Matti Pitkanen (matpitka@pcu.helsinki.fi)
Sun, 3 Oct 1999 13:36:13 +0300 (EET DST)
On Sun, 3 Oct 1999, Hitoshi Kitada wrote:
> Dear Matti,
>
> Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
>
> Subject: [time 869] Re: [time 865] 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,
> > >
> > > Your observation in the following is correct.
> > >
> > > Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
> > >
> > > Subject: [time 865] Re: [time 861] Re: [time 860] Re: [time 855] Re: [time
> > > 847]Unitarity of S-matrix
> > >
> > >
> > > >
> > > >
> > > >
> > > > I noticed what might be the reason for the paradoxal conclusion
> > > > about the triviality of S-matrix.
> > > >
> > > >
> > > > The expression of S-matrix is
> > > >
> > > > <m_0|Sn> = <m_0| P*(1/(1+X)|n_0>
> > > >
> > > > Expand this to geometric series to get
> > > >
> > > > ...= delta (m,n) + sum_n <m_0| X^n|n_0>
> > > >
> > > > = delta (m,n) + (1/i*epsilon) sum_n <m_0| L_0(int) X^(n-1)|n_0>
> > > >
> > > > Here I have used X= (1/L_0(free)+iepsilon)L_0(int) to the first
> > > > X in the expansion in powers of X.
> > > >
> > > > The point is that formula contains 1/epsilon factor!!
> > > >
> > > > Thus the limit is extremely delicate. S-matrix is notrivial
> > > > if L_0(int)|m_0> is of order epsilon and goes to zero at
> > > > the limit epsilon->0.
> > > >
> > > >
> > > > This is dangerously delicate but I think that similar problems
> > > > must be encountered with ordinary time dependent scattering theory
> > > > when one restricts to 'energy shell' E=constant.
> > >
> > > Also in time dependent expression, taking the limit t -> \infty requires a
> > > delicate argument and as well dangerous (;-)
> > >
> > >
> > > > The task would
> > > > be to find proper formulation or possibly understand why p-adics
> > > > save the situation.
> > >
> > > Before going to p-adics, there is a possibility to be checked: If
> standpoint
> > > of real numbers works or not?
> >
> > Your are right.
> >
> > There are also mathematical challenges related to the localization
> > in zero modes occurring for final states of quantum jump.
>
> What is "zero modes" and how is it related with quantum jump?
Zero modes emerge in the construction of configuration space geometry.
a) Configuration space of 3-surfaces is union of spaces having coset
decomposition U_(zero modes)G/H and for G/H, "fiber" the metric is
nontrivial. G/H is infinite-dimensional counterpart of symmetric spaces
having G as isometry group and left invariant metric.
b) Metric of configuration space is degenerate in zero modes: this means
that line element in these degrees of freedom divergences.
c) Contravariant Kahler metric in turn defines defines propagator
in perturbation theoretic calculation of functional integral.
What is done is to take vacuum funtional exp(K), K Kahler function
and develop it around maximum of K as perturbative Gaussian integral.
This is exactly the same which is done in quantum field theories.
The propagator does not couple to zero modes since metric has
no components in zero modes. Hence these degrees of freedom
do not describe quantum fluctuating degrees of freedom. They
are 'classical'.
d) Zero modes turn out to correspond to variables characterizing
size and shape of 3-surface and classical induce Kahler field
associated with 3-surface. Classical em field is closely related
to classical Kahler field.
Zero modes are very much like vacuum expectation values
of Higgs fields or magnetization vector in thermodynamics, except that
there are infinitely many of them. They can be regarded as fundamental
order parameters.
e) The classicality of zero modes suggests that quantum jump
involves complete localization in zero modes. Same is suggested
by standard QFT and thermodynamics where localization in zero modes
is assumed (no quantum superposition over worlds with different
Higgs vacuum expectation values).
f) Localization in zero modes
means that the universe looks classical: spacetime surfaces
appearing in superposition of spacetime surfaces in
any final state of quantum jump are macroscopically equivalent.
g) In TGD the requirement that quantum jump is quantum measurement,
leads to the conclusion that localization in zero modes must
occur in quantum jump. This if entanglement coefficients
associated with fiber degrees of freedom do not depend on fiber
degrees of freedom. If they depend then localization also in
fiber degrees of freedom must occur. This is however in contradiction
with basic symmetries. That entanglement coefficients do
not depend on fiber degrees of freedom must follow from
Super Virasoro invariance and related conditions but I have
no proper proof for this.
g) Localization in zero modes reduces dramatically the complexity
of theory since calculations reduce to those in quantum fluctuating
fiber degrees of freedom.
>
> >
> > The final proof would be precise Feynmann rules
> > yielding S-matrix which is unitary order by order. BTW, I remember
> > having years ago looked the sketch of the perturbative proof of
> > unitarity. Analyticity and cuts of scattering amplitudes were
> > somehow involved.
>
> In the field theoretical proofs, cut-offs may be necessary. This is seen also
> in recent researches.
>
Best,
MP
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