Abstract
In theories of the Kaluza-Klein kind there are spins or total angular moments in higher dimensions which manifest as charges in the observable $d=(3+1)$. The charge conjugation requirement, if following the prescription in ($3+1$), would transform any particle state out of the Dirac sea into the hole in the Dirac sea, which manifests as an anti-particle having all the spin degrees of freedom in $d$, except $S^{03}$, the same as the corresponding particle state. This is in contradiction with what we observe for the anti-particle. In this paper we redefine the discrete symmetries so that we stay within the subgroups of the starting group of symmetries, while we require that the angular moments in higher dimensions manifest as charges in $d=(3+1)$. We pay attention on spaces with even $d$.
Highlights
The definition of the discrete symmetries in the (3 + 1) dimensions after letting a series or rather a group of Killing transformations to manifest the corresponding Noether’s charges in (3 + 1), we shall denote these symmetries by CN, PN and TN, which means that we analyse the type of symmetries in the extra dimensional space leading to observed symmetries in (3 + 1)
(first to SO(7, 1) × U(1)II × SU(3) and ) to SO(3, 1) × SU(2)I × SU(2)II ×U(1)II ×SU(3), leaving all the family members massless (in the toy model case we found the solution for the compactification of the (x5, x6) surface into an almost S2 for particular spin connections and vielbeins) ensure that the spins in d > 4 manifest in d = (3 + 1) all the observed charges
We define in this paper the discrete symmetries, CN, PN and TN (eqs. (3.1), (3.4)) in even dimensional spaces leading in d = (3 + 1) to the experimentally observed symmetries, if the Kaluza-Klein kind of a theory [33,34,35,36,37,38,39] with d > (3 + 1) determining charges in d = (3 + 1)
Summary
Following the procedure used in the previous case of d = (5 + 1), the operator CH transforms, let say the first state, which represents due to its quantum numbers the right handed (with respect to d = (3 + 1)) u-quark with spin up, weak chargeless, carrying the colour charge ( This state solves the Weyl equation for the negative energy and inverse momentum, carrying all the eigenvalues of the Cartan subalgebra operators (S12, S56, S78, S9 10, S11 12, S13 14), except S03, of the opposite values than the starting state The crucial point really is that the N -indexed operators CN , PN(d−1) and TN with their associated x-transformations do not transform the extra (d − 4) coordinates so that background fields depending on these extra dimension coordinates do not matter
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