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Quantization without quantization
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no,location=no, scrollbars=yes,resizable=yes,status=no,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.0409212059050.4633-100000@feynman.harvard.edu>...\n\n> Quantization without quantization - prejudices and reality\n> ==========================================================\n\nLet me first state that I agree fully with almost all that you say.\nNot unexpectedly, the only one point which I disagree upon is the\ndiscussion on gauge anomalies, and even there I only disagree\npartially. What you say on this issue makes sense and is the\nstandard lore, but it needs elaboration.\n\n> Reality IVa: Chiral gauge theories and similar theories in 2k dimensions\n> typically lead to gauge anomalies. Gauge anomalies imply that the theory\n> is no longer consistent because the unphysical (and negative-norm)\n> polarizations of the gauge bosons (or graviton) no longer decouple. The\n> structure of anomalies can be calculated if one tries to regularize\n> certain UV divergences, but the result is purely reflected by the low\n> energy (IR) spectrum of the theory. There is no way to avoid the\n> conclusion about gauge anomalies except for cancelling them.\n>\n> Anomalies in global symmetries do not imply an inconsistency, but they\n> drastically influence the physics of a given theory.\n>\n> Central charge (or Weyl anomaly) is a specific example of an anomalous\n> term that can be calculated in many ways, and there is no consistent way\n> to avoid its nonzero value.\n\nWe should first define what we mean by a gauge theory. Since not\nall quantum theories have well-defined classical counterparts, we\nneed a definition which is intrinsically quantum. I propose to\ndefine a gauge symmetry as a symmetry with a well-defined,\nnilpotent BRST operator. In that case the symmetry is a redundancy\nof the description, because we can define the physical Hilbert\nspace as the space of BRST cohomology classes.\n\nWith this definition, it is tautologically true that no consistent\ngauge anomalies exist. Not because an anomaly would necessarily\nbe inconsistent, but because it would ruin nilpotency, making the\nsymmetry into a global symmetry ("global" is a terribly confusing\nword in this context, btw. I would prefer the word "non-gauge").\nThere is nothing intrinsically wrong with anomalous global\nsymmetries whose non-anomalous part is isomorphic to a gauge\nsymmetry. The canonical example is the minimal models in CFT with\ncentral charge 1/2 <= c <= 1. They are physically consistent\nin the strong sense that they are realized (and measured!) in\nexperimentally accessible systems. And still the anomaly-free part\nof the symmetry algebra is isomorphic to the Weyl gauge symmetry of\nstring theory.\n\nIf we could take the classical limit of such a system, it would seem\nto have a gauge symmetry. Namely, the anomaly vanishes in the\nclassical limit, and we can write down a classical BRST operator\nwhich is nilpotent, and the symmetry is gauge on the classical\nlevel. There is no classical way to distinguish between such a\n"fake" gauge symmetry and a genuine gauge symmetry which extends to\nthe quantum level. The quantum world is what it is, and classical\nintuition can often go wrong.\n\nUnfortunately, we cannot check this argument for the minimal models,\nbecause they don\'t seem to have a good classical limit. Some aspects\ncan be captured by Landau-Ginzburg models, but others are totally\nopaque in the LG picture, like the supersymmetry of the c = 7/10\nmodel.\n\nThus some anomalous gauge symmetries (= anomalous global symmetries\nwhose non-anomalous part is isomorphic to a gauge symmetry) may be\nconsistent, but all are not. It must be realized that gauge\nsymmetries have two qualitatively different types of anomalies:\n\n1. Anomalies seen in field theory, related to the existence of\nchiral fermions. This class include the ABJ anomalies in the\nstandard model and the Green-Schwartz mechanism. There are two good\nreasons to expect that such anomalies are inconsistent: Nature\navoids them in the standard model, and the corresponding algebra\ndoes not seem to have any good representations.\n\n2. Anomalies like the Virasoro and affine Kac-Moody algebra, and\ntheir higher-dimensional analogues. These algebras have interesting\nunitary representations, but cannot be seen in field theory because\nthey involve the observer\'s trajectory. There is no reason to\nexpect such anomalies to be inconsistent, especially since they do\narise in condensed matter models like the 2D Ising model.\n\nThe different extensions can be illustrated for the current algebra\non the 3D torus. Use a Fourier basis with momenta m = (m_i) in Z^3,\nstructure constants f^abc, second Casimir delta^ab and third Casimir\nd^abc. The Mickelsson-Faddeev algebra describes the ABJ anomaly:\n\n[J^a(m), J^b(n)] = f^abc J^c(m+n)\n\n+ d^abc epsilon^ijk m_i n_j A^c_k(m+n),\n\n[J^a(m), A^b_k(n)] = f^abc A^c_k(m+n) + delta^ab m_k delta(m+n),\n\n[A^a_i(m), A^b_j(n)] = 0.\n\nA^a_i(m) are the Fourier components of the gauge connection.\n\nThe "central" extension (which commutes with gauge transformations\nbut not with diffeomorphisms):\n\n[J^a(m), J^b(n)] = f^abc J^c(m+n) + delta^ab m_i S^i(m+n),\n\n[J^a(m), S^i(n)] = [S^i(m), S^j(n)] = 0,\n\nm_i S^i(m) = 0.\n\nThese two extensions of the current algebra in 3D have thus very\ndifferent properties, and to conclude that inconsistincy of the\nformer implies inconsistency of the latter is simply wrong.\n\nFinally, we must define exactly what we mean by consistency. At the\nmost basic level, a quantum theory is defined by a Hilbert space\nand a unitary time evolution. If the theory has some symmetries,\nthey must be realized as unitary operators acting on this Hilbert\nspace as well. If time translation is included among the symmetries,\nwhich is the case for the Poincare algebra (and more subtly for\ndiffeomorphisms), requiring a unitary representation of the\nsymmetry algebra seems to be enough for consistency.\n\nFrom this viewpoint, there is a 1-1 correspondence between general-\ncovariant quantum theories (GCQT) and unitary representations of the\ndiffeomorphism group on a conventional Hilbert space. Namely, if we\nhave a GCQT, its Hilbert space carries a unitary rep of the diffeo\ngroup. And if we have a unitary rep of the diffeo group, the Hilbert\nspace on which it acts can be interpreted as the Hilbert space of\nsome GCQT. Since all unitary quantum irreps of the diffeo group are\nanomalous, apart from the trivial one, all interesting GCQTs carry\nanomalous reps of the diffeo group. So rather than being\ninconsistent, the second type of gauge anomaly is in fact a\nnecessary condition for non-trivial consistency.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Lubos Motl <motl@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.0409212059050.46...arvard.edu>...
> Quantization without quantization - prejudices and reality
> ==========================================================
Let me first state that I agree fully with almost all that you say.
Not unexpectedly, the only one point which I disagree upon is the
discussion on gauge anomalies, and even there I only disagree
partially. What you say on this issue makes sense and is the
standard lore, but it needs elaboration.
> Reality IVa: Chiral gauge theories and similar theories in 2k dimensions
> typically lead to gauge anomalies. Gauge anomalies imply that the theory
> is no longer consistent because the unphysical (and negative-norm)
> polarizations of the gauge bosons (or graviton) no longer decouple. The
> structure of anomalies can be calculated if one tries to regularize
> certain UV divergences, but the result is purely reflected by the low
> energy (IR) spectrum of the theory. There is no way to avoid the
> conclusion about gauge anomalies except for cancelling them.
>
> Anomalies in global symmetries do not imply an inconsistency, but they
> drastically influence the physics of a given theory.
>
> Central charge (or Weyl anomaly) is a specific example of an anomalous
> term that can be calculated in many ways, and there is no consistent way
> to avoid its nonzero value.
We should first define what we mean by a gauge theory. Since not
all quantum theories have well-defined classical counterparts, we
need a definition which is intrinsically quantum. I propose to
define a gauge symmetry as a symmetry with a well-defined,
nilpotent BRST operator. In that case the symmetry is a redundancy
of the description, because we can define the physical Hilbert
space as the space of BRST cohomology classes.
With this definition, it is tautologically true that no consistent
gauge anomalies exist. Not because an anomaly would necessarily
be inconsistent, but because it would ruin nilpotency, making the
symmetry into a global symmetry ("global" is a terribly confusing
word in this context, btw. I would prefer the word "non-gauge").
There is nothing intrinsically wrong with anomalous global
symmetries whose non-anomalous part is isomorphic to a gauge
symmetry. The canonical example is the minimal models in CFT with
central charge [itex]1/2 <= c <= 1[/itex]. They are physically consistent
in the strong sense that they are realized (and measured!) in
experimentally accessible systems. And still the anomaly-free part
of the symmetry algebra is isomorphic to the Weyl gauge symmetry of
string theory.
If we could take the classical limit of such a system, it would seem
to have a gauge symmetry. Namely, the anomaly vanishes in the
classical limit, and we can write down a classical BRST operator
which is nilpotent, and the symmetry is gauge on the classical
level. There is no classical way to distinguish between such a
"fake" gauge symmetry and a genuine gauge symmetry which extends to
the quantum level. The quantum world is what it is, and classical
intuition can often go wrong.
Unfortunately, we cannot check this argument for the minimal models,
because they don't seem to have a good classical limit. Some aspects
can be captured by Landau-Ginzburg models, but others are totally
opaque in the LG picture, like the supersymmetry of the [itex]c = 7/10[/itex]
model.
Thus some anomalous gauge symmetries (= anomalous global symmetries
whose non-anomalous part is isomorphic to a gauge symmetry) may be
consistent, but all are not. It must be realized that gauge
symmetries have two qualitatively different types of anomalies:
1. Anomalies seen in field theory, related to the existence of
chiral fermions. This class include the ABJ anomalies in the
standard model and the Green-Schwartz mechanism. There are two good
reasons to expect that such anomalies are inconsistent: Nature
avoids them in the standard model, and the corresponding algebra
does not seem to have any good representations.
2. Anomalies like the Virasoro and affine Kac-Moody algebra, and
their higher-dimensional analogues. These algebras have interesting
unitary representations, but cannot be seen in field theory because
they involve the observer's trajectory. There is no reason to
expect such anomalies to be inconsistent, especially since they do
arise in condensed matter models like the 2D Ising model.
The different extensions can be illustrated for the current algebra
on the 3D torus. Use a Fourier basis with momenta [itex]m = (m_i)[/itex] in [itex]Z^3,[/itex]
structure constants [itex]f^{abc},[/itex] second Casimir [itex]\delta^ab[/itex] and third Casimir
[itex]d^{abc}[/itex]. The Mickelsson-Faddeev algebra describes the ABJ anomaly:
[itex][J^a(m), J^b(n)] = f^{abc} J^c(m+n)+ d^{abc} \epsilon^ijk m_i n_j A^{c_k}(m+n),[J^a(m), A^{b_k}(n)] = f^{abc} A^{c_k}(m+n) + \delta^ab m_k \delta(m+n),[A^{a_i}(m), A^{b_j}(n)] =[/itex] .
[itex]A^{a_i}(m)[/itex] are the Fourier components of the gauge connection.
The "central" extension (which commutes with gauge transformations
but not with diffeomorphisms):
[itex][J^a(m), J^b(n)] = f^{abc} J^c(m+n) + \delta^ab m_i S^i(m+n),[J^a(m), S^i(n)] = [S^i(m), S^j(n)] = 0,m_i S^i(m) =[/itex] .
These two extensions of the current algebra in 3D have thus very
different properties, and to conclude that inconsistincy of the
former implies inconsistency of the latter is simply wrong.
Finally, we must define exactly what we mean by consistency. At the
most basic level, a quantum theory is defined by a Hilbert space
and a unitary time evolution. If the theory has some symmetries,
they must be realized as unitary operators acting on this Hilbert
space as well. If time translation is included among the symmetries,
which is the case for the Poincare algebra (and more subtly for
diffeomorphisms), requiring a unitary representation of the
symmetry algebra seems to be enough for consistency.
From this viewpoint, there is [itex]a 1-1[/itex] correspondence between general-
covariant quantum theories (GCQT) and unitary representations of the
diffeomorphism group on a conventional Hilbert space. Namely, if we
have a GCQT, its Hilbert space carries a unitary rep of the diffeo
group. And if we have a unitary rep of the diffeo group, the Hilbert
space on which it acts can be interpreted as the Hilbert space of
some GCQT. Since all unitary quantum irreps of the diffeo group are
anomalous, apart from the trivial one, all interesting GCQTs carry
anomalous reps of the diffeo group. So rather than being
inconsistent, the second type of gauge anomaly is in fact a
necessary condition for non-trivial consistency.
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