Noncommutative spectral geometry and the deformed Hopf algebra

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Noncommutative spectral geometry and thedeformed Hopf algebra structure of quantum fieldtheoryTo cite this article Mairi Sakellariadou et al 2013 J Phys Conf Ser 442 012016

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Noncommutative spectral geometry and the

deformed Hopf algebra structure of quantum field

theory

Mairi Sakellariadou

Department of Physics Kingrsquos College University of London Strand WC2R 2LS LondonUK

E-mail mairisakellariadoukclacuk

Antonio Stabile

Dipartimento di Fisica ldquoER Caianiellordquo and Gruppo Collegato INFN Universita diSalerno I-84100 Salerno Italy

E-mail anstabilegmailcom

Giuseppe Vitiello


E-mail vitiellosainfnit

Abstract We report the results obtained in the study of Alain Connes noncommutativespectral geometry construction focusing on its essential ingredient of the algebra doubling Weshow that such a two-sheeted structure is related with the gauge structure of the theory itsdissipative character and carries in itself the seeds of quantization From the algebraic pointof view the algebra doubling process has the same structure of the deformed Hops algebrastructure which characterizes quantum field theory

1 Noncommutative spectral geometry and the standard modelOne may assume that near the Planck energy scale the geometry of space-time ceases to havethe simple continuous form we are familiar with At high enough energy scales quantum gravityeffects turn on and they alter space-time One can thus assume that at high energy scales space-time becomes discrete and the coordinates do not longer commute Such an approach could apriori be tested by its phenomenological and cosmological consequences

Combining noncommutative geometry [1 2] with the spectral action principle led toNoncommutative Spectral Geometry (NCSG) used by Connes and collaborators [3] in anattempt to provide a purely geometric explanation for the Standard Model (SM) of electroweakand strong interactions In their approach the SM is considered as a phenomenological modelwhich dictates the geometry of space-time so that the Maxwell-Dirac action functional leads tothe SM action The model is constructed to hold at high energy scales namely at unification

DICE2012 IOP PublishingJournal of Physics Conference Series 442 (2013) 012016 doi1010881742-65964421012016

Content from this work may be used under the terms of the Creative Commons Attribution 30 licence Any further distributionof this work must maintain attribution to the author(s) and the title of the work journal citation and DOI

Published under licence by IOP Publishing Ltd 1

scale to get its low energy consequences which will then be tested against current data oneuses standard renormalization techniques Since the model lives at high energy scales it canbe used to investigate early universe cosmology [4]-[11] 1 The purpose of this contribution istwofold firstly to investigate the physical meaning of the choice of the almost commutativegeometry and its relation to quantization [12] and secondly to explore the relation of NCSGwith the gauge structure of the theory and with dissipation [12] We will show that Connesconstruction is intimately related with the deformed Hopf algebra characterizing quantum fieldtheory (QFT) [13] and therefore the seeds of quantization are built in such a NCSG construction

We start by summarizing the main ingredients of NCSG composed by a two-sheeted spacemade from the product of a four-dimensional smooth compact Riemannian manifold M witha fixed spin structure by a discrete noncommutative space F composed by only two pointsThus geometry is specified by the product of a continuous manifold for space-time times aninternal geometry for the SM The noncommutative nature of the discrete space F is denoted bya spectral triple (AH D) The algebra A = Cinfin(M) of smooth functions onM is an involutionof operators on the finite-dimensional Hilbert space H of Euclidean fermions it acts on H bymultiplication operators The operator D is the Dirac operator partM =

radicminus1γmicronablasmicro on the spin

Riemannian manifold M D is a self-adjoint unbounded operator in H The space H is theHilbert space L2(M S) of square integrable spinors S onM Thus one obtains a model of puregravity with an action that depends on the spectrum of the Dirac operator which provides the(familiar) notion of a metric

The most important ingredient in this NCSG model is the choice of the algebra constructedwithin the geometry of the two-sheeted space MtimesF it captures all information about spaceThe product geometry is specified by

A = A1 otimesA2 H = H1 otimesH2 (1)

as a consequence of the noncommutative nature of the discrete space F composed by only twopoints In other words Eq (1) expresses at the algebraic level and at the level of the space of thestates the two-sheeted nature of the geometry spaceMtimesF Assuming A is symplectic-unitaryit can be written as [14]

A = Ma(H)oplusMk(C) (2)

with k = 2a and H being the algebra of quaternions which encodes the noncommutativity ofthe manifold The first possible value for the even number k is 2 corresponding to a Hilbertspace of four fermions but this choice is ruled out from the existence of quarks The nextpossible value is k = 4 leading to the correct number of k2 = 16 fermions in each of the threegenerations Thus we will consider the minimal choice that can accommodate the physics of theSM certainly other choices leading to larger algebras which could accommodate particle beyondthe SM sector could be possible

The second basic ingredient of the NCSG is the spectral action principle stating that withinthe context of the product MtimesF the bare bosonic Euclidean action is given by the trace ofthe heat kernel associated with the square of the noncommutative Dirac operator and is simply

Tr(f(DΛ)) (3)

where f is a cut-off function andΛ fixes the energy scale This action can be seen a la Wilson asthe bare action at the mass scale Λ The fermionic term can be included in the action functionalby adding (12)〈JψDψ〉 where J is the real structure on the spectral triple and ψ is a spinorin the Hilbert space H of the quarks and leptons

1 See the contribution of M Sakellariadou in the same conference


2

Since we are considering a four-dimensional Riemannian geometry the trace Tr(f(DΛ)) canbe expressed perturbatively as [15]-[18]

Tr(f(DΛ)) sim 2Λ4f4a0 + 2Λ2f2a2 + f0a4 + middot middot middot+ Λminus2kfminus2ka4+2k + middot middot middot (4)

in terms of the geometrical Seeley-deWitt coefficients an known for any second order ellipticdifferential operator It is important to note that the smooth even test function f which decaysfast at infinity appears through its momenta fk

f0 equiv f(0)

fk equivint infin

0f(u)ukminus1du for k gt 0

fminus2k = (minus1)kk

(2k)f (2k)(0)

Since the Taylor expansion of the cut-off function vanishes at zero the asymptotic expansion ofEq (4) reduces to

Tr(f(DΛ)) sim 2Λ4f4a0 + 2Λ2f2a2 + f0a4 (5)

Hence the cut-off function f plays a role only through its three momenta f0 f2 f4 which arethree real parameters related to the coupling constants at unification the gravitational constantand the cosmological constant respectively More precisely the first term in Eq (5) which is inin Λ4 gives a cosmological term the second one which is in Λ2 gives the Einstein-Hilbert actionfunctional and the third one which is Λ-independent term yields the Yang-Mills action for thegauge fields corresponding to the internal degrees of freedom of the metric

The NCSG offers a purely geometric approach to the SM of particle physics where thefermions provide the Hilbert space of a spectral triple for the algebra and the bosons areobtained through inner fluctuations of the Dirac operator of the product M times F geometryThe computation of the asymptotic expression for the spectral action functional results to thefull Lagrangian for the Standard Model minimally coupled to gravity with neutrino mixing andMajorana mass terms Supersymmetric extensions have been also considered

In this report we closely follow the presentation of Ref [12] (see also [19]) The relationbetween the algebra doubling and the deformed Hopf algebra structure of QFT are discussed inSection 2 dissipation the gauge structure and quantization in Section 3 Section 4 is devotedto conclusions where we also mention about the dissipative interference phase arising in thenoncommutative plane in the presence of algebra doubling

2 Noncommutative spectral geometry and quantum field theoryOur first observation is that the doubling of the algebra A rarr A1 otimes A2 acting on theldquodoubledrdquo space H = H1 otimes H2 which expresses the two sheeted nature of the NCSG (cfEq (1)) is a key feature of quantum theories As observed also by Alain Connes in Ref [1]already in the early years of quantum mechanics (QM) in establishing the ldquomatrix mechanicsrdquoHeisenberg has shown that noncommutative algebras governing physical quantities are at theorigin of spectroscopic experiments and are linked to the discretization of the energy of theatomic levels and angular momentum One can convince himself that this is the case byobserving that in the density matrix formalism of QM the coordinate x(t) of a quantumparticle is split into two coordinates x+(t) (going forward in time) and xminus(t) (going backwardin time) The forward in time motion and the backward in time motion of the density matrixW (x+ xminus t) equiv 〈x+|ρ(t)|xminus〉 = ψlowast(xminus t)ψ(x+ t) where xplusmn = x plusmn y2 is described indeed byldquotwo copiesrdquo of the Schrodinger equation respectively

ihpartψ(x+ t)

partt= H+ψ(x+ t) minusihpartψ

lowast(xminus t)

partt= Hminusψ

lowast(xminus t) (6)


3

which can be written as

ihpart〈x+|ρ(t)|xminus〉

partt= H 〈x+|ρ(t)|xminus〉 (7)

where H is given in terms of the two Hamiltonian operators Hplusmn as

H = H+ minusHminus (8)

The introduction of a doubled set of coordinates (xplusmn pplusmn) (or (x px) and (y py)) and the useof the two copies of the Hamiltonian Hplusmn operating on the outer product of two Hilbert spacesH+ otimes Hminus thus show that the eigenvalues of H are directly the Bohr transition frequencieshνnm = En minus Em which are at the basis of the explanation of spectroscopic structure Wehave observed elsewhere that the doubling of the algebra is implicit also in the theory of theBrownian motion of a quantum particle [20] (see also [12] and the references there quoted) andthe doubled degrees of freedom are known [13] to allow quantum noise effect

It is thus evident the connection with the two sheeted NCSG Moreover it has been shown [12]that as a consequence of the algebra doubling Eq (1) the NCSG construction has an intrinsicgauge structure and is a thermal dissipative field theory As we will discuss below this suggeststhat Connes construction carries in itself the seeds of quantization namely it is more than justa classical theory construction

Let us start by discussing the simple case of the massless fermion and the U(1) local gaugetransformation group We will see how in this case the doubling of the algebra is related tothe gauge structure of the theory Extension to the massive fermion case the boson case andnon-Abelian gauge transformation groups is possible [21 22] The system Lagrangian is

L = Lminus L = minusψγmicropartmicroψ + ψγmicropartmicroψ (9)

The fermion tilde-field ψ(x) which satisfies the usual fermionic anticommutation relations andanticommutes with the field ψ(x) is a ldquocopyrdquo (with the same spectrum and couplings) of theψ-system We thus ldquodoublerdquo the field algebra by introducing such a tilde-field ψ(x)

For simplicity no coupling term of the field ψ(x) with ψ(x) is assumed in L In the quantized

theory let adaggerk and adaggerk denote the creation operators associated to the quantum fields ψ and

ψ respectively (all quantum number indices are suppressed except momentum) The vacuum|0(θ)〉 of the theory is

|0(θ)〉 =prodk

[cos θk + sin θka

daggerkadaggerk

]|0〉 (10)

namely a condensate of couples of adaggerk and adaggerk modes Here |0〉 denotes the vacuum |0 0〉 equiv|0〉 otimes |0〉 with |0〉 and |0〉 the vacua annihilated by the annihilation operators ak and akrespectively On the other hand |0(θ)〉 is the vacuum with respect to the fields ψ(θx) andψ(θx) which are obtained by means of the Bogoliubov transformation

ψ(θx) = Bminus1(θ)ψ(x)B(θ) (11a)

ψ(θx) = Bminus1(θ)ψ(x)B(θ) (11b)

where B(θ) equiv eminusiG with the generator G = minusisum

k θk(adaggerkadaggerkminusakak) For simplicity θ is assumed

to be independent of space-time Extension to space-time dependent Bogoliubov transformationscan be done [22] |0(θ)〉 is a SU(2) generalized coherent state [23]

The Hamiltonian for the ψ(x) ψ(x) system is H = H minus H (to be compared with Eq (8))

and is given by H =sum

k h ωk(adaggerkak minus adaggerkak) The θ-vacuum |0(θ)〉 is the the zero eigenvalue


4

eigenstate of H The relation [adaggerkak minus adaggerkak]|0(θ)〉 = 0 for any k (and any θ) characterizes the

θ-vacuum structure and it is called the θ-vacuum conditionThe space of states H = HotimesH is constructed by repeated applications of creation operators

of ψ(θx) and ψ(θx) on |0(θ)〉 and is called the θ-representation |0(θ)〉 θ-representationscorresponding to different values of the θ parameter are unitarily inequivalent representationsof the canonical anti-commutation relations in QFT [13] The state |0(θ)〉 is known to bea finite temperature state [13 21 22 24 25] namely one finds that θ is a temperature-dependent parameter This tells us that the algebra doubling leads to a thermal field theoryThe Bogoliubov transformations then induce transition through system phases at differenttemperatures

We now consider the subspace Hθcsub|0(θ)〉 made of all the states |a〉θc including |0(θ)〉such that the θ-state condition

[adaggerkak minus adaggerkak]|a〉θc = 0 for any k (12)

holds in Hθc (note that Eq (12) is similar to the Gupta-Bleuler condition defining the physicalstates in quantum electrodynamics (QED)) Let 〈〉θc denote matrix elements in Hθc we have

〈jmicro(x)〉θc = 〈jmicro(x)〉θc (13)

where jmicro(x) = ψγmicroψ and jmicro(x) = ψγmicroψ Equalities between matrix elements in Hθc say〈A〉θc = 〈B〉θc are denoted by A sim= B and we call them θ-w-equalities (θ-weak equalities) Theyare classical equalities since they are equalities among c-numbers Hθc is invariant under thedynamics described by H (even in the general case in which interaction terms are present in Hprovided that the charge is conserved)

The key point is that due to Eq (13) the matrix elements in Hθc of the Lagrangian Eq (9)are invariant under the simultaneous local gauge transformations of ψ and ψ fields given by

ψ(x)rarr exp [igα(x)]ψ(x) ψ(x)rarr exp [igα(x)] ψ(x) (14)

ie〈L〉θc rarr 〈Lprime〉θc = 〈L〉θc in Hθc (15)

under the gauge transformations (14) The tilde term ψγmicropartmicroψ thus plays a crucial role in the

θ-w-gauge invariance of L under Eq (14) Indeed it transforms in such a way to compensatethe local gauge transformation of the ψ kinematical term ie

ψ(x)γmicropartmicroψ(x)rarr ψ(x)γmicropartmicroψ(x) + gpartmicroα(x)jmicro(x) (16)

This suggests to us to introduce the vector field Aprimemicro by

gjmicro(x)Aprimemicro(x) sim= ψ(x)γmicropartmicroψ(x) micro = 0 1 2 3 (17)

Here and in the following the bar over micro means no summation on repeated indices Thus thevector field Aprimemicro transforms as

Aprimemicro(x)rarr Aprimemicro(x) + partmicroα(x) (18)

when the transformations of Eq (14) are implemented In Hθc Aprimemicro can be then identified withthe conventional U(1) gauge vector field and can be introduced it in the original Lagrangianthrough the usual coupling term igψγmicroψAprimemicro

The position (17) does not change the θ-vacuum structure Therefore provided that onerestricts himselfherself to matrix elements in Hθc matrix elements of physical observables


5

which are solely functions of the ψ(x) field are not changed by the position (17) Ouridentification of Aprimemicro with the U(1) gauge vector field is also justified by the fact that observables

turn out to be invariant under gauge transformations and the conservation laws derivable from Lnamely in the simple case of Eq (9) the current conservation laws partmicrojmicro(x) = 0 and partmicrojmicro(x) = 0are also preserved as θ-w-equalities when Eq (17) is adopted Indeed one obtains [21 22]partmicrojmicro(x) sim= 0 and partmicrojmicro(x) sim= 0 One may also show that

partνF primemicroν(x) sim= minusgjmicro(x) partνF primemicroν(x) sim= minusgjmicro(x) (19)

in Hθc In the the Lorentz gauge from Eq (19) we also obtain the θ-w-relations partmicroAprimemicro(x) sim= 0

and part2Aprimemicro(x) sim= gjmicro(x)

In conclusion the ldquodoubled algebrardquo Lagrangian (9) for the field ψ and its ldquodoublerdquo ψ canbe substituted in Hθc by

Lgsim= minus14F primemicroνF primemicroν minus ψγmicropartmicroψ + igψγmicroψAprimemicro in Hθc (20)

where remarkably the tilde-kinematical term ψγmicropartmicroψ is replaced in a θ-w-sense by the couplingterm igψγmicroψAprimemicro between the gauge field Aprimemicro and the matter field current ψγmicroψ

Finally in the case an interaction term is present in the Lagrangian (9) Ltot = L+LI LI =LIminusLI the above conclusions still hold providedHθc is an invariant subspace under the dynamicsdescribed by Ltot

Our discussion has thus shown that the ldquodoublingrdquo of the field algebra introduces agauge structure in the theory Connes two sheeted geometric construction has intrinsic gaugeproperties

The algebraic structure underlying the above discussion is recognized to be the one of thenoncommutative q-deformed Hopf algebra [24] We remark that the Hopf coproduct mapA rarr Aotimes 1 + 1otimesA equiv A1 otimesA2 is nothing but the map presented in Eq (1) which duplicatesthe algebra On the other hand it can be shown [24] that the Bogoliubov transformationof ldquoanglerdquo θ relating the fields ψ(θx) and ψ(θx) to ψ(x) and ψ(x) Eqs (11) are obtainedby convenient combinations of the q-deformed Hopf coproduct ∆adaggerq = adaggerq otimes q12 + qminus12 otimes adaggerqwith q equiv q(θ) the deformation parameters and adaggerq the creation operators in the q-deformedHopf algebra [24] These deformed coproduct maps are noncommutative All of this signals adeep physical meaning of noncommutativity in the Connes construction since the deformationparameter is related to the condensate content of |0(θ)〉 under the constrain imposed by theθ-state condition Eq (12) Actually such a state condition is a characterizing condition forthe system physical states The crucial point is that a characteristic feature of quantum fieldtheory [13 24] is that the deformation parameter labels the θ-representations |0(θ)〉 and asalready mentioned for θ 6= θprime |0(θ)〉 and |0(θprime)〉 are unitarily inequivalent representationsof the canonical (anti-)commutation rules [13 25] In turn the physical meaning of this isthat an order parameter exists which assumes different θ-dependent values in each of therepresentations Thus the q-deformed Hopf algebra structure of QFT induces the foliation ofthe whole Hilbert space into physically inequivalent subspaces From our discussion we concludethat this is also the scenario which NCSG presents to us

One more remark in this connection is that in the NCSG construction the derivative in thediscrete direction is a finite difference quotient [1 2 12] and it is then suggestive that the q-derivative is also a finite difference derivative This point deserves further formal analysis whichis in our plans to do

We thus conclude that Connes NCSG construction is built on the same noncommutativedeformed Hopf algebra structure of QFT In the next Section we show that it is also relatedwith dissipation and carries in it the seeds of quantization


6

3 Dissipation gauge field and quantizationIn the second equation in (19) the current jmicro act as the source of the variations of the gaugefield tensor F primemicroν We express this by saying that the tilde field plays the role of a ldquoreservoirrdquoSuch a reservoir interpretation may be extended also to the gauge field Aprimemicro which is known toact indeed in a way to ldquocompensaterdquo the changes in the matter field configurations due to thelocal gauge freedom

When we consider variations in the θ parameter (namely in the q-deformation parameter)induced by the Bogoliubov transformation generator we have (time-)evolution over the manifoldof the θ-labeled (ie q-labeled) spaces and we have dissipative fluxes between the doubledsets of fields or in other words according to the above picture between the system and thereservoir We talk of dissipation and open systems when considering the Connes conctructionand the Standard Model in the same sense in a system of electromagnetically interacting matterfield neither the energy-momentum tensor of the matter field nor that of the gauge fieldare conserved However one verifies in a standard fashion [26] that partmicroT

microνmatter = eFmicroνjmicro =

minuspartmicroTmicroνgauge field so that what it is conserved is the total Tmicroνtotal = Tmicroνmatter + Tmicroνgauge field namely

the energy-momentum tensor of the closed system matter field electromagnetic field Asremarked in Ref [12] each element of the couple is open (dissipating) on the other one althoughthe closeness of the total system is ensured Thus the closeness of the SM is not spoiled in ourdiscussion

In order to further clarify how the gauge structure is related to the algebra doubling andto dissipation we consider the prototype of a dissipative system namely the classical one-dimensional damped harmonic oscillator

mx+ γx+ kx = 0 (21)

and its time-reversed (γ rarr minusγ) (doubled) image

my minus γy + ky = 0 (22)

with time independent m γ and k needed [27] in order to set up the canonical formalism foropen systems The system of Eq (21) and Eq (22) is then a closed system described by theLagrangian density

L(x y x y) = mxy +γ

2(xy minus yx) + kx y (23)

It is convenient to use the coordinates x1(t) and x2(t) defined by

x1(t) =x(t) + y(t)radic

2 x2(t) =

x(t)minus y(t)radic2

(24)

The motion equations are rewritten then as

mx1 + γx2 + kx1 = 0 (25a)

mx2 + γx1 + kx2 = 0 (25b)

The canonical momenta are p1 = mx1 + (12)γx2 p2 = minusmx2minus (12)γx1 the Hamiltonian is

H = H1 minusH2 =1

2m(p1 minus

γ

2x2)2 +

k

2x2

1 minus1

2m(p2 +

γ

2x1)2 minus k

2x2

2 (26)

We then recognize [28 29 21 22] that

Ai =B

2εijxj (i j = 1 2) εii = 0 ε12 = minusε21 = 1 (27)


7

with B equiv c γe acts as the vector potential and obtain that the system of oscillators Eq (21)and Eq (22) is equivalent to the system of two particles with opposite charges e1 = minuse2 = e inthe (oscillator) potential Φ equiv (k2e)(x1

2 minus x22) equiv Φ1 minus Φ2 with Φi equiv (k2e)xi

2 and in theconstant magnetic field B defined as B = nablatimesA = minusB3 The Hamiltonian is indeed

H = H1 minusH2 =1

2m(p1 minus

e1

cA1)2 + e1Φ1 minus

1

2m(p2 +

e2

cA2)2 + e2Φ2 (28)

and the Lagrangian of the system can be written in the familiar form

L =1

2m(mx1 +

e1

cA1)2 minus 1

2m(mx2 +

e2

cA2)2 minus e2

2mc2(A1

2 +A22)minus eΦ

=m

2(x2

1 minus x22) +

e

c(x1A1 + x2A2)minus eΦ (29)

Note the ldquominusrdquo sign of the Lorentzian-like (pseudoeuclidean) metric in Eq (29) (cf alsoEqs (8) (9) and(28)) not imposed by hand but derived through the doubling of the degreesof freedom and crucial in our description (and in the NCSG construction)

The doubled coordinate x2 thus acts as the gauge field component A1 to which the x1

coordinate is coupled and vice versa The energy dissipated by one of the two systems isgained by the other one and viceversa in analogy to what happens in standard electrodynamicsas observed above The picture is recovered of the gauge field as the bath or reservoir in whichthe system is embedded [21 22]

Our toy system of harmonic oscillators Eq (21) and Eq (22) offers also an useful playgroundto show how dissipation is implicitly related to quantization rsquot Hooft indeed has conjecturedthat classical deterministic systems with loss of information might lead to a quantumevolution [30 31 32] provided some specific energy conditions are met and some constraintsare imposed Let us verify how such a conjecture is confirmed for our system described byEq (21) and Eq (22) We will thus show that quantization is achieved as a consequence of

dissipation We rewrite the Hamiltonian Eq (26) as H = HI minusHII with

HI =1

2ΩC(2ΩC minus ΓJ2)2 HII =

Γ2

2ΩCJ2

2 (30)

where C is the Casimir operator and J2 is the (second) SU(1 1) generator [29]

C =1

4Ωm

[(p2

1 minus p22

)+m2Ω2

(x2

1 minus x22

)] J2 =

m

2

[(x1x2 minus x2x1)minus Γr2

] (31)

C is taken to be positive and

Γ =γ

2m Ω =

radic1

m(k minus γ2

4m) with k gt

γ2

4m

This H belongs to the class of Hamiltonians considered by rsquot Hooft and is of the form [33 34]

H =2sumi=1

pi fi(q) (32)

with p1 = C p2 = J2 f1(q) = 2Ω f2(q) = minus2Γ Note that qi pi = 1 and the other Poissonbrackets are vanishing The fi(q) are nonsingular functions of qi and the equations for theqrsquos qi = qi H = fi(q) are decoupled from the conjugate momenta pi A complete set of


8

observables called beables then exists which Poisson commute at all times Thus the systemadmits a deterministic description even when expressed in terms of operators acting on somefunctional space of states |ψ〉 such as the Hilbert space [31] Note that this description in termsof operators and Hilbert space does not imply per se quantization of the system Physicalstates are defined to be those satisfying the condition J2|ψ〉 = 0 This guaranties that H is

bounded from below and H|ψ〉 = HI |ψ〉 = 2ΩC|ψ〉 HI thus reduces to the Hamiltonian forthe two-dimensional ldquoisotropicrdquo (or ldquoradialrdquo) harmonic oscillator r + Ω2r = 0 Indeed puttingK equiv mΩ2 we obtain

H|ψ〉 = HI |ψ〉 =

(1

2mp2r +

K

2r2)|ψ〉 (33)

The physical states are invariant under time-reversal (|ψ(t)〉 = |ψ(minust)〉) and periodical withperiod τ = 2πΩ HI = 2ΩC has the spectrum Hn

I= hΩn n = 0plusmn1plusmn2 since C has been

chosen to be positive only positive values of n are considered Then one obtains

〈ψ(τ)|H|ψ(τ)〉h

τ minus φ = 2πn n = 0 1 2

Using τ = 2πΩ and φ = απ where α is a real constant leads to

HnIeffequiv 〈ψn(τ)|H|ψn(τ)〉 = hΩ

(n+

α

2

) (34)

HnIeff

gives the effective nth energy level of the system namely the energy given by HnI

correctedby its interaction with the environment We conclude that the dissipation term J2 of theHamiltonian is responsible for the zero point (n = 0) energy E0 = (h2)Ωα In QM thezero point energy is the ldquosignaturerdquo of quantization since it is formally due to the nonzerocommutator of the canonically conjugate q and p operators Our conclusion is thus that the(zero point) ldquoquantum contributionrdquo E0 to the spectrum of physical states is due and signalsthe underlying dissipative dynamics dissipation manifests itself as ldquoquantizationrdquo

We remark that the ldquofull Hamiltonianrdquo Eq (32) plays the role of the free energy F and 2ΓJ2

represents the heat contribution in H (or F) Indeed using S equiv (2J2h) and U equiv 2ΩC andthe defining relation for temperature in thermodynamics partSpartU = 1T with the Boltzmannconstant kB = 1 from Eq (32) we obtain T = hΓ ie provided that S is identified with theentropy hΓ can be regarded as the temperature Note that it is not surprising that 2J2h behavesas the entropy since it controls the dissipative part of the dynamics (thus the irreversible loss ofinformation) It is interesting to observe that this thermodynamical picture is also cofirmed bythe results on the canonical quantization of open systems in quantum field theory [27]

On the basis of these results (confirming rsquot Hooftrsquos conjecture) we have proved that theNCSG classical construction due to its essential ingredient of the doubling of the algebra isintrinsically related with dissipation and thus carries in itself the seeds of quantization [12]

4 ConclusionsIn Ref [35] it has been shown that a ldquodissipative interference phaserdquo also appears in theevolution of dissipative systems Such a feature exhibits one more consequence of the algebradoubling which plays so an important role in the Connes NCSG construction In this reportwe have discussed how such a doubling process implies the gauge structure of the theory itsthermal characterization its built in dissipation features and the consequent possibility to exhibitquantum evolution behaviors Here we only mention that the doubling of the coordinates alsoprovides a relation between dissipation and noncommutative geometry in the plane We refer


9

to Ref[12] for details with reference to Connes construction (see also [19]) We only recall thatin the (x+ xminus) plane xplusmn = x plusmn y2 (cf Section 2) the components of forward and backwardin time velocity vplusmn = xplusmn given by

vplusmn =partH

partpplusmn= plusmn 1

m

(pplusmn ∓

γ

2x∓

) (35)

do not commute [v+ vminus] = ih γm2 Thus it is impossible to fix these velocities v+ and vminus asbeing identical [35] Moreover one may always introduce a set of canonical conjugate positioncoordinates (ξ+ ξminus) defined by ξplusmn = ∓L2K∓ with hKplusmn = mvplusmn so that [ξ+ ξminus] = iL2 whichcharacterizes the noncommutative geometry in the plane (x+ xminus) L denotes the geometriclength scale in the plane

One can show [35] that an AharonovndashBohm-type phase interference can always be associatedwith the noncommutative (XY ) plane where

[XY ] = iL2 (36)

Eq (36) implies the uncertainty relation (∆X)(∆Y ) ge L22 due to the zero point fluctuationsin the coordinates Consider now a particle moving in the plane along two paths starting andfinishing at the same point in a forward and in a backward direction respectively forming aclosed loop The phase interference ϑ may be be shown [35] to be given by ϑ = AL2 where Ais the area enclosed in the closed loop formed by the paths and Eq (36) in the noncommutativeplane can be written as

[XPX ] = ih where PX =

(hY

L2

) (37)

The quantum phase interference between two alternative paths in the plane is thus determinedby the length scale L and the area A In the dissipative case it is L2 = hγ and thequantum dissipative phase interference ϑ = AL2 = Aγh is associated with the two pathsin the noncommutative plane provided x+ 6= xminus When doubling is absent ie x+ = xminusthe quantum effect disappear It is indeed possible to show [20] that in such a classical limitcase the doubled degree of freedom is associated with ldquounlikely processesrdquo and it may thus bedropped in higher order terms in the perturbative expansion as indeed it happens in Connesconstruction At the Grand Unified Theories scale when inflation took place the effect of gaugefields is fairly shielded However since these higher order terms are the ones responsible forquantum corrections the second sheet cannot be neglected if one does not want to precludequantization effects as it happens once the universe entered the radiation dominated era

References[1] Connes A 1994 Noncommutative Geometry (New York Academic Press)[2] Connes A and Marcolli M 2008 Noncommutative Geometry Quantum Fields and Motives (India Hindustan

Book Agency)[3] Chamseddine A H Connes A and Marcolli M 2007 Adv Theor Math Phys 11 991[4] Nelson W and Sakellariadou M 2010 Phys Rev D 81 085038[5] Nelson W and Sakellariadou M 2009 Phys Lett B 680 263[6] Marcolli M and Pierpaoli E Preprint hep-th09083683[7] Buck M Fairbairn M and Sakellariadou M 2010 Phys Rev D 82 043509[8] Nelson W Ochoa J and Sakellariadou M 2010 Phys Rev Lett 105 101602[9] Nelson W Ochoa J and Sakellariadou M 2010 Phys Rev D 82 085021

[10] Sakellariadou M 2011 Int J Mod Phys D 20 785[11] Sakellariadou M 2010 PoS CNCFG 2010 028 [arXiv11012174 [hep-th]][12] Sakellariadou M Stabile A and Vitiello G 2011 Phys Rev D 84 045026[13] Blasone M Jizba P and Vitiello G 2011 Quantum Field Theory and its macroscopic manifestations (London

Imperial College Press)


10

[14] Chamseddine A H and Connes A 2007 Phys Rev Lett 99 191601[15] Chamseddine A H and Connes A 2006 J Math Phys 47 063504[16] Chamseddine A H and Connes A 1996 Phys Rev Lett 77 4868[17] Chamseddine A H and Connes A 1977 Comm Math Phys 186 731[18] Chamseddine A H and Connes A 2010 Comm Math Phys 293 867[19] Sakellariadou M Stabile A and Vitiello G 2012 J of Phys Conf Series 361 012025[20] Blasone M Srivastava Y N Vitiello G and Widom A 1998 Annals Phys 267 61[21] Celeghini E Graziano E Nakamura K and Vitiello G 1992 Phys Lett B285 98[22] Celeghini E Graziano E Nakamura K and Vitiello G 1993 Phys Lett B304 121[23] Perelomov A M 1986 Generalized coherent states and their applications (Berlin Springer-Verlag)[24] Celeghini E De Martino S De Siena S Iorio A Rasetti M and Vitiello G 1998 Phys Lett A244 455[25] Umezawa H Matsumoto H and Tachiki M 1982 Thermo Field Dynamics and condensed states (Amsterdam

North-Holland)[26] Landau L D and Lifshitz E M 1980 The Classical Theory of Fields (Oxford Butterworth-Heinemann)[27] Celeghini E Rasetti M and Vitiello G 1992 Annals Phys 215 156[28] Tsue Y Kuriyama A and Yamamura M 1994 Prog Theor Phys 91 469[29] Blasone M Graziano E Pashaev O K and Vitiello G 1996 Annals Phys 252 115[30] rsquot Hooft G 1999 Class Quant Grav 16 3263[31] rsquot Hooft G 1999 Basics and Highlights of Fundamental Physics (Erice) [arXivhep-th0003005][32] rsquot Hooft G 2007 J Phys Conf Series 67 012015[33] Blasone M Jizba P and Vitiello 2001 Phys Lett A 287 205[34] Blasone M Celeghini E Jizba P and Vitiello G 2003 Phys Lett A 310 393[35] Sivasubramanian S Srivastava Y Vitiello G and Widom A 2003 Phys Lett A311 97


11

Noncommutative spectral geometry and the

deformed Hopf algebra structure of quantum field

theory

Mairi Sakellariadou

Department of Physics Kingrsquos College University of London Strand WC2R 2LS LondonUK

E-mail mairisakellariadoukclacuk

Antonio Stabile


E-mail anstabilegmailcom

Giuseppe Vitiello


E-mail vitiellosainfnit

Abstract We report the results obtained in the study of Alain Connes noncommutativespectral geometry construction focusing on its essential ingredient of the algebra doubling Weshow that such a two-sheeted structure is related with the gauge structure of the theory itsdissipative character and carries in itself the seeds of quantization From the algebraic pointof view the algebra doubling process has the same structure of the deformed Hops algebrastructure which characterizes quantum field theory

1 Noncommutative spectral geometry and the standard modelOne may assume that near the Planck energy scale the geometry of space-time ceases to havethe simple continuous form we are familiar with At high enough energy scales quantum gravityeffects turn on and they alter space-time One can thus assume that at high energy scales space-time becomes discrete and the coordinates do not longer commute Such an approach could apriori be tested by its phenomenological and cosmological consequences

Combining noncommutative geometry [1 2] with the spectral action principle led toNoncommutative Spectral Geometry (NCSG) used by Connes and collaborators [3] in anattempt to provide a purely geometric explanation for the Standard Model (SM) of electroweakand strong interactions In their approach the SM is considered as a phenomenological modelwhich dictates the geometry of space-time so that the Maxwell-Dirac action functional leads tothe SM action The model is constructed to hold at high energy scales namely at unification


Content from this work may be used under the terms of the Creative Commons Attribution 30 licence Any further distributionof this work must maintain attribution to the author(s) and the title of the work journal citation and DOI

Published under licence by IOP Publishing Ltd 1











Tr(f(DΛ)) (3)




2




f0 equiv f(0)

fk equivint infin



(2k)f (2k)(0)







ihpartψ(x+ t)


lowast(xminus t)

partt= Hminusψ



3














|0(θ)〉 =prodk

[cos θk + sin θka

daggerkadaggerk

]|0〉 (10)











4






















5















6







mx+ γx+ kx = 0 (21)








2 x2(t) =


(24)


mx1 + γx2 + kx1 = 0 (25a)

mx2 + γx1 + kx2 = 0 (25b)


H = H1 minusH2 =1

2m(p1 minus

γ

2x2)2 +

k

2x2

1 minus1

2m(p2 +

γ

2x1)2 minus k

2x2

2 (26)


Ai =B



7




H = H1 minusH2 =1

2m(p1 minus

e1

cA1)2 + e1Φ1 minus

1

2m(p2 +

e2

cA2)2 + e2Φ2 (28)


L =1

2m(mx1 +

e1

cA1)2 minus 1

2m(mx2 +

e2

cA2)2 minus e2

2mc2(A1

2 +A22)minus eΦ

=m

2(x2

1 minus x22) +

e







HI =1


Γ2

2ΩCJ2

2 (30)


C =1

4Ωm

[(p2

1 minus p22

)+m2Ω2

(x2

1 minus x22

)] J2 =

m

2


] (31)


Γ =γ

2m Ω =

radic1

m(k minus γ2

4m) with k gt

γ2

4m


H =2sumi=1

pi fi(q) (32)



8



H|ψ〉 = HI |ψ〉 =

(1

2mp2r +

K

2r2)|ψ〉 (33)








(n+

α

2

) (34)

HnIeff








9


vplusmn =partH


m

(pplusmn ∓

γ

2x∓

) (35)



[XY ] = iL2 (36)



(hY

L2

) (37)







10




11











Tr(f(DΛ)) (3)




2




f0 equiv f(0)

fk equivint infin



(2k)f (2k)(0)







ihpartψ(x+ t)


lowast(xminus t)

partt= Hminusψ



3














|0(θ)〉 =prodk

[cos θk + sin θka

daggerkadaggerk

]|0〉 (10)











4






















5















6







mx+ γx+ kx = 0 (21)








2 x2(t) =


(24)


mx1 + γx2 + kx1 = 0 (25a)

mx2 + γx1 + kx2 = 0 (25b)


H = H1 minusH2 =1

2m(p1 minus

γ

2x2)2 +

k

2x2

1 minus1

2m(p2 +

γ

2x1)2 minus k

2x2

2 (26)


Ai =B



7




H = H1 minusH2 =1

2m(p1 minus

e1

cA1)2 + e1Φ1 minus

1

2m(p2 +

e2

cA2)2 + e2Φ2 (28)


L =1

2m(mx1 +

e1

cA1)2 minus 1

2m(mx2 +

e2

cA2)2 minus e2

2mc2(A1

2 +A22)minus eΦ

=m

2(x2

1 minus x22) +

e







HI =1


Γ2

2ΩCJ2

2 (30)


C =1

4Ωm

[(p2

1 minus p22

)+m2Ω2

(x2

1 minus x22

)] J2 =

m

2


] (31)


Γ =γ

2m Ω =

radic1

m(k minus γ2

4m) with k gt

γ2

4m


H =2sumi=1

pi fi(q) (32)



8



H|ψ〉 = HI |ψ〉 =

(1

2mp2r +

K

2r2)|ψ〉 (33)








(n+

α

2

) (34)

HnIeff








9


vplusmn =partH


m

(pplusmn ∓

γ

2x∓

) (35)



[XY ] = iL2 (36)



(hY

L2

) (37)







10




11




f0 equiv f(0)

fk equivint infin



(2k)f (2k)(0)







ihpartψ(x+ t)


lowast(xminus t)

partt= Hminusψ



3














|0(θ)〉 =prodk

[cos θk + sin θka

daggerkadaggerk

]|0〉 (10)











4






















5















6







mx+ γx+ kx = 0 (21)








2 x2(t) =


(24)


mx1 + γx2 + kx1 = 0 (25a)

mx2 + γx1 + kx2 = 0 (25b)


H = H1 minusH2 =1

2m(p1 minus

γ

2x2)2 +

k

2x2

1 minus1

2m(p2 +

γ

2x1)2 minus k

2x2

2 (26)


Ai =B



7




H = H1 minusH2 =1

2m(p1 minus

e1

cA1)2 + e1Φ1 minus

1

2m(p2 +

e2

cA2)2 + e2Φ2 (28)


L =1

2m(mx1 +

e1

cA1)2 minus 1

2m(mx2 +

e2

cA2)2 minus e2

2mc2(A1

2 +A22)minus eΦ

=m

2(x2

1 minus x22) +

e







HI =1


Γ2

2ΩCJ2

2 (30)


C =1

4Ωm

[(p2

1 minus p22

)+m2Ω2

(x2

1 minus x22

)] J2 =

m

2


] (31)


Γ =γ

2m Ω =

radic1

m(k minus γ2

4m) with k gt

γ2

4m


H =2sumi=1

pi fi(q) (32)



8



H|ψ〉 = HI |ψ〉 =

(1

2mp2r +

K

2r2)|ψ〉 (33)








(n+

α

2

) (34)

HnIeff








9


vplusmn =partH


m

(pplusmn ∓

γ

2x∓

) (35)



[XY ] = iL2 (36)



(hY

L2

) (37)







10




11














|0(θ)〉 =prodk

[cos θk + sin θka

daggerkadaggerk

]|0〉 (10)











4






















5















6







mx+ γx+ kx = 0 (21)








2 x2(t) =


(24)


mx1 + γx2 + kx1 = 0 (25a)

mx2 + γx1 + kx2 = 0 (25b)


H = H1 minusH2 =1

2m(p1 minus

γ

2x2)2 +

k

2x2

1 minus1

2m(p2 +

γ

2x1)2 minus k

2x2

2 (26)


Ai =B



7




H = H1 minusH2 =1

2m(p1 minus

e1

cA1)2 + e1Φ1 minus

1

2m(p2 +

e2

cA2)2 + e2Φ2 (28)


L =1

2m(mx1 +

e1

cA1)2 minus 1

2m(mx2 +

e2

cA2)2 minus e2

2mc2(A1

2 +A22)minus eΦ

=m

2(x2

1 minus x22) +

e







HI =1


Γ2

2ΩCJ2

2 (30)


C =1

4Ωm

[(p2

1 minus p22

)+m2Ω2

(x2

1 minus x22

)] J2 =

m

2


] (31)


Γ =γ

2m Ω =

radic1

m(k minus γ2

4m) with k gt

γ2

4m


H =2sumi=1

pi fi(q) (32)



8



H|ψ〉 = HI |ψ〉 =

(1

2mp2r +

K

2r2)|ψ〉 (33)








(n+

α

2

) (34)

HnIeff








9


vplusmn =partH


m

(pplusmn ∓

γ

2x∓

) (35)



[XY ] = iL2 (36)



(hY

L2

) (37)







10




11






















5















6







mx+ γx+ kx = 0 (21)








2 x2(t) =


(24)


mx1 + γx2 + kx1 = 0 (25a)

mx2 + γx1 + kx2 = 0 (25b)


H = H1 minusH2 =1

2m(p1 minus

γ

2x2)2 +

k

2x2

1 minus1

2m(p2 +

γ

2x1)2 minus k

2x2

2 (26)


Ai =B



7




H = H1 minusH2 =1

2m(p1 minus

e1

cA1)2 + e1Φ1 minus

1

2m(p2 +

e2

cA2)2 + e2Φ2 (28)


L =1

2m(mx1 +

e1

cA1)2 minus 1

2m(mx2 +

e2

cA2)2 minus e2

2mc2(A1

2 +A22)minus eΦ

=m

2(x2

1 minus x22) +

e







HI =1


Γ2

2ΩCJ2

2 (30)


C =1

4Ωm

[(p2

1 minus p22

)+m2Ω2

(x2

1 minus x22

)] J2 =

m

2


] (31)


Γ =γ

2m Ω =

radic1

m(k minus γ2

4m) with k gt

γ2

4m


H =2sumi=1

pi fi(q) (32)



8



H|ψ〉 = HI |ψ〉 =

(1

2mp2r +

K

2r2)|ψ〉 (33)








(n+

α

2

) (34)

HnIeff








9


vplusmn =partH


m

(pplusmn ∓

γ

2x∓

) (35)



[XY ] = iL2 (36)



(hY

L2

) (37)







10




11















6







mx+ γx+ kx = 0 (21)








2 x2(t) =


(24)


mx1 + γx2 + kx1 = 0 (25a)

mx2 + γx1 + kx2 = 0 (25b)


H = H1 minusH2 =1

2m(p1 minus

γ

2x2)2 +

k

2x2

1 minus1

2m(p2 +

γ

2x1)2 minus k

2x2

2 (26)


Ai =B



7




H = H1 minusH2 =1

2m(p1 minus

e1

cA1)2 + e1Φ1 minus

1

2m(p2 +

e2

cA2)2 + e2Φ2 (28)


L =1

2m(mx1 +

e1

cA1)2 minus 1

2m(mx2 +

e2

cA2)2 minus e2

2mc2(A1

2 +A22)minus eΦ

=m

2(x2

1 minus x22) +

e







HI =1


Γ2

2ΩCJ2

2 (30)


C =1

4Ωm

[(p2

1 minus p22

)+m2Ω2

(x2

1 minus x22

)] J2 =

m

2


] (31)


Γ =γ

2m Ω =

radic1

m(k minus γ2

4m) with k gt

γ2

4m


H =2sumi=1

pi fi(q) (32)



8



H|ψ〉 = HI |ψ〉 =

(1

2mp2r +

K

2r2)|ψ〉 (33)








(n+

α

2

) (34)

HnIeff








9


vplusmn =partH


m

(pplusmn ∓

γ

2x∓

) (35)



[XY ] = iL2 (36)



(hY

L2

) (37)







10




11







mx+ γx+ kx = 0 (21)








2 x2(t) =


(24)


mx1 + γx2 + kx1 = 0 (25a)

mx2 + γx1 + kx2 = 0 (25b)


H = H1 minusH2 =1

2m(p1 minus

γ

2x2)2 +

k

2x2

1 minus1

2m(p2 +

γ

2x1)2 minus k

2x2

2 (26)


Ai =B



7




H = H1 minusH2 =1

2m(p1 minus

e1

cA1)2 + e1Φ1 minus

1

2m(p2 +

e2

cA2)2 + e2Φ2 (28)


L =1

2m(mx1 +

e1

cA1)2 minus 1

2m(mx2 +

e2

cA2)2 minus e2

2mc2(A1

2 +A22)minus eΦ

=m

2(x2

1 minus x22) +

e







HI =1


Γ2

2ΩCJ2

2 (30)


C =1

4Ωm

[(p2

1 minus p22

)+m2Ω2

(x2

1 minus x22

)] J2 =

m

2


] (31)


Γ =γ

2m Ω =

radic1

m(k minus γ2

4m) with k gt

γ2

4m


H =2sumi=1

pi fi(q) (32)



8



H|ψ〉 = HI |ψ〉 =

(1

2mp2r +

K

2r2)|ψ〉 (33)








(n+

α

2

) (34)

HnIeff








9


vplusmn =partH


m

(pplusmn ∓

γ

2x∓

) (35)



[XY ] = iL2 (36)



(hY

L2

) (37)







10




11




H = H1 minusH2 =1

2m(p1 minus

e1

cA1)2 + e1Φ1 minus

1

2m(p2 +

e2

cA2)2 + e2Φ2 (28)


L =1

2m(mx1 +

e1

cA1)2 minus 1

2m(mx2 +

e2

cA2)2 minus e2

2mc2(A1

2 +A22)minus eΦ

=m

2(x2

1 minus x22) +

e







HI =1


Γ2

2ΩCJ2

2 (30)


C =1

4Ωm

[(p2

1 minus p22

)+m2Ω2

(x2

1 minus x22

)] J2 =

m

2


] (31)


Γ =γ

2m Ω =

radic1

m(k minus γ2

4m) with k gt

γ2

4m


H =2sumi=1

pi fi(q) (32)



8



H|ψ〉 = HI |ψ〉 =

(1

2mp2r +

K

2r2)|ψ〉 (33)








(n+

α

2

) (34)

HnIeff








9


vplusmn =partH


m

(pplusmn ∓

γ

2x∓

) (35)



[XY ] = iL2 (36)



(hY

L2

) (37)







10




11



H|ψ〉 = HI |ψ〉 =

(1

2mp2r +

K

2r2)|ψ〉 (33)








(n+

α

2

) (34)

HnIeff








9


vplusmn =partH


m

(pplusmn ∓

γ

2x∓

) (35)



[XY ] = iL2 (36)



(hY

L2

) (37)







10




11


vplusmn =partH


m

(pplusmn ∓

γ

2x∓

) (35)



[XY ] = iL2 (36)



(hY

L2

) (37)







10




11




11

Noncommutative spectral geometry and the deformed Hopf algebra

Documents

Transcript of Noncommutative spectral geometry and the deformed Hopf algebra