Lie sphere geometry in nuclear scattering processes

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Abstract

The Lie sphere geometry is a natural extension of the Möbius geometry, where the 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 of Möbius geometries, which has been investigated recently in a bicomplex matrix representation. When higher dimensional space-time geometries are invoked by inverse projections starting from an originating point geometry, the Lie sphere scheme provides a more natural structure of the involved Clifford algebras compared to the previous representation. The spin structures resulting from the generated Clifford algebras can potentially be used for the geometrization of internal particle symmetries. A simple model, which includes the electromagnetic spin, the weak isospin, and the hadronic isospin, is suggested for further verification.

Introduction

The spin of a particle can be understood in terms of irreducible representations of the Poincaré group. Thus, it has a clear geometric background. Since the success of the concept of isospin in the middle of the last century physicists and mathematicians consider a possible geometrization of the internal symmetries of a quantum particle, see for example Hermann [21]. Today there is the hope that the geometrization problem can be addressed by string theory and the AdS/CFT correspondence [20], [31], [46]. The acronym CFT indicates that the conformal symmetry plays a substantial role in these considerations. However, in principle other geometries and symmetries could be considered. Even if one takes a step back into a simplified world which consists only of two space dimensions, there are overall 23 families of Klein geometries [26].

Among these geometries one finds the Lie sphere geometry [28], which has been considered in detail for example by Cecil and Chern [8], [9], Cecil and Ryan [10], Sharpe [40], Benz [3], and Jensen et al. [22]. There is also the software library of Kisil [24] for graphical representation. The Lie sphere geometry has been applied in the last decades in the study of Dupin submanifolds [16]. This research has been initiated by the thesis of Pinkall [34]. The Lie sphere geometry considers points and lines as special cases of spheres, which can again be understood as points in an ambient space. Lie spheres have been applied in physics by Bateman, Timerding, and Cunningham within optics and electrodynamics [1], [2], [13], [42]. The radius of the Lie sphere is identified in these applications with r=ct. This connection between geometry and relativistic physics has been investigated also by Cartan [7]. Thus, it seems to be worth to consider the Lie spheres for further applications. The following discussion therefore suggests to generalize the sequence of Möbius geometries introduced in [45] and apply the Lie sphere geometry instead of the Möbius geometry. From the perspective of physics one can then address the question mentioned in the beginning of this introduction and try to identify the isospin symmetries of electroweak and strong interactions within the algebraic spin representations of a hierarchy of Lie sphere geometries.

The importance of projective geometry in relativistic physics has been noted already by Klein [25]. Today string theory and the AdS/CFT holography point towards the importance of projective geometry in the representation of the fundamental geometries in physics. Furthermore, one may assume that the dimensionality of physical space is not given a priori and the emergence of space-time has to be considered in a generalized geometric context [38], [47]. In AdS/CFT and string theory the geometry is generalized to higher dimensional spaces by adding space dimensions. A constraint on the total number of dimensions has been provided by supergravity [14], [33]. In [45] and in the following discussion the next level of dimensionality is reached by adding one space and one time dimension, which is in line with the conformal compactification. This guarantees that the concepts of projective geometry can be applied without modification within the series of geometric spaces of different dimension. However, the special importance of Minkowski space-time cannot be derived in a natural way at this level of investigation.

Compared to its predecessor [45] the following discussion, which is based on the Lie sphere geometry instead of the Möbius geometry, provides a few advantages. As shown in Section 4 the generalized approach is more natural with respect to the signature of the applied Clifford algebras. The number of basis elements with positive square is equal to the number of basis elements with negative square. In addition, the geometry originates by inverse projections out of a line instead of a plane, which is discussed in Section 5. The line can be identified in its projective context 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 dimensionality. Section 7 outlines how the content of [45] is included as a subalgebra in the generalized representation. In Section 9 the algebraic representation is connected with the Lie sphere geometry. The mass formula for electrons and protons, which has been discussed already in [45], indicates the emergence of particle configurations from their geometric context. Finally, Section 12 shows that the approach is, compared to [45], more consistent with nuclear physics, because the electromagnetic spin, the weak isospin, and the hadronic isospin can be assigned to the Clifford matrix representation of the proposed geometry. The approach therefore appears to be more suitable than [45] for the geometrization of internal particle symmetries.

Section snippets

Bicomplex numbers

The bicomplex numbers have been discussed by Segre [37] in 1892. An equivalent number system called tessarines has been introduced by Cockle [12] even in 1848. Detailed introductions with more information on the history of bicomplex numbers can be found for example in [29], [36]. The bicomplex number can be further extended to multicomplex numbers, see Price [35]. An application of multicomplex numbers in physics has been proposed recently in [11]. The bicomplex number has one real and three

Hypercomplex units for the representation of the Lie algebra sl(2,R)

In [45] it turned out to be useful to represent the Lie algebra sl(2,R) with non-commutative hypercomplex units. This provides additional insights into the structure of higher dimensional Clifford algebras and their geometries. The matrices are introduced ası=(0110),ȷ=(0110). The two generating matrices are denoted by ı and ȷ. Multiplication of the two elements results inıȷ=(1001)=ȷı. The three matrices correspond to the Lie algebra of the special linear group SL(2,R). They will be used as

Hypercomplex representation of the Maks periodicity of Clifford algebras

The Clifford algebra periodicity, which has been investigated from a mathematical point of view by Maks [30], will be applied in a hypercomplex representation similar to [45] in order to generate higher dimensional geometries by inverse projections. However, compared to [45] a different set of Clifford algebras is involved. The base geometry has now an odd number of 2m+1=n dimensions. The n1 basis elements of the paravector Clifford algebra Rm,m are transformed to the basis elements of the

Start with the Clifford algebra R0,0

In [45] the series of projective spaces introduced in the previous section started from the complex numbers, referring to the Clifford algebra R0,1. However, from a conceptual point of view, it is more appealing to start from a null algebra consisting of the empty set. This null algebra can be assigned to the Clifford algebra R0,0. Nevertheless, one can construct a paravector algebra based on R0,0, which is made up of the trivial identity basis element e0=1 alone. The paravector algebra is thus

From the Clifford algebra R1,1 to R2,2

With the Clifford algebra R1,1 in place, Eqs. (14) and (15) can be applied again to generate the next higher dimensional geometry. One arrives now at the Clifford algebra R2,2 consisting of four matrices. Equation (14) results in the first two basis elements of R2,2e1=ıȷe1,e2=ıȷe2. The elements e1 and e2 on the left-hand side of the equations refer to the higher dimensional geometry. The elements e1 and e2 in the tensor product refer to the basis elements of the Clifford algebra R1,1. Two

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 contributionseμe¯ν=gμν+σμν. Here the bar symbol e¯ν refers to conjugation of the considered basis element. The symmetric contributions of the product are represented in terms of the metric tensor gμν. The anti-symmetric contributions are identified with the spin tensor σμν, see [45] for more details. The right hand side of Eq.

Commutation relations

The spin angular momentum operator corresponds to the spin tensor divided by a factor of twosμν=σμν2. The commutation relations of the spin angular momentum operator can be taken from the literature. They are summarized in the following sum of four terms[sμν,sρσ]=gμσsνρgμρsνσgνσsμρ+gνρsμσ. If one uses this formula and inserts the spin angular momentum operator given by Eqs. (20) and (21) the corresponding metric tensor can be calculated. One findsgμν=(1000001000001000001000001). Thus the

Lie sphere geometry

The space R3,2 can be represented as a paravector algebra with the Clifford algebra R2,2. It is this space, which is used to represent Lie sphere geometry in two-dimensional manifolds like R2, the sphere S2 or the hyperbolic space H2. One finds that a line in the Lie quadric of R3,2 corresponds to a pencil of oriented spheres in R2. Coordinates in the space R3,2 thus represent these oriented spheres. The representation includes also point spheres and planes as limit cases. For more details it

Group limit and momentum operators

The Poincaré group can be obtained from the Anti-de Sitter group SO(3,2,R) as a group limit. Following the discussion at the end of Section 7 the third row and the third column is used to define the spin representation of the momentum operators in the limit of small ϵ as pμ=ϵsμ2. The spin angular momentum can be restricted to the remaining Minkowski subspacelimϵ0(0s01ϵs02s03s04s100ϵs12s13s14ϵs20ϵs210ϵs23ϵs24s30s31ϵs320s34s49s41ϵs42s430)=(0s01p0s02s03s100p1s12s13p0p10p2p3s20s21p20s23s30s31p

Extension to higher dimensional spaces

With the scheme given by Eqs. (14) and (15) one can extend to higher dimensional geometries. One arrives at the Clifford algebra R3,3, which can be used as a paravector model for R4,3eμ=(1,e1,e2,e3,e4,e5,e6). The space R4,3 can be used to represent a Lie quadric referring to oriented hyperboloids, oriented point hyperboloids, and oriented hyperplanes in Minkowski space R3,1. Thus, the center of the Lie sphere in Eq. (24) is an element of Minkowski space. The Lie spheres as they have been

Application to nuclear physics

The basis elements of the Clifford algebra R3,3 are referring now to 8×8 matrices. As these matrices are generated over the bicomplex numbers they are in fact equivalent to 16×16 complex matrices. The matrix structure can be used to incorporate Pauli spin × weak isospin × hadronic isospin × Dirac anti-particles states. Each of the mentioned participants contributes with a factor of two to the overall spin structure. Thus, the model is eligible to consider for example the hadronic level

Scattering amplitudes in a Klein-Gordon theory for fermions

Beside such fundamental considerations one can investigate the question, whether it is possible to reparametrize existing experimental data with the algebraic expressions introduced in the previous sections. The intention is to work with a Klein-Gordon theory for fermions. For spinless particles the Klein-Gordon theory provides scattering amplitudes of the form [4], [32]f|Jμ|i=piμ+pfμ. Here pi is a Minkowski space vector, which describes the momentum of the initial state and pf the momentum

Summary

The modulo (1,1) periodicity of Clifford algebras has been applied starting from the initial Clifford algebra R0,0. The sequence of higher dimensional Clifford algebras Rn,n, generated by inverse projections, is related to Lie sphere geometries, which naturally include Möbius geometries as their subgeometries. An essential part in this representation play the bicomplex numbers, which stand in relation to S3 in the same way as the complex numbers stand in relation to S1. The diffeomorphism of

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