Non-Commutative Geometry, Non-Associative Geometry and the Standard Model of Particle Physics

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by  Latham Boyle

Non-Commutative Geometry, Non-Associative Geometry
and the Standard Model of Particle Physics
Latham Boyle and Shane Farnsworth
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
(Dated: January 2014)
Connes’notionofnon-commutativegeometry(NCG)generalizesRiemanniangeometryandyields
a striking reinterepretation of the standard model of particle physics, coupled to Einstein gravity.
Wesuggestasimplereformulationwithtwokeymathematicaladvantages: (i)itunifiesmanyofthe
traditionalNCGaxiomsintoasingleone;and(ii)itimmediatelygeneralizesfromnon-commutative
to non-associative geometry. Remarkably, it also resolves a long-standing problem plaguing the
NCGconstruction of thestandardmodel, byprecisely eliminating from theaction thecollection of
4 7unwantedtermsthatpreviouslyhadtoberemovedbyanextra,non-geometric,assumption. With
1 this problem solved, the NCG algorithm for constructing the standard model action is tighter and
0 more explanatory than thetraditional one based on effective field theory.
2

c INTRODUCTION constructing the SM which is tighter and more explana-
e
D torythanthe traditionalonebasedoneffective fieldthe-
Since the early 1980s, Connes and others have been ory(EFT).IntheusualEFTconstruction,onemustgive
1 developing the subject of non-commutative geometry three independent inputs: (i) the symmetries of the ac-
1
(NCG) [1, 2]. Its mathematical interest stems from the tion; (ii)the fermionsandtheirrepresentations;and(iii)
] fact that it provides a natural generalization of Rieman- the scalars and their representations. In the NCG con-
h nian geometry (much as Riemannian geometry, in turn, struction,oneonlyneedstwoinputs: (i’)thechoiceofal-
t
- provides a natural generalization of Euclidean geome- gebra A (which determines the symmetry of the action);
p
try). Itsphysicalintereststemsfromthefactthatitsug- and (ii’) the representation of A (which determines the
e
h gests an elegant geometric reinterpretation of the stan- fermions and their representations). Note that the third
[ dardmodel(SM)ofparticlephysics(coupledtoEinstein EFT input (the scalarsand their representations),is not

gravity) [3–11]. For an introduction, see Refs. [12, 13]. required in the NCG approach: i.e. the number of higgs
3
v Hereweproposeasimplereformulationofthetraditional bosons and their representations are a predicted output,
3 NCG formalism that has fundamental advantages from once the symmetries and the fermionic representations
8 both the mathematical and physical standpoint. are specified! Furthermore, the NCG construction ex-
0 Our idea, in brief, is as follows. In the tradi- plains various aspects of the SM fermionic representa-
5
tional NCG formalism, a geometry is described by a so- tions which are unexplained by the usual EFT construc-
.
1 called real spectral triple A,H,D,J,γ (again, see e.g. tion(suchaswhyallSMfermionstransformineitherthe
0 Refs. [12, 13] for an intro{duction). Th}e essence of our trivialorfundamentalrepresentationofthegaugegroups
4
1 reformulationisthe observationthattheseelements nat- SU(2)andSU(3)–see[10,11,16]formoredetails). This
: urally fuse to form a new algebra B; and that many of striking situation is reviewed at greater length in [15].
v
the traditional NCG axioms may then be recovered by
i
X thesinglerequirementthatB isanassociative -algebra.
∗ REFORMULATION OF CONNES’ FRAMEWORK
r From the mathematical standpoint, this has two ad-
a
vantages. First, it unifies many of the traditional NCG
axiomsintoasingleone,therebyclarifyingtheirmeaning. Inthissection,wepresentourformulationintwosteps.
Second, it naturally generalizesConnes’ frameworkfrom Inthefirststep,weexplainhowtoextendA(a -algebra)
∗
non-commutative to non-associative geometry. Remark- toΩA(thedifferentialgraded -algebraofformsoverA),
∗
ably, it also has an unexpected physical consequence: it even when A is non-associative. In the second step, we
solvesa keyproblem(highlightede.g. in[9]andthe con- explain how H may be promoted to a bimodule over A
cluding section of [10]) which has plaguedthe NCG con- by defining a new algebra B0 = A H; and similarly,
⊕
struction of the SM action functional. It does so by pre- H may be promoted to a bimodule over ΩA by defining
cisely eliminating from the action the collection of 7 un- a new algebra B = ΩA H. In the process, we obtain
⊕
wanted terms that previously had to be removed by an a new view of the operator J, and Connes’ “order-zero”
extra (empirically-motivated, non-geometrical) assump- and“order-one”axioms. Again,akeyvirtueofourrefor-
tion (called the “massless photon” assumption in [10]), mulationisthatitnaturallyextendstothecasewhereA
in order to obtain agreement between the usual SM ac- is non-associative. (The mathematical and physical mo-
tionandthe actionobtainedfromtheNCGconstruction tivations for generalizing non-commutative geometry to
[9–11, 13]. non-associativegeometryareexplainedin[14];forearlier
Withthis problemsolved,NCGgivesanalgorithmfor work related to non-associative geometry, see [17–21].)
2
Step 1: Promoting A ΩA. Let A be a unital and b′ =a′+h′) given by
0
→
-algebra over a field F. (A may be non-commutative,
∗orevennon-associative.) We introduce,foreachelement b0b′0 =aa′+ah′+ha′, (2)
a A, a corresponding formal symbol δ[a]. ΩA is the
∈ where aa′ A is the product inherited from A, while
algebrageneratedbyAandthese differentialsδ[a], mod- ∈
ah′ H and ha′ H are precisely the left- and right-
ulo the relations δ[fa] = fδ[a], δ[a+a′] = δ[a]+δ[a′], ∈ ∈
actions defined above. In this language, two familiar ax-
δ[aa′] = δ[a]a′+aδ[a′] (with f F, and a,a′ A) and,
∈ ∈ ioms of non-commutative geometry – namely, (i) the as-
in addition, modulo appropriate associativity relations.
sociativity of A and (ii) the order-zero condition – are
Forexample,intheusualcasewhereAisassociative,we
take ΩA to be associative as well, and impose relations condensed into the single assumption that B0 is an as-
sociative algebra. Furthermore, the familiar definition
like (aδ[b])c = a(δ[b]c), etc; in this way, we recover the
usualalgebraΩAdefined,e.g. inSection6.1ofRef.[22]. of right-action in terms of left action, Ra = JLa∗J∗, is
reinterpreted as the statement that the map
More generally, when A is non-associative, ΩA must be
equipped with compatible associativity relations (for ex- b0 =a+h b∗0 =a∗+Jh (3)
ample, if A is an alternative algebra, then ΩA could be →
taken to be alternative as well; other examples are given is an anti-automorphism of B0, with period 2 when the
in Ref. [25]). KOdimensionis0,1,6or7mod8(i.e.whenJ2 =1)and
We can write ΩA = Ω0A Ω1A Ω2A ... where period4whentheK0dimensionis2,3,4or5mod8(i.e.
⊕ ⊕ ⊕
ΩmA is the subspace of ΩA consisting of linear combina- when J2 = 1). In particular, when J2 = 1, B0 is a -
− ∗
tions of terms containing m differentials δ[a]: for exam- algebra,with operationgivenby(3). Theadvantageof
∗
ple, (a1δ[a2])(δ[a3]δ[a4])+(δ[a5](δ[a6]δ[a7]))a8 is an ele- thisreformulationisthatitcontinuestomakesensewhen
ment of Ω3A. In particular, Ω0A = A; and if ωm ΩmA A is non-associative: in this case, B0 is non-associative,
and ωn ΩnA then ωmωn Ωm+nA, so ΩA is gra∈ded. If too, and the familiar order zero condition is replaced by
∈ ∈
wedefineδ[a∗]= δ[a]∗,thenΩAisa -algebra,withits a compatible restriction on the associativity properties
− ∗
-operationnaturallyinheritedfromA. Notethatwecan of B0. For example, if A is an alternative algebra, like
i∗nterpret δ as a linear map from Ω0A Ω1A; and this the octonions, we can require B0 to be an alternative
→
may, in turn, be promoted to a linear map d : ΩmA algebra, too. The interpretation of J in terms of the
Ωm+1A which (even in the non-associative case) may b→e anti-automorphism (3) is unaffected.
defined recursively by requiring it to satisfy a graded Justaswegiveabi-representationofAonH bydefin-
Leibniz rule d[ωmωn]=d[ωm]ωn+( 1)mωmd[ωn], along inganewalgebraB0 =A H,wegiveabi-representation
withtheconditionsd[a]=δ[a]andd−2[a]=0(ωm ΩmA, of ΩA on H by defining a⊕new algebra
ω ΩnA, a A). It follows that d2[ω] = 0 (ω∈ ΩA),
n
∈ ∈ ∈
so that (ΩA,d) is a differential graded -algebra. B =ΩA H (4)
Step 2: Promoting Ω0A B0 ∗and ΩA B. ⊕
→ → with the product between two elements of B (b=ω+h
Usually, in non-commutative geometry, one starts by
and b′ =ω′+h′) given by
defining (in two steps) a bi-representation of A on H,
so that H becomes a bi-module overA. In the firststep, bb′ =ωω′+ωh′+hω′, (5)
one defines a left actionof A onH – i.e. a bilinear prod-
uct ah = Lah H between elements a A and h H. whereωω′ ΩAisthe productinheritedfromΩA,while
In the second s∈tep one uses J, an anti-u∈nitary ope∈rator ωh′ H an∈d hω′ H are bilinear products that define
on H, to define a corresponding right-action of A on H the l∈eft-action and∈right-action of ΩA on H[30]. Thus,
– i.e. another bilinear product ha ≡ Rah ∈ H given by just as A = Ω0A is a subalgebra of ΩA, B0 is a corre-
Ra JLa∗J∗. The left and right action are required to sponding subalgebra of B. The elements ω ΩA are
sati≡sfy the so-called order-zero condition (ah)b = a(hb) linear combinations of products of a’s and δ[a]∈’s: having
or, equivalently, [La,Rb]=0 ( a,b A). already introduced the left- and right-action of a on H
∀ ∈
Wewouldliketoreformulatethisconstructioninaway (ah = L h and ha = R h), we now obtain the left- and
a a
that makes sense evenwhen A is non-associative. Fortu- right-action of δ[a] on H by regarding D, a hermitian
nately, the natural definition of a “bi-representation” of operator on H, as the representation on H of the map
a non-associativealgebra(or,equivalently,a “bimodule” δ : Ω0A Ω1A, and requiring that it satisfies the corre-
over a non-associativealgebra) was found long ago (per- sponding→Leibnizrule: D(ah)=δ[a]h+a(Dh). Thisgives
haps by Samuel Eilenberg [23]), and is explained simply δ[a]h=[D,L ]h andhδ[a]=J[D,L ]∗J∗h. We see that
a a
and succinctly in Ch. II.4 ofRef. [24]. The idea is that a Connes’ order-zero and order-one conditions are reinter-
bimodule H over A is nothing but a new algebra preted here as associativity conditions on B: in particu-
lar, the order zero condition [L ,R ] = 0 is the require-
B0 =A⊕H, (1) mentthattheassociator[ω0,h,ωa0′]vbanishes,whiletheor-
withtheproductbetweentwoelementsofB0 (b0 =a+h der one conditions [La,[D,Rb]]=0 and [[D,La],Rb]=0
3
are the requirements that the associators [ω0,h,ω1] and and qλ is the corresponding diagonal embedding of C in
[ω1,h,ω0]vanish. Inthe casewhereAisnon-associative, H. Then La (the left action of a on ) is given by
H
the familiar order-zero and order-one conditions are re-
placed by compatible associativity constraints on B. LaLR = qλLR, LaLL = qLL,
L Q = q Q , L Q = qQ ,
a R λ R a L L
(8)
APPLICATION TO THE STANDARD MODEL LaL¯R = λI2×2L¯R, LaL¯L = λI2×2L¯L,
L Q¯ = mQ¯ , L Q¯ = mQ¯ .
a R R a L L
In this section, we first review the traditional formu-
lation of the standard model in non-commutative geom- where q, qλ and λI2×2 acton the doubletindex, while m
acts on the color index.
etry, and then explain how our reformulation naturally
D obeys the following four geometric constraints:
yields a new constraint that resolves a well-known puz-
D† =D, D,γ =0, [D,J]=0 and [[D,L ],R ]=0. In
zle that arisesin the traditionalformulation. Forclarity, thebasis{L ,Q} ,L ,Q ,L¯ ,Q¯ ,L¯ ,Q¯ a,thbeseimply
we will deal in this section with a single-generation of { R R L L R R L L}
standard model fermions; the extension to the full set of 0 0 y† 0 m† n† 0 0
three generations is straightforward. l
 0 0 0 y† n¯ 0 0 0 
The standard model is described by a finite- q
gwmdeihembenrersaneiosoCnifonqi6sau.ltahAtreeeraianlslisgoapenbes∗cr,-taaarlnaogldfebtcMrroiapm3l(gepCilv{e)exAnis,nHbutyhm,eDbCae,lr⊕Jgs,e,γHbH}ra⊕oisfoMftKh330e(×Cda)il3--, D = myn0l ny00Tq 0000 0000 0000 0000 y000lT y000T  (9)
complexmatrices. H isa32-dimensionalcomplexHilbert  q 
 
space(32isthenumberoffermionicdegreesoffreedomin  0 0 0 0 y¯l 0 0 0 
 
astandardmodelgeneration,including the right-handed  0 0 0 0 0 y¯ 0 0 
 q 
neutrino). To describe the action of γ and J on H, it is
convenient to split H into four 8-dimensional subspaces where
H =H H H¯ H¯ . HereH andH containthe
right-haRn⊕dedLan⊕dleRft⊕-hanLdedparticRles,whilLeH¯ andH¯ y = yl,11 yl,12 and y = yq,11 yq,12 (10)
contain the corresponding anti-particles. If h R H iLs l (cid:18)yl,21 yl,22 (cid:19) q (cid:18)yq,21 yq,22 (cid:19)
R R
a right-handed particle (with h¯R H¯R the corr∈espond- arearbitrary2 2matricesthatactonthedoubletindices
ing anti-particle) and hL HL is∈a left-handed particle in the lepton a×nd quark sectors, respectively, while
(withh¯ H¯ thecorresp∈ondinganti-particle),thenthe
L L
∈
helicityoperatorγ andtheanti-linearchargeconjugation a b ~c d~
m= and n= (11)
operator J act as follows: (cid:18) b 0 (cid:19) (cid:18)~0 ~0 (cid:19)
γh = h , γh =h , γ¯h =h¯ , , γ¯h = h¯ ,
R − R L L R R L − L (6) are 2 2 and 6 2 matrices, respectively; and in n we
JhR=h¯R, JhL=¯hL, Jh¯R=hR, Jh¯L=hL. haveu×sedvector×notationtoemphasizethat~c,d~and~0are
3 1 columns. Of the 8 complex parameters a,b,~c,d~ ,
To describe the action of A on H, it is convenient to × { }
further split each of the four spaces (H ,H ,H¯ ,H¯ ) only a is present in the standard model (where it corre-
R L R L
sponds to the right-handed neutrino’s majorana mass).
into a lepton and quark subspace: H = L Q ,
H = L Q , H¯ = L¯ Q¯ , and HR¯ = L¯R ⊕ Q¯R. The remaining 7 parameters b,~c,d~ present a puzzle –
L L L R R R L L L { }
Eachofthe⊕fourleptonspace⊕s L ,L ,L¯ ,L¯ isa⊕copy they are an unwanted blemish that must be removed in
R L R L
ofC2;anelementofanyofthe{sefourspacesco}rrespond- order to match observations. Traditionally, they are re-
movedbyintroducinganextraassumption(namely,that
ingly carries a doublet (neutrino vs. electron) index.
Eachofthefourquarkspaces Q ,Q ,Q¯ ,Q¯ isacopy D commutes with La for a = λ,qλ,0 A) [9]; but, as
R L R L { }∈
of C2 C3; an element of an{y one of these fo}ur spaces emphasizedby Chamseddine and Connes (see e.g. Sec. 5
⊗ ofRef.[10]),thisadhocsolutionisunsatisfying,andcries
correspondinglycarriestwo indices: a doublet (up quark
out for a better understanding.
vs. down quark) index and a triplet (color) index. Now
consideranelementa= λ,q,m ,whereλ Cisa Our reformulation yields a simple and satisfying solu-
F
complex number, q H{is a qua}te∈rnAion,and m ∈M3(C) tion to this puzzle. We have seen that the associativ-
∈ ∈ ity of B = ΩA H implies the usual order zero and
is a 3 3 complex matrix, and write
⊕
× order one constraints ([L ,R ] = 0 and [[D,L ],R ] =
a b a b
α β λ 0
q = , q = , (7) 0); but notice that it also implies a new constraint:
(cid:18) β¯ α¯ (cid:19) λ (cid:18) 0 λ¯ (cid:19) [[D,L ],[D,R ]] = 0. This constraint may be satisfied
− a b
where α andβ are complex numbers. Here q is the stan- in four different ways, by setting (i) b =~c = d~= 0; (ii)
dard2 2complexmatrixrepresentationofaquaternion, yq,11 = yq,21 = b = 0; (iii) yl,11 = yl,21 =~c = d~= 0; or
×
4
(iv) yl,11 = yl,21 = yq,11 = yq,21 =~c = 0. Note, in par- to an algebra B that is associative. But, to evaluate the
ticular, that solution (i) precisely corresponds to setting spectralaction,onethentensorsthisfinitegeometrywith
the 7 unwanted parameters (b,~c,d~) to zero, without the a continuous geometry to form a new geometry T′, and
additional ad hoc assumption described above! one can check that the corresponding algebra B′ is not
We can go further by noting that the general embed- associative(whenonegoesbeyondtheorderoneassocia-
dingofCinHisgivenbyqλ(nˆ)=Re(λ)I2×2+Im(λ)nˆ ~σ, tors). In this sense, non-associativity already appears in
·
where ~σ are the three Pauli matrices, and nˆ is a unit the traditional NCG embedding of the standard model.
3-vector specifying the embedding direction. Since all of Itisinterestingtoconsiderwhetherthisnon-associativity
theseembeddingsareequivalent,thediagonalembedding might be connected to the generalizedinner fluctuations
q =q (zˆ)inEq.(8)wasarbitrary,andmaybereplaced consideredin [27, 28], which bear a striking resemblance
λ λ
by the more general possibility L L = q (nˆ )L and to the inner derivations of a non-associative(and in par-
a R λ l R
L Q = q (nˆ )Q . If we redo the preceding analysis ticular, an alternative) algebra [14]. (iv) In a follow-up
a R λ q R
withthis modification,the four solutionsfor D aremod- paper [29], we have shown how the formalism presented
ifiedaccordingly: inparticular,insolution(i), D isgiven here yields a new perspective on the symmetries of a
by Eq. (9), where the 2 2 matrices y and y are arbi- non-commutative geometry and, in particular, when we
× l q
trary, the 6 2 matrix n vanishes, and the 2 2 matrix apply this new understanding to the spectral triple tra-
× ×
m is given by m = PTMP, with M an arbitrary 2 2 ditionally used to describe the standard model (i.e. the
symmetric matrix and P = (I2×2+nˆl ~σ)/√2 a pro×jec- spectraltriple discussedin the previous section), we find
·
tion operator. Then, one can check the following result: thatitactuallypredictsaslightextensionofthestandard
given the arbitrary 2 2 matrices y , y and M, there model, with two new particles: a U(1) gauge boson,
l q B−L
×
is a preferredchoice for the embedding directions nˆ and and a complex scalar field which carries B L charge
l −
nˆq such that, after a change of basis on H, La is given andisresponsiblefor“higgsing”thenewU(1)B−L gauge
by Eq. (8) [with the diagonal embedding q = q (zˆ)], D symmetry. These two new particleshave importantphe-
λ λ
is given by Eq. (9), m and n are given by Eq. (11) with nomenological and cosmological consequences that will
b=~c=d~=0, while y and y are given by: be analyzed in subsequent work.
l q
yl=(cid:18)yyννϕϕ12 −+yyeeϕϕ¯¯21(cid:19), yq=(cid:18)yyuuϕϕ21 +−yyddϕϕ¯¯12(cid:19), (12) rieAdcokuntoawtltehdeg“eNmoennctosm.muPtaarttiveofGtehoims ewtroyrkanwdasPacratri--
cle Physics”Workshopat the Lorentz Center, in Leiden;
with {yν,ye,yu,yd,ϕ1,ϕ2} ∈ C. This is precisely the we thank the organizers and participants, and particu-
finite geometry that (after fluctuation and substitution
larly Ali Chamseddine and Alain Connes for helpful in-
intothespectralaction)generatesthestandardmodelof
put. Research at the Perimeter Institute is supported
particle physics (see Refs. [9–11, 13])!
by the Governmentof CanadathroughIndustry Canada
Thisstrikinglysuccessfulmatchbetweenthegeometric
and by the Province of Ontario through the Ministry of
structure on the one hand, and the standard model La-
Research & Innovation. LB also acknowledges support
grangianon the other, appears to provide significantad-
from an NSERC Discovery Grant.
ditional evidence: (i) for the suitability of Connes’ NCG
frameworkfordescribingthestandardmodel,and(ii)for
the appropriateness of the reformulation presented here.
[1] A. Connes, Non-Commutative Geometry, Academic
DISCUSSION
Press, Boston MA (1994) ISBN-9780121858605.
[2] A. Connes Publ. Math. de l’IHES [0073-8301], 62, 41
We end with a few brief remarks. (i) In this paper, (1985).
although we have developed a formalism suited to non- [3] A. Connes and J. Lott, Nucl. Phys. Proc. Suppl. 18B,
29 (1991).
associative geometry, our main application has been to
[4] A. Connes, Commun. Math. Phys. 182, 155 (1996)
the associative finite geometry that describes the stan-
[hep-th/9603053].
dard model of particle physics. In a forthcoming paper [5] A.H.ChamseddineandA.Connes,Phys.Rev.Lett.77,
[25] we present a family of geometries (and their asso- 4868 (1996).
ciated spectral actions) that provide a nice illustration [6] A. H. Chamseddine and A. Connes, Commun. Math.
of our formalism in the fully non-associative case. (ii) Phys. 186, 731 (1997) [hep-th/9606001].
Refs. [10, 11] observe that the standard model algebra [7] J. W. Barrett, J. Math. Phys. 48, 012303 (2007)
Amo=reCde⊕epHly⊕asMa3s(uCb)alagneablryazeodfAab′o=veMc2a(nHb)⊕e uMn4d(eCrs)t(osoede [[98]] AA[h..epCH-ot.hnC/n0he6sa,0m8Js2He2dE1dP].in0e6,1A1.,C08o1nn(2es00a6n)d[hMep.-tMh/a0rc6o0l8li2,2A6]d.v.
also [26]). What new light does our formalism shed on Theor. Math. Phys. 11, 991 (2007) [hep-th/0610241].
this observation? (iii) We have seen that the finite ge- [10] A. H. Chamseddine and A. Connes, J. Geom. Phys. 58,
ometry T that encodes the standard model corresponds 38 (2008) [arXiv:0706.3688 [hep-th]].
5
[11] A.H.ChamseddineandA.Connes,Phys.Rev.Lett.99, [22] G. Landi, hep-th/9701078.
191601 (2007) [arXiv:0706.3690 [hep-th]]. [23] S. Eilenberg, Ann.Soc. Polon. Math. 21, 125 (1948).
[12] A. H. Chamseddine and A. Connes, arXiv:1008.0985 [24] R.D. Schafer, An Introduction to Nonassociative Alge-
[hep-th]. bras, Academic Press, NewYork NY (1966).
[13] K.vandenDungenandW.D.vanSuijlekom,Rev.Math. [25] S. Farnsworth and L. Boyle, in preparation.
Phys.24, 1230004 (2012) [arXiv:1204.0328 [hep-th]]. [26] A. Devastato, F. Lizzi and P. Martinetti, JHEP 1401,
[14] S.Farnsworth and L. Boyle, arXiv:1303.1782 [hep-th]. 042 (2014) [arXiv:1304.0415 [hep-th]].
[15] L. Boyle and S. Farnsworth, in preparation. [27] A.H.Chamseddine,A.ConnesandW.D.vanSuijlekom,
[16] T. Krajewski, J. Geom. Phys. 28, 1 (1998) J. Geom. Phys. 73, 222 (2013) [arXiv:1304.7583 [math-
[hep-th/9701081]. ph]].
[17] R. Wulkenhaar, J. Geom. Phys. 25, 305 (1998) [28] A.H.Chamseddine,A.ConnesandW.D.vanSuijlekom,
[hep-th/9607086]. JHEP 1311, 132 (2013) [arXiv:1304.8050 [hep-th]].
[18] R. Wulkenhaar, Phys. Lett. B 390, 119 (1997) [29] S. Farnsworth and L. Boyle, arXiv:1408.5367 [hep-th].
[hep-th/9607096]. [30] For brevity, we are skipping over the issue of junk
[19] R. Wulkenhaar, J. Math. Phys. 38, 3358 (1997) forms,andthecorrespondingdistinctionbetweenΩAand
[hep-th/9607094]. ΩAD =ΩA/J (seeSec.6.2ofRef.[22]),sincethisnuance
[20] R. Wulkenhaar, Int. J. Mod. Phys. A 13, 2627 (1998) is not important for our present purposes; but we note
[hep-th/9607237]. thattheymaybereadily encorporatedinourapproach.
[21] S.E. Akrami and S. Majid, J. Math. Phys. 45, 3883
(2004).