**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Sun, 10 Oct 1999 08:30:05 +0300 (EET DST)*

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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
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*> representation" seems to be promising object to study.
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*>
*

*> T= X/(1+X) = (1/L_0+iepsilon)V /(1+X).
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*>
*

*> T(x,y), that is T in "position representation" would be what I have
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*> earlier considered "scattering kernel".
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*>
*

*>
*

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
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*> and has purely geometric interpretation as a square
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*> of configuration space Dirac operator.
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*>
*

*> 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**
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*> would reflect the fact that *all rows of T* have vanishing p-adic norm
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*> (is like unitary matrix but zero norm rows).
*

*> PL_0T=0 for two-point function would give my p-adic cohomology
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*> geometrical meaning. In real context PL_0T=0 would presumably have no
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*> 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
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*> not generalize to p-adic context. Therefore PL_0T=0 instead
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*> of equation containing delta function on right hand side is
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*> p-adically natural.
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*>
*

*>
*

Best,

MP

**Next message:**Hitoshi Kitada: "[time 930] Re: [time 928] Re: [time 923] Unitarity"**Previous message:**Matti Pitkanen: "[time 928] Re: [time 923] Unitarity"**In reply to:**Hitoshi Kitada: "[time 927] Re: [time 923] Unitarity"**Next in thread:**Hitoshi Kitada: "[time 930] Re: [time 928] Re: [time 923] Unitarity"

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