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Theoretical particle physics works by formulating the Lagrangian function for a system; the Lagrangian L being the difference between kinetic energy K and potential energy V; i.e.,
From the Lagrangian equations can be established that give the dynamics of the system. Then transformation of the equation are considered that leave the Lagrangian unchanged. These are the symmetries of the system. For examples of Lagrangian analysis see Particle Physics.
Noether's theorems are the most beautiful results in mathematical physics. The most famous says that if the Lagrange function is invariant to a continous transformation then there is a corresponding conserved quantity.
The Lagrange function is indepedent of a shift in the time variable then energy is conserved. Likewise shifts in the spatial coordinates leave the Lagrange function unaffected. Consequently linear momentum in each of the three directions is conserved. Shifts in the angle of orientation leave the Lagrangian unaffected and angular momentum is conserved.
For discrete transformations such as parity (right/left) shifts there are no conserved quantities.
It should be noted that there is definitely a relationship between fermions and bosons. The electrostatic charge of an electron is manifested as an electromagnetic field and a photon is a vibration in that field. Another way of saying this is that the force between electrostatically charged particles is carried by photons. The same applies for the nuclear weak and strong forces and the W and Z bosons. And also for the Higgs field and the Higgs boson. So there does exist fermion-boson partnerships but they are not necessarily unique; i.e., the partners of the electron and the proton are both photons.
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