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1. Parity (physics) – In quantum mechanics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is often described by the simultaneous flip in the sign of all three spatial coordinates, P, ↦. It can also be thought of as a test for chirality of a physical phenomenon, a parity transformation on something achiral, on the other hand, can be viewed as an identity transformation. All fundamental interactions of particles, with the exception of the weak interaction, are symmetric under parity. The weak interaction is chiral and thus provides a means for probing chirality in physics, in interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions. A matrix representation of P has determinant equal to −1, and hence is distinct from a rotation, in a two-dimensional plane, a simultaneous flip of all coordinates in sign is not a parity transformation, it is the same as a 180°-rotation. Under rotations, classical geometrical objects can be classified into scalars, vectors, in classical physics, physical configurations need to transform under representations of every symmetry group. Quantum theory predicts that states in a Hilbert space do not need to transform under representations of the group of rotations, the projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. For example, projective representations of the 3-dimensional rotation group, which is the orthogonal group SO, are ordinary representations of the special unitary group SU. Projective representations of the group that are not representations are called spinors. If one adds to this a classification by parity, these can be extended, for example, vectors and axial vectors which both transform as vectors under rotation. One can define reflections such as V x, ↦, which also have negative determinant, then, combining them with rotations one can recover the particular parity transformation defined earlier. The first parity transformation given does not work in an number of dimensions, though. In odd number of only the latter example of a parity transformation can be used. Parity forms the abelian group Z2 due to the relation P2 =1, all Abelian groups have only one-dimensional irreducible representations. For Z2, there are two representations, one is even under parity, the other is odd. These are useful in quantum mechanics, newtons equation of motion F = ma equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, however, angular momentum L is an axial vector, L = r × p, P = × = L

2. T-symmetry – In theoretical physics, T-symmetry is the theoretical symmetry of physical laws under a time reversal transformation, T, t ↦ − t. Although in restricted contexts one may find this symmetry, the universe itself does not show symmetry under time reversal. Hence time is said to be non-symmetric, or asymmetric, except for equilibrium states when the law of thermodynamics predicts the time symmetry to hold. However, quantum measurements are predicted to violate time symmetry even in equilibrium, contrary to their classical counterparts. Time asymmetries are generally distinguished as between those intrinsic to the physical laws, those due to the initial conditions of our universe. Physicists also discuss the time-reversal invariance of local and/or macroscopic descriptions of physical systems and our daily experience shows that T-symmetry does not hold for the behavior of bulk materials. Of these macroscopic laws, most notable is the law of thermodynamics. Many other phenomena, such as the motion of bodies with friction, or viscous motion of fluids, reduce to this. The question of whether this time-asymmetric dissipation is really inevitable has been considered by many physicists, the name comes from a thought experiment described by James Clerk Maxwell in which a microscopic demon guards a gate between two halves of a room. It only lets slow molecules into one half, only fast ones into the other, by eventually making one side of the room cooler than before and the other hotter, it seems to reduce the entropy of the room, and reverse the arrow of time. Many analyses have made of this, all show that when the entropy of room and demon are taken together. Modern analyses of this problem have taken into account Claude E. Shannons relation between entropy and information, many interesting results in modern computing are closely related to this problem — reversible computing, quantum computing and physical limits to computing, are examples. These seemingly metaphysical questions are today, in ways, slowly being converted into hypotheses of the physical sciences. The current consensus hinges upon the Boltzmann-Shannon identification of the logarithm of phase space volume with the negative of Shannon information, in this notion, a fixed initial state of a macroscopic system corresponds to relatively low entropy because the coordinates of the molecules of the body are constrained. As the system evolves in the presence of dissipation, the coordinates can move into larger volumes of phase space, becoming more uncertain. One can, however, equally well imagine a state of the universe in which the motions of all of the particles at one instant were the reverse, such a state would then evolve in reverse, so presumably entropy would decrease. Why is our state preferred over the other, one position is to say that the constant increase of entropy we observe happens only because of the initial state of our universe. Other possible states of the universe would actually result in no increase of entropy, in this view, the apparent T-asymmetry of our universe is a problem in cosmology, why did the universe start with a low entropy

3. CP violation – In particle physics, CP violation is a violation of CP-symmetry, the combination of C-symmetry and P-symmetry. CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle while its spatial coordinates are inverted. The discovery of CP violation in 1964 in the decays of neutral kaons resulted in the Nobel Prize in Physics in 1980 for its discoverers James Cronin and Val Fitch. It plays an important role both in the attempts of cosmology to explain the dominance of matter over antimatter in the present Universe, and in the study of weak interactions in particle physics. The strong interaction and electromagnetic interaction seem to be invariant under the combined CP transformation operation, historically, CP-symmetry was proposed to restore order after the discovery of parity violation in the 1950s. The idea behind parity symmetry is that the equations of physics are invariant under mirror inversion. This leads to the prediction that the image of a reaction occurs at the same rate as the original reaction. Parity symmetry appears to be valid for all reactions involving electromagnetism, until 1956, parity conservation was believed to be one of the fundamental geometric conservation laws. They proposed several possible direct experimental tests, overall, the symmetry of a quantum mechanical system can be restored if another symmetry S can be found such that the combined symmetry PS remains unbroken. Simply speaking, charge conjugation is a symmetry between particles and antiparticles, and so CP-symmetry was proposed in 1957 by Lev Landau as the symmetry between matter and antimatter. In other words, a process in which all particles are exchanged with their antiparticles was assumed to be equivalent to the image of the original process. Direct CP violation is allowed in the Standard Model if a complex phase appears in the CKM matrix describing quark mixing, a necessary condition for the appearance of the complex phase is the presence of at least three generations of quarks. The reason why such a complex phase causes CP violation is not immediately obvious, consider any given particles a and b, and their antiparticles a ¯ and b ¯. Now consider the processes a → b and the corresponding antiparticle process a ¯ → b ¯, before CP violation, these terms must be the same complex number. We can separate the magnitude and phase by writing M = | M | e i θ, if a phase term is introduced from the CKM matrix, denote it e i ϕ. Note that M ¯ contains the conjugate matrix to M, so it picks up a phase term e − i ϕ, however, consider that there are two different routes for a → b. In 1964, James Cronin, Val Fitch and coworkers provided clear evidence from kaon decay that CP-symmetry could be broken and this work won them the 1980 Nobel Prize. This discovery showed that weak interactions violate not only the charge-conjugation symmetry C between particles and antiparticles and the P or parity, but also their combination, the discovery shocked particle physics and opened the door to questions still at the core of particle physics and of cosmology today

4. Chirality (physics) – A chiral phenomenon is one that is not identical to its mirror image. The spin of a particle may be used to define a handedness, or helicity, for that particle, a symmetry transformation between the two is called parity. Invariance under parity by a Dirac fermion is called chiral symmetry, an experiment on the weak decay of cobalt-60 nuclei carried out by Chien-Shiung Wu and collaborators in 1957 demonstrated that parity is not a symmetry of the universe. The helicity of a particle is right-handed if the direction of its spin is the same as the direction of its motion and it is left-handed if the directions of spin and motion are opposite. By convention for rotation, a clock, with its spin vector defined by the rotation of its hands. Mathematically, helicity is the sign of the projection of the vector onto the momentum vector, left is negative. The chirality of a particle is more abstract and it is determined by whether the particle transforms in a right- or left-handed representation of the Poincaré group. For massive particles—such as electrons, quarks, and neutrinos—chirality and helicity must be distinguished, with the discovery of neutrino oscillation, which implies that neutrinos have mass, the only observed massless particle is the photon. The gluon is also expected to be massless, although the assumption that it is has not been conclusively tested, hence, these are the only two particles now known for which helicity could be identical to chirality, and only one of them has been confirmed by measurement. All other observed particles have mass and thus may have different helicities in different reference frames and it is still possible that as-yet unobserved particles, like the graviton, might be massless, and hence have invariant helicity like the photon. Only left-handed fermions and right-handed antifermions interact with the weak interaction, Chirality for a Dirac fermion ψ is defined through the operator γ5, which has eigenvalues ±1. Any Dirac field can thus be projected into its left- or right-handed component by acting with the projection operators /2 or /2 on ψ, the coupling of the charged weak interaction to fermions is proportional to the first projection operator, which is responsible for this interactions parity symmetry violation. A common source of confusion is due to conflating this operator with the helicity operator, since the helicity of massive particles is frame-dependent, it might seem that the same particle would interact with the weak force according to one frame of reference, but not another. The resolution to this paradox is that the chirality operator is equivalent to helicity for massless fields only. A theory that is asymmetric with respect to chiralities is called a chiral theory, many pieces of the Standard Model of physics are non-chiral, which is traceable to anomaly cancellation in chiral theories. Quantum chromodynamics is an example of a theory, since both chiralities of all quarks appear in the theory, and couple to gluons in the same way. The electroweak theory, developed in the mid 20th century, is an example of a chiral theory, originally, it assumed that neutrinos were massless, and only assumed the existence of left-handed neutrinos. However, it is still a chiral theory, as it does not respect parity symmetry, vector gauge theories with massless Dirac fermion fields ψ exhibit chiral symmetry, i. e. rotating the left-handed and the right-handed components independently makes no difference to the theory

5. Pin group – In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the group, just as the spin group maps 2-to-1 to the special orthogonal group. In general the map from the Pin group to the group is not onto or a universal covering space. The non-trivial element of the kernel is denoted −1, which should not be confused with the transform of reflection through the origin. The pin group of a definite form maps onto the orthogonal group, the pin groups for a positive definite quadratic form Q and for its negative −Q are not isomorphic, but the orthogonal groups are. In terms of the forms, O = O. Using the + sign convention for Clifford algebras, one writes Pin +, = Pin Pin −, = Pin and these both map onto O = O = O. By contrast, we have the natural isomorphism Spin ≅ Spin, there are as many as eight different double covers of O, for p, q ≠0, which correspond to the extensions of the center by C2. Only two of them are pin groups—those that admit the Clifford algebra as a representation and they are called Pin and Pin respectively. Every connected topological group has a universal cover as a topological space. In 2001, Andrzej Trautman found the set of all 32 inequivalent double covers of O x O, the compact subgroup of O. The two pin groups correspond to the two central extensions 1 → → Pin ± → O →1, the group structure on Spin is already determined, the group structure on the other component is determined up to the center, and thus has a ±1 ambiguity. The two extensions are distinguished by whether the preimage of a reflection squares to ±1 ∈ Ker, and the two pin groups are named accordingly. Concretely, in Pin+, r ~ has order 2, in 1 dimension, the pin groups are congruent to the first dihedral and dicyclic groups, Pin + ≅ C2 × C2 = Dih 1 Pin − ≅ C4 = Dic 1. In 2 dimensions, the distinction between Pin+ and Pin− mirrors the distinction between the group of a 2n-gon and the dicyclic group of the cyclic group C2n. In 3 dimensions the situation is as follows, the Clifford algebra generated by 3 anticommuting square roots of +1 is the algebra of 2×2 complex matrices, and Pin+ is isomorphic to SO × C4. The Clifford algebra generated by 3 anticommuting square roots of -1 is the algebra H ⊕ H and these groups are nonisomorphic because the center of Pin+ is C4 while the center of Pin− is C2 × C2. The center of Pin = Pin+ is C2 when n is even, C2 × C2 when n =1 mod 4, and C4 when n =3 mod 4