Lie sphere geometry in nuclear scattering processes
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 . 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
In [45] it turned out to be useful to represent the Lie algebra with non-commutative hypercomplex units. This provides additional insights into the structure of higher dimensional Clifford algebras and their geometries. The matrices are introduced as The two generating matrices are denoted by ı and ȷ. Multiplication of the two elements results in The three matrices correspond to the Lie algebra of the special linear group . 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 dimensions. The basis elements of the paravector Clifford algebra are transformed to the basis elements of the
Start with the Clifford algebra
In [45] the series of projective spaces introduced in the previous section started from the complex numbers, referring to the Clifford algebra . 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 . Nevertheless, one can construct a paravector algebra based on , which is made up of the trivial identity basis element alone. The paravector algebra is thus
From the Clifford algebra to
With the Clifford algebra in place, Eqs. (14) and (15) can be applied again to generate the next higher dimensional geometry. One arrives now at the Clifford algebra consisting of four matrices. Equation (14) results in the first two basis elements of The elements and on the left-hand side of the equations refer to the higher dimensional geometry. The elements and in the tensor product refer to the basis elements of the Clifford algebra . 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 contributions Here the bar symbol refers to conjugation of the considered basis element. The symmetric contributions of the product are represented in terms of the metric tensor . The anti-symmetric contributions are identified with the spin tensor , 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 two 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 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 finds Thus the
Lie sphere geometry
The space can be represented as a paravector algebra with the Clifford algebra . It is this space, which is used to represent Lie sphere geometry in two-dimensional manifolds like , the sphere or the hyperbolic space . One finds that a line in the Lie quadric of corresponds to a pencil of oriented spheres in . Coordinates in the space 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 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 . The spin angular momentum can be restricted to the remaining Minkowski subspace
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 , which can be used as a paravector model for The space can be used to represent a Lie quadric referring to oriented hyperboloids, oriented point hyperboloids, and oriented hyperplanes in Minkowski space . 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 are referring now to matrices. As these matrices are generated over the bicomplex numbers they are in fact equivalent to 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] Here is a Minkowski space vector, which describes the momentum of the initial state and the momentum
Summary
The modulo periodicity of Clifford algebras has been applied starting from the initial Clifford algebra . The sequence of higher dimensional Clifford algebras , 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 in the same way as the complex numbers stand in relation to . The diffeomorphism of
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