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:
>
snip
> [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.
>
>
Best,
MP
This archive was generated by hypermail 2.0b3 on Sun Oct 17 1999 - 22:40:47 JST