aa r X i v : . [ phy s i c s . g e n - ph ] J u l Lie sphere geometry in nuclear scattering processes
S. Ulrych
Wehrenbachhalde 35, CH-8053 Z¨urich, Switzerland
Abstract
The Lie sphere geometry is a natural extension of the M¨obius geometry, wherethe latter is very important in string theory and the AdS/CFT correspondence.The extension to Lie sphere geometry is applied in the following to a sequence ofM¨obius geometries, which has been investigated recently in a bicomplex matrixrepresentation. When higher dimensional space-time geometries are invoked byinverse projections starting from an originating point geometry, the Lie spherescheme provides a more natural structure of the involved Clifford algebras com-pared to the previous representation. The spin structures resulting from the gen-erated Clifford algebras can potentially be used for the geometrization of internalparticle symmetries. A simple model, which includes the electromagnetic spin,the weak isospin, and the hadronic isospin, is suggested for further verification.
Keywords:
Lie sphere geometry, nuclear physics, Clifford algebras, bicomplexnumbers, AdS/CFT
1. Introduction
The spin of a particle can be understood in terms of irreducible representa-tions of the Poincar´e group. Thus, it has a clear geometric background. Since thesuccess of the concept of isospin in the middle of the last century physicists andmathematicians consider a possible geometrization of the internal symmetries ofa quantum particle, see for example Hermann [1]. Today there is the hope thatthe geometrization problem can be addressed by string theory and the AdS/CFTcorrespondence [2, 3, 4]. The acronym CFT indicates that the conformal symme-try plays a substantial role in these considerations. However, in principle othergeometries and symmetries could be considered. Even if one takes a step backinto a simplified world which consists only of two space dimensions, there areoverall 23 families of Klein geometries [5].Among these geometries one finds the Lie sphere geometry [6], which has beenconsidered in detail for example by Cecil and Chern [7, 8], Cecil and Ryan [9],
Preprint submitted to Journal of Mathematical Analysis and Applications March 1, 2020
Sharpe [10], Benz [11], and Jensen et al. [12]. There is also the software library ofKisil [13] for graphical representation. The Lie sphere geometry has been appliedin the last decades in the study of Dupin submanifolds [14]. This research hasbeen initiated by the thesis of Pinkall [15]. The Lie sphere geometry considerspoints and lines as special cases of spheres, which can again be understood aspoints in an ambient space. Lie spheres have been applied in physics by Bateman,Timerding, and Cunningham within optics and electrodynamics [16, 17, 18, 19].The radius of the Lie sphere is identified in these applications with r = ct . Thisconnection between geometry and relativistic physics has been investigated alsoby Cartan [20]. Thus, it seems to be worth to consider the Lie spheres forfurther applications. The following discussion therefore suggests to generalizethe sequence of M¨obius geometries introduced in [21] and apply the Lie spheregeometry instead of the M¨obius geometry. From the perspective of physics onecan then address the question mentioned in the beginning of this introductionand try to identify the isospin symmetries of electroweak and strong interactionswithin the algebraic spin representations of a hierarchy of Lie sphere geometries.The importance of projective geometry in relativistic physics has been notedalready by Klein [22]. Today string theory and the AdS/CFT holography pointtowards the importance of projective geometry in the representation of the fun-damental geometries in physics. Furthermore, one may assume that the dimen-sionality of physical space is not given a priori and the emergence of space-timehas to be considered in a generalized geometric context [23, 24]. In AdS/CFTand string theory the geometry is generalized to higher dimensional spaces byadding space dimensions. A constraint on the total number of dimensions hasbeen provided by supergravity [25, 26]. In [21] and in the following discussionthe next level of dimensionality is reached by adding one space and one timedimension, which is in line with the conformal compactification. This guaranteesthat the concepts of projective geometry can be applied without modificationwithin the series of geometric spaces of different dimension. However, the specialimportance of Minkowski space-time cannot be derived in a natural way at thislevel of investigation.Compared to its predecessor [21] the following discussion, which is based on theLie sphere geometry instead of the M¨obius geometry, provides a few advantages.As shown in Section 4 the generalized approach is more natural with respect tothe signature of the applied Clifford algebras. The number of basis elements withpositive square is equal to the number of basis elements with negative square. Inaddition, the geometry originates by inverse projections out of a line instead of aplane, which is discussed in Section 5. The line can be identified in its projectivecontext with a point. One may think here of the point geometry as an origin,which is inherent to the physical space with an a priori undefined dimensional-ity. Section 7 outlines how the content of [21] is included as a subalgebra in thegeneralized representation. In Section 9 the algebraic representation is connectedwith the Lie sphere geometry. The mass formula for electrons and protons, whichhas been discussed already in [21], indicates the emergence of particle configura-tions from their geometric context. Finally, Section 12 shows that the approachis, compared to [21], more consistent with nuclear physics, because the electro-magnetic spin, the weak isospin, and the hadronic isospin can be assigned to theClifford matrix representation of the proposed geometry. The approach thereforeappears to be more suitable than [21] for the geometrization of internal particlesymmetries.
2. Bicomplex numbers
The bicomplex numbers have been discussed by Segre [27] in 1892. An equiv-alent number system called tessarines has been introduced by Cockle [28] even in1848. Detailed introductions with more information on the history of bicomplexnumbers can be found for example in [29, 30]. The bicomplex number can befurther extended to multicomplex numbers, see Price [31]. An application of mul-ticomplex numbers in physics has been proposed recently in [32]. The bicomplexnumber has one real and three imaginary parts ς = ς + iς + jς + ijς . (1)Both complex units of the bicomplex numbers are defined to square to minus one i = j = − , ij = ji = √ . (2)The two commutative bicomplex units are distinguished by their behavior withrespect to the Clifford conjugation, which can be denoted in a physical contextas spin conjugation. The complex unit i changes sign, the bicomplex unit j not¯ ς = ς − iς + jς − ijς . (3)There is a second Clifford involution denoted as reversion. In physical applica-tions one can identify this involution with Hermitian conjugation. The reversionapplied to the bicomplex number is changing the sign of both complex units ς † = ς − iς − jς + ijς . (4)The notation is inspired by the conjugation of Dirac spinors ¯ ψ = ψ † γ . Twocomplex units with different behavior with respect to conjugation can be appliedpotentially in the description of the CP violation, where weak and strong phaseshave to be represented with exactly this property [33, 34, 35].The sign change of the hypercomplex units with respect to conjugation andreversion is consistent with previous publications of the author, which were basedon hyperbolic numbers, see for example [36]. The hyperbolic complex numbersare defined slightly different, but they are in fact congruent. The hyperbolic unitcorresponds in terms of the bicomplex numbers to j HC ≡ ij BC . (5) HC refers here to hyperbolic complex numbers and BC to bicomplex numbers.In the context of the following discussion it is worth to note that the Lie sphere ge-ometry in combination with hyperbolic numbers has been considered by Bobenkoand Schief [37].The bicomplex numbers stand in relation to the sphere S as the complexnumbers are in relation to S . In the following the radius of the bicomplexnumber will be set to r = 1. This shortens the notation and takes into accountthat in the projective context the length is an irrelevant quantity. The radiuscan be reintroduced if necessary. Due to the diffeomorphism between S and thegroup SU (2 , C ) it is possible to represent the bicomplex number in terms of themost general form of a group element of SU (2 , C ) ς = e iϕ cos ϑ + je iη sin ϑ . (6)In order to further study the properties of the bicomplex numbers one can alsochoose the spin representation of SO (3 , R ) to parametrize the bicomplex number ς ( αβγ ) = e − i ( β + γ ) cos α/ je i ( β − γ ) sin α/ . (7)One can now calculate the impact of conjugation and reversion on this represen-tation. Especially interesting is the squared modulus, which is usually leading tothe squared radius of a complex number under the standard complex conjuga-tion. For conjugation of the bicomplex number given by Eq. (7) one finds withthe help of Eq. (3) ς ¯ ς = cos α + j sin α cos β . (8)The map ς ¯ ς does not provide the squared radius and is even not a real number.A corresponding calculation can be done with the reversion. One finds by meansof Eq. (4) ςς † = 1 + ij sin α sin β . (9)Here the squared radius appears as the real part, which corresponds in this caseto r = 1. The remaining three terms in Eqs. (8) and (9) can be understood asthe result of the Hopf map S → S . They form a parametrization of the sphere S . The expressions do not depend anymore on γ , which refers to the S fiber ofthe Hopf map.The relation between Pauli spin and the bicomplex numbers has been discussedbefore for example by Smirnov [38], see also the references in this article for someapplications in physics. In this sense the representation of the bicomplex numbersas given by Eq. (7) can be used to define the Pauli spinor as χ = ς ( θϕ . (10)As discussed by Smirnov [38] the non-commutative Pauli matrices can be replacedby a new type of operators acting on the commutative bicomplex numbers. Asa consequence, the Pauli spinor with negative magnetic quantum number canbe deduced from the first Pauli spinor by means of Hermitian conjugation andmultiplication by the unit j χ − = jχ † . (11)One could think here of resolving all common spin structures, like the Diracspin or the isospin, into commutative number systems. If necessary, furthermulticomplex units can be introduced. The following discussion will not followthis route, but uses a mixture of bicomplex numbers and conventional matrixrepresentations, which are displayed in terms of non-commutative units, in orderto be aligned with [21].In the terminology of Clifford algebras the bicomplex numbers can be under-stood as the complexification C , of the hyperbolic numbers R , . The basis ofthe real algebra is given by e µ = (1 , e ) = (1 , ij ). The algebra is complexified bymeans of either i or j . Alternatively, the bicomplex numbers can be interpretedas the complexification C , of the complex numbers R , . The basis of the realalgebra can be chosen as e µ = (1 , e ) = (1 , i ). The complexification is then donewith the unit j .
3. Hypercomplex units for the representation of the Lie algebra sl (2 , R ) In [21] it turned out to be useful to represent the Lie algebra sl (2 , R ) withnon-commutative hypercomplex units. This provides additional insights into thestructure of higher dimensional Clifford algebras and their geometries. The ma-trices are introduced as ı = (cid:18) − (cid:19) , = (cid:18) (cid:19) . (12)The two generating matrices are denoted by ı and . Multiplication of the twoelements results in ı = (cid:18) − (cid:19) = − ı . (13)The three matrices correspond to the Lie algebra of the special linear group SL (2 , R ). They will be used as the building blocks for the sequence of Cliffordalgebras, which will be applied in the following sections.
4. Hypercomplex representation of the Maks periodicity of Cliffordalgebras
The Clifford algebra periodicity, which has been investigated from a mathemat-ical point of view by Maks [39], will be applied in a hypercomplex representationsimilar to [21] in order to generate higher dimensional geometries by inverse pro-jections. However, compared to [21] a different set of Clifford algebras is involved.The base geometry has now an odd number of 2 m + 1 = n dimensions. The n − R m,m are transformed to thebasis elements of the ambient Clifford algebra by e k = ı ⊗ e k , k = 1 , . . . , n − . (14)On the right-hand side of the equation are the basis elements of the source ge-ometry. The two additional basis elements of the ambient Clifford algebra areconstructed by means of the first element of the paravector Clifford algebra e = 1 e n = i ⊗ , e n +1 = ıj ⊗ . (15)The resulting basis elements generate the Clifford algebra R m +1 ,m +1 . In contrastto [21] the representation is given in terms of a tensor product. This is necessaryto avoid ambiguities in the representation, which will become obvious in Section 7.
5. Start with the Clifford algebra R , In [21] the series of projective spaces introduced in the previous section startedfrom the complex numbers, referring to the Clifford algebra R , . However, froma conceptual point of view, it is more appealing to start from a null algebraconsisting of the empty set. This null algebra can be assigned to the Cliffordalgebra R , . Nevertheless, one can construct a paravector algebra based on R , ,which is made up of the trivial identity basis element e = 1 alone. The paravectoralgebra is thus representing the real numbers R as the elementary geometry, whichcan be identified with an originating point in a projective context. The schemeintroduced in the previous section can be applied to the paravector algebra R , .There are no basis elements e k available in Eq. (14). However, two basis elementscan be derived by means of Eq. (15) e = i ⊗ (cid:18) ii (cid:19) , e = ıj ⊗ (cid:18) j − j (cid:19) . (16)Thus one arrives at a representation of the Clifford algebra R , . The two basiselements can be used together with the identity to form a paravector model e µ = (1 , e i ) of the space R , . The complex numbers, which form the initialalgebra in [21], are included as the subalgebra e µ = (1 , e ) representing the space R , .
6. From the Clifford algebra R , to R , With the Clifford algebra R , in place, Eqs. (14) and (15) can be appliedagain to generate the next higher dimensional geometry. One arrives now at theClifford algebra R , consisting of four matrices. Equation (14) results in the firsttwo basis elements of R , e = ı ⊗ e , e = ı ⊗ e . (17)The elements e and e on the left-hand side of the equations refer to the higherdimensional geometry. The elements e and e in the tensor product refer to thebasis elements of the Clifford algebra R , . Two further basis elements for R , can be generated by means of Eq. (15). Now the identity element refers to atwo-dimensional matrix e = i ⊗ , e = ıj ⊗ . (18)The four 4 × R , and can be considered as a paravector model for the geometric space R , .The scheme given by Eqs. (14) and (15) can then be applied again to generate8 ×
7. The spin tensor
One can consider the algebra introduced in the previous section in more detail.The product of two basis elements can be written in terms of symmetric and anti-symmetric contributions e µ ¯ e ν = g µν + σ µν . (19)Here the bar symbol ¯ e ν refers to conjugation of the considered basis element. Thesymmetric contributions of the product are represented in terms of the metrictensor g µν . The anti-symmetric contributions are identified with the spin tensor σ µν , see [21] for more details. The right hand side of Eq. (19) does not expose anycomplex or hypercomplex units, which differs from the representation in previouspublications of the author, e.g., in [40].One can calculate now the spin tensor σ µν of the Clifford algebra R , . Theresult is displayed in Eq. (20). The four elements in the first column of the matrix,which is given the index 0, can be identified with the basis elements of the algebra σ i = e i . The other elements can be calculated by direct matrix multiplication.Further insight is obtained if one breaks up the spin matrices into the basiselements of the base geometry by means of Eqs. (17) and (18). The calculationcan be performed even more easily than with matrices. One then obtains thespin tensor σ µν of the Clifford algebra R , in terms of the basis elements of theClifford algebra R , and the set of four hypercomplex units introduced in theprevious sections σ µν = − ı ⊗ e − ı ⊗ e − i ⊗ − ıj ⊗ ı ⊗ e ⊗ ıij − ıi ⊗ e − j ⊗ e ı ⊗ e − ⊗ ıij − ıi ⊗ e − j ⊗ e i ⊗ ıi ⊗ e ıi ⊗ e ıij ⊗ ıj ⊗ j ⊗ e j ⊗ e − ıij ⊗ . (20)The result can be compared with the corresponding matrix for the Clifford algebra R , in [21]. One finds that the third row σ ν as well as the third column σ µ areinserted and the notation is enriched by the tensor product. The tensor productis necessary, because the two diagonal matrices σ = 1 ⊗ ıij and σ = ıij ⊗
8. Commutation relations
The spin angular momentum operator corresponds to the spin tensor dividedby a factor of two s µν = σ µν . (21)The commutation relations of the spin angular momentum operator can be takenfrom the literature. They are summarized in the following sum of four terms[ s µν , s ρσ ] = g µσ s νρ − g µρ s νσ − g νσ s µρ + g νρ s µσ . (22)If one uses this formula and inserts the spin angular momentum operator givenby Eqs. (20) and (21) the corresponding metric tensor can be calculated. Onefinds g µν = − − . (23)Thus the matrix generators are referring to the Anti-de Sitter group SO (3 , , R ).Compared to [21], which referred at this stage to the Lorentz group SO (3 , , R ),the third column g µ and the third row g ν have been added.
9. Lie sphere geometry
The space R , can be represented as a paravector algebra with the Cliffordalgebra R , . It is this space, which is used to represent Lie sphere geometryin two-dimensional manifolds like R , the sphere S or the hyperbolic space H .One finds that a line in the Lie quadric of R , corresponds to a pencil of orientedspheres in R . Coordinates in the space R , thus represent these oriented spheres.The representation includes also point spheres and planes as limit cases. For moredetails it is referred to Cecil [8] and Jensen et al. [12].In the considered five-dimensional space the metric consists of two timelikecoordinates assigned to e and e . In order to match the metric with the Cliffordalgebra and identify the M¨obius geometry of the limit r = 0 correctly, one has toshuffle the coordinates compared to Cecil [8]. The two orientations of the spherein R with center p and unsigned radius r > (cid:20)(cid:18) p, ± r, − p · p + r , p · p − r (cid:19)(cid:21) . (24)Here the notation has been chosen exactly as in [8] to allow for a straightforwardcomparison.Clifford paravectors can now be formed by contracting the basis elements ofthe Clifford algebra R , with the above projective points. The correspondingvector of basis elements can be written out in detail as e µ = (1 , e , e , e , e ) . (25)With this procedure the Lie sphere geometry is assigned to elements of the con-sidered Clifford algebra.
10. Group limit and momentum operators
The Poincar´e group can be obtained from the Anti-de Sitter group SO (3 , , R )as a group limit. Following the discussion at the end of Section 7 the third rowand the third column is used to define the spin representation of the momentumoperators in the limit of small ǫ as p µ = ǫs µ . The spin angular momentum canbe restricted to the remaining Minkowski subspacelim ǫ → s ǫs s s s ǫs s s ǫs ǫs ǫs ǫs s s ǫs s s s ǫs s = s p s s s p s s − p − p − p − p s s p s s s p s . (26)The considered groups have been applied also in extensions of general relativityto the Anti-de Sitter group SO (3 , , R ), see the comments and reprints in the0corresponding section of Blagojevic and Hehl [41]. The momentum p should notbe confused with the center of the Lie sphere in Eq. (24).One should remember that now the Anti-de Sitter group is used to parametrizethe Lie Sphere geometry in the two-dimensional Euclidean plane R . For example,the momenta in the ambient space correspond to the generators, which change theradius of the Lie spheres within the two-dimensional base geometry. Whereas themomenta in the base geometry are a linear combination of the following angularmomentum operators in ambient space p = − s − s , p = − s − s . (27)These relations have been used before for example by Kastrup [42] and they havebeen applied also in [21]. The Lie sphere geometry in the plane R may findapplications for example in condensed matter physics [24].
11. Extension to higher dimensional spaces
With the scheme given by Eqs. (14) and (15) one can extend to higher dimen-sional geometries. One arrives at the Clifford algebra R , , which can be used asa paravector model for R , e µ = (1 , e , e , e , e , e , e ) . (28)The space R , can be used to represent a Lie quadric referring to oriented hy-perboloids, oriented point hyperboloids, and oriented hyperplanes in Minkowskispace R , . Thus, the center of the Lie sphere in Eq. (24) is an element ofMinkowski space. The Lie spheres as they have been originally considered byBateman and Timerding [17, 19] are thus a subgeometry.
12. Application to nuclear physics
The basis elements of the Clifford algebra R , are referring now to 8 × ×
16 complex matrices. The matrix structure can be usedto incorporate Pauli spin × weak isospin × hadronic isospin × Dirac anti-particlesstates. Each of the mentioned participants contributes with a factor of two tothe overall spin structure. Thus, the model is eligible to consider for examplethe hadronic level including protons, neutrons, neutrinos, and electrons. Withthe scheme given by Eqs. (14) and (15) it is possible to invoke further higherdimensional geometries. String theory even invokes a ten-dimensional space tohave enough structure to cover the Standard Model with its three generations ofquarks and leptons [43]. However, in order to proof feasibility of the concept it1seems to be better to take a step back and investigate nuclear processes withineffective quantum field theories on the hadron level [44].The motivation to work in spaces of a priori undetermined dimensionality isgiven by the hope that the method can identify the gauge symmetries in thesequence of geometric spaces to be of projective nature [45], whereas global sym-metries like the chiral symmetry are related to coordinate transformations ofthe geometrized internal symmetries in a space of fixed dimension. Note thatthe weak isospin is related to a local gauge symmetry, whereas the hadronicisospin is described by a global symmetry. In this context one could try to relatethe corresponding symmetry breaking mechanisms, which lead to either Higgs orGoldstone bosons, with a geometric perspective in order to derive particle masses.This thought can be motived by a hypothetical mass formula for electrons andprotons [21] 4 π exp (4 π ) = (cid:18) m p m e (cid:19) . (29)One may assume that the left-hand side of the equation is related to an angularmomentum. The right-hand side is given in terms of a squared mass ratio. Theformula is thus inspired by the relation between angular momentum and squaredmasses given by the Regge trajectories. The experimental proton to electronmass ratio is calculated from this mass formula with a deviation of 3 . π being related to the surface area of a unit sphere, a minordeformation towards an ellipsoid can change the surface area in a way that thevalues on both sides of Eq. (29) match exactly.
13. Scattering amplitudes in a Klein-Gordon theory for fermions
Beside such fundamental considerations one can investigate the question, whetherit is possible to reparametrize existing experimental data with the algebraic ex-pressions introduced in the previous sections. The intention is to work with aKlein-Gordon theory for fermions. For spinless particles the Klein-Gordon theoryprovides scattering amplitudes of the form [46, 47] h f | J µ | i i = p iµ + p fµ . (30)Here p i is a Minkowski space vector, which describes the momentum of the initialstate and p f the momentum of the outgoing state. In a fermion theory based onthe Klein-Gordon equation the matrix elements of Eq. (30) will be replaced by acurrent of the form h f | J µ | i i = ¯ u f ( p νi e ν ¯ e µ + e µ ¯ e ν p νf ) u i . (31)2This expression has been applied in [40] to Mott scattering and it has beenshown that the scattering amplitude is equivalent to the corresponding amplitudeobtained in the Dirac theory.The correspondence to the Dirac theory can be formally confirmed quite easily.Equation (31) can be transformed with the help of Eq. (19) into a representation,which is proportional to the Gordon decomposition of the Dirac theory h f | J µ | i i = ¯ u f ( d µ + σ µν q ν ) u i . (32)The initial and final momenta are encoded in the following expressions d = p f + p i , q = p f − p i . (33)One can now generalize these expressions to higher dimensional geometries. Theinitial and final momenta are now supposed to live in the momentum space cor-responding to the Lie sphere geometry in R , . The u i and u f are then eight-component spinors. As the spin tensor σ µν is evaluated with the basis elementsof the Clifford algebra R , given by Eq. (28), the matrix elements of the spintensor are bicomplex 8 × σ µν to further evaluate its substructure. Now the subcomponents arereferring to the basis elements of the Clifford algebra R , .One can now make first attempts to reparametrize existing experimental datain terms of the spin group of SO (4 , , R ) and interpret the results within Minkowskispace R , . Consider again the discussion in Section 10 within the correspondinglower dimensional geometry, where SO (3 , , R ) is replaced now by SO (4 , , R )and R by R , .
14. Summary
The modulo (1 ,
1) periodicity of Clifford algebras has been applied startingfrom the initial Clifford algebra R , . The sequence of higher dimensional Clif-ford algebras R n,n , generated by inverse projections, is related to Lie spheregeometries, which naturally include M¨obius geometries as their subgeometries.An essential part in this representation play the bicomplex numbers, which standin relation to S in the same way as the complex numbers stand in relation to S . The diffeomorphism of the bicomplex numbers to SU (2 , C ) and the relationto spinors is briefly discussed.It is indicated how the higher dimensional Clifford algebras can be assignedon a qualitative level to the internal particle symmetries. In an initial modelthe hadronic isospin can be combined with the weak isospin, the electromagneticspin, and the Dirac negative energy contributions to form a geometrized modelof the overall spin and isospin structure. The considered Clifford algebra is R , .3As the sequence of Clifford algebras is unlimited further higher dimensional ge-ometries can be invoked with the given scheme to include further internal particlesymmetries. References [1] R. Hermann,
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