Quantum Field Theory

Conservation Rules

		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 180o 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 [view large image]	Figure 01eb Conservation [view large image]

Foundation of theoretical physics rests on the Lagrangian L (or Lagrangian Density in case of field equations) from which the equations of motion (or field equations) are derived. The very specific examples in Figure 01ea are taken from Newtonian mechanics (for symmetry in temporal and spatial translation). The case on free fall in gravity is valid only near the surface of the Earth (or any spherically solid body), while the one for cruising rocket can be applied to any force free region (such as the inter-stellar or inter-galactic space). The space-time symmetry in General Relativity and Standard Model (for elementary particles) is more general, but may not be valid in very early universe or at Planck scale. Anyway, the following is a short list of some familiar conservation laws :


• Energy - This is a consequence of symmetry in temporal translation t' = t + t₀. This law is valid in Newtonian mechanics.


• Momentum - This is a consequence of symmetry in spatial translation x' = x + x₀. 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 = t₃ + (Y/2),
  where Q is the electric charge, t₃ 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 180o 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.
Figure 01fa Parity [view large image]	Figure 01fb Parity Violation	However, it was demonstrated conclusively 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 intrinsic parity.



The parity operator P belong to the so-called Z2 group, which is defined by IP=PI=P, and I2=P2=I, where I is the identity operator. Successive operations produce the multiplicative quantum numbers, e.g., Ptotal = (Pa)(Pb)... 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.

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

A Table of Parity for some classical variables and elementary particles :



As mentioned in the above table, parity has no effect on scalar since it has zero dimension, there is no direction to re-orient. Some vector such as the angular momentum L = r x p has even parity (null effect) because the change of parity in the two defining vectors canceled out (-x- = +). Other even parity vector such as magnetic field is the consequence of demanding the physical equation to be either even or odd parity (see Ampere's Law in classical electromagnetism). On the other hand, a scalar variable such as the helicity can have odd parity because it is the inner product of two vectors with opposite 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	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 following table :
  
  
  
  
• Charge Conjugation - The C operation is another example of the Z² group. It changes the sign of quantum charges including the electrical charge, baryon number , lepton number, and the 3rd isospin (t₃), 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 CPT Theorem states that "Quantum Field Theory (QFT)" is invarient under the combined operation of CPT, where
  C (e -e), P (x -x), and T (t -t) (see Figure 01fe), if the following conditions are met :
  
  

  The theory must possess a Hermitian Lagrangian. It is invariant under Lorentz transformation. Quantization of integer spin quantum field must use commutation relations, while anticommutation relations are used for the half integer spin quantum field. See "Second Quantization". Particles and antiparticles have identical masses and lifetimes. All other internal quantum numbers of antiparticles are opposite to those of the particles. See "Charge Conjugation".

  Figure 01fe CPT for Electron [view large image]

  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 01ff, 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

  Figure 01ff Test for CPT Invariance with Cyclotron [view large image]

  ratio is 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".

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