Operational Discrete Symmetries and CP-Violation

The discrete symmetries of the Lorentz group are on the one hand a ‘complex’ interplay between linear and anti-linear operations on spinor fields and on the other hand simple linear reflections of the Minkowski space. We define operations for T , CP and CPT leading to both kinds of actions. These operations extend the action of SL(2, CI ), representing the action of the proper orthochronous Lorentz group SO(1, 3) on the Weyl spinors, to an action of the full group O(1, 3). But it is more instructive to reverse the arguments. The action of O(1, 3) is the natural way how SL(2, CI ) together with its conjugation structure acts on Minkowski space. Focusing on the symmetries of these (anti-)linear operations we can for example distinguish between CP -invariant and CP -violating symmetries. This is important if gauge symmetries are included. It turns out that, contrary to the general belief, CP and T are not compatible with SU(n) for n ≥ 3, especially with SU(3)colour or with the U(3)-Cabibbo-Kobayashi-Maskawa matrix. 1 e-mail adresses: M.H.: [email protected] H.S.: [email protected] The history of the discrete symmetries was a history of surprises. For example when C.S. Wu discovered parity violation (after theoretical advice given by Lee and Yang), Wolfgang Pauli wrote to his former assistant Viktor Weisskopf: ”What shocks me is not the fact that ‘God is just left-handed’ but the fact that in spite of this He exhibits Himself as left/right symmetric when He expresses Himself strongly. In short, the real problem now is why the strong interactions are left/right symmetric. How can the strength of an interaction produce or create symmetry groups, invariances or conservation laws? This question prompted me to my premature and wrong prognosis. I don’t know any good answer to that question but one should consider that already there exists a precedent: the rotational group in isotopic spin-space, which is not valid for the electromagnetic field. One does not understand either why it is valid at all. It seems that there is a certain analogy here!” [1]. Even more unexpected was the discovery of CP -violation by Finch et al. Why was there such a surprise? Beginning with the discovery of spin by Stern and Gerlach and with the theoretical work of Dirac and Weyl the ‘real version’ of the Lorentz group, i.e. O(1, 3), lost more and more of its fundamental meaning and should have been replaced by SL(2, CI ). But therein parity is not defined. Only after the discovery of parity violation the Weyl theory became familiar. Nowadays the Standard Model is written in Weyl spinors. Is there a similar hint for CP -violation? In order to answer this question one has to take a look at the representation structure of the Lorentz group on the Weyl spinors. Is it possible to understand the discrete part of the Lorentz group like the continuous part? In the first three sections we define the discrete symmetries as (anti-)linear operations within the different kinds of Weyl spinors. Their action on the Cartan (bispinor) representation of the Minkowski space is the familiar action of the discrete symmetries on Minkowski space. Next we show that these operations lead also to the discrete symmetry operations on Dirac spinor fields. The discrete symmetry operations on the Weyl spinors are deeply connected with the spin and boost structure of SL(2, CI ). Including also inner symmetries we show that the discrete symmetries are not compatible with every representation of the inner symmetry groups. This holds especially for CP and T in an SU(3) gauge theory. It is interesting that for the same reason the U(3)-Cabibbo-Kobayashi-Maskawa matrix breaks CP invariance.