﻿ Principles of Invariance in Physics
San José State University

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Principles of Invariance
in Physics

## Background

Theoretical particle physics works by formulating the Lagrangian function for a system; the Lagrangian L for a system being the difference between kinetic energy its K and its potential energy V; i.e.,

#### L = K − V

From the Lagrangian equations can be established that give the dynamics of the system. Then transformations 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.

# Examples of Principles of Invariance in Physics

• ## Noether's Theorem(s)

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.

If the Lagrange function is independent of a shift in the time variable then energy is conserved. Likewise if shiftsn in the spatial coordinates leave the Lagrange function unaffected then the linear momentum in each of the three directions is conserved. If shifts in the angle of orientation leave the Lagrangian unaffected then angular momentum is conserved.

For discrete transformations such as parity (right/left) shifts there are no conserved quantities.

• ## Dirac's Antiparticles

Paul Dirac hypothesized the changing the signs of particles' charges should leave the physics of particle systems unaffected. From this he deduced the existence of anti-particles, which were subsequently found, the first being the positron. Subsequently a negatively charged particle with all other characteristics identical to the proton was found. Neutral particles may be their own antiparticle.

Instead of considering changes in the charges of particles this procedure has come to be considered as an interchange of particles and their anti-particles. The success of Dirac's speculation motivated the theoretical search for other symmetries involving interchanges of particles,

• ## The Higgs Boson and the Mass of Particles

The interchanges of any particles having mass, such as the quarks, leptons and the W and Z bosons, would seem to require that they have the same mass in order for their interchange to leave the Lagrangian unaffected. This is the case if the particles do not have an interaction with the Higgs boson. In fact, due to other considerations without that interaction their masses would have to be zero, as is the case photons For more on this see Higgs physics.

• ## Interchange of Spin Types

The next speculation was that for all particles there exists a partner particle differing only in spin. Bosons have integral spins and fermions half-integral spins. The Lagrangian has to be such that it would be unaffected by an interchange of particles and their partners. This is called supersymmetry.

At first .supersymmetry was considered just an interesting speculation but it was realized that the reality of supersymmetry would solve certain problems with the Standard Model. For more on this matter see Benefits of Supersymmetry.

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 samels thought to apply 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.