PX156 - I4 - internal symmetries

these are known as gauge symmetries
in quantum mechanics, a particles is described using a complex valued wavefunction, $ψ$
it holds all the information about the state of the particle
probability of the particle being in the state is:

P = | ψ |^{2} = ψ^{*} ψ

the probability is an observable but $ψ$ can be changed without changing the probability
define: $ψ^{'} = e^{i χ} ψ$ , where, $χ =$ constant
but:

\begin{aligned} P (ψ^{'}) & = | ψ^{'} |^{2} \\ = ψ^{' *} ψ \\ = (e^{- i χ} ψ^{*}) (e^{i χ} ψ) \\ = ψ^{*} ψ \\ = | ψ |^{2} \\ ∴ P (ψ^{'}) & = P (ψ) \end{aligned}

changing the global phase doesn't affect the physics
this is a 'global' symmetry as it appears at every point in space-time by the same amount, $χ$
global changes do not agree with special relativity
want to change the phase by different amounts at different space-time points and have no effect on the physics, ie: $χ \to χ (\vec{x}, t)$
the physics should be unchanged when a change, $ψ^{'} = e^{i χ (\vec{x}, t)} ψ$ , is made
this is called 'local' symmetry
this will still satisfy $| ψ^{'} |^{2} = | ψ |^{2}$
$ψ$ obeys the the schrodinger equation, so should $ψ^{'}$

\begin{aligned} \vec{\nabla} ψ & = - i h \frac{\partial ψ}{\partial t} \\ \vec{\nabla} ψ^{'} & = \vec{\nabla} (e^{i χ (\vec{x}, t)} ψ) \\ = e^{i χ (\vec{x}, t)} \vec{\nabla} ψ + ψ e^{i χ (\vec{x}, t)} \vec{\nabla} χ (\vec{x}, t) \end{aligned}

the second term spoils the invariance of the symmetry
to make the physics invariant to local changes, new things need to be added to the theory, which cancel the effect of the $\vec{\nabla} χ (\vec{x}, t)$ term
these new things are boson fields $⟹$ bosons are needed $⟹$ interactions of particles and bosons $⟹$ charge conservation
charge conservation comes from symmetries to phase changes
the standard model is a "gauge theory"
this idea lies at the heart of theoretical particle physics