aa r X i v : . [ phy s i c s . g e n - ph ] D ec Synopsis of a Unified Theory for All Forces and Matter
Zeng-Bing Chen
National Laboratory of Solid State Microstructures,School of Physics, Nanjing University, Nanjing 210093, China (Dated: December 20, 2018)Assuming the Kaluza-Klein gravity interacting with elementary matter fermions in a (9 + 1)-dimensional spacetime ( M ), we propose an information-complete unified theory for all forcesand matter. Due to entanglement-driven symmetry breaking, the SO (9 ,
1) symmetry of M is broken to SO (3 , × SO (6), where SO (3 ,
1) [ SO (6)] is associated with gravity (gauge fields ofmatter fermions) in (3+1)-dimensional spacetime ( M ). The informational completeness demandsthat matter fermions must appear in three families, each having 16 independent matter fermions.Meanwhile, the fermion family space is equipped with elementary SO (3) gauge fields in M ,giving rise to the Higgs mechanism in M through the gauge-Higgs unification. After quantumcompactification of six extra dimensions, a trinity—the quantized gravity, the three-family fermionsof total number 48, and their SO (6) and SO (3) gauge fields—naturally arises in an effective theoryin M . Possible routes of our theory to the Standard Model are briefly discussed. PACS numbers: 04.50.+h, 12.10.-g, 04.60.Pp
The tendency of unifying originally distinct physicalsubjects or phenomena has profoundly advanced modernphysics. Newton’s law of universal gravitation, Maxwell’stheory of electromagnetism, and Einstein’s relativitytheory are among the most outstanding examples forsuch a unification. The tremendous successes of modernquantum field theory, i.e., the Standard Model (SM), mo-tivate the ambition of unifying all the four forces knownso far—a kind of “theory of everything”. The superstringtheory and quantum gravity (particularly, loop quantumgravity—LQG [1–4]), both with remarkable results, aretwo tentative proposals. Here we take a more “orthodox”viewpoint following the SM and LQG, rather than thesuperstring theory.The tradition of physics, initiated from Newton, is todescribe a physical system by dynamical laws, usuallyin terms of differential equations. To determine the(classical or quantum) state of the system, the initial (aswell as boundary) conditions have to be given regardlessof dynamical laws. Different allowed initial conditionslead to different solutions to dynamical laws. Sucha tradition is called Newton’s paradigm [5], which isbelieved not to apply to the whole Universe. For theunique Universe the initial conditions must themselves bea part of physical laws. Thus, if any form of the theory ofeverything is conceivable at all, it must unify dynamicallaws and states, i.e., break down the distinction betweendynamical laws and initial conditions [6] such that itapplies to the Universe as a whole.Recently, we attempted another unification by unifyingspacetime and matter as information via an information-complete quantum field theory, which describes elemen-tary fermions, their gauge fields and spacetime (gravity)as a trinity [7, 8]. Therein, complete physical informationof the trinity is encoded in dual entanglement—spacetime-matter entanglement and “programmed” en-tanglement between elementary fermions and their gauge fields (together as matter), a fact called the information-completeness principle (ICP). The basic state-dynamicspostulate [8] is that the Universe is self-created into astate | e, ω ; A..., ψ... i of all physical contents (spacetimeand matter), from no spacetime and no matter, with theleast action ( ~ = c = 1) | e, ω ; A..., ψ... i = e iS GM ( e,ω ; A...,ψ... ) |∅i ,δS GM ( e, ω ; A..., ψ... ) | e, ω ; A..., ψ... i = 0 . (1)Here |∅i ≡ |∅ G i ⊗ |∅ M i is the common vacuum stateof matter (the matter vacuum |∅ M i ) and geometry(the empty-geometry state |∅ G i in LQG [2]); A... ( ψ... )represent all gauge (matter fermions) fields; gravity isdescribed by the tetrad field e aµ and the spin connection ω abµ , to be specified below. Note that the dynamicallaw and states always appear jointly in the postulate[Eq. (1)]. This is in sharp contrast to the tradition wherethe dynamical law [i.e., δS GM ( e, ω ; A..., ψ... ) = 0] andstates are given separately. Thus, besides the conceptualadvantages as shown previously [8], the theory unifies thedynamical law and states.The conceptual advantages of the information-complete quantum description motivate us to considerthe ultimate unification of all known forces in this Letter.As is well-known, consistent superstring theory (or itsupdated version, the M -theory) exists only in (9 + 1)-or (10 + 1)-dimensional spacetime. Here we assume a(9 + 1)-dimensional spacetime ( M ) instead of theusual (3+1)-dimensional one ( M ), but do not assumestring and supersymmetry. Then the trinary fields of ourtheory consist of the (9 + 1)-dimensional Kaluza-Kleingravity [9] interacting with elementary matter fermionsand an elementary gauge field. The ICP predicts that(1) matter fermions must appear in three families, each ofwhich has 16 independent fermions, and (2) there are the SO (3) family gauge fields. A new, entanglement-inducedsymmetry breaking mechanism and compactifying sixextra dimensions then lead to the (3 + 1)-dimensionaltrinity, in which the Higgs fields naturally arise. Information-complete unification .—Matter fermions inthe SM are six quarks ( u , d ; c , s ; t , b ) and six leptons(electron e , electron neutrino ν e ; muon µ , muon neutrino ν µ ; tau τ , tau neutrino υ τ ). They can be preciselygrouped into three “families” (or “generations”) as ( υ e , e , u , d ), ( υ µ , µ , c , s ), and ( υ τ , τ , t , b ). Togetherwith the corresponding antiparticles, totally we have 45matter fermions if each neutrino is merely left-handed.The three families have identical properties, except fordistinct mass patterns. The origin of this amazingstructure is a long-standing puzzle in the SM, which,together with the mass-generating Higgs mechanism,describes very successfully these fermions interactingvia SU (3) × SU (2) × U (1) gauge fields (the strong,weak, and electromagnetic forces). The particular SMgroup structure was discovered empirically; there is nofundamental principle dicatating why we should choosethis particular group, but not others [10]. Some GrandUnification Theories (GUTs) extended the SM group intolarger groups, such as SU (5) [11] and SO (10) [12, 13].Here we assume a (9 + 1)-dimensional spacetime (acurved manifold M ) with an SO (9 ,
1) symmetry,where the coordinates x = ( x A ) = ( x µ , x ¯ µ ) with “curvedindices” µ = 0 , , , µ = 4 , , ...,
9. A Minkowskivector is denoted by y = ( y I ) = ( y a , y ¯ a ) with “flatindices” a = 0 , , , a = 4 , , ...,
9; the Minkowskimetric η IJ has signature [ − , + , ..., +]. Gravity in M isthen described by the tetrad field ˆ e IA ( x ) (with the inverseˆ e AI ), which relates the (9 + 1)-dimensional metric ¯ g AB ( x )via ¯ g AB ( x ) = ˆ e IA ( x )ˆ e JB ( x ) η IJ . As the SO (9 ,
1) grouphas 45 generators, gravity in M has 45 independentfield components and thus 90 independent internal statesprovided that the polarization degree of freedom (DoF)is taken in account.Matter in M is assumed to be an elementaryfermion field ψ in the spinorial representation of SO (9 , = 32 dimensions. In terms of the Diracmatrices γ a and γ for M and ˜ γ ¯ a for six extradimensions ( M ), one can construct 10 Γ-matrices [9]Γ a = γ a ⊗ I, Γ ¯ a = iγ ⊗ ˜ γ ¯ a − , (2)which satisfy { Γ I , Γ J } = Γ I Γ J + Γ J Γ I = 2 η IJ I . TheseΓ-matrices then form the spinorial representation of SO (9 ,
1) as[Π IJ , Π KL ] = i ( η IL Π JK + η JK Π IL − η IK Π JL − η JL Π IK ) , (3)with 45 generators Π IJ = i [Γ I , Γ J ]. The 32-dimensionalrepresentation implies that the matter fermions wouldhave 32 independent internal states if no furtherconstraint is required. Recall that the SM describeschiral fermions such that fermions of different chiralities transform differently under SU (3) × SU (2) × U (1). Inparticular, each SM fermion family has only 30 internalstates (i.e., 15 independent matter fermions).Let us consider the input arising from our information-completeness trinary description. According to thegeneral formalism of an information-complete quantumfield theory [8], all physical predictions are encodedby spacetime-matter entanglement, which can be de-composed in a Schmidt form such that spacetime (i.e.,gravity) and matter are mutually defined to acquire infor-mation. For our description to be information-complete,the number of independent gravity field components andthe number of independent gauge field components (ifany) must match the number of independent matterfermions. As the spinorial representation of SO (9 ,
1) hasonly 16 elementary fermions, there must be three and onlythree fermion families (duplicates), each of which belongsto the spinorial representation of SO (9 , ψ α withthe family indices α = 1 , ,
3. Then the number ofindependent gauge field components is determined to be16 × −
45 = 3. Note that the family indices label a newinternal space of fermions, or a new quantum number,which is physically different from the quantum numbercharacterizing fermions of each family. If we assume thatthe (elementary) gauge field acts on this family space (thefamily triplet), it is natural to require the family gaugefields to be SO (3) gauged, namely, the family triplet ofthe matter fermions form a basis for a 3-dimensionalirreducible representation of SO (3): [ t α , t β ] = iε αβγ t γ .Here t α are the 3 generators of SO (3), ( t α ) βγ = − iε αβγ for the adjoint representation, and the SO (3) structureconstants ε αβγ are totally antisymmetric and ε = 1.Now it is ready to write down the total action of gravityand matter in M . We can use the beautiful languageof differential forms to express the relevant geometry.For instance, ˆ e I ( x ) = ˆ e IA ( x ) dx A represents 1-form. Theinfinitesimal rotation of ˆ e I is then a 2-form d ˆ e I = − ˆ ω IJ ∧ ˆ e J (Note that d ˆ e I + ˆ ω IJ ∧ ˆ e J ≡ ˆ T I is the torsion; herewe consider therefore a torsion-free manifold), where anantisymmetric 1-form ˆ ω IJ = − ˆ ω JI is the so-called spinconnection. With the connection 1-form, a tensor 2-formˆ R IJ = d ˆ ω IJ + ˆ ω IK ∧ ˆ ω KJ = ˆ R IJAB dx A dx B can be definedand is known as the curvature. The scalar curvaturereads ˆ R = ˆ R IJAB ˆ e AI ˆ e BJ . The total action then readsˆ S (9+1)GM (ˆ e, ˆ ω ; ψ ; z ) = ˆ S (9+1)G (ˆ e, ˆ ω ) + ˆ S (9+1)G+M (ˆ e, ˆ ω ; ψ ; z ) =ˆ S (9+1)G + ˆ S (9+1)G+g + ˆ S (9+1)G+D+g , whereˆ S (9+1)G = 116 πG Z M dx ˆ e ( ˆ R + Λ ) , ˆ S (9+1)G+g = − Z M dx ˆ ef αAB f AB,α , ˆ S (9+1)G+D+g = Z M dx ˆ e ¯ ψ β Γ I ˆ e AI iD βγA ψ γ . (4)Here G (Λ ) represents the Newton (cosmological)constant in M , ˆ e = (cid:12)(cid:12) det ˆ e AI (cid:12)(cid:12) ; ψ = ( ψ α ), ¯ ψ = ψ † Γ , the covariant derivative of Dirac’s spinors reads D βγA = δ βγ ∂ A + iδ βγ ˆ ω IJA Π IJ − ig ( t α ) βγ z αA , where thefield strengths of the SO (3) family gauge fields are f αAB = ∂ A z αB − ∂ B z αA + g ε αβγ z βA z γB with coupling constant g .The theory is not a pure Kaluza-Klein theory, but requires/predicts the existence of the explicit SO (3)family gauge fields even in M , showing the necessityof our trinary description at the most fundamentallevel . The presence of the elementary gauge fields isadvantageous [9] to compactify the extra dimensions, toprovide an origin of the Higgs fields, and to avoid thedifficulty of obtaining chiral fermions in M for pureKaluza-Klein theories (For chirality in M , see also[14]). The state and dynamics of the elementary trinityare then given by | ˆ e, ˆ ω ; ψ ; z i = e i ˆ S (9+1)GM (ˆ e, ˆ ω ; ψ ; z ) |∅i ,δ ˆ S (9+1)GM (ˆ e, ˆ ω ; ψ ; z ) | ˆ e, ˆ ω ; ψ ; z i = 0 . (5)The above considerations fix our information-completeGUT with gravity (or the Grand Integration Theory,GIT). Now it is amazing to see that our theory is uniquely determined by the spacetime symmetry, thegauge principle, and the ICP. The GIT, however, doesnot assume a single gauge group as in the usual GUT.Here the physical predictions of the GIT are the dualentanglement [7, 8] in | ˆ e, ˆ ω ; ψ ; z i , i.e., entanglementbetween (9 + 1)-dimensional Kaluza-Klein gravity andmatter, as well as entanglement, programmed by gravity,between the three-family fermions and the SO (3) familygauge fields. The second line of Eq. (5) gives theequations of motion, conservation laws, and constraintsof the GIT, all acting on | ˆ e, ˆ ω ; ψ ; z i . Below we considerthe effective theory of the GIT in M . Entanglement-driven symmetry breaking .—Now let usconsider the physical consequences of gravity-matterentanglement. After gravity-matter entangling, theoriginal internal space of the Kaluza-Klein gravity will bephysically differentiated by the family quantum numberand as such, the original SO (9 ,
1) symmetry of theKaluza-Klein gravity must be broken to a lower sym-metry. This new mechanism of symmetry breaking canthus be called entanglement-driven symmetry breaking.Mathematically, the maximal subgroup of SO (9 ,
1) is SO (3 , × SO (6) which has the same rank (namely,5) as SO (9 , SO (3 , × SO (6), resulting inthe conventional M (or the usual gravity) and 6-dimensional compactified space M ; other choices leadto inconsistencies this way or another by following theprocedure given below. As a result of symmetry breakingfrom SO (9 ,
1) to SO (3 , × SO (6), ψ α for given familyshould be the spinorial representations of SO (3 ,
1) andof SO (6) simultaneously , which are 4- and 8-dimensional, respectively. The total dimensions of ψ α are again 32.Meanwhile, symmetry breaking singles out M gravityas the “programming system” [7, 8].Then, according to the usual Kaluza-Klein mechanism,one can associate the Lorentz group SO (3 ,
1) with gravityin M and SO (6) with the gauge field of matterfermions. Namely, we can expand ¯ g AB ( x ) with thefollowing ansatz [9, 15]¯ g AB = (cid:18) g µν ( x µ ) + ˜ g ¯ µ ¯ ν ( x ¯ µ ) W ¯ µµ W ¯ νν W ¯ νµ W ¯ µν ˜ g ¯ µ ¯ ν ( x ¯ µ ) (cid:19) (6)with W ¯ νµ = ξ ¯ ν ¯ a ¯ b ( x ¯ µ ) Z ¯ a ¯ bµ ( x µ ), where ξ ¯ ν ¯ a ¯ b ( x ¯ µ ) is theKilling vectors appearing in the infinitesimal isometry I + iε ¯ a ¯ b Σ ¯ a ¯ b : x ¯ ν → x ′ ¯ ν = x ¯ ν + ε ¯ a ¯ b ( x µ ) ξ ¯ ν ¯ a ¯ b ( x ¯ µ ); theinfinitesimal parameters ε ¯ a ¯ b ( x µ ) are independent of x ¯ µ .The ansatz (6), after integrating the extra dimensionsin ˆ S (9+1)GM (ˆ e, ˆ ω ; ψ ; z ), results in the (3 + 1)-dimensionalactions for gravity and the SO (6) gauge field Z ¯ a ¯ bµ (withcoupling constant g ) S ¯ΛG = 116 πG Z M dx e ( R + ¯Λ) ,S gG = − Z M dx e F ¯ a ¯ bµν F µν, ¯ a ¯ b . (7)Here e = | det e µa | = p − det g µν , R stands for the scalarcurvature in M , G is the usual Newton constant,and the cosmological term ¯Λ gathers also the effects ofcompactified space on gravity in M [15]; the SO (6)field strengths F ¯ a ¯ bµν = ∂ µ Z ¯ a ¯ bν − ∂ ν Z ¯ a ¯ bµ + g ( Z ¯ a ¯ cµ Z ¯ b ¯ cν − Z ¯ a ¯ cν Z ¯ b ¯ cµ ). Thus, by treating M and M on an equalfooting, non-Abelian gauge fields naturally arise from ahigher-dimensional gravity.The M action for the family gauge fields reads S fG = − Z M dx ef αµν f µν,α . (8)Following the idea of gauge-Higgs unification proposedlong ago [16, 17], the extra components of z αA living in M are interpreted as the Higgs fields φ α . We denote theaction of the Higgs fields interecting with z αµ and gravityby S HfG . The action of the Dirac sector (matter fermionsinteracting with gravity, the gauge fields, and φ α ) is S DHgfG = S DHG + S DgfG , (9)where S DgfG = R M dx e ¯ ψ β γ a e µa iD βγµ ψ γ with D βγµ = δ βγ ( ∂ µ + iω abµ Π ab − ig Z ¯ a ¯ bµ Π ¯ a ¯ b ) − ig ( t α ) βγ z αµ ; S DHG is theaction for the Higgs fields coupling with the Dirac fields.Here we do not give the explicit forms of S HfG and S DHG and leave them for a future work.
Quantum compactification of extra dimensions .—Usually, compactification of extra dimensions in theKaluza-Klein and superstring theories is a complicatedand unfinished issue. Below we further show that we caninsist on the information-complete trinary descriptioneven in M as a robust theoretical structure and assuch, we could vastly simplify the problem of uncoveringthe reliable feature of the compactification without goinginto its detailed physics.Let us first focus on M . It is well-known thatthe Lorentz algebra SO (3 ,
1) is locally isomorphic to SU (2) × SU (2). Here one SU (2) generates the rotationsand another the boosts. In LQG [1–4], gravity in M possesses only one SU (2) gauge structure related to therotations, while the boost DoFs are not dynamical [18].After the 3 + 1 spacetime decomposition and taking thetime gauge, one arrives at the Hamiltonian formalism[2, 4]. The dynamical variables of the SU (2) gravity,in terms of e µa and ω abµ , are the connection field ˆ A rm [defined on M ; m : spatial indices and r : SU (2)-valued],whose conjugate variable is the “gravitational electricfield” ˆ E sn . The canonical dynamical variables for Z ¯ a ¯ bµ , z αµ , and ψ α , whose explicit forms are not importantin subsequent discussions, can also be obtained withinLQG. A remarkable result of LQG is to identify thestate space of quantized gravity, spanned by a completeorthogonal basis {| Γ , { j l } , { i n }i} consisted of the spin-network states with respect to an abstract graph Γ(with nodes n and oriented links l ) in three-dimensionalregion R with boundary ∂ R . Here j l is an irreducible j representation of SU (2) for each link l and i n the SU (2)intertwiner for each node n .As we discussed above, there are three families ofmatter fermions. Quantum mechanically, the familyinformation can be encoded by introducing three internalstates | f α i attached on the fermion state space; {| f α i} forms a complete orthogonal basis. Then accordingto the ICP, we have to introduce another quantumsystem interacting/entangling with the fermion familyspace such that the family information can be acquired.Such a quantum system can only be attributed tothe compactified space, which must be effectively asimple three-state system. The three (the number isrequired by the ICP to be the same as the familynumber) states are denoted by | α, i ≡ | α i ( α =1 , , M arequantized, resulting in a complicated state. Yet, in the“quantum compactification” mechanism proposed here,the compactified M has only three states accessible tophysical DoFs in M .After such a compactification, all the remaining forcesfields and the fermion fields are confined in M suchthat their field quanta are massless as protected bythe relevant symmetries. Meanwhile, each | α i encodesall information for matter and gravity (including thegauge fields with mixed indices of M and M )related to M although the quantized M itself ismerely an effective three-state system. This immediatelymeans that information encoded in | α i (except for the three states), as well as the related matter, is “dark”with respect to the gauge fields in M . Thus, ourGIT enables a quantum information definition of (non-Abelian) dark matter, which is dark to non-Abeliangauge fields survived in M ; for dark energy, see [8].In terms of the spin-network basis, following theformalism in Ref. [8] we propose the quantum-compactification ansatz for quantized spacetime M X l ∈ Γ ∩ ∂ R n ∈ Γ ∩R ,α S Γ ( l, n, α ) | Γ , j l , i n ; α i | α i ≡ X α s Γ ( α ) | Γ , α i | α i , (10)where three | Γ , α i form an orthogonal basis and arethe “three-world” states corresponding to | α i and thethree families of matter fermions. Entanglement betweenquantized spacetime M and M , as shown inEq. (10), suggests interaction between them. To beconsistent with the Einstein equations in M , the onlypossible form of the interaction would be S G , = 116 πG X α Z M dx e ˜Λ α | α i h α | . (11)Namely, our GIT predicts the cosmological constantΛ ≡ ¯Λ + P α ˜Λ α | α i h α | to be an operator in {| α i} ;the total cosmological term contains the remnant effectof the compactified space. Similarly to the ansatz inEq. (10), we conjecture that there is also interaction(entanglement) between the fermion family space andthe compactified space [see Eq. (13) below]. Thecorresponding action is denoted by S DG , , whoseexplicit form is a future work. Total action and dynamics .—To sum up the aboveresults, the total action in M is S GIT = S ΛG + S gG + S fG + S DHgfG + S HfG + S DG , , (12)with S ΛG = S ¯ΛG + S G , . While we leave physics of theHiggs mechanism for future work, it should be noted thatin our GIT, each family of matter fermions is labeledby one Higgs field and entangled with one of the threefamily gauge fields after symmetry breaking of the familysymmetry; the SO (6) gauge fields act universally uponthe three families. Thus, for each family the totalnumber of gauge fields is 15 + 1, matching the numberof elementary fermions. This shows that our GIT isinformation-complete even in M . Below we considerthe dynamics of the physical Universe described by S GIT .The information-complete trinary description allowsus to adopt either a “global view” or a “local view”of the dynamics. The global view is reflected bythe fact that all physical predictions about the wholeUniverse are encoded in gravity-matter entanglement |A , α, ( ψ, z , Z ) i = e iS GIT |∅i⊗| α i for given “initial state” | α i of M . Thanks to the spin-network basis, |A , { α } , ( ψ, φ, z , Z ) i = X l ∈ Γ ∩ ∂ R n ∈ Γ ∩R ,α S Γ ( l, n, α ) | Γ , j l , i n ; α ; t i⊗ | α i ⊗ | ( ψ, φ, z , Z ); Γ , l, n ; α ; t i . (13)Here t denotes time; S Γ must be time-independent as |A , { α } , ( ψ, φ, z , Z ) i is annihilated by, among others, thetotal Hamiltonian, known as the Hamiltonian constraint.Because a spin-network state | Γ , j l , i n ; α ; t i definesspacetime and thus, is spacetime [2], we can include t explicitly in | Γ , j l , i n ; α ; t i . | ( ψ, φ, z , Z ); Γ , l, n ; α ; t i ≡| ( ψ, φ, z , Z ); Γ , k l , F n , S n , w n ; α ; t i ( F n , S n , w n : number offermions, number of Higgs scalars, and the field strengthat node n , respectively, k l : flux of the electric gauge fieldsacross surface l ; see Refs. [2, 4]), programmed by a givenspin-network state and by | α i , encodes entanglementbetween (i.e., all physical predictions about) matterfermions and their gauge fields (the Higgs scalars arethe extra components of the family gauge fields). Sucha particular entanglement structure of the Universe iscalled dual entanglement [7, 8]. Following Ref. [8], theevolution operator U GM ≡ exp[ iS GIT ] has a factorizablestructure. In this way, the dynamics of the Universe canbe cast into a dual form without the notorious “problemof time” [2, 4] in quantum gravity, thus recovering adescription of the local view.Note that all states for gravity, matter fermions, andgauge fields, once Schmidt-decomposed in the dual form,are physical predictions of the theory and thus must beannihilated automatically by any constraints appearingin the theory. This fact embodies again that ourformalism unifies the dynamical law and states.
Possible routes to the SM .—To show more predictivepower of the GIT, let us briefly consider chirality, whichis a remarkable feature of the SM. For pure Kaluza-Klein theories, it is difficult to obtain chiral fermions[9, 14]. But in our theory, the situation is dramaticallydifferent. In M , the chirality operator χ = iγ ⊗ Γ five6D ,where γ (Γ five6D ≡ ˜ γ ˜ γ ... ˜ γ ) is the chirality for M ( M ); the M chirality and the M chirality arecorrelated [9, 14, 17]. Now if the elementary SO (3) [or itsisomorphic group, SU (2)] gauge fields have topologicallynon-trivial background (e.g., the monopole or instantonconfiguration) in M , one can have chiral zero modesof fermions forming a complex representation of thesymmetry group [9, 19–21]. In this case, these masslessfermions of definite chirality may correspond to thefermion spectrum as described by the SM.To facilitate our discussion on possible routes to theSM, note that SO (6) [ SO (3)] is isomorphic to SU (4)[ SU (2)]. We can thus replace the Z -fields ( z -fields)with SU (4)-gauged [ SU (2)-gauged] ones in the above M effective theory. Now the Dirac fields are therepresentations of SU (4) and of SU (2) simultaneously, represented by ψ ηv = (cid:0) Q η v l η (cid:1) , where η = 1 , SU (2)-valued and v = 1 , , , SU (4)-valued, e.g., ψ η v = Q η v (quarks of three colors indexed by v ), ψ η = l η (leptons). Here, the lepton number in our GIT worksalso as the fourth color, similarly to the Pati-Salammodel [22]. In the Γ five6D -diagonal basis, the SO (6)generators split into two 4 × ψ ηv for given η has two four-component spinors,each of which corresponds either to the four-dimensionalcomplex representation of SU (4) or to its conjugate [23].Note that the index η doubles the total number ofelementary fermions, which however is halved by therestriction of definite M chirality in our theory. Thisleaves the total number of elementary fermions invariant.Now each family of matter fermions is labeled by one ofthe three SU (2) gauge fields (as well as the correspondingHiggs field) after symmetry breaking of the SU (2)symmetry. The SM symmetry SU (3) × SU (2) × U (1) canbe obtained if symmetry breaking SU (4) → SU (3) × U (1)occurs in our theory. The resulting SM is informationallyincomplete and effectively has 16 − (8 + 3 + 1) = 4 matterfermions dark to the SU (3) × SU (2) × U (1) gauge fields.The SM symmetry might of course arise when M is acoset space; for details see, e.g., [9]. Conclusions and discussions .—To summarize, insist-ing on the information-complete trinary description ofnature, we have described very briefly a unified theoryfor all forces and matter. Seen from our discussion, thefollowing two facts are strictly related or even mutuallyexplained, namely, (1) matter fermions have threefamilies, each having 16 independent matter fermions,and (2) spacetime is (9 + 1)-dimensional and displaysentanglement-driven symmetry breaking from SO (9 , SO (3 , × SO (6). The latter fact in turnexplains why the observed spacetime is M andthe gauge group for matter fermions is SO (6). Theinformation-complete trinary description of our theorynaturally arises, thus demonstrating that the overallpicture is consistent with our previous insistence onthe information-completeness. Indeed, the ICP, whichis essential for a new quantum formalism without themeasurement postulate [7, 8], works here as the funda-mental principle restricting not only the required gaugegroup, but more, such as the dimensions of spacetimeand the number of elementary matter fermions. Dueto the discreteness of spacetime acting as a naturalregulator [24] and severe limitation of information bythe information-complete trinary description, our theoryshould be free of singularities. Assuming M and theICP, the theory seems to be unique.So far, our GIT predicts, in certain sense, some basicfacts that are used in the SM without any fundamentalexplanations. These include the dimensions of spacetime,the three-family pattern, the number of elementarymatter fermions, and the possible origin of the Higgsmechanism. No previously known single theory could“predict” all these facts simultaneously. According to ourGIT there is one and only one matter fermion (a right-handed neutrino) that remains to be discovered for eachfamily. If the three-family structure of the cosmologicalconstant [see Eq. (11)] survives, it would be very excitingto see its cosmological test.Is it possible to introduce supersymmetry/superstringin an information-complete formalism? 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