A very beautiful result..
Not without troubles though!
In QM, when we write
\overline{|\Psi>}=U|\Psi>,
\bar{A}=UAU^{\dagger}
we mean[all books say] that U connects states that belong to an irreducible representation of some symmetry group G. And [only 1 or 2 book say] that implicit in the above two equations is the
invariance of the
vacuum under G.
Indeed, \overline{|\Psi>} and |\Psi> must be connected to the vacuum through some appropriate creation operators;
\overline{|\Psi>} = \bar{A^{\dagger}}|0>
|\Psi> = A^{\dagger}|0>
so it is easy to see that the top two equations are consistent, only if
U|0> = |0>
or, by writing, U=1 + i\epsilon_{a}Q_{a} ;
Q^{a}|0> = 0, \forall{a}
So, what does this mean in english?
We say that the symmetry is
manifest and
unitarily implimented on the Hilbert space of states,
only when Noether charges
annihilate the vacuum.
I.e. only when
Q|0>=0, the invariance of the Hamiltonian is manifest in the degeneracies of the energy eigenstates(particles) corresponding to the irreducible representation of G;
E(\bar{\Psi}) = <\bar{\Psi}|UHU^{\gagger}|\bar{\Psi}> = <\Psi|H|\Psi> = E(\Psi)
When Q_{a}|0> \neq 0 the particles are nolonger degenerate for, in this case we have
E(\bar{\Psi}) \neq E(\Psi)
and say that the G-symmetry spontaneously broken (hidden) eventhough H is still invariant under G:
"
Spontaneous breakdown of symmetry is the
lack of degeneracy in the particle spectra in a
symmetric theory"
We said that
Q|0> = 0 means that the symmetry is manifest!
So, is it possible to show that
Q_{a}|0> = 0 \Rightarrow \partial_{\mu}J_{a}^{\mu} = 0
Yes, it is and this is exactly what Coleman proved:
If, for any 4-vector J_{\mu}(x)
Q(t)= \int d^3x J^{0}(x)
is well defined operator on the H-space, and Q(t)|0> = 0, then
\partial_{\mu}J^{\mu}= 0
and Q(t) is time independent.
However, the converse need not be true! (
Q|0>=0, is sufficient but not necessary condition for exact symmetry), To see this, consider a Poincare'-invariant theory with internal symmetry group G (
[G,Poincare]=0),
Noether theorem then leads to a conserved G-currents \partial_{\mu}J^{\mu}_{a} = 0 and time-independent charges;
Q_{a}= \int d^3x J_{a}^{0}(x)
Now, let us calculate the norm of Q_{a}|0>
<0|Q_{a}Q_{a}|0> = \int d^3x <0|J_{a}^{0}(x) Q_{a}|0>
From the invariance under translations:
J^{0}_{a}(x)= e^{-i\hat{P_{\mu}}x^{\mu}}J_{a}(0) e^{i\hat{P_{\mu}}x^{\mu}}
[P_{\mu},Q_{a}]=0
it follows that
<0|Q_{a}^{2}|0> = \int d^3x <0|J_{a}^{0}(0)Q_{a}|0> \rightarrow \infty,
unless Q_{a}|0> = 0.
I.e, either the symmetry is exact and unitarly implemented on the H-Space (
Q|0>=0), or it is spontaneously broken (
Q|0> does not exist in the H-Space because its norm is infinite).
So, the behavior of the vacuum under a symmetry transformations is crucial in categorising how that symmetry is implemented in the H-Space of the quantum theory.
Coleman summarizes this by saying:
"
symmetries of the vacuum are symmetries of the world"
OK, this seems to paint a nice picture, so where are the troubles that I mentioned in the beginning of this post?
The troubles show up when we consider a gauge-constraint systems (I am not talking about the well-known QFT problems)
Take, for example, the EM gauge transformation;
A'_{\mu} = A_{\mu} + \partial_{\mu}\Lambda
Everything in nature indicates that this is an exact symmetry. So we expect to find a unitary operator U, such that
U|0>=|0>
and,
A'_{\mu} = UA_{\mu}U^{\dagger} = A_{\mu} + \partial_{\mu}\Lambda
are satisfied in the quantum theory of free EM field. But this leads to the contradictory statement;
<0|A'_{\mu}|0> = <0|A_{\mu}|0> = <0|A_{\mu}|0> + \partial_{\mu}\Lambda
or,
\partial_{\mu}\Lambda = 0
(this can not be right because \Lambda is an arbitrary function)
So, one is led to believe that the EM vacuum is not invariant under the gauge transformation. i.e
U|0> \neq |0>
or in terms of Q
Q_{\Lambda}|0> \neq 0
This means that the gauge symmetry is spontaneously broken!
i.e
<0|[iQ_{\Lambda},A_{\mu}]|0> = \partial_{\mu}\Lambda \neq 0
but this is equivalent to the statement that the field operator has non-vanishing vacuum expectation value;
<0|A_{\mu}|0> \neq 0
which is wrong because of Poincare' invariance.
So, I am baffled!
As far as I know, this problem remains unsolved even when we choose a small(PHYSICAL) subspace of the H-Space by postulating appropriate subsidiary condition.
If you people want a PhD, just solve the above problem
regards
sam
