tulip said:
Thank you everyone for the replies, they've got things a bit clearer in my head. I'll take a look at the book recommendations when I get to the library tomorrow.
xepma, I really liked the way you described things, but there's just one thing I'm still confused about. If U(1) is a group that acts to change the phase of the wavefunction, and the phase is a non-physical property of a particle, doesn't that mean that systems of any type of particle will have a symmetry under U(1)? Noether's theorem tells us that symmetries lead to conserved charges - so what is the conserved charge associated with this phase change symmetry?
A very astute observation! The U(1) symmetry associated with the
global phase invariance is indeed valid for all sorts of particles. The conserved quantity associated with this symmetry is the number of particles.
As I understood it, a U(1) symmetry exists in the theory of QED (which applies to charged particles only) where it gives rise to the conservation of electric charge - is this correct?
My first comment is that this type of symmetry is a different type of symmetry then the U(1) global phase invariance. This really is the domain of gauge theory.
The idea is as follows: the representations I mentioned above are associated with the global properties of the wavefunction / particle.
Recall that when we act on the wavefunction with a symmetry operation h, we actuall mean that we act on the representation space V through the mapping F(h) (where F is the representation of the action of h on V). The wavefunction is said to live in the representation space V. We talk of a global symmetry, since the entire wavefunction is an element of one representation space.
In gauge theory this idea is extended. The global symmetry is turned into a
local symmetry. You can imagine that we can assign a representation space V to each spatial point x, giving V(x). Performing a symmetry operation means that we perform a symmetry operation at each point x. We can perform a different symmetry operation at each point x, i.e. the group "element" is x-dependent: h(x). For instance, a local U(1) phase transformation now looks like e^{ie\phi(x)}.
I assume here that although we have an infinite number of representation spaces, V(x), we still have "the same" type of representation at each point x (i.e. same dimension etc). This representation is labeled by the
charge e of the particle with respect to the symmetry group G. This is the mathematical meaning of the charge: it labels the kind of representation the particle carries (for both local and global symmetries).
For a U(1) symmetry we simply have the electrical charge. For non-Abelian local symmetries, like SU(3), it leads to the notion of color charge. Quarks carry red, green or blue charge, meaning they live in the 3-dimensional representation of SU(3), called the fundamental representation. Gluons, on the other hand, live in the 8-dimensional representation space of SU(3), called the adjoint representation.
The fact that we now have a local symmetry has major consequences. We know now that we change the phase of the wavefunction in a space-dependent way, i.e.
\Psi(x,t) \rightarrow e^{ie\phi(x)}\Psi(x,t)
If we have an action of the field \Psi we always deal with the kinetic term. The kinetic term compares the value of the field at different points x. Put differently, the kinetic term is evaluated through the derivative of the field. To make sense of this term we take the function \phi(x) to be a smooth and differentiable.
If you evaluate a kinetic term such as: \partial_{\mu}\Psi^* \partial^{\mu}\Psi(x,t) you'll notice that the differential operator creates a term proportional to \partial_\mu \phi(x). This means that the action of a free field, containing a kinetic term is not invariant under local gauge transformations! Local gauge symmetry does not exist in a free theory because of the kinetic term.
To solve this one introduces another field, A, which also transforms under gauge transformations (but differently). The transformation is "defined" such that the overall action is invariant under local gauge transformations again. This field is called a gauge field, and for a gauge group U(1) this field is identified with the Electromagnetic vector potential.
The moral of the story is that a non-interacting theory is not locally gauge invariant, only globally. However, we can impose local gauge invariance at the cost of introducing a coupling to another field, the gauge field. The gauge field cancels the anomalous terms, thus generating local gauge invariance. In addition, this new gauge field can have dynamics as well.
A final remark: At first instance it looks like that we have changed the spacetime manifold M into the larger product manifold, MxV, where V is the representation space. But this structure is too simple and does not encapture all the physics. For that you need the notion of a fiber bundle. The particle fields then live in the space E, which is a fiber bundle. Locally, the bundle looks like the product space MxV, but globally the space may have non-trivial topological properties (this is the geometrical origin of the Aharanov-Bohm effect).
