# [time 904] Re: [time 901] Re: [time 899] Virasoro conditions,renormalizationgroupinvariance,unitarity,cohomology

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 7 Oct 1999 20:26:09 +0300 (EET DST)

On Fri, 8 Oct 1999, Hitoshi Kitada wrote:

> Dear Matti,
>
> Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
>
> Subject: [time 901] Re: [time 899] Virasoro
> conditions,renormalizationgroupinvariance,unitarity,cohomology
>
>
> >
> >
> > On Fri, 8 Oct 1999, Hitoshi Kitada wrote:
> >
> > > Dear Matti,
> > >
> > > Thanks for posting your paper. I read it but before going to physical
> > > justification part I again stumbled on mathematical part: the proof that
> (19)
> > > vanishes. As I reread your [time 894], I found it is interesting idea but
> does
> > > not seem to work. I calculated like a blind mathematician:
> > >
> >
> >
> >
> > > m_1=Ym_0, Y=\sum_{k>0}(-X)^k =X(1+Y),
> > >
> >
> > You srat from state |m_1> and manipulate it.
> >
> >
> > > (BTW note (1+r)^{-1}=\sum_{k=or>0} (-r)^k not r^k !)
> > >
> > > = Xm_0 + Xm_1 (1)
> > >
> > > = Xm_0 + XPm_1 + X(1-P)m_1
> >
> > OK
> > >
> > > XPm_1 = 0 yields
> > >
> > > m_1 = Xm_0 + X(1-P)m_1
> > >
> > You are manipulating the state m_1 in the following. OK.
> >
> >
> > > = Xm_0 + Zm_1, Z=X(1-P),
> > >
> > > = Xm_0 + ZXm_0 + ZXm_1 (by (1))
> > >
> > > = (1+Z)Xm_0 + ZXPm_1 + ZX(1-P)m_1
> > >
> > > =(1+Z)Xm_0 + ZX(1-P)m_1 (by XPm_1=0)
> > >
> > > = ....
> > >
> > > = (1+Z+Z^2+Z^3+...)Xm_0 +lim_{k->infty}Z^kXm_1
> > >
> > > =(1-Z)^{-1}Xm_0 + lim_{k->infty}Z^kXm_1.
> > >
> > > This does not seem to vanish in general.
> > >
>
> My point is that this formula uses your decomposition X=XP+X(1-P), but this
> does not seem to give
>
> <m_0|P|n_1>+<m_1|P|n_0>=0.

But as I see it, the last formula above is for |m_1>
rather than this quantity. Perhaps there is some mistyping
or confusion on my side.

Perhaps we should leave for a moment and look the consequences.
the construction of the p-adic cohomology.

Best,

MP

>
>
> >
> >
> > This is state |m_1> as far as I can understand and should not vanish.
> >
> >
> > > If this does vanish, your T is 0, not only that sum T+T^dagger=0: One
> would
> > > need to use cancellation.
> >
> > I must admit that I cannot follow what you mean. You demonstrate that
> >
> > |m_1> can be written as |m_1> = (1-Z)^(-1)X|m_0>
> > OK?
> >
> > Certainly, if T is zero if |m_1> vanishes. But why |m_1> should vanish?
> > If your argument shows that |m_1> vanishes then I am in trouble
> > but your argument seems to show that |m_1> does NOT vanish??
> > It seems that I do not understand your point!
> >
> > Best,
> > MP
> >
> >
>
>
> Best wishes,
> Hitoshi
>
>
>

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