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

*Sun, 3 Oct 1999 10:56:14 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**Previous message:**Hitoshi Kitada: "[time 867] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**In reply to:**Matti Pitkanen: "[time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Next in thread:**Hitoshi Kitada: "[time 870] Re: [time 868] Re: [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,
*

*>
*

*> 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
*

*> > >
*

*> > > 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
*

*> > 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 .
*

*>
*

[Hitoshi]

*> 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).
*

Let me think: in TGD framework iepsilon is added to the propagators

1/p^2-m^2 +iepsilon to define momentum space integrals. Masses

get tiny imagy component and planewaves get small exponential factor

making them divergence at t-->infity. Therefore one goes outside the state

space. This would mean in TGD context that one consider eigenstates

of Diff^4 invariant momentum generator with slightly imaginary eigen

values: they are not normalizable configuration space spinor fields.

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.
*

*>
*

OK

*> 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.
*

*>
*

I think I understand this.

*> 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
*

You

*>
*

*> 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.
*

I think I understand this. You have continuous spectrum

and projector E_0 is replaced by dE_0/dE and integration yields

Int dEdE_0/dE = E_(E)-E_0(F).

*>
*

*> 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_-.
*

*>
*

Can one explain verbally HH_+, HH and HH_-? Or actually 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.
*

*>
*

OK. I think that I understand this. R_0(z) is defined in

HH_+. Therefore V must map HH_- to HH_+ in order that everything is

well defined.

*> 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.
*

*>
*

*>
*

This means that one gets S-matrix without any reference to

unitary time evolution if this kind of condition holds true?

*> >
*

*> > 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!
*

*>
*

Glad to hear that. I should be familiar with these delicacies but I am

not. It took 21 years before I ended at the concrete level

of worrying details of this kind(;-)!

I remember had a course on Hilbert spaces and resolutions

of identity something like 25 years ago: it seems that wheel must

be reinvented again and again(;-).

Best,

MP

**Next message:**Matti Pitkanen: "[time 869] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**Previous message:**Hitoshi Kitada: "[time 867] Re: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847]Unitarity of S-matrix"**In reply to:**Matti Pitkanen: "[time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time 847] Unitarity of S-matrix"**Next in thread:**Hitoshi Kitada: "[time 870] Re: [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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