Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 11 Oct 1999 13:42:35 +0300 (EET DST)
On Mon, 11 Oct 1999, Hitoshi Kitada wrote:
> Dear Matti,
>
> Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
>
> Subject: [time 933] Re: [time 931] Re: [time 928] Re: [time 923] Unitarity
>
>
> >
> >
> > On Mon, 11 Oct 1999, Hitoshi Kitada wrote:
> >
> > > Dear Matti,
> > >
> > > Matti Pitkanen <matpitka@pcu.helsinki.fi>
> > >
> > > Subject: [time 931] Re: [time 928] Re: [time 923] Unitarity
> > >
> > >
> > > >
> > > >
> > > > Dear Hitoshi,
> > > >
> > > > Still one question, can you tell in five words(;-) what is the
> difference
> > > > between QFT and wave mechanics approches? One might think that
> > > > basically there can be no difference if Hamiltonian quantization really
> > > > works. Didn't Schwinger follow the Hamiltonian quantization?
> > > >
> > > > MP
> > >
> > > My understanding is that QFT uses the operator-valued distribution which
> is
> > > the basic quantity. Standard approach uses state function. The
> formulations
> > > would be different in their interpretaion and applicability of standard
> > > scattering theory to QFT seems small.
> >
> > The introduction of field operators is new element and Fock space replaces
> > the Hilbert space of wave functions. One can generalize
> > Lippmann-Scwinger to abstract Hilbert space and presumably does so.
>
> Yes. If one can formulate in Hamiltonian formalism in QFT, it would give a
> similar structure to standard one, and one can argue in a similar way. There
> is a possibility here, but Hamiltonian formalism would be possible only in
> Euclidean metrics and usually it is not taken seriously. Of course there is a
> method of introducing Euclidean metric by replacing i*t by a new real variable
> x^0. My assertion is not in this Euclidetization, but in using genuine
> Euclidean metric as the basic metric.
Hamiltonian formalism is also possible in Minkowski metric but one
has to prove Lorentz invariance separately and this makes formalism
awkward. Functional integral formalism is invariant but as far
as I have understood (after five years of attempts to
construct functional integral formalism for TGD)
it is difficult to understand how unitarity would result
in this approach when action is extremely nonlinear as in my case.
Yes: I have understood the basic idea of your approach.
>
> >
> > You are right that the applicability at practical level is small.
> >
> >
> > In any case, in TGD configuration space spinor field is formally in
> > same position as classical Dirac spinor. No second quantization is
> > performed for it although spinor components correspond to
> > Fock states generated by second quantized induced spinor fields
> > on spacetime surfaces.
>
> In this respect, there is a possibility of applying standard method, but the
> metric might be a problem.
And the unitarity...
Funny thing, how very few theoreticians after all pay attention
for these problems. They just calculate.
In perturbative QFT one can express
discontinuities (T^dagger_n-T)_n= (T^daggerT)_n the r.h.s
being expressed in terms formed of lower order amplitudes T_k, k<n
and from these deduce amplitude T_n using dispersion relations.
In this approach QFT would give only the lowest order term from which one
starts iteration and it becomes questionable whether QFT makes
sense anymore.
Strongest counter argument against p-adic approach is that
T^daggerT=0 so that *p-adic* cross sections vanish. Real
scattering rates obtained from p-adic ones by canonical
identification are nonvanishing but are very probably
not deducible from real S-matrix. But can anyone claim that *real*
unitarity of QFT has been experimentally verified? I remember chapter
about dispersion relations for pion in Bjorken Drell but who takes it
seriously now when quarks have been discovered?
Best,
MP
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