Quaternionic Gauge Transformations and Yang-Mills Fields in Weyl Type Geometries
QQuaternionic Gauge Transformations and Yang-Mills fieldsin Weyl Type Geometries
J.E. Rankin a) San Miguel Road, Concord, CA (Dated: 7 October 2019)
This elementary discussion generalizes a Weyl geometry to allow quaternion valuedgauge transformations and classical Yang-Mills geometric fields. This developmentwill assume that the symmetric metric tensor is real in some gauge, and will de-velop the left and right handed approaches to quaternionic gauge transformations.Quaternionic gauge transformations are shown actually to require the shifting ofsome of Weyl’s nonmetricity into torsion to define a properly transforming gaugefield full curvature tensor, which is constructed as an asymmetric sum of left andright handed forms. Natural, gauge invariant, dimensionless variables are definedsuitable for physics, and for use as a general formalism to describe these geome-tries, including General Relativity, in rather general circumstances. The geometry“self measures” these variables. Weyl’s original action principle provides an ex-ample of an action rephrased in these gauge invariant variables, along with someunexpected possible insights on mechanics promoted by such a formulation of thataction. Those include the torsion tensor and nonmetricity being constructed frommechanical energy-momentum. The Weyl form of action is then generalized to aquaternionic gauge field. The insights on mechanics now include spin 1 / I. INTRODUCTION
This work introduces a quaternion valued generalization of the Weyl geometry which caninclude General Relativity in a natural combination with quaternionic, classical Yang-Millsfields. Since quaternions have four components, and conventional SU (2) Yang-Mills fieldsoperate with three , this is a more general Yang-Mills structure which actually containsthe more standard SU (2) case as the subset with no real component. The extension of theWeyl geometry into quaternions will be seen to require the shifting of some nonmetricityinto torsion in order to produce a nontrivial, correctly transforming, curvature tensor formassociated with the Yang-Mills gauge field.Exposition will be elementary such that anyone who can understand the original worksof Weyl and Eddington , the basics of classical Yang-Mills fields , and elementaryquaternions can follow this. Indeed, the core equations are (26), (29), (30), (33), and(34), and those can probably be previewed and understood by many readers without readingthe rest of this paper.Additionally, when the scalar curvature is nonzero, the structure itself defines intrinsic,simple, gauge invariant, dimensionless variables that make it possible to carry through anexposition relatively simply in the quaternions. The structure literally measures itself via a) [email protected] a r X i v : . [ g r- q c ] O c t these variables. i Without them, the division into right and left handed expressions, and thepresence of noncommuting, quaternionic quantities will produce cumbersome expressions,or quantities that are difficult to interpret. These gauge invariant variables will be seenalways to obey a necessary kinematic constraint imposed by the geometry. Substantialsimilarities will be found between that constraint equation, and well known equations fromclassical and quantum particle mechanics, including those of spin 1 / x µ is simply denoted by “ ,µ ”, and x µ itself will be considered to be rendered dimensionlessthrough the introduction of a universal scale factor b . That scale factor relates dimension-less Lorentzian coordinates x µ to lab unit Lorentzian coordinates x µLAB via x µ = √ b x µLAB .The constant b is an inverse length squared, one that is not assigned a specific value ini-tially, but is assumed to be definite. Ideally, an obvious value would eventually emergethrough comparison of equations to physics. II. QUATERNIONIC GAUGES AND CURVATURES IN WEYL-LIKE GEOMETRIES
In order to make sense of Weyl-like geometries allowing quaternionic gauge transforma-tions, one foundation rule is necessary. There must exist a gauge or gauges in which themetric tensor is real (with signature (+ − −− )). Such a metric is considered to be a given.That simplifies the model enough to proceed fairly easily.
A. Quaternions
In this presentation, the basic quaternions will be taken as abstract mathematical objectssimilar to the number 1 and the imaginary unit ı in complex numbers. Specifically, theyare taken to be the four quantities Q µ , where the subscript does not imply the quantitiesare a four vector, and where Q = σ (1)and Q k = − ıσ k (2)for k = 1 , ,
3, where the σ k are the standard Pauli spin matrices , and σ is the unit matrix.The basic properties of quaternions are reviewed in many references, such as Adler , andMorse and Feshbach , and there are many representations of them which may differ fromthose of equations (1) and (2), yet which are algebraically isomorphic to those quantities.Such possibly isomorphic representations will be denoted here by Q (cid:48) µ if needed, and theycould be given particular coordinate transformation properties for convenience, unlike the Q µ , which are mathematical invariants.The Q k have an obvious vector form (cid:126)Q = (cid:88) k =1 ˆ e k Q k (3) i This use of the word “measure” does not include the eventual reduction of a superposition of states intoan eigenstate. It refers only to the fact that the geometry intrinsically defines the overall quantities to beeventually measured experimentally in the laboratory. where the ˆ e k are the Cartesian unit vectors. A completely analogous form exists for the Q (cid:48) k , (cid:126)Q (cid:48) = (cid:88) k =1 ˆ e (cid:48) k Q (cid:48) k (4)although now the ˆ e (cid:48) k may be unit vectors in one of the curvilinear coordinate systems incommon use . A general quaternion A is given by A = Q A + (cid:126)Q · (cid:126)A where A is called thereal part, and the vector (cid:126)A is called the imaginary part. These quaternion forms are calledreal quaternions because A and (cid:126)A are themselves real quantities. B. Basics of the Geometry
The basic Weyl-like geometry is understood to be based on a symmetric metric tensorand a Weyl four vector combined into a more general affine connection . The reason forthe term “Weyl-like” here in place of just “Weyl” will become apparent shortly, when itwill be seen that the connection must include some torsion through an unbalancing of theoriginal Weyl connection in order for the overall geometry to be nontrivial and reasonablyneat in the quaternions.In the gauge in which the metric is real, g µν will be denoted by ˜ g µν . Tensor indices arelowered and raised using g µν , and its inverse.Since g µν is real in some gauge, it is always possible to write g µν = ˜ g µν γ (5)where γ is some quaternionic scalar. This allows the various operations with g µν to be setup through easy correspondence with operations on real forms. For example, g µν is easilycalculated through the standard requirement that g µα g αν = δ µν (6)where δ µν is the Kronecker delta. In fact, g µν = γ − ˜ g µν (7)Furthermore, equations (5) and (7) give g ατ g µν = γ − ˜ g ατ ˜ g µν γ = ˜ g ατ ˜ g µν = ˜ g µν ˜ g ατ = ˜ g µν γγ − ˜ g ατ = g µν g ατ (8)That metric combination is always real, gauge invariant, and commutes as a unit witheverything.Since gauge transformations are to be quaternionic, they can generally be applied to g µν from either the left or right, as denoted by¯ g µν = λg µν ρ (9)where λ is the left gauge transformation, and ρ is the right. However, more restricted caseswhere one of these two multipliers is always taken to be 1 will be most useful here. Sinceequation (9) can always be written as ¯ g µν = λ ˜ g µν γρ (10)according to equation (5), and since ˜ g µν commutes with every scalar multiplier, restrictingeither λ or ρ to be 1 may not be a serious restriction in practice. C. Left and Right Covariant Derivatives / Christoffel Symbols
Since g µν can now have a limited quaternionic nature, the expression g µν ; γ = g µν,γ − g αν (cid:8) αµγ (cid:3) − g µα (cid:8) ανγ (cid:3) = 0 (11)is not necessarily the same as g µν ; γ = g µν,γ − (cid:2) αµγ (cid:9) g αν − (cid:2) ανγ (cid:9) g µα = 0 (12)Thus, equation (11) defines the covariant derivative of g µν with respect to the “right handedChristoffel symbol” (cid:8) αµν (cid:3) , while equation (12) defines the covariant derivative of g µν withrespect to the “left handed Christoffel symbol” (cid:2) αµν (cid:9) . Both equations actually define theassociated Christoffel symbols, giving (cid:8) αµν (cid:3) = 12 g ατ ( g µτ,ν + g ντ,µ − g µν,τ ) (13)and (cid:2) αµν (cid:9) = 12 ( g µτ,ν + g ντ,µ − g µν,τ ) g ατ (14)respectively.Furthermore, equation (6) and δ µν,τ = 0 give g βµ,τ = − g βν g να,τ g αµ (15)This and equations (8), (11), and (12) then give g βξ ; τ ≡ g βξ,τ + (cid:8) βατ (cid:3) g αξ + (cid:8) ξατ (cid:3) g βα = 0 (16)and g βξ ; τ ≡ g βξ,τ + g αξ (cid:2) βατ (cid:9) + g βα (cid:2) ξατ (cid:9) = 0 (17)This indicates that the contravariant metric’s indices interact with their associated Christof-fel symbols on the opposite side from the covariant metric’s indices in the definition of thecovariant derivative of the metric. That same positioning convention for covariant andcontravariant indices and their associated connection is now adopted here for all covariantderivatives of any tensor quantity as well, and also for more general affine derivatives, wherethe affine derivative simply uses the full affine connections which also include the Weyl fourvector , R Γ αµν and L Γ αµν in place of the Christoffel symbols (cid:8) αµν (cid:3) and (cid:2) αµν (cid:9) . The moregeneral affine connections are prefixed with the lowered “R” and “L” to maintain differentright and left handed forms on the more general level, just as there are right and left handedChristoffel symbols, and superficially they differ only in their Christoffel portion. They arediscussed in more detail shortly.For completeness, now define the “ ˜; ” derivative as the reversal of the convention justgiven for the “ ; ” derivative. That means that all the tensor - Christoffel symbol positionsare reversed for each term for each tensor index in the covariant derivative expression. Forexample, g µν ˜ ; γ = g µν,γ − (cid:8) αµγ (cid:3) g αν − (cid:8) ανγ (cid:3) g µα ( (cid:54) = 0) (18)By definition, the Christoffel symbols are always defined using a “normal” covariant deriva-tive of the metric tensor, “ ; ”. Clearly, tensor - Christoffel symbol positions in theseexpressions are determined both by the type of Christoffel symbol (“ { ]” or “[ } ”), and theuse of “ ; ” or “ ˜; ” in the derivative.With these facts established, the left handed Christoffel symbols and more general affineconnection L Γ αµν will now be arbitrarily dropped, and the right handed cases used in whatfollows. However, it should also be noted that any action principle in this extended structurewill eventually involve an added quaternion conjugate (“ QC ”) term to keep the action real. ii That quaternion conjugate term will tend to involve left handed forms to balance the righthanded forms that are now being chosen in the first part of the action. Because of that,the basic left hand should not be suppressed or shortchanged overall, although a somewhatdifferent type of right or left handedness will also arise at the next level to be examined.There, the right and left handed forms will be found necessarily to enter asymmetrically,but they should still tend to be balanced overall by corresponding opposite handed formsin the QC terms in the action.For general quaternionic M µ and N ν ,( M µ N ν ) ; τ (cid:54) = M µ ; τ N ν + M µ N ν ; τ (19)Additionally, contraction on tensor indices inside an already evaluated covariant derivativewill not necessarily equal the covariant derivative of the contracted quantity. Examplessuch as these limit the usefulness of these gauge varying, generalized covariant derivativesoutside of gauges in which quantities commute easily in products. However, in other cases,these covariant derivatives will still be helpful, and will be used. On the other hand, anygenuine physics in this structure will require gauge invariant constructions, including gaugeinvariant covariant derivatives. Those will be developed later, and they will involve real,gauge invariant Christoffel symbols that thus avoid these limitations just noted. D. Weyl-Like Connections, Gauge Properties, and Curvatures
Since right handed forms are now chosen, such as equation (13), specialize equation (9)to λ = 1, or ¯ g µν = g µν ρ (20)Corresponding to that, ¯ g µν = ρ − g µν (21)These together with equation (13) then give that¯ { αµν ] = ρ − { αµν ] ρ + δ αµ ρ − ρ ,ν + δ αν ρ − ρ ,µ − ρ − g ατ g µν ρ ,τ = ρ − { αµν ] ρ + δ αµ ρ − ρ ,ν + δ αν ρ − ρ ,µ − g ατ g µν ρ − ρ ,τ (22)where the second equation follows from equation (8).Weyl’s original theory would now suggest an affine connection R Γ αµν (hereafter denotedby Γ αµν ) Γ αµν = { αµν ] + δ αµ v ν + δ αν v µ − g ατ g µν v τ (23)where v µ is the Weyl four vector, and¯ v µ = ρ − (cid:0) v µ − ρ ,µ ρ − (cid:1) ρ = ρ − v µ ρ − ρ − ρ ,µ (24)These give ¯Γ αµν = ρ − Γ αµν ρ (25) ii If a quaternion is written as a 2 × QC term (the transpose of the complex conjugate)gives the same result as taking the trace of the matrix, which is what is done in Yang-Mills theoryactions . as the quaternionic analog of the gauge invariance of the Weyl connection in his originalgeometry.Equation (24) will indeed be adopted, but in place of equations (25) and (23), adopt¯Γ αµν = ρ − Γ αµν ρ + kδ αµ ρ − ρ ,ν (26)where k is a real constant, andΓ αµν = { αµν ] + nδ αµ v ν + δ αν v µ − g ατ g µν v τ (27)Here n is also a real constant, and k = 1 − n n (cid:54) = 1, some amount of torsion appears, and furthermore, n = 0 gives a vanishing affinederivitive for the metric, a “metric compatible” case. Note that the cases n (cid:54) = 1 producea form of Einstein’s “lambda transformation” of the connection in equation (26), whichas he notes, leaves the curvature tensor invariant when the curvature tensor and lambdatransformation involve real (or complex) quantities.Now both of the connections Γ αµν and { αµν ] have associated curvature tensors, but thosehave both a “right handed” and a “left handed” form themselves corresponding to use ofnormal or reversed covariant derivative conventions such as those used in “ ; ” or “ ˜; ”,irrespective of the right-left nature of the underlying connection used in them. Specifically, R B γµτσ = Γ γµσ,τ − Γ γµτ,σ + Γ γητ Γ ηµσ − Γ γησ Γ ηµτ (29) L B γµτσ = Γ γµσ,τ − Γ γµτ,σ + Γ ηµσ Γ γητ − Γ ηµτ Γ γησ (30) R R γµτσ = { γµσ ] ,τ − { γµτ ] ,σ + { γητ ] { ηµσ ] − { γησ ] { ηµτ ] (31)and L R γµτσ = { γµσ ] ,τ − { γµτ ] ,σ + { ηµσ ] { γητ ] − { ηµτ ] { γησ ] (32)When equation (26) is substituted into equations (29) and (30) to obtain the gaugeproperties of those curvature tensors, generally neither curvature form transforms in aparticularly neat manner by itself. Surprisingly however, the combination B γµτσ = k + 12 k R B γµτσ + k − k L B γµτσ (33) does have neat gauge transformation properties. iii Specifically¯ B γµτσ = ρ − B γµτσ ρ (34)which is basically the same form as the gauge transformation of a Yang-Mills field in SU (2)gauge theory .Furthermore, for real (or complex) quantities instead of quaternionic quantities, productslike Γ γητ Γ ηµσ commute internally, and B γµτσ clearly reduces to the usual curvature tensorform, and becomes gauge invariant like Weyl’s curvature tensor . Thus, it becomes theappropriate generalization of the curvature tensor in the quaternions. To facilitate its use iii Reverse roles of R B γµτσ and L B γµτσ if left handed forms with ρ = 1 and λ (cid:54) = 1 are adopted initially ratherthan right. in what follows, the coefficients in the definition of this tensor in equation (33) are giventheir own symbols, k + = k + 12 k (35)and k − = k − k (36)Note that k + + k − = 1, and k + − k − = 1 /k .Now note that if one attempts to use a full Weyl connection analog by setting n = 1 inequation (27), that causes k to become zero in equation (26), and no suitable generalizedcurvature B γµτσ emerges at all. Rather, as k approaches 0 in equation (33), the coefficientsof R B γµτσ and L B γµτσ approach equal but opposite infinite values. Essentially, equation (33)must then be replaced by B γµτσ = R B γµτσ − L B γµτσ (37)or any simple multiple of the right side of this equation, but that will lead to the dis-appearance of the derivatives of Γ αµν from the result. Without the derivative terms, thisquantity cannot reduce to anything at all like the curvature tensor of a Weyl geometrywhen quantities commute, reducing to zero instead. In other words, this structure actuallydiscriminates against the exact quaternionic analog of Weyl’s original theory , and favorsthe cases which have some torsion. This is one primary reason the original Weyl connectionmust be unbalanced in order to generalize to the quaternions. Clearly n (cid:54) = 1 is necessaryfor a nontrivial structure.Finally, the four vector v µ of equation (24) has its own directly associated Yang-Millsfield tensor . The gauge properties give that y µν = v ν,µ − v µ,ν + 2( v ν v µ − v µ v ν ) (38)gauge transforms just as B γµτσ does, or¯ y µν = ρ − y µν ρ (39)Comparison of the transformation rules of equation (24) and the corresponding equationin Guidry then relates his A µ to v µ via v µ = − ı g A µ (40)where g is the coupling constant, and the − ı is absorbed into the σ k of equation (2) thatare embedded within the A µ , producing the imaginary basis quaternions Q k in their place.Additionally, his U = ρ − to complete the matchup with his SU (2) Yang-Mills theory. iv Asa bonus, one sees that the absorption of ( g/
2) into A µ is what renders it dimensionless (ifit is not already dimensionless), and suitable for this structure. E. The Makeup of the Curvature Tensor
Equation (27) can be written Γ αµν = { αµν ] + U αµν (41) iv Guidry’s potential is totally imaginary in the quaternions, and as long as the gauge transformation ρ isunitary, or a real, nonzero constant times a unitary transformation, Guidry’s potential remains totallyimaginary. where U αµν = nδ αµ v ν + δ αν v µ − g ατ g µν v τ (42)Substituting these into equations (29) and (30) then gives the surprisingly neat results R B γµτσ = R R γµτσ + U γµσ ; τ − U γµτ ; σ + U γητ U ηµσ − U γησ U ηµτ (43)and L B γµτσ = L R γµτσ + U γµσ ˜ ; τ − U γµτ ˜ ; σ + U ηµσ U γητ − U ηµτ U γησ (44)These equations are one example (perhaps the best) in which the “ ; ” and “ ˜; ” covariantderivatives give results that are both compact, and express useful information.However, in order to proceed further with the evaluation of B γµτσ via equations (33), (43),and (44), the covariant derivatives should be written out as partial derivatives and Christof-fel symbol terms, and equation (42) should be substituted into the result. Furthermore,any resulting partial derivatives of g µν or g µν should be evaluated using equations (11) and(16) to substitute terms with Christoffel symbols in place of the partial derivative terms. Inpractice, the combination ( g γη g µσ ) ,τ always appears as a unit, and can be eliminated using( g γη g µσ ) ,τ = g γη g ασ { αµτ ] + g γη g µα { αστ ] −{ γατ ] g αη g µσ − { ηατ ] g γα g µσ (45)keeping equation (8) in mind for the result. The fact that the g γη g µσ terms are real thenallows equation (45) to have more than one valid form simply by varying the position ofsuch terms in its products. However, the same form should consistently be chosen internallythroughout evaluation of either one of the separate tensors in the pair R B γµτσ or L B γµτσ toavoid possibly encountering extraneous terms that should evaluate to zero with some effort,but are more easily avoided from the outset. v Additionally, the full expression of equation(45) itself should be real, and could be moved around as a unit in products in its containingequation if necessary. However, all this flexibility leads to more than one expansion of B γµτσ in gauge varying quantities like v µ , although all the expansions are equivalent, and all willlead to the same, unique, gauge invariant result in what follows. Since the gauge invariantresult contains any real physics, its uniqueness is what is important. v Extraneous terms can be a problem particularly when verifying overall gauge transformation propertiesand complete gauge balancing of the sum of all terms after the substitutions.
The result of the above substitutions and expansions gives B γµτσ = k + R R γµτσ + k − L R γµτσ + ( k + v µ ; τ + k − v µ ˜ ; τ ) δ γσ − ( k + v µ ; σ + k − v µ ˜ ; σ ) δ γτ − ( k + v η ; τ + k − v η ˜ ; τ ) g ηγ g µσ + ( k + v η ; σ + k − v η ˜ ; σ ) g ηγ g µτ + 1 k ([ v η , { γατ ] ] g ηα g µσ − [ v η , { γασ ] ] g ηα g µτ )+ ( k + v σ v µ + k − v µ v σ ) δ γτ − ( k + v τ v µ + k − v µ v τ ) δ γσ − v η v β g ηβ ( g µσ δ γτ − g µτ δ γσ )+ ( k + v α v τ + k − v τ v α ) g αγ g µσ − ( k + v α v σ + k − v σ v α ) g αγ g µτ + n [ k + ( v σ ; τ − v τ ; σ ) + k − ( v σ ˜ ; τ − v τ ˜ ; σ )] δ γµ − nk (cid:0)(cid:2) v σ , { γµτ ] (cid:3) − (cid:2) v τ , { γµσ ] (cid:3)(cid:1) − n k ( v σ v τ − v τ v σ ) δ γµ − nk [( v µ v τ − v τ v µ ) δ γσ − ( v µ v σ − v σ v µ ) δ γτ ]+ nk ( v α v τ − v τ v α ) g αγ g µσ − nk ( v α v σ − v σ v α ) g αγ g µτ (46)where the “[ , ]” terms are conventional commutators. Those commutators will clearlyvanish in gauges in which { γασ ] is real.Now B γµτσ can be contracted to give B µτ = B ωµτω (47)and clearly equation (34) gives that ¯ B µτ = ρ − B µτ ρ (48)The similarity between this equation and equation (39) might then raise expectations thatthe antisymmetric part of B µτ will be proportional to y µτ once some method is found forcontracting the right side of equation (46) to a neat expression. However, this will generallynot be quite true. A check reveals that the antisymmetric part of k + R R ωµτω + k − L R ωµτω equals { − [1 / (4 k )] } [ γ − γ ,µ , γ − γ ,τ ] where γ is the gauge function in g µν = ˜ g µν γ . Since thiscommutator does not generally vanish, then the antisymmetric part of k + R R ωµτω + k − L R ωµτω is generally not zero, and that antisymmetric component must be gauge balanced elsewhereby antisymmetric terms, even though it vanishes in gauges in which g µν is real, and alsowhen γ remains in the complex plane. However, the obvious exception is the case k = 1 / k corresponds to n = 1 /
2. The consequences of all this will become clearer in the nextsection where a simple method is developed to express contractions of the expansion of B γµτσ given by equation (46).Finally define the scalar curvature B = B µτ g µτ (49)Since ¯ g µτ = ρ − g µτ (50)0then equations (48) and (49) give ¯ B = ρ − B (51)Thus B is the key quantity needed to define gauge invariant variables. As conceived by Weyland Eddington , it is basically an intrinsic yardstick provided by the spacetime structureitself to reduce equations to dimensionless, gauge invariant quantities that can correspondto actual physics. It allows the structure to measure itself. For this purpose, it is assumedto be nonzero. F. Gauge Invariant Variables and Their Fundamental Identity
Define ˆ g µν = g µν ( B/C ) (52)and its inverse ˆ g µν = CB − g µν (53)where C is a constant to be explained shortly. Using equations (20) and (51), these areseen to have the same value in all gauges. They are gauge invariant forms of the metric andits inverse. As such, they are immediately assumed to be real, giving a real metric tensorthat can be used in physics. The constant C is necessary if all quantities are real, and thescalar curvature B < g µν and ˆ g µν from beingopposite, C = − vi Thus having established its right to exist, C isretained as a (real, dimensionless) constant in general.There is a gauge invariant ˆ { αµν } based on the real ˆ g µν , and it is real,ˆ { αµν } = 12 ˆ g ατ (ˆ g µτ,ν + ˆ g ντ,µ − ˆ g µν,τ ) (54)without a right-left nature any longer. The covariant derivative with respect to it is indicatedby “ (cid:107) ”, and it is now quite well behaved, including obeying the product rule since ˆ { αµν } commutes with everything. Both the “ ˜; ” and the “ ; ” covariant derivative conventionswill reduce to it. If equations (52) and (53) are substituted into equation (54), the resultgives ˆ { αµν } = B − { αµν ] B + δ αµ B − B ,ν + δ αν B − B ,µ − B − g ατ g µν B ,τ = B − { αµν ] B + δ αµ B − B ,ν + δ αν B − B ,µ − g ατ g µν B − B ,τ (55)These real, commuting Christoffel symbols now give us a normal, real, gauge invariantRiemannian geometry on which we can impose a form of General Relativity. They define aRiemann curvature tensor ˆ R γµτσ , and both the conventions of R R γµτσ in equation (31), andof L R γµτσ in equation (32) reduce to it. There is a (now symmetric) ˆ R µτ = ˆ R ωµτω , and ascalar ˆ R = ˆ g µτ ˆ R µτ . vi My thanks to Daniel Galehouse for correctly insisting that cases with negative values of B , thus implyingnegative C , are legitimate . v µ = B − (cid:0) v µ − B ,µ B − (cid:1) B = B − v µ B − B − B ,µ (56)which is fully quaternionic generally. Then in analogy to equation (27), define the gaugeinvariant ˆΓ αµν = ˆ { αµν } + nδ αµ ˆ v ν + δ αν ˆ v µ − ˆ g ατ ˆ g µν ˆ v τ (57)Note that since ˆΓ αµν is fully quaternionic, the full affine derivative of a quantity using ˆΓ αµν is not as well behaved as the covariant derivative using only the real ˆ { αµν } .Now substituting into equation (57) from equations (55) and (56), and using equation(27), one sees ˆΓ αµν = B − Γ αµν B + kδ αµ B − B ,ν (58)But this is exactly the same form as a gauge transformation on Γ αµν as defined in equation(26). Thus, if one defines R ˆ B γµτσ and L ˆ B γµτσ using ˆΓ αµν in full analogy to the use of Γ αµν in R B γµτσ and L B γµτσ of equations (29) and (30), the result gives finally thatˆ B γµτσ = k + R ˆ B γµτσ + k − L ˆ B γµτσ = B − B γµτσ B (59)This can be expanded just like equation (46), but now with so many quantities real, themuch simpler result isˆ B γµτσ = ˆ R γµτσ + ˆ v µ (cid:107) τ δ γσ − ˆ v µ (cid:107) σ δ γτ − ˆ v η (cid:107) τ ˆ g ηγ ˆ g µσ + ˆ v η (cid:107) σ ˆ g ηγ ˆ g µτ + ( k + ˆ v σ ˆ v µ + k − ˆ v µ ˆ v σ ) δ γτ − ( k + ˆ v τ ˆ v µ + k − ˆ v µ ˆ v τ ) δ γσ − ˆ v η ˆ v β ˆ g ηβ (ˆ g µσ δ γτ − ˆ g µτ δ γσ )+ ( k + ˆ v α ˆ v τ + k − ˆ v τ ˆ v α ) ˆ g αγ ˆ g µσ − ( k + ˆ v α ˆ v σ + k − ˆ v σ ˆ v α ) ˆ g αγ ˆ g µτ + n (cid:0) ˆ v σ (cid:107) τ − ˆ v τ (cid:107) σ (cid:1) δ γµ − n k (ˆ v σ ˆ v τ − ˆ v τ ˆ v σ ) δ γµ − nk [(ˆ v µ ˆ v τ − ˆ v τ ˆ v µ ) δ γσ − (ˆ v µ ˆ v σ − ˆ v σ ˆ v µ ) δ γτ ]+ nk (ˆ v α ˆ v τ − ˆ v τ ˆ v α ) ˆ g αγ ˆ g µσ − nk (ˆ v α ˆ v σ − ˆ v σ ˆ v α ) ˆ g αγ ˆ g µτ (60)Much of the right-left distinction of equation (46), along with the commutators, is nowgone. The main left-right distinction remaining is in the terms involving products of ˆ v µ ,because that quantity is fully quaternionic still.2Now using equations (28), (35), and (36), equation (60) and equation (59) then contractto give ˆ B µτ = ˆ B ωµτω = B − B µτ B = ˆ R µτ + (cid:0) ˆ v µ (cid:107) τ + ˆ v τ (cid:107) µ (cid:1) + ˆ v α (cid:107) α ˆ g µτ − (ˆ v µ ˆ v τ + ˆ v τ ˆ v µ ) + 2ˆ v α ˆ v α ˆ g µτ + (1 + n ) (cid:0) ˆ v µ (cid:107) τ − ˆ v τ (cid:107) µ (cid:1) + 4 − n − n − n (ˆ v µ ˆ v τ − ˆ v τ ˆ v µ ) (61)where the symmetric and antisymmetric parts have been clearly separated with the antisym-metric part all on the last two lines. Because ˆ v µ (cid:107) τ − ˆ v τ (cid:107) µ = ˆ v µ,τ − ˆ v τ,µ , that antisymmetricpart is − ˆ w µτ = (1 + n ) (cid:0) ˆ v µ (cid:107) τ − ˆ v τ (cid:107) µ (cid:1) + 4 − n − n − n (ˆ v µ ˆ v τ − ˆ v τ ˆ v µ )= (1 + n ) (ˆ v µ,τ − ˆ v τ,µ )+ 4 − n − n − n (ˆ v µ ˆ v τ − ˆ v τ ˆ v µ )= − (1 + n ) [ˆ v τ,µ − ˆ v µ,τ + 2 (ˆ v τ ˆ v µ − ˆ v µ ˆ v τ )+ 2 − n − n (ˆ v τ ˆ v µ − ˆ v µ ˆ v τ ) (cid:21) = − (1 + n ) (cid:20) ˆ y µτ + 2 − n − n (ˆ v τ ˆ v µ − ˆ v µ ˆ v τ ) (cid:21) (62)where ˆ y µτ = ˆ v τ,µ − ˆ v µ,τ + 2(ˆ v τ ˆ v µ − ˆ v µ ˆ v τ )= B − y µτ B (63)by equations (38) and (39), since equation (56) has the same form as the gauge trans-formation of v µ in equation (24). Additionally, the distinct alternate contraction of ˆ B γµτσ gives ˆ B γγτσ = 4 (cid:20) n ˆ y τσ + 1 − n − n (ˆ v σ ˆ v τ − ˆ v τ ˆ v σ ) (cid:21) (64)This contraction of the curvature tensor is also important in Weyl’s original theory .We have ˆ w µν = (1 + n ) (cid:20) ˆ y µν + 2 − n − n (ˆ v ν ˆ v µ − ˆ v µ ˆ v ν ) (cid:21) (65)For n (cid:54) = 1 /
2, the quantity ˆ w µν appears to have a tail on it in addition to ˆ y µν . This wasanticipated above when the antisymmetric part of k + R R ωµτω + k − L R ωµτω was noted to requireadditional antisymmetric terms to gauge balance it unless n = 1 /
2. This is the form thoseextra terms take in ˆ w µν .Given that B = B µτ g µτ , ˆ g µτ = CB − g µτ , and ˆ B µτ = B − B µτ B , thenˆ B = ˆ B µτ ˆ g µτ = CB − B µτ g µτ = C (66)This is a fundamental, kinematic identity the gauge invariant variables must satisfy byvirtue of their definitions, and the geometry’s kinematics. Substituting from equations3(61), and (62) for the expansion of ˆ B µτ , equation (66) becomesˆ R + 6ˆ v µ (cid:107) µ + 6ˆ v µ ˆ v µ = C (67)As an additional point that will be of use later, note that if all the derivatives ( ,µ ) andcoordinates in this result are reexpressed using (locally Lorentzian) standard lab coordinates x µLAB = x µ / √ b which have standard dimensions or units, then equation (67) becomesˆ R + 6ˆ v µ (cid:107) µ + 6ˆ v µ ˆ v µ = b C (68)The second term in equation (56) is the most useful one in understanding how ˆ v µ is affectedby this reversion to lab coordinates with dimensions.Obviously the case n = 1 / w µν in equation (65).Furthermore, for n = 1 /
2, and only for this value of n , ˆ B γγτσ given by equation (64) isproportional to ˆ w τσ given by equation (62). That proportionality is a property that is truein Weyl’s original theory , so the case n = 1 / n = 1, a value not allowed in this model). One canfairly say that the n = 1 / SU (2) Yang-Mills field (withoptional additional real part) as a unique antisymmetric tensor in the structure, therebyunifying it with the framework of Einstein’s Riemannian General Relativity naturally.The cost of all these simplifications introduced by choosing n = 1 /
2, is that the structurenow has an equal mix of Weyl’s nonmetricity with torsion, rather than insisting on justnonmetricity or torsion alone. That may seem unusual for a model in which the non-Riemannian behavior is primarily based on a Weyl-like four vector. Nevertheless, it doesachieve a notable reduction in the complexity of the results. It’s interesting to see thatquaternionic curvatures in this model seem not only to reject the quaternionic generalizationof the pure Weyl model, as noted after equation (37), but that they also preferentially selectthis case with an equal balance of torsion and nonmetricity. That preference is expressedby the overall simplicity of this case. No such preferential selection between torsion andnonmetricity would appear with purely real or complex gauges and curvatures.However, note that it is also true that effective nonmetricity may not vanish even when n = 0, even though the full affine derivative of ˆ g µν using connection ˆΓ αµν vanishes then,implying metric compatibility. This effective nonmetricity can be seen by looking at equa-tion (64) in the n = 0 case, and noting that the change in length of a vector transportedusing the full affine connection around a closed loop involves this quantity . This may stillbe nonzero even in the n = 0 case here, because ˆ v µ is fully quaternionic. Thus, it appearsthat there may be no quaternionic models in this family which are completely devoid of allaspects of nonmetricity. This result appears to follow from the fact that covariant and con-travariant vectors interact with the affine connection on opposite sides of the (quaternionic)connection, and the length of a vector is a contraction of a covariant and a contravariantvector. To put this another way, the affine derivative using the full ˆΓ αµν , no longer obeys theproduct rule of differentiation because ˆΓ αµν is quaternionic, not real. Thus, the calculationsof Weyl and Eddington which would give the change in a parallel transported vector’slength around a closed loop in terms of the affine derivative of the metric (which vanishesgiven metric compatibility), would no longer be completely valid. III. THE ORIGINAL, REAL VARIABLE WEYL ACTION
At this point, the kinematical framework of a quaternionic Weyl-like geometry is in place.The use of gauge invariant variables allows definition of a real metric tensor suitable forGeneral Relativity, and also produces an SU (2) Yang-Mills field, with an added possiblereal component as well, since the quaternions also have a real component. Quaternions arevisualized as a four dimensional Euclidian space which is SU (2) × SU (2), or SO (4).4Beyond this, the framework is so far a general, blank slate, since an action principle shoulddefine a particular dynamics to proceed further. In that regard, it is perfectly legitimateto phrase the action in terms of the gauge invariant variables, and indeed, it may be theonly easy way to formulate an action without having to keep track of left and right handedparts, and other complications. However, if the gauge invariant variables are used, equation(67) must be included in the action as a constraint with a Lagrange multiplier, since thegauge invariant variables are not independent. That leads to some surprises.To best illustrate this, if everything is restricted to the real numbers, and the original Weylaction is translated into gauge invariant variables , then generalized slightly, it becomesfor this structure I = (cid:90) (cid:112) − ˆ g (cid:20)(cid:16) ˆ R − σ (cid:17) − j (ˆ y µν ˆ y µν ) + 6ˆ v µ ˆ v µ + (cid:16) ˆ R + 6ˆ v µ (cid:107) µ + 6ˆ v µ ˆ v µ − C (cid:17) ˆ β (cid:105) d x (69)where j is a dimensionless gravitational constant, and ˆ β is the Lagrange multiplier. Thefact that this is a modified, unbalanced Weyl structure does not affect this form. Thecontraction of the gauge invariant torsion with itself has the value (3 / v µ ˆ v µ , but additionalterms based separately on that are not included here. For σ = and C = 1, this is theoriginal Weyl action , a fact which also can be seen by substituting for ˆ R in it everywhereby using the constraint, discarding any total divergence terms, and expanding survivingterms into the unhatted Weyl variables. The Yang-Mills tensor reduces to the curl of theWeyl vector here, or essentially what Weyl called the electromagnetic field. In practice,the constraint of equation (67) is used to determine B , given ˆ g µν , and a value of v µ in anyparticular gauge .Among other results, vii this action will give that ˆ β = 1 − [(4 σ ) /C ], and that the elec-tromagnetic four current is proportional to ˆ v µ , thus giving ˆ v µ (cid:107) µ = 0 from conservation ofcharge. In the original Weyl case, ˆ β = 0, and there is no contribution to the stress tensorfrom the constraint terms. Because ˆ v µ = v µ − [(1 /
2) ln B ] ,µ here, if C − ˆ R ≈ C , the con-straint itself clearly takes the form of the Hamilton-Jacobi equation for a nonzero rest masscharged particle in the combined gravitational and electromagnetic field . The vanishingof the covariant divergence of the stress tensor required by Einstein’s equations (using the“ (cid:107) ” derivative), also gives results consistent with this interpretation, but with a twist notedbelow.In order to make sense of the large dimensionless cosmological constant of 1 /
4, it appearsthe scale factor b = Λ /σ where Λ is the usual cosmological constant in laboratory units.Thus, it seems b must be tiny to make sense of the original Weyl action, viii although a muchsmaller value for σ would seem to relieve that difficulty, while only deviating slightly fromWeyl’s original action . However, σ is eventually found to be irrelevant for this action,as will be noted in comments on σ at the end of this section. The bigger factor here isthat if the charge in the coupling constant is electronic, the 6ˆ v µ ˆ v µ in the action with nogravitational constant in front of it will then restrict the value of b to be near Planck scalevalues. That assumes 6ˆ v µ ˆ v µ should be the same order of magnitude as the j (ˆ y µν ˆ y µν ) termin the action, where j = (cid:0) /g (cid:1) G (cid:0) b /c (cid:1) (70)Here, G is Newton’s constant, and for standard electromagnetism coupled through an elec-tronic charge, g = (2 e ) / ( (cid:126) c ) in Gaussian units, and A µ = (Φ , − (cid:126)A ) . This seems to be amore likely assumption for the value of b . vii Full details are available in section V by simply retricting the results there to be real, reversing the signused there for j , and setting τ = 1. viii Actually, Weyl’s original treatment uses coordinates that still have dimensions when taking derivatives,introducing possible confusion on scales. v µ ) moves like a particle whose charge is fixed bythe coupling constant absorbed into v µ in equation (40), and whose rest mass is that of therest mass term in the Hamilton-Jacobi equation in the constraint. That rest mass term isnear Planck scale rest mass for near Planck scale b , unless the ˆ R term is large enough tochange that. Evidently, the sign of the charge is effectively reversed by reversing the signof the energy (four vector) in the Hamilton-Jacobi equation. In fact, the normal Lorentzforce law seems to correspond to negative energy solutions, oddly enough. Otherwise acheck shows the sign would be wrong for normal Lorentz force when using the result fromthe vanishing of the covariant divergence of the stress tensor mentioned above.However, this action is dependent upon the term 6ˆ v µ ˆ v µ outside the constraint to achievethose neat results, ix and that term is unusual. If it is omitted (or if 16 times the contractionof the torsion with itself, is subtracted from the action density), then that is equivalent toinjecting the negative of this term times B into the original Weyl action, and expandingthat into the gauge dependent Weyl variables. That’s then a higher order action, and itgenerally produces (via the constraint term contributions in the gauge invariant variablesaction) both positive energy density fields, plus additional, ghost, negative energy densityfields x with identical equations of motion to those of the positive energy density fields, xi butwith opposite signed contributions to both the stress tensor and the current four vector .On the one hand, the singularity theorems that afflict classical General Relativity thenno longer need apply, but on the other hand, it may be an understatement to say thatthis is uncommon , even if some papers are exploring negative energy density fields . Aperhaps more intriguing result is that the constraint now takes on the form of the Klein-Gordon equation from wave mechanics in an electromagnetic and gravitational field (bysubstituting B = ψ − ), with the standard added conformal term and a nonzero rest mass,rather than a Hamilton-Jacobi type form. Correctly formed wave function stress tensorand four current terms also appear, and appear consistent with atomic scale Klein-Gordonequation physics if b is set using atomic instead of Planck level scales. Furthermore withan added self dual, antisymmetric part to the metric, the second order Dirac equation formcan be seen, and in quaternions, there are enough correct degrees of freedom to correspondto a two row, complex spinor . Additionally, in a little known paper, Soviet physicistYuri Usachev shows that the Dirac spinor can also be successfully treated as a scalar (whichcould be consistent with a quaternionic scalar), and yet still represent spin 1 / . These points will be elaborated further below as part of the results from a fullyquaternionic action.Before continuing to that quaternionic action, more could be said here about the abovemodel resulting from Weyl’s original action principle. For example, Maxwell’s equations inˆ v µ seem to be a modified Proca equation with nonlinear modifications introduced from theRicci tensor. Additionally, the C term in the constraint effectively renders the value of σ almost irrelevant because it allows σ to be eliminated from the gravitational equations witha little effort. Moreover, because some of the terms in the stress tensor have no gravitationalconstant as a factor in their value, it’s not clear if the approximation C − ˆ R ≈ C is reallyvalid in many cases, although if C − ˆ R is greater than zero and approximately constant,that would not significantly modify the above points. However, the primary purpose ofthis entire example has been to provide a simple introductory illustration of some of thebasic features of actions formed from the gauge invariant variables, not to do a detailedexamination of this limited, although historically important particular case. Therefore,that case will not be detailed further here. ix The terms ˆ R + 6ˆ v µ ˆ v µ are essentially just ˆ B µν ˆ g µν , ignoring a noncontributing, divergence term. x My thanks to Jim Wheeler for making these observations to me privately. xi There are subtle differences in higher order terms however, once ˆ R is replaced with its value. IV. COMMENT ON TORSION AND TRANSLATIONS
The presence of gauge potentials in torsion has been criticized on general grounds asbeing contrary to the geometric relation between torsion and translations, and thus toenergy-momentum . The model above achieves its generalization of Weyl’s geometry intothe quaternions in a neat manner by necessarily shifting some of Weyl’s nonmetricity intotorsion, and so appears to conflict with this general criticism. This requires an additionalcomment.As Einstein noted, torsion is not invariant under his “lambda transformation” , whichis essentially a treatment of the affine connection as a gauge potential rather than as thegauge field that would be the case in equation (25). This difference is crucial to generalizingWeyl’s structure into quaternions. Thus it seems odd that a gauge potential could not bepart of the torsion to correspond to this fact.Fortunately, a straightforward resolution of this conflict is already suggested in sectionIII using the simple real variable action given in equation (69), and its resulting dynamics.As noted there, under simple assumptions, the constraint then becomes the mathemati-cal Hamilton-Jacobi equation for a (nonzero rest mass) charged particle in the combinedelectromagnetic and gravitational fields. A moment’s reflection then reveals that the gaugeinvariant potential ˆ v µ is essentially the mechanical portion of the four momentum of the“particle” . Thus the gauge invariant torsion will be a three index tensor antisymmetricin the two covariant indices, directly constructed from that four momentum, and the Kro-necker delta, times a constant. Then the torsion is indeed related to energy-momentum,and that energy-momentum also appears as a source of the gravitational field through thestress tensor. This would seem to resolve the conflict in a novel manner, one which actuallyreaffirms the criteria specified by Hehl and Gronwald in their criticism . The spirit ofthat observation survives when other actions are chosen , such as in the next section, soit seems likely to be a more general property of this kinematical structure. V. A QUATERNIONIC ACTION
Weyl’s original, real action provides a quick survey of the mathematics of this structurewith its use of gauge invariant variables and their associated constraint in the action. How-ever, a much more general, quaternionic action is needed to discover how a fully quaternionicstructure affects that mathematics. Now the action of equation (69) was not just a simpleextension of a standard action for gravitation and electromagnetism because it containedan additional 6ˆ v µ ˆ v µ term. As noted then, if 16 times the contraction of the torsion withitself is subtracted from that action density, then those extra terms are removed. However,if instead 16 κ times the contraction of the torsion with itself is subtracted, then the actioncontains an additional term 6 τ ˆ v µ ˆ v µ where τ = 1 − κ , and κ and τ are real constants. Usingthese parameters, both the original Weyl action form, and the simpler standard Einsteinand Yang-Mills action are special cases of the results, for τ = 1 and τ = 0 respectively.Thus this τ parametrized action will now be used, with the additional step of including thefull, quaternionic Yang-Mills field.Since it was already noted in section III that the τ = 0 case for the action includesnegative energy density fields for solutions in the complex plane , this case may seem tobe an odd choice to include. However, even the ( τ = 1) generalized, original, real Weylaction (equation (69)) ultimately produces a term ( C/ g µν for the cosmological constantterm in the gravitational equation, because ˆ β = 1 − [(4 σ ) /C ] = 0. By substituting for ˆ v µ ˆ v µ from the constraint, an additional ( C/ g µν also will appear (from the 3 τ ˆ v µ ˆ v µ ˆ g µν termin equation (76)). This might be interpreted either as part of the cosmological constantterm, or as a “rest mass” related stress-energy tensor term. Either way, clearly one signchoice for C will produce a negative overall energy density, even in that simple model. And,a no-frills, quaternionic action form without unfamiliar extra terms beyond the constraintterms should be instructive, even if ultimately exotic in some respects. Furthermore, the7 τ = 1 original Weyl choice for the action will be seen to produce results that are no longer assimple when extended into quaternions, and it will even produce additional negative energydensities in terms not associated with the constraint. Thus that case will lose some of itsappealing simplicity.Therefore, adopt the quaternionic action I = (cid:90) (cid:112) − ˆ g (cid:26)(cid:16) ˆ R − σ (cid:17) + 12 (cid:20) j (ˆ y µν ˆ y µν )+ (cid:16) ˆ R + 6ˆ v µ (cid:107) µ + 6ˆ v µ ˆ v µ − C (cid:17) ˆ β + 6 τ ˆ v µ ˆ v µ + QC ] } d x (71)where the QC term is the quaternion conjugate of everything preceding it in that level ofbrackets. The reversal in the sign of the gravitational constant j will be explained shortly.The variation of this action is most simply carried out by varying the individual realcomponents of each of the quaternionic quantities involved. To facilitate that, adopt a com-ponent notation in which any quaternionic quantity Z is written out in its real componentsas Z = Q Z + (cid:88) k =1 Q k Z k = Z + (cid:126)Q · (cid:126)Z (72)where the Z a are all real, and a is the quaternion component index, with a ranging from 0to 3. The matching vector notation should be self explanatory. Any additional lower indicessuch as tensor indices, will physically follow the quaternion component index if lower tensorindices are present for the quantity in question.In terms of those real components of quaternionic quantities, equation (71) now becomes I = (cid:90) (cid:112) − ˆ g (cid:26)(cid:16) ˆ R − σ (cid:17) + 12 j (ˆ y µν ˆ y µν − (cid:126) ˆ y µν · (cid:126) ˆ y µν (cid:17) + (cid:16) ˆ R + 6ˆ v µ (cid:107) µ + 6ˆ v µ ˆ v µ − (cid:126) ˆ v µ · (cid:126) ˆ v µ − C ) ˆ β − (cid:16) (cid:126) ˆ v µ (cid:107) µ + 2ˆ v µ (cid:126) ˆ v µ (cid:17) · (cid:126) ˆ β + 6 τ ˆ v µ ˆ v µ − τ(cid:126) ˆ v µ · (cid:126) ˆ v µ (cid:111) d x (73)where the added quaternion conjugate terms have already been explicitly included in thisresult. This action is used in conjunction with the component form of equation (63)ˆ y aµν = ˆ v aν,µ − ˆ v aµ,ν + 4 (cid:88) k =1 3 (cid:88) n =1 (cid:15) akn ˆ v kν ˆ v nµ (74)where lowercase Latin indices range from 0 to 3 unless otherwise noted, and (cid:15) abcd is thetotally antisymmetric unit symbol with (cid:15) = 1. This action is a functional of ˆ g µν , ˆ v aµ ,and ˆ β a .The variation of the components ˆ β a yields four equations which reassemble into thequaternionic form of the required constraint of equation (67). Just as with the real casein section III, this equation in principle gives B when ˆ g µν and v µ are already known, thelatter being obtained in some arbitrary gauge. The solution for B then gives ˆ v µ from v µ by equation (56). However, this can be a challenging problem to solve in the full quater-nions compared to cases in the complex plane . Nevertheless, in the free space case ofLorentzian spacetime and vanishing Yang-Mills field, the constraint has simple solutionsconstructed from solutions to linear wave equations, including the standard solutions for afree Dirac particle. This will be shown shortly.8 A. Equations of Motion
The variation of the components ˆ v aµ yields four, four-vector equations, which reassembleinto the quaternionic form of the Yang-Mills field equations. That gives the deceptivelysimple looking equation ˆ y µν (cid:107) ν + 2 (ˆ y µν ˆ v ν − ˆ v ν ˆ y µν ) = 3 j (cid:104) ˆ g µν ˆ β ,ν − (cid:16) ˆ v µ ˆ β + ˆ β ˆ v µ (cid:17) − τ ˆ v µ (cid:105) (75)with the second and third terms on the left or field side forming a commutator, and thesecond and third terms on the right or “source” side forming an anticommutator. xii Wheneverything is real in the τ = 0 case, and the sign reversal adopted for j in equation (71) isremembered, this reduces exactly to equation (3.8) in an earlier paper .The variation of the components ˆ g µν yields the gravitational Einstein equations. Forthese, quaternions will remain stated in components, and that givesˆ R µν −
12 ˆ R ˆ g µν + σ ˆ g µν = j (cid:20)(cid:18) ˆ y α µ ˆ y αν + 14 ˆ g µν ˆ y αρ ˆ y αρ (cid:19) − (cid:18) (cid:126) ˆ y αµ · (cid:126) ˆ y αν + 14 ˆ g µν (cid:126) ˆ y αρ · (cid:126) ˆ y αρ (cid:19)(cid:21) − (cid:18) ˆ R µν −
12 ˆ R ˆ g µν (cid:19) ˆ β − ˆ β (cid:107) µ (cid:107) ν + ˆ β (cid:107) α (cid:107) ρ ˆ g αρ ˆ g µν + 3 (cid:16) ˆ β ,µ ˆ v ν + ˆ β ,ν ˆ v µ (cid:17) − β ,α ˆ v α ˆ g µν − v µ ˆ v ν ˆ β + 3ˆ v α ˆ v α ˆ g µν ˆ β +6 (cid:126) ˆ v µ · (cid:126) ˆ v ν ˆ β − (cid:126) ˆ v α · (cid:126) ˆ v α ˆ g µν ˆ β − C ˆ g µν ˆ β − (cid:16) (cid:126) ˆ β ,µ · (cid:126) ˆ v ν + (cid:126) ˆ β ,ν · (cid:126) ˆ v µ (cid:17) + 3 (cid:126) ˆ β ,α · (cid:126) ˆ v α ˆ g µν +6 (cid:16) ˆ v µ (cid:126) ˆ v ν + ˆ v ν (cid:126) ˆ v µ (cid:17) · (cid:126) ˆ β − g µν ˆ v α (cid:126) ˆ v α · (cid:126) ˆ β − τ [2ˆ v µ ˆ v ν − ˆ v α ˆ v α ˆ g µν − (cid:126) ˆ v µ · (cid:126) ˆ v ν + (cid:126) ˆ v α · (cid:126) ˆ v α ˆ g µν (cid:105) (76)This immediately illustrates why j has been given a reversed sign in the action. Clearly theYang-Mills field stress tensor term containing ˆ y µν has a sign opposite to the stress tensorterm for (cid:126) ˆ y µν contributed by the purely imaginary part of the Yang-Mills field. xiii Thus, thesign of j has been chosen so that the totally imaginary portion of the field has a stresstensor term which behaves like a “normal” or positive energy density, while the real portionproduces negative energy density. Since standard Yang-Mills theory works with the totallyimaginary subset of all quaternion valued Yang-Mills fields, those fields have been assignedpositive energy density. xiv xii As a side observation, equations (56) and (63) give that ˆ y µν (cid:107) ν +2(ˆ y µν ˆ v ν − ˆ v ν ˆ y µν ) = B − [ y µν (cid:107) ν +2( y µν v ν − v ν y µν )] B . xiii Given quaternions Z and ˆ Z such that ˆ Z = B − ZB , where Z may have tensor indices, then ( ˆ Z + QC ) = B − ( Z + QC ) B = ( Z + QC ). Then ˆ y aµν → y aµν is allowed in the j stress-energy tensor terms. xiv The real part ˆ y µν could be constrained to vanish so that the reversed sign energy density term is absentfrom the outset. The term ˆ α µν ˆ y µν is simply added to the action density, and the real, antisymmetricquantity ˆ α µν is also varied in the action. τ (cid:54) = 0, a similar situation also arises with some reversed sign energy densityterms in the τ terms in the stress tensor, and if the terms for the imaginary part of ˆ v µ aredominant, only τ ≤ τ ≤ not include the generalization of the original Weyl action ( τ = 1)into the complex and quaternion numbers. Thus, the generalization of the original Weylaction into the quaternions already appears to allow negative energy density from the τ (cid:54) = 0terms in the stress tensor, even without considering the stress tensor terms arising from theconstraint.Now two more equations can be derived immediately from equations (63), (67), (75), and(76), together with ˆ y µν (cid:107) ν (cid:107) µ = 1 √− ˆ g (cid:16)(cid:112) − ˆ g ˆ y µν (cid:17) ,ν,µ = 0 (77)This last equation combines with the “ (cid:107) µ ” divergence of equation (75), together with equa-tion (63), and equation (75), to give thatˆ g µν ˆ β (cid:107) µ (cid:107) ν − ˆ v µ (cid:107) µ ˆ β − ˆ β ˆ v µ (cid:107) µ − v µ ˆ β ,µ + ˆ β ,µ ˆ v µ − β ˆ v µ ˆ v µ + 2ˆ v µ ˆ v µ ˆ β = 2 τ ˆ v µ (cid:107) µ (78)This can be considered the conservation equation for quaternionic charge, and is the firstof the two additional derived equations mentioned above. Notice that for ˆ β equal to a realconstant other than − τ , it gives ˆ v µ (cid:107) µ = 0, while a nonconstant ˆ β generally gives no suchsimple result.Using equation (67) judiciously on this equation to substitute for some terms or portionsof terms (a term may be split into the sum of two equal halves), allows this equationeventually to be rewritten as 32 (cid:104) ˆ g µν ˆ β (cid:107) µ (cid:107) ν − ˆ v µ (cid:107) µ ˆ β − v µ ˆ β ,µ +ˆ v µ ˆ v µ ˆ β − (cid:16) C − ˆ R (cid:17) ˆ β (cid:21) − (cid:104) ˆ β (cid:107) µ (cid:107) ν ˆ g µν − ˆ β ˆ v µ (cid:107) µ − β ,µ ˆ v µ + ˆ β ˆ v µ ˆ v µ −
16 ˆ β (cid:16) C − ˆ R (cid:17)(cid:21) = 2 τ ˆ v µ (cid:107) µ (79)The left side is an asymmetric combination of a right handed Klein-Gordon equation form(if the square brackets vanish) and a left handed Klein-Gordon equation form, with − ˆ v µ inplace of v µ , ˆ β in place of the wavefunction, and with the addition of the standard (real)(1 /
6) ˆ R “conformal” term coefficient times ˆ β in each case. The − (1 / C ˆ β term acts asthe “rest mass” squared term, and has the proper sign if C <
0. If ˆ β commutes with ˆ v µ andits derivatives, such as when everything is in a complex plane, or when ˆ β is real, then thetwo parts merge into a single, Klein-Gordon equation form with the reversed sign, gaugeinvariant potential, although for nonzero τ , it has an additional, inhomogenous term in thatequation. This result becomes the equation of motion of ˆ β , and immediately illustrates whynegative energy densities might arise from the constraint terms in this set of equations.Particularly for τ = 0, equation (79) appears to be linear in ˆ β , while ˆ β appears linearlyin its stress tensor terms in equation (76) (and the quaternionic “four-current sources” inequation (75)), rather than quadratically as a square of either a real number or an absolutevalue of a quaternion. xv That means sign reversals in the stress tensor terms may occur. xv Note that ˆ β = 0 is always a possible solution to equation (79) if τ = 0. β . That will simplify the trace equation to(1 − τ ) ˆ R − σ = C (cid:16) ˆ β − τ (cid:17) (80)For most values of τ , including zero, this relates ˆ R and ˆ β . However for the special value τ = 1, this can be solved instead for ˆ β = 1 − σ/C ), which is constant. However eventhen, only ˆ β is set, while the three imaginary components of ˆ β are not determined throughthis equation. Even in the complex plane, the one imaginary component is not set. Outsidethe pure real numbers, ˆ β is not necessarily constant for τ = 1, and thus generally ˆ v µ (cid:107) µ (cid:54) = 0even for τ = 1. That suggests the τ = 1 case is no longer so simple in the complex andquaternion numbers, unlike the case of real numbers in section III. Since as already noted,this τ = 1 case may allow negative energy density in the τ terms in the stress tensor, the τ = 0 (Einstein / Yang-Mills) case may be simpler and no more exotic in nature overallthan the τ = 1 case now. Thus, τ = 0 will be the case assumed hereafter unless otherwisenoted.Since τ = 0, notice that none of the non-field terms (terms not containing ˆ y µν ) on theright hand side of equation (76) involve a gravitational constant, but they all do include ˆ β .Its magnitude will serve in place of a gravitational constant, xvi and equation (80) directlyreflects that. The action of equation (71) no longer shares the problem of the action ofequation (69) in which terms appear with no moderating effects of either a gravitationalconstant, or ˆ β . Thus the current action avoids the problem of section III in which b had tobe set such that j (defined by equation (70)) was of order unity. That was necessary therein order to obtain consistency of stress tensor term magnitudes. For the current action, b (and thus j ) is still free at this point. Furthermore once the term − (cid:16) ˆ R µν − ˆ R ˆ g µν (cid:17) ˆ β is moved from the right side of equation (76) to the left side of that equation, it becomesclear that eventually the quantity (1 + ˆ β ) will wind up in the denominator of the right sideof Einstein’s equation, as well as the cosmological constant term. Then in regions in whichˆ β → −
1, the gravitational coupling constant j and cosmological constant σ might be(incorrectly) interpreted as “running” to ever larger magnitudes, and the same might applyto the other “source” terms remaining on the right side of Einstein’s equation. Note that anegative ˆ β is “normal” positive energy density in the term (1 / C ˆ g µν ˆ β in equation (76)if C is negative, so this case need not be exotic. However, this analysis so far applies onlyto the coupling of the gravitational field to the stress-energy tensor, and does not includethe running of any other coupling constants, such as is encountered in standard quantumfield theory . xvii B. Interpreting the Constraint and Its Effects
As expected, the action of equation (71) has produced the familiar Einstein and Yang-Mills equations, with an additional real component in the Yang-Mills field, and the entireYang-Mills field appearing in the stress-energy tensor in Einstein’s equation. However, theadded constraint terms in the action also produce possible “sources” in both field equations,and introduce the constraint itself as another equation to solve. In that regard, equation(79) makes it clear that one solution for the Lagrange multiplier ˆ β is ˆ β = 0, and thatillustrates immediately that all those additional “source” terms may vanish. That produces xvi Attempts to relate ˆ β to actual physics would be expected to incorporate Newton’s constant G into themagnitude of ˆ β explicitly at some point . xvii On the other hand, this result does demonstrate the type of relatively simple mathematics that willproduce apparent “running” of any coupling constant. An electromagnetic coupling example in thecomplex plane appears in earlier work . y µν ˆ v ν − ˆ v ν ˆ y µν ) in equation (75) acts as a source forthe Yang-Mills field. The constraint equation (67) in principal gives ˆ v µ by solving for B ,given ˆ g µν , and v µ in any arbitrary gauge. When working with fully quaternionic quantities,this is nontrivial. This subsection will analyze a very simple case which can be reasonablywell handled analytically, the free case in which v µ as well as y µν vanish in some gauge.Additionally, the metric ˆ g µν will be assumed to be Lorentzian with signature (+ − −− ). Ifthe analog between the constraint and mechanics noted in section III is maintained, thiscould be called the “free particle” case.However, even if that analogy is inappropriate, the constraint and its associated terms inthe equations of motion remain an integral part of the mathematical structure of the gaugeinvariant variables used to describe the system, and that will be true of all actions expressedin those variables. That means the constraint and its results must still be investigated, andan effort made to interpret the results as part of the predicted physics for any such actions,including the action of equation (71). This is an automatic part of the use of these gaugeinvariant variables. Obviously, cases that produce ˆ β = 0 will minimize, but still not entirelyremove this concern, as the action of section III illustrates when Weyl’s original parametersare used. Any further insights require additional examples, including the one generated bythe action of equation (71) (assuming τ = 0) that will now be examined.As a preliminary, note that equations (67) and (56) can be used to show that B (cid:16) ˆ v µ (cid:107) µ + ˆ v µ ˆ v µ (cid:17) B − = 16 (cid:16) C − ˆ R (cid:17) (81)and thus 16 (cid:16) C − ˆ R (cid:17) = −
32 ˆ g µν B ,µ B − v ν + 12 ˆ g µν v ν B ,µ B − + 1 √− ˆ g (cid:16)(cid:112) − ˆ g ˆ g µν v ν (cid:17) ,µ + ˆ g µν v µ v ν + 34 ˆ g µν B ,µ B − B ,ν B − −
12 1 √− ˆ g (cid:16)(cid:112) − ˆ g ˆ g µν B ,ν (cid:17) ,µ B − (82)Because of the quaternionic nature of this equation, the substitution B = ψ − will notlinearize this equation as it does in the complex plane . Therefore, assume B = χ − ψ − (83)where χ ,µ χ − = ψ − ψ ,µ (84)2Provided this equation has a solution for χ and ψ in terms of each other, xviii this reducesequation (82) to 16 (cid:16) C − ˆ R (cid:17) = 3ˆ g µν χ − χ ,µ v ν − ˆ g µν v ν χ − χ ,µ + 1 √− ˆ g (cid:16)(cid:112) − ˆ g ˆ g µν v ν (cid:17) ,µ + ˆ g µν v µ v ν + χ − √− ˆ g (cid:16)(cid:112) − ˆ g ˆ g µν χ ,ν (cid:17) ,µ (85)which can be rewritten as0 = 32 (cid:26) χ − √− ˆ g (cid:16)(cid:112) − ˆ g ˆ g µν χ ,ν (cid:17) ,µ + 1 √− ˆ g (cid:16)(cid:112) − ˆ g ˆ g µν v ν (cid:17) ,µ +2 ˆ g µν χ − χ ,µ v ν + ˆ g µν v µ v ν − (cid:16) C − ˆ R (cid:17)(cid:27) − (cid:26) χ − √− ˆ g (cid:16)(cid:112) − ˆ g ˆ g µν χ ,ν (cid:17) ,µ + 1 √− ˆ g (cid:16)(cid:112) − ˆ g ˆ g µν v ν (cid:17) ,µ +2 ˆ g µν v ν χ − χ ,µ + ˆ g µν v µ v ν − (cid:16) C − ˆ R (cid:17)(cid:27) (86)Obviously if everything commutes, this will reduce to a Klein-Gordon equation (for C < τ = 0, although the currentequation is even less neat in form. However, if v µ = 0, it does produce the quaternionicKlein-Gordon “free particle” equation (with the added conformal term containing (1 /
6) ˆ R )1 √− ˆ g (cid:16)(cid:112) − ˆ g ˆ g µν χ ,ν (cid:17) ,µ = 16 (cid:16) C − ˆ R (cid:17) χ (87)where the metric is assumed to be a flat spacetime metric (which sets ˆ R = 0 if that’sglobal). This is the promised simple, linear equation for the “free particle” solutions to theconstraint equation, provided of course that equation (84) is also solved for ψ . If v µ = 0is assumed at the outset, then ˆ v µ = − (1 / B − B ,µ directly from equation (56), and thenequations (83) and (84) give ˆ v µ = ψ ,µ ψ − . Expanding the constraint of equation (67) fromthat gives 1 √− ˆ g (cid:16)(cid:112) − ˆ g ˆ g µν ψ ,ν (cid:17) ,µ = 16 (cid:16) C − ˆ R (cid:17) ψ (88)However, this only indicates that ψ and χ both obey the Klein-Gordon equation. It doesnot yet give ψ in terms of χ by using equation (84). C. Solving Equation (84)
If equation (84) is differentiated using the “ ,ν ” derivative, this gives integrability condi-tions necessary in order for the equation even to have solutions. Requiring χ ,µ,ν = χ ,ν,µ , and ψ ,µ,ν = ψ ,ν,µ gives a pair of equations which may be subtracted to give the two integrabilityconditions by also using equation (84) itself. They are (written as one equation) (cid:2) ψ − ψ ,µ , ψ − ψ ,ν (cid:3) = 0 = (cid:2) χ ,µ χ − , χ ,ν χ − (cid:3) (89) xviii At this point, quaternionic equation (84) will be considered to hold only for a specific gauge for v µ . , ]” terms are conventional commutators. These may be simplified to ψ ,µ ψ † ψ ,ν − ψ ,ν ψ † ψ ,µ = 0 (90) χ ,µ χ † χ ,ν − χ ,ν χ † χ ,µ = 0 (91)where the “ † ” indicates the quaternion conjugate of the quantity it follows. xix If the quaternion conjugate of equation (84) is taken, and added to the original equation,the result can be written as (cid:0) χ † χ (cid:1) ,µ χ † χ = (cid:0) ψ † ψ (cid:1) ,µ ψ † ψ (92)This states that if equation (84) has a solution, then the squared real norms of χ and ψ arealways proportional. The constant of proportionality may be chosen as one without loss ofgenerality, so that they both have exactly the same real norms, or amplitudes.Generally, if ψ ,µ = λ µ ψ and λ µ λ ν = λ ν λ µ (“four momentum” left eigenvalues exist andcommute), or if ψ ,µ = ψα µ and α µ α ν = α ν α µ (“four momentum” right eigenvalues existand commute), then the integrability conditions specified by equation (90) are satisfied.Similar conditions on left or right eigenvalues of χ ,µ will satisfy the integrability conditionsof equation (91). Furthermore, all two component or two dimensional cases are easily shownalways to satify the integrability conditions. These are cases in which both ψ and χ can bewritten in terms of quaternions with only two nonzero components in common directions outof the four possible components in a general quaternion. These then have simple generalsolutions to equation (84) relating χ and ψ to each other. In the subset of these caseswhere one of the two components is real and the other is in a common fixed direction inthe imaginary three-space, this is trivial, since then both χ and ψ lie in the same complexplane, all the quantities in equations (84), (91), and (90) commute, and the obvious generalsolution to equation (84) is χ = ψ .If such a two dimensional solution does not lie in a complex plane, then it lies in a planein the totally imaginary three-space, and the axes of that space can be chosen such that χ = Q χ + Q χ = Q ( χ + Q χ ) = ( χ − Q χ ) Q (93)and ψ = Q ψ + Q ψ = Q ( ψ + Q ψ ) = ( ψ − Q ψ ) Q (94)Since these will have the same amplitude, χ + χ = ψ + ψ and equation (84) becomes χ ,µ χ † = ψ † ψ ,µ = ( χ − Q χ ) ,µ Q ( − Q ) ( χ + Q χ )= ( ψ − Q ψ ) ( − Q ) Q ( ψ + Q ψ ) ,µ = ( χ − Q χ ) ,µ ( χ + Q χ )= ( ψ − Q ψ ) ( ψ + Q ψ ) ,µ = ( ψ + Q ψ ) ,µ ( ψ − Q ψ ) (95)where the last step follows from the fact that all the quantities on the last three linescommute with each other. By inspection, the general solution to this equation is χ = − ψ ,and χ = ψ . The integrability conditions are then also shown to be true just as easily asthis solution was found. xix Some of the references use a reversed notation which places the “ † ” and other similar operators to theleft of the symbol affected. χ and ψ lie in the same two dimensional plane in the imaginary three-space, and they arereflections of each other through an arbitrary plane containing the origin and the normalvector at the origin to the plane containing χ and ψ . This last generalization follows sincethe original choice of space axes was made arbitrarily to simplify the form of χ and ψ , andto make the plane of reflection contain both the Q and Q axes as well. That specific casecan be rotated back into a general plane in the imaginary three-space. It should be notedthat all the two dimensional cases for χ and ψ produce linear relations between them, alongwith the two linear Klein-Gordon forms xx of equations (87) and (88). Thus, the principle ofsuperposition applies to these solutions, allowing wave packets to be formed by conventionalsuperposition of plane waves .Now after setting C = −
1, use equation (68) to convert back to standard lab coordinateswith dimensions using b = 6[( m c ) / (cid:126) ] for some reference “rest mass” m , along with thespeed of light c and the reduced Planck’s constant (cid:126) . Then typical plane wave solutionsin the imaginary three-space might be ψ = Q [ Ae Q ( kx − ωt ) ] with χ = Q [ Ae − Q ( kx − ωt ) ],where A , k , and ω are all constants, and k = ω c − m c (cid:126) (96)The fact that ( b / C = − [( m c ) / (cid:126) ] has been used with both equations (87) and (88)here.All these two dimensional forms will now be shown to be adequate to include the set ofstandard Dirac “free particle” solutions for spin 1 /
2, so they are potentially not just ofacademic interest.
D. Spinor - Quaternion Equivalences
A two row complex spinor ψ has four real components, ψ R , ψ I , ψ R , and ψ I , wherethe first row of the spinor is ψ R + ıψ I , and the second row is ψ R + ıψ I . A real quaternionalso has four real components, so define the quaternion equivalent version of ψ as ψ = ψ R Q − ψ I Q + ψ R Q − ψ I Q (97)In matrix form, this quaternion becomes ψ = (cid:18) ψ R + ıψ I − ψ R + ıψ I ψ R + ıψ I ψ R − ıψ I (cid:19) (98)The first column of this is simply the original spinor, and the second column has the standardconjugate relationships to the first column common to all quaternions , so this is indeeda quaternion equivalent to the original spinor. In both cases, ψ † ψ has the same value. Ifthere are two different original spinors ζ and ψ , the translation of the spinor inner product ζ † ψ into quaternions will not be simply ζ † ψ , but rather the translation becomes ζ † ψ → (cid:0) ζ † ψ − Q ζ † ψQ (cid:1) (99)as may be verified by direct calculation. This is the complex projection of a quaternionicvalue into the Q , Q complex plane . Thus, the translated inner product of two differentspinor wavefunctions is still the same complex number in the quaternion formulation that itis in spinors. This actually becomes the definition of the spinor equivalent inner product of xx These Klein-Gordon forms may become nonlinear in strong gravitational fields with significant ˆ R mag-nitudes, something that also happens in standard Klein-Gordon field theory with the added conformalterm . ζ = ψ , this reduces to ψ † ψ , as it must. It shouldbe noted that this quantity (which might well be called the “spinner” product) differs fromthe standard inner product of two quaternions, which is merely the dot product of the twoquaternion space four component vectors that each original quaternion represents. Thatstandard inner product can be expressed as (1 / ζ † ψ + ψ † ζ ), and it is actually the realpart of the spinor equivalent inner product.One additional feature is necessary to convert many spinor equations into their quaternionequivalent equations. Spinors are frequently multiplied by ı = √− ı . But to multiply thesame spinor rows by ı in a quaternion equation, it is also necessary to multiply the two rowsof the second quaternion column by − ı as part of the same operation. But right multiplyingany product by − Q produces exactly that result. The notation adopted to express this isthat in the translation from spinor to quaternion equations, ı → − ( | Q ) (100)The operator “ | ” is called the barred operator, and it indicates that the quantity associatedwith it (the Q ) multiplies the product containing it from the far right . Thus, itproduces the quaternion equivalent of multiplying the original spinor expression by ı . Inthat regard, note also that [ − ( | Q ) ψ ] † = [ − ψQ ] † = Q ψ † , so that the Q must multiplythe quaternion conjugate result from the left in order to maintain the correct equivalence tospinor results after multiplication. This should be expected, and can be facilitated furtherif necessary by defining a reversed barred operator that specifies multiplication from thefar left in a product. That reversed barred operation is specified by ( Q (cid:107) ) when necessary,in obvious analogy to − ( | Q ). E. “Free Particle” Dirac Solutions and Spin
The standard solutions for the Dirac free particle are well known in a standard Diracrepresentation . Those solutions may easily be reexpressed in a chiral representation inwhich the Dirac four row spinor is separated into a pair of two row spinors , either ofwhich alone can describe the system by using the second order form of Dirac’s equation,and using the appropriate first order Dirac equation to define the other two row spinor ifit is needed. If the upper two rows are chosen as the basic spinor, the standard free particleforms become ξ u + = 1 √ e − ıωt (cid:20) (cid:21) , ξ d + = 1 √ e − ıωt (cid:20) (cid:21) ,ξ u − = 1 √ e ıωt (cid:20) (cid:21) , ξ d − = 1 √ e ıωt (cid:20) (cid:21) (101)The corresponding quaternionic forms of ξ are ξ u + = 1 √ e Q ωt , ξ d + = 1 √ Q e Q ωt ,ξ u − = 1 √ e − Q ωt , ξ d − = 1 √ Q e − Q ωt (102)where in both sets of equations, the subscripts “ u ” and “ d ” refer to spin up or down, andthe “+” and “ − ” refer to positive or negative energy. Clearly the quaternionic forms of thisset all satisfy equation (88) (with ψ = ξ ) in the flat space “free particle” case with k = 0,and ω then given by equation (96). They consist of two dimensional solutions in every case,and thus each has a corresponding χ given by equation (84). Therefore they form a validset of solutions to the constraint of equation (67). It should also be noted that this entireset can be generalized to represent free particle motion rather than just the case of the free6particle at rest. A substitution such as ωt → ωt − kz in all cases in the set generalizes it toa moving particle.Now in standard Dirac theory, these solutions may be superposed to create new solutionssuch as superpositions of spin up and down. In quaternions, some of these combinations arestill two dimensional with a known matching χ from the rules already given, but some arefully four dimensional in quaternions, such as the superposition ξ = aξ u + + bξ d + . However,this case can also easily be shown to have commuting right eigenvalues, which satisfies theintegrability conditions. Indeed, the matching χ for this case is easily shown to be the sumof the individual χ ’s for each part. Other superpositions, such as aξ u + + bξ d − will have a χ made up of the difference of the individual χ ’s. In these ways, these linear superpositions arealso solutions to the constraint equation (67) in the free space or “free particle” case. Thus,the constraint continues to show resemblances to particle mechanics just as it did with theoriginal Weyl action in section III. In this case, the similarity includes some properties ofspin 1 / xxi The simplestgauge invariant result that is available is the calculation of ˆ v µ = ψ ,µ ψ − = ξ ,µ ξ − . For a“particle” at rest ( k = 0), only ˆ v (cid:54) = 0, andˆ v u + 0 = Q ωc , ˆ v d + 0 = − Q ωc , ˆ v u − = − Q ωc , ˆ v d − = Q ωc (103)Thus as far as these gauge invariant quantities are concerned, a flip in “spin” directionis equivalent to a flip in the sign of the “energy”. Also note that the similar eigenvaluesin the Dirac spinors are taken from ξ ,µ = λ µ ξ , where the λ µ are the eigenvalues. In thespinors, the eigenvalues do not flip sign under a flip in spin direction. Thus differenceswith Dirac formalism do appear. However, this particular difference comes directly fromthe translation from equation (101) to equation (102), since Q does not commute with Q , but rather, anticommutes. The sign reversal vanishes if right eigenvalues α µ are used inthe quaternion case in the equation ξ ,µ = ξα µ instead of left eigenvalues, but the equationˆ v µ = ξ ,µ ξ − selects the left eigenvalue, not the right. Nevertheless, it will now be shownthat superpositions of the ξ spin up and spin down cases (for the same energy), tell a muchmore detailed story than these deceptively few examples have, and will produce a ˆ v thatbehaves much as spin 1 / ξ = cos ( θ/ ξ u + + sin ( θ/ ξ d + e − Q φ (104)where θ is the spherical coordinate polar angle measured from the positive Q axis, and φ isthe azimuthal angle measured counterclockwise from the positive Q axis toward the positive Q axis, looking down from the positive Q axis. xxii Then a straightforward calculation ofˆ v = ξ , ξ − gives ˆ v = ωc ( Q sin θ cos φ + Q sin θ sin φ + Q cos θ ) (105)The zeroth component of ˆ v µ actually sweeps out a continuous change in direction in thepurely imaginary quaternions to whatever direction the spherical coordinate angles specify.That is also the spin up direction in regular three space if the eigenspinors of σ z are trans-formed and superposed via the exact spinor analog of the transformation and superpositionof ξ u + and ξ d + in equation (104) above . Thus, the quaternion imaginary three space di-rection of ˆ v exactly tracks the spin up direction in ordinary three space. This is evidently xxi The wavefunction is not directly measurable, being part of the intrinsic standard of self measure (literallythe “gauge”) in the structure. xxii
This is based on the treatment of spin 1 / , after it is correctedfor consistency with the (easily verified) correct results of page 406 by letting φ → − φ on page 410.
7a direct way to visualize spin direction via the quaternion space direction of this eigenvalue.In contrast, the ( φ = 0 case) superposition ξ = cos ( θ/ ξ u + + sin ( θ/ ξ u − fails even toproduce any well defined eigenvalue except at the extreme values of θ = 0 and θ = π , whichare the cases without superposition. xxiii That shows that the spin and energy cases are ac-tually quite different, in spite of the agreement in eigenvalues at the extreme polar angles.At the same time, the rest of the material above shows that clear, gauge invariant spin 1 / not just a mathemat-ical formality. The surprise is that the spin 1 / , but rather associated with energy-momentum.Here, that “energy-momentum” set of eigenvalues remains, xxiv but is directly combinedwith the spin. Thus, the resemblance between the constraint and particle mechanics clearlycontinues, and is even enhanced with the quaternionic action of this section. VI. SELF MEASUREMENT AND MECHANICS
The gauge invariant variables of this model embody the self measuring properties of Weyl-like geometries with scalar curvature B (cid:54) = 0. The scalar curvature becomes an intrinsicruler by which the structure creates gauge invariant quantities suitable for physics. Thegeometry is thus literally, “Self gauging,” and displays that property through the intrinsicconstraint of equation (67) between the gauge invariant variables. That constraint veryclosely resembles known equations from particle mechanics in combined gravitational andgauge fields, including classical mechanics, and even spin 1 / xxv Neither has yet been demonstrated that I know of.The second obvious problem is that the similarities to mechanics are for single particlemechanics. Current physics is built upon quantum field theory which is a many bodytheory . The only theory of many body relativistic quantum mechanics that I know ofthat might fit naturally into this geometry is the, “Many-amplitudes,” theory of relativisticquantum mechanics explored by Egon Marx . That might suggest a path to overcome thisobjection.Third, particles come with many different rest masses, and the second order Dirac equa-tion has a spin interaction term missing from the current structure . These tie togetherbecause at least in the complex plane, or for quaternionic cases projected into the complexplane, the introduction of ˆ a µν , a self dual, antisymmetric component to the metric tensormay address both problems at once . The full metric tensor becomes ˆ m µν = ˆ g µν + ˆ a µν ,while its inverse ˆ M µν is defined by ˆ M µλ ˆ m λν = δ µν . These give ˆ M µν = (ˆ g µν − ˆ a µν ) / [1+(ˆ a/ g µν , and ˆ a = ˆ a µν ˆ a µν . In the resulting general-ization of the constraint of equation (67), a spin interaction term is added, and the quantity C → C [1 + (ˆ a/ a to control the“rest mass”, and to play much the same role as a Higgs Field in standard quantum field xxiii This absence of a precise eigenvalue because of oscillating cross terms and even infinite values in ˆ v µ = ξ ,µ ξ − , is the common case when there are superpositions inside ξ . The unusual, well defined eigenvaluesof the spin superpositions indicate that the spin superpositions still represent an eigenstate. xxiv True energy-momentum in this structure requires collaboration between these eigenvalues, and the ap-propriate terms in the stress tensor. A full example set is mapped out elsewhere for the case restrictedto the complex plane with C = −
1. When that example set is restricted to the purely positive energydensity field, the eigenvalues and stress tensor correlations match standard Klein-Gordon theory. Forˆ β = 0, the “vacuum” becomes more of a “virtual particle” which can possess “spin,” but not real energy. xxv The Newtonian gravitational field is a negative energy density field that successfully matches observedphysics in its realm of applicability. . Indeed, the zero order approximation is ˆ a = − K where K is a real constant,and this is also the case that produces the correct Dirac-like spin interaction term. Tofit that term, the value K = 4 seems to be the appropriate base Dirac case, as detailedin a separate paper . In the same framework, C = 1, since 1 + (ˆ a/
4) = −
3. This zeroorder approximation for ˆ a would generalize to higher orders via ˆ a = − ˆ λ , where ˆ λ is a realvariable field, xxvi one that obviously will now affect the “rest mass,” allowing for more thana single “rest mass” value. xxvii However, whatever objections like these are raised, it’s also still true that multiple coin-cidences in form and content appear between the internal constraint of these self measuringWeyl-like geometries, and known equations of classical and quantum particle mechanics inthe gravitational and gauge fields of this structure. This includes close parallels in the formof the stress tensors and four currents of the gravitational and gauge fields . Further-more, these familiar wave mechanical equation forms are known to produce sharp laboratoryspectral lines that can used in lab measurements . Thus an abstract theory of a self measur-ing geometry points directly to a recognizable laboratory phenomenon, spectral lines whichcan be used in actual lab measurements. This is unlikely to be just coincidence. The scaleof some predicted general relativistic effects will now be seen to add to the novelties. VII. SCALE OF CURVATURE RELATED EFFECTS
As long as general relativistic effects involve only the (dimensionless) metric tensor, thismodel introduces no new scale for those effects. However, general relativistic quantitiescan also involve a nonzero curvature tensor ˆ R γµτσ . Since that includes derivatives withrespect to the coordinates, the dimensionless quantity used in this model is not quite thesame as the usual expressions for curvature which still have units. Thus, this model willallow simple, new estimates to be made of scales at which this dimensionless curvaturetensor cannot be ignored, and since it is dimensionless, those scales may have physicalsignificance. But, it’s also true that some important equations involving physical effects ofthe Riemann tensor will not affected by this scale factor dependent, dimensionless result.The scale factor completely cancels from the equation of geodesic deviation when it isfactored out to reexpress quantities in laboratory units, so geodesic deviation is clearly notaffected. What follows will attempt to clarify these different possibilities. A. Ricci Tensor
Einstein’s equation (76) gives the Ricci tensor via the Einstein tensor, which is deter-mined by the stress-energy tensor. The traditional Reissner-Nordstr¨om solution for a pointelectronic charge xxviii of magnitude e , easily fits into the complex plane in this model (with the real part ˆ y µν = 0), and gives immediate estimates for the scale of ˆ R νµ . It iseasily shown that the diagonal components of ˆ R νµ are of unit magnitude at a radius fromthe central charge (in laboratory, Gaussian units) xxix r L = (cid:18) e (cid:126) c (cid:19) / (cid:115) L P √ b (106)In this, L P = (cid:112) ( (cid:126) G ) /c , the well-known Planck length. Thus, the Ricci tensor is significantat a radius which is roughly the geometric mean of the Planck length, and the inverse ofthe square root of the scale factor b . xxvi The antisymmetric part to the metric may not have a large amount of freedom if the structure is to beconsistent in its self measurement. This is detailed in a separate paper . xxvii Also, for ˆ λ <
4, inflationary cosmologies are possible . xxviii The problem of confining an actual charge in a volume of very small radius is not addressed here. xxix Einstein’s equations for the Reissner-Nordstr¨om solution give a unit magnitude for diagonal componentsof ˆ R νµ when j ( g / e / (2 b r L )] = 1, where j is given by equation (70). Notice that g eventuallycancels from this condition. . However ,if the scale factor is atomic scale, then the Ricci tensor becomes significant at a much largerradius, a perhaps unexpected result. If b = (2 m c ) / (cid:126) , and m is set to the electron mass(which would be consistent with the similarities to mechanics noted in section VI when C = 1 and ˆ a = − r L ≈ . · − cm. This radius sweeps out a circle of crosssectional area roughly 10 − cm. around the central charge as viewed from any direction.This is approximately the lower bound of observed neutrino interaction cross sections ,again perhaps unexpected. The independence of the scale factor from the Planck lengthproduces this separate, intermediate scale constructed from both.However, as noted above, some important equations involving physical effects of theRiemann tensor are not affected by this scale factor dependent result. The most immediateexample of an equation that does reveal effects would be the covariant divergence of thestress tensor, specifically the terms (cid:104) ˆ β (cid:107) γ (cid:107) γ δ νµ − ˆ β (cid:107) ν (cid:107) µ (cid:105) (cid:107) ν = ˆ R νµ ˆ β ,ν (107)The coupling of those (force density) terms to the covariant divergence of the rest of thestress tensor appears very dependent upon the magnitude of the Ricci tensor. They aredecoupled when it vanishes, even though ˆ β is the same quantity that appears elsewherein the stress tensor. xxx Only the dimensionless quantity ˆ R νµ clearly quantifies this generalrelativistic variation in coupling. B. Scalar Curvature
The scale at which ˆ R becomes significant can be estimated through the portion of thestress-tensor in equation (76) generated by the constraint terms. This estimate appears inanother paper assuming a case limited to the complex plane. Positive energy density isassumed, and it is assumed that the wavefunction for the designated “rest mass” can becontained in a small enough (spherical) volume to reach the peak magnitude necessary toproduce a unit magnitude ˆ R inside that volume. The calculated radius of such a volumeis within a few orders of magnitude of the radius above for significant ˆ R νµ , and is smaller,at least when the electron rest mass is the mass involved throughout. The most obviouseffect a significant scalar curvature has is through the ˆ R term in the constraint , whichplays the same role as the added conformal term in a Klein-Gordon equation . That termis otherwise ignored. VIII. SUMMARY
The well known Weyl geometry can be successfully extended to quaternionic gauge trans-formations and Yang-Mills fields provided half the nonmetricity is shifted into torsion. Thisis the primary finding of this paper. For nonvanishing scalar curvature, the geometry pro-vides intrinsic gauge invariant variables which can correspond to measured quantities inphysics, and which facilitate the extension to the quaternions by minimizing the number ofnoncommuting quantities in the equations. This is the second main topic of this paper.Those gauge invariant variables are rendered dimensionless by expressing all coordinatesas dimensionless quantities by use of a scale factor b with dimensions of inverse length xxx However, any additional ˆ α ˆ R term in the action density for any other scalar field ˆ α , generates similarterms and coupling in the stress tensor and its covariant divergence. / .These are additional main points of this paper.Besides the constraint, the full equations of motion are developed given an action principlewith a fully quaternionic Yang-Mills field. The action used is basically an Einstein/Yang-Mills action with some optional additional terms suggested by Weyl’s original action. Caseslimited to real variables, or the complex plane, are included. The constraint is necessarilyincluded in the action, and generates field sources for both the Einstein and Yang-Millsfields. The resulting equations for an isolated electronic charge and mass suggest thatunlike the metric tensor deviations from flat space, the (newly dimensionless) Ricci tensor inGeneral Relativity may become significant at much greater distances than the Planck length,depending on the value assigned to the scale factor. However, many general relativisticresults such as geodesic deviation are unaffected by this scaling.Overall, this model extends Weyl’s work into the quaternions, and that facet should notbe controversial. It then offers tantalizing hints of deep links between the internal structureof Weyl-like geometries, classical and quantum particle mechanics, spin 1 /
2, and the self-measuring property of this type of geometry. In the Weyl-like geometry, these are directlyintegrated with the Einstein and Yang-Mills fields that share the unified structure, plus apossible self dual antisymmetric part to the metric to provide both a spin interaction, anda controller of “rest mass.” Nevertheless, substantial obstacles remain to such an approachoverall, such as the necessity for a “many body” version of the formalism. The frameworkalso is too narrow to embrace the full SU (3) × SU (2) × U (1) structure of the currentStandard Model , although it does contain the SU (2) and U (1) gauge fields as subsetsof the quaternionic Yang-Mills field. The quaternionic Yang-Mills field is SU (2) × SU (2)structure. ACKNOWLEDGMENTS
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