Signatures of Noncommutative Geometry in Muon Decay for Nonsymmetric Gravity

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by  Dinesh Singh

Signatures of Noncommutative Geometry in Muon Decay for Nonsymmetric Gravity
Dinesh Singh,∗ Nader Mobed,† and Pierre-Philippe Ouimet‡
Department of Physics, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada
(Dated: January 27, 2009)
It is shown how to identify potential signatures of noncommutative geometry within the decay
spectrumofamuoninorbitneartheeventhorizonofamicroscopic Schwarzschildblackhole. This
possibility follows from a re-interpretation of J.W. Moffat’s nonsymmetric theory of gravity, first
published in Phys. Rev. D 19, 3554 (1979), where the antisymmetric part of the metric tensor
manifests the hypothesized noncommutative geometric structure throughout the manifold. It is
further shown that for a given sign convention, the predicted signatures counteract the effects of
curvature-induced muon stabilization predicted by D. Singh and N. Mobed in Phys. Rev. D 79,
024026(2009). Whileitisunclearwhetherevidencefornoncommutativegeometrymaybefoundat
the Large Hadron Collider (LHC) anytime soon, this approach at least provides a useful direction
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for future quantumgravity research based on theideas presented here.
0
0
PACSnumbers: 02.40.Gh,04.60.Bc,04.62.+v,11.30.Cp,13.35.Bv
2

n
a Introduction.–Efforts to find a unified theory of par- group for classifying elementary particles according to
J
ticle physics beyond the Standard Model (SM) have led their mass and spin angular momentum, for situations

7 researcherstoconsidervariousapproachestowardsreach- involving the non-inertial motion of spin-1/2particles in
2
ingthisgoal. Onepossibilitythathasgainedwidespread space with respect to a global laboratory frame in a flat

attention over the past two decades is string theory [1], space-time background. This is a relevant issue given
]
c primarily because of the claim that it includes a quan- that the Poincar´e group is well-defined only for strictly
q
tum theory of gravity within its framework. However, inertialmotion[6],leadingtoconceptualchallengeswhen
-
r therearenumerouscompetingformulations,suchasloop applied to an elementary particle’s accelerated motion
g
quantum gravity [2] and others, with radically different in flat space-time or geodesic motion in curved space-
[
foundations and implications as they relate to Beyond- time. By describing the Poincar´e group generators for

1 Standard-Model (BSM) physics. This is due to the fact the canonical momentum P in terms of curvilinear spa-
v
thattheSMoffersnoinsightsabouthowgravitybehaves tial co-ordinates to accommodate the symmetries of the
8
at quantum length scales. particle’sclassicalspatialtrajectory,it is shownthat the
0
3 A recent approach in quantum gravity of serious con- corresponding Pauli-Lubanskispin four-vector W yields
4 sideration is noncommutative geometry [3], which as- aCasimirscalarW W thatisno longerthe Lorentzin-
1. sumes that pairs of space-time co-ordinates violate com- variant 12 12 +1 m20·,wherem20 =P·P. Rather,W·W
0 mutationatsomescale¯kwithdimensionsofarea,andin- isnowfra(cid:0)me dep(cid:1)endent dueto anadditivetermcoupling
9 steadbehavelikenoncommutatingoperators. Thisprop- the Pauli spin operator σ with a Hermitian three-vector
:0 erty is described by Ri = 2i~ ǫijk[Pj,Pk], knownas the non-inertial dipole
v operat(cid:0)or [(cid:1)6, 7] because it generates an interaction term
i [Xµ,Xν] = i¯kJµν(Xα), (1) analogous to a dipole interaction with a magnetic field.
X
For r the particle trajectory’s local radius of curvature
ar where Xµ is a Hermitian co-ordinate operator and Jµν with respect to the laboratory frame, the dimensionality
isadimensionlessantisymmetricHermitianoperator,re- of R is P /r, with the property that R 0 for fixed
| | →
sembling orbital angular momentum generators with an momentum as r , while for Cartesian co-ordinates
SU(2) Lie algebra structure. Historically, this proposal torepresentstrict→lyr∞ectilinearmotion,R=0identically.
was introduced several decades ago [4] to truncate ul-
traviolet divergences in the early developments of QED, More recently, this computation was repeated in the
but lost favour after successful techniques of regulariza- presenceofacurvedspace-timebackground,describedin
tion and renormalization were introduced. Nonetheless, terms of Fermi normal co-ordinates. Upon applying the
noncommutativegeometryhasmadearesurgenceoflate, formalismtothespecificexampleofmuondecaywhileor-
in part because of its capacity to offset curvature singu- bitingamicroscopicKerrblackhole[8],itwasshownthat
larities inside black holes [5] by truncating the curvature themuondecayspectrumbecomessignificantlydistorted
fieldstrengthinmuchthesamewayaswasdonetoover- due to both non-inertial effects from R and curvature-
come perceived deficiencies in quantum field theory. induced contributions, ultimately leading to a predicted
In contrast to taking a direct approach for quantum stabilization of the muon when approaching an orbital
gravityresearch,arecentinvestigation[6]focussedonthe radius close to the particle’s Compton wavelength. This
more intermediate problem of understanding the appli- by itself is a highly significant result, since it suggests
cability of the Poincar´egroup, the space-time symmetry that the intersection between quantum mechanics and
2
gravitation becomes relevant at a length scale over 20 connection. An orthonormal vierbein set e¯µ (X)
αˆ
{ }
orders of magnitude larger than the Planck scale for this and its inverse e¯αˆ (X) can be obtained to define
µ
application, and suggests non-trivial behaviour of ele- a local Lorentz (cid:8)frame, d(cid:9)enoted by co-ordinates with
mentary particles in curvedspace-time apparently unac- hatted indices. The metric tensor is then described by
counted for by current explorations of quantum gravity Fg (X)=η e¯αˆ (X)e¯βˆ (X), such that
µν αˆβˆ µ ν
research. Italsocorroboratesanearlierclaim[9]thatpre-
dictscurvature-inducedchangestotheelectricfieldatthe Fg00(X) = 1 FRl00m(T)XlXm+ , (3a)
− ···
Compton scale generatedby a charged microscopic Kerr 2
Fg (X) = FR (T)XlXm+ , (3b)
black hole with the same mass and spin as an electron. 0j −3 l0jm ···
In addition, this paper noted an interesting observation 1
Fg (X) = η FR (T)XlXm+ , (3c)
that if the metric tensor in Fermi normal co-ordinates ij ij − 3 lijm ···
Fg (X)wereallowedtobecomenonsymmetricwhilesi-
muµlνtaneously allowing for Xµ Xµ in accordance with whereηµν istheMinkowskimetricandFRµαβν(T)isthe
→ projection of the Riemann tensor for general relativity
(1),thenitistheoreticallypossibletoidentifynon-trivial
(GR) onto the Fermi frame.
signaturesofnoncommutativegeometryin the muonde-
Astraightforwardconversionof(2)tomutuallyorthog-
cayspectrum. However,apreliminarycomputationwith
onal spatial curvilinear co-ordinates is then introduced,
this extension yielded no contribution whatsoever due
definedbyUµ = T,ui ,whereXi =Xi(u). Subsequent
to noncommutative geometry, a highly surprising result
projection of (2)(cid:0)onto t(cid:1)he local Lorentz frame leads to
with no obvious explanation forthcoming.
Nonetheless,sincethis firstinvestigationintononcom- γµˆ(P ~Γ (U)) m ψ(U) = 0, (4)
µˆ µˆ
mutativegeometry,itwassoonrealizedthatamorecom- − −
(cid:2) (cid:3)
plete computation to yield a signature wouldrequire the where
full details of a suitably-defined nonsymmetric theory of iΓ = Γ¯(S)+γˆlγmˆ Γ¯(T) δˆ0 , (5)
gravity. It so happens that a theory of nonsymmetric µˆ µˆ ˆ0[ˆlmˆ] µˆ
gravity developed years ago by Moffat [10, 11], origi-
and
nally derivedas a purely classicaltheorywith a different
1
motivation in mind, may satisfy this requirement if the Γ¯(S) = FRm (T)XjXk, (6a)
ˆ0 12 jmk,0
antisymmetric part of the metric tensor is identified ex-
1 1
clusively with the noncommutative geometric extension Γ¯(S) = FR (T)+ FR (T)Xl Xm, (6b)
of space-time, as originally proposed [8]. The purpose ˆ −(cid:20)2 j00m 3 jl0m (cid:21)
of this paper is to present the outcome of this compu- Γ¯(T) = 1 FR (T)XjXk
tation in a microscopic Schwarzschild space-time back- ˆ0[ˆlmˆ] 12 k[lm]j,0
ground using Moffat’s nonsymmetric gravity theory to 1
+ FR (T)+FR (T) Xk. (6c)
incorporateadditionalcurvaturetermscoupledtok¯ that 3 lm0k k[lm]0
(cid:2) (cid:3)
werepreviouslynotconsidered. As well, this paper com- After defining D =P ~Γ and using the identity
µˆ µˆ µˆ
pares two possible formulations of noncommutative ge- −
ometry,as determined by the choice forJµν(Xα)in (1). γµˆγνˆγρˆ = ηνˆρˆγµˆ 2γ[νˆηρˆ]µˆ iγ5γσˆεµˆνˆρˆ , (7)
σˆ
− −
While k¯ may have theoretical physical constraints on its
magnitude, in this paper k¯ is treated as a free parame- where εµˆνˆρˆσˆ is the Levi-Civita symbol with εˆ0ˆ1ˆ2ˆ3 =1, it
ter to be determined by observation. Geometric units of follows that
G=c=1 are assumed throughout, where the curvature D = P ~ γ5Γ¯(C) iΓ¯(S) , (8)
tensors presented here satisfy the conventions of MTW µˆ µˆ− (cid:16) µˆ − µˆ (cid:17)
[12] but with 2 metric signature. Γ¯(C) = εˆ0ˆlmˆ Γ¯(T) , (9)
Formalism.–−While details of the formalism employed µˆ µˆ ˆ0[ˆlmˆ]
herearegivenelsewhere[8],it isimportantto offersome where “S” gives the symmetric part of the spin connec-
relevant highlights to justify its later extension to in- tion under chiral symmetry and “C” implies chiral de-
clude noncommutative geometry. Given Fermi normal pendence by its coupling with γ5.
co-ordinates Xµ = T,Xi defined in a local neighbour- Following standard definitions, the Pauli-Lubanski
hood about a spin-(cid:0)1/2 pa(cid:1)rticle’s worldline, where T is vector in a local Lorentz frame is
the proper time and Xi is the local Cartesianspatialco-
1
ordinateintheFermiframe,thecovariantDiracequation Wµˆ = −4εµˆαˆβˆγˆσαˆβˆDγˆ. (10)
with mass m and ∂ =∂/∂Xµ is
µ
Using both (7) and the identity ε σρˆσˆ = 2iγ5σ ,
[iγµ(X)(∂ +iΓ (X)) m/~]ψ(X) = 0. (2) µˆνˆρˆσˆ − µˆνˆ
µ µ − its squared magnitude is
The set of gamma matrices γµ(X) satisfy 3 i
{γµ(X),γν(X)} = 2gFµν(X) and Γµ{(X) is}the spin WαˆWαˆ = −4DαˆDαˆ + 4σαˆβˆhDαˆ,Dβˆi (11)
3
and leads to connection
3 ~2 ~ 2
WαˆWαˆ = −4(cid:20)m20+ 6 FRαˆαˆ (cid:21)+ 2 (σ·R) Wλµν = Γλµν − 3δλµWν, (16)
(cid:0) (cid:1)
~σαˆβˆQ + 3~ γ5Γ¯(C) iΓ¯(S) Pαˆ where Wµ = Wλ[µλ] is a vector field expressed in terms
− 4 αˆβˆ 2 (cid:16) αˆ − αˆ (cid:17) of g[µν]. The modified tensor components in the Fermi
+ 3~2∇αˆ Γ¯(S)+iγ5Γ¯(C) , (12) framearethenintroducedintotheorthonormalvierbeins
4 (cid:16) αˆ αˆ (cid:17) e¯µαˆ(X) and inverse vierbeins e¯αˆµ(X) [8]. Moffat’s
{ }
where m2 = PαˆP , the Casimir invariant for mo- theory is particularly useful in th(cid:8)e linear(cid:9)approximation
mentum,0∇ is the gαˆradient operator in curvilinear co- about the Minkowski metric [11], since the symmetric
µˆ
and antisymmetric field equations decouple to first or-
ordinates,andQ ispurelycurvature-dependent. When
αˆβˆ der. Given that Fg (X) according to (3) is expressible
FR (T) 0, (12) reduces to its flat space-time ex- µν
presµsνiαoβn [6],→where ~(σ R) is the non-inertial dipole precisely as a linearized expansion about ηµν, this par-
2 · ticular form of Moffat’s nonsymmetric gravity theory is
interaction term to justify the given name for R.
especially useful in this context.
Extension for Noncommutative Geometry.–To ensure
Noncommutative Geometry in Muon Decay.–For this
that (3) is truly symmetric requires assuming that the
paper, the decay process occurs for a muon in circular
Riemann tensor is also symmetric in the middle two
orbit around a microscopic Schwarzschild black hole [8]
indices. Since this is generally not true, the symmet-
ofmassM =3 10−12 cmwith (classical)orbitalradius
ric properties of Fg (X) follow directly from assuming ×
µν r 6M, where Xj represents spatial quantum fluctua-
the condition that ds2 = Fg (X)dXµdXν, where Xµ 0 ∼
µν tions about r . The matrix element [6, 8] for the muon
and dXµ are ordinary c-numbers. This is an important 0
decay reaction µ− e−+ν¯ +ν is
observation given that the antisymmetric combination → e µ
FRµ[αβ]ν(T)= 12 FRµαβν(T)−FRµβαν(T) leadstothe 2 = G2F L(µ)Mµˆνˆ, (17)
appearance of ter(cid:2)ms like (cid:3) |M| 2 µˆνˆ (e)
FR (T)XµXν = 1FR (T)[Xµ,Xν] . (13) where
µ[αβ]ν −4 αβµν
Lµˆνˆ = Tr p/ γµˆ D/ +m γ5n/ γνˆ 1 γ5 ,(18a)
Assuming that Xµ are ordinary c-numbers, it fol- (µ) h νµ (cid:16) µ µ µ(cid:17) (cid:0) − (cid:1)i
lows that (13) automatically vanishes. However, if Mµˆνˆ = Tr D/ +m γ5n/ γµˆp/ γνˆ 1 γ5 , (18b)
Xµ Xµ, then (13) is nonzero from (1), which results (e) (cid:2)(cid:0) e e e(cid:1) νe (cid:0) − (cid:1)(cid:3)
in d→s2 1Fg (X)(dXµ dXν +dXν dXµ) + and nµˆ is the polarization vector for the charged lep-
→ 2 (µν) ⊗ ⊗
o12pFegra[µtνo]r(Xde)fidnXedµa∧tdtXheνk,¯-wshcaelree,Fagn[dµν](X)isaHermitian titonc.anFbroemshtohweno[r8t]hotghoantaln/ity=conn/d(iSt,iRoen)+nµˆin/D(Sµˆ,Im=) +0,
γ5 n/(C,Re)+in/(C,Im) , eventually(cid:2)leading to (cid:3)
i¯k 1
Fg (X) = δ0 FR (T)+ δi FR (T) (cid:2) (cid:3)
[µj] 6 J(cid:20)kl(µX).0jkl 2 µ ijkl ((cid:21)14) |M|2 = 32G2F(cid:16)PανˆeD¯µαˆ(cid:17)(cid:16)PβνˆµD¯eβˆ(cid:17) , (19)
×
whereD¯ =P ~ Γ¯(C) iΓ¯(S) +m n(C) n(S) .
Thisleadstoanimmediateextensionofthe spinconnec- αˆ αˆ− (cid:16) αˆ − αˆ (cid:17) 0(cid:16) αˆ − αˆ (cid:17)
tion, in the form Both the gravitational and R-dependent contributions
to the muon decay rate are additive corrections to Γ
0
i¯k ≈
Γ¯(µˆS) → Γ¯(µˆS)+ 12FRlm0µ,0(T)Jlm, (15a) G2Fm5µ/(192π3)≈2.965×10−16 MeV [8]. As well, |M|2
formally contains new terms due to noncommutative ge-
Γ¯(C) Γ¯(C)+ i¯k εˆ0ˆkˆ FR (T)Jlm, (15b) ometrywhen accountingfor (15) andthe additionalcur-
µˆ → µˆ 48 µˆ lmjk,0 vature terms due to Moffat’s theory.
which subsequently introduces non-trivial terms to (10). Forthispaper,Jµν isdefinedaccordingtotwopossible
Moffat’s Nonsymmetric Gravity Theory.–It is reason- choices [13]:
able to surmise that metric corrections due to noncom-
Jµν(X) = iCµν, (Case 1) (20a)
mutative geometry, as proposed by (14), are very small. (0)
tNheavteartgheenleesrsa,libzyatiaolnloawwinagyffroormFGg[µRν]i(sXst)ill6=re0q,uiirtedistcoleaacr- = ri C(µ1ν)γXγ, (Case 2) (20b)
0
commodate a nonsymmetric description of gravity. The
whereCµν andCµν aredimensionlessreal-valuedstruc-
theoryputforwardbyMoffat[10,11]proposesthatboth (0) (1)γ
the metric tensor gµν and the connection Γλµν are non- ture constants chosen with C(i0j) = C(i1j)k = +1 for
symmetricintheirlowerindices. Furthermore,themodi- ij = 12,23,31 ,and all others equal to zero. When ap-
{ }
fiedcurvaturetensorisdescribedusingthenonsymmetric plied to Moffat’s nonsymmetric theory [11], it is shown
4
1.60 1.60 1.60
1.40 CCuurrvvaattuurree + NI 1.40 CCuurrvvaattuurree + NI 1.40 CCuurrvvaattuurree + NI
Non-Inertial (NI) Curvature + NI + NG Curvature + NI + NG
1.20 Flat Background 1.20 Curvature + NG 1.20 Curvature + NG
1.00 1.00 1.00
ˆxΓ() 0.80 ˆxΓ() 0.80 ˆxΓ() 0.80
1Γ0 1Γ0 1Γ0
0.60 0.60 0.60
0.40 0.40 0.40
0.20 0.20 0.20
0.000 0.1 0.2 0.3 0.4 0x.5 0.6 0.7 0.8 0.9 1 0.000 0.1 0.2 0.3 0.4 0x.5 0.6 0.7 0.8 0.9 1 0.000 0.1 0.2 0.3 0.4 0x.5 0.6 0.7 0.8 0.9 1
(a) M =3 10−12 cm(r0 6M) (b) √¯k=5 10−9 cm(Case1) (c) √¯k=2 10−8 cm(Case2)
× ∼ × ×
FIG. 1: Michel spectrum for muon decay while in circular orbit around a microscopic Schwarzschild black hole of mass
M = 3×10−12 cm and orbital radius r0 ∼ 6M. Fig. 1(a) shows the various profiles available, including both curvature
and R-dependent non-inertial contributions (NI) [8], minus the predictions due to noncommutative geometry (NG). Fig. 1(b)
shows thecontributions due to noncommutative geometry for Case 1, given by(20a), while Fig. 1(c) shows thecorresponding
description for Case 2, given by (20b).
thatthecomponentsforW (asc-numbers)intheFermi noncommutative geometry requires a curved space-time
µ
frame for Case 1 are backgroundtomanifestitsexistence,itisunclearhowthe
empirically-motivated choices for √¯k in Figs. 1(b) and
¯k
FW (X) = FR (T)Ckl , (21) 1(c) relate with the theoretical motivations given else-
µ 3 kl0µ,0 (0)
where. IrrespectiveofwhethertheLargeHadronCollider
while the corresponding components for Case 2 are (LHC)canproduceSchwarzschildblackholeswithoutin-
voking large extra spatial dimensions, it is also unclear
FW (X) = ¯k FR (T)Cklm, (22a) whether direct observational evidence for noncommuta-
0 3r0 klm0 (1) tive geometry can be gained by this method. Nonethe-
¯k less,thisapproachatleastprovidesaninterestingavenue
FW (X) = FR (T)Ckl Xm
j 3r kl0j,0 (1)m for further exploration.
0 h
1
+ FR (T)Cklm . (22b)
2 klmj (1) (cid:21)
AnalysisandConclusion.–Allcomputationsinthispa-
∗ Electronic address: [email protected]
per are performed in the muon’s local rest frame. Fig-
† Electronic address: [email protected]
ure 1 shows the Michel spectrum Γˆ(x)=dΓ/dx in units ‡ Electronic address: [email protected]
of Γ0 as a function of the outgoing electron energy frac- [1] J.Polchinski,StringTheory(CambridgeUniversityPress
tionx. ItisclearfromFig.1(a)intheabsenceofnoncom- 1998); M.B.Green,J.H.Schwarz,andE.Witten,Super-
mutativegeometrythatspace-timecurvaturereducesthe string Theory (Cambridge University Press 1987).
decay rate [8]. Figs. 1(b) and 1(c) revealthe presence of [2] A. Ashtekar, Phys. Rev. D 36, 1587 (1987); Phys. Rev.
¯k due to Cases 1 and 2, respectively, where the change Lett. 57, 2244 (1986); C. Rovelli, Quantum Gravity
(Cambridge University Press, 2004).
of profile occurs for x & 0.65. It is interesting to note
[3] A. Connes, Commun. Math. Phys. 192, 155 (1996);
that this contribution serves to counteract the emergent
P.Aschieri,M.Dimitrijevic,F.MeyerandJ.Wess,Class.
curvature-induced stabilization at the muon’s Compton Quant. Grav. 23, 1883 (2006).
wavelength scale for the adopted sign convention of the [4] H.S. Snyder,Phys. Rev. 71, 38 (1947).
structure constants. As well, it is surprisingto note that [5] P. Nicolini, arXiv:0807.1939v1 [hep-th].
asignatureappearsonlyfortermsquadraticincurvature [6] D.SinghandN.Mobed,Class.QuantumGrav.24,2453
but linear in k¯, as all such linear curvature terms iden- (2007).
[7] D. Singh, Phys. Rev. D 71, 105003 (2005); D. Singh, N.
tically vanish when time-averaged over an orbital cycle.
Mobed, and G. Papini, J. Phys. A 37, 8329 (2004).
The reasons for its occurrence are not known. [8] D.SinghandN.Mobed,Phys.Rev.D79,024026(2009).
Identification of a discernable signal requires √¯k [9] K. Rosquist, Class. QuantumGrav. 23, 3111 (2006).
∼
10−9 cmforCase1,aboutanorderofmagnitudesmaller [10] J.W. Moffat, Phys.Rev. D 19, 3554 (1979).
than for Case 2. While this length scale for √¯k is [11] J.W. Moffat, Phys.Lett. B355, 447 (1995).
within the realm of possibility and is strictly speaking [12] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravita-
an unknown quantity, its value is at least ten orders of tion (W.H. Freeman and Company 1973).
[13] J. Madore et al., Eur. Phys. J. C16, 161 (2000).
magnitude larger than the theoretical upper bound of
[14] M. Chaichian, M.M. Sheikh-Jabbari, and A. Tureanu,
√¯k ∼10−18 cm [14] predicted in the context of noncom- Phys. Rev.Lett. 86, 2716 (2001).
mutativeQEDinflatspace-time. Sincethistreatmentof