopf) algebra. The next step is the description of the Poincaré-like symmetries which will be implemented as invariances of the action. One of course wants to introduce a description of translations, space-rotations and boosts that follows as closely as possible the analogy with the well-established descriptions that apply in the commutative limit λ 0. Since functions in κ-minkowski can be fully described in terms of the Weyl maps, and since the Weyl maps are fully specified once given on Fourier exponentials, one can, when convenient, confine the discussion to the Fourier exponentials. Since in classical Minkowski the translation generator acts according to P µ (e ikx ) = k µ e ikx (29) in an analysis of κ-minkowski based on the time-to-the-right Weyl map it is natural to define translations as generated by the operators Pµ R such that P R µ Ω R (e ikx ) = k µ Ω R (e ikx ). (30) Since, as mentioned, the exponentials e i k ˆx e ik 0ˆx 0 form a basis of κ-minkowski, in order to establish the form of the action of these translation generators on products of function of the κ-minkowski coordinates, the structure which is codified in the coproduct Pj R, one can simply observe that P R j Ω R (e ikx )Ω R (e ipx ) = iω R ( j e i(k +p)x ) = iω R ((k +p) j e i(k +p)x ) = [P R j Ω R (e ikx )][Ω R (e ipx )] + [e λp R 0 ΩR (e ikx )][P R j Ω R (e ipx )], (31) where p +q (p 0 + q 0, p 1 + q 1 e λp 0, p 2 + q 2 e λp 0, p 3 + q 3 e λp 0 ). This is conventionally described by the symbolic notation Following an analogous procedure one can derive P R j = P R j 1 + e λp R 0 P R j (32) P R 0 = P R P R 0 (i.e., while for space translations one has a nontrivial coproduct, for time translations the coproduct is trivial). Using the full machinery of the mathematics of Hopf algebras one can verify that the quadruplet of operators Pµ R does give rise to a genuine Hopf algebra of translation-like symmetry transformations. The next step is to obtain a 7-generator Hopf algebra, describing four translationlike operators and three rotation-like generators. For what concerns the translations we have found that an acceptable Hopf-algebra description is obtained by straightforward 8

9 quantization of the classical translations: the Pµ R translations were just obtained from the commutative-spacetime translations through the Ω R Weyl map. Also for rotations this strategy turns out to be successful: M R j Ω R (f) = Ω R (M j f) = Ω R ( iɛ jkl x k l f) (33) And, while for the (spatial) translations one finds a nontrivial coproduct, the coproduct of rotations is trivial ( primitive ): It is also straightforward to verify that M j = M j M j (34) [M j, M k ] = iε jkl M l (35) Therefore the triplet M j forms a 3-generator Hopf algebra that is completely undeformed (classical) both in the algebra and in the coalgebra sectors. (Using the intuitive description introduced earlier this is a trivial rotation Hopf algebra, whose structure could be equally well captured by the standard Lie algebra of rotations.) There is therefore a difference between the translations sector and the rotations sector. Both translations and rotations can be realized as straightforward (up to ordering) quantization of their classical actions, but while for rotations even the coalgebraic properties are classical (trivial coalgebra) for the translations we found a nontrivial coalgebra sector. Our translations and rotations can be put together straightforwardly to obtain a 7-generator translations-rotations symmetry Hopf algebra. It is sufficient to observe that [M j, Pµ R,S ] Ω(e ikx ) = ε jkl Ω([ x k µ + µ x k ] l e ikx ) = δ µk ε jkl Ω( l e ikx (36) ) from which it follows that [M i, P R,S j ] = iε ijk P R,S k, [M i, P R,S 0 ] = 0 (37) i.e. the action of rotations on energy-momentum is undeformed. Accordingly, the generators M j can be represented as differential operators over energy-momentum space in the familiar way: M j = iε jkl P k Pl. In the analysis of translations and rotations in κ-minkowski we have already encountered two different situations: rotations are essentially classical in all respects, while translations have a classical action (straightforward Ω-map quantization of the corresponding classical action) but have nontrivial coalgebraic properties (nonprimitive coproduct). Of course, the fact that some symmetry transformations in a noncommutative spacetime allow classical description (through the Weyl map) is not to be expected in general. In general one can only require that the results should reproduce the familiar ones for commutative Minkowski in the limit of vanishing noncommutativity parameters (λ 0 in κ-minkowski), when the limit of vanishing noncommutativity parameters is taken consistently throughout. The description of boosts in κ-minkowski is of this more general type: it cannot be obtained by quantization of the classical action. 9

10 The classical boosts N R j should have action N R j Ω R (f) = Ω R (N j f) = Ω R (i[x 0 j x j 0 ]f]). (38) And actually it is easy to see (and it is obvious) that these boosts combine with the rotations M j to close the (undeformed) Lorentz algebra, and that adding also the translations Pµ R one obtains the undeformed Poincaré algebra. However, these algebras cannot be extended (by introducing a suitable coalgebra sector) to obtain a Hopf algebra of symmetries of theories in our noncommutative κ-minkowski spacetime. In particular, one finds an inconsistency in the coproduct of these boosts Nj R, which signals an obstruction originating from an inadequacy in the description of the action of boosts on (noncommutative) products of κ-minkowski functions. The problem is that (Nj R ) would not be an element of the algebraic tensor product, i.e. it is not a function only of the elements M, N, P. Since the classical choice Nj R is inadequate there are two possible outcomes: either there is no 10-generator symmetry-algebra extension of the 7-generator symmetry algebra (Pµ R,M j ) or the 10-generator symmetry-algebra extension exists but requires nonclassical boosts. The latter is true. The generators of the needed modified boost action, N j, are found through a rather tedious analysis which is not appropriate to report in detail here. One starts by observing that, by imposing that the deformed boost generator N j (although possibly having a nonclassical action) transforms as a vector under spatial rotations, the most general form of N j is N j Ω(φ(x)) = Ω{[ix 0 A( i x ) j + λ 1 x j B( i x ) λx l C( i x ) l j iɛ jkl x k D( i x ) l ]φ(x)} (39) where A, B, C, D are unknown functions of P R µ (in the classical limit A = i, D = 0; moreover, as λ 0 one obtains the classical limit if λc 0 and B λp 0 ). Imposing that in the formula N R j [Ω(e ikx )Ω(e ipx )] = [N R (1),jΩ(e ikx )][N R (2),jΩ(e ikp )] (40) it should be possible to write N(1),j R and N (2),j R in terms of generators of the Hopf algebra, one clearly obtains some constraints on the functions A, B, C, D. The final result is N R j Ω R (f) = Ω R ([ix 0 j + x j ( 1 e2iλ 0 2λ λ 2 2 ) λx l l j ]f) (41) It is easy to verify that the Hopf algebra (Pµ R, M j, Nj R ) satisfies all the requirements for a candidate symmetry algebra for theories in κ-minkowski spacetime. Of course, the fact that one replaces the classical Poincaré Lie algebra with the quantum deformed-poincaré Hopf algebra has some striking consequences. Perhaps most notably, one finds that the mass Casimir in the quantum version has the form ( 2 λ sinh λp 0 R 2 ) 2 e λ(p R 0 )2 ( P R ) 2. (42) 10

11 This also suggests a choice of action for free scalar fields in κ-minkowski: S(Φ) = d 4 x Φ( λ 2 M 2 )Φ, with 2 λ = ( 2 λ sinh λp 0 R 2 ) 2 e λp 2 0 P 2 R (43) This action is invariant under the quantum symmetry in the same (at least technical) sense that, as discussed in the previous subsection, the ordinary action for scalar fields in commutative Minkowski spacetime is invariant under classical-poincaré transformations. Aside on the Weyl-map-choice ambiguity In order to keep these notes at a relatively elementary level I set aside in the analysis reported in the preceding subsection an issue which is at the center of a rather intense scientific debate: the (apparent?) Weyl-map-choice ambiguity that affects the symmetry analysis. While it would be here inappropriate to go too deep into the discussion of this issue, let me just stress that the problem originates from the description of translations. Let us in fact compare the description of translations given above, which is most intuitive when adopting the Ω R Weyl map, with the description of translations which would be most intuitive if one instead adopted the Ω S Weyl map: P S µ Ω S (e ikx ) = k µ Ω S (e ikx ). (44) Also in introducing the rule of integration we had relied on the Weyl map; however, in that case we happily found that the two rules of integration, in spite of the different way in which they are introduced, are actually equivalent. Instead in the case that we are now considering, the one of the description of translations, the ambiguity is genuinely there: the Pµ R and Pµ S descriptions of translations are truly different! This can be easily verified using the fact that the same function Φ of κ-minkowski coordinates can be of course expressed in time-symmetrized form as well as in time-to-the-right form: Φ = d 4 p f(p)ω R (e ipx ) d 4 p f(p 0, pe λp0/2 )e 3λp0/2 Ω S (e ipx ) (45) The fact that P R µ and P S µ are different operators than follows straightforwardly from observing that P R µ (e i k ˆx e ik 0ˆx 0 ) = P R µ Ω R (e ikx ) = k µ Ω R (e ikx ) = k µ (e i k ˆx e ik 0ˆx 0 ) = k µ (e ik 0ˆx 0 /2 e ie λ 2 k 0 k ˆx e ik 0ˆx 0 /2 ) e λ 2 k 0 k µ (e ik 0ˆx 0 /2 e ie λ 2 k 0 k ˆx e ik 0ˆx 0 /2 ) = P S µ (e i k ˆx e ik 0ˆx 0 ). (46) In the end one finds, on the basis of the Ω S Weyl map, a description of the symmetries (P S µ, M j, N j ), which in all other ways is completely analogous to the description 11

12 (P R µ, M j, N j ) which we derived earlier. Using technical terminology these are just two different bases for the description of the same abstract Hopf algebra; however, as the community attempts to develop physical applications of this mathematics the differences between alternative bases do appear to pose a challenge, which has not yet been overcome. Quantum fields in canonical noncommutative spacetime Building a field theory on these spacetimes can be challenging. Some proposals have been put forward, but none of the present formulations is immune to criticism. And as usual the situation for Lie-algebra spacetime (such as κ-minkowski) is even more intricate that in the case of canonical spacetimes. For the purposes of these notes it will be sufficient to discuss here briefly the case of a scalar theory with quartic interaction ( λφ 4 theory ) in the case of canonical noncommutativity. We start with a generating functional for Green functions (partition function): Z [J] = ( D [φ] exp i d 4 x [ 1 2 φ(x) 2 φ(x) m2 2 φ2 (x) λ 24 φ4 (x) J(x)φ(x) φ(x)j(x) ]). (47) From this I intend to obtain energy-momentum-space Feynman rules, so we must rewrite the partition function in energy-momentum space, using the Fourier transform. As I develop the analysis it will also turn out to be necessary to introduce an integral representation of the delta function. We denote the delta function by δ (4) θ (k) (where the subscript θ is reminder of the fact that we are working in the case of canonical spacetime), and we introduce it as follows: δ (4) θ (k) = d 4 x Ω(exp(ikx)), (48) (2π) 4 in very close analogy with the familiar commutative spacetime delta function. Equipped with these tools we can get started on our first objective, which should be the analysis of the structure of the two-point function at tree-level. For this result we can of course switch off the coupling λ. By performing Fourier transforms of all fields in the action, then combining the products of exponentials into a single exponential, and finally using (48), one obtains from (47) a simple formula for the momentum-space deformed partition function of the free theory: Z 0 [J] = where C(k) is the undeformed Mass Casimir, ( i D [φ] exp d 4 k [ φ( k) [ C(k) m 2] φ(k) 2 +J(k)φ( k) + φ(k)j( k)]), (49) C(k) = k 2 0 k 2. (50) 12

13 In fact, following the same analysis done in the previous subsection for the κ-minkowski case, one finds that in canonical spacetimes the symmetries are also deformed, but in a different way, which in particular does not affect the form of the Mass Casimir. It is convenient to introduce the normalized partition function Z 0 [J] Z 0 [J] /Z 0 [0], and from (49) with simple manipulations (and performing the elementary functional integration) one finds that Z 0 [J] = exp ( i 2 d 4 k J(k)J( ) k) C D (k) m 2. (51) To obtain Green functions from (51) we must now simply perform some functional derivatives. In particular, from the general formula for the two-point function at treelevel G (2) 0 (p, p ) = δ2 Z0 [J] δj( p)δj(p (52) ) J=0 one easily obtains G (2) 0 (p, p ) = i δ(4) (p p ) C(p) m 2. (53) In the case of the canonical noncommutative spacetimes the two-point function at tree-level is unmodified. 2 In order to be able to investigate the properties of the two-point function beyond tree level and in order to establish the form of the tree-level vertex we must now analyze the O(λ) contributions to the Green functions. For this we must of course reinstate λ 0, i.e. we need to analyze Z (1) [J] rather than Z 0 [J]. It turns out to be useful to rely on a simple relationship between Z (1) [J] and Z 0 [J], which one can obtain with manipulations analogous to the ones described above: Z (1) [J] = i λ 24 ( 4 ) 4 δ (4) k i i=1 j=1 d 4 k j 2π exp i 2 1 i<j 4 k i k j δ δj( k j ) Z θ 0 [J], (54) where I introduced the notation p q = p µ θ µν q ν. The O(λ) contribution to the two-point function can be written as ( ) G (2) λ (p, p )= δ2 Z1 [J] δj( p)δj(p. (55) ) J=0 connected It is easy to verify that the 24 connected elements of the one loop deformed two-point function split into two different classes: 16 planar contributions and 8 non-planar 2 This could be guessed already at the level of the generating functional, using the fact that the antisymmetry of θ µν leads to cyclic dependence of the integrals of product of fields on the order of the fields, and in particular the product of two fields under integral is undeformed. 13

14 contributions. Planar contributions are associated with the 16 possibilities for attaching the external momenta to consecutive internal lines (e.g. to k 1, k 2 ). The difference between these planar diagrams and the diagrams, which can be described as nonplanar, that correspond to the remaining 8 permutations (in which instead the external lines are attached to non-consecutive lines), is meaningful in our noncommutative spacetimes. As one might have already expected on the basis of the form of the product, the order of the lines coming out of a vertex is reflected in the structure of some θ-dependent phase factors. An example of planar contribution is d 4 ( k 4 j 2π δ(4) i λ 4 24 j=1 δ 2 Z0 [J] δj( p)δj( k 2 ) J=0 and an example of non-planar one is i λ 24 4 j=1 d 4 ( k 4 j 2π δ(4) δ 2 Z0 [J] δj( p)δj( k 2 ) J=0 ) k i exp i=1 δ 2 Z0 [J] δj(p )δj( k 3 ) ) k i exp i=1 δ 2 Z0 [J] δj(p )δj( k 4 ) [ i ] 2 (p p + 2p k 1 + 2p k 1 ) J=0 δ 2 Z0 [J], (56) δj( k 1 )δj( k 4 ) J=0 [ i ] 2 (p p + 2p k 1 ) J=0 δ 2 Z0 [J]. (57) δj( k 1 )δj( k 3 ) J=0 From (52) and (53) it is possible to see that these expressions contain the term δ (4) (p p ), which, in light of the antisymmetry of θ µν, allows one to ignore terms like p p and 2p k 1 + 2p k 1, and leads to the conclusion that planar terms do not involve any nontrivial θ-dependent phase factors. Nonplanar terms instead do involve nontrivial phase factors of the type exp(±ip q), where p and q represent one external and one internal momentum, and the sign of the exponent depends on the specific momenta involved. Using these observations, and the result on the tree-level two-point function reported above, it is easy to combine all the O(λ) (tadpole) contributions (planar and nonplanar) to the full two-point function in the canonical theory. For the truncated two-point function the result is i λ 6 d 4 k i (2 + cos (p k)), (58) (2π) 4 k 2 m 2 where p is the external momentum (the propagation momentum). One much-studied consequence of this modification of the tadpole is the IR/UV mixing : the θ-dependent modification of the tadpole is such that when one goes through the renormalization procedure the UltraViolet (high-energy) divergences can only be eliminated at the cost of replacing them with some corresponding InfraRed (low-energy) divergences. The Wilson decoupling between IR and UV degrees of freedom is a crucial ingredient of most applications of field theory in physics, and it is probably incompatible with canonical noncommutativity: the associated uncertainty principle of the type x µ x ν θ µν implies that it is not possible to probe short distances (small, say, x 1 ) without probing simultaneously the large-distance regime ( x 2 θ 21 / x 1 ). 14

15 References [1] NO REFERENCE LIST 15