**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Sun, 3 Oct 1999 16:06:37 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 867] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**Previous message:**Matti Pitkanen: "[time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**In reply to:**Matti Pitkanen: "[time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Next in thread:**Matti Pitkanen: "[time 868] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"

Dear Matti,

Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:

Subject: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time

847]Unitarity of S-matrix

*>
*

*>
*

*> On Sun, 3 Oct 1999, Hitoshi Kitada wrote:
*

*>
*

*> > Dear Matti,
*

*> >
*

*> > My question in the following is that:
*

*> >
*

*> > You stated the scattering space \HH_s is the same as the free space P\HH.
*

This

*> > means
*

*> >
*

*> > \HH_s = P\HH,
*

*> >
*

*> > hence
*

*> >
*

*> > any free state u=|n_0> in P\HH=\HH_s satisfies
*

*> >
*

*> > u = Pu.
*

*> >
*

*> > Thus
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*> >
*

*> > 1/(1+X) |n_0> = 1/(1+X) P|n_0> = |n_0>.
*

*> >
*

*> > The last equality here follows from
*

*> >
*

*> > 1/(1+X) = 1-\sum_{n=1}^\infty (-X)^n
*

*> >
*

*> > and
*

*> >
*

*> > X P = 0
*

*> >
*

*> > by
*

*> >
*

*> > X = X= 1/(L_0 +i*epsilon)*L_0(int)
*

*> >
*

*> > and L_0(int) P|n_0> = 0.
*

*>
*

*>
*

*> Yes. Suppose that every state L_0(int)P|n> vanishes and states P|n> span
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*> the space spanned by |n_0>. Then one can express |n_0> as superposition
*

*> of P|n>:s and conclude that L_0(int)|n_0> vanishes and S is trivial.
*

*>
*

*> I am really amazed! How it is possible to obtain nontrivial S-matrix
*

*> at all in QM? Same kind proof for unitarity should hold also there!
*

*> The point is that I "know" that the expansion yields S-matrix. There
*

*> is now doubt about that. But how on Earth can I demonstrate the unitarity?
*

*> There is something which I do not understand but what is it?
*

*>
*

*> Could this paradox be related to the taking of epsilon-->0 limit?
*

*> Condition
*

*>
*

*> L_0(int)* P*(1/(1+X)) |m_0> =0
*

*>
*

*> holds true only at in the sense of limit epsilon-->0 .
*

Yes, your guess is correct. The limit as epsilon -> 0 corresponds to taking

the limit t -> \infty in time dependent expression. Originally scattering

theory started with stationary theory, just from the equation you proposed:

\Psi = \Psi_0 -R_0(z)V \Psi.

(Sommerfeld, etc.) Time dependent method is later introduced, and it is only

recently that the method is recognized powerful.

When taking the limit of e.g. R_0(z) = R_0(E+i\epsilon) = (H_0 - E -

i\epsilon)^{-1} as \epsilon -> 0, R_0(z) does not remain a bounded operator

from \HH into itself anymore (although this is the case when \epsilon > 0),

and lim_{\epsilon->0}R_0(z) must be considered a (bounded) operator from \HH_+

to \HH_-, where \HH_+ \subset \HH \subset \HH_-, a Gel'fand triple. This is

because H_0 (or H) has continuous spectra (we consider general case without

super Virasoro condition). There are three cases:

1) If E (real) is not a spectrum of H_0 then R_0(E) = (H_0-E)^{-1} exists as a

bounded operator from \HH into itself.

2) If E is a point spectrum (of finite degree) then

-(2i\pi)^{-1}\int R_(z) dz

(integration is on a circle rounding E counter clockwise)

gives the projection P(E) onto the eigenspace corresponding to eigenvalue E.

3) But when a closed interval [F,E] (F<E) is included in the continuous

spectrum of H_0, none of the above holds, and we have

E_0(E) - E_0(F) = -(2i\pi)^{-1} \int R(z) dz

= -(2i\pi)^{-1} lim_{\epsilon->0}\int R(z) dz (*)

(z=\lam+i\epsilon or \lam-i\epsilon)

(integration is around a path which passes point E and F counter clockwise).

Here E_0(E) is called "resolution of the identity" that expresses H_0 as

H_0 = \int_{-\infty}^\infty \lam dE_0(\lam).

E_0(E) is a kind of operator-valued measure.

In this way, in the continuos spectra, one cannot take the limit

lim_{\epsilon->0} R_0(E+i\epsilon) directly but the limit has meaning only in

the sense of mean as (*) above. If one wants to get a pointwise limit

lim_{\epsilon->0} R_0(E+i\epsilon), one has to take a smaller space \HH_+ as

its domain and a larger space \HH_- as its range. This is the cause that we

have to consider Gel'fand triple (\HH_+,\HH,\HH_-), \HH_+ \subset \HH \subset

\HH_-.

Resolvent equation

R(z) - R_0(z) = -R_0(z)VR(z) = -R(z)VR_0(z)

and

R_0(z): \HH_+ -> \HH_-

R(z): \HH_+ -> \HH_-

say

R_0(z)VR(z): \HH_+ -> \HH_-.

But the range of R(z) is included in \HH_- and the domain of R_0(z) is \HH_+

that is smaller than \HH_-. If the resolvent equation holds at the limit

\epsilon->0, V must be an operator

V: \HH_- -> \HH_+.

This means that V needs to "decay" in some sense, and if this is satisfied the

resolvent equation holds at the limit \epsilon -> 0.

Equations in the previous TeX file can be justified if V satisfies this type

of assumption. Here is a possibility that S-matrix S(E) is well-defined with

super Virasoro conditions: (H-E)\Psi=0 and (H_0-E)\Psi_0=0.

*>
*

*> Limit of this equation would not be same as equation
*

*> obtained putting epsilon=0 from the beginning to get L_0(int|m_0>=0?
*

*> This would not be surprising since epsilon prescription can be seen
*

*> as a manner to make propagators well defined. I do not know.
*

Your thought is on the right track!

*>
*

*>
*

*> Best,
*

*>
*

*> MP
*

Best wishes,

Hitoshi

**Next message:**Hitoshi Kitada: "[time 867] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**Previous message:**Matti Pitkanen: "[time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**In reply to:**Matti Pitkanen: "[time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Next in thread:**Matti Pitkanen: "[time 868] Re: [time 864] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"

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