Electroweak, strong interactions and Higgs fields as components of gravity in noncommutative spacetime
EELECTROWEAK, STRONG INTERACTIONS AND HIGGS FIELDS ASCOMPONENTS OF GRAVITY IN NONCOMMUTATIVE SPACETIME
NGUYEN AI VIET
Department of Physics, College of Natural Sciences and Information Technology Institute,Vietnam National University, E3 Building 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
A new noncommutative spacetime of structure M × Z × Z is proposed. The generalizedHilbert-Einstein action contains gravity, all known interactions and Higgs field. This theorycan also provide a unified geometric framework for multigravity, which might explain theexistence of dark matter and inflationary cosmology. In other words, high energy physics haslaid out the crude shape of the new spacetime, while cosmology will shed light to the moredetails of it. Between 1921-1948, Kaluza, Klein and Thiry had shown that the Hilbert-Einstein actionin the spacetime extended with a circle M × S consisted of gravity, electromagnetism anda Brans-Dicke scalar. In 1968, R.Kerner had generalized the Kaluza-Klein theory to includenonabelian gauge fields. Today, the multi-dimensional theories are studied widely as candidatesof unified theories of interactions. However, these theories have a weakness of containing aninfinite tower of massive fields leading to theoretical and observational obstacles.In the 1980’s, Connes had put forward the new concept of spacetime based on noncommuta-tive geometry(NCG) .In 1986, Connes and Lott applied the idea to the spacetime extended bytwo discrete points M × Z and shown that Higgs fields emerged naturally in a gauge theorywith a quartic potential. The most attractive feature of NCG with discrete extra dimensions isthat it does not contains an infinte tower of massive fields.In 1993, Chamsedine, Felder and Fr¨ohlich made the first attempt to generalize the Hilbert-Einstein action to NCG, leading to no new physical content. In 1994, G.Landi, N.A.Viet andK.C.Wali had overcome this no-go result and derived the zero mode sector of the Kaluza-Kleintheory from the generalized Hilbert-Einstein’s action. Viet and Wali have generalized thismodel further and obtained a full spectum consisting of bigravity, bivector and biscalar. In eachpair, one field is massless and the other one is masive.The incorporation of the nonabelian gauge fields in Viet-Wali’s model is a not trivial task.Recently, Viet and Du have successfully derived the nonabelian gauge interaction from the a r X i v : . [ g r- q c ] O c t ilbert-Einstein’s action. However, it is possible to do so only in two following cases:i. The gauge vector fields must be abelian on one sheet of spacetime and nonabelian onthe other one. This is exactly the case of the electroweak gauge fields on the two copies ofConnes-Lott’s spacetime of chiral spinors.ii. The gauge vector field must be the same on both copies of spacetime of chiral spinors.This is also the case of QCD of strong interaction.So, NCG can ”explain” the specific gauge symmetry structure of the Standard Model.In this article, we propose a new noncommutative spacetime structure M × Z × Z , whichis the ordinary spacetime extended by two discrete extra dimension, each consists of two discretepoints. In other words, this noncommutative spacetime consists of two copies of Connes-Lott’sspacetime. The generalized Hilbert-Einstein action in this new spacetime contains all the knowninteractions of Nature and the observed Higgs field. In a more general case, this theory can alsolead to multigravity, which might be necessary to explain the dark matter and inflationarycosmology related observations. The noncommutative spacetime M × Z × Z is the usual four dimensional spinor manifoldextended by two extra discrete dimensions given by two differential elements DX and DX in addition to the usual four dimensional ones dx µ . Each extra dimension consists of only twopoints. This structure can also be viewed as four sheeted space-time having a noncommutativedifferential structure with the following spectral triplet:i) The Hilbert space H = H v ⊕ H w which is a direct sum of two Hilbert spaces H u = H uL ⊕ H uR , u = v, w , which are direct sums of the Hilbert spaces of left-handed and right-handedspinors. Thus the wave functions Ψ ∈ H can be represented as followsΨ( x ) = (cid:18) Ψ v ( x )Ψ w ( x ) (cid:19) , Ψ u ( x ) = (cid:18) ψ vL ( x ) ψ wR ( x ) (cid:19) ∈ H u ; u = v, w, (1)where the functions ψ uL,R ( x ) ∈ H uL,R are defined on the 4-dimensional spin manifold M .ii) The algebra A = A v ⊕ A w ; u = A uL ⊕ A uR contains the 0-form FF ( x ) = (cid:18) F v ( x ) 00 F w ( x ) (cid:19) , F u ( x ) = (cid:18) f uL ( x ) 00 f uR ( x ) (cid:19) ∈ A u , (2)where f uL,R ( x ) are real valued function operators defined on the ordinary spacetime M andacting on the Hilbert spaces H uL,R .iii) The Dirac operator D = Γ P D P = Γ µ ∂ µ + Γ D + Γ D , P = 0 , , , , , D = (cid:18) D m θ m θ D (cid:19) , D = (cid:18) d m θ m θ d (cid:19) , d = γ µ ∂ µ (3) D µ F = (cid:18) ∂ µ F v ( x ) 00 ∂ µ F w ( x ) (cid:19) , ∂ µ F u = (cid:18) ∂ µ f uL ( x ) 00 ∂ µ f uR ( x ) (cid:19) (4) D F = m ( F v − F w ) r , D F u = m ( f uL − f uR ) r , r = (cid:18) − (cid:19) (5)where θ , θ are Clifford elements θ = θ = 1, m , m are parameters with dimension of mass.The construction of noncommutative Riemannian geometry in the Cartan formulation isgiven in in a perfect parallelism with the ordinary one. Here we will use the following flat andcurved indices to extend the 4 dimensions with 5-th and 6-th dimensions. E, F, G = A, ˙6 , A, B, C = a, ˙5 , a, b, c = 0 , , , P, Q, R = M, , M, N, L = µ, , µ, ν, ρ = 0 , , , . (7)he construction of noncommutative Riemannian geometry is in a perfect parallelismwith the ordinary one. The starting point is the locally flat reference frame, which is a lineartransformation of the curvilinear one with the vielbein coefficients. For transparency, let uswrite down the vielbein in 4,5 and 6 dimensions as follows e a = dx µ e aµ ( x ) , E A = DX M E AM ( x ) , E E = DX P E EP ( x ) , (8)where e aµ ( x ) , E AM ( x ) , E EP are 4,5 and 6-dimensional vielbeins. The Levi-Civita connection 1-formsΩ † EF = − Ω F E are introduced as a direct generalization of the ordinary case. With a condition , which is a generalization of the torsion free condition one can determine the Levi-Civitaconnection 1-forms and hence the Ricci curvature tensor from the generalized Cartan structureequations T E = DE E + E F Ω EF (9) R EF = D Ω EF + Ω EG ∧ Ω GF (10)Then we can calculate the Ricci scalar curvature R = η EG η F H R EF GH .The construction of our model is carried out in two subsequent steps. First, we constructthe 6-dimensional Ricci curvature with an ansatz containing one 5-dimensional gravity and two5-dimensional vectors fields, where one is abelian and the other is nonabelian to use Viet-Du’sresults. Then R = R − G MN G MN = R + L g (5) (11)where G MN ; M, N = 0 , , , , SU (2) × U (1) gauge fields.In the second step, the gravity sector is reduced further to 4-dimensional gravity nonabeliangauge SU (3) vector of strong interaction R = R − T rH µν H µν , H µν = ∂ µ B ν − ∂ ν B µ + ig S [ B µ , B ν ] , (12)where B µ = B iµ ( x ) λ i are the gluon field and λ i , i = 1 , .., L g (5) to the 4-dimensional electroweak gauge-Higgs sector of the Standard Model as follows L g (5) = −
14 ( F µν F µν + G µν G µν ) + 12 ∇ µ ¯ H ∇ µ H + V ( ¯ H, H ) , (13)where H is a Higgs doublet, ∇ µ is the gauge covariant derivative and V ( ¯ H, H ) is the usualquartic potential of the Higgs field.
In Section 2, we have presented the minimal ansatz to include the all known interactions andthe Higgs fields. New cosmological observations might shed light to more detailed structure ofthe new commutative spacetime. In principles, in a more general case, our model can adopt upto 4 gravitational fields, one of those is massless while the other ones are massive.From theoretical points of view, this model can provide a geometric construction approachto the massive gravity, which has recently attracted a lot of attention as a candidate theoryof modified gravity . From the viewpoints of modern cosmology, multigravity might give newexplanations the existence of dark matter and inflationary cosmology. Summary and discusions
We have presented a new noncommutative spacetime M × Z × Z , which can unify all the knowninteractions and Higgs field on a geometric foundation. This is very similar to the foundation ofEinstein’s general relativity. This model unifies all forces in nature without resorting to infinitetower of massive fields.In the most general case, this theory can contain more (but still a finite number) degreesof freedom, including four different massless and massive gravity fields, Brans-Dicke scalars andmore gauge fields. The model can provide a geometric foundation to the theories of massiveand modified gravity. The reality might be just a special case of the most general theory. Thecosmological observations might help us to see more details of this theory.There are some issues, at the moment we are not able to answer such as the physical meaningof the sixth dimension and the energy scale of this theory. It is worth to quote the followingrelation from the work by Viet and Du g = 8 m (cid:112) πG N , (14)where g the weak coupling constant and G N is the Newton constant. This relation must holdwhen the theory becomes valid. One can speculate this might happen at an energy scale, whichis million times lower that the Planck scale. That might be the case in some evolution stage ofour universe after the Big Bang. All the above perspectives would merit more research. Acknowledgments
The discussions with Pham Tien Du, Do Van Thanh (College of Natural Sciences, VNU) andNguyen Van Dat (ITI-VNU) are greatly appreciated. The author would also like to thank JeanTran Thanh Van for the hospitality at Quy Nhon and supports. The work is partially supportedby ITI-VNU and Department of Physics, College of Natural Sciences, VNU.
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