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

*Sun, 26 Sep 1999 21:23:52 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"**Previous message:**Hitoshi Kitada: "[time 815] A summary on [time 814] Still about construction of U"**In reply to:**Matti Pitkanen: "[time 814] Re: [time 812] Re: [time 811] Re: [time 810] Re: [time 809] Stillabout construction of U"**Next in thread:**Hitoshi Kitada: "[time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"

Thank you for good posting. Your are right in that Hilbert space

is extended. One however obtains S-matrix for which other half

of unitary condition with summation over intermediate states of

extended Hilbert space is satisfied and this makes

S-matrix physical. Other half of unitarity conditions

involving sum over the intermediate states in smaller Hilbert space is

lost.

See below.

On Mon, 27 Sep 1999, Hitoshi Kitada wrote:

*> Dear Matti,
*

*>
*

*> Thank you for your answers. Rather sufficient data about your theory are
*

*> collected. I understand your theory as follows. I omit the detailed structure of
*

*> TGD, and summarize an abstract structure to see the essence:
*

*>
*

*> You define the following operators whose domains are included in a Hilbert space
*

*> \HH:
*

*>
*

*> H = L_0(tot),
*

*>
*

*> H_0 = L_0(free) = \sum_n L_0(n),
*

*>
*

*> L_0(n) = p^2(n) - L_0(vib,n),
*

*>
*

*> V = H - H_0 = L_0(int).
*

*>
*

*> You expect that these operators can be extended to selfadjoint operators defined
*

*> in \HH.
*

*>
*

*> (Note: The difference between symmetric (or Hermitian) operators and selfadjoint
*

*> operators is if their adjoint operators are the same as the original operators.
*

*> If the adjoint operator H^* is the same as H, H is called selfadjoint. There are
*

*> examples that are not selfadjoint, but symmetric. E.g., let T be a Laplacian
*

*> with domain C consisting of infinitely differentiable functions on R^n with
*

*> compact supports, and consider T in the usual L^2(R^n) space with inner product:
*

*> (f,g) = \int_{R^n} f(x) g(x)^* dx (g(x)^* is complex conjugate).
*

*>
*

*> Then clearly T is symmetric: (Tf,g) = (f,Tg) for any f, g in C. But the adjoint
*

*> T^* is also defined for distributions h satisfying the condition:
*

*>
*

*> ||h|| = (\int_{R^n} \sum_{0=<|k|=<2} |D^k h(x)|^2 dx)^{1/2} < \infty.
*

*>
*

*> Here D = (d/dx_1, ..., d/dx_n), k = (k_1, ..., k_n) with k_j being nonnegative
*

*> integer, D^k =(d/dx_1)^{k_1}...(d/dx_n)^{k_n}. The space of h above is called
*

*> Sobolev space of order 2, and is denoted H^2(R^n). This space is larger than C.
*

*> Thus T^* is a true extension of T, and T is symmetric but not selfadjoint.
*

*> However T^* satisfies (T^*)^* = T^*, thus is selfadjoint. Usually one considers
*

*> the maximal extension of T, and it is usually a unique closed extension of T. In
*

*> this case T is called essentially selfadjoint. Once one knows T can be extended
*

*> to a selfadjoint operator, one usually uses the same notation T to denote its
*

*> extension.)
*

*>
*

*> Returning to your operators, I proceed without assuming super Virasoro condition
*

*> for the time being. The reason will be clear in the following.
*

*>
*

*> Let z=E+ie, e>0, E: real. Since H and H_0 are selfadjoint, their spectra are
*

*> confined in the real line. Thus the following definition makes sense:
*

*>
*

*> R(z)=(H-z)^{-1}, R_0(z)=(H_0-z)^{-1}.
*

*>
*

*> These are called resolvents, and are continuous (i.e. bounded) operators with
*

*> domain \HH.
*

*>
*

*> (Note that for symmetric operators, the domains of their resolvents are not
*

*> necessarily equal to \HH.)
*

*>
*

*> Resolvents satisfy the resolvent equation:
*

*>
*

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

*>
*

*> for any non-real number z. Set for f in \HH
*

*>
*

*> \Psi = R(z)f, \Psi_0 = R_0(z)f.
*

*>
*

*> Then resolvent equation gives
*

*>
*

*> \Psi = \Psi_0 - R_0(z)V\Psi (1)
*

*>
*

*> and
*

*>
*

*> \Psi = \Psi_0 - R(z)V\Psi_0. (2)
*

*>
*

*> The former equation is your equation (1) in [time 798] and is equivalent to (2).
*

*> Namely (2) gives a solution of (1).
*

*>
*

*> If we assume super Virasoro condition on \Psi and \Psi_0:
*

*>
*

*> H\Psi = 0, H_0\Psi_0 = 0,
*

*>
*

*> we have from
*

*>
*

*> (H-z)\Psi = f = (H_0-z)\Psi_0,
*

*>
*

*> that
*

*>
*

*> -z\Psi = -z\Psi_0.
*

*>
*

*> Thus
*

*>
*

*> \Psi = \Psi_0
*

*>
*

*> if z not = 0. Thus the scattering operator U in your notation satisfies
*

*>
*

*> \Psi = U\Psi_0 = \Psi_0
*

*>
*

*> and
*

*>
*

*> U = I.
*

*>
*

*> This is not your expectation. Why this happened? There are two possible reasons:
*

*>
*

*> 1) The first is that we have assumed that both of \Psi and \Psi_0 are in the
*

*> Hilbert space \HH. If we assume \Psi_0 is in \HH, then \Psi must be outside \HH.
*

This is certainly the case since Psi contains superposition of

off mass shell states. p^2-L_0(vib)=0 is not satisfied for Psi.

If this were not the case, the entire equation would be nonsensical

since right hand side would be of form (L_0(int)/+ie)Psi.

Thus we have Hilbert spaces which we could call Hilb_0 and Hilb.

One the other hand. Psi is image of on mass shell state under Psi_0-->Psi

and S-matrix is defined as matrix elements

SmM== <Psi_0(m),Psi (M)>.

One restricts outgoing momenta to on mass shell momenta in inner product.

This means projection of Psi (m) to the space Hilb_0 spanned by Psi_0:s

when one calculates inner products defining S-matrix.

One obtains unitarity relations

sum_N SmN (SnN)^* = delta (m,n)

from completeness in Hilb: sum_N |N> <N|=1

but NOT

sum_m smM (SmN)*.

since Hilb_0 completeness relation sum_m |m><m|=1 are not true in Hilb

but become sum_m |m><m>= P, P projector to Hilb_0.

But this seems to be enough! One obtains S-matrix with orthogonal

rows: this gives probability conservation plus additional conditions.

Colums are however not orthogonal.

*>
*

*> 2) The second is that we assumed z not = 0. If z = 0, then both of \Psi and
*

*> \Psi_0 might be expected to be in \HH.
*

*>
*

*> The first case contains the problem to which space \Psi belongs. The expected
*

*> space is a Hilbert space larger than \HH, whose norm is smaller than that of
*

*> \HH.
*

*>
*

*> The second case must be considered as a result of taking the limit e->+0, so it
*

*> would contain some difficult topological problems; contrary to the expectation
*

*> above, \Psi would also be outside \HH.
*

*>
*

*> 1) corresponds to the case E not = 0, and 2) to E = 0.
*

*>
*

*> Now a brief comment based on ordinary quantum scattering theory:
*

*>
*

*> 1) If we consider general E, we would also have to consider \Psi_0 moving in a
*

*> Hilbert space larger than \HH. (This is suggested by (4) below.)
*

*>
*

*> 2) is anticipated to be more difficult than 1). The study of this case would
*

*> tell that your universe is in a resonance state.
*

*>
*

*>
*

*> I conclude with a comment on a relation between (1) and the equation in [time
*

*> 804]:
*

*>
*

*> S = (W_+)^* (W_-) = lim_{t->\infty} exp(itH_0) exp(-2itH) exp(itH_0)
*

*>
*

*> = I+2i\pi \int_{-\infty}^\infty E'_0(\lam)V R(\lam+i0)V E'_0(\lam)d\lam.
*

*>
*

*> -2i\pi \int_{-\infty}^\infty E'_0(\lam)VE'_0(\lam)d\lam
*

*>
*

*> An iteration of resolvent equation gives
*

*>
*

*> R(z)V = - \sum_{n=1}^\infty (-R_0(z)V)^n.
*

*>
*

*> This and (2) give a solution of (1):
*

*>
*

*> U(z) = I + \sum_{n=1}^\infty (-R_0(z)V)^n. (3)
*

*>
*

*> (Here the variable z is attached as we consider general case without super
*

*> Virasoro condition.)
*

*>
*

*> Similarly, S becomes
*

*>
*

*> S
*

*>
*

*> = I + 2i\pi \int_{-\infty}^\infty E'_0(\lam)V (R(\lam+i0)V-I) E'_0(\lam)d\lam
*

*>
*

*> = I + 2i\pi \int_{-\infty}^\infty E'_0(\lam) VU(\lam+i0) E'_0(\lam)d\lam. (4)
*

*>
*

*> This gives a relation of your scattering operator U(z) with time dependent
*

*> method.
*

*>
*

*> In the treatment of the equation (1), it is helpful to consider general context
*

*> allowing z to take general values. Let me explain.
*

*>
*

*> The factor V in front of U(\lam+i0) in (4) is interaction term L_0(int). This
*

*> seems to decay as a -> \infty, as you stated in [time 808]:
*

*>
*

*> [MP] [time 808]
*

*> > Diff^4 invariant momentum generators are defined in the following manner.
*

*> > Consider Y^3 belonging to delta M^4_+xCP_2 ("lightcone boundary").
*

*> > There is unique spacetime surface X^4(Y^3) defined as absolute minimum
*

*> > of Kaehler action.
*

*> >
*

*> > Take 3-surface X^3(a) defined by the intersection of lightcone
*

*> > proper time a =constant hyperboloidxCP_2 with X^4(Y^3). Translate it
*

*> > infinitesimal amount to X^3(a,new)and find the new absolute minimum
*

*> > spacetime surface goinb through X^3(a,new). It intersectors
*

*> > lightcone at Y^3(new). Y^3(new) is infinitesimal translate
*

*> > of Y^3: it is not simple translate but slightly deformed surface.
*

*> >
*

*> > In this manner one obtains what I called Diff^4 invariant infinitesimal
*

*> > representation of Poincare algebra when one considers also infinitesimal
*

*> > Lorentz transformations. These infinitesimal transformations need
*

*> > *not* form closed Lie-algebra for finite value a of lightcone proper time
*

*> > but at the limit a--> the breaking of Poincare invariance is expected
*

*> > to go to zero and one obtains Poincare algebra since the distance to
*

*> > the lightcone boundary causing breaking of global Poincare invariance
*

*> > becomes infinite. The Diff^4 invariant Poincare algebra p_k(a--> infty)
*

*> > defines momentum generators appearing in Virasoro algebra.
*

*>
*

*> If this is the case, V would work to damp the behavior of U(\lam+i0)\Psi_0,
*

*> which would result in that
*

*>
*

*> VU(\lam+i0)\Psi_0 belongs to a good Hilbert space, possibly to a space smaller
*

*> than \HH (as experienced in ordinary quantum scattering).
*

*>
*

*> This would make the treatment of scattering operator and S-matrix possible:
*

*> \Psi= U\Psi_0 with super Virasoro condition would be understood by considering
*

*> the neighborhood of the desired eigenvalue 0 or E.
*

Best,

MP

**Next message:**Hitoshi Kitada: "[time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"**Previous message:**Hitoshi Kitada: "[time 815] A summary on [time 814] Still about construction of U"**In reply to:**Matti Pitkanen: "[time 814] Re: [time 812] Re: [time 811] Re: [time 810] Re: [time 809] Stillabout construction of U"**Next in thread:**Hitoshi Kitada: "[time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"

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