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Symmetry means that something is unchanged after some sort of operation. In the case of geometrical symmetry it is the configuration that remains the same after rotating 180^{o} as shown in Figure 01ea. Whereas in theoretical physics symmetry the invariance is about the form of the equations before and after the transformation. Symmetry in mathematical expressions constrains the form from an infinite number of choices, it also allows extraction of information without deluging in detailed computation. A good example is the conservation laws emerging from the global symmetry of the equations. Some people see beauty in such formulation. | ||

## Figure 01ea Symmetry |
## Figure 01eb Conservation |

- Foundation of theoretical physics rests on the Lagrangian L (or Lagrangian Density
- Energy - This is a consequence of symmetry in temporal translation t' = t + t
_{0}. This law is valid in Newtonian mechanics. - Momentum - This is a consequence of symmetry in spatial translation x' = x + x
_{0}. This law is valid in Newtonian mechanics. - Angular Momentum - This is a consequence of symmetry in spatial rotation such as
_{}. This law is valid in Newtonian mechanics. - Energy-momentum - This follows the newer symmetrical operation - the space-time translation x'
_{k}= x_{k}+ a_{k}, where k runs from 0 to 3 (with the 0 index denotes the time component ct), the a_{k}'s are constants. This conservation rule is valid in Relativity and Standard Model. - Angular Momentum (in the new paradigm) - It is the product of symmetry under the Lorentz transformation
_{}in 3-D space. In quantum theory, the spin of the particle is added to the orbital angular momentum to obtain the total angular momentum. - Electric Charge - This quantity is conserved in EM (electromagnetism), Strong, and Weak interactions. The theoretical derivation is known as the Noether's Theorem. It is an additive quantum number as shown in Figure 01eb.
- Baryon Number - It is one of the eight conserved quantities via Noether's Theorem in QCD. It is valid in all the interactions (e.g., EM, Strong, and Weak) of the Standard Model, and computed by the formula B = (n
_{q}- n_{q'})/3, where q denotes quark, q' for anti-quark. The denominator of 3 is to compensate for the fact that every baryon is composed with three quarks. - Lepton Number - Similar to the conservation of baryon number, this conservation rule is also derived from the global U(1) symmetry.
It is conserved in all the interactions of the Standard Model, and computed by the formula L = (n
_{l}- n_{l'}), where l denotes lepton, l' for anti-lepton. The separate conservation rules for each lepton family are broken as the neutrinos are now found to have tiny non-zero mass (although the conservation of L still holds). The diagram below illustrates the various lepton numbers in a muon decay :

- Isospin 3rd component (in weak interaction) - This quantity is conserved in general, but has nothing to do with angular momentum. It is a label to distinct the various states in weak interaction, and somehow considered as some sort of weak charge by the relation :

Q = t_{3}+ (Y/2),

where Q is the electric charge, t_{3}is the third component of the isospin, and Y is the "hypercharge". The table below illustrates the assignment of these quantum numbers to various particles in weak interaction :

- Parity - Parity is the operation P that reverses the coordinate (x,y,z) to (-x,-y,-z) of a system. It is equivalent to a mirror reflection, i.e., (y) to (-y) followed by a 180
^{o}rotation through an axis perpendicular to the mirror, i.e., (x,z) to (-x,-z) (Figure 01fa). It has been long held that parity is conserved before and after all particle interactions. However, it was demonstrated conclusively - The parity operator P belong to the so-called Z
^{2}group, which is defined by IP=PI=P, and I^{2}=P^{2}=I, where I is the identity operator. Successive operations produce the multiplicative quantum numbers, e.g., P_{total}= (P_{a})(P_{b})... - There is a state vector associated with the P operator with eigenvalue +1 or -1 corresponding to either the symmetric (no change) or anti-symmetric (change by a minu sign only) state after the parity operation (Figure 01fc).
- Parity either with eigen-value +1 or -1, is conserved before and after an electromagnetic or strong interaction; but not in the weak interaction. For example, in the weak interaction process of beta decay:
**n****p**+**e**+^{-}_{}, P(before) = +1 which is not equal to P(after) = (+1)(+1)(-1) = -1 as demonstrated in the experiment. - For the ferminons with spin-1/2, the particles (electron, neutrino, and quark) have positive parity, while antiparticles (positron, anti-neutrino, and anti-quark) have negative parity.
- Bosons have same intrinsic parity for both particles and antiparticles. Spin-0 with positive parity such as the Higgs boson is called a scalar. Spin-0 particle with negative parity is called a pseudoscalar such as the and K mesons. A vector boson (photon) has spin-1 and negative parity; while a pseudovector boson has positive parity.
- Time Reversal - Most theoretical formulations are symmetrical under the t -t operation. However, the observable universe does not show such symmetry, primarily due to the "second law of thermodynamics".

- T violation can occur in three levels as shown below (from intrinsic to man-made) :
- Theory on the weak interaction.
- Second law of thermodynamics.
- Quantum non-invasive measurements.

#### Figure 01fd Time Reversal, Symmetry and Violation [view large image]

Figure 01fd shows the time reversal invariance in physical law and quantum process with T represents the operation. It also portrays a sequence of time irreversible implosion.

- The effect of time reversal on some classical variables are listed in the followings :
- T Parity = +1 - position, acceleration, force, energy, mass, electric potential, field, charge density, polarization, and all physical constants (except those associated with weak interaction).
- T Parity = -1 - time, velocity, linear momentum, angular momentum, power, electromagnetic vector potential, electric current density, magnetization, magnetic field and induction.

- Charge Conjugation - The C operation is another example of the Z
^{2}group. It changes the sign of quantum charges including the electrical charge, baryon number , lepton number, and the 3rd isospin (t_{3}), but doesn't affect the mass, linear momentum or spin of the particles.

- Only the strong and electromagnetic interactions obey charge conjugation symmetry. The weak interaction does not follow as C would transform a right-handed anti-neutrino to a right-handed neutrino, which does not exist.
- Since its eigen-value of +1 demands same charges before and after the conjugation operation, only truly neutral systems (those where all quantum charges and the magnetic moment are zero) are eigenstates of charge conjugation. The eligible ones : such as photon (, , , , Y) have c = -1; while those similar to
^{o}(, ') have c = +1. - It was believed for awhile that the separate violations of P symmetry and C symmetry would cancel out to preserve CP symmetry until 1964 when CP violation was revealed by the experiment with the neutral K meson decay (Figure 01eb). Note that the K meson does not have a definite value of c, i.e., it is not in either a state with c = +1 or -1; in other words, it is in a mixed states of both.

- CPT - The combined operations of C (e -e), P (
**x**-**x**), and T (t -t) preserve all theories including the classical, relativistic, and quantum. Thus, the CP violation in weak interaction is supposed to be compensated by a time reversal transformation.

There were many tests of CPT invariance over the years. The latest one in 2015 achieved an accuracy of several parts per trillion in its chance of being wrong. The test compares the cyclotron frequencies of antiproton p ^{-}and protron p^{+}(in the form of H^{-}atom for technical reason) by running them alternatively in the machine. As the formula shown in Figure 01fe, for a constant magnetic field B the cyclotron frequency is proportional to the charge to mass ratio q/m. After correcting the effect of the electrons in the H^{-}atom, the cyclotron frequency ratio is#### Figure 01fe Test for CPT Invariance with Cyclotron [view large image]

calculated to have the value : [(q/m) _{p-}/(q/m)_{H-}]_{cal}= 1.001089218754(2) if CPT invariance holds. Results from 6500 measurements yield : [(q/m)_{p-}/(q/m)_{H-}]_{exp}- [(q/m)_{p-}/(q/m)_{H-}]_{cal}= 1(69)x10^{-12}.

See detail in original article "High-precision comparison of the antiproton-to-proton charge-to-mass ratio".

## Figure 01fa Parity |
## Figure 01fb Parity Violation |
in 1957 by an experiment that parity conservation is violated in weak interaction (Figure 01fb). |

- In particle physics, parity is a property of particles called the

## Figure 01fc Parity, Even and Odd [view large image] |

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