C-symmetry

An early surprise appeared in the 1950s, when Chien Shiung Wu demonstrated that the weak interaction violated P-symmetry.For several decades, it appeared that the combined symmetry CP was preserved, until CP-violating interactions were discovered.It is currently believed that CP-violation during the early universe can account for the "excess" matter, although the debate is not settled.routinely suggested that perhaps distant galaxies were made entirely of anti-matter, thus maintaining a net balance of zero in the universe.The various fundamental particles can be classified according to behavior under charge conjugation; this is described in the article on C-parity.Gauge symmetry, in this geometric setting, is a statement that, as one moves around on the circle, the coupled object must also transform in a "circular way", tracking in a corresponding fashion.More formally, one says that the equations must be gauge invariant under a change of local coordinate frames on the circle.In (vastly) simplified terms, it is a technique for performing calculations to obtain solutions for a system of coupled differential equations via perturbation theory.A key ingredient to this process is the quantum field, one for each of the (free, uncoupled) differential equations in the system.The quantum field plays a central role because, in general, it is not known how to obtain exact solutions to the system of coupled differential questions.The quantum field provides exactly this: it enumerates all possible free-field solutions in a vector space such that any one of them can be singled out at any given time, via the creation and annihilation operators.This prescription for a quantum field naturally generalizes to any situation where one can enumerate the continuous symmetries of the system, and define duals in a coherent, consistent fashion.They are naturally anti-commuting (this follows from the Clifford algebra), which is exactly what one wants to make contact with the Pauli exclusion principle.piece is associated with the determinant bundle of the spin structure, effectively tying together the left and right-handed spinors through complex conjugation.To tie all of these together into a knot, one finally has the concept of transposition, in that elements of the Clifford algebra can be written in reversed (transposed) order.The other is to realize that, in low dimensions (in low-dimensional spacetime) there are many "accidental" isomorphisms between various Lie groups and other assorted structures.The laws of electromagnetism (both classical and quantum) are invariant under the exchange of electrical charges with their negatives.interpreted as an anti-particle field, satisfying the complex-transposed Dirac equation Note that some but not all of the signs have flipped.that transposes the gamma matrices to insert the required sign-change: The charge conjugate solution is then given by the involution The 4×4 matrixThis matrix is representation dependent due to a subtle interplay involving the complexification of the spin group describing the Lorentz covariance of charged particles.[4] This can be interpreted as saying that this phase may vary along the fiber of the spinor bundle, depending on the local choice of a coordinate frame.Put another way, a spinor field is a local section of the spinor bundle, and Lorentz boosts and rotations correspond to movements along the fibers of the corresponding frame bundle (again, just a choice of local coordinate frame).Because C-symmetry is a discrete symmetry, one has some freedom to play these kinds of algebraic games in the search for a theory that correctly models some given physical reality.It is very tempting, but not quite formally correct to just multiply these out, to move around the location of this minus sign; this mostly "just works", but a failure to track it properly will lead to confusion.If CP is combined with time reversal (T-symmetry), the resulting CPT-symmetry can be shown using only the Wightman axioms to be universally obeyed.Note, however, spinors as defined abstractly in the representation theory of Clifford algebras are not fields; rather, they should be thought of as existing on a zero-dimensional spacetime.The relationship to the P and T symmetries for a fermion field on a spacetime manifold are a bit subtle, but can be roughly characterized as follows.P and T operations applied to the spacetime manifold can then be understood as also flipping the coordinates of the tangent space as well; thus, the two are glued together.Under a change of basis of the tangent space, elements of the Clifford algebra transform according to the spin group.The Weyl spinors, together with their complex conjugates span the tangent space, in the sense that The alternating algebra
physicstransformationparticlesantiparticleschargeselectric chargeP-symmetryT-symmetryfundamental forceselectromagnetismgravitystrongweak interactionsnaturecontinuous symmetriesmotiondiscrete symmetriesChien Shiung WuCP-violatingNobel Prizesmatteranti-mattercosmologyDirac equationquantum field theoryfundamental particlesC-parityKlein–Gordon equationRiemannian geometrycomplex conjugationelectrical chargeNoether currentsclassical electrodynamicsMaxwell's equationsfiber bundlecircle bundleelectromagnetic potentialgauge connectionEhresmann connectiongauge-invariantcoordinate framesanti-particlesperturbation theoryquantum fieldcreation and annihilation operatorsladder operatorscanonical commutation relationsLorentz invarianceframe bundlecolor chargefundamental representationsCasimir invariantsuniversal enveloping algebraspacetimemanifoldline bundledeterminant bundleRiemannianpseudo-Riemannian manifoldstangent bundlecotangent bundlemetricdifferential equationstangentcotangent spacescalculus on manifoldsLaplaciansymplectic manifoldscanonical coordinatesspin structurespinorsMinkowski spacetimeClifford algebraClifford bundlespin manifoldWeyl spinorsPauli exclusion principlechiral elementgamma matrixpseudo-orthogonal groupLorentz groupspin groupdouble coveringDirac operatorstranspositionLie groupsclassicalquantuminvariantelectronsquarksfundamental particlefermiongamma matricesinvolutiongamma groupcomplex numberchiralityangular momentum operatortotally antisymmetric tensorhelicitychiral operatorWeyl equationPauli matricesparityMajorana conditionspinor bundlesectionMajorana spinorssecond-quantizedunitary operatorneutrinoantineutrinoStandard Modelleft-right modelsscalar fieldscomplex planeCartesian productdiscrete symmetryCP-symmetrymesonsCP violationCKM matrixCPT-symmetryWightman axiomshigher-dimensional gamma matricesWeyl–Brauer matricesClifford algebrastangent spacetangent manifoldvector spacetensor algebrainner productprinciple fiber bundleC parityG-parityAnti-particleAntimatterTruly neutral particleCPT symmetryPin groupSymmetry (physics)