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\today \hfill LBNL-53835 \\
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{\large \bf Correspondence and Analyticity.}
\footnote{This work is supported in part by the Director, Office of Science,
Office of High Energy and Nuclear Physics, Division of High Energy Physics,
of the U.S. Department of Energy under Contract DE-AC03-76SF00098}
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Henry P. Stapp\\
{\em Lawrence Berkeley National Laboratory\\
University of California\\
Berkeley, California 94720}
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\begin{abstract}
The analyticity properties of the S-matrix in the physical region are
determined by the correspondence principle, which asserts that the
predictions of classical physics are generated by taking the classical limit
of the predictions of quantum theory. The analyticity properties deducible
in this way from classical properties include the locations
of the singularity surfaces, the rules for analytic continuation around
these singularity surfaces, and the analytic character (e.g., pole,
logarithmic, etc.) of these singularities. These important properties
of the S-matrix are thus derived without using stringent locality assumptions,
or the Schroedinger equation for temporal evolution, except for freely
moving particles. Sum-over-all-paths methods that emphasize paths of
stationary action tend to produce the quantum analogs of the contributions
from classical paths. These quantum analogs are derived directly from the
associated classical properties by reverse engineering the
correspondence-principle connection.
(This article is an invited contribution to a special issue of Publications
of RIMS commemorating the fortieth anniversary of the founding of the
Reseach Institute for Mathematical Science.)
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{\bf 1. Introduction.}
The S matrix was introduced by Wheeler[1]. It specifies the amplitude for
the scattering of any set of originally noninteracting initial particles
to any set of eventually noninteracting final particles. The full physical
content of relativistic quantum theory resides in the S matrix: any two
formulations that give the same S matrix are considered to be physically
equivalent.
The S matrix is a function of the momentum-energy four-vectors
of the initial and final particles. The law of conservation of momentum-energy
entails that the term of the S matrix that describes the scattering of any
specified set of initial particles to any specified set of final particles
must have a momentum-energy conservation-law delta function that constrains
the sum of the momentum-energy vectors of the final particles to be
equal to the sum of the momentum-energy vectors of the initial
particles. The remaining factor, which is defined only at points that satisfy
this conservation-law condition, is called a scattering function. It is finite
at almost all points in its domain of definition. This is important because
computations starting from the Schroedinger equation tend to give scattering
functions that are everywhere infinite. Thus Heisenberg[2] and others [3] have
proposed an approach to relativistic quantum theory that avoids the infinities
that arise from the Schroedinger equation by discarding that equation
altogether, and computing the S matrix directly from certain of its general
properties. In this approach one never specifies the (Schroedinger-equation-
induced) temporal evolution that takes initial states continuously to final
states, but which, according to the basic philosophy, lacks physical
significance. The S-matrix method works very well for simple cases. It may
work in general, but new computational techniques would be needed to achieve
this.
A key property of the scattering functions is that each of them is
analytic (holomorphic) at almost every point of its original (real) domain
of definition. This property was originally deduced from an examination of
Feynman's formulas for these functions, which are derived essentially from
the (relativistic) Schroedinger equation. Landau[4] and Nakanishi[5]
independently deduced the very restrictive necessary conditions for the
occurrence of singularities of these functions. Coleman and Norton[6] then
noted that these Landau-Nakanishi conditions are precisely the conditions
for the existence of a {\it classical} physical process that has the same
topological structure --- i.e., has the same arrangement of line segments
connected at vertices --- as the Feynman graph with which it is associated.
A Feynman graph is topological structure of line segments joined at vertices.
It was used by Feynman to specify a corresponding mathematical contribution
to the S matrix. The associated Landau-Nakanishi diagram is a diagram in
four-dimensional space-time that has the same topological structure, but
moreover satisfies all of the conditions of a corresponding process in
classical physics. Thus a Landua-Nakanishi diagram can be regarded as a
representation of a process in classical-physics that consists of a network
of point particles that interact only at point vertices, and that propagate
between these vertices as freely moving particles.
The rules of (relativistic) classical particle physics assign a momentum-energy
four-vector to each line of the diagram, and impose the conservation-law
condition that the energy-momentum flowing into the diagram along the
initial incoming lines must be able to flow along the lines of the graph,
and then out along the final outgoing lines {\it with energy-momentum
conserved at each vertex.} This conservation-law condition is imposed also
by the Feynman rules. But the Landau-Nakanishi (i.e., classical-physics)
diagram is required to satisfy also the ``classical physics'' requirement
that each line of the spacetime diagram be a {\it straight-line segment that
is parallel to the momentum-energy carried that line.} [In classical
relativistic particle physics each freely-moving particle moves in space-time
in the direction of its momentum-energy four-vector ($p=mv,v^2=1$), but this
property is not imposed in quantum theory: it would conflict with the
uncertainty principle, and, likewise, with the Fourier-transformation
connection between space-time displacements and momentum-energy that
constitutes the foundation of quantum theory.]
The Landau-Nakanishi diagram is, then, the picture of a possible classical
process, involving point particles interacting at points, and conforming to
the conditions of relativistic classical-particle physics. These conditions
were shown by Landau and Nakanishi to specify the {\it location (in the space
of the momentum-energy four vectors of the initial and final particles) of
a singularity---failure of analyticity---of the contribution to the S matrix
corresponding to the associated Feynman graph.}
The purpose of this article is to highlight the fact that although this
important connection between the physical-region singularities of the quantum
scattering functions and associated classical scattering processes was
originally derived from very strong quantum assumptions involving the
concepts of point interactions and continuous Schroedinger evolution in
time, the result is actually a consequence of much less. It is a consequence
of the ``correspondence principle'' connection between relativistic quantum
physics and relativistic classical-particle physics. This principle asserts
that the predictions of classical physics emerge from quantum theory
in the ``classical limit'' in which all effects due to the nonzero value
of Planck's constant become negligible.
The correspondence principle entails, however, much more than just the
analyticity of the S matrix at all points that do not correspond to a
classical-physics process. It entails also that, in a real neighborhood
of almost every real singular point, the scattering function is the limit
of a function analytic in the interior of a certain cone-like domain that
extends some finite distance into the complex domain from its tip in the
real neighborhood. This means that {\it each physical scattering function
is a limit of single analytic function.} That feature of the S matrix is
one of the key general properties upon which the S-matrix approach is based.
Its derivation from the correspondence principle was given by Chandler and
Stapp[7] and by Iagolnitzer and Stapp[8]. The first of these two papers sets
out the general framework, but is formulated within a distribution-analytic
framework in which the wave functions are, apart from mass-shell-constraint
delta functions, infinitely differentiable functions of compact support.
Consequently, it achieves analyticity only modulo infinitely differentiable
background terms. The second of these papers uses essentially Gaussian
wave functions to obtain full analyticity.
It is worth noting that Sato [9] independently constructed a mathematical
machinery called the sheaf of microfunctions, which can be used to describe
the same cone-like domain when applied to the S matrix.
The correspondence principle entails even more. It specifies also the
{\it nature} of these singularities: whether they are, for example, pole,
or logarithmic singularities. This means that the quantum effects
closely associated with these classical-physics processes are determined
already by the correspondence principle, without appeal the notion of
true point interactions or of the Schroedinger equation. That is, the
correspondence principle, which is a condition on the classical limit of
quantum theory, can be ``reverse engineered'' to deduce those features of the
quantum S matrix that produce the classical result in the classical limit.
And these feature include the analytic character of the the S Matrix
scattering functions in their original (real) domains of definition.
{\bf 2. An Asymptotic Fall-Off Property.}
The papers with Chandler and Iagolnitzer just cited deal exclusively with
particles of non-zero rest mass. The momentum-space wave function of particle
$i$ then has, due to the mass-shell condition, the form
$$
\Psi_i (p_i) = \psi_i (p_i) 2\pi \delta (p_i^2 -m_i^2), \eqno (2.1)
$$
where $p_i^2$ is the Lorentz inner product of $p_i$ with itself, with metric
$(1,-1,-1,-1)$, and $m_i$ is the (nonzero) rest-mass of particle $i$.
Quantum theory is characterized, fundamentally, by the Fourier-transform
link between momentum-energy and space-time. Thus the spacetime form of
this momentum-energy wave function is given by the Fourier transform:
$$
\widetilde\Psi_i (x_i) = \int (2\pi)^{-4} d^4 p_i \exp (-ip_ix_i)\Psi_i (p_i).
\eqno (2.2)
$$
The spacetime wave function has important asymptotic fall-off properties.
In Appendix A of reference [13] it is shown that if $\psi_i(p_i)$ has compact
support and is continuous, together with its first and second derivatives,
and if $u$ is any positive time-like four-vector satisfying $v^2 =1$, then
$$
\lim_{\tau \rightarrow \infty} f(m_i,\tau)\widetilde\Psi_i(v\tau)
=\psi_i(m_iv), \eqno (2.3)
$$
where
$$
f(m_i,\tau)=2m_i (2\pi i\tau/m_i)^{2/3} \exp (im_i\tau). \eqno (2.4)
$$
In the formula (2.2) the expression $p_ix_i$ in the exponent is originally
divided by Planck's constant over $2\pi$. But that factor has been removed by
choosing units of space and time so that Planck's constant (divided by $2\pi$)
and the velocity of light are both unity. But then letting $\tau$ go to
infinity is effectively equivalent to letting Planck's constant go to zero: the
expansion of the spacetime scale is mathematically equivalent to going to the
classical limit. Formula (2.3) shows that in this limit the probability
distribution in spacetime for a freely moving particle is specified by the
momentum-space distribution function $\psi_i(p_i)$ in accordance with the
relativistic classical physics formula $p_i=m_iv$.
The fall-off property described above was derived from quantum theory.
Later I shall derive it from classical physics.
The correspondence principle asserts that the classical-physics results
hold not only for these free-particle states but also for processes
corresponding to networks of locally interacting particles that propagate
freely over the asymptotically large distances between their interactions:
the classical physics probabilities emerges from the quantum probabilities
in the asymptotic $\tau \rightarrow \infty$ limit. This
correspondence-principle requirement determines not only the locations
and natures of the singulaties of the quantum momentum-space scattering
functions, but normally entails also that, in a real neighborhood of a
singular point $P$, the scattering function is a limit of a function
analytic in the intersection of a complex neighborhood of $P$ with the
interior of a cone that extends from the real domain in a set of directions
that is specified by the structures of the classical scattering diagrams
associated with that singular point $P$. This connection between
{\it directions of analyticity} at singularities and classical spacetime
diagrams is made via a $4n$-dimensional displacement vector $U$ introduced
in reference [7].\\
{\bf 3. The $4n$-dimensional displacement vector $U$.}
Consider a spacetime diagram $D$ that describes a possible network of
classical particles with a total of $n$ initial and final particles. This
diagram $D$ determines (via the directions of the initial and final lines)
a set $P=(p_1, ... ,p_n)$ of initial and final momentum-energy vectors.
It is convenient to introduce in addition to the {\it physical}
momentum-energy vectors $p_i$, which have positive energy components, also the
{\it mathematical} momentum-energy vectors $k_i$, where $k_i=p_i$ for initial
particles, and $k_i= -p_i$ for final particles. Then the law of conservation
of energy momentum reads $\sum k_i = 0$.
The $4n$-dimensional displacement vector $U$ is defined as follows.
From any arbitrarily chosen origin $O$ in spacetime draw, for
each initial and final particle $i$, a vector $u_i$ from $O$ to some point
on the straight-line that contains the initial or final line $i$. Define
$$
U = (u_1, ... , u_n). \eqno (3.1)
$$
For a fixed spacetime diagram D this 4n-dimensional displacement vector $U$
is not uniquely fixed: one can add to $U$ any vector of the form
$$
U_0=(a+b_1k_1, a+b_2k_2, ... , a+b_nk_n), \eqno (3.2)
$$
where a is a real spacetime vector, and for each $i$ the parameter
$b_i$ is a real number. Changing $a$ just shifts the location of $D$
relative to the origin $O$, and changing $b_i$ just slides the tip of
$u_i$ along the straight line $i$.
Notice that the combination of the four conservation-law delta functions and
the $n$ mass-shell delta functions restricts the relevant set of points in
the $4n$-dimensional space of points $K= (k_1, ... ,k_n)$ to a surface of
co-dimension $4+n$, and that the $4+n$ dimensional set of vectors $U_0$ spans
the set of normals to that co-dimension $4+n$ surface: the contravarient
vectors formed by taking linear combinations of the gradients to the arguments
of the $4+n$ delta functions constitute the set of vectors $U_0$. This is the
simplest example of the important fact that the set of vectors $U$ associated
with a singular point $K$ generally span the space defined by the set of
normal vectors to the surface of singular points passing though $K$.
This normality of the vectors $U$ associated with diagrams of classical
physics to the surfaces of singularities of the S matrix provides the link
between relativistic classical physics and domains of analyticty of
scattering functions in relativistic quantum physics.
{\bf 4. Another Asymptotic Fall-Off Property.}
If the wave function $\psi_i(p_i)$ in Eq. (2.1) is infinitely differentiable
and of compact support, and if $V$ is the associated velocity (double) cone
consisting of all lines through the origin ($p_i=0$) that intersect the
compact support (in the mass shell $p_i^2=m_i^2$) of $\psi_i(p_i)$ then, for
all $u$ in any compact set that does not intersect $V$, the function
$\widetilde\Psi (u\tau)$ uniformly approaches zero faster than any inverse
power of the scale parameter $\tau$: for any integer $N$
$$
\lim_{\tau\rightarrow\infty}\tau^N \widetilde\Psi_i(u\tau)=0. \eqno (4.1)
$$
This is a standard result (cf. ref[8], Eqn. (28)),
and it allows one to prove the weaker analyticity properties that hold
modulo infinitely differential back-ground terms. (See ref. [7]). But
to derive full analyticity from the correspondence principle a stronger
fall-off property is needed.
This stronger asymptotic fall-off property is obtained by introducing into
the wave functions $\psi_i(p_i)$ an exponential factor that
shrinks in width as $\tau$ tends to infinity. Specifically, one
introduces free-particle momentum-space wave functions of the form
$$
\psi_{\tau,\gamma,\bar p}(p) = \chi(p) \exp (-(p-\bar p)^2 \gamma\tau).
\eqno (4.2)
$$
and also requires the infinitely differential function $\chi(p)$
(of compact support) to be analytic at $p=\bar p$, where
$p^2 = \bar p^2 = m^2.$ Then the following fall-off property holds:
for all $4$-vectors $u$ in any compact set that does not intersect the line
through the origin containing $\bar p$, and for all $\gamma \geq 0$ smaller
than some fixed $\gamma_0$, there is a pair of finite numbers $(C,\alpha)$
such that for all $\tau$
$$
|\widetilde\Psi_{\tau,\gamma,\bar p}(u\tau)|<
C\exp -\alpha\gamma\tau. \eqno (4.3)
$$
Classical and quantum proofs of this fall-off property will be described
below. But let us first show how this property of the free-particle
coordinate-space wave functions is used to deduce, from the correspondence
principle, domains of analyticity for the momentum-space scattering function.
{\bf 5. Kinematics and Probabilities.}
The connection to the correspondence principle is obtained by using initial
and final wave functions $\Psi_i(p_i,u_i)$ of the form
$$
\Psi_i(\tau,\gamma,\bar p_i;p_i,u_i)=
\Psi_i(\tau,\gamma,\bar p_i;p_i)\exp iu_ip_i \eqno (5.1)
$$
where, for any $i$, in accordance with (2.1) and (4.2),
$$
\Psi(\tau,\gamma,\bar p;p)=\psi_{\tau,\gamma,\bar p}(p)2\pi\delta (p^2-m^2)
$$
The wave function (5.1) represents the particle state
obtained by translating the state represented by $\Psi_i$ by the spacetime
displacement $u_i$. The parameters $\gamma$ are taken to be the same
for all $i$. It is convenient to use henceforth real $\chi_i(p_i)$, each of
which is equal to one (unity) in some finite neighborhood of $\bar p_i$.
The correspondence-principle results are obtained by examining the
$\tau \rightarrow \infty$ behaviour of the transition amplitude
$$
A(\tau)=S[\{\Psi_i(\tau,\gamma,\bar p_i; p_i, u_i\tau)\}] \eqno (5.2)
$$
where the right-hand side is
$$
\left[\prod_i \int (2\pi)^{-4} d^4k_i
\Psi_i(\tau,\gamma,\bar k_i;k_i)\right]S(K)\exp iKU\tau.
$$
The absolute value squared of the complex number $A(\tau)$, times $f(\tau)$,
is the transition probability associated with these states of the initial
and final particles, and $f(\tau)$ is the inverse of the square of
the product of the norms of the wave functions $\psi_i$ of (4.2). This factor
grows like $(\tau)^{3n}$, but this growth can be absorbed into a bound of the
form $C exp -\alpha\gamma\tau$ by a slight adjustment of $C$ and $\alpha$.
{\bf 6. The Correspondence-Principle Condition.}
For any fixed $\bar K$ (with $\sum \bar k_i =0$ and, for each $i$,
$\bar{k}_i^2 = m_i^2)$ there is a set $C(\bar K)$ of vectors $U$ such that each
pair of $4n$-dimensional vectors $(\bar K, U)$ satisfies the Landau-Nakanishi
conditions. This set $C(\bar K)$ includes the set $C_0(\bar K)$ consisting of
all of the vectors $U_0$ of the form (3.2): each of these vectors $U_0$
specifies a classical-physics diagram D in which all of the initial and
final particles pass through a single common point. Each of these vectors
$U_0$ has a null (Lorentz) inner product with every tangent vector to
--- i.e., with every infinitesimal displacement in --- the surface at
$\bar K$ of singularities generated by the mass-shell and overall
conservation-law delta functions.
Suppose $C(\bar K) = C_0(\bar K)$. That would mean that, on the one hand,
there are for the set $\{\bar k_i\}$ of initial and final (mathematical)
momentum-energy vectors specified by $\bar K$ no classical-physics diagrams
except the trivial ones in which all the initial and final particles pass
through a common point, and, on the other hand, according to the Feynman
rules, no singularity of the quantum scattering function. But from the
S-matrix point of view the Feynmam rules are suspect, because they come
essentially from the physically meaningless continuous time evolution,
and also lead to infinities. However, the general correspondence principle
condition that the predictions of classical physics should emerge in the
limit where Planck's constant goes to zero, or, equivalently, where $\tau$
goes to infinity, would seem to be an exceedingly plausible and secure
condition. The analyticty of the scattering function at this point $\bar K$
is, in fact, a consequence of that correspondence condition.
For any point $\bar K$ such that $C(\bar K) = C_0(\bar K)$
consider any $U$ that does not belong $C(\bar K)$. If $U$ does not belong
to $C(\bar K)=C_0(\bar K)$ then for at least one of the $n$ particles
$i$ the component vector $U_i$ is not parallel to $\bar k_i$. But then
the amplitude $A(\tau)$ will pick up an exponential fall-off factor of
the kind shown in (4.3). These vectors $U$ cover a unit sphere in the
$3n-4$-dimensional subspace that is normal to the $n+4$-dimentional
subspace $C(\bar K)$. Thus there will be a {\it least value} of $\alpha$
for the $U$'s on this (compact) unit sphere.
This uniform exponential fall-off over this unit sphere arises,
in the classical computation, from the exponential fall off of the overlap
of the probability functions of the initial and final particles:
i.e., from the exponentially decreasing probability, as $\tau$ increases,
for {\it all} of the initial and final particles to be in any single
finite region of space-time that grows like the square root of $\tau$.
In classical physics such an exponential decrease in this probability,
coupled with the fact that the only classical scattering process that
can carry the initial momentum-energies to the final momentum-energies is
one where all the initial and final particle trajectories pass through
some such growing space-time region entails a similar fall off of the
transition probabilities: the probability for this kind of classical process
to occur cannot grow faster than the product of the probabilities that
the particle can all be in any such growing region. Thus the
correspondence principle requires that transition amplitude
$A(\tau)$ have the same sort of fall off as the one arising from the
overlap of the wave functions. It will now be shown that this condition
entails the analyticity of the scattering function at this point
$\bar K$ where $C(\bar K) = C_0(\bar K)$.
{\bf 7. Derivation of analyticity at trivial points.}
By a ``trivial point'' I mean a point $\bar K$ such that
$C(\bar K) = C_0(\bar K)$: the only classical processes with
external momenta specified by $\bar K$ are the trivial single-vertex
diagrams.
The set of Landau-Nakanishi surfaces that enter any bounded region of
$K$ space has been shown to be finite [Ref. 10]. And each such surface is
confined to a co-dimension-one analytic manifold. Consequently, each trivial
point $\bar K$ lies in an open neighborhood of such points.
Introduce a set of analytic coordinates $q$ in the $3n-4$-dimensional
manifold in $K$-space restricted by the mass-shell and conservation-law
conditions near $\bar K$. Let the $q$ be a subset of the space components of
the set of vectors $(k_i-\bar k_i)$, and let the $v$ associated with
any $q(K)$ in the neighborhood of $q(\bar K)=0$ be the corresponding
$3n-4$ components of $U\tau$ mod $C_0(K)$, so that $KU\tau$ in (5.2)
becomes $(-qv -\bar kv)$, where the metric $(1,1,1)$ is now used, and $v$
represents displacements away from the displacements that generate the trivial
single-vertex processes. Then the $A(\tau)$ in (5.2), times the (unimportant)
phase factor $\exp(i\bar kv)$. can be written as
$$
T(v,r)=\int dq F(q)\exp(-r\mu(q)) \exp(-iqv), \eqno (7.1)
$$
where
$$
\mu(q) = \sum_i (k_i(q)-k_i(0))^2, \eqno (7.2)
$$
$r= \gamma \tau$, and $F(q)$ is the scattering function times a factor that is
real, infinitely differentiable of compact support, and analytic at $q=0$,
which is the $q$-space image of $\bar K$. A fall-off property of the form
(4.3) is required to hold for all $\tau$ and all $0\leq\gamma \leq \gamma_0$,
with $r=\gamma \tau$, and all $v=\hat{v}\tau$ with $|\hat{v}|=1$ . What needs
to be proved is that this fall-off condition, together with the analogous
rapid (faster than any power of $\tau$) fall off at $\gamma = 0$, entails the
analyticity of $F(q)$ at $q=0$.
This rapid fall off of the bounded $T(v,0)=T(\hat{v}\tau,0)$ for all
unit vectors $\hat{v}$ means that $F(q)$ is the well-defined and infinitely
differentiable Fourier transform:
$$
F(q)=(2\pi)^l\int dv\exp(iqv) T(v,0), \eqno (7.3)
$$
where $l=3n-4$. To show that $F(q)$ is analytic at $q=0$ re-write this
equation in the form
$$
(2\pi)^l F(q)=\int dv\exp(iqv)\times
$$
$$
\left[T(v,\gamma_0|v|)exp(\gamma_0|v|\mu(q))
-\int_0^{\gamma_0|v|} dr \frac{\partial}{\partial r}
[T(v,r)\exp(r\mu(q))]\right]. \eqno (7.4)
$$
Consider first the first term in the big brackets. The correspondence
principle requires the factor $T(v,\gamma_0|v|)$ to be bounded by
$C\exp(-\alpha\gamma_0|v|)$. The function $\mu(q)$ is zero at $q=0$,
and hence the associated exponential growth is dominated by the fall-off
factor for $q$ in a sufficiently small neighborhood of $q=0$, Indeed, this
bound keeps the integral well defined and analytic for all $q$ in a small
complex neighbor of $q=0$. Thus the contribution $F_1(q)$ to $F(q)$ coming
from the first term in the big brackets is analytic at $q=0$.
To prove that this property holds also for the other contribution, $F_2(q)$,
substitute (7.1) into the second term in the big brackets. The
$\partial/\partial r$ can be moved under the integral over $dq$ because
$F(q)$ is infinitely differentiable of compact support. This gives for the
integrand
$$
\exp(iqv)\frac{\partial}{\partial r}[T(v,r)\exp(r\mu(q))]=
$$
$$
\int dq' F(q')\exp(i(q-q')v)\exp(r(\mu(q)-\mu(q')))[\mu(q)-\mu(q')]
\eqno (7.5)
$$
Hefer's theorem [8] allows one to write
$$
\mu(q)-\mu(q')=\rho(q,q')\cdot(q-q'), \eqno (7.6)
$$
where $\rho$ is a vector whose the components $\rho_j$ (j=1,... ,3n-4)
are analytic in a product of domains around $q=0$, and $q'=0$. Then (7.5)
becomes
$$
\exp(iqv)\frac{\partial}{\partial r}[T(v,r)\exp(r\mu(q))]=
Div_v [\exp(iqv)\exp(r\mu(q))H(q,v,r)], \eqno (7.7)
$$
where $H(q,v,r)$ is the vector
$$
H(q,v,r)= -i\int dq' F(q')\exp(-iq'v)exp(-r\mu(q'))\rho(q,q').
\eqno (7.8)
$$
We may thus write
$$
(2\pi)^l F_2(q)=-\lim_{R\rightarrow \infty}
\left[\int_{|v| 0$ for all $v$ in a closed cone $V$ that contains the
closure of $V(H(\bar K))$ in its interior, apart from the origin $v=0$.
For these $q$ the exponential factor $\exp iqv$ in (7.3) get from $Im$ $q$
a factor $\exp -\alpha |Im$ $ q||v|$, where $\alpha > const > 0$. This means,
because $T(v,0)$ is bounded, that the integral is absolutely converent,
and hence that $F_H(q)$ is analytic near $q=0$ for $Im$ $q$ in $Q$.
Most of the real points $q$ very near to $q=0$ are ``trivial'' points,
of the kind considered in the preceding section. At those trivial points
$q'$, the function $F(q') = F_H(q') + F_A(q')$ is analytic. These two terms
are taken at these points $q'$ to be just the contributions to $F(q)$
specified in (7.4) and (7.10) restricted to the regions $V(H(\bar K))$ and
$V(A(\bar K))$ respectively. Both of these contributions are analytic in
the intersection of some neighborhood of $q$ with the cone $Q$. Thus one
can stay in the domain of analyticity by moving $Im$ $q$ slightly into
the cone $Q$ in order pass to the other side of the surface of
singularities that passes through $q=0$.
A more elaborate presentation of this argument, and of its generalizations
to more complex cases, can be found in references [7] and [8], and also
in Iagolnitzer's book [11].
{\bf 9. Correspondence-Principle Asymptotic Fall Off.}
I have described some of the analytic consequences of the fall-off properties
(2.3), (4.1), and (4.3). I turn now to a fuller discussion of the roots
of these fall-off properties in the correspondence to classical properties.
The statistical predictions of quantum mechanics correspond, at least in a
formal way, to the predictions of classical {\it statistical} mechanics.
In the latter theory one describes a system of $n$ particles at any time $t$
in terms of a function $\rho (x,p,t)$, which specifies how the probability
is distributed over the points $(x,p)$ of ``phase space,'' where $x$
specifies the $3n$ coordinate variables and $p$ specifies the $3n$
momentum-space variables. Free-particle evolution keeps $p$ fixed and
shifts the location $x_i$ of a particle of (rest) mass $m_i$ during a
time interval $t$ to the location $x_i + tp_i/m_i$. For large $t$ the second
term dominates, and the coordinate-space probability function goes over to
the momentum-space probability function, properly scaled to account for the
diverging directions of the different momentum vectors. This classical
kinematics entails that for free particles the classical distribution
$\rho (x,p,t)$ at large times $t$ becomes a product over $i$ of functions
$$
\rho (u_it, p_i,t)= |\rho (u_im_i) \rho(p_i)/f(m_i,t)^2|, \eqno (9.1)
$$
where
$$
\rho(p_i)=\int d^3x_i \rho(x_i,p_i, t'), \eqno (9.2)
$$
is independent of $t'$, and $f(m_i,t)$ is the function defined in (2.4).
Here I am, for simplicity, assuming that the momenta are small enough so
that the non-relativistic formulas (where $t=\tau$ and $p_0=m$) are adequate.
(The fully covariant formulation gives the same results.) The factor
$(m_i/t)^3$ coming from $f(m_i,t)^{-2}$ compensates for the linear spreading
out of the probability distribution in coordinate space, and the $1/(2m_i)^2$
comes from the normalization in (2.1). This equality of the classically-derived
and quantum-mechanically derived limits constitutes, in this case,
{\it part of} the correspondence-principle relationship between the
asymptotic properties in classical and quantum theory: both theories give
the same asymptotic form for the probability distribution in $(x,p)$,
for the case $\gamma =0$.
There is no conflict here with the uncertainty principle limitation on the
idea of a distribution in both $x$ and $p$ simultaneously: the huge
spreading out of the coordinate-space distribution eliminates any such
conflict.
But what is the rate of approach to this limit?
The probability distribution in coordinate space at $t=0$ for the function
in (4.2), at $\gamma=0$, would be given by the (absolute value squared of the)
Fourier transform of $\chi(p)$. This transform of the infinitely
differentiable compactly supported $\chi(p)$ falls off faster than any power
of $|x|$. This leads to the quantum mechanical prediction (4.1). Classically,
this original $x$-space distribution is the constant (non-expanding)
background to the $t$-dependent diverging trajectories. If this non-expanding
background falls off faster than any power of $x$ then its contribution at
points $x=u\tau$ will fall off faster than any power of $\tau$. Hence the
approach to the large-$t$ limit computed classically, by using the
straight-line trajectories in space-time, also exhibits the faster than
any power fall off specified in (4.1): the classical and quantum predictions
agree about both the limit and the rate of approach to this limit.
But what is the rate of fall off for the case $\gamma >0$?
To show that the fall off in this case conforms to (4.3) it is sufficient to
go to the frame where $\bar p$ is pure spacelike and the space part of
$u$ is nonzero. Then
$$
|\widetilde{\Psi}_{\tau,\gamma,\bar p}(u\tau)|=
|\int d^3q/(2\pi)^{-3}\chi(q)\exp(-\tau[q^2\gamma+i(qu-u_0(q_0-\bar p_0))])|,
\eqno (9.3)
$$
where I again use the metric (1,1,1) for the $3$-vector products $qu$
and $q^2$, and $q_0-\bar p_0 =(q^2+m^2)^{1/2} -m$.
To get the quantum prediction, consider a distortion of the $q$-space
contour that is parameterized by a scalar $\alpha$. For $q^2 > \alpha$
there is no distortion. For $Re$ $q^2 < \alpha$ the component of
$Im$ $q$ that is directed along $u$ is
shifted (keeping real the other two components of the $3$-vector $q$) so that
$$
Re[q^2\gamma+i(qu-u_0(q_0-\bar p_0))]=\alpha\gamma. \eqno (9.4)
$$
Distort the contour from $\alpha =0$ to a value such that all real $q$
in $q^2 \leq \alpha$ lie inside the open set where $\chi$ is one,
and such that $|Im$ $q|$ remains less than $m$.
Then for all real points $q$ with $ q^2>\alpha$ one has an exponential
fall-off factor $\exp -\alpha\gamma\tau$. For real $q$ such that
$q^2 < \alpha$ the condition (9.4) gives a factor $\exp -\alpha\gamma\tau$.
One can obtain a bound like this for every four vector $u$ on the unit
(Euclidean) sphere, minus small open holes around the rays along the
positive and negative time axis (along which $\bar p$ has been taken to lie).
These holes can be defined by conditions on the three-vector part $\vec{u}$
of $u$: $|\vec{u}|< \epsilon$. The only singularity that could block this
continuation is the singularity of $q_0$ at $q^2+m^2=0$, and this is
prevented by our condition $|Im$ $q|