=\th(q)\<\eta+|\s B(D+p\s q\s Q,q^{-1} P+p,q\s\s Q)\s|\eta+\>\\ &&\hskip1cm +\th(-q)\<\eta-|\s B(D+p\s (-q)\s Q,(-q)^{-1} P+p,(-q)\s\s Q)\s|\eta-\>\;.\e Introducing $\<(\s\cdot\s)\>_\pm\equiv \<\eta\pm|(\s\cdot\s)\s|\eta\pm\>$, it follows that \b\

=\th(q)\s q\s\ = \th(q)\s q-\th(-q)\s (-q)\equiv q\;, \e
a desirable value indeed. Additionally, for whatever values we assign to $B$ and $C$ it follows
that
\b \ = q+O(\hbar)\;, \e
and, furthermore, it is not difficult to show that
\b \ =q^r+O(\hbar)\;, \e
for all $r$.
In addition,
\b \ \hskip-1.3em&&=\th(q)[\s p\s q\s\ =|q|\s\<\s Q\s\>=0 \e
for all $(p,q)$, and likewise $\ =0$ for any odd power.
However, let us instead consider
\b \ =q^2\,\<\s Q^2\s\>\;, \e
which, along with the modest requirement that $\<\s Q^2\s\>=1$, leads to
\b \ =q^2 \e
from which we can conclude that $q^2$ is the mean value of $Q^2$.
By a suitable choice of the fiducial vector $|\eta\>$, we can also arrange that
\b \ =q^{2r}+O(\hbar)\;. \e
In the same states $|p,q\>$, it follows that
\b \ =\<\s B(D+p\s|q|\s Q,|q|^{-1}P+p,|q|\s Q)\s\>\;.\e
If the function $\eta(x)=\ =0\;,\hskip1cm \ =p\;,\e
and thus implies that $p$ is the mean value of $P$ in the coherent state $|p,q\>$.
Additionally, we see that
\b \ =\<\s (|q|^{-1}\s P+p)^2\s\>=p^2+q^{-2}\s\<\s P^2\s\>\;,\e
where the factor $\ =\hbar^2\s c$ for a dimensionless constant $c>0$.
With the foregoing expectation values, we can argue that
the expression given by
\b \ \hskip-1.2em&&\equiv\ \\
&&=\half\s(p^2+\hbar^2\s c/q^2+\omega\s q^2)+\lambda_0 \s q^4\\
&&=\half(p^2+\omega\s q^2)+\lambda\s q^4+O(\hbar) \e
seems to provide a reasonable connection between suitable quantum and
classical Hamiltonians.
Traditionally, when dealing with coherent states, one also speaks about a resolution of unity in the form
\b \one=\int |p,q\>\ $ serves as a reproducing kernel for a reproducing kernel Hilbert
space representation of the underlying abstract Hilbert space $\frak{H}$; see \cite{aron}.
\subsection*{Many dimensional affine coherent states}
We now extend the preceding analysis to a discussion of many dimensional affine coherent states. Consider
a set of $J$ ($J$ being {\it odd}) independent affine fields such as $Q_j$ and $D_j$, $j=0,\pm1,\pm2,\ldots,\pm J^*$, where $J^*\equiv(J-1)/2$, and the only nonvanishing commutator is given by
\b [Q_l,D_j]=i\hbar\s\delta_{l,j}\s Q_l\;, \e
and for each $j$ the spectrum of $Q_j$ is the entire real line save for zero. The coherent states for this system are taken as
\b |p,q\>\equiv e^{\t i\s \Sigma_j p_j\s Q_j/\hbar}\,e^{\t -i\Sigma_j\s \ln(|q_j|)\s D_j/\hbar}\,|\eta\>\;. \e
In terms of the states $|x\>$, where $Q_j\s|x\>=x_j\s|x\>$, for all $j$, it follows that
\b \ ={\t\prod}_{j=-J^*}^{J^*}\frac{[2\s|q'_j|^{-1}\s|q_j|^{-1}\s]^{1/2}}{[\s q'^{-2}_j+q_j^{-2}\s]^{1/2}}
\,e^{\t -(1/2\omega)(p'_j-p_j)^2/[\s q'^{-2}_j+q_j^{-2}\s]}\;.\e
Let us extend our present example to an infinite number of degrees of freedom, for which $J\ra\infty$.
This leads to the expression
\b \ ={\t\prod}_{j=-\infty}^\infty\frac{[2\s|q'_j|^{-1}\s|q_j|^{-1}\s]^{1/2}}{[\s q'^{-2}_j+q_j^{-2}\s]^{1/2}}
\,e^{\t -(1/2\omega)(p'_j-p_j)^2/[\s q'^{-2}_j+q_j^{-2}\s]}\;,\e
which provides a well-defined product representation for affine coherent states
provided that the variables $\{p'_j,q'_j\}_{j=-\infty}^\infty$ and $\{p_j,q_j\}_{j=-\infty}^\infty$
are well chosen.
\subsubsection*{Representation for a field}
A one-dimensional affine field theory involves field operators that satisfy
the affine commutation relation
\b [\s\varphi(x),\rho(y)\s]=i\s\hbar\s\delta(x-y)\s\varphi(x)\;, \hskip1cm x,y\in\mathbb{R}\;,\e
where $\varphi(x)$ is the generalization of $Q$ and $\rho(y)$ generalizes $D$. Let us regularize this field
formulation by introducing a one-dimensional lattice space with $J<\infty$ lattice points (again with $J$
conveniently chosen as an {\it odd} number) each separated by a
lattice spacing $a>0$, which leads to a regularized affine field representation given by fields
$\varphi_k$ and $\rho_k$, where $x$ has been replaced by the integer $k$ and $x=k\s a$. Here,
$-J^*\le k\le J^*$, and $J^*\equiv(J-1)/2$ as before. These operators
obey the affine commutation relation given in the form
\b [\s\varphi_j,\rho_k\s]=i\s\hbar\s a^{-1}\s\delta_{j,k}\s\varphi_j\;.\e
The coherent states for such a regularized field are given by
\b |p,q\>\equiv e^{\t i\Sigma_j p_j\varphi_j\s a/\hbar}\,e^{\t-\Sigma_k\s\ln(|q_k|)\s \rho_k\s a/\hbar}\,|\eta\>\;.\e
For our {\bf first example}, we choose $|\eta\>$ so that (with $\hbar=1$)
\b \<\phi|\eta\>=M\,e^{\t -\half\s\omega\s\Sigma_k\s\phi_k^2\s a}\;; \e
here, the vector $|\p\>$ replaces $|x\>$ as used before, where $\varphi(x)\s|\p\>=\p(x)\s|\p\>$. In the present
case the overlap function of two coherent states becomes
\b \ ={\t\prod}_{j=-J^*}^{J^*}\frac{[2\s|q'_j|^{-1}\s|q_j|^{-1}\s]^{1/2}}{[\s q'^{-2}_j+q_j^{-2}\s]^{1/2}}
\,e^{\t -(1/2\omega)(p'_j-p_j)^2\s a/[\s q'^{-2}_j+q_j^{-2}\s]}\;.\e
Next we investigate a possible continuum limit in which $J^*\ra\infty$, $a\ra0$, and initially we require that
$(2\s J^*+1)\s a=J\s a\equiv X$ may be large but finite; a subsequent limit in which $X\ra\infty$ is taken later.
Moreover, in this limit we also insist that $p_j\ra p(x)$and $q_j\ra q(x)$, and that both functions are {\it continuous}. To ensure that we focus on the representations induced by the given fiducial vector,
we restrict attention to functions $p(x)$ that have compact support and functions $q(x)$ such that
$\ln|q(x)|$ also has compact support, or stated otherwise, $|q(x)|=1$ outside a compact region. It is clear that the exponential factor exhibits a
satisfactory continuum limit given by
\b \Sigma_j\s(p'_j-p_j)^2\s a/[\s q'^{-2}_j+q_j^{-2}\s]\ra \tint\s[p'(x)-p(x)]^2/
[q'(x)^{-2}+q(x)^{-2}]\,dx\;.\e
However, the continuum limit of the prefactor turns out to be
{\it identically zero} unless $|q'(x)|=|q(x)|$ for all $x$! This implies that $\ =0$ whenever
$|q'(x)|\not\equiv |q(x)|$ (as befits a {\it non$\s$}separable Hilbert space!), and thus the operator
representation with this fiducial vector has turned out to be highly singular. Stated otherwise, the affine field algebra fails to have an acceptable continuum limit for a strictly Gaussian fiducial vector with the indicated form.
As a {\bf second example}, suppose that
\b \eta(\p)=\<\p|\eta\>=N\frac{e^{\t -\half\s\omega \Sigma_j\s \p^2_j\s a}}{[\s\Pi_j\s|\p_j|\s]^{(J-1)/2J}}\;. \e
This unusual expression leads to a normalizable fiducial vector, i.e.,
\b N^2 \int \frac{e^{\t -\omega \Sigma_j\s \p^2_j\s a}}{[\s\Pi_j\s|\p_j|\s]^{(J-1)/J}}\,\Pi_j\s d\p_j=1\;,\e
the finiteness of which is clear.
For the second example, we find that
\b \<\p|p,q\>=N\s[\s\Pi_j\s|q_j|^{-1}\s]^{1/2J}\, \frac{e^{\t i\Sigma_k p_k\s \p_k\s a -\half\omega \Sigma_j\s (\p^2_j/q_j^2)\s a}} {[\s\Pi_j\s|\p_j|\s]^{(J-1)/2J}}\;, \e
and the coherent state overlap function $\ $ for the second example is given by
the integral
\b \ \hskip-1.4em&&=N^2\s[\Pi_k|q'_k|^{-1}\s|q_k |^{-1}]^{1/2J}\,\int e^{\t -i\Sigma_k(p'_k-p_k)\p_k\s a}\\ &&\hskip1em \times e^{\t -
\half\s\omega\s\Sigma_k\s \p_k^2\s[\s q'^{-2}_k+q^{-2}_k]\s a}\,\frac{1}{[\s\Pi_j\s|\p_j|\s]^{(J-1)/J}}
\,\Pi_kd\p_k\\
&&= N^2\s{\t\prod_k}\frac{[2\s|q'_k|^{-1}\s|q_k |^{-1}]^{1/2J}} {[\s q'^{-2}_k+q^{-2}_k\s]^{1/2J}}\\ &&\hskip1em\times\int\frac{e^{\t -i\Sigma_k(p'_k-p_k)\p_k\s a/[(\s q'^{-2}_k+q^{-2}_k\s)/2]^{1/2}}
e^{\t-\omega\s\Sigma_k\s \p^2_k\s a}} {[\s\Pi_j\s|\p_j|\s]^{(J-1)/J}}\,\Pi_kd\p_k\;. \e
In considering the continuum limit, we again focus on
continuous functions $p(x)$ and $\ln|q(x)|$ that have compact support, and we initially restrict attention to a finite overall interval $X=J\s a$. In this case, the new prefactor involves the $1/J^{\rm th}$ root of the previous prefactor, and this new version leads to an acceptable continuum limit. In particular,
it is possible to evaluate the prefactor itself exactly, which also equals the limited expression
$\ $, as
\b \ \hskip-1.5em&&= \lim_{a\ra0}{\t\prod_j}\frac{[2\s|q'_j|^{-1}\s|q_j |^{-1}]^{1/2J}} {[\s q'^{-2}_j+q^{-2}_j\s]^{1/2J}}\\
&& \hskip-2em= \exp[-(1/2X)\tint \{\s\ln|q'(x)|+\ln|q(x)|+\ln[(q'(x)^{-2}+q(x)^{-2})/2]\s\}\,dx]\;. \e
For arguments of compact support, it follows in the limit that $X\ra\infty$, that the prefactor
becomes {\it unity} and thus $\ =1$ for all arguments! While unusual, this result is
perfectly acceptable from a reproducing kernel point of view. (Indeed, this result is surely more acceptable
than the conclusion for the first example where $\ $ was identically zero except when
$|q'(x)|=|q(x)|$ for all $x$.) Moreover, for the second example, the expression $\ $ is well defined and is continuous in its labels in the continuum limit when $X$ is finite as well as in the
further limit that $X\ra\infty$. Thus, although we can not explicitly evaluate the coherent state
overlap function in the general case for the second example, it is clearly a continuous function of positive type suitable to be a reproducing kernel for the (separable) Hilbert space of interest.
\subsection*{Construction of field theoretic affine coherent states}
In light of the foregoing discussion, it is but a small step to introduce the affine coherent
states of interest in the study of nonrenormalizable scalar field models. In the rest of this section,
we introduce the ground state of our proposed models and define the associated affine coherent states
using that ground state as the fiducial vector; in the following section we give a brief discussion that
outlines the motivation for the particular choice of the ground state for these models.
Our analysis is aimed at a field theory model regularized by a spacetime lattice that has one
time dimension and $s$ space dimensions. Focusing on the spatial aspects, we put $L<\infty$ lattice points in each of the $s$ directions
with a lattice spacing of $a>0$. Each lattice point is labeled by a multi-integer $k=(k_1,k_2,\ldots,k_s)$,
where each $k_j\in\mathbb{Z}=\{0,\pm1,\pm2,\cdots\}$. The total number of lattice points in a spatial slice of the lattice is given by $N'\equiv L^s$, a factor which plays the role that $J$ played in examples one and two above. In the representation in which the field operator
is diagonal, there is a field associated with each lattice point, $\p_k$, much as in examples one and two above,
save for the fact that now the label $k$ is $s$-dimensional. In the present case, the ground state of the system
Hamiltonian is chosen (with $\hbar=1$) as
\b \Psi_0(\p)= M \frac{e^{\t -\half \Sigma'_{k,l}\p_k\s A_{k-l}\s \p_l\s a^{2s}-\half\s W(\p\s a^{(s-1)/2})}}{\Pi'_k[\s\Sigma'_l\s
J_{k,l}\s\p^2_l\s]^{(N'-1)/4N'}}\;, \e
where primes on products and sums signify they apply only to a given spatial slice.
In this expression, $A_{k-l}$ is a numerical matrix proportional to $a^{-(1+s)}$ and $J_{k,l}=1/(2s+1)$ for
$l=k$ and when $l$ is any of the $2s$ nearest neighbors of $k$ in the same spatial slice; otherwise $J_{k,l}=0$.
The role of $J_{k,l}$ is to provide a local average of field values in the sense that ${\o{\p^2_k}}\equiv
\Sigma_l\s J_{k,l}\s\p^2_l$, which means that the denominator exhibits an integrable singularity for the
ground state distribution even in the limit $N'\ra\infty$. The $A$ factor in the exponent and the $J$ factor in the denominator well
represent the functional form of the ground state for large and small field values,
respectively; the unspecified
function $W$ (discussed in the following section) is needed to modify intermediate field values.
The affine coherent states for this fiducial vector are given by
\b \<\p|p,q\>\hskip-1.5em&&=M\s\frac{ \Pi'_k[\s|q_k|^{-1/2}\s]}{\Pi'_k[\s\Sigma'_l\s
J_{k,l}\s(\p^2_l/q^2_l)\s]^{(N'-1)/4N'}}\\
&&\hskip-2em\times {e^{\t i\Sigma'_k p_k \p_k\s a^s-\half\Sigma'_{k,l}(\p_k/|q_k|)\s A_{k-l}\s
(\p_l/|q_l|)\s a^{2s}-\half\s W((\p/|q|)\s a^{(s-1)/2})}}\;, \e
which leads to the overlap expression
\b \ \hskip-1.5em&&=M^2\,\Pi'_k\{[\s|q'_k|^{-1/2}\s][\s|q_k|^{-1/2}\s]\}\,
\int \Pi'_k\s d\p_k\,e^{\t-i\Sigma'_k(p'_k-p_k)\p_k\s a}\\
&&\times{e^{\t -\half\Sigma'_{k,l}(\p_k/|q'_k|)\s A_{k-l}\s (\p_l/|q'_l|)\s a^{2s}
-\half\s W((\p/|q'|)\s a^{(s-1)/2})}}\\
&&\times {e^{\t -\half\Sigma'_{k,l}(\p_k/|q_k|)\s A_{k-l}\s (\p_l/|q_l|)\s
a^{2s}-\half\s W((\p/|q|)\s a^{(s-1)/2})}}\\
&&\times \frac{1}{{\Pi'_k[\s\Sigma'_l\s
J_{k,l}\s(\p^2_l/q'^2_l)\s]^{(N'-1)/4N'}}\;{\Pi'_k[\s\Sigma'_l\s
J_{k,l}\s(\p^2_l/q^2_l)\s]^{(N'-1)/4N'}}}\;. \e
In the present case, and when $p'_k=p_k$ for all $k$, observe that a simple change of variables does
not eliminate the $q'$ and $q$ variables from the integrand as was the case for example two. However, we do
expect that in the continuum limit, the prefactor will again cancel with the coefficients $q'$ and $q$
from the denominator factor for the simple reason that in the continuum limit, all neighboring $q'$ and
$q$ values will become equal to ensure a continuous label function $q'(x)$ and $q(x)$, and as such they will
emerge from the denominator and join the prefactor leading, in the case
of an infinite spatial volume, to a factor of unity, much as was the case for example two. Thus we expect the
continuum limit for an infinite volume also to be given by the expression
\b \ \hskip-1.5em&&=\lim_{a\ra0}\,M^2\s\int \Pi'_k\s d\p_k\,e^{\t-i\Sigma'_k(p'_k-p_k)\p_k\s a}\\
&&\hskip1em\times{e^{\t -\half\Sigma'_{k,l}(\p_k/|q'_k|)\s A_{k-l}\s (\p_l/|q'_l|)\s a^{2s}-\half\s W((\p/|q'|)\s a^{(s-1)/2})}}\\
&&\hskip1em\times {e^{\t -\half\Sigma'_{k,l}(\p_k/|q_k|)\s A_{k-l}\s (\p_l/|q_l|)\s a^{2s}-\half\s W((\p/|q|)\s a^{(s-1)/2})}}\\
&&\hskip1em\times\frac{1}{[\Pi'_k[\s\Sigma'_l\s J_{k,l}\s\p^2_l)\s]^{(N'-1)/2N'}}\;. \e
On the other hand, in a finite spatial volume, the preceding expression for the coherent state overlap function would be multiplied by the factor
\b e^{\t -(1/2V')\tint[\s\ln|q'(x)|+\ln|q(x)|\s]\,d^s\!x}\;,\e
where $V'\equiv N'\s a^s=(L\s a)^s<\infty$ is the volume of the spatial slice.
This concludes our discussion of the affine coherent states relevant for
the field theory models of interest.
\section{Brief Review of the Author's Approach to\\Nonrenormalzable Models}
We now outline the author's program to study the Euclidean-space formal functional integral given (for $\hbar=1$ and $n\ge5$) by
\b S(h)={\cal N}\int \exp(\s \tint\{h\p-\half[(\nabla\p)^2+m_0^2\s\p^2]-g_0\s\p^4\s\}\,d^n\!x)\,
{\cal D}\p\;. \label{e1}\e
A study of such an expression via conventional perturbation theory is not acceptable because of the
unlimited number of new counterterms needed such as $\p^6,\, \p^8,\, \p^{10},$ $\ldots$, etc., as well as
higher derivative terms as well. We need a radically new approach.
\subsubsection*{A lattice framework to build on}
We now sketch our alternative approach. First, we adopt a finite, periodic, hypercubic, spacetime
lattice with $L<\infty$ sites on a side, a lattice spacing of $a>0$, and lattice sites labeled by a
multi-integer $k=\{k_0,k_1,k_2,\ldots,\k_s\}$, $k_j\in\mathbb{Z}$, where $s=n-1$ is the number of spatial dimensions and $k_0$
denotes the Euclidean time direction, which will become the true time direction after a Wick rotation. Second,
we approximate the former equation by a lattice functional integral given by
\b N\!\int\exp(\s\Sigma_k\{h_k\p_k\s a^n-\half(\p_{k^*}-\p_k)^2\s a^{n-2}-\half\s m_0^2\s\p^2_k\s a^n-g_0\s \p^4_0\s a^n\s\}\s)\,\Pi_k\s d\p_k\;, \e
where $k^*$ denotes each of the $n$ next nearest neighbors in a positive sense and a summation over such points
is implicit. As it stands, this lattice expression represents a lattice cutoff of the formal continuum
functional integral. We need to introduce a counterterm into this expression to account for the needed
renormalizations that will appear in the continuum limit. Choosing counterterms on the basis of perturbation
theory is inappropriate, and we need an alternative principle to choose the counterterms.
\subsubsection*{Hard core interactions}
There are strong reasons to believe that a perturbation theory {\it about the free theory} does not hold true for nonrenormalizable models. In fact, the author has long argued \cite{kla17} that the nonlinear interaction term for nonrenormalizable models acts partially like a hard core, which, in a functional integral formulation, acts to project out certain field histories that would otherwise be allowed by the free theory alone. An argument favoring this explanation is provided by the Sobolev-type inequality
\b \{\tint \p(x)^4\,d^n\!x\}^{1/2}\le C\{\tint[\s(\nabla\p(x))^2+m^2\s\p(x)^2]\,d^n\!x\}\;, \e
which for $n\le4$ holds for $C=4/3$, while for $n\ge5$ holds only for $C=\infty$ \cite{book}. In the latter case, this
means that there are functions $\p(x)$ such that the left hand side diverges while the right hand side is finite. One example of such a function is given by
\b \p_{singular}(x)=|x|^{-p}\,e^{\t-x^2}\;,\hskip1.7cm n/4\le p < n/2-1\;. \e
The issues discussed above are even more self evident for a one dimensional system with the classical action
\b I=\tint[\s\half({\dot{x}}(t)^2-x(t)^2)-g\s x(t)^{-4}]\,dt\;, \e
for which, when $g>0$, the paths are unable to penetrate the barrier at $x=0$, and thus the family of theories for $g>0$ also exhibit a hard-core behavior and they do {\it not} converge to the free theory as $g\ra0$. This hard-core behavior holds in both the classical and quantum theories for this example.
\subsubsection*{Pseudofree models}
If the interacting theories do not pass to the free theory as the nonlinear coupling constant
reduces to zero, to what limit do they converge? We have introduced the term {\it pseudofree model}
to label the limit of the interacting theories when the coupling constant goes to zero. It is the pseudofree theory about which a perturbation exists, if one exists at all, and not about the usual free theory. Our initial goal
in understanding nonrenormalizable theories is to get a handle on the pseudofree theory, a theory that
is fundamentally different from the usual free theory.
As a result of the lack of any connection of the interacting theories with the usual free theory, our procedure to choose the proper counterterm for nonrenormalizable models will turn out to be
somewhat circuitous.
\subsubsection*{Role of sharp time averages}
Let us consider the average of powers of the expression
\b \Sigma_{k_0}\s F(\p,a)\,a \;,\e
where $F(\p,a)$ is a function of lattice points all at a fixed value of $k_0$,
in any lattice spacetime, based on the distribution generated by the exponential of the action, which
we denote by $\<\s(\,\cdot\,)\s\>$. We write the
average of the $p^{\rm th}$ power of this expression as
\b \<\s[\Sigma_{k_0}\,F(\p,a)\,a]^p\s\>=\Sigma_{k_{0_1},k_{0_2},\ldots,k_{p_0}}\,a^p\,\<\s F(\p_1,a)\,
F(\p_2,a)\,\cdots\,F(\p_p,a)\s\>\;,\e
where $\p_j$ here refers to the fact that ``$k_0=j$" in this term. A straightforward inequality leads to
\b &&|\,\<\s F(\p_1,a)\,F(\p_2,a)\,\cdots\,F(\p_p,a)\s\>\,|\no\\
&&\hskip1cm \le |\,\<\s [F(\p_1,a)]^p\s\>\,\<\s [F(\p_2,a)]^p\s\>\,\cdots\,\<\s [F(\p_p,a)]^p\s\>\,|^{1/p}\;, \e
which casts the problem into one at a sharp time. For sufficiently large $L$, it follows that this sharp
time expression may be given by
\b \<\s [F(\p,a)]^p\s\>=\int [F(\p,a)]^p\,\Psi_0(\p)^2\,\Pi'_k\s d\p_k\;, \e
where the integral is taken over fields at a fixed value of $k_0$, $\Psi_0(\p)^2$ denotes the ground state distribution, and $\Pi'_k$ denotes a product over the spatial lattice at a fixed value of
$k_0$. Thus we have arrived at the
important conclusion that if the sharp time average is finite, then the full spacetime average is also finite.
Both the inequality
noted above and the argument involving the ground state distribution may be found in
\cite{kla1,kla2,kla3}.
\subsubsection*{Choosing the ground state}
Attention now turns to finding the ground state, or more to the point, {\it choosing} the ground
state so that expressions we desire to be finite actually become finite. In particular, this remark means that the ground state is {\it tailored}
or {\it designed} so
that those quantities that are divergent in the usual perturbation theory are in fact rendered finite.
Once the ground state is chosen, one defines the associated lattice Hamiltonian for the system by means of
the expression
\b \H\equiv-\half\s\hbar^2\Sigma'_k\,\d^2/\d\p_k^2+\half\s\hbar^2\,[1/\Psi_0(\p)]\,\Sigma'_k\,
\d^2\s\Psi_0(\p)/\d\p_k^2\;, \e
where $\Sigma'_k$ denotes a sum over a spatial lattice at a fixed value of $k_0$. Note the appearance
of $\hbar^2$ in both factors. In turn, the lattice Hamiltonian readily leads to the lattice
action. In summary, we focus first on the desired modification of the ground state for the system,
which leads to the lattice Hamiltonian, and finally to the desired lattice action.
It has been argued that the ground state for a free system ($g_0=0$) and with no counterterm
is clearly a Gaussian. Moreover, such a function leads to divergences for several quantities of
interest, such as those expressions basic to a mass perturbation, namely, when $F(\p,a)=\Sigma'_k\s\p^2_k\s a^s$.
It has also been argued that the {\it source} of those divergences can be traced to a
specific, single factor when the integrals involved are reexpressed in hyper-spherical
coordinates, which are defined by
\b &&\p_k\equiv\kappa\s\eta_k\;,\hskip.5cm \Sigma'_k\p^2_k=\kappa^2\;,\hskip.5cm \Sigma'_k\eta^2_k=1\\
&&\hskip2.6em \kappa\ge0\;,\hskip.5cm -1\le\eta_k\le1\;. \e
As an example, consider a typical (Gaussian) integral of interest given by
\b &&K\int [\Sigma'_k\s\p^2_k\s a^s]^p\;e^{\t-\Sigma'_{k,l}\s\p_k\s A_{k-l}\s \p_l\,a^{2s}}\;\Pi'_kd\p_k\no\\
&&\hskip.6cm=2K\int\s \k^{2p}\s a^{sp}\;e^{\t-\k^2\Sigma'_{k,l}\eta_k\s A_{k-l}\s\eta_l\,s^{2s}}
\;\k^{(N'-1)}\,d\k\,\delta(1-\Sigma'_k\eta^2_k)\,\Pi'_kd\eta_k\;. \e
Here, $A_{k-l}\propto a^{-(1+s)}$ is a matrix responsible for the spatial gradient and other suitable quadratic terms in the lattice Hamiltonian, and $N'$ is the total number of lattice sites in a
spatial slice of the lattice. In the continuum limit, in which $a\ra0$, it follows that
$L\ra\infty$ in such a way, initially, that $V'\equiv(L\s a)^s=N'\s a^s<\infty$. Later, one may
extend the continuum limit procedure so that $V'\ra\infty$ as well. In short, in the
continuum limit, it follows that $N'\ra\infty$. It is not difficult to show that in the
foregoing integral, as displayed in hyper-spherical coordinates, it is the term
$N'$ in the measure factor $\k^{(N'-1)}$ that is {\it the very source of the divergences!}
If one could change that factor to one that remains {\it finite} in the continuum limit,
{\it the divergences would disappear!}
How is one to change a factor that has arisen from a bona-fide coordinate transformation?
The answer is: It cannot be done {\it directly}, but it can be done {\it indirectly}. The way
to do so is to choose a different, non-Gaussian ground state distribution, one that has roughly the form
\b K'\,\frac{1}{\k^{(N'-1)}}\,e^{\t-\Sigma'_{k,l}\,\p_k\s A_{k-l}\s \p_l\,a^{2s}}\;, \e
a form that has an additional factor in the denominator to cancel the measure factor $\k^{(N'-1)} $
altogether; in fact, it doesn't need to cancel it all, but it is a reasonable place to
begin. Such a ground state distribution arises from a Hamiltonian that is not quadratic
but contains another component that we identify as the desired counterterm. Finally,
that counterterm is taken over to the lattice action, and thereby we have determined
the counterterm in this convoluted manner!
There are many ways to choose the Hamiltonian so that the counterterm leads to a modification
of the ground state of the desired form. A large class of ground state modifications may be given by
\b M\frac{1}{\Pi'_k[\Sigma'_l\s J_{k,l}\s\p^2_l\s]^{(N'-1)/4\s N'}}
\;e^{\t-\half\Sigma'_{k,l}\,\p_k\s A_{k-l}\s\p_l-\half\s W(\p\s a^{(s-1)/2})} \e
for various choices of the constant coefficients $J_{k,l}$. For example, one may choose $J_{k,l}=
\delta_{k,l}$, and this is appropriate to discuss ultralocal models, which may be
described as relativistic models with their spatial gradient terms removed.
As we have shown elsewhere, such a choice leads to a Poisson ground state distribution,
which, although appropriate and correct for ultralocal models, is not desirable for
truly relativistic models. For relativistic models, on the other hand, it has been
proposed \cite{klaold} to choose the expression
\b J_{k,l}=\frac{1}{2 s+1}\delta_{k,l\s\in\s\{k\s\cup\s {\rm nn}\}}\;, \e
where the expression $l\in\{k\s\cup \s{\rm nn}\}$ means that $l=k$ and all the spatial nearest neighbors to $k\s$; $J_{k,l}=0$ elsewhere in the spatial slice, and $\Sigma'_l\s J_{k,l}=1$.
With $J_{k,l}$ so chosen, it follows that the Hamiltonian does not represent a local
interaction in the continuum limit due to cross terms coming from one derivative each of
the $A$ and $J$ terms. To fix that, the unwanted cross terms are removed from the lattice
Hamiltonian by means of a suitable, auxiliary term $W(\p\s a^{(s-1)/2})$ in the ground state that
effects mid-level field values. Readers who may be interested in what form
of counterterm for the lattice action such a modified ground state leads to should consult
\cite{kla1,kla2,kla3}.
This is not the place to debate the merits of the suggested proposal for the relativistic models
and their proposed ground state wave functions. Rather, in this paper, we accept the suggested
form of the ground state, and, consequently, we are then led to the
set of affine coherent states with the indicated
ground state chosen as the fiducial vector that was discussed in the previous section.
\section{Commentary}
The discussion of affine coherent states for a single degree of freedom emphasized the difficulty in
ensuring that $\ =q$ for all $q\in \mathbb{R}-0$. The solution involved a superposition
of disjoint states. When generalized to infinitely many degrees of freedom, this disjoint feature would have
led to an infinite number of unitarily inequivalent representations of the affine variables. To avoid this
situation, we instead accepted the requirement that $\ =q^2$, a compromise which eventually led to a suitable representation for infinitely many degrees of freedom. (It is interesting to observe
that this modification has some similarities with the Wilson construction for wavelets which also involves
symmetric fiducial vectors; e.g., see \cite{dau91}.)
For quantum gravity, which is the principal nonrenormalizable model of interest, it is noteworthy that the
classical field variable $g_{a\s b}(x)$, the $3\times3$ spatial metric, forms a positive-definite matrix and
thus it is possible to define affine coherent states for quantum gravity such that
\b \<\pi,g|\s \hat{g}_{a\s b}(x)\s|\pi,g\>=g_{a\s b}(x) \e
with an irreducible representation of the appropriate affine variables; see \cite{klaAA}. This fact
means that for the gravitational field we have the best situation we could hope for!
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{\it Ann. Phys.} {\bf 322}, 2569-2602 (2007).
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\end{thebibliography}
\end{document}
---------------0911220935845--
_++\th(-q)\s(-q)\s\

_-\;, \e
and we note that if we choose the parameters $B$ and $C$ such that $\

_+=1$, it follows that $\

_-=-1$, and we find that
\b \

_++\

_-+\