[time 929] Re: [time 928] Re: [time 923] Unitarity

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 10 Oct 1999 08:30:05 +0300 (EET DST)

Dear Hitoshi,

Slight correction to the spontaneous digression.

On Sun, 10 Oct 1999, Matti Pitkanen wrote:
> On Sun, 10 Oct 1999, Hitoshi Kitada wrote:

> [Spontaneuous digression:
> It just occurred to me that S-matrix should be formulated
> in terms of the configuration counterpart of Green function G(r,r')
> at least in zero modes.
> a) In ordinary QM G(r,r',E=0) satifies
> H_0G(r,r',0)= -2*pidelta (r,r').
> b) The counterpart of matrix T in "position
> representation" seems to be promising object to study.
> T= X/(1+X) = (1/L_0+iepsilon)V /(1+X).
> T(x,y), that is T in "position representation" would be what I have
> earlier considered "scattering kernel".

Here the previous version contained error. Let us try again!

 c) By acting to this by PL_0 one obtains
 PL_0T|m_0> =0 for any state |m_0>
since the action of PL_0T on any state |m_0> gives

P V/(1+X)|m_0> = PV|m>

by using the definition of T.

This vanishes because PL_0(tot)|m>=0
and gives PL_0(free)|m>+PV|m> =0 which in turn gives PV|m>=0.
VP|m_1>=0 IS NOT used as I claimed first.
Only Virasoro condition for L_0(tot).

> d) PL_0 is analogous to nabla^2 in ordinary scattering theory
> and has purely geometric interpretation as a square
> of configuration space Dirac operator.
> e) In standard acattering theory one would have something like
> nabla^2 G(r,r',E)= -4*pi delta (r,r').
> for the Green function.
> f) The **absence of delta function on the right hand side of PL_0T=0**
> would reflect the fact that *all rows of T* have vanishing p-adic norm
> (is like unitary matrix but zero norm rows).
> PL_0T=0 for two-point function would give my p-adic cohomology
> geometrical meaning. In real context PL_0T=0 would presumably have no
> solutions (just like nabla^2 phi=0 has no bounded solutions in E^3.

g) Note that nonexistence of bounded solutions to nabla^2 Phi=0
in real context can be shown using inner product

<Phi_1,Phi_2> = Int nabla Phi_1 *nabla Phi_1 dV

The norm for square integrable solutions of nabla^2 Phi=0 would obviously
vanish since boundary terms would cancel in partial integration. The
vanishing of norm in real context shows that there are no
bounded square integrable solutions of nabla^2 Phi=0. In p-adic context
situation is different. Similar proof does not work since
zero norm of state does not imply its vanishing.

> h) It should be also noticed that the definition of p-adic delta
> function migth well be impossible since standard integration does
> not generalize to p-adic context. Therefore PL_0T=0 instead
> of equation containing delta function on right hand side is
> p-adically natural.


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