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      A quantum theory of spacetime in spinor formalism and the physical reality of cross-ratio representation: the equation of density parameters of dark energy, matter, and ordinary matter is derived: Ω M 2 = 4 Ω b Ω Λ

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            ABSTRACT

            By theorizing the physical reality through the deformation of an arbitrary cross-ratio, we leverage Galois differential theory to describe the dynamics of isomonodromic integratable system. We found a new description of curvature of spacetime by the equivalency of isomonodromic integratable system and Penrose’s spinor formalism of general relativity. Using such description, we hypothetically quantize the curvature of spacetime (gravity) and apply to the problem of the evolution of the universe. The Friedmann equation is recovered and compared so that the mathematical relationship among dark energy, matter (dark matter + ordinary matter), and ordinary matter, ΩM24ΩbΩΛ, is derived; the actual observed results are compared to this equation (calculated ΩM = 0.33 vs. observed ΩM = 0.31); the model might explain the origin of dark energy and dark matter of the evolution of the universe.

            Main article text

            INTRODUCTION

            We looked for the simplest mathematical object to identify the underlying reality of nature, and we found it to be cross-ratio. By defining cross-ratio over Riemann sphere, infinity is just another usual point; similarly, there shouldn’t be any point in the universe more special than others. However, the variety of nature must be realized as a condition for such an underlying object.

            In “Cross-ratio arbitrariness and the constraint to the parameter space of projective space basis” section, the article explains how potential physical varieties come from different representations of the same thing – cross-ratio deformation. So, the cross-ratio consists of both simplicity and variety. A successful example is like Einstein’s masterpiece – general theory of relativity. Although Einstein’s field equation is simple, many interesting solutions emerged.

            In “Cassidy’s work on isomonodromic system” section, we introduce Galois differential theory and related Cassidy’s work. It is a mathematical machinery to manifest the deformation of cross-ratio. Cassidy’s work consists of introducing a 2 by 2 matrix differential equation and related isomonodromic integratable system, so it can describe the deformation. By such machinery, we formulate an alternative theory of the dynamics of curvature of spacetime to recover the spinor general relativity equivalent counterpart (for which a brief introduction is given in “Brief overview of spinor formulation of general relativity” section). By such connection, we hypothetically claim the origin of spacetime is from the isomonodromic integratable system, and spacetime is more fundamentally described by the curvature rather than metric or coordinated mathematical framework, that is, spinor formulation of general relativity might be more fundamental than classical general relativity; a similar argument was postulated by Penrose (1960).

            In “As an application to the problem of modeling the universe evolution” section, we apply the calculation of the deformation of the isomonodromic integratable system with certain simplified conditions so a solution is found. The solution is used to recover Friedmann equation and related density parameters such that observed and calculated results are compared. This proposes an explanation of the origin of dark matter and dark energy without new kind of matter or energy, as they are new kind of gravitational field of spacetime’s curvature.

            BRIEF OVERVIEW OF SPINOR FORMULATION OF GENERAL RELATIVITY

            Penrose’s spinor approach to general relativity

            Spinor formulaism of general relativity (i.e., Spinor GR) (Penrose, 1960) adopted a coordinate-free approach. The correspondence between tensors and spinors is obtained by the use of a hybrid spinor ΣΝCD′ (2 × 2 Hermitian matrix per CD index, denoted as ΣΝ in short) (Einstein’s summation notation is used in this article):

            ΣΜACΣΝBC+ΣΝACΣΜBC=gΜΝεAB,forA,B,C=0,1;Μ,Ν=0,1,2,3,  (1)

            where εBC represents 2 × 2 skew-symmetric “metric” spinor (components’ values are 0 or ± 1). To raise or lower spinor index, either one of εBCεBC, εBC, and εBC is used:

            ΞA=εABΞB,ΞB=ΞAεAB  (2)

            where Ξ denotes spinor. Primed indexes refer to the complex conjugate spin space. Please note that

            εBDεDF=ΔBF,ΔBF=0(if BF),or 1(ifB=F).  (3)
            ΣΜAB=defgΜΝΣΝCDεCAεDB  (4)

            is Hermitian because gΜΝ (g is the metric) is real.

            Spinor GR requires the covariant derivatives of ΣΝCD , gΜΝ, and (εCAεDB) to be zero.

            The spinor equivalent of tensor GΜΝ is stated as

            GACBD=ΣΜACΣΝBDGΜΝ,GΜΝ=ΣΜACΣΝBDGACBD  (5)

            Curvature tensors, symmetry and Einstein tensors in spinor general relativity

            By RAEBFCGDH (spinor version of Riemann-Christoffel tensor RΜΝΡΣ), the curvature in spinor GR is defined by curvature spinors - ΧABCD and ΦABGH (Penrose, 1960):

            RAEBFCGDH=12(ΧABCDεEFεGH+εCDΦABGHεEF+εABΦEFCDεGH+εABεCDΧEFGH)

            ( indicates complex conjugate), which satisfy the following equations (Penrose, 1960):

            (AHBH+BHAH)ΞF=ΧABFEΞE(ADAC+ACAD)ΞF=ΦFEDCΞE  (6)

            where AG and BG are spinor covariant derivatives and raised spinor derivatives, respectively. They can be modified to

            εGF(AHBH+BHAH)ΞF=εGFΧABFEΞE(AHBH+BHAH)ΞG=ΧABGEΞEεGF(ADAC+ACAD)ΞF=εGFΦFEDCΞE=εGF(ΦDCFE)ΞE(ADAC+ACAD)ΞG=ΦDCGEΞE  (7)

            for which we use the symmetry property of ΧAB(FE) and ΦDC(FE), ΦDCFE=ΦFEDC (Penrose,1960).

            By Bianchi identity of Riemann curvature,

            ΤRΜΝΡΣ+ΡRΜΝΣΤ+ΣRΜΝΤΡ=0

            it implies the relationship of curvature spinors, which is called Bianchi identity of spinor GR (Penrose, 1960):

            BDΧFEAB=ACΦFEDC  (8)

            One can further modify it by the symmetry of (Penrose, 1960)

            ΧFEAB=ΧABFE,ΦFEDC=ΦDCFE

            to

            BDΧABFE=ACΦDCFEBDΧABFEεGF=ACΦDCFEεGFBDΧABGE=ACΦDCGE

            Note that we can express Bianchi identity as

            BDΧABGE=ACΦDCGE  (9)

            where ΧABGE and ΦDCGE are 2 × 2 traceless matrices if considering G and E as the matrix indexes because of the symmetry property of ΧAB(FE) and ΦDC(FE) (Penrose, 1960) (trivially proved by contracting ΧABFEεGF and ΦDCFEεGF). The author claims this form (or in matrix equivalent), and Equation (7) is more fundamental to the original version of Equations (6) and (8).

            By the symmetry of Riemann curvature RΜΝΡΣ, the curvature spinor ΧFEAB needs to obey (Penrose, 1960)

            Λ=def12ΧABAB is real.  (10)

            Penrose (1960) mentioned that the geometry of a Riemannian space (with signature +,−,−,−) can be described by spinor terms with the role of curvature tensor being taken over spinors ΧFEAB and ΦFEDC satisfying their symmetry property, Bianchi identity (Equation 8), and Equation (6).

            Finally, Ricci spinor is equivalent to Ricci tensor (Penrose, 1960):

            RACBD=ΛεABεCDΦABCD  (11)

            Ricci scalar is (Penrose, 1960)

            R=4Λ  (12)

            Einstein spinor is equivalent to Einstein tensor (Penrose, 1960):

            GACBD=ΛεABεCDΦABCD  (13)

            APPLICATION OF GALOIS DIFFERENTIAL THEORY TO SPINOR FORMULATION OF GENERAL RELATIVITY

            Cassidy’s work on isomonodromic system

            For a commuting set of derivations zi (i=0,...,m), there exists an integratable (parameterized second-order linear differential) system [Cassidy & Singer, 2005: Prop6.3(3)]

            ziY=BziY  (14)

            with integratability condition: [Bzi,Bzj]=ziBzjzjBzi(i,j=0,...,m), where Y is in GL2(f(zi)), Bz0=sl2(f(zi)), Bzk=gl2(f(zi)) (k=1,…,m), f(zi) is a set of functions which zi acts on. The variable set {zk}, for k=1,…,m, is called parameters-set. This system is called isomonodromic system. Furthermore, the related Gal-group is a proper Zariski dense subgroup of SL2(cz0), where Gal-group is the parameterized Picard–Vessiot group (PPV-group) and cz0 is the differential constant of z0 (but function of parameters-set) or called z0-invariant. Y is called the fundamental solution matrix.

            The PPV-group is a -automorphism group such that Gal ∈ PPV-group if Gal:Y → Y and Gal = Gal . Therefore, Gal(Y) is another fundamental solution matrix to the system.

            Let us define the Lie algebra corresponding to one-parameter subgroup of Gal by G through exponential map:

            Gal=Exp[Κ G],hereΚ is a real parameter

            By expanding the exponential map with parameter Κ → 0 and commuting property of Gal (i.e., Gal = Gal), the commuting property of Lie algebra G and exists:

            G=G,  (15)

            It is obvious that G ∈ sl2(cz0).

            There is a property of integratability condition that change of variables from parameter-set zi to zΜ preserves this property (by change of variables) if there exists smooth maps zizΜ for zi(zΜ). The transformed new integratable system (together with z0Y=Bz0Y) is

            zΜY=BzΜY,whereBzΜ=izizΜBziforΜ=1,...,n;i=1,...,m  (16)

            note that n is not necessarily equal to m.

            Cross-ratio arbitrariness and the constraint to the parameter space of projective space basis

            A cross-ratio (https://en.wikipedia.org/wiki/Cross-ratio) can be defined by a function of four points on Riemann sphere:

            Cross-Ratio(z0z2)(z1z3)(z1z2)(z0z3)

            If we allow any arbitrary cross-ratio fixed value to be assigned, those four points are then constrained (only three points are free to move). If we further assume any arbitrary reference fixed point to be assigned, say for z0, then those other points are also constrained by four real parameters (x0, x1, x2, x3), that is,

            CrossRatio=(z0z2(x0,x1,x2,x3))(z1(x0,x1,x2,x3)z3(x0,x1,x2,x3))(z1(x0,x1,x2,x3)z2(x0,x1,x2,x3))(z0z3(x0,x1,x2,x3))

            So, the arbitrariness of description of a cross-ratio allows four real parameters. Physically, we mean the principle of cross-ratio arbitrariness is natural because no particular point over Riemann sphere is more special and no reference point is more special than others.

            By Yoshida (Chapters I and IV), the realization of cross-ratio has direct relationship with hypergeometric function (three regular singular points):

            z(1z)d2ydz2+(c(a+b+1)y)dydzaby=0,where a,b,c are complex numbers

            Since hypergeometric function is a solution of related hypergeometric differential equation, so if we deform the hypergeometric differential equation, we also deform the related hypergeometric function and its underlying cross-ratio realization.

            Referring to Hypergeometric function (https://en.wikipedia.org/wiki/Hypergeometric_function), any second-order differential equation with three regular singular points can be converted to hypergeometric differential equation by change of variables, so we can describe the hypergeometric differential equation by second-order differential equation. Leveraging Cassidy’s work in previous section, we can deform the hypergeometric differential equation by the deformation of second-order differential equation. Physically, we mean we can deform the realization of cross-ratio by deformation of related isomonodromy integratable system by a deformation space of four real parameters:

            ΜY=xΜY=(Μ Id)Y=BΜYor ΜΔEFYFG=DΜEFYFG=BΜEFYFG  (17)

            the later is in spinor form, where BΜ(z0,xΝ)=izi(xΜ)xΜBzi(z0,zj) for Μ, Ν = 0,…,3;i,j = 1,…,3; xΝ ∈ Reals.

            Id is 2 × 2 identity matrix; D^ΜEFΜΔEF; note that BΜ(z0, xΝ) and Bzi(z0,zj) are matrices gl2, and D^ΜEF,BΜEF are hybrid spinors like ΣΜAB′ in spinor GR (Robert, Chapter 13).

            Operators of integratability condition and quantized Bianchi equation for curvature spinors

            Based on the isomonodromy integratable system formulated in previous section, we look for a set of the Lie algebras associated with Gal that can produce the integratability. In order to do so, some fundamental operators, such as derivation operator and integratability operator, are introduced. They are related to mathematical objects in spinor GR so that Bianchi identity is recovered by an operator equation. Finally, the equivalency of spinor GR and this theory is summarized.

            Let us define spinor (derivation) operator:

            D^ABEFΣΜABD^ΜEF  (18)

            and D^ΜEF is the differential hybrid spinor operator from previous section (so it is associated with an integratable system), not covariant derivative.

            We demand ΣΜAB (denoted as ΣΜ in short) is the differential constant of D^Ν in the sense that D^ΜEFinvariant|D^ΜEF(ΣΝABX)=ΣΝAB(D^ΜEFX) for any spinor X, so it is consistent with spinor GR requirement (covariant derivative of ΣΝAB is zero) except D^. ΣΝAB defined here satisfies the same relationship in Equation (4).

            Note that commutative Μ implies commutative D^ABEF:D^ABEFD^CDFG=D^CDEFD^ABFG.

            By Equations (16) and (18) as well as ΣΜAB being D^ΜEFinvariant, there is a commuting set of spinor operator for an integratable system:

            D^ABEFYFG=H ABEFYFGD^BAEFYFG=HBAEFYFG,  (19)

            where

            H ABEFΣΜABBΜEF

            (↔ indicates complex conjugate, so prime and unprime indices are interchanged), and note that the ordering between prime and unprime indices are irrelevant (Robert, Chapter 13).

            The integratability condition is used to facilitate a definition on a set of spinor operator.

            J^ABCDEF is called integratability operator (or, in short, int operator), acting on Y |

            J^ABCDEFYFG=(H ABEGH CDGFHCDEGH ABGF)YFG=H [ABEGH CD]GFYFG,

            and

            J^BADCEFYFG=H[BAEGHDC]GFYFG

            [AB’ and CD’] are anti-commutating indices notation as usual in spinor GR. So, by integratability condition (14), this implies

            J^ABCDEFYFG=(D^[ABEG(HCD]GF))YFG,J^BADCEFYFG=(D^[BAEG(HDC]GF))YFG  (20)

            There are a total of four sets of spinor operators from the contraction of J^ and J^, which have a close relationship with int operators and curvature spinors:

            J^ACEFJ^ABCBEF=εBDJ^ABCDEF,J^ACEF=J^BABCEF=εBDJ^BADCEF,M^BDEFJ^ABADEF=εACJ^ABCDEF,M^BDEF=J^BADAEF=εACJ^BADCEF,  (20.b)

            Specifically defining a bar-spinor: J¯BDEFJ^BADAEF=M^BDEF, one can see the J¯BDEF has symmetric contraction over prime indices of J^ACEF except over complex space: that is, YFH.

            Note that J^(AC)EF and M^(BD)EF have symmetric property over their first two indices because

            J^ACEF=εBDJ^ABCDEF=εDBJ^CDABEF=J^CAEF
            ,

            where we use the property of anti-symmetry of εDB and J^ABCDEF=J^CDABEF.

            Let us look for an integratable system with the Lie algebra of PPV-group generated by spinor operators J^ACEF,M^BDEF on YFG:

            J^ACEFYFG=JACEFYFG,

            where JACEF is a spinor without derivation;

            M^BDEFYFG=MBDEFYFG

            where MBDEF is a spinor without derivation;

            such that JACEF and MBDEF are matrices (if considering E and F as the matrix indices) of Gal Lie algebra G. By requirements in Equation (15), but with association to D^C,DFG instead of , commuting property of G(J^,M^) against D^ reads:

            (J^ABEFD^CDFG)YGH=(D^CDEFJ^ABFG)YGH,(M^BDEFD^ACFG)YGH=(D^ACEFM^BDFG)YGH

            Let’s define an fundamental equation as:

            εBC(J^ABEFD^CDFG)YGH=εBC(M^BDEFD^ACFG)YGH  (21)

            or equivalently:

            (J^ABEFD^BDFG)YGH=(J¯DBEFD^BAFG)YGH

            after applying the commuting property of (J^,M^) against D^ above and using symmetry of M^(BD)FG:

            (D^BDEFJ^ABFG)YGH=(D^ACEFM^DCFG)YGH

            This is the quantized version (i.e., operator-wise equivalent) of Bianchi identity which describes the differentiation on curvature spinors Φ, Χ (compared to Equation (9) BDΧABGE=ACΦDCGE)), while the spinor covariant derivative is replaced by spinor derivation operator D^, and Χ, Φ are replaced by operators J^ and M^, respectively. The quantization here refers to changing the classical equation of physical observable (spinor derivative of curvature in spinor GR) to operator equation (e.g., operator D^BDEFJ^ABFG acting on YGH), like Schrodinger’s equation that KE + PE = E →(p^22m+V)Ψ=EΨ.

            As a short summary, we look for an integratable system, where G is generated by the int operator J^, and G is a solution to the fundamental (equation 21). This system leads to quantized Bianchi identity.

            Let us define a set of spinor operators Δ^ABCDEF such that

            (Δ^ABCDEFΔ^CDABEF)YFH=(D^[ABEG(HCD]GF))YFH  (22)

            By contracting over B’ and D’, we get

            εBD(Δ^ABCDEFΔ^CDABEF)YFH=(Δ^ABCBEF+Δ^CDADEF)YFH=εBD(D^[ABEG(H*CD]GF))YFH.

            Using the integratability condition property (Equation 20) for J^ operator on the right-hand side, we get

            (Δ^ABCBEF+Δ^CDADEF)YFH=εBD(J^ABCDEF)YFH

            The right-hand side is just J^ACEF (Equation 20b), and the equation becomes

            (Δ^ABCBEF+Δ^CDADEF)YFH=J^ACEFYFH  (23)

            Let us do the same for (Equation 22), but contracting over A and C:

            εAC(Δ^ABCDEFΔ^CDABEF)YFH=εAC(D^[ABEG(HCD]GF))YFH

            After the same operations, we get

            (Δ^ABADEF+Δ^ADABEF)YFH=M^BDEFYFH  (24)

            By decomposing YFH as matrix into ΞF and ΗH (both are column vectors of dimension 2) into Equations (23) and (24): Y → ΞΗ, we get: (Δ^ABCBEF+Δ^CDADEF)ΞF=JACEFΞF, (Δ^ABADEF+Δ^ADABEF)ΞF=MBDEFΞF (and same for Η). They are equivalent to the equations of curvature spinors in Equation (7) if

            Δ^ABCBEF,Δ^ABADEFA,BCB,A,BAD  (25)

            and

            JACEF=ΧACEF,MBDEF=ΦBDEF(not the operators)  (26)

            In conclusion, spinor GR requires a well-defined spinor covariant derivative ∇∇. ∇∇ gives raise to the relationship as Equation (7), so that curvature spinors Χ and Φ are well established, which obey Bianchi identity (Equation 9). These conditions are said to be equivalent to “generate” spacetime manifold of {+,–,–,–} (Penrose, 1960) (except certain torsionness symmetric properties). In this article, the author claims that the spinor covariant derivative ∇∇ is corresponding to the operator Δ^; curvature spinors Φ and Χ are generated by J^ and M^ operators on Y; Bianchi identity is satisfied by the operators J^, M^ in such quantized form.

            To illustrate the correspondences between spinor GR and this part of the theory (i.e., isomonodromic system):

            ○Spinor covariant dervatives: Δ^ABCDEF  (27)
            ○Curvatures:Φ,ΧJ^,M^  (28)
            ○Bianchi identity (dynamics of curvatures):BDΧABGE=ACΦDCGE(D^BDEFJ^ABFG)YGH=(D^ACEFM^DCFG)YGH  (29)

            Note that J^ and M^ are just linear combinations of int operators, which must exist in an integratable system, while Y is a special solution to quantized Bianchi equation, which is a result of commuting relationship of J^, M^ and D^. The consequence is that, we can only claim that the equivalence of spinor GR to this theory is up to particular set of solution Y and its associated values of J^ and M^, that is, spinor GR is equivalent to this part of the theory in such quantized sense.

            The symmetry of curvature spinor

            Certain symmetries of curvature spinors are described by Penrose in previous sections, and those symmetries are consequence of classical general relativity. We show that certain symmetries are naturally arisen by the operators introduced in previous section, while some others are torsion related and needed to be met additionally. However, those additional conditions are not considered as mandatary because the key objective is to describe that spacetime’s curvature can be manifested by the deformation of isomonodromy integrable system, not to explain the naturalness of torsion.

            From Equation (26) and the traceless property of matrices generated from int operators (Equation 20), it is concluded that ΧABGE and ΦDCGE are traceless matrices (if considering G and E as the matrix indices), and

            int operatorsJ^symmetry of   Χ,ϕ:ΧAB(FE) and ΦDC(FE)  (30)

            Because of the symmetry properties of J^(AC)EF and M^(BD)EF, it implies trivially

            Χ(AB)FE  and   Φ(DC)FE  (31)

            Besides these two symmetries, spinor GR also requires symmetry:

            ΧABFE=ΧFEAB  and   ΦDCFE=ΦFEDC
            ,

            and the Λ reality (Equation 10):

            Λ=ΧABAB2=12ΧABCD εACεBD=12εBDΧABAD=12εBDJABAD  (32)

            By Gomez-Lobo and Martin-Garcia (2012), GR symmetry (Equation 31) and Λ reality are consequences of the torsion-free property of original GR theory. However, torsion free is known that it is not a mandatory requirement; for example, Einstein–Carton theory removes such constraints.

            In conclusion, to recover GR symmetry ΧABFE=ΧFEAB and ΦDCFE=ΦFEDC and Λ reality, there are conditions for spinors generated by operators J^ and M^ to meet. However, the discussion of sufficient and necessary conditions for J^ and M^ to meet these symmetries is not covered yet in this paper.

            Translation to ordinary Ricci, Einstein tensors, and Bianchi equation

            The purpose of this section is to show the explicit links between related mathematical objects spinor GR (spinors of Ricci and Einstein tensors) and the isomonodromy integratable counterparts. Because those are key components of classical general relativity, showing explicit relationships helps visualize the equivalency and claim the origin of related tensors.

            Recall the spinor equivalences of Ricci tensor (Equation 11), Ricci scalar (Equation 12), and Einstein tensor (Equation 13): RACBD=ΛεABεCDΦABCD, R = 4 Λ, and GACBD=ΛεABεCDΦABCD. The following are J and M equivalences:

            RAC BD=(12εFHJGFGH)εAB εCD(MABEεEC)  (33)
            R=2εBDJABAD  (34)
            GAC BD=(12εFHJGFGH)εAB εCD(MABEεEC)  (35)

            Follow the ordinary way of spinor GR to translate from spinor to tensor (Equation 5), Einstein tensor is equivalent to

            GΜΝ=ΣΜACΣΝBD((12εFHJGFGH)εAB εCD(MABEDεEC)).  (36)

            If we use the property of MABEDεEC=MBAEDεEC, and MABEDεEC=MABECεED, for which the symmetries are proved in the previous section, then

            GΜΝ=ΣΜACΣΝBD((12εFHJGFGH)εAB εC D(MBAEεED))=ΣΝBDΣΜAC((12εFHJGFGH)εAB εC D(MBAEεED))=GΝΜ

            (where, in the last right-hand side, we rename the dummy indices). So, it is a clear symmetric for G(ΜΝ) as expected.

            Bianchi identity (Equation 9) in spinor GR describes the dynamics of curvature. Unlike Einstein field equation, which the dynamic of curvature is described by solving (up to) second-order differential equation of metric tensor, spinor GR handles the same problem by first-order differential equation – Bianchi identity (Penrose, 1960). However, spinor GR is coordinate-free yet a classical theory; there is no metric differential analog. Nevertheless, quantized Bianchi equation do provide quantized analog to spinor GR as

            BDΧABGE=ACΦDCGE(D^BDEFJ^ABFG)YGH=(D^ACEFM^DCFG)YGH.

            Once we solve the operators Δ^, D^, J^, M^, and fundamental solution matrix Y, the dynamics of curvature spinors Χ and Φ can be found by equations Equation (26). The ordinary curvature components such as Ricci tensors and Ricci scalar as well as Einstein tensors can be found.

            From last three chapters of this section, the overall approach of spacetime dynamics is done by first identifying related classical curvature spinors and the derivatives equations (i.e., Bianchi identity), then manifesting the spacetime by a deeper structure from isomonodromy system; finally, illustrating related tensors such as Ricci and Einstein tensors equivalence by associated spinors by associated objects from isomonodromy system. Physically, the approach claims a more fundamental reality from isomonodromy system over classical spacetime just like Penrose mentioned that spinor GR may be more deep-rooted than tensors (Penrose, 1960).

            Einstein condition and the connection to quantum operators

            It is well known that Einstein’s theory (theory of general relativity) does not assign a definite stress–energy tensor to the gravitational field.

            When solving Einstein field equation, we guess the form of metric and energy–momentum–stress tensor TΑΒ, because the physical interpretation to define TΑΒ is not yet known without knowing the metric first. In this article, the author uses the same approach to define TΑΒ but claims that the curvature is caused by Δ^ on Y, which constitutes energy–momentum–stress distribution. As required, the energy–momentum conservation is guaranteed in “quantized sense” by quantized Bianchi equation (just as in classical GR, Bianchi identity guarantees conservation). However, it is expected that there must exist a condition that local energy/mass and momentum density do contribute proportional to GΑΒ as Einstein field equation: GΑΒ=8ΠGTΑΒ (G is gravitational constant).

            As GΑΒ is related to ΣΑEGΣΒFH multiplying linear combination of Λ (by constraction of J) and enties of M (Equation 36), let’s examinate how to find J and M. To find J and M, which are contractions of int operators J^ on Y (Equation 20), Equation (22) explains that we can get them by Δ^ABCDEF. Considering the form of the operator Δ^:

            Δ^ABCDEF=D^ABEGD^CDGF+ΣΜABΣΝCD(ΓΜΝEGD^ΝGF),  (37)

            where ΓΜΝEG is the unknown hybrid-spinor. This implies that int operator J^ can be found by

            J^ABCDEFYFH=(Δ^[ABCD]EF)YFH=ΣΜABΣΝCD(ΓΜΝEGD^ΝGFΓΝΜEGD^ΜGF)YFH  (38)

            for which we use the fact that Μ and Ν are dummy indexes, and D^ΜEG and D^ΝGF commute; this implies J and M are related to terms of (ΣΜABΣΝCD(ΓΜΝEGD^ΝGFΓΝΜEGD^ΜGF)YFH)(Y1)HK, where (Y−1)HK is the inverse of YFH. The introduction of Δ^ (Equation 37) is to change from dealing with HABEG toΓΜΝEG, while this form Δ^ seems not unique.

            Let BΝEF is invertible (if considering E and F are considered as the matrix indices), that is, in GL, we may naturally define the hybrid-spinor ΓΜΝEG by

            ΓΜΝEG=(D^ΜEF(BΝFH))(BΝ1)HG.

            So it implies

            ΣΜABΣΝCD(ΓΜΝEGD^ΝGFΓΝΜEGD^ΜGF)YFH=ΣΜABΣΝCD(ΓΜΝEGBΝGFΓΝΜEGBΜGF)YFH=ΣΜABΣΝCD(D^[ΜEG(BΝ]GF))YFH=(ΣΜABD^[ΜEG(ΣΝCDBΝ]GF))YFH=(D^[ABEG(HCD]GF))YFH

            where we use core equation (Equation 17) and condition, that is, ΣΜABis D^-invariant. So, we get the int operator working as desired.

            For scalar case of D^ΜEF, we currently consider (D^ΜEFΜΔEF as previously stated):

            D^ΜEFYFH=D^ΜYEH=p^ΜiYFH  (39)

            where p^Μ is the usual quantum energy, momentum operator, while, for simplicity, pi (i=1,2,3) is negative to usual momentum; we can plug it into the following expression and get the relationship:

            ΣΑEGΣΒFHΣΜABΣΝCDi(ΓΜΝEGp^ΝΓΝΜEGp^Μ)=ΣΑEGΣΒFH(Σ[ΜABΣΝ]CD)iΓΜΝEGp^Ν~entry of GΑΒ~entry of8ΠGTΑΒ

            note that we choose to ignore the complex conjudge sign of M and detail of the indexes per entity as we just want to illustrate the possible condition to match TΑΒ (as operator-sense) of Einstein field equation.

            It motivates the definition of a hybrid-spinor density to match the unit of Einstein field equation:

            ΓΜΝEGihG[Density]  (40)

            such that Δ^ with ΓΜΝEG density generates the curvature by the source of local energy–momentum from related ordinary quantum mechanics operators (iħ∂Μ). The detail of ΣΜAB,ΣΝCD and ΓΜΝEG results in the freedom that one can assume the structure of TΑΒ as we normally solve problems in classical general relativity (demonstrated in later sections). Nevertheless, one must solve the operators and required equations in section 3.3, in order to find ΓΜΝEG density for a complete solution.

            The generalization to gauge operators for spacetime

            The gauge theory adopted in this theory is, as usual, to allow certain transformation of mathematical objects including the solution (Y here) such that same set of equations are invariant. So, if we found those gauge transformation, a new dynamic description of the curvature of spacetime exists, which we call it generalization to the gauge operators (or called fields).

            The core theory of this article naturally suggests the spinor operation over 2 × 2 matrix (the base mathematical field) instead of scalar because of core equation (Equation 17). So in this section (ONLY), the author illustrates the equations in spinor over 2 × 2 matrix, that is, core equation reads from DΜEFYFG=BΜEFYFG to DΜY=BΜY, so it is easy to see the natural simplicity.

            Gauge operators (or fields) =def operators used to define the transformation of following objects as well as Y simultaneously while keeping following equations invariant:

             Objects to transform by gauge fields: D^Μ, ΣΜAB, HAB, BΜ, J^AB

             Equations to be invariant:

             commutativity of D^:  [D^AB,D^CD]Y=0

            D^ invariant:  D^ΝΣΜABX=ΣΜABD^ΝX (ΣΜAB denotes as generalizedΣΜAB, so it is 2 × 2 matrix in general indices of A,B’, Μ)

             Equations of system (Equation 19):  D^ABY=HABY, D^ΜY=BΜY

             Integratability: [BΜ,BΝ] = D^ΜBΝD^ΝBΜ

             Γ operator (if any): ΓΜΝ=(D^ΜBΝ)BΝ1

             commutativity of J^ and M^:   [J^ABD^CD]Y=0, [M^ABD^CD]Y=0

             Fundamental equation (Equation 21): εBC(J^ABD^CD)Y=εBC(M^BDD^AC)Y or

            (J^ABD^BD)Y=(J¯DBD^BA)Y(original form)

            One can easily observe that only certain transformation is allowed because the equations above must be simultaneously satisfied. Physically, it means only certain gauge field related to transformation of curvature of spacetime is allowed. In the next section, we will exercise this generalization in certain way to apply to evolution model of the universe.

            A key symmetric structure of fundamental equation in second form (J^ABD^BD)Y=(J¯DBD^BA)Y is obvious that the author defines this in the original version. As it has Hermitian conjugate structure if considering the indices of A’D and some J¯ and D^.

            AS AN APPLICATION TO THE PROBLEM OF MODELING THE UNIVERSE EVOLUTION

            In this section, some forms of operators and solution Y are deployed such that a simple solution of equation (21) is obtained. An equation explaining the evolution model of the universe is derived; usual Friedmann equation is compared as well as density parameters of dark matter, ordinary matter, and dark energy is proposedly originated; furthermore, the prediction of density parameter of ordinary matter is compared to the observation of Planck data (Carroll & Ostlie, 2007; Density data: http://hyperphysics.phy-astr.gsu.edu/hbase/astro/denpar.html). At last, certain equations of unsolved fields are discussed for the consistency of the theory.

            Basic cosmological model

            There are some assumptions on the form of YGH, D^ΜEF, and ΓΜΝEF as well as the new gauge field – hybrid spinor ΛΜEF to the application of a cosmological model we interest:

            YGHYGH(t)=(eΑ(t)00eΑ(t)),(Α(t)is a scalar function of t)
            D^ΝEFΝΔEF+aΛΝEF, Λ0EFis tindependent,
            parameter space are t = x0, x = {x1, x2, x3}; ΛΝEF(t,x) are 2 × 2 commutating matrices (if considering E and F as the matrix indices) such that [ΛΜ, ΛΝ] = 0, and a(t) is a real scalar field for t-dependent;

            ΓΜΝEFa(t)aX(x)ΓcΜΝEF,  (41)

            where aX(x) is the real scalar field for x-dependent only, ΓcΜΝEF is a constant spinor such that

            ΓcΜΝEF=0 for Ν0,ΓcΜΝEF=ΓcΔEFis a real scalar constant ifΝ=0.
            Finally, we assume BΜEF is an invertible matrix (if considering E and F as the matrix indices).

            Let’s examinate J^EFGHIKYKL from Equation (38):

            J^EFGHIKYKL=ΣΜEFΣΝGH(ΓΜΝIGD^ΝGKΓΝΜIGD^ΜGK)YKL=(aaX(ΣΜEFΣΝGH)(a(ΓcΜΝIGΛΝGKΓcΝΜIGΛΜGK)+ΓcΜΝIKΝΓcΝΜIKΜ)YKL

            where we use the commutativity property of D^, form of D^, form of ΓΜΝEF (Equation 41), and ΣΜEF is D^-invariant. We further apply the form of ΓcΜΝEF (Equation 41) that ΓcΜΝEF=0 for Ν0 and YKL is t-dependent only:

            J^EFGHIKYKL=a2aX(x)Γc((Σ3i=1ΣiEF)Σ0GHΣ0EF(Σ3i=1ΣiGH))Λ0IK(x)YKL(t).

            Let us study the overall t-scaling (i.e., t-dependent) effect of the equation by introducing length scaling factor l from a:

            al3[Density   scaling   factor]  to   [length scaling   factor].J^EFGHIKYKL=ΓcΣEFGHaX(x)Λ0IK(x)l(t)6YKL(t),  (42)

            where a new spinor from combination of ΣΜAB:ΣEFGH(Σ3i=1ΣiEF)Σ0GHΣ0EF(Σ3i=1ΣiGH).

            Then we get J and M by J^,M^ and the inverse of YGH:

            JACEF=(J^ACEGYGH)(Y1)HF=(εBDJ^ABCDEGYGH)(Y1)HF=(εBDΓcΣABCDaX(x)Λ0EG(x)l(t)6YGH)(Y1)HF=ΣAC(ΓcaX(x)Λ0EF(x)l(t)6),MBDEF=(M^BDEGYGH)(Y1)HF=(εACJ^ABCDEGYGH)(Y1)HF=(εACΓcΣABCDaX(x)Λ0EG(x)l(t)6YGH)(Y1)HF=ΣBD(ΓcaX(x)Λ0EF(x)l(t)6),

            where ΣACεBDΓcΣABCD,ΣBDεACΓcΣABCD.

            Let’s examinate the Einstein spinor GACBD (Equation 35) based on these assumptions and the calculation above. GACBD is contributed by Λ and MABCE; Λ is contributed by JABCB Equation 32, and Einstein tensor is derived by GΜΝ=ΣΜEFΣΝGHGEFGH. The t-dependency of JACEF and MBDEF is clearly proportional to l−6. Refering to the Equation (1 and 4), ΣΜ and ΣΝ balance each others (because of the metric and its inverse) so we can neglect the overall t-scaling effect of ΣΜEFΣΝFG×ΣAC, or ΣΜEFΣΝFG×ΣBD from GΜΝ; it makes sense, as in spinor approach of GR, the curvature dynamics is reflected by Χ, Φ curvature spinors, notΣΜA,C,ΣΝE,F. One can also choose Σ0GH and ΣiEF to be the near constants to make computation easy; however, one must check the consistency of the choice with the core equations (refer to section: the generalization to gauge operators for spacetime).

            In conclusion, GΜΝ is contributed by the term

            ΓcaX(x)Λ0EF(x)l(t)6,

            and by the following definition (to be consistent with Einstein equation):

            Ordinary matter density tensor=defordinary stress-energy denisity tensor=defGΜΝ8ΠG

            the t-dependency of ordinary matter density in this case is proportional to 1l(t)6. Note that, the ordinary matter density defined here is to match Einstein equation for this model only.

            Calculation of scaling factor and comparison to Friedmann equation

            Solving Equation (21, the fundamental equation) is the core task to get the dynamics of the cosmological model, or more generally the physical structure of spacetime. The first step is to calculate the term, J^CDEFD^ABFG, which is left side of the fundamental equation. After using the form of D^, ΓΜΝEF (41) and D^-invariant, it implies:

            J^CDEFD^ABFGYGH=εFH(ΣΓABΣΜCFΣΝDH)×aaX(a2ΓcΜΝEFΛΓFIΛΝIGa2ΓcΝΜEFΛΜFIΛΓIG+aΓcΜΝEFΛΓFGΝ+aΓcΜΝEFΛΝFGΓ+aΓcΜΝEFΝ(ΛΓFG)aΓcΝΜEFΛΓFGΜaΓcΝΜEFΛΜFGΓaΓcΝΜEFΜ(ΛΓFG)+Ν(a)ΓcΜΝEFΛΓFGΜ(a)ΓcΝΜEFΛΓFG+ΓcΜΝEGΓ,ΝΓcΝΜEGΓ,Μ)YGH.

            Applying ΓcΜΝEF for Ν ≠ 0 (condition 41), YGH is t-dependent only, commuting property of ΛΜEF (if considering E and F as the matrix indices), and the following equations from later discussion (Equation 45):

            aΛiEF=ai(Λ0EF)a0(ΛiEF)aΛiEGΛ0GF=i(Λ0EF)

            for i= 1,2,3 (a′ is t-derivative of a), we get:

            J^CDEFD^ABFGYGH=aaXΓcεFH(ΣCFDH)(a2Σ0ABΛ0EFΛ0FGYGH+aΣ0ABΛ0EGYGH+2aΣ0ABΛ0EG(YGH)+aΣiABΛiEG(YGH)+aΣ0AB0(Λ0EG)YGH+Σ0AB(YEH)),

            where Y′,Y′ denote first and second t-derivatives of Y.

            By right side of fundamental equation:

            (M^BDEFD^CDFG)YGH=εDA(M^BDEFD^CAFG)YGH.

            Combing both sides, we get εDA (J^CDEFD^ABFG)YGH= εAD(M^BDEFD^C,AFG)YGH as

            εDAaaXΓcεFH(ΣCFDH)(a2Σ0ABΛ0EFΛ0FGYGH+aΣ0ABΛ0EGYGH+2a Σ0ABΛ0EG(YGH)+aΣiABΛiEG(YGH)+aΣ0AB0(Λ0EG)YGH+Σ0AB(YEH))=εDAaaXΓcεFH(ΣFBHD)(a2Σ0CAΛ0EFΛ0FGYGH+aΣ0CAΛ0EGYGH+2a Σ0CAΛ0EG(YGH)+aΣiCAΛiEG(YGH)+aΣ0CA0(Λ0EG)YGH+Σ0CA(YEH))

            Using the form of YGH=(eΑ(t)00eΑ(t)),

            differentiating it results in

            (YGH)YGHϕ,(YGH)YGH(ϕ2+ϕ),

            where Α′ ≡ ϕ. After substituting these results to the equation above, all terms are matrices operating on YGH only (if considering G,H as the matrix indices), so we can take it out as YGH is in GL. The equation becomes

            εDAaaXΓcεFH(ΣCFDH)(Σ0AB(a2Λ0EFΛ0FG+Λ0EG(a+2aϕ)+a0(Λ0EG)+(ϕ2+ϕ)ΔEG)+ΣiABa(ϕΛiEG))=εDAaaXΓcεFH(ΣFBHD)(Σ0CA(a2Λ0EFΛ0FG+Λ0EG(a+2aϕ)+a0(Λ0EG)+(ϕ2+ϕ)ΔEG)+ΣiCAa(ϕΛiEG)).

            Finally, we further assume the following conditions in order to look for a simple solution:

            {ϕΛiEG0}(ΛiEGcoupling withϕcontributing very little to the overall dynamics),

            and note that Λ0FG is t-independent. Then, there exists a solution:

            0=a2Λ0EFΛ0FG+(a+2aϕ)Λ0EG+(ϕ2+ϕ)ΔEG.

            Let’s pick an abelian matrix structure for Λ0FG (if considering F and G as the matrix indices)below:

            Λ0FG=(s(x)y(x)y(x)s(x)),

            and we have the set of differential equations for the solution:

            0=2ay(a s+ϕ)+ya0=a2 s2+a2 y2+2a sϕ+ϕ2+sa+ϕ,

            and we pick a solution that a ≠ 0:

            a=2a(as+ϕ),ϕ=a2(s2y2)ϕ2

            To get an equation to compare with usual Friedmann equation, let’s converting the solution to length scaling factor l and getting a scalar Hll:

            l=2(l3ϕ+s)3l2,H=2(l3ϕ+s)3l3

            The equation to be compared with Friedmann equation is

            (H(l,ϕ)H0)2=4 s29l6H02+8sϕ9l3H02+4ϕ29H02,  (43)

            while Friedmann equation is (spatial curvature k is assumed to be zero, Λcos is usual cosmological constant, Ωn are the usual density parameters for different types of matter):

            (H(l)H0)2=nΩnln=8ΠG3H02(nΡn(ln))+Λcos3H02

            Densities of dark energy, matter, and ordinary matter (ΩΛ, ΩM, Ωb)

            As usual in Λ CDM model, we identify the zeroth and negative power terms of scaling factor (l0, l−3 and l−6), in equation (H(l,ϕ)H0)2 (Equation 43), so there are three types of matter to be responsible for the evolution of the scaling factor, and their density parameters are namely ΩΛ, ΩM, and Ωb, respectively. The Ωb is proposedly to be responsible for ordinary matter density as the t-dependency of ordinary matter density is proportional to l−6 in preceding discussion:

            Ωb=4s29H02,ΩΛ=4ϕ29H02,ΩM=8sϕ9H02.

            ΩM24ΩbΩΛ (approximation − equal sign because of the approximation of the model)

            To compare observation results from observation data (Carroll & Ostlie, 2007; Density data: http://hyperphysics.phy-astr.gsu.edu/hbase/astro/denpar.html), ΩΛ and Ωb data are used:

            ΩΛ0.69,Ωb0.04;

            the calculated and observed ΩM are

            Calculated  ΩM=0.33        vs       Observed  ΩM=0.31(upper   bound   for   best   fit)

            The observation results of Ωb, ΩΛ, ΩM motivate the speculation for this model to claim the relationship of densities; it is similar that Maxwell speculated that electromagnetic wave is the light itself because the speed of electromagnetic wave from his equations was found accidentally the same as light speed.

            The author claims that, more accurate relationship among those sources of evolution of the universe should be manageable after certain assumption is removed such as time-independent of Λ0. Moreover, complete proof of calculation of Ωb requires particle theory, which is also not covered in this article.

            Spinor gauge field-ΛΜEF dynamics

            In preceding discussion of calculation of the solution for fundamental equation (Equation 21), a set of PDE related to ΛΜEF is used. In fact, they are the results of core constraints of the theory.

            Let us look at the commutativity of D^ after appying gauge fieldΛΜEF:

            D^A,BEFD^C,DFGYGH=D^C,DEFD^A,BFGYGH
            ,

            there exist a solution

            [ΝΔEF+aΛΝEF,Μ+aΛΜEF]YFH=0

            so([Ν(aΛΜEF)YFH+a2ΛΝEFΛΜ]FGYGH)=0;

            by ΛΜEF is abelian (if considering E and F as the matrix indices), it implies

            [Ν(aΛΜ]EG)YGH=0.  (44)

            Then, let us look at integratability of the system: B[ΜEFBΝ]FG=D^[ΜEFBΝ]FG. As ΓΜΝEF=0 for Ν ≠ 0 , and it provides D^ΜEFBΝFG by Equation (38) ΓΜΝEG=(D^ΜEF(BΝFH))(BΝ1)HG, so we get

            D^ΜEFBiFG=0,B[iEFBj]FG=0,B[0EFBi]FG=aaXΓcB0EG(for  i,j=1,2,3;Μ=0,1,2,3).

            By the core equation D^iEFYFG=Bi EFYFG, BiEF is solved by(iΔEF+aΛiEF)Y(t)FG=Bi EFY(t)FG; exist a solution

            BiEF=aΛiEF,

            it is consistent with B[iEFBj]FG=0 and ΛiEF abelian property (if considering E and F as the matrix indices). D^ΜEFBi FG=0, equation leads to:

            a  ΛjEFΛiFG+j(ΛiEG)==0a2Λ0EFΛiFG+aΛiEG+a0(ΛiEG)==0.

            Combining the above equation and Equation (44), the dynamics of gauge field ΛΜEF is got:

            (aj(ΛiEF)=ai(ΛjEF)a 0(ΛiEF)=aΛiEF+a i(Λ0EF)i(ΛjEG)=a ΛjEFΛiFGi(Λ0EG)=a ΛiEFΛ0FG)   (45)

            for i,j=1,2,3(a′ is t-derivative of a).

            Please note that the calculation of cosmological model in this section does not include solving gauge field-ΛΜEF; however, the theory requires that the dynamics of ΛΜEF must exist in order to be consistent.

            DISCUSSION

            This article provides another approach of describing the physical nature by simplicity of cross-ratio. However, the complete correspondence between cross-ratio deformation and isomonodromic integratable system used in this article is not yet covered. Potentially, additional property beyond our spacetime might emerge.

            The description of quantum mechanical wave function is also not discussed, and this is what this theory should extend to. With such description and spacetime description in this article, a more complete picture of matter and spacetime should be reviewed.

            Finally, the author proposes that the physical reality is the general deformation of same cross-ratio.

            References

            1. Carroll B. W, Ostlie D. A. 2007. An introduction to modern astrophysics. 2nd ed. Pearson Education. USA:

            2. Cassidy P, Singer M. 2005. Galois theory of parameterized differential equations and linear differential algebraic groups. arXiv:math/0502396arxiv.org

            3. Gomez-Lobo A. G.-P, Martin-Garcia J. M. 2012. Spinors: A Mathematica package for doing spinor calculus in general relativity. Computer Physics Communications. Vol. 183:2214–2225

            4. Penrose R. 1960. Chapters 1–3 of “A spinor approach to general relativity”. Annals of Physics. Vol. 10:171–201

            5. Robert M. W. General Relativity. Chicago, IL: University of Chicago Press. 1984

            6. Yoshida M. Hyper geometric functions, my love: Modular interpretations of configuration spaces. Springer Fachmedien Wiesbader (1997). Germany:

            Competing interests

            The authors declare no competing interests.

            Publishing notes

            © 2016 Liu. This work has been published open access under Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com.

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            27 August 2016
            25 September 2016
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            Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, USA
            Author notes
            [* ]Corresponding author’s e-mail address: jackieliu0@ 123456gmail.com
            Article
            3802:XE
            10.14293/S2199-1006.1.SOR-PHYS.A4HPPH.v1
            27ffb33f-f6eb-4638-8316-9f74d68ebae1
            © 2016 J. C. H. Liu

            This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .

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