Symmetry (physics)

2024-09-1611:08 Status:PHYS106 Tags: Group theory

The environment of a physical system has a symmetry if there is some operation we can perform on the system that gives us an equivalent environment/configuration.

Translational symmetry Time translation symmetry = doing something later Rotational symmetry Combinations of symmetry Reflection symmetry*** Boost symmetry = constant velocity + Relativistic mechanics

For each symmetry in the environment of a physical system, there will be a conserved quantity, some physical entity that cannot change with time. = Noether’s theory

= Unchanging time entails conservation of energy - can associate a quantity called energy to a system, and this doesn’t change with time.

• Throwing up a backpack - hand = environment, backpack = system. As soon as the environment interacts with the system, there is no longer a guarantee of time symmetry
• Einstein: The quantity we call mass quantifies the energy of an object when it is stationary: $E=\Delta m{c}^2$ Conservation of energy implies the mass of an isolated object cannot change.

= For an environment that unchanged if we shifted come direction (e.g. moves to the left) there is a conservation of momentum in that direction which at low velocities (relative to the speed of light) is $p\approx mv$, otherwise ($p=\frac{mv}{\sqrt{1-\frac{v_x^2+v_y^2+v_z^2}{c^2}}}$). This is often for perfect, frictionless vacuums (air breaks conservation of momentum). Additionally, momentum is additive.

For an environment with translational symmetry in some direction, the component of total momentum in that direction is conserved.

• e.g. outer space $p$ is constant for single objects

$p_n^{before}=p_n^{after}$ where $n$ are the dimensions, $x,y,z$

For an isolated object in outer space energy conservation (mass is constant), momentum in a

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