Gauge Invariance and Conserved Quantities

Gauge Invariance and Conserved Quantities“Noether's theorem” was proven by German mathematician, Emmy Noether in 1915 and published in 1918. Noether's theorem has become a fundamental tool of quantum field theory – and has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics".

Amalie Emmy Noether 1882-1935

Consider the charged scalar field,

and the following transformation on , where is a constant.

Now suppose that can be varied continuously – so that we havean infinite number of small, continuous for which exp(i ) = ’ and L L. The set of all these transformations, U = exp(i ), form a group ofoperators. It is called U(1), a unitary group (because U*U = 1)

Here the Hermitian conjugate is the complex conjugatebecause U is not a matrix.

=1

Now, we have a more astounding result: we can vary the (complex) phaseof the field operator, , everywhere in space by any continuous amount and not affect the “laws of physics” (that is the L) which govern the system! Note that everywhere in space the phase changes by the same . This is called a global symmetry.

’

Remember Emmy Noether!

With the help of Emmy Noether, we can prove that charge is conserved!

Deriving the conserved current and the conserved charge:

Euler-Lagrange equation conserved current

But our Lagrangian density also contains a *, so we obtain additional termslike the above, with replaced by *. In each case the Euler –Lagrange equationsare satisfied. So, the remaining term is as follows:

The conserved current condition is

To within an overall constant the conserved current is :

a four – vector!

Now we need to evaluate .

The great advantage of being a continuous constant is that thereare an infinite number of very small which carry with them all the physics of the “continuity”. That is, with no loss of rigor we can assume is small!

Finally, the conserved current operator (to within an overall constant) for the charged, spin = 0, particle is

Since we may adjust the overall constant to reflect the charge of the particle, we can replace with q. The formalism givesoperator for charge, but not the numerical value.

The value of the charge is calculated from:

pincoming outgoing

incomingparticle

outgoingparticle integrate over all space

integrate over timeS0(t)

One obtains a number!

In this calculation

‘ Note Dirac delta function in k’ and p

Note Dirac delta function in k and p

‘

Next, we can do the integrations over d3x. Each gives a Dirac delta function in k and k’.

The time disappears! Q is time independent.

The integration over k’ is done with the Dirac delta function from the d3x integration.

The remaining integration over k will be done with the Dirac delta functions from the commutation relations.

Note: + and/or – must be together.