We present a deformation of the Minkowski space as embedded into the
conformal space (in the formalism of twistors) based in the quantum versions of
the corresponding kinematic groups. We compute explicitly the star product,
whose Poisson bracket is quadratic. We show that the star product although
defined on the polynomials can be extended differentiably. Finally we compute
the Eucliden and Minkowskian real forms of the deformation.Comment: Presented at XVII European Workshop on String Theory 2011. Padova
  (Italy) September 05-09; Fortschr. Phys. 1-7 (2012

Cervantes, D.

Fioresi, R.

Lledo, M. A.

Nadal, F. A.

English

arXiv

D. Cervantes

R. Cervantes

M.A. Lledó

F.A. Nadal

Crossref

Fortschritte der Physik

Quadratic deformation of Minkowski space

We present a deformation of the Minkowski space as embedded into the conformal space (in the formalism of twistors) based in the quantum versions of the corresponding kinematic groups. We compute explicitly the star product, whose Poisson bracket is quadratic. We show that the star product although defined on the polynomials can be extended differentiably. Finally we compute the Eucliden and Minkowskian real forms of the deformation

Cervantes, Dalia

Fioresi, Rita

Lledó Barrena, María Antonia

Nadal, Felip A.

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:1207.1316v1  [hep-th]  5 Jul 2012Fortschritte der Physik, 6 July 2012Quadratic deformation of Minkowski spaceDalia Cervantes1,∗, Rita Fioresi2,∗∗, Marı´a A. Lledo´3,∗∗∗, and Felip A. Nadal3,†1 Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de Me´xico, Circuito Exterior Me´xicoD.F. 04510, Me´xico.2 Dipartimento di Matematica, Universita` di Bologna. Piazza di Porta S. Donato, 5. 40126 Bologna. Italy.3 Departament de Fı´sica Teo`rica, Universitat de Vale`ncia and Institut de Fı´sica Corpuscular (CSIC-UVEG).C/Dr. Moliner, 50, E-46100 Burjassot (Vale`ncia), Spain.Key words Deformations, star product, Minkowski spacetime.We present a deformation of the Minkowski space as embedded into the conformal space (in the formalismof twistors) based in the quantum versions of the corresponding kinematic groups. We compute explicitlythe star product, whose Poisson bracket is quadratic. We show that the star product although defined on thepolynomials can be extended differentiably. Finally we compute the Eucliden and Minkowskian real formsof the deformation.Copyright line will be provided by the publisher1 The Grassmannian and the Minkowski spaceWe consider the Grassmannian G(2, 4), the set of two-planes inside C4. A plane π ∈ G(2, 4) is given bytwo linearly independent vectors or by any two linear combinations of them that are independent, soπ = span(a, b) = span(a′, b′), (a′, b′) = (a, b)h, h ∈ GL(2,C).There is a transitive action of GL(4,C) (or SL(4,C)) ) on G(2, 4).gπ = span{ga, gb}, g ∈ GL(4,C).If we consider a particular point π0π0 = span1 00 10 00 0 , with stability groupP0 ={(L M0 R)∈ SL(4,C)}, and G(2, 4) = SL(4,C)/P0.We notice that the conformal group of space time, SO(2, 4), has spin group SU(2, 2). Its complexifica-tion, SO(6,C), has spin group SL(4,C).How to extract the Minkowski space from G(2, 4)? Notice that since the two vectors are independent,π =a1 b1a2 b2a3 b3a4 b4 ,∗ E-mail: daliac@nucleares.unam.mx∗∗ E-mail: fioresi@dm.unibo.it∗∗∗ Corresponding author E-mail: maria.lledo@ific.uv.es, Phone: +00 34 9635 43252, Fax: +00 34 9635 43381† E-mail: Felip.Nadal@ific.uv.esCopyright line will be provided by the publisher2 D. Cervantes, R. Cervantes, M. A. Lledo´, and F. A. Nadal: Deformation of Minkowski spaceat least one of the 2× 2 determinants in this matrix is 6= 0. The space is covered by the atlasUij ={(a, b) ∈ C4 × C4 / aibj − biaj 6= 0}, i < j, i, j = 1, . . . 4.U12 is the big cell, and using the GL(2,C) freedom, a plane in U12 can be represented byπ =1 00 1t31 t32t41 t42 =(11t),with the entries of t totally arbitrary. So U12 ≈ C4, and it is a good candidate for the Minkowski space.What about the group action? U12 is left invariant by the lower parabolic subgroup of SL(4,C),Pl ={(x 0Tx y), / detx · det y = 1},and it acts on t ast 7→ ytx−1 + T.The group is SL(2,C) × SL(2,C) × C× ⋉ T 4, so it is the Poincare´ group where instead of the Lorentzgroup we have put its double cover.t belongs to the twistor space associated to spacetime. Using the Pauli matrices, we can revert to thespacetime notation and obtain the standard action of the Poincare´ group on Minkowski spacetimet = xµσµ =(x0 + x3 x1 − ix2x1 + ix2 x0 − x3).Also the spacetime metric has an interpretation in the twistor formalism,det t = (x0)2 − (x1)2 − (x2)2 − (x3)2.2 Algebraic approachQuantization of spacetime means to deform the commutative algebra of functions (can be polynomials orsmooth functions) to a non commutative algebra. Other properties that we want to consider in the quantumsetting have to be first defined in the algebraic formalism and then ’quantized’. This is the case of thegroup actions. The respective algebras areO(SL(4,C)) = C[gAB]/(det g − 1), A,B = 1, . . . , 4.O(Pl) = C[xij , yab, Tai]/(detx · det y − 1), i, j = 1, 2, a, b = 3, 4.O(M) = C[t31, t32, t41, t42].The group law is expressed in terms of a coproductO(SL(4,C))∆c−−−−→ O(SL(4,C))⊗O(SL(4,C))gAB −−−−→∑C gAC ⊗ gCB,C = 1, . . . , 4,with the propertyµG ◦ (∆cf)(g1, g2) = f(g1g2), f ∈ O(GL(4,C)).The action on the Minkowski space is a coactionO(M)∆˜−−−−→ O(Pl)⊗O(M)tai −−−−→ yabS(x)ji ⊗ tbj + Tai ⊗ 1.Copyright line will be provided by the publisherfdp header will be provided by the publisher 33 The quantum Minkowski spaceIn Refs. [2, 3] one substitutes the group SL(4,C) by SLq(4,C)in the twistor construction. All the schemeof coaction and big cell can be repeated in the quantum case, which gives a quantization for the Minkowskispace as a big cell inside a quantum conformal space (a quantum Grassmannian).We just state the result: The quantum Minkowski space is a quantum matrix algebra with the rowsinterchanged. The correspondenceMq(2)→ Oq(M) is given in terms of the respective generators:(aˆ11 aˆ12aˆ21 aˆ22)⇄(tˆ32 tˆ31tˆ42 tˆ41).This means that the commutation relations among the quantum generators are the followingtˆ42 tˆ41 = q−1tˆ41tˆ42, tˆ31tˆ41 = q−1 tˆ41tˆ31, tˆ32 tˆ41 = tˆ41tˆ32 + (q−1 − q)tˆ42tˆ31,tˆ31 tˆ42 = tˆ42tˆ31, tˆ32tˆ42 = q−1 tˆ42tˆ32, tˆ32 tˆ31 = q−1tˆ31tˆ32.What happened to the groups? They have become quantum groups with a non commutative product anda coproduct that is the same than the one we had before. This means that the group law has not changed,nor the coaction on the Grassmannian and the Minkowski space. The only change is that all these varietieshave become non commutative. It is a remarkable property of matrix quantum groups that the coproduct iscompatible with both, the commutative product and the non commutative one.In the quantum version Oq(Pl), the sets of generators x, y and T are separately isomorphic to 2 ×2 matrix algebras, but while x and y commute among them, T does not commute with the rest of thegenerators.4 Algebraic star productA quantum matrix algebra is an algebra over Cq = C[q, q−1], where q is a parameter. Moreover, as amodule over Cq, it is a free module, which means that it has a basis. It is well known that there is at leastone ordering among the generators such that the standard monomials associated to this ordering are a basisof the quantum matrix algebra. (This is a non trivial property).The ordering is the followingtˆ41 < tˆ42 < tˆ31 < tˆ32,and then there is an isomorphism (ordering rule or quantization map) between Oq(M) and O(M)[q, q−1]:O(M)[q, q−1]QM−−−−→ Oq(M)ta41tb42tc31td32 −−−−→ tˆa41tˆb42tˆc31 tˆd32.With the quantization map we can pull back the non commutative product toO(M)[q, q−1]. This definesa star product,f ⋆ g = Q−1M(QM(f)QM(g)), f, g ∈ O(M)[q, q−1],which can be computed explicitly(ta41tb42tc31td32) ⋆ (tm41tn42tp31tr32) = q−mc−mb−nd−dpta+m41 tb+n42 tc+p31 td+r32+µ=min(d,m)∑k=1q−(m−k)c−(m−k)b−n(d−k)−p(d−k)Fk(q, d,m)ta+m−k41 tb+k+n42 tc+k+p31 td−k+r32Fk(q, d,m) are numerical factors defined recursively. We recover the semiclassical interpretation of thealgebra being an algebra of functions, but with a star product.Copyright line will be provided by the publisher4 D. Cervantes, R. Cervantes, M. A. Lledo´, and F. A. Nadal: Deformation of Minkowski space5 Differential star productThe previous formula for the star product is nice and compact, but can only be computed on polynomials.Can we extend it to smooth functions? Not obvious. We prove that there exists a (unique) differential starproduct that coincides with the one given above on polynomials.Change of the parameter: q = eh. We expand in powers of h so we obtain a star product of the formf ⋆ g = fg +∞∑j=1hjCj(f, g),withf = ta41tb42tc31td32, g = tm41tn42tp31tr32.At each order, we have contributions from each of the terms with different kCn(f, g) =µ=min(d,m)∑k=0C(k)n (f, g).We want to write Cn as a bidifferential operator. But this is not trivial because all the dependence in theexponents should cancel.C1 =C(0)1 + C(1)1 = −(t41t31∂31 ⊗ ∂41 + t42t41∂42 ⊗ ∂41+t32t42∂32 ⊗ ∂42 + t32t31∂32 ⊗ ∂31 + 2t42t31∂32 ⊗ ∂41),Antisymmetrizing and changing variables we obtain the Poisson bracket{f, g} = i((x0)2 − (x3)2)(∂1f∂2g − ∂1g∂2f) + x0x1(∂0f∂2g − ∂0g∂2f)− x0x2(∂0f∂1g − ∂0g∂1f)− x1x3(∂2f∂3g − ∂2g∂3f) + x2x3(∂1f∂3g − ∂1g∂3f))Notice that if the x’s are real, then the Poisson bracket is pure imaginary. Also, it is quadratic.We have computed explicitly up to the order h2, but the expression is already too big to display it here.We looked for an argument to show it at arbitrary order. This can be done by careful inspection. The proofthat it is differential at each order is rather technical and we do not reproduce it here [4] But having theexplicit formula for the polynomials is essential to apply the argument.An example of the possible difficulty: suppose that we want to reproduce xm−1 as the result of applyinga differential operator to xm. We have several choices,xm−1 =1xxm, xm−1 =1m∂xxm, ...But none of them is both, independent on the exponent m and polynomial in the variable. So the rightcombination of coefficients should appear in order to cancel the factors that appear when differentiating.For example, if the result were mxm−1, then we have a differential operatormxm−1 = ∂xxm.Since we have recovered the interpretation of ’functions’ for the non commutative algebra, we can tryto express the coaction as an action of this space of functions. Remember that, formally, for the generatorsthe coaction is the same than in the commutative algebra. We just need to pull it back to the star productalgebra.We define the transformed variables (no translations are considered here)τij ≡ µG×M ◦ ∆˜⋆(tij) = yabtbjS(xji); soCopyright line will be provided by the publisherfdp header will be provided by the publisher 5µG×M ◦∆⋆(ta41tb42tc31td32) = τ⋆a41 ⋆G×M τ⋆b42 ⋆G×M τ⋆c31 ⋆G×M τ⋆d32 .One just has to expand the star products in the right hand side. Up to order h we have computed it in termsof a differential operator,12D1(τ41, τ41)∂2τ41+D1(τ41, τ42)∂τ41∂τ42 +12D1(τ42, τ42)∂2τ42+D1(τ42, τ31)∂τ42∂τ31+12D1(τ31, τ31)∂2τ31+D1(τ31, τ32)∂τ31∂τ32 +12D1(τ32, τ32)∂2τ32+D1(τ41, τ31)∂τ41∂τ31+D1(τ41, τ32)∂τ41∂τ32 +D1(τ42, τ32)∂τ42∂τ32 .and the coefficients are polynomials of order 6 in the variables x, y, t.6 Real forms: Euclidean and Minkowski quantum spacesLet A be a commutative algebra over C. An involution ι of A is an antilinear map satisfying, for f, g ∈ Aand α, β ∈ Cι(αf + βg) = α∗ιf + β∗ιg, (antilinearity)ι(fg) = ι(f)ι(g), (automorphism)ι ◦ ι = 11.Let us consider the set of fixed points of ι,Aι = {f ∈ A / ι(f) = f}.It is easy to see that this is a real algebra whose complexification is A. Aι is a real form of A.Classical Minkowski space:(ιM(t31) ιM(t32)ιM(t41) ιM(t42))=(t31 t41t32 t42), ιM(t) = tT .The combinationsx0 =12(t31 + t42), x1 =12(t32 + t41),x2 =12i(t41 − t32), x3 =12(t31 − t42),are fixed points of the involution. One hasO(M)ιM = R[x0, x1, x2, x3].Classical Euclidean space:(ιE(t31) ιE(t32)ιE(t41) ιE(t42))=(t42 −t41−t32 t31), ιE(t) = cof(t).The commbinationsz0 =12(t31 + t42), z1 =i2(t32 + t41),z2 =12(t41 − t32), z3 =i2(t31 − t42),Copyright line will be provided by the publisher6 D. Cervantes, R. Cervantes, M. A. Lledo´, and F. A. Nadal: Deformation of Minkowski spaceare fixed points of ιE, and as before,O(M)ιE = R[z0, z1, z2, z3].Formally, the same expressions on the generators as in the classical case are involutions in the quantumalgebra. A few things change:• Checking with the commutation relations they are antiautomorphisms, this isιM(fg) = ιM(g)ιM(f),ιE(fg) = ιE(g)ιE(f).• This discards the interpretation of the real form of the non commutative algebra as the set of fixedpoints of the involution.• When pulling back to the star product algebra, the Poisson bracket is purely imaginary.Finally one finds also the corresponding involutions in the group,ιPl,M(x) = S(y)T , ιPl,M(y) = S(x)T , ιPl,M(T ) = TT ;ιPl,E(x) = S(x)T , ιPl,E(y) = S(y)T , ιPl,E(T ) = cof(T ),It is not difficult to realize that in the Minkowskian case the real form of the Lorentz group (correspondingto the generators x and y) is SL(2,C)R and in the Euclidean case is SU(2)× SU(2).AcknowledgmentsD. Cervantes wants to thank the Departament de Fı´sica Teo`rica, Universitat de Vale`ncia for the hospitalityduring the elaboration of this work.Felip A. Nadal wants to thank CSIC for a JAE-predoc grant.This work has been supported in part by grants FIS2008-06078-C03-02 and FIS2011-29813-C02-02 ofMinisterio de Ciencia e Innovacio´n (Spain) and ACOMP/2010/213 of Generalitat Valenciana.References[1] D. Cervantes, R. Fioresi, M. A. Lledo´ and F. A. Nadal. arXiv:1109.4438 [hep-th].[2] R. Fioresi, Rev. Math. Phy., Vol. 9, n. 4, 453-465, (1997); Rev. Math. Phy. 11, 25-40 (1999).[3] D. Cervantes, R. Fioresi, M. A. Lledo´, To appear in Advances in Mathematical Physics. arXiv:1007.4469[math.QA]; Lect.Notes Math. 2027 (2011) 69-99[4] D. Cervantes, R. Fioresi, M. A. Lledo´, arXiv:1109.4438 [hep-th]Copyright line will be provided by the publisher

We present a deformation of the Minkowski space as embedded into the conformal space (in the formalism
of twistors) based in the quantum versions of the corresponding kinematic groups. We compute explicitly
the star product, whose Poisson bracket is quadratic. We show that the star product although defined on the
polynomials can be extended differentiably. Finally we compute the Eucliden and Minkowskian real forms
of the deformation

FIORESI, RITA

M. A. Lledo

F. A. Nadal

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

Quadratic deformation of Minkowski space

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Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna