## Diffeomorphisms, LQG, and positive energy

<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>Everybody has presumably already noticed\nhttp://www.arxiv.org/abs/hep-th/0409182, where Robert Helling and\nGiuseppe Policastro compare Fock and LQG quantization of a string.\nIn particular, they notice that one can do LQG quantization of\nthe harmonic oscillator, and obtain an energy spectrum which is\nnot bounded from below. Needless to say, this is bad news for LQG.\n\nThis paper gives a good background for a general discussion of\ncanonical quantization of general-covariant theories like general\nrelativity. At some stage in this process, we want to find a\nunitary representation of the diffeomorphism generators on some\nHilbert space. Ideally, we want our representations of the\ndiffeomorphism group to be non-trivial, irreducible, unitary,\nanomaly-free, and of lowest-energy type. Unfortunately, a theorem\nstates that no such representation exists, which is major\ncomplication.\n\nThis point is best illustrated by quantization of the string. The\nrelevant group here is the conformal group rather than the group of\ndiffeomorphisms, but the math is the same since the infinite-\ndimensional conformal group in 2D is isomorphic to two copies of\nthe full diffeo group in 1D. In bosonic string theory, one starts\nwith 26 coordinate fields. The Virasoro algebra acts on these, but\nsince the central charge c = 26 is non-zero this rep is not\nanomaly-free. To remedy that, one introduces a ghost with c = -26.\nThe total rep is anomaly-free but not irreducible, being a direct\nsum. More importantly, it is not unitary since the ghost has\nnegative central charge. Finally one factors out states with\nnegative and zero norm by passing to BRST cohomology. The resulting\nrepresentation is unitary and has c = 0, and is thus trivial, in\naccordance with the theorem.\n\nThomas Thiemann does something different. He does indeed obtain an\nirrep which is both non-trivial, unitary and anomaly-free. To avoid\nconflict with the classification of Virasoro irreps, this "polymer"\nrepresentation must therefore not be of lowest-energy type.\nHelling-Policastro makes this explicit when they calculate the\nspectrum of the polymer harmonic oscillator, and find that the\nenergy is not bounded from below.\n\nContrary to the moderators of sps (especially one), I think that I\nhave some credibility when I say that I am not biased in favor of\nstring theory. Nevertheless, I believe that the Helling-Policastro\npaper, and previous arguments by Urs Schreiber, are fatal to LQG.\nIt is sad to see the good guys lose, but having lowest-energy\nrepresentations is the very essence of quantum theory from my\npoint of view.\n\nHowever, the LQG string was only a toy problem. The really\ninteresting situation is in canonical quantization of 3+1D gravity,\nwhich should proceed in analogy with the 1+1D case. It does not\nreally matter whether we deal with field theory, string theory, or\nLQG. In particular, even if only the first-quantized formulation\nof string theory is well understood, we may assume that some\nnon-perturbative Hamiltonian formulation of string/brane field theory\nexists. If such a Hamiltonian formulation (perhaps in some\ngeneralized sense) does not exist even in principle, we will be in\ntrouble anyway because we will have to rethink the foundations of\nquantum mechanics.\n\nThus assume that we have some general-covariant theory, and that\nwe need a non-trivial, unitary, lowest-weight representation of the\n3D diffeomorphism generators on a separable Hilbert space. Such\na representation is necessarily anomalous.\n\nProof: Well-known in 1D, where the only anomaly-free unitary irrep\nis the trivial one. In higher dimensions, the diffeomorphism\nalgebra has many subalgebras generated by vector fields of the form\nf(y)d/dy, where y = k_i x^i for some numbers k_i. A unitary irrep\nof a big algebra must restrict to a unitary rep (in general\nreducible) of each subalgebra labelled by k_i. But we know that\nevery restriction must be a direct sum of trivial irreps. Only the\ntrivial rep has only trivial restrictions. QED.\n\nThe next natural step, if we want to proceed in analogy with the\nstring, is to mod out diffeomorphisms, e.g. by a BRST procedure.\nHowever, this meets with a serious technical obstacle, which has\nled me to assert that one must live with the anomaly. The problem\nis that there is no well-defined BRST operator, AFAIU. This is a\ngenuinely new problem which does not appear in 1D. In fact, there\nare three qualitatively different situations:\n\n1. For finite-dimensional algebras, the BRST operator is always\nwell-defined and nilpotent.\n\n2. For diffeomorphisms in 1D, the BRST operator is always\nwell-defined, but nilpotent only for some special parameter value\nsuch as c = 26.\n\n3. For diffeomorphisms in two and more dimensions, the BRST\noperator is apparently ill defined. The problem is that normal\nordering introduces an unrestricted sum over transverse modes,\nwhich gives an infinity not present in 1D. This can be avoided in\nthe algebra generators by starting from a Taylor expansion around a\nmarked 1D curve ("the observer\'s trajectory") rather than from the\nfields themselves, but I don\'t see how to extend this trick to the\nBRST charge. Maybe one can nevertheless define a good BRST\noperator. If so, one may pass to BRST cohomology in the same way as\nin string theory.\n\n(There are apparently also string theorists who consider anomalous\nWeyl symmetry - there is a large literature on subcritical strings\nand Liouville theory, about which I unfortunately know nothing. I am\nunsure about the status of Liouville theory - GSW say that it is\nprobably inconsistent and Polchinski that it is equivalent to the\ncritical string.)\n\nBut here is the punchline. The 3+1D diffeomorphism anomaly, which\nmust exist in the Hamiltonian formalism for the same reason that\nthere is a central charge in string theory, is a physical effect.\nAs such, it must be visible also in path-integral quantization;\nother anomalies manifest themselves already in low-order\nperturbation theory. And even if the anomaly cancels against ghosts\nfor some models and is therefore invisible to path integrals, there\nshould be similar models, analogous to subcritical strings, which\nare truly anomalous. For such theories the anomaly must be present\nin any formalism.\n\nAlas, it is well known in field theory (and string theory, they are\nnot different in this respect), that there are no gravitational\nanomalies in 4D whatsoever, see e.g. Weinberg, Chapter 23. So if\nsuch theories are quantized canonically, the diffeomorphism\ngenerators cannot be unitarily and non-trivially represented on a\nweakly-continuous Hilbert space! And if these theories don\'t admit\na canonical quantization, even in principle, something very weird\nis going on.\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>Everybody has presumably already noticed
http://www.arxiv.org/abs/http://www....ep-th/0409182, where Robert Helling and
Giuseppe Policastro compare Fock and LQG quantization of a string.
In particular, they notice that one can do LQG quantization of
the harmonic oscillator, and obtain an energy spectrum which is
not bounded from below. Needless to say, this is bad news for LQG.

This paper gives a good background for a general discussion of
canonical quantization of general-covariant theories like general
relativity. At some stage in this process, we want to find a
unitary representation of the diffeomorphism generators on some
Hilbert space. Ideally, we want our representations of the
diffeomorphism group to be non-trivial, irreducible, unitary,
anomaly-free, and of lowest-energy type. Unfortunately, a theorem
states that no such representation exists, which is major
complication.

This point is best illustrated by quantization of the string. The
relevant group here is the conformal group rather than the group of
diffeomorphisms, but the math is the same since the infinite-
dimensional conformal group in 2D is isomorphic to two copies of
the full diffeo group in 1D. In bosonic string theory, one starts
with 26 coordinate fields. The Virasoro algebra acts on these, but
since the central charge $c = 26$ is non-zero this rep is not
anomaly-free. To remedy that, one introduces a ghost with $c = -26$.
The total rep is anomaly-free but not irreducible, being a direct
sum. More importantly, it is not unitary since the ghost has
negative central charge. Finally one factors out states with
negative and zero norm by passing to BRST cohomology. The resulting
representation is unitary and has $c = 0,$ and is thus trivial, in
accordance with the theorem.

Thomas Thiemann does something different. He does indeed obtain an
irrep which is both non-trivial, unitary and anomaly-free. To avoid
conflict with the classification of Virasoro irreps, this "polymer"
representation must therefore not be of lowest-energy type.
Helling-Policastro makes this explicit when they calculate the
spectrum of the polymer harmonic oscillator, and find that the
energy is not bounded from below.

Contrary to the moderators of sps (especially one), I think that I
have some credibility when I say that I am not biased in favor of
string theory. Nevertheless, I believe that the Helling-Policastro
paper, and previous arguments by Urs Schreiber, are fatal to LQG.
It is sad to see the good guys lose, but having lowest-energy
representations is the very essence of quantum theory from my
point of view.

However, the LQG string was only a toy problem. The really
interesting situation is in canonical quantization of $3+1D$ gravity,
which should proceed in analogy with the $1+1D$ case. It does not
really matter whether we deal with field theory, string theory, or
LQG. In particular, even if only the first-quantized formulation
of string theory is well understood, we may assume that some
non-perturbative Hamiltonian formulation of string/brane field theory
exists. If such a Hamiltonian formulation (perhaps in some
generalized sense) does not exist even in principle, we will be in
trouble anyway because we will have to rethink the foundations of
quantum mechanics.

Thus assume that we have some general-covariant theory, and that
we need a non-trivial, unitary, lowest-weight representation of the
3D diffeomorphism generators on a separable Hilbert space. Such
a representation is necessarily anomalous.

Proof: Well-known in 1D, where the only anomaly-free unitary irrep
is the trivial one. In higher dimensions, the diffeomorphism
algebra has many subalgebras generated by vector fields of the form
$f(y)d/dy,$ where $y = k_i x^i$ for some numbers $k_i$. A unitary irrep
of a big algebra must restrict to a unitary rep (in general
reducible) of each subalgebra labelled by $k_i.$ But we know that
every restriction must be a direct sum of trivial irreps. Only the
trivial rep has only trivial restrictions. QED.

The next natural step, if we want to proceed in analogy with the
string, is to mod out diffeomorphisms, e.g. by a BRST procedure.
However, this meets with a serious technical obstacle, which has
led me to assert that one must live with the anomaly. The problem
is that there is no well-defined BRST operator, AFAIU. This is a
genuinely new problem which does not appear in 1D. In fact, there
are three qualitatively different situations:

1. For finite-dimensional algebras, the BRST operator is always
well-defined and nilpotent.

2. For diffeomorphisms in 1D, the BRST operator is always
well-defined, but nilpotent only for some special parameter value
such as $c = 26$.

3. For diffeomorphisms in two and more dimensions, the BRST
operator is apparently ill defined. The problem is that normal
ordering introduces an unrestricted sum over transverse modes,
which gives an infinity not present in 1D. This can be avoided in
the algebra generators by starting from a Taylor expansion around a
marked 1D curve ("the observer's trajectory") rather than from the
fields themselves, but I don't see how to extend this trick to the
BRST charge. Maybe one can nevertheless define a good BRST
operator. If so, one may pass to BRST cohomology in the same way as
in string theory.

(There are apparently also string theorists who consider anomalous
Weyl symmetry - there is a large literature on subcritical strings
and Liouville theory, about which I unfortunately know nothing. I am
unsure about the status of Liouville theory - GSW say that it is
probably inconsistent and Polchinski that it is equivalent to the
critical string.)

But here is the punchline. The $3+1D$ diffeomorphism anomaly, which
must exist in the Hamiltonian formalism for the same reason that
there is a central charge in string theory, is a physical effect.
As such, it must be visible also in path-integral quantization;
other anomalies manifest themselves already in low-order
perturbation theory. And even if the anomaly cancels against ghosts
for some models and is therefore invisible to path integrals, there
should be similar models, analogous to subcritical strings, which
are truly anomalous. For such theories the anomaly must be present
in any formalism.

Alas, it is well known in field theory (and string theory, they are
not different in this respect), that there are no gravitational
anomalies in 4D whatsoever, see e.g. Weinberg, Chapter 23. So if
such theories are quantized canonically, the diffeomorphism
generators cannot be unitarily and non-trivially represented on a
weakly-continuous Hilbert space! And if these theories don't admit
a canonical quantization, even in principle, something very weird
is going on.



> Alas, it is well known in field theory (and string theory, they are > not different in this respect), that there are no gravitational > anomalies in 4D whatsoever, see e.g. Weinberg, Chapter 23. So if > such theories are quantized canonically, the diffeomorphism > generators cannot be unitarily and non-trivially represented on a > weakly-continuous Hilbert space! And if these theories don't admit > a canonical quantization, even in principle, something very weird > is going on. Weird indeed, even if I just understood the sketch of the argument, I have a question right from the beginning though. I understand lowes energy type simply means there is a lowest energy in the system? Quick googling doesn't lead to any more precise definitions, could you point me the right way if I'm far off? "In particular, they notice that one can do LQG quantization of the harmonic oscillator, and obtain an energy spectrum which is not bounded from below. Needless to say, this is bad news for LQG." In diffeomorphism invariant context we have no good definition of energy to begin with even in classical physics. Is this result obtained here then truely such a critical problem for LQG? It's not expected that there is a physical energy observable in QG to begin with. --- frank

 Recognitions: Gold Member Science Advisor Thomas Larsson wrote: >... At some stage in this process, we want to find a unitary >representation of the diffeomorphism generators on some >Hilbert space. Ideally, we want our representations of the >diffeomorphism group to be non-trivial, irreducible, unitary, >anomaly-free, and of lowest-energy type... Just a technical point. The diffeomorphisms are a subgroup of the group in question. In recent LQG papers a larger group is used consisting of homeomorphisms which are smooth at all but a finite number of points. See Smolin's paper http://arxiv.org/hep-th/0408048 especially page 9 where this paper by Rovelli is cited: http://arxiv.org/gr-qc/0403047

## Diffeomorphisms, LQG, and positive energy

<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>thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:&lt;24a23f36.0409220742.157cfce3@posting.google.com&gt;...\n&gt; Everybody has presumably already noticed\n&gt; http://www.arxiv.org/abs/hep-th/0409182, where Robert Helling and\n&gt; Giuseppe Policastro compare Fock and LQG quantization of a string.\n&gt; In particular, they notice that one can do LQG quantization of\n&gt; the harmonic oscillator, and obtain an energy spectrum which is\n&gt; not bounded from below. Needless to say, this is bad news for LQG.\n&gt;\n\nHelling-Policastro showed that canonical quantization a la LQG is\nstrange. However, it might be worth pointing out that string theory\nadmits no canonical quantization at all in more than two dimensions.\nEverything is done using path integrals. Usually one justifies formal\nmanipulations with path integrals by referring to the well-defined\nHamiltonian formalism. This seems somewhat dubious if no Hamiltonian\nformalism exists.\n\nThis puts some perspective on LQG\'s achievements. It is not obvious\nto me that strange canonical quantization is so much worse than no\ncanonical quantization at all.\n\nThe secret reason why canonical quantization of diff-invariant\ntheories in more than 2D fails is that the relevant diffeomorphism\ngroup anomaly is little known. The diffeomorphism generators should\nbe represented by unitary operators on a conventional Hilbert space,\nand all non-trivial such representations are anomalous. Since neither\nthe string theory nor LQG camps care about these anomalies in 4D,\nthey cannot do canonical quantization.\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">&nbsp;&nbsp;View this Usenet post in original ASCII form </a></div><P></jabberwocky>thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0409220742.157cfce3@p...google.com>...
> Everybody has presumably already noticed
> http://www.arxiv.org/abs/http://www....ep-th/0409182, where Robert Helling and
> Giuseppe Policastro compare Fock and LQG quantization of a string.
> In particular, they notice that one can do LQG quantization of
> the harmonic oscillator, and obtain an energy spectrum which is
> not bounded from below. Needless to say, this is bad news for LQG.
>

Helling-Policastro showed that canonical quantization a la LQG is
strange. However, it might be worth pointing out that string theory
admits no canonical quantization at all in more than two dimensions.
Everything is done using path integrals. Usually one justifies formal
manipulations with path integrals by referring to the well-defined
Hamiltonian formalism. This seems somewhat dubious if no Hamiltonian
formalism exists.

This puts some perspective on LQG's achievements. It is not obvious
to me that strange canonical quantization is so much worse than no
canonical quantization at all.

The secret reason why canonical quantization of diff-invariant
theories in more than 2D fails is that the relevant diffeomorphism
group anomaly is little known. The diffeomorphism generators should
be represented by unitary operators on a conventional Hilbert space,
and all non-trivial such representations are anomalous. Since neither
the string theory nor LQG camps care about these anomalies in 4D,
they cannot do canonical quantization.



> The diffeomorphism generators should > be represented by unitary operators on a conventional Hilbert space, > and all non-trivial such representations are anomalous. Could you explain this a little more? In what sense are they anomalous? What do you need the non-trivial representations for and which ones are they?



daniel@elit.net (Daniel Elander) wrote in message news:<37d84b42.0409290028.1b1dd162@p...google.com>... > > The diffeomorphism generators should > > be represented by unitary operators on a conventional Hilbert space, > > and all non-trivial such representations are anomalous. > > Could you explain this a little more? In what sense are they > anomalous? What do you need the non-trivial representations for and > which ones are they? Let us first consider quantization of the string. If you do it the string theory way, there is a conformal Weyl symmetry acting on the world sheet. In 2D the conformal group is infinite-dimensional $- it$ consists of transformations which are analytic in $z = x+iy$ and $z* =x-iy$. If we concentrate on the former, we see that an infinitesimal conformal transformation is generated by $$L_m = z^{m+1} d/dz,$$ and they obey the algebra $$[L_m, L_n] = (n-m) L_{m+n}[/itex]. However, it is easy to see that this is the algebra of vector fields (= infinitesimal diffeomorphisms) on the circle, vect(1). If the circle coordinate is x, the generators are $L_m = -i \exp(imx) d/dx$. vect(1) has a central extension, known as the Virasoro algebra: $[L_m, L_n] = (n-m) L_{m+n} - c/12 (m^3 - m) \delta_{m+n,0},$$ where c is a c-number known as the central charge or conformal anomaly. This means that the Virasoro algebra is still a Lie algebra - anti-symmetry and the Jacobi identities still hold. The term linear in m is unimportant, because it can be removed by a redefinition [itex]of L_0$. The $m^3$ term is a genuine quantum effect, which simply is there when you quantize a string. When string theorists criticize Thiemann's LQG string, they are basically complaining that he does not get this term, which simply must be there. There is some confusion about anomaly freedom here, because at the end people want to eliminate the conformal symmetry. The nice way to do this is to introduce a ghost pair $b_m, c_n,$ satisfying fermionic brackets $${ b_m, c_n } = \delta_{m+n,0}[/itex]. One can now write down the BRST operator, which looks something like (double dots denote normal ordering) $Q = sum_m :L_{-m} c_m: + 1/2 sum_{m,n} (m-n) :b_{-m-n} c_m c_n:$$ If the BRST operator is nilpotent, [itex]Q^2 = 0, we$ can identify physical states with BRST cohomology. A state is physical if it is BRST closed, Q|phys> $= 0,$ and two states are equivalent if they differ by a BRST exact term, |phys> ~ |phys'> if Q( |phys> - |phys'> ) = . It turns out that the ghost has central charge $c = -26,$ so the BRST operator is nilpotent if the $L_m's$ have $c = 26;$ this is where the 26 dimensions of they bosonic string comes from. However, the important thing from my viewpoint is not that the end result is anomaly free, but that an anomaly exists, even if only in intermediate calculations. Thiemann does not have an anomaly even intermediately, and from this string theorists (and myself) conclude that LQG is wrong. Let us now return to the math. A lowest-weight representation (LWR) of the Virasoro algebra is characterized by a lowest-weight state $|h,c>$ satisfying $$L_0 |h,c> = h |h,c>,L_{-m} |h,c> = 0,[/itex] for all $-m <$ . It is known that the only unitary LWR of vect(1) is the trivial one. However, the Virasoro algebra has many unitary LWRs: the discrete series with $<= c < 1,$ where $c = 1 - 6/m(m+1), m$ positive integer and $h = h_{p,q}(c) = (pm^2 - q(m+1)^2) / 4m(m+1)$$ (or something similar, I'm quoting from memory), and also all c > 1, h > . Anyway, the important thing is that the only acceptable value of (h,c) with c = is h = - this is the trivial representation. That Thiemann obtains a non-trivial unitary representation of the 1D diffeomorphism group with c = is thus very strange. It is hard to see how that could be compatible with quantum theory. The Virasoro algebra can be generalized to several dimensions - an extension of the diffeomorphism algebra on the N-dimensional torus, say. The generators are $$L_k(m) = i \exp(im.x) d/dx^k,$$ where [itex]m = (m_i)$ and m.$x = m_i x^i$ and I use the summation convention. The algebra depends on two parameters (abelian charges) $c_1$ and $c_2,[L_i(m), L_j(n)] = n_i L_j(m+n) - m_j L_i(m+n)+ (c_1 m_j n_i + c_2 m_i n_j) m_k S^k(m+n),[L_i(m), S^j(n)] = n_i S^j(m+n) + \delta^j_i m_k S^k(m+n),[S^i(m), S^j(n)] = 0,m_k S^k(m) =$ . This is the Virasoro algebra in 1D because the last condition then becomes m S(m) $= 0,$ which only has the solution S(m) $~ \delta(m)$. The Virasoro extension is not central (does not commute with everything) except in 1D$. It$ is straightforward but somewhat tedious to check that these relations indeed do define a Lie algebra. Just as the Virasoro algebra is anomalous when $c != 0,$ its higher- dimensional sibling is anomalous unless both $c_1 = c_2 =$ . And just as for the Virasoro algebra, there is no non-trivial, unitary LWRs unless the algebra is anomalous. The correct definition of lowest-weight is more subtle in several dimensions. Let me just say that the right definition does not introduce any anisotropy. Now, if you do canonical quantization of a conformal or diffeomorphism invariant theory, then the conformal or diffeomorphism algebras acts on the Hilbert space of the theory. Due to quantum effects (normal ordering), anomalies arise - one must deal with the Virasoro algebras, in one or several dimensions. Perhaps the anomalies can be cancelled in the end (I doubt this, though), but it is absolutely clear the they must arise intermediately. This is true in canonical quantization of the string, which string theorists are eager to point out, since this observation hurts LQG. But it is equally true if you quantize a diff-invariant theory in $3+1D,$ for the same reasons. However, string theory can only see diff anomalies if the number of dimensions equals $4k+2,$ and even then it is another kind of anomaly, related to chiral fermions rather than to the Virasoro algebra. Consequently, string theorists cannot do canonical quantization in more than 2D. Urs Schreiber explicitly pointed this out to me; it is kind of obvious but I had not thought of this formulation before. I think that it is kind of interesting to know the obstruction to canonical quantization in several dimensions, and the way around this obstruction. Alas, it is abundantly clear that most physicists disagree with me.



"Thomas Larsson" schrieb im Newsbeitrag news:24a23f36.0409300116.739d3a1b@posting.google.com... > The Virasoro algebra can be generalized to several dimensions - > an extension of the diffeomorphism algebra on the N-dimensional > torus, say. The generators are > > $L_k(m) = i \exp(im.x) d/dx^k,$ [...] > Now, if you do canonical quantization of a conformal or > diffeomorphism invariant theory, then the conformal or > diffeomorphism algebras acts on the Hilbert space of the theory. Due > to quantum effects (normal ordering), anomalies arise - one must > deal with the Virasoro algebras, in one or several dimensions. Or so it might seem at first sight. I am not so sure about this step. What you know is the algebra of the $L_k(m)$. What you have not shown is that these generators show up all by themselves when quantizing gravity. You mention the string as the lowest-dimensional example of your approach, but it is not really, for the following reason: For 2 dimensional (conformal) gravity coupled to matter the modes of the Hamiltonian constraint H are $$H_n = L_n + \bar L_-n$$ and the modes of the spatial diffeomorphism constraints D are $$D_n = L_n - \bar L_-n .$$ These are the generators that you have to think about when talking about canonically quantizing gravity. They don't seem to be special cases of the algebra of the $L_k(m)$ in two dimensions, though. One reason for that is that $H_n$ and $D_n$ are *spatial* modes only. That's due to the very nature of canonical quantization of diff-invariant theories: There you have to choose a foliation of your parameter space into spacelike hyperslices, states are functionals of data on any one hyperslice and the constraints act on these states. The constraints are defined for every point of the given hyperslice, not for every point of the full parameter space. This is best visualized for 1+0 dimensional diff-invariant theories (e.g. the KG particle). The $D_n$ generate a Virasoro algebra with c=0. (As a side note: This means they can all be imposed at once and the states annihilated by them are known as boundary states, up to some details.) The $H_n$ don't generate any closed algebra at all. It is true that the $H_n$ and the $D_n$ together generate an algebra isomophic to two copies of the algebra generated by $L_n,$ but this is an artifact of 2 dimensions and does not have higher dimensional counterparts. Therefore this fact does not play any role when you want to apply the $L_k(m)$ to the general quantization of gravity. Even in 2 dimensions it implies that (a single copy of) the algebra of the $L_k(m)$ in 2D does not incorporate the algebra of the canonical constraints of the theory, which are $H_n$ and $D_n$ for all integer n.



thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0409300116.739d3a1b@p...google.com>... > is BRST closed, Q|phys> $= 0,$ and two states are equivalent if > they differ by a BRST exact term, > > |phys> ~ |phys'> if Q( |phys> - |phys'> ) = . > Yuck! The text was right but the formula is not. |phys> ~ |phys'> if |phys> - |phys'> $= Q |\phi>$ for some arbitrary state $|\phi>$. Btw, the cohomology stuff is a nice generalization of ideas learnt in classes on vector calculus. In that case, Q stands for the $grad,$ curl and div operators, and $Q^2 =$ translates into curl $grad = div$ curl = . In electromagnetism, the electric and magnetic fields are closed: curl $E = 0, div B =$ . One solution is that they are exact, i.e. $E = grad V, B =$ curl A. This does not hold globally in topologically non-trivial situations.



"Urs Schreiber" wrote in message news:... > "Thomas Larsson" schrieb im Newsbeitrag > news:24a23f36.0409300116.739d3a1b@posting.google.com... > > > The Virasoro algebra can be generalized to several dimensions - > > an extension of the diffeomorphism algebra on the N-dimensional > > torus, say. The generators are > > > > $L_k(m) = i \exp(im.x) d/dx^k,$ > > [...] > > > Now, if you do canonical quantization of a conformal or > > diffeomorphism invariant theory, then the conformal or > > diffeomorphism algebras acts on the Hilbert space of the theory. Due > > to quantum effects (normal ordering), anomalies arise - one must > > deal with the Virasoro algebras, in one or several dimensions. > > Or so it might seem at first sight. I am not so sure about this step. What > you know is the algebra of the $L_k(m)$. What you have not shown is that these > generators show up all by themselves when quantizing gravity. > > You mention the string as the lowest-dimensional example of your approach, > but it is not really, for the following reason: > > For 2 dimensional (conformal) gravity coupled to matter the modes of the > Hamiltonian constraint H are > > $H_n = L_n + \bar L_-n$ > > and the modes of the spatial diffeomorphism constraints D are > > $D_n = L_n - \bar L_-n .$ > > These are the generators that you have to think about when talking about > canonically quantizing gravity. They don't seem to be special cases of the > algebra of the $L_k(m)$ in two dimensions, though. Yes, they are! We can embed two commuting 1D Virasoro algebras into the 2D Virasoro in several ways. They simplest is to take $$L_n = L_1(m_1), \bar L_n = L_2(m_2)[/itex]. where $m = (m_1, m_2)$. More physically, conformal transformations are generated by vector fields of the form $L_n = z^(n+1) d/dz, \bar L_n = (z*)^(n+1) d/d(dz*)$. They generate two subalgebras of the algebra of general Laurent polynomial vector fields in 2D, which is spanned by vector fields of the form $L_1(m) = x^(m_1) y^(m_2) x d/dx, L_2(m) = x^(m_1) y^(m_2) y d/dy$. In the previous post I considered Fourier polynomials rather than Laurent polynomials, but it makes no difference. The algebra and the extension are the same. Incidentally, this observation gives us an interesting class of representations of the 2D Virasoro algebra. Given any Lie algebra g, a Lie subalgebra $h \subset g,$ and a h representation R, we can construct the induced g representation Ind R. Mathematicians write this as Ind $R = U(g)\tensor_U(h) R$. In words, it means that we start from the universal enveloping algebra U(g), and factor out the relations in U(h) that come from R. This is in a sense the biggest g rep whose restriction to h is R. In particular, if we take g = Vir(2) (2D diffeomorphisms) and $h = Vir(1)+Vir(1)$ (conformal transformations), then we get a Vir(2) rep for each minimal model. The induced rep does not need to be unitary or irreducible, and we don't really gain anything since everything of interest is already encoded in the conformal algebra. Nevertheless, it shows that Vir(2) reps exist which contain the information in conformal field theory. The only crucial point is that $Vir(1)+Vir(1),$ including the central charge, is a subalgebra of Vir(2). > > One reason for that is that $H_n$ and $D_n$ are *spatial* modes only. That's due > to the very nature of canonical quantization of diff-invariant theories: > There you have to choose a foliation of your parameter space into spacelike > hyperslices, states are functionals of data on any one hyperslice and the > constraints act on these states. The constraints are defined for every point > of the given hyperslice, not for every point of the full parameter space. > This is best visualized for 1+0 dimensional diff-invariant theories (e.g. > the KG particle). > > The $D_n$ generate a Virasoro algebra with c=0. (As a side note: This means > they can all be imposed at once and the states annihilated by them are known > as boundary states, up to some details.) The $H_n$ don't generate any closed > algebra at all. > > It is true that the $H_n$ and the $D_n$ together generate an algebra isomophic > to two copies of the algebra generated by $L_n,$ but this is an artifact of 2 > dimensions and does not have higher dimensional counterparts. Therefore this > fact does not play any role when you want to apply the $L_k(m)$ to the general > quantization of gravity. > > Even in 2 dimensions it implies that (a single copy of) the algebra of the > $L_k(m)$ in 2D does not incorporate the algebra of the canonical constraints > of the theory, which are $H_n$ and $D_n$ for all integer n. As I mentioned elsewhere, the Dirac algebra of ADM constraints is physically equivalent to the 4D diffeo algebra. Phase space is a covariant concept $- it$ is the space of solutions to the classical equations of motion. We may label a solution by the values of the positions and momenta at time t=0. However, this is only a coordinatization of phase space, and it is not preserved by all 4D diffeomorphisms. If we insist on keeping the standard coordinatization, we must add compensating transformations which bring us back to it. The combination of 4D diffeomorphism and compensating transformations generate the Dirac algebra of constraints. However, I propose to do canonical quantization directly in the covariant phase space, where the constraint algebra is exactly the 4D diffeomorphism algebra vect(4). I believe that Rovelli has made similar attempts, but he has of course not the relevant anomaly. A rather remarkable fact is that the 4D diffeomorphism algebra seems to know about, and resolve, the various problems of time. Let us again recall that in order to build lowest-energy reps of the diffeomorphism algebra, we must first expand all fields in a Taylor series around a marked curve, "the observer's trajectory". We thus write $\phi(x) = sum_m \phi_m(t) (x - q(t))^m,$$ where the sum runs over multi-indices m. Actually, this expression defines a field [itex]\phi(x,t)$. To ensure that the fields are independent of the parameter t, we must demand that $d \phi(x,t)/dt = .$ vect(4) acts not only on the $\phi(x)'s,$ but also non-linearly on the p-jet space spanned by q(t), $\phi_0(t), \phi_1(t), \phi_2(t), .$.$. , \phi_p(t)$ This looks one-dimensional, but with multi-indices the formulas make sense in higher dimensions as well. q(t) is on the same footing as the $\phi_m(t);$ it is thus a physical field, has a canonical momentum, and is represented on the Fock space. The crucial observation is that we now have a realization of vect(4) on finitely many functions of a *single* variable t. This is the situation where normal ordering works without giving infinities. But we now get an action of an extra Virasoro algebra "for free", namely reparametrizations generated by vector fields $f(t)d/dt$. The full constraint algebra thus becomes $vect(4)+vect(1).$ Here is how one can address some of the problems of time in this setup: $* We$ can define a local energy operator $H = -i d/dt$. It affects the observer's trajectory q(t) but not the field $\phi(x),$ since we demanded that they be independent of t. IOW, H moves the fields relative to the physical point q(t). It is thus not a Hamiltonian constraint which generates coordinate transformations, but a genuine local energy operator which moves physical objects relative to each other. * Energy, like time, has both vector and scalar properties; it is the zeroth component $P_0$ of four-momentum, but also the Hamiltonian which is bound from below by the scalar mass. These two notions correspond to different operators: 4-momentum is the diffeomorphism $d/dx^u,$ whereas the Hamiltonian is the reparametrization $-i d/dt$. * One problem with a background reference metric is that the causal structure of spacetime changes: two events may be spacelike separated wrt the reference Minkowski metric but timelike wrt the full metric. This is a problem because a correlator is identically zero to all orders in perturbation theory if the events are outside the Minkowski lightcone, even if they are inside the true lightcone. This problem is evaded by formulating everything in terms of data $\phi_m(t)$ living on the observer's trajectory, where everything is causally related. Classically at least, there is no problem in principle (although many in practice) with this: the Euler-Lagrange equations are replaced by a hierarchy of equations for the Taylor coefficients. Spacelike separation only reappears in the limit $p ->$ infinity (which is subtle). IMO, providing resolutions of the paradoxes of time is one of the most important things that quantum gravity should address. If we combine the fact that the diffeomorphism algebra answers these questions, with the fact that we need the anomalies for unitarity, I don't see how vect(4) can fail to be relevant to quantum gravity. There are simply too many pieces that fall into place.



thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0410022329.38dc3c7b@p...google.com>... > "Urs Schreiber" wrote in message news:... > > "Thomas Larsson" schrieb im Newsbeitrag > > news:24a23f36.0409300116.739d3a1b@posting.google.com... > ... a great deal of good stuff skipped ... > IMO, providing resolutions of the paradoxes of time is one of the most > important things that quantum gravity should address. If we combine the fact > that the diffeomorphism algebra answers these questions, with the fact that > we need the anomalies for unitarity, I don't see how vect(4) can fail to be > relevant to quantum gravity. There are simply too many pieces that fall into > place. Carlo Rovelli, in his new book Quantum Gravity says this about unitarity and general relativity (and by extension, quantum gravity): "In conventional QM and QFT unitarity is the consequence of the time translation symmetry of the dynamics. In GR there isn't, in general, an analogous notion of time translation symmetry. Therefore there is no sense in which conventional unitarity is necessary in the theory. One often hears that without unitarity a theory is inconsistant. This is a misunderstanding that follows from the erroneous assumption that all physical theories are symmetric under time translations." (Section 10.1.3, Draft $of 12/30/2003)$ Could you comment on this? Is it relevant to your insights?



DickT wrote: > Carlo Rovelli, in his new book Quantum Gravity says this about > unitarity and general relativity (and by extension, quantum gravity): > > "In conventional QM and QFT unitarity is the consequence of the time > translation symmetry of the dynamics. In GR there isn't, in general, > an analogous notion of time translation symmetry. Therefore there is > no sense in which conventional unitarity is necessary in the theory. > One often hears that without unitarity a theory is inconsistant. This > is a misunderstanding that follows from the erroneous assumption that > all physical theories are symmetric under time translations." > (Section 10.1.3, Draft $of 12/30/2003)$ Unitarity has nothing to do with time trasnslation invariance. It is needed to preserve probability: The sum of absolute squares of S-matrix elements with fixed input and an orthonormal basis of output states must sum to 1. This easily translates into $S^*S=1$. Thus giving up unitarity is giving up the probability interpretation of the S-matrix. Of course, the concept of an S-matrix, which gives transitions between $t=-inf$ and $t=inf$ states, is itself questionable in a theory of gravity which allows a big bang, so that $t=-inf$ is forbidden. And without S-matrix there is no problem with unitarity... Arnold Neumaier



rthompson10@new.rr.com (DickT) wrote in message news:<93eecb7c.0410121507.5b7e39f4@p...google.com>... > Carlo Rovelli, in his new book Quantum Gravity says this about > unitarity and general relativity (and by extension, quantum gravity): > > "In conventional QM and QFT unitarity is the consequence of the time > translation symmetry of the dynamics. In GR there isn't, in general, > an analogous notion of time translation symmetry. Therefore there is > no sense in which conventional unitarity is necessary in the theory. > One often hears that without unitarity a theory is inconsistant. This > is a misunderstanding that follows from the erroneous assumption that > all physical theories are symmetric under time translations." > (Section 10.1.3, Draft $of 12/30/2003)$ > > Could you comment on this? Is it relevant to your insights? Unitarity is a tricky subject in general relativity. By definition, a unitary matrix preserves the inner product, but it is difficult to write down an inner product which is invariant under arbitrary diffeomorphisms. Classically, the only field types which allow an invariant inner product are half-forms, and n/2-forms if the spacetime dimension n is even. Then you can define the inner product by (u, $v) = \int u v,$ because u v is a volume form. Otherwise, the definition will involve the metric, which is against the spirit of diffeomorphism invariance. Without an inner product, it is hard to say what unitarity means. Nevertheless, I believe that Rovelli is wrong about unitarity. Physically it means that probability is preserved - all probabilities must sum up to 1, and there are no negative probabilities. This is such a fundamental property that I don't see how it can ever be wrong. In fact, I don't understand what not preserving probabilities would mean. Fortunately, we can say something quite general, even if we don't know exactly how the unitary representations look. Assume that some group G has a unitary representation R and a subgroup H. Then the restriction of R to H is still unitary, and this must hold for every subgroup H of G. In particular, let G be the diffeomorphism group in n dimensions and H the diffeomorphism group in 1D. There are infinitely many such subgroups, and the restriction of R to each and every one of them must be unitary. Fortunately, the unitary irreps of the diffeomorphism group in 1D (or its Lie algebra, at least) are known. The result is that the only proper unitary irrep is the trivial one, but there are many unitary irreps if you have a diffeomorphism anomaly. From this it follows that the trivial rep is the only unitary rep also in n dimensions. Well, maybe one should be content with the trivial representation. After all, general covariance is a gauge symmetry (more about that below), and we are told that a gauge symmetry is a redundancy of the description. Well, in the case of general covariance, this leads to severe conceptual difficulties, mainly because the Hamiltonian is a generator of the symmetry. In other words, the Hamiltonian is a redundancy of the description, so there is no energy and no time; the Hamiltonian is replaced by a Hamiltonian constraint H = . This has kept people confused for a long time, myself not the least, but I am in good company: Einstein himself considered abandoning general covariance for this reason. A lot of people (in fact, most people) argue that this is not a problem. They are wrong. It is very closely related to locality. How can you have locality in a theory where there is no a priori notion of distance? This is really a serious difficulty, and it had made some deep thinkers (read: 't Hooft) so desperate that they are even willing to abandon quantum mechanics. In the beginning of the talk at http://online.kitp.ucsb.edu/online/kitp25/thooft/ , 't Hooft explains that he simply believes in locality, and that this is why he explores these weird ideas about hidden variables. It is interesting that the inventor of holography has such a strong belief in locality. However, there is one well-known case where we know how to combine locality and diff invariance with quantum theory: conformal field theory. CFT is usually thought of as a theory of conformally invariant QFT in 2D, but since the local conformal group is the same as (twice) the diffeomorphism group in 1D, it is also about diff invariant QFT in 1D. Locality means that the correlation functions depend on separation. For two points z and w in R or C, the correlator is $$< \phi(z) \phi(w) > ~ 1/(z-w)^{2h} +[/itex] more, where "more" stands for less singular terms when $z -> w$. That the correlation function has this form is a diff-invariant statement. The "more" terms will change under an arbitary diffeomorphism, but the leading singularity will always have the same form, and in particular the anomalous dimension h is well defined. We can phrase this slightly differently. The short-distance singularity only depends on two points being infinitesimally close. This is good, because we cannot determine the finite distance between two points without knowing about the metric. GR does not have a background metric structure, but it does have a background differentiable structure (locally at least), and that is enough for defining anomalous dimensions. Diffemorphisms move points around, but they don't separate two points which are infinitesimally close. There is a well-known theorem which states that all correlation functions in general relativity are trivial, which is easy to prove by moving points. How can CFT by compatible with this theorem? Well, it isn't. The relevant algebra in CFT is not really the 1D diffeomorphism algebra (or the 2D conformal algebra), but its central extension known as the Virasoro algebra: $[L_m, L_n] = (n-m) L_m+n - c/12 (m^3 - m) \delta_m+n$. A lowest-energy representation is characterized by a vacuum satisfying $L_0$ |vac> = h |vac>, $L_m$ |vac> = for all m < . In particular, the lowest $L_0$ eigenvalue can be identified with the anomalous dimension h in the correlation function above. This means that locality, in the sense of correlation functions depending on separation, requires that h > . However, the central extension c is non-zero for any non-trivial, unitary irrep with $h !=$ . This leads to the main observation: ------------------------------------------------------------------- | Locality and unitarity are compatible with diffeomorphism (and | | local conformal) symmetry only in the presence of an anomaly. | ------------------------------------------------------------------- This is true in higher dimensions as well. Consider the correlator $<\phi(x) \phi(y)>,$ where x and y are points in $R^n$. We could take some 1D curve q(t) passing through x and y, such that $x = q(t)$ and $y = q(t').$ Then the short-distance behaviour is of the form $< \phi(x) \phi(y) > ~ 1/(t - t')^{2h} +$ more, and h is independent of the choice of curve (there might be some regularity condition). The group of diffeomorphisms which preserve q(t) is a Virasoro algebra, so h > implies that c > . Note that what I'm saying is completely standard in the application of CFT to statistical physics in 2D. The simplest example of a unitary model is the Ising model, which consists of three irreps, with $c = 1/2$ and $h = 0, h = 1/16, h = 1/2$. The Ising model is perfectly consistent despite the anomaly, both mathematically (unitarity), and more importantly physically (it is realized in nature, in soft condensed matter systems). But why then do people say that general covariance is a gauge symmetry, and that gauge anomalies are inconsistent? Well, depending on what they really mean, this statement is either trivial or wrong. Let us look a little closer to gauge symmetries in a Hamiltonian setting. Let us assume that we have some phase space, and a Lie algebra g with generators $J^a,$ satisfying $[J^a, J^b] = f^{ab_c} J^c,$$ acts on this phase space. If the bracket with the Hamiltonian gives us a new element in g, $$[J^a, H] = C^{a_b} J^b$$ we say that g is a *symmetry* of the Hamiltonian system. If g in addition contains arbitary functions of time, the symmetry is a *gauge symmetry*. In this case, a solution to Hamilton's equations depends on arbitrary functions of time and is thus not fully specified by the positions and momenta at time t = . The standard example is electromagnetism, where the zeroth component [itex]A_0$ of the vector potential is arbitrary, because its canonical momentum $F^{00} = .$ An arbitary time evolution is of course not acceptable. The reason why this seems to happen is that a gauge symmetry is a redundancy of the description; the true dynamical degrees of freedom are fewer than what one naively expects. In electromagnetism, the gauge potential has four components but the photon has only two polarizations. There are various ways to handle quantization of gauge systems. One is to eliminate the gauge degrees of freedom first and then quantize. This is cumbersome and people usually prefer to quantize first and then eliminate the gauge symmetries. The simplest way, although not the best, is to require that the gauge generators annihilate physical states, $$J^a[/itex] |phys> $= 0,$$ and also that two physical states are equivalent if the differ by some gauge state, [itex]J^a |>$. This procedure gives you a nice Hilbert space of physical states. However, one thing may go terribly wrong. Upon quantization, a symmetry may acquire some quantum corrections, so that g is replaced by $[J^a, J^b] = f^{ab_c} J^c + \hbar D^{ab} + O(\hbar^2)$. The operator $D^{ab}$ is called an *anomaly*. We can also have anomalies of the type $[J^a, H] = C^{a_b} J^b + \hbar E^a + O(\hbar^2)$. If we now try to keep the definition of a physical state, we see that we must also require that $D^{ab}$ |phys> = . This implies further reduction of the Hilbert space. In the case that $D^{ab}$ is invertible, it means that there are no physical states at all, so the Hilbert space is empty. This is obviously not good. However, it does not necessarily mean that the anomaly by itself is inconsistent, only that our definition of physical states is. In the presence of an anomaly, additional states become physical. So our Hilbert space becomes bigger, containing some, or even all, of the previous gauge degrees of freedom. It is important to realize that such a "fake" gauge symmetry may well be consistent. The Virasoro algebra is obviously anomalous, with the central charge playing the role of the $D^{ab},$ and still it has unitary representations with non-zero c. Of course, a gauge anomaly *may* be inconsistent, if the anomalous algebra does not possess any unitary reps. This is apparently what happens for the chiral-fermion type anomaly which is relevant e.g. in the standard model. With diffemorphism symmetry, the extra degrees of freedom are the components of the observer's trajectory. To learn something about a quantum system, we must interact with it. So we inject a test particle, or observer, into the system. In classical physics this does not affect the system. E.g., a test particle in GR will move along geodesics but it will not change the solution to Einstein's equation. In quantum physics, the test particle perturbs the system. I find it quite remarkable that the mathematics knows that we need an observer, because the anomaly is a functional of the observer's trajectory. In field theory one does not introduce such a test particle, so the anomaly is invisible there. In 1D, only one trajectory is possible, which is why the Virasoro extension is central in 1D but not otherwise. At some point mathematical physics boils down to belief systems. I simply believe that both quantum mechanics, general covariance, unitarity, and locality are fundamental properties of reality. In particular, it should be possible to give a local definition of energy, which is bounded from below by the mass of a particle. As I showed above, combining these desiderata is only possible if general covariance acquires an anomaly analogous to the central charge. This observation led me to look for, and discover, the higher-dimensional analogue of the Virasoro algebra and to work out some of its representation theory. So it was physics intuition that led to the mathematical discovery, and not the other way around.



Thomas Larsson wrote: > rthompson10@new.rr.com (DickT) wrote in message news:<93eecb7c.0410121507.5b7e39f4@p...google.com>... > >>Carlo Rovelli, in his new book Quantum Gravity says this about >>unitarity and general relativity (and by extension, quantum gravity): >> >>"In conventional QM and QFT unitarity is the consequence of the time >>translation symmetry of the dynamics. In GR there isn't, in general, >>an analogous notion of time translation symmetry. Therefore there is >>no sense in which conventional unitarity is necessary in the theory. >>One often hears that without unitarity a theory is inconsistant. This >>is a misunderstanding that follows from the erroneous assumption that >>all physical theories are symmetric under time translations." >>(Section 10.1.3, Draft $of 12/30/2003)$ >> >>Could you comment on this? Is it relevant to your insights? > > Unitarity is a tricky subject in general relativity. By definition, a > unitary matrix preserves the inner product, but it is difficult to > write down an inner product which is invariant under arbitrary > diffeomorphisms. Classically, the only field types which allow an > invariant inner product are half-forms, and n/2-forms if the spacetime > dimension n is even. Then you can define the inner product by > > (u, $v) = \int u v,$ > > because u v is a volume form. Otherwise, the definition will > involve the metric, which is against the spirit of diffeomorphism > invariance. Without an inner product, it is hard to say what unitarity > means. > > Nevertheless, I believe that Rovelli is wrong about unitarity. > Physically it means that probability is preserved - all probabilities > must sum up to 1, and there are no negative probabilities. This is such > a fundamental property that I don't see how it can ever be wrong. In > fact, I don't understand what not preserving probabilities would mean. There seems to be a misunderstanding here, 'unitarity' being used in two different meanings. I think Rovelli means with unitarity the fact that in QM and QFT the S-matrix comes out unitary. This is nontrivial - in QM it requires asymptotic completeness which is not easy to prove for Coulomb interactions; in QFT it is guaranteed by canonical quantization, together with an argument showing that renormalization does not destroy it. You mean with unitarity a 'unitary representation' of a symmetry group. Unitary representations are of course necessary to have a consistent quantum mechanics, but this is an issue completely separate from properties of the S-matrix. So Rovelli's statement '' there is no sense in which conventional unitarity is necessary in the theory'' and your statements, amounting to ''unitary representations are needed to have a sound probability interpretation'' are not contradictory. Arnold Neumaier



thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0410140745.3184eb8c@posting.google.com>.. .... > However, there is one well-known case where we know how to combine > locality and diff invariance with quantum theory: conformal field > theory. CFT is usually thought of as a theory of conformally invariant > QFT in 2D, but since the local conformal group is the same as (twice) > the diffeomorphism group in 1D, it is also about diff invariant QFT in > 1D. Locality means that the correlation functions depend on separation. > For two points z and w in R or C, the correlator is > > $< \phi(z) \phi(w) > ~ 1/(z-w)^{2h} +$ more, > > where "more" stands for less singular terms when $z -> w$. That the > correlation function has this form is a diff-invariant statement. The > "more" terms will change under an arbitary diffeomorphism, but the > leading singularity will always have the same form, and in particular > the anomalous dimension h is well defined. > > We can phrase this slightly differently. The short-distance singularity > only depends on two points being infinitesimally close. This is good, > because we cannot determine the finite distance between two points > without knowing about the metric. GR does not have a background metric > structure, but it does have a background differentiable structure > (locally at least), and that is enough for defining anomalous > dimensions. Diffemorphisms move points around, but they don't separate > two points which are infinitesimally close. > > There is a well-known theorem which states that all correlation > functions in general relativity are trivial, which is easy to prove by > moving points. How can CFT by compatible with this theorem? Well, it > isn't. The relevant algebra in CFT is not really the 1D diffeomorphism > algebra (or the 2D conformal algebra), but its central extension known > as the Virasoro algebra: > > $[L_m, L_n] = (n-m) L_m+n - c/12 (m^3 - m) \delta_m+n$. > > A lowest-energy representation is characterized by a vacuum satisfying > > $L_0$ |vac> = h |vac>, $L_m$ |vac> = for all m < . > > In particular, the lowest $L_0$ eigenvalue can be identified with the > anomalous dimension h in the correlation function above. So the lowest energy value is well defined as a diff invariant by the leading term of the Laurent expansion above. This is pretty cool. I wonder whether the lowest energy value may correspond to a correlator's singularity diff invariant in higher dimensions too. Or maybe requiring invariance imposes a constraint on "admissible" diffeomorphisms in higher dimensions? > .... > There are various ways to handle quantization of gauge systems. One is > to eliminate the gauge degrees of freedom first and then quantize. This > is cumbersome and people usually prefer to quantize first and then > eliminate the gauge symmetries. I doubt that the two approaches are physically equivalent. > The simplest way, although not the best, > is to require that the gauge generators annihilate physical states, > > $J^a$ |phys> $= 0,$ > > and also that two physical states are equivalent if the differ by some > gauge state, $J^a |>$. This procedure gives you a nice Hilbert space of > physical states. > > However, one thing may go terribly wrong. Upon quantization, a > symmetry may acquire some quantum corrections, so that g is replaced by > > $[J^a, J^b] = f^{ab_c} J^c + \hbar D^{ab} + O(\hbar^2)$. > > The operator $D^{ab}$ is called an *anomaly*. We can also have anomalies of > the type > > $[J^a, H] = C^{a_b} J^b + \hbar E^a + O(\hbar^2)$. > > If we now try to keep the definition of a physical state, we see that > we must also require that > > $D^{ab}$ |phys> = . > > This implies further reduction of the Hilbert space. In the case that > $D^{ab}$ is invertible, it means that there are no physical states at all, > so the Hilbert space is empty. This is obviously not good. I wonder what's the physical meaning of $D^{ab}'s$ invertibility or lack thereof. Is it a thermodynamic condition on quantisation (i.e. on measurement)? > However, it > does not necessarily mean that the anomaly by itself is inconsistent, > only that our definition of physical states is. In the presence of an > anomaly, additional states become physical. So our Hilbert space > becomes bigger, containing some, or even all, of the previous gauge > degrees of freedom. > .... > > With diffemorphism symmetry, the extra degrees of freedom are the > components of the observer's trajectory. To learn something about a > quantum system, we must interact with it. So we inject a test particle, > or observer, into the system. In classical physics this does not affect > the system. E.g., a test particle in GR will move along geodesics but > it will not change the solution to Einstein's equation. In quantum > physics, the test particle perturbs the system. I find it quite > remarkable that the mathematics knows that we need an observer, because > the anomaly is a functional of the observer's trajectory. This sounds like recovering physical states as degree's of freedom of observers' trajectories, i.e. as measurement outcomes. I like it. Anyways, great post for this reader. IV



Arnold Neumaier wrote in message news:<416EBC5F.3080903@univie.ac.at>... > You mean with unitarity a 'unitary representation' of a symmetry group. > Unitary representations are of course necessary to have a consistent > quantum mechanics, but this is an issue completely separate from > properties of the S-matrix. > > So Rovelli's statement '' there is no sense in which conventional > unitarity is necessary in the theory'' and your statements, amounting to > ''unitary representations are needed to have a sound probability > interpretation'' are not contradictory. Hm, I don't understand the distinction. I'm mainly thinking of groups of spacetime symmetries which contain the Hamiltonian as a generator, especially the Poincare and diffeomorphism groups. A unitary representation of such a group means that time evolution is unitary, no? Isn't this more or less the same thing as preserving probabilities?



Thomas Larsson wrote: > Arnold Neumaier wrote in message news:<416EBC5F.3080903@univie.ac.at>... > > >>You mean with unitarity a 'unitary representation' of a symmetry group. >>Unitary representations are of course necessary to have a consistent >>quantum mechanics, but this is an issue completely separate from >>properties of the S-matrix. >> >>So Rovelli's statement '' there is no sense in which conventional >>unitarity is necessary in the theory'' and your statements, amounting to >>''unitary representations are needed to have a sound probability >>interpretation'' are not contradictory. > > Hm, I don't understand the distinction. I'm mainly thinking of groups > of spacetime symmetries which contain the Hamiltonian as a generator, > especially the Poincare and diffeomorphism groups. A unitary representation > of such a group means that time evolution is unitary, no? Isn't this > more or less the same thing as preserving probabilities? Yes, given a unitary representation of the time translations, time evolution is unitary. This does not yet guarantee a unitary S-matrix, which is a matter of asymptotic completeness of scattering states. In nonrelativistic QM, it is known that there are Hamiltonians for which asymptotic completeness is violated, although their dynamics is, of course, unitary. Arnold Neumaier



thomas_larsson_01@hotmail.com (Thomas Larsson) wrote in message news:<24a23f36.0410140745.3184eb8c@p...google.com>... > rthompson10@new.rr.com (DickT) wrote in message news:<93eecb7c.0410121507.5b7e39f4@p...google.com>... > > Carlo Rovelli, in his new book Quantum Gravity says this about > > unitarity and general relativity (and by extension, quantum gravity): > > > > "In conventional QM and QFT unitarity is the consequence of the time > > translation symmetry of the dynamics. In GR there isn't, in general, > > an analogous notion of time translation symmetry. Therefore there is > > no sense in which conventional unitarity is necessary in the theory. > > One often hears that without unitarity a theory is inconsistant. This > > is a misunderstanding that follows from the erroneous assumption that > > all physical theories are symmetric under time translations." > > (Section 10.1.3, Draft $of 12/30/2003)$ > > > > Could you comment on this? Is it relevant to your insights? > > Nevertheless, I believe that Rovelli is wrong about unitarity. > Physically it means that probability is preserved - all probabilities > must sum up to 1, and there are no negative probabilities. This is such > a fundamental property that I don't see how it can ever be wrong. In > fact, I don't understand what not preserving probabilities would mean. > I think the problem might be in a naive interpretation of probabilities. The observables and associated probabilities attain their physical interpretation by serving as building blocks for $H_{int}$ between the physical system and the observer system in a setting where we can modell the time evolution through $i d/dt \psi = H \psi$. Or alternatively $C =$ in the language of constraints with the constraint simply given by: $C = p_t + H$. (stealing this from urs Schreibers Time in Qm http://www-stud.uni-essen.de/~sb0264/TimeInQM.html) If, however the constraint does not decompose like this, and as I understand it Rovellis argument is that in a diffeomorpohism invariant context it wont, we'd need to redo the analysis of the interaction with an observer system. The usual argument of requiring a hermitian H generating a unitary "time flow" in order to have conserved probabilities breaks down because the concept of a time flow becomes more subtle and the relation between H, the observables, and probabilities changes accordingly. The later point was argued by Kuchar at the recent Isham60 conference, and he mentioned a forthcoming paper that uses decoherent histories to redo this analysis. A reference he gave me, which supposedly is looking at similar things for the string is this: http://arxiv.org/abs/http://www.arxi.../gr-qc/0108022 I'm not certain whether decoherent histories is general enough to look at this question, particularly if the time evolution is deeply encoded in the quantum state. We'll see what it shows. --- frank