View Full Version : Q: Harmonic oscillator and Virasoro algebra
<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>Is it possible to have a representation of the Virasoro algebra of a\ncertain central charge as the Hilbert space of a harmonic oscillator, such\nthat L_0 = a^\\dagger a + \\half, L_1 = a^2, L_{-1} = (a^\\dagger)^2 (with\nproper normalization) ?\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>Is it possible to have a representation of the Virasoro algebra of a
certain central charge as the Hilbert space of a harmonic oscillator, such
that L_0 = a^\dagger a + \half, L_1 = a^2, L_{-1} = (a^\dagger)^2 (with
proper normalization) ?
Lubos Motl
Nov10-04, 03:51 PM
<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>On Wed, 10 Nov 2004, Xi Yin wrote:\n\n> Is it possible to have a representation of the Virasoro algebra of a\n> certain central charge as the Hilbert space of a harmonic oscillator, such\n> that L_0 = a^\\dagger a + \\half, L_1 = a^2, L_{-1} = (a^\\dagger)^2 (with\n> proper normalization) ?\n\nHi Xi!\n\nThat\'s a very interesting question. Of course, the commutators between\nL0,L1,L-1 work out fine, and there is no obvious way to rule your CFT out.\nNevertheless it looks rather strange. ;-)\n\nAll operators on the harmonic oscillator Hilbert space can be written as\nfunctions of "a" and "a^\\dagger". It is easy to see that L_n must have\nterms with "n+m" a\'s and "m" a^\\dagger\'s, so that the difference is n.\n\nMoreover, your conjectured Virasoro representation simply has one\neigenstate for each positive half-integer eigenvalue of L_0 - well, L_0 is\nthe usual Hamiltonian of the harmonic oscillator.\n\nIt does not look like a usual CFT if you have one operator at every level.\nUsing some heuristic intuition, I think that your potential representation\nis a unitary representation - L_1 is the hermitean conjugate of L_{-1},\nand it will probably generalize to L_n - and it has a positive definite\nscalar product.\n\nBut from the density of states, that does not grow at all with L_0, it\nlooks like your central charge is zero (or "infinitesimal"). But an empty\nCFT is the only unitary CFT with c=0, so I would expect that no solution\nto your problem exists.\n\nIt\'s just an expectation at this state. However, it looks pretty clear to\nme that assuming the Virasoro algebra - commutators with L_0 - you can\nwrite L_n as\n\nf_n(L_0) times a^n\n\nup to some permutations of the factors simply because all the pairs of "a"\nand "a^\\dagger" can be combined into a function of L_0. But then I guess\nthat even the Poisson brackets - which is the object to look at that does\nnot care about the permutations of the factors (ordering) - will just work\nout incorrectly. The commutator of a^n and adagger^m seems to be too\ncomplicated. ;-)\n\nYou can try to solve your problem recursively. Try to find L_2 so that the\ncommutator of L_2 with L_{-1} gives you three times L_1, as desired.\n\nWell, that already looks pretty bad. L_{-1} contains two "adaggers", and\ntherefore its commutator with L_2 (which you can always write in the\nnormal ordered form) will contain some terms with at least one "adagger".\nBut L_{1}, which you wanted to get, has no "adagger", which already looks\nbad.\n\nDo you agree that this rules out your hypothetical Virasoro rep? ;-)\n\nAll the best\nLubos\n_____________________________________ _________________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\nWebs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\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>On Wed, 10 Nov 2004, \Xi Yin wrote:
> Is it possible to have a representation of the Virasoro algebra of a
> certain central charge as the Hilbert space of a harmonic oscillator, such
> that L_0 = a^\dagger a + \half, L_1 = a^2, L_{-1} = (a^\dagger)^2 (with
> proper normalization) ?
Hi \Xi!
That's a very interesting question. Of course, the commutators between
L0,L1,L-1 work out fine, and there is no obvious way to rule your CFT out.
Nevertheless it looks rather strange. ;-)
All operators on the harmonic oscillator Hilbert space can be written as
functions of "a" and "a^\dagger". It is easy to see that L_n must have
terms with "n+m" a's and "m" a^\dagger's, so that the difference is n.
Moreover, your conjectured Virasoro representation simply has one
eigenstate for each positive half-integer eigenvalue of L_0 - well, L_0 is
the usual Hamiltonian of the harmonic oscillator.
It does not look like a usual CFT if you have one operator at every level.
Using some heuristic intuition, I think that your potential representation
is a unitary representation - L_1 is the hermitean conjugate of L_{-1},
and it will probably generalize to L_n - and it has a positive definite
scalar product.
But from the density of states, that does not grow at all with L_0, it
looks like your central charge is zero (or "infinitesimal"). But an empty
CFT is the only unitary CFT with c=0, so I would expect that no solution
to your problem exists.
It's just an expectation at this state. However, it looks pretty clear to
me that assuming the Virasoro algebra - commutators with L_0 - you can
write L_n as
f_n(L_0)[/itex] times [itex]a^n
up to some permutations of the factors simply because all the pairs of "a"
and "a^\dagger" can be combined into a function of L_0. But then I guess
that even the Poisson brackets - which is the object to look at that does
not care about the permutations of the factors (ordering) - will just work
out incorrectly. The commutator of a^n and adagger^m seems to be too
complicated. ;-)
You can try to solve your problem recursively. Try to find L_2 so that the
commutator of L_2 with L_{-1} gives you three times L_1, as desired.
Well, that already looks pretty bad. L_{-1} contains two "adaggers", and
therefore its commutator with L_2 (which you can always write in the
normal ordered form) will contain some terms with at least one "adagger".
But L_{1}, which you wanted to get, has no "adagger", which already looks
bad.
Do you agree that this rules out your hypothetical Virasoro rep? ;-)
All the best
Lubos
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
<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>Okay, I think the construction I had in mind probably turns out to be\nnon-unitary. The harmonic oscillator has W_\\infty symmetry, which has\nVirasoro subalgebras, but I guess none of the them are represented\nunitarily?\n\nOn a related issue, I\'m now very puzzled by how one can find CFT1 dual to\nstring theory on AdS2. It\'s very unlikely for the ordinary conformal\nquantum mechanics involving finite number of fields to have exponential\ngrowth of states. So either we somehow loose the full Virasoro symmetry,\nor the conformal QM has to involve infinite number of fields, like the\ntype 0 matrix model. I\'m very confused.\n\n-Xi\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>Okay, I think the construction I had in mind probably turns out to be
non-unitary. The harmonic oscillator has W_\infty symmetry, which has
Virasoro subalgebras, but I guess none of the them are represented
unitarily?
On a related issue, I'm now very puzzled by how one can find CFT1 dual to
string theory on AdS2. It's very unlikely for the ordinary conformal
quantum mechanics involving finite number of fields to have exponential
growth of states. So either we somehow loose the full Virasoro symmetry,
or the conformal QM has to involve infinite number of fields, like the
type matrix model. I'm very confused.
-\Xi
Matti Pitkanen
Nov10-04, 10:15 PM
<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.4.31.0411101639510.9771-100000@feynman.harvard.edu>...\n\n> On Wed, 10 Nov 2004, Xi Yin wrote:\n>\n> > Is it possible to have a representation of the Virasoro algebra of a\n> > certain central charge as the Hilbert space of a harmonic oscillator, such\n> > that L_0 = a^\\dagger a + \\half, L_1 = a^2, L_{-1} = (a^\\dagger)^2 (with\n> > proper normalization) ?\n>\n> Hi Xi!\n>\n> That\'s a very interesting question. Of course, the commutators between\n> L0,L1,L-1 work out fine, and there is no obvious way to rule your CFT out.\n> Nevertheless it looks rather strange. ;-)\n\n\nThese operators define a unitary representation of SL(2,R). The\nproblem is however that you have only one creation operator a^\\dagger\nwith conformal weight 1 rather than infinite number of them with all\npossible conformal weights. Hence you cannot build L_n, n>=1 as\nbilinears in any manner.\n\nMatti Pitkanen\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.4.31.0411101639510.9771-100000@feynman.harvard.edu>...
> On Wed, 10 Nov 2004, \Xi Yin wrote:
>
> > Is it possible to have a representation of the Virasoro algebra of a
> > certain central charge as the Hilbert space of a harmonic oscillator, such
> > that L_0 = a^\dagger a + \half, L_1 = a^2, L_{-1} = (a^\dagger)^2 (with
> > proper normalization) ?
>
> Hi \Xi!
>
> That's a very interesting question. Of course, the commutators between
> L0,L1,L-1 work out fine, and there is no obvious way to rule your CFT out.
> Nevertheless it looks rather strange. ;-)
These operators define a unitary representation of SL(2,R). The
problem is however that you have only one creation operator a^\dagger
with conformal weight 1 rather than infinite number of them with all
possible conformal weights. Hence you cannot build L_n, n>=1 as
bilinears in any manner.
Matti Pitkanen
Lubos Motl
Nov14-04, 03:31 PM
<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>Is it OK to say that you have not yet found a harmonic oscillator\nrepresentation of the Virasoro algebra?\n\nThe Virasoro algebra generates diffeomorphisms of a circle, and therefore,\nnaively and classically, you may also try to extend these diffeomorphisms\nof a circle into diffeomorphisms of the phase space, and take\n(essentially; classically; ordering issues neglected)\n\nL_m = (a/adagger)^m adagger.a.\n\nBut the question is whether this can be defined. Note that classically\nit\'t not well-defined at adagger=0 or a=0. Quantum mechanically, this is a\nwhole new task because both "a" as well as "adagger" are singular, and new\nordering subtleties must be taken into account. Both "a" and "adagger" can\nbe given many left inverses or right inverses (which are not unique), but\nnot the other inverses.\n\nIf you want to hope that meaningful operators like that may be defined,\nyou would like to define\n\n1/a := adagger (adagger.a + c)^{-1}\n\nwhere you must hope that the calculation will tell you that "c" is\nnonzero, so that the inverse is well-defined on the harmonic oscillator\'s\nHilbert space.\n\nNo one has given the construction yet, and so far I continue to believe\nthat the harmonic oscillator\'s Hilbert space with L_0 = adagger.a+1/2\ncannot be a representation of the Virasoro algebra.\n________________________________________ ______________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)\nWebs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\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>Is it OK to say that you have not yet found a harmonic oscillator
representation of the Virasoro algebra?
The Virasoro algebra generates diffeomorphisms of a circle, and therefore,
naively and classically, you may also try to extend these diffeomorphisms
of a circle into diffeomorphisms of the phase space, and take
(essentially; classically; ordering issues neglected)
L_m = (a/adagger)^m[/itex] adagger.a.
But the question is whether this can be defined. Note that classically
it't not well-defined at adagger=0 or a=0. Quantum mechanically, this is a
whole new task because both "a" as well as "adagger" are singular, and new
ordering subtleties must be taken into account. Both "a" and "adagger" can
be given many left inverses or right inverses (which are not unique), but
not the other inverses.
If you want to hope that meaningful operators like that may be defined,
you would like to define
1/a := adagger (adagger[itex].a + c)^{-1}
where you must hope that the calculation will tell you that "c" is
nonzero, so that the inverse is well-defined on the harmonic oscillator's
Hilbert space.
No one has given the construction yet, and so far I continue to believe
that the harmonic oscillator's Hilbert space with L_0 = adagger.a+1/2
cannot be a representation of the Virasoro algebra.
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/384-9488 home: +1-617/868-4487 (call)
Webs: http://schwinger.harvard.edu/~motl/ http://motls.blogspot.com/
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Matti Pitkanen
Nov15-04, 08:47 AM
<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.4.31.0411141624200.17591-100000@feynman.harvard.edu>...\n\n> Is it OK to say that you have not yet found a harmonic oscillator\n> representation of the Virasoro algebra?\n\nA defining representation for what might be called quantum Virasoro\nalgebra is in question. L_n =z^n zd/dz being replaced with a^dagger^n\nL_0. String world sheet parametrized by z becomes quantum world sheet\nif you want to put it in this manner.\n\nThe point in my proposal is to notice that Virasoro algebra allows\nalso a representation in which 1/z is replaced with zbar and zd/dz\nwith id/dphi: n becomes angular momentum rather than conformal weight.\nAt unit circle z-->1/z is equivalent with z-->zbar but not outside.\nClassically you get the same algebra as using the usual definition.\nYou cannot define what 1/z means quantally, but you can define what\nzbar means and negative powers of z are not needed near origin to\nTaylor expand.\n\nz--> a^dagger, zbar-->a, zd/dz --> Hamiltonian gives what might be\ncalled quantum Virasoro. Only the subalgebras V_+ (n>=0) and V_-\n(n<=0) close. [V_+,V_-] extends to a larger algebra since operators\nof form (a^dagger)^ma^n L_0 appear in commutators. Additional\noperators are of form L_n= (a^dagger)^nL_0,n>=0 in V_+. Restriction to\nV_+ is analogous to considering only holomorphic functions having\nTaylor series in z^n, n>=0.\n\nHarmonic oscillator ground state is annihilated by the quantum\ncounterparts of antiholomorphic functions expressible as power series\nof a^n<-->zbar^n, n<0 annihilate the ground state. The action of the\ncommutators in [V_+,V_-] on ground state gives a c-number term\nresulting from normal ordering: a good guess is that it corresponds to\nthe standard central extension for Virasoro algebra.\n\n\nMatti Pitkanen___\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.4.31.0411141624200.17591-100000@feynman.harvard.edu>...
> Is it OK to say that you have not yet found a harmonic oscillator
> representation of the Virasoro algebra?
A defining representation for what might be called quantum Virasoro
algebra is in question. L_n =z^n zd/dz being replaced with a^{dagger}^nL_0. String world sheet parametrized by z becomes quantum world sheet
if you want to put it in this manner.
The point in my proposal is to notice that Virasoro algebra allows
also a representation in which 1/z is replaced with zbar and zd/dz
with id/dphi: n becomes angular momentum rather than conformal weight.
At unit circle z-->1/z is equivalent with z-->zbar but not outside.
Classically you get the same algebra as using the usual definition.
You cannot define what 1/z means quantally, but you can define what
zbar means and negative powers of z are not needed near origin to
Taylor expand.
z--> a^{dagger}, zbar-->a, zd/dz --> Hamiltonian gives what might be
called quantum Virasoro. Only the subalgebras V_+ (n>=0) and V_-(n<=0) close. [V_+,V_-] extends to a larger algebra since operators
of form (a^{dagger})^ma^n L_0 appear in commutators. Additional
operators are of form L_n= (a^{dagger})^nL_0,n>=0 in V_+. Restriction to
V_+ is analogous to considering only holomorphic functions having
Taylor series in z^n, n>=0.
Harmonic oscillator ground state is annihilated by the quantum
counterparts of antiholomorphic functions expressible as power series
of a^n<-->zbar^n, n<0 annihilate the ground state. The action of the
commutators in [V_+,V_-] on ground state gives a c-number term
resulting from normal ordering: a good guess is that it corresponds to
the standard central extension for Virasoro algebra.
Matti Pitkanen___
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