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\section{\usemenu{slacpub7205::context::slacpub7205005}{The HandyDandy Formula}}\label{section::slacpub7205005}
One essential ingredient missing from current elementary particle physics
is a {\it nonperturbative} connection between masses and coupling constants.
We believe that one reason that contemporary conventional approaches to
relativistic quantum mechanics fail to produce a {\it simple}
connection between these two basic classes of parameters is that they start
by quantizing classical, manifestly covariant continuous field
theories. Then the uncertainty principle necessarily produces infinite
energy and momentum at each spacetime point. While the renormalization
program initiated by Tomonoga, Schwinger, Feynman and Dyson succeeded
in taming these infinities, this was only at the cost of relying on
an expansion in powers of the coupling constant. Dyson\cite{19} showed in
1952 that this series cannot be uniformly convergent, killing
his hope that renormalized QED might prove to be a fundamental theory
\cite{85}. Despite the technical and phenomenological successes
of nonAbelian gauge theories, this difficulty remains unresolved
at a fundamental level. What we propose here
as a replacement is an expansion in {\it particle number} rather than
in coupling constant. The first step in this direction already yields
a simple formula with suggestive phenomenological applications, as we
now show.
We consider a twoparticle system with energies $e_a,e_b$ and masses
$m_a,m_b$ which interact via the exchange of a composite state
of mass $\mu$. We assume that the exterior scattering state is in a coordinate
system in which the particles have momenta of equal magnitude $p$
but opposite direction. The conventional SMatrix approach starts on
energyshell and on 3momentum shell with the algebraic
connections\cite{6}
\begin{eqnarray}
e_a^2m_a^2&=&p^2=e_b^2m_b^2\nonumber\\
M^2&=&(e_a+e_b)^2\vec p_a +\vec p_b^2\\
\vec p(M;m_a,m_b)&=&{[(M^2(m_a+m_b)^2)
(M^2(m_am_b)^2)]^{{1\over 2}}\over 2M}\nonumber
\end{eqnarray}
but then requires an analytic continuation in $M^2$ off mass shell.
Although this keeps the problem finite in a sense, it leads
to a nonlinear selfconsistency or {\it bootstrap} problem from which
a systematic development of {\it dynamical} equations has yet to
emerge.
We take our clue instead from nonrelativistic multiparticle scattering theory
\cite{27,92,93,94,28,3,97}
in which once a twoparticle bound state vertex opens up,
at least one of the constituents must interact with a third particle
in the system before the bound state can reform. This eliminates the
singular ``self energy diagrams'' of relativistic quantum field theory
from the start. Further, the algebraic structure of the Faddeev
equations automatically guarantees the unitarity of the three particle
amplitudes calculated from them \cite{31}. The proof only requires
the unitarity of the twobody input\cite{54,57}.
This suggests that it might be possible to develop an ``onshell''
or ``zero range'' multiparticle scattering theory starting from
some twoparticle scattering amplitude formula which guarantees swave
onshell unitarity.
In order to implement our idea, rather than use Eq.15 we define the
parameter $k^2$, which on shell is the momentum of either particle
in the zero 3momentum frame, in terms of the variable $s$ which in the physical
region runs from $(m_a+m_b)^2$ (i.e. elastic scattering threshold)
to the highest energy we consider by
\begin{equation}
k^2(s;m_a+m_b) = s(m_a+m_b)^2
\end{equation}
Then we can insure onshell unitarity for the scattering amplitude
$T(s)$ with the normalization $Im \ T(s) =\sqrt{s(m_a+m_b)^2}T^2$ in the
physical region by
\begin{eqnarray}
T(s)&=&{e^{i\delta (s)}sin \ \delta (s)\over\sqrt{s(m_a+m_b)^2}}
={1\over k\ ctn \ \delta (s) i\sqrt{s(m_a+m_b)^2}}\nonumber\\
&=&{1\over \pi}\int_{(m_a+m_b)^2}^{\infty}ds'{\sqrt{s'(m_a+m_b)^2}T(s')^2
\over s' si\epsilon}=
{2\over \pi}\int_0^{\infty}dk'{sin^2\delta(k')\over k^2 (k')^2i\epsilon}
\end{eqnarray}
We arrived at this way of formulating the twobody input for
multiparticle dynamical equations in a rather circuitous way.
It turns out that this representation does indeed lead to well defined
and soluble {\it zero range} three and four particle equations of the
FaddeevYakubovsky
type\cite{57,75}, and that {\it primary singularities} corresponding
to bound states and CDD poles \cite{17} can be introduced and fitted
to low energy two particle parameters without destroying the unitarity
of the three and four particle equations. However, if we adopt the SMatrix
point of view
which suggests that elementary particle exchanges should appear in this
nonrelativistic model as ``left hand cuts'' starting at $k^2 = 
\mu_x^2/4$, where $\mu_x$ is the mass of the exchanged quantum
\cite{72}, then we discovered\cite{57} that
the unitarity of the 3body equations can no longer be
maintained; our attempt to use this model as a starting point for
doing elementary particle physics was frustrated.
We concluded that a more fundamental approach was required, in the
pursuit of which\cite{11,73} the nonperturbative
formula which is the subject of this paper was
discovered\cite{51}. However the reasoning was
considered so bizarre as, according to one referee, not even to
qualify as science. This paper aims to rectify that deficiency
by carrying through the derivation in the context of a relativistic
scattering theory, which we will call {\it TMatrix theory} in order
to keep it distinct from the more familiar SMatrix theory from which it
evolved. Thanks to a comment by Castillejo\cite{16}
in the context of our treatment of the fine structure of the spectrum
of hydrogen\cite{51},
we finally realized that the success of our new approach required us
from the start to view our {\it TMatrix} as embedded in a
multiparticle space. This can be accomplished using the relativistic
kinematics of Eq.16 rather than of Eq.15 for the offshell extension which leads to
dynamical equations.
As we know from earlier work on partial wave dispersion
relations\cite{72}, if we know that the scattering
amplitude has a pole at $s_{\mu}=\mu^2$, or equivalently
at $k^2+\gamma^2=0$ where $\gamma =+\sqrt{(m_a+m_b)^2\mu^2}$
then a subtraction in the partial wave dispersion relation given by
Eq.17 easily accommodates the constraint while preserving onshell
unitarity in the physical region. This allows us to define the
dimensionless {\it coupling constant} $g^2$ as the ``residue at the
bound state pole'' with appropriate normalization. We choose to
do this by the alternative definition of $T(s)$ given below:
\begin{eqnarray}
T(s;g^2,\mu^2) &=& {g^2\mu\over s\mu^2}={g^2\mu\over
k^2(s)+\gamma^2}\nonumber\\
&=&{1\over k\ ctn \ \delta(s) +ik(s)}
\end{eqnarray}
Consistency with the dispersion relation, assuming a constant value for
$g^2$, then requires that at $k^2=0$
\begin{eqnarray}
T((m_a+m_b)^2);g^2,\mu^2) &=& {1\over \gamma}={g^2\mu\over \gamma^2}\nonumber\\
k\ ctn \ \delta((m_a+m_b)^2)&=& \gamma
\end{eqnarray}
Consequently $g^2\mu=\gamma$ and by taking $\gamma ^2$ also from Eq. 190
we obtain our
desired result, the {\it handydandy formula} connecting masses and
coupling constants:
\begin{equation}
(g^2\mu)^2 =(m_a+m_b)^2  \mu^2
\end{equation}
In the nonrelativistic context where
$\gamma_{NR}^2=2m_{ab}\epsilon_{ab}$, $m_{ab}=m_am_b/(m_a+m_b)$,
$\epsilon_{ab} =m_a+m_b\mu$, this evaluation of the value of
$k \ ctn \ \delta$ at low energy is equivalent to assuming that
the phase shift is given by the {\it mixed effective range
expansion}\cite{53}:
\begin{equation}
k \ ctn \ \delta = \gamma + k^2/\gamma= \gamma +(k^2 +\gamma^2)/\gamma
\end{equation}
corresponding to the {\it zero range} bound state wave function $r\psi (r)=
e^{\gamma r}$ which assumes its asymptotic form very close to point
where the positions of the two particles coincide. As Weinberg
discusses in considerable detail in his papers on the quasiparticle approach
\cite{92,93,94}, this constraint requires
the bound state to be purely composite  i.e. to contain precisely
two particles with no admixture of effects due to other degrees of
freedom. We believe that his analysis supports our contention that we can claim
the same interpretation for our relativistic model of a bound state,
and hence that we have derived the proper twoparticle input for relativistic
dynamical nparticle equations of the FaddeevYakubovsky type. These equations,
which are readily solved for three and four particle systems, will
be presented on another occasion\cite{75}.
What follows next is an unsystematic presentation of results,
some of which were initially obtained using the {\it combinatorial
hierarchy}\cite{11,73}, but which we now
claim to have placed firmly within
at least the phenomenology of standard elementary particle physics.....
\begin{center}
{\bf\Large THE TICTOC LABORATORY: A Paradigm for BitString Physics}
\end{center}
Just prior to ANPA 19 I will be attending a conference organized by
Professor Zimmermann entitled {\it NATURA NATURANS: Topoi of
Emergence}. The following notes are intended to serve as raw
material for my presentation there. Some of these ideas came out of
extensive correspondence I have had with Ted and Clive
following ANPA 18, and owe much to their comments. In particular
the section 6 should be compared to Clive's discussion of a
scientific investigation in his paper in these
proceedings\cite{39}.
I would also like to remind you before we start of Eddington's
parable that if we set out to measure the length of the fish in the
sea, and we find that they are all greater than one inch long we
have the option of concluding (a) that all the fish in the sea
are greater than one inch long or (b) that we are using a net with a one inch mesh.
Thinking of my approach in this way, I seem to be finding out that
{\it because} I insist on finite and discrete measurement accuracy
together with standard methodological principles. I am bound
to end up with something that looks like a finite and discrete
relativistic quantum mechanics that has the ``universal constants''
we observe in the laboratory. Whether the cosmology we observe
is also constrained to the same extent is an interesting question.
My guess is that we will find that historical contingency
plays a significant role.
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