Canonical quantization and covariance
- From: thomas_larsson_01@xxxxxxxxxxx (Thomas Larsson)
- Date: Sun, 19 Jun 2005 05:44:46 +0000 (UTC)
The primary purpose with this thread is to convince Urs Schreiber and
others that the group of 4-diffeomorphisms is the correct constraint
algebra of GR, provided that we use a covariant formalism. Let me start
with a quote from http://www.arxiv.org/abs/gr-qc/0202079 by Carlo Rovelli:
"There are several ways in which a field theory can be cast in hamiltonian
form. One possibility is to take the space of the fields at fixed time as
the (nonrelativistic) configuration space Q. This strategy badly breaks
special, and, in a general covariant theory, general relativistic
invariance. Lorentz covariance is broken by the fact that one has to
choose a Lorentz frame for the t variable. I find far more disturbing the
conflict with general covariance. The very foundation of general covariant
physics is the idea that the notion of a simultaneity surface all over the
universe is devoid of physical meaning. Seems to me that it is better not
to found hamiltonian mechanics on a notion devoid of physical
significance.
A second alternative is to formulate mechanics on the solutions of the
equations of motion. The idea goes back to Lagrange. In the generally
covariant context, a symplectic structure can be defined over this space
using a spacelike surface, but one can show that the definition is surface
independent ant therefore it is well defined. This strategy has been
explored by Witten, Ashtekar and several others. The structure is viable
in principle, but very difficult to work with in practice. The reason is
that we do not know the space of solutions of an interacting theory.
Therefore we must either work over a space with the initial data on some
instantaneity surface, and therefore, effectively, go back to the
conventional fixed time formulation. Thus this formulation has the merit
of telling us that the hamiltonian formalism is actually intrinsically
covariant, but it does not really indicate how to effectively deal with it
in a covariant manner."
Rovelli then proceeds to a third alternative, which is not relevant to the
present discussion.
Postponing the construction of the covariant phase space to later in this
post, let me discuss how the constraint algebra acts on it. We know that
the Einstein action is invariant under 4-diffeos acting on the dynamical
fields alone (so there is no background metric). Hence 4-diffeos map
histories into each other in a well-defined way. A given 4-diffeo maps any
point on history 1 into some point on history 2; if it mapped different
points on history 1 into different histories it would not be a symmetry.
So in the covariant formulation of phase space, which we know exists but
is difficult to construct in practice, the constraints generate the
4-diffeo group.
However, if we choose to coordinatize phase space by the intersection of
histories with the t=0 surface, we run into a problem: 4-diffeos with a
non-zero temporal component do not preserve the surface. To return to the
standard coordinatization, we must add a compensating transformation,
which modifies the constraint algebra. The resulting Dirac algebra is a
mess; it is an open algebra rather than an honest Lie algebra, it singles
out the time direction in way which breaks covariance, and it might even
be conceptually flawed when the notion of spacelikeness is quantized.
This does not mean that the 4-diffeo algebra is easily understood; on the
contrary, the quantum representations are only partially understood. But
to work with the Dirac algebra is probably an order of magnitude more
complicated, without any gains in principle. I have thought about
translating the 4-diffeo anomalies into the Dirac algebra, but not really
found the motivation to give it a serious try. Perhaps somebody else will
succeed.
Instead, I find it more attractive to construct the covariant phase space
that Rovelli alludes to. To that end, I want to draw attention to the
similarity with the situation in gauge theories, where we want to compute
quantities which are gauge invariant, i.e. which only depend on orbits. We
can coordinatize the space of gauge orbits by fixing a gauge; this is
analogous to coordinatize the space of histories which solve the
Euler-Lagrange (EL) equations by their intersections with the t=0 surface.
Gauge fixing leads to formulations which are complicated, ugly, and
sometimes flawed (Gribov ambiguities).
Another alternative is to describe the space of gauge orbits directly, but
this is very complicated. Holonomies (Wilson loops) are gauge invariant,
but a loop basis is highly overcomplete. Three loops are linearly
dependent if one of them is the concatenation of the two others. To
disentangle an independent and complete set of loops seems like a hopeless
task. This is quite analogous to explicitly describing the space of
solutions of interacting theories.
However, there is a third way to treat gauge symmetries, namely the BRST
method, and this is the preferred method nowadays. To describe the space
of gauge orbits is very difficult, but we are really interested in the
dual space of functions of gauge orbits; that will become our Hilbert
space after quantization. Fortunately, this dual space can be described
quite easily, as the cohomology of the BRST complex. In addition to the
physical fields, we introduce "ghosts", which are unphysical fermionic
scalar fields; they disobey the spin-statistics theorem.
Let V* be the extended space of both fields and ghosts, and C(V*) the
space of functions over V*. C(V*) decomposes into subspaces of fixed ghost
number, C(V*) = \sum_k=0^\infty C_k. We can define the BRST charge Q: C_k
-> C_k+1, which is fermionic and nilpotent, Q^2 = 0. This allows us to set
up the BRST complex,
0 --> C_0 --> C_1 --> C_2 --> ...
The cohomology groups are defined as usual, H_k = (ker Q/im Q)_k. One can
show that H_k = 0 if k > 0 and that H_0 is the space of functions over
gauge orbits, which is exactly the object we were looking for. The
situation is completely analogous to the de Rham complex, where C_k are
spaces of k-forms and Q is the exterior derivative.
This story is well known, and the big advantage is that we describe the
very complicated space H_0 in terms of the spaces C_k and the operator Q,
which are easy to compute, and we never have to explicitly coordinatize
orbit space by choosing a gauge. AFAIU, the BRST method has no
disadvantages.
My idea now is to extend the analogy with canonical quantization by giving
a cohomological description of phase space. That this should work is not
surprising; after all, the histories are orbits of the abelian
one-parameter group of time translation. Similar idea were employed by
Batalin and Vilkovisky 25 years ago, so let me describe that first,
following the standard reference
Henneaux, M. and Teitelboim, C.: Quantization of gauge systems.
Princeton Univ. Press (1992)
Assume that we have a dynamical system with action S and configuration
space V, with basis q^i. The EL equations read
E_i = dS/dq^i = 0.
All derivatives are functional derivatives. For simplicity, we assume that
there are no gauge degrees of freedom, so these equations specify q^i(t)
uniquely in terms of q^i(0). Introduce the extended configuration space V*
by adding fermionic antifields q*_i. We can define the Koszul-Tate (KT)
derivative Q, which acts on C(V*), by
Q q^i = 0
Q q*_i = E_i.
Clearly Q^2 = 0, so this defines a KT complex, whose zeroth cohomology
group H^0(Q) is the space of functions over the space of solutions to the
EL equations, and H^k(Q) = 0 otherwise.
However, the BV approach only applies to the Lagrangian formulation and
path integrals, and it is not suited for canonical quantization. One can
define the antibracket by (q^i, q^_j) = delta^i_j, but in order to
quantize canonically we need an honest Poisson bracket. My idea (and this
is where the new stuff begins), was to consider not only the space of
functions C(V*), but all differential operators D(V*) over V*. So this is
the space of functions over the phase space corresponding to V*, with
coordinates (q^i, q*_i, p_i, p*^i). In terms of the non-zero canonical
(anti-)commutation relations
[p_j, q^i] = {p*^i, q*_j} = delta^i_j,
the KT derivative can explicitly be written as
Q = \sum_i E_i p*^i.
This operator acts in a well-defined manner on any element in
D(V*) = sum_k=-\infty^\infty D_k, giving rise to a two-sided KT complex.
... <-- D_-2 <-- D_-1 <-- D_0 <-- D_1 <-- D_2 <-- ...
Under the additional assumption that the Hessian (d^2 S/dq^i dq^j) is
non-singular, one can show that the H^0(Q) is the same physical phase
space as above. The space of functions over the physical phase space has
been reformulated as cohomology of the complex above. The surface t=0 does
not play a distinguished role anymore, and we are done.
Some comments:
1. In conventional canonical quantization, the momenta are defined by
p_i = dL/d(\dot q^i).
So where did that equation go? The EL equation E_i = 0 acts like a
first-class constraint, whereas the equation for p_i is like a gauge
fixing condition, making the constraints second class. If we want to
eliminate the constraints using Dirac's method, the constraints should be
second class. In cohomological methods, the first-class constraints "hit
twice", and the equation for p_i is not needed. Which is good, since it
breaks covariance, effectively taking us back to a preferred time
direction.
2. Upon quantization, we replace the Poisson brackets by commutators.
Conventionally, only equal-time commutators give rise to delta-functions.
Writing out time dependence explicitly, the CCR above read
[p_j(t), q^i(t')] = {p*^i(t), q*_j(t')} = delta^i_j delta(t-t'),
which contains an addition delta-function in time. But these relations
hold on the auxiliary spaces D_k, not on the physical phase space H^0(Q).
That these auxiliary CCRs are local in spacetime rather than just in space
is a major simplification.
3. So this is the quantization step: replace the Poisson brackets above by
(anti-)commutators and represent the resulting graded Heisenberg algebra
on a Hilbert space with energy bounded from below. This reverses the order
of things: usually, one introduce dynamics first and quantize afterwards,
whereas I quantize in the space of arbitary histories first and impose
dynamics afterwards.
4. What is the Hamiltonian? It can be described very simply one the
auxiliary space D_k as the generator of rigid t translations, H = -id/dt,
i.e. the Schroedinger equation. This operator encodes dynamics on the
physical phase space H^0(Q), because the EL equations hold there.
5. What about the harmonic oscillator? This does not quite
work out, which is a major embarrassment. The problem is that the Hessian
is singular. The EL equations read in Fourier space
(k^2 - w^2) q_k = 0,
and the Hessian is
(k^2 - w^2) delta(k,k'),
which clearly vanishes when k^2 = w^2, i.e. when the EL equations hold.
However, this is a purely classical problem. Quantization dutifully
eliminates half of the classical oscillators (those with negative energy),
as expected.
I don't really understand why the harmonic oscillator goes wrong. Maybe
somebody has an idea how to fix it.
.
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