Re: Gauge Transformations in Momentum Space?
- From: thomas_larsson_01@xxxxxxxxxxx
- Date: Sat, 17 Sep 2005 16:18:41 +0000 (UTC)
mikem@xxxxxxxxxxxxx skrev:
> Most textbook treatments of gauge transformations do it in
> position space. So far, I haven't found any that discuss
> in detail what they look like in momentum space, and what
> issues arise in the QFT Fock space.
>
[...]
>
> So old positive-energy modes in the energy range 0 to w get
> transformed into negative-energy modes. In the 2nd-quantized Fock
> space this means we're mixing some of the annihilation and creation
> operators - because they were defined in terms of the original +ve
> and -ve energy modes. Such mixing usually means that we're mapping
> between unitarily inequivalent representations, i.e: between
> orthogonal Fock spaces.
>
Your statement is incorrect. Negative-energy gauge transformations,
like all negative-energy operators, annihilate the vacuum, and
thus we stay within the Hilbert space. However, this is closely
related to an interesting and difficult set of questions, which I
have spent quite a lot of time thinking about.
Let us first consider the free Maxwell field. The kinematical
Hilbert space is spanned by canonical variables E_u(x) and A_v(x),
where x is a point in 3-space, Greek indices u,v = 0,1,2,3 and
Latin indices i,j = 1,2,3. The constraints are
E_0(x) = 0, G(x) = d_i E_i(x) = 0.
The latter is Gauss' law, which is the generator of gauge
transformations. The Maxwell field is just a bunch of harmonic
oscillators a_u(k), a*_u(k), where (up to numerical factors)
A_u(x) = int dk ( a_u(k) exp(ikx) + a*_u(k) exp(-ikx) )
E_u(x) = int dk ( a_u(k) exp(ikx) - a*_u(k) exp(-ikx) )
The oscillators are labelled by 3-momenta k, and their energy is
w_k = |k| and -|k|, respectively. The normal-ordered Hamiltonian
is
H = int dk w_k a^u(k) a*_u(k),
and the Fock vacuum is annilated by the annihition operators
a_u(k),
a_u(k)|vac> = 0.
In particular, the negative-energy part of the constraints
annihilate the vacuum. So the answer to your question is negative:
the gauge generators do not create states of negative energy,
because such states are annilated by the vacuum.
Physical quantitities should be gauge invariant, so we want to mod
out the constraints. The physical Hilbert space is spanned by
states satisfying
E_0(x)|phys> = 0, G(x)|phys> = 0.
The crucial property that makes this possible is that the
kinematical Hilbert space carries a quantum representation of the
constraint algebra. The representation is of lowest-energy type
(any operator O with negative H eigenvalue annihilates the
vacuum), and the Gauss law algebra holds without anomalies,
[G(x), G(y)] = 0. The physical Hilbert space has a positive-definite
inner product, so time evolution is unitary.
Interacting theories such as non-abelian Yang-Mills or QED with
matter are more subtle. E.g., in Yang-Mills the canonical
variables are E^a_u(x) and A^b_v(x), and the constraints read
E^a_0(x) = 0,
G^a(x) = d_i E^a_i(x) + f^abc A^b_i(x) E^c_i(x) = 0,
where f^abc are the structure constants. The last term is not
linear, which makes a huge difference. In order to obtain a
well-defined action on the Fock vacuum, we must normal order the
bilinear term. If we try to calculate the bracket of two
normal-ordered Gauss constraints, we will find something like
[G^a(x), G^b(y)] = f^abc G^c(x) delta(x-y)
+ f^acd f^bcd delta(x-y) delta(y-x).
The second term is proportional to delta(0)! The same kind of
extension arises if we consider QED with matter. In general the
extension is proportional to the value of the second Casimir
operator. It vanishes only in the free Maxwell theory, because the
adjoint rep of U(1) is trivial.
The appearence of the infinities in interacting field theories is
hardly surprising. We can render some observables finite by
renormalization, but I think that in this process G^a(x) ceases to
be a well-defined operator. The renormalized operators presumably
violate associativity, e.g.
(G^a(x))_ren (G^b(x))_ren != (G^a(x) G^b(x))_ren,
which means that they are not operators at all. AFAIU, every
renormalization scheme more complicated that normal ordering will
have such problems. If the renormalized generators were operators,
they would obey some extension of the Gauss law algebra, and the
kinematical Hilbert space would carry a lowest-energy
representation of the extended gauge algebra. The rep would not
need to be unitary, but the Hamiltonian must be bounded from below
by construction, i.e. the rep must be of lowest-energy type. It is
known how to construct lowest-energy reps of anomalous gauge
algebras, and subtracting off counterterms is not how you do it.
To understand what is going on, it is useful to consider a simpler
model, which is not quite as trivial as the free Maxwell field,
but where constraint generators remain operators after
renormalization: the free bosonic string in D dimensions. String
theory is a complex subject, but my point only requires that we
know that it has a gauge symmetry with generators L_m, m in Z, and
brackets
[L_m, L_n] = (n-m) L_m+n.
It is important that the gauge generators carry energy, because
the Hamiltonian H = L_0 does not commute with them;
[H, L_m] = m L_m. Actually, we have two commuting sets of
gauge generators and the Hamiltonian is the sum of two terms,
but this is irrelevant here.
When we quantize the theory, we demand that the kinematical
Hilbert space should have a lowest energy state |vac>, and thus
L_m |vac> = 0, for all m < 0.
So negative energy states are not allowed to arise even before
gauges are modded out! Next we want to pass to the physical
Hilbert space, by restriction to physical states satisfying
L_m |phys> = 0, for all m, positive or negative.
However, it is well known that there is an obstruction to
implementing this condition - the constraint algebra acquires a
central extension, the conformal anomaly c:
[L_m, L_n] = (n-m) L_m+n - (c/12) (m^3 - m) delta_m+n,
where delta_m is the Kronecker delta. Only if c = 0, which happens
if D = 26, can we impose the physical state condition. If we try
to impose it for non-zero c, we find that
[L_-m, L_m] |phys> = (c/12) (m^3 - m) |phys> = 0,
so the physical Hilbert space would be completely empty - not a
very interesting theory. This is the origin of the special role of
D = 26 in bosonic string theory. However, the free string defines
a consistent theory even if D < 26. This is because the inner
product in the kinematical Hilbert space is already
positive-definite (or at least can be made so by without factoring
out much less than all constraints).
Everything up til now is, I believe, completely undisputable.
However, the remainder of this post is not. I believe that it will
be possible to construct a formulation where the constraints are
well-defined operators on the kinematical Hilbert space, perhaps
not for all non-abelian gauge theories but at least for the
standard model. The Gauss law constraints will then be anomalous,
but the full algebra must be a well-defined Lie algebra extension.
This is a very powerful condition. In terms of the smeared gauge
generators
G(X) = \int X^a(x) G^a(x),
the only extension for which representations of lowest-energy type
are known is given by
[G(X), G(Y)] = G([X,Y]) +
+ k \int dt dq^i/dt <d_i X(q(t)),Y(q(t))>,
where <.,.> denotes the Killing form. Here q^i(t) is a privileged
curve in space.
.
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