[time 903] Re: [time 901] Re: [time 899] Virasoro conditions,renormalizationgroupinvariance,unitarity,cohomology


Hitoshi Kitada (hitoshi@kitada.com)
Fri, 8 Oct 1999 02:17:15 +0900


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.

>
>
> 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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