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As part of the on-going discussion of Thiemann's "Loop-String" paper, the following was posted by Thomas Larsson, on 21 February at SPR (sci.physics.research) and also in an earlier version at Jacques Distler's board, the String Coffee Table.
Today I checked both places---SPR and Distler's board---and did not find any response. Maybe it is too early. Or perhaps Larsson's post was overlooked.
At String Coffee it is about halfway down a rather long page
http://golem.ph.utexas.edu/string/archives/000300.html
and possible to miss (I found it only on the second pass, scrolling
down that page).
I'm hoping for some comment.
---------Larsson's post---------
This is an expanded version of a post to the string coffee table,
http://golem.ph.utexas.edu/string/archives/000300.html . It is in
response to a post by K-H Rehren, who pointed out the crucial
algebraic difference between LQG representations and lowest-energy
representations. This explains the absense of anomalies in Thiemann's
approach and IMO settles the status of LQG as a quantum theory.
K-H Rehren:
>Dorothea Bahns has shown in her diploma thesis, that if one quantizes
>classical invariant observables (Pohlmeyer charges) by embedding them
>into the oscillator algebra via normal odering (N.O.), then the N.O.
>invariants fail to commute with N.O. Virasoro constraints, and
>commutators of N.O. invariants among themselves yield other N.O.
>invariants plus "quantum corrections" which are #not# quantized
>classical invariants. Thus, the quantum algebra has not only
>relations differing from the classical ones by hbar corrections
>(which everybody expects), but it would have #more generators# than
>the classical algebra. This feature ("breakdown of the principle of
>correspondence") is worse than a central extension, because the
>latter is a multiple of one, and as such #is# a quantized classical
>observable, suppressed by hbar. This feature is a property of the
>quantization, i.e., the very choice of the quantum algebra by
>replacing classical invariants by N.O. ones. One may or may not
>appreciate the oscillator quantization with features like this.
The correspondence principle is not necessarily violated. To
construct extensions of the diffeomorphism algebra in more than 1D,
one must first expand all fields in a Taylor series around some point
q. There are no conceptual problems to express classical physics in
terms of Taylor data (q and the Taylor coefficients) rather than in
terms of fields, although there may be problems with convergence.
The reason why such a trivial reformulation is necessary is that the
higher-dimensional generalizations of the Virasoro cocycle (there are
two of them) depend on the expansion point q. The relevant extensions
are thus non-linear functions of data already present classically,
which seems consistent with the correspondence principle.
>A no-go theorem (V. Kac) states that a unitary positive-energy (L0>0)
>representation of Diff without central charge must be trivial
>(one-dimensional). Put otherwise: c=0 is only possible if one
>abandons the positive-definite Hilbert space metric (ghosts), or
>positive energy, or unitarity.
Hence we can not maintain non-triviality, anomaly freedom, positive
energy, unitarity and ghost freedom at the same time. It seems to me
that giving up anomaly freedom makes least damage, especially since
we know that anomalous conformal symmetry is important in 2D
statistical models, such as the Ising and tricritical Ising models.
It is important to realize that such models have been realized
experimentally (e.g. in a monolayer of argon atoms on a graphite
substrate) and that the non-zero conformal anomaly has been measured
(perhaps only in computer experiments). Hence anomalous conformal
symmetry is not intrinsically inconsistent.
A clarification is in order at this point. The multi-dimensional
Virasoro algebra is a kind of gravitational anomaly, but no such
anomalies exist in 4D within a field theory framework. However, it
turns out that the phrase "within a field theory framework" is a
critical assumption. As I explained above, the relevant cocycles
depend on the expansion point q, and thus they can not be expressed
in terms of the fields which are independent of q.
>Thiemann seeks for the quantum algebra within an LQG type auxiliary
>algebra which has a unitary representation of Diff. The latter is
>#not# subject to the positive-energy condition.
The passage to lowest-energy representations is the crucial
quantization step. If such representations do not appear in LQG, then
it cannot IMO be a genuine quantum theory. If LQG is essentially
classical it explains why there is no anomaly.
Let me elaborate on the difference between lowest-energy (LE) reps
and the LQG type of rep, as described in section 2.6 of gr-qc/9910079
by Gaul and Rovelli. The canonical variables are A(x) and E(x) and
the CCR read
[E(x), A(y)] = delta(x-y), [E(x),E(y)] = [A(x),A(y)] = 0.
In the paper, the operators are smeared over suitable submanifolds
and have some index decorations, but that is irrelevant for the
argument. The canonical variables act of functionals Psi(A), with
E(x) = d/dA(x), and the constant functional 1 can be identified with
a vacuum |vac> by
E(x) |vac> = 0 for all x.
Bilinears of the form
A(x)E(y)
generate a gl(infinity), and we can embed diffeomorphism and current
algebras into this gl(infinity). The key point to observe is that
these bilinears are already normal ordered w.r.t. the vacuum |vac>.
Normal ordering means by definition that things that annihilate the
vacuum are moved to the right, and E(y) does annihilate the vacuum.
Since no further normal ordering is necessary, no anomalies arise.
However, this is not what I would call a LE rep. Rather, I would call
it a "lowest-A-number" rep; the A-number operator \int dx A(x)E(x) is
always positive. This space is essentially classical in nature, so it
is not so surprising that there is no anomaly.
Let us contrast this type of rep with LE reps. For simplicity, let us
assume that x and y are points in 1D; the higher-dimensional case
requires a passage to jet space which complicates things, although
not in an essential way. We can now expand A(x) and E(x) in a Fourier
series, and the Fourier components A_m and E_m satisfy the CCR
[E_m, A_n] = delta_m+n,0 , [E_m, E_n] = [A_m, A_n] = 0.
The LQG vacuum satisfies
E_m |vac> = 0 for all m.
The LE vacuum |0>, OTOH, is defined by
E_-m |0> = A_-m |0> = 0 for all -m < 0.
In other words, it is the modes of negative frequency, i.e. those
that travel backwards in time, that annihilate this vacuum.
The bilinears that generate gl(infinity),
A_m E_n ,
are normal ordered w.r.t. the LQG vacuum |vac> but not w.r.t. the
LE vacuum |0>. To normal order w.r.t. the latter, we need to move
negative-frequency modes to the right:
:A_m E_n: = A_m E_n m >= n
E_n A_m m < n
This is the main algebraic difference between the LQG "lowest-A-number" reps and LE reps. This is an essential difference; in infinite
dimension, there is no Stone-von Neumann theorem that makes different
vacua equivalent.
--------------end of post-----------------
Today I checked both places---SPR and Distler's board---and did not find any response. Maybe it is too early. Or perhaps Larsson's post was overlooked.
At String Coffee it is about halfway down a rather long page
http://golem.ph.utexas.edu/string/archives/000300.html
and possible to miss (I found it only on the second pass, scrolling
down that page).
I'm hoping for some comment.
---------Larsson's post---------
This is an expanded version of a post to the string coffee table,
http://golem.ph.utexas.edu/string/archives/000300.html . It is in
response to a post by K-H Rehren, who pointed out the crucial
algebraic difference between LQG representations and lowest-energy
representations. This explains the absense of anomalies in Thiemann's
approach and IMO settles the status of LQG as a quantum theory.
K-H Rehren:
>Dorothea Bahns has shown in her diploma thesis, that if one quantizes
>classical invariant observables (Pohlmeyer charges) by embedding them
>into the oscillator algebra via normal odering (N.O.), then the N.O.
>invariants fail to commute with N.O. Virasoro constraints, and
>commutators of N.O. invariants among themselves yield other N.O.
>invariants plus "quantum corrections" which are #not# quantized
>classical invariants. Thus, the quantum algebra has not only
>relations differing from the classical ones by hbar corrections
>(which everybody expects), but it would have #more generators# than
>the classical algebra. This feature ("breakdown of the principle of
>correspondence") is worse than a central extension, because the
>latter is a multiple of one, and as such #is# a quantized classical
>observable, suppressed by hbar. This feature is a property of the
>quantization, i.e., the very choice of the quantum algebra by
>replacing classical invariants by N.O. ones. One may or may not
>appreciate the oscillator quantization with features like this.
The correspondence principle is not necessarily violated. To
construct extensions of the diffeomorphism algebra in more than 1D,
one must first expand all fields in a Taylor series around some point
q. There are no conceptual problems to express classical physics in
terms of Taylor data (q and the Taylor coefficients) rather than in
terms of fields, although there may be problems with convergence.
The reason why such a trivial reformulation is necessary is that the
higher-dimensional generalizations of the Virasoro cocycle (there are
two of them) depend on the expansion point q. The relevant extensions
are thus non-linear functions of data already present classically,
which seems consistent with the correspondence principle.
>A no-go theorem (V. Kac) states that a unitary positive-energy (L0>0)
>representation of Diff without central charge must be trivial
>(one-dimensional). Put otherwise: c=0 is only possible if one
>abandons the positive-definite Hilbert space metric (ghosts), or
>positive energy, or unitarity.
Hence we can not maintain non-triviality, anomaly freedom, positive
energy, unitarity and ghost freedom at the same time. It seems to me
that giving up anomaly freedom makes least damage, especially since
we know that anomalous conformal symmetry is important in 2D
statistical models, such as the Ising and tricritical Ising models.
It is important to realize that such models have been realized
experimentally (e.g. in a monolayer of argon atoms on a graphite
substrate) and that the non-zero conformal anomaly has been measured
(perhaps only in computer experiments). Hence anomalous conformal
symmetry is not intrinsically inconsistent.
A clarification is in order at this point. The multi-dimensional
Virasoro algebra is a kind of gravitational anomaly, but no such
anomalies exist in 4D within a field theory framework. However, it
turns out that the phrase "within a field theory framework" is a
critical assumption. As I explained above, the relevant cocycles
depend on the expansion point q, and thus they can not be expressed
in terms of the fields which are independent of q.
>Thiemann seeks for the quantum algebra within an LQG type auxiliary
>algebra which has a unitary representation of Diff. The latter is
>#not# subject to the positive-energy condition.
The passage to lowest-energy representations is the crucial
quantization step. If such representations do not appear in LQG, then
it cannot IMO be a genuine quantum theory. If LQG is essentially
classical it explains why there is no anomaly.
Let me elaborate on the difference between lowest-energy (LE) reps
and the LQG type of rep, as described in section 2.6 of gr-qc/9910079
by Gaul and Rovelli. The canonical variables are A(x) and E(x) and
the CCR read
[E(x), A(y)] = delta(x-y), [E(x),E(y)] = [A(x),A(y)] = 0.
In the paper, the operators are smeared over suitable submanifolds
and have some index decorations, but that is irrelevant for the
argument. The canonical variables act of functionals Psi(A), with
E(x) = d/dA(x), and the constant functional 1 can be identified with
a vacuum |vac> by
E(x) |vac> = 0 for all x.
Bilinears of the form
A(x)E(y)
generate a gl(infinity), and we can embed diffeomorphism and current
algebras into this gl(infinity). The key point to observe is that
these bilinears are already normal ordered w.r.t. the vacuum |vac>.
Normal ordering means by definition that things that annihilate the
vacuum are moved to the right, and E(y) does annihilate the vacuum.
Since no further normal ordering is necessary, no anomalies arise.
However, this is not what I would call a LE rep. Rather, I would call
it a "lowest-A-number" rep; the A-number operator \int dx A(x)E(x) is
always positive. This space is essentially classical in nature, so it
is not so surprising that there is no anomaly.
Let us contrast this type of rep with LE reps. For simplicity, let us
assume that x and y are points in 1D; the higher-dimensional case
requires a passage to jet space which complicates things, although
not in an essential way. We can now expand A(x) and E(x) in a Fourier
series, and the Fourier components A_m and E_m satisfy the CCR
[E_m, A_n] = delta_m+n,0 , [E_m, E_n] = [A_m, A_n] = 0.
The LQG vacuum satisfies
E_m |vac> = 0 for all m.
The LE vacuum |0>, OTOH, is defined by
E_-m |0> = A_-m |0> = 0 for all -m < 0.
In other words, it is the modes of negative frequency, i.e. those
that travel backwards in time, that annihilate this vacuum.
The bilinears that generate gl(infinity),
A_m E_n ,
are normal ordered w.r.t. the LQG vacuum |vac> but not w.r.t. the
LE vacuum |0>. To normal order w.r.t. the latter, we need to move
negative-frequency modes to the right:
:A_m E_n: = A_m E_n m >= n
E_n A_m m < n
This is the main algebraic difference between the LQG "lowest-A-number" reps and LE reps. This is an essential difference; in infinite
dimension, there is no Stone-von Neumann theorem that makes different
vacua equivalent.
--------------end of post-----------------
Last edited: