=p$ and $\

=q$, and thus the physical significance of the $c$-numbers $p$ and $q$ is that of {\it mean values in the coherent states}; they decidedly are {\it not} sharp eigenvalues for either $P$ or $Q$. Additionally, we assume that $|\eta\>$ is chosen so that $\<\eta|\s[\s P^2+Q^2\s]\s|\eta\>\ra0$ as $\hbar\ra0$, and that all necessary domain conditions hold. Next, recall that Schr\"odinger's equation may be derived from an abstract variational principle given---the subscript $Q$ stands for quantum--- by \bn I_Q=\tint\<\psi(t)|[\s i\hbar\d/\d t-\H(P,Q)\s]|\psi(t)\>\,dt\;, \en which under independent variations of $\<\psi(t)|$ and $|\psi(t)\>$ lead to Schr\"odinger's equation, \bn i\s\hbar\d\,|\psi(t)\>/\d t=\H(P,Q)\s|\psi(t)\>\;, \en along with its adjoint. This is the quantum side of this variational principle. Now consider a {\it macro}scopic experimenter limited in her study of a {\it micro}scopic one-degree of freedom system so that she is vastly restricted in the actual variations of $|\psi(t)\>$ that she is able to make. Indeed, let us assume she can only move the system to a different location and/or change its velocity by a constant amount [N.~B.: We say ``velocity'' noting that $p\equiv\d\s L({\dot q},q)/\d{\dot q}\equiv p({\dot q},q)$ (with $L$ being the Lagrangian), and thus the velocity ${\dot q}={\dot q}(p,q)$]. The experimentalist is unable to probe the system at the microscopic level so that she can not make any changes to the state vector $|\psi(t)\>$ other than those regarding location and velocity. In stating these restrictions, we have limited the experimenter to the variational set of states for which $|\psi(t)\>\ra|p(t),q(t)\>$ for some fixed fiducial vector, namely, we have limited her just to the set of coherent states. In this case it follows that \bn I_Q\ra I_{Q\,restricted}\equiv\int\

\,dt\;,\en which readily leads to \bn I_{Q\,restricted}\equiv\tint[p(t){\dot q}(t)-H(p(t),q(t))\s]\,dt\;, \en an expression that has all the appearance of being the action functional for a classical system, and the stationary variation of which, accounting for the proper boundary conditions, leads to the equations \bn {\dot q}(t)=\d\s H(p,q)/\d p(t)\;,\hskip2em {\dot p}(t)=-\d\s H(p,q)/\d q(t)\;, \en two equations that have all the appearance of being Hamilton's dynamical equations of motion. Let us examine this alleged relationship with a classical theory more closely. For one thing, the proposed Hamiltonian $H(p,q)$ is given by \bn H(p,q)\hskip-1.2em&&\equiv\

\no\\&&=\<\eta|\H(P+p,Q+q)|\eta\>\no\\
&&=\H(p,q)+\<\eta|[\s\H(P+p,Q+q)-\H(p,q)\s]|\eta\>\;. \label{you}\en
In the last line of this expression, and apart from the first term $\H(p,q)$, the second term is $O(\hbar;p,q)$ so that in the limit $\hbar\ra0$,
we find that $H(p,q)=\H(p,q)$. In short, the classical-looking system that has arisen from the restricted version
of the quantum action functional is {\it the very classical system associated with the given quantum
system}. There is only one additional point to clarify. Normally, we say for a classical system that the variables
$p$ and $q$ are the {\it exact values} of the momentum and position, implying that these values are absolutely sharp
values in the classical view. On the other hand, before $\hbar\ra0$, the meaning of $p$ and $q$ is that of
{\it mean values} and not of sharp values. In the world in which we all live, $\hbar$ is {\it not} zero and
therefore the classical and quantum systems must {\it coexist}. Thus it makes sense to assert that the
restricted quantum action functional which has the form of a classical system is in fact the correct
action functional for the
classical system associated with the given quantum system and that in fact---still referring to the real
world---it is consistent to assume that the true classical variables are mean values of some other variables.
After all, who has ever measured the classical values of $p$ and $q$ to, say, $10^{137}$ decimal places
to verify that they really are the exact momentum and position as hypothesized? In summary, we are
led to propose that {\it the restricted variational form of the quantum action
functional is the true classical action functional and its limited variation leads to the true
classical equations of motion}.
The quantum corrections arising from the second term in ({\ref{you})
may vary depending on different choices of the fiducial vector $|\eta\>$; this property simply reflects the fact
that the restricted action functional involves a projection from a larger space, and different projections
can lead to differing elements. Of course, these quantum corrections
are generally extremely tiny and almost always can be neglected; when that is the case, we may say that
the resultant equations of motion are strictly classical with no dependence
on $\hbar$ whatsoever. However, as one may imagine, there are some exceptional
systems where these corrections play a significant qualitative role. Even when that is the case, these
terms may be just nuisance factors that can be safely ignored, or they may act to change the physics
in significant ways.\v
{\it Affine Variables:} For this example, we choose a different set of basic operators, namely $Q$ and $D$,
where $[\s Q,D\s]=i\hbar Q$, which is called the {\it affine commutation relation}. This equation is
surprisingly close to the canonical commutation relation when we observe that $[ Q,P]Q=i\hbar Q$, which
on bringing the extra $Q$ on the left side inside the commutator, leads to $[Q,D]=i\hbar Q$, with $D\equiv
\half(QP+PQ)$. Clearly, $D$ has the dimensions of $\hbar$. From a representation point of view,
there is, up to unitary equivalence, just one
inequivalent representation of the canonical commutation relation for self-adjoint operators, while there are
three inequivalent
representations of the affine commutation relation for self-adjoint operators, distinguished by the fact
that $Q>0$, $Q<0$, and $Q=0$ (strictly speaking, all these uniqueness results apply to unitary operators generated by the self-adjoint operators). For classical systems for which $q>0$ is the physical realm---such as cases where the potential is of the form $V(q)=c|q|^{-\beta}\,,\beta>0$---it is natural
to use affine kinematical variables with elements $D$ and $Q>0$ since they can both be realized as self-adjoint operators (in contrast to canonical operators). For systems where $-\infty \hskip-1.3em&&=q^2\<\eta|Q^2|\eta\>=q^2\no\\
\ \hskip-1.3em&&=\<\eta|(D+pqQ^2)|\eta\>=pq\no\\
\ \hskip-1.3em&&=\<\eta|(P/|q|+p|q|Q/q)^2|\eta\>=p^2+\ /q^2\;,\en
which implies that the physical meaning of $q^2$ and $pq$ are
the mean values of $Q^2$ and $D$, respectively. Let us now apply these states.
Just like the canonical case, we assert that the quantum action functional for affine variables is given by
\bn I_Q=\tint\<\psi(t)|[i\hbar\d/\d t-\H(P,Q)]|\psi(t)\>\,dt\;, \en
and its stationary variation leads to Schr\"odinger's equation
\bn i\hbar\d\s|\psi(t)\>/\d t=\H(P,Q)\s|\psi(t)\> \en
and its adjoint. We focus on Hamiltonians such that
\bn \H(P,Q)=\half m^{-1}P^2+V(Q)\equiv \half m^{-1}P^2+\Sigma_{j=0}^J\s c_j\s Q^{2j} \;. \en
As with the canonical case, we now restrict our variations and consider
\bn I_{Q\s restricted}=\tint\ \,dt \;,\en
now for the affine coherent states. To proceed further, we first note that
\bn i\hbar\ =\<\eta|\s[-\half({\dot{p/q}})\s q^2Q^2+(\dot{\ln|q|})D]|\eta\>=
\half(p{\dot q}-q{\dot p})\;.\en
For the Hamiltonian part, we observe that
\bn H(p,q)\hskip-1.3em&&\equiv\ \no\\
&&=\<\eta|[\half m^{-1}(P/|q|+p|q|Q/q)^2+\Sigma_{j=0}^J\s c_j\s q^{2j} Q^{2j})]|\eta\>\no\\
&&=\half { m}^{-1} p^2+\Sigma_{j=0}^J\s c_j\s \ \s q^{-2}\no\\
&&\equiv\half { m}^{-1} p^2+\Sigma_{j=0}^J\s c'_j\s q^{2j}+\half m^{-1}\ \s q^{-2}\;, \en
where we have set
$c'_j\equiv c_j\s \ /\hbar^2$, which, in the present units, is dimensionless.
Stationary variation of this action with respect to $p$ and $q$, subject to appropriate
boundary conditions, leads to the associated Hamilton equations of motion. Apart from
typically small $\hbar$ changes of the various constants, there appears an additional,
unexpected force (proportional to $\hbar^2$) that prohibits solutions from crossing
$q=0$. This situation qualitatively changes the solutions from those based on the true
classical theory in which we let $\hbar\ra0$ before deriving the equations of motion.
However, if we are permitted to take the strict classical limit $\hbar\ra0$ before deriving
the equations of motion by the variational principle, then our classical theory
would lead to all the expected solutions.
\subsection{Coherent States for Scalar Fields}
By generalizing the one-dimensional example above based on affine coherent states, we now study the coherent states
for covariant scalar fields; in this effort we are partially guided by an analogous story for ultralocal
fields that appears in \cite{book} as well as a preliminary study of these questions
in \cite{hindawi}. We start with the lattice-regularized, covariant pseudofree theory, and we deliberately
choose the ground state for this model as the fiducial vector. Thus we are led to consider (for $\hbar=1$)
the set of states
\bn &&\hskip-3em\<\p|p,q\>=K\s \Pi'_k \s|q_k|^{-1/2}
\frac{e^{\t i\Sigma'_k (p_k/2q_k)\s \p_k^2\s a^s
-\half\Sigma'_{k,l}(\p_k/|q_k|)\s A_{k-l}\s(\p_l/|q_l|)\s a^{2s}}}
{\Pi'_k[\s\Sigma'_l\s J_{k,l}\s(\p_l^2/q_l^2)\s]^{(1-2ba^s)/4}}\no\\
&&\hskip13em\times e^{\t -\half\s W(\s(\p/|q|)\s a^{(s-1)/2}/\hbar^{1/2})}\;. \en
The coherent state overlap function $\ $ is given by
\bn \ =\int \ \<\p|p,q\>\,\Pi'd\p_k\;, \en
which is represented by
\bn &&\hskip-3em\ =K^2\s\Pi'_k(|q'_k|\s|q_k|)^{-1/2}\no\\
&&\hskip2em\times\int \frac{e^{\t -i\Sigma'_k (p'_k/2q'_k)\s \p_k^2\s a^s
-\half\Sigma'_{k,l}(\p_k/|q'_k|)\s A_{k-l}\s(\p_l/|q'_l|)\s a^{2s}}}
{\Pi'_k[\s\Sigma'_l\s J_{k,l}\s(\p_l^2/{q'_l}^2)\s]^{(1-2ba^s)/4}}\no\\
&&\hskip9em\times e^{\t -\half\s W(\s(\p/|q'|)\s a^{(s-1)/2}/\hbar^{1/2})}\no\\
&&\hskip3em\times\frac{e^{\t i\Sigma'_k (p_k/2q_k)\s \p_k^2\s a^s
-\half\Sigma'_{k,l}(\p_k/|q_k|)\s A_{k-l}\s(\p_l/|q_l|)\s a^{2s}}}
{\Pi'_k[\s\Sigma'_l\s J_{k,l}\s(\p_l^2/q_l^2)\s]^{(1-2ba^s)/4}}\no\\
&&\hskip9em\times e^{\t -\half\s W(\s(\p/|q|)\s a^{(s-1)/2}/\hbar^{1/2})} \,\Pi'd\p_k\;. \en
As we approach the continuum limit in this expression, and restricting attention to
continuous functions $p_k\ra p({\bf x})$ and $q_k\ra q({\bf x})$, it follows that
\bn &&\hskip-3em\ =K^2\s\Pi'_k(|q'_k|\s|q_k|)^{-ba^s}\no\\
&&\hskip2em\times\int \frac{e^{\t -i\Sigma'_k (p'_k/2q'_k)\s \p_k^2\s a^s
-\half\Sigma'_{k,l}(\p_k/|q'_k|)\s A_{k-l}\s(\p_l/|q'_l|)\s a^{2s}}}
{\Pi'_k\s[\Sigma'_l\s J_{k,l}\s\p_l^2\s]^{(1-2ba^s)/4}}\no\\
&&\hskip9em\times e^{\t -\half\s W(\s(\p/|q'|)\s a^{(s-1)/2}/\hbar^{1/2})}\no\\
&&\hskip3.5em\times\frac{
{e^{\t i\Sigma'_k (p_k/2q_k)\s \p_k^2\s a^s
-\half\Sigma'_{k,l}(\p_k/|q_k|)\s A_{k-l}\s(\p_l/|q_l|)\s a^{2s}}}}
{\Pi'_k[\s\Sigma'_l\s J_{k,l}\s\p_l^2\s]^{(1-2ba^s)/4}}\no\\
&&\hskip9em\times e^{\t -\half\s W(\s(\p/|q'|)\s a^{(s-1)/2}/\hbar^{1/2})}\,\Pi'_kd\p_k\;,\label{eee}\en
where we taken advantage of the fact that for continuous functions we can bring the
$q'$ and $q$ factors out of the denominators in the former expression.
Although we can not write an analytic expression for the entire continuum limit of this
expression, we note that the new prefactor, $\Pi'_k(|q'_k|\s|q_k|)^{-ba^s}$, has a
continuum limit given by
\bn \Pi'_k(|q'_k|\s|q_k|)^{-ba^s}\ra e^{\t-b\tint[\s\ln|q'({\bf x})|+\ln|q({\bf x})|]\,d{\bf x}}\;. \en
{\it This meaningful partial result for the continuum limit holds only for the affine coherent states;
it would decidedly not have led to meaningful results for canonical coherent states \cite{hindawi}. In other words, measure mashing has had the effect of changing a canonical system into an affine system!} This
result also favors the choice $R=2ba^sN'$ as it is compatible with an infinite spacial volume.
The rest of the coherent state overlap integral in (\ref{eee}) is too involved to be simplified, but if we ask only for
$\ $ then some progress can be made. In this case we have
\bn \ \hskip-1.3em&&=K^2\Pi'_k\s|q_k|^{-1}\no\\
&&\hskip1em\times\int \frac{e^{\t -i\Sigma'_k (p'_k/2q_k)\s \p_k^2\s a^s
-\half\Sigma'_{k,l}(\p_k/|q_k|)\s A_{k-l}\s(\p_l/|q_l|)\s a^{2s}}}
{\Pi'_k[\s\Sigma'_l\s J_{k,l}\s(\p_l^2/{q_l}^2)\s]^{(1-2ba^s)/4}}\no\\
&&\hskip5em\times e^{\t -W(\s(\p/|q|)\s a^{(s-1)/2}/\hbar)/2}\no\\
&&\hskip3em\times\frac{e^{\t i\Sigma'_k (p_k/2q_k)\s \p_k^2\s a^s
-\half\Sigma'_{k,l}(\p_k/|q_k|)\s A_{k-l}\s(\p_l/|q_l|)\s a^{2s}}}
{\Pi'_k[\s\Sigma'_l\s J_{k,l}\s(\p_l^2/q_l^2)\s]^{(1-2ba^s)/4}}\no\\
&&\hskip5em\times e^{\t -W(\s(\p/|q|)\s a^{(s-1)/2}/\hbar)/2} \,\Pi'd\p_k\no\\
&&=K^2 \int \frac{e^{\t -i\Sigma'_k ((p'_k-p_k)q_k/2)\s \p_k^2\s a^s
-\Sigma'_{k,l}\p_k\s A_{k-l}\s\p_l\s a^{2s}}}
{\Pi'_k[\s\Sigma'_l\s J_{k,l}\s\p_l^2\s]^{(1-2ba^s)/2}}\no\\
&& \hskip5em\times e^{\t -W(\s\p\s\s a^{(s-1)/2}/\hbar)}\,\Pi'd\p_k \;.\label{e77} \en
In this form, we see that $\ =\ $,
from which we learn
that the expression $\ \equiv \ $ contains the same information
as contained in $\ $; we also learn that $\ $ achieves a
meaningful continuum limit provided that $\ $ already achieves one, and it is
clear from the form of (\ref{e77}) that such a continuum limit holds.
\subsection{Quantum/Classical Connection}
In the same spirit as the single affine degree of freedom, we seek to connect the classical
action functional with a restricted form of the quantum action functional. Thus we proceed
directly to the expression
\bn I_{Q\s restricted}=\int[\ \,dt\;, \label{www}\en
based on the Hamiltonian operator $\H$ for the lattice covariant pseudofree model and
the associated coherent states. Following the calculation of the affine model, we are led to
the expression (with $\hbar=1$, ${\hat\p}_k=\p_k$, and assuming units are chosen so that $\<\eta|\p_k^2|\eta\>=\ell^2=1$) for (\ref{www}) given by
\bn I_{Q\s restricted}\hskip-1.3em&&=\int\{\s\half\Sigma'_k (p_k{\dot q}_k-q_k{\dot p}_k)\s a^s-\<\eta|\s [\s\half \Sigma'_k(P_k/|q_k|+p_k|q_k|\p_k/q_k)^2\s a^s\no\\
&&\hskip4em +\half\Sigma'_k(|q_{k^*}|\p_{k^*}-|q_k|\p_k)^2\s a^{s-2}+\half s(L^{-2s}a^{-2})\Sigma'_kq_k^2\p_k^2\s a^s
\no\\&&\hskip4em +\half\Sigma'_k\F_k(|q|\s\p)\s a^s\s]|\eta\>-E_{pf}\}\,dt\no\\
&& =\int\{\half\Sigma'_k(p_k{\dot q}_k-q_k{\dot p}_k)\s a^s -\half\Sigma'_k( p_k^2 +\ \s q_k^{-2})\s a^s\no\\ &&\hskip3em -\half\Sigma'_k\Sigma'_{k^*}\<\s(|q_{k^*}|\p_{k^*}-|q_k|\p_k)^2\>\s a^{s-2}-\half s(L^{-2s}a^{-2})\s \Sigma'_kq_k^2\s a^s
\no\\&&\hskip3em -\half\Sigma'_k\<\F_k(\p)\>\s q_k^{-2}\s a^s-E_{pf}\}\,dt \;, \en
where in the last line we have made the sum over $k^*$ explicit.
This equation has all the expected ingredients apart from the rather unusual term
$\half\Sigma'_k\Sigma'_{k^*}\<\s(|q_{k^*}|\p_{k^*}-|q_k|\p_k)^2\>a^{s-2}$ that we investigate next. We observe that
\bn &&\<\s(|q_{k^*}|\p_{k^*}-|q_k|\p_k)^2\>\no\\
&&\hskip3em=q_{k^*}^2\<(\p_{k^*}-\p_k)^2\>+(|q_{k^*}|
-|q_k|)^2\<\p_k^2\>\no\\
&&\hskip8em+[(q_{k^*}^2-q^2_k)+(|q_{k^*}|-|q_k|)^2]\<(\p_{k^*}-\p_k)\p_k\>\no\\
&&\hskip3em\equiv C_1\s q_{k^*}^2+(|q_{k^*}| -|q_k|)^2+C_2\s[(q_{k^*}^2-q^2_k)+(|q_{k^*}|-|q_k|)^2]\;,\no\\ \en
where $C_j$, $j=1,2$, are constants due to translation invariance of the ground state.
The first term contributes to the mass, the second term and the latter part of the third term
contribute to the lattice derivative, $(1+C_2)(q_{k^*}-q_k)^2$, since for continuous functions
the sign of $q_{k^*}$ and $q_k$ are identical except for the possible exception when they are both infinitesimal, in which
case they make a negligible contribution to the sum.
Moreover, in the continuum limit $C_2\ra0$, and compared to unity it may be omitted from that factor.
The initial part of the last factor sums to
zero thanks to the periodic boundary conditions. We observe that $\ \hskip-1.3em&&=\<\s[\s\half\Sigma'_k(P_k+p_kQ_k)^2a^s+V(Q)-E_{pf}\s]\s\>\no\\
&&=\half\Sigma'_k \s p^2_k\s a^s+\<[\s\half\Sigma'_k P_k^2\s a^s+V(Q)-E_{pf}\s]\s\>\no\\
&&=\half\Sigma'_k\s p^2_k\s a^s\;, \en
meaning that the final expression for the pseudofree
restricted action (with $\hbar=1$) is given by
\bn I_{Q\s restricted}\hskip-1.3em&&=\int\{\half\Sigma'_k(p_k{\dot q}_k-q_k{\dot p}_k)\s a^s -\half\Sigma'_k[ p_k^2 +C_3\s (q_k^{-2}-1)]\s a^s\no\\ &&\hskip1em -\half\Sigma'_k\Sigma'_{k^*}\s(q_{k^*}-q_k)^2\s a^{s-2}-\half s(L^{-2s}a^{-2})\s \Sigma'_k(q_k^2-1)\s a^s
\no\\&&\hskip1em -\half\Sigma'_k C_4\s (q_k^{-2}-1)\s a^s\s\}\,dt\;. \en
If instead of the pseudofree model we dealt with an
interacting model, then the ground state would have a different form as would the form of the Hamiltonian operator; the restricted action would reflect this difference by adding a nonlinear term
to the functional form of the action.
\subsection{Coherent States for Multi-Component Scalar \\(Higgs-like) Models}
To deal with multi-component scalar fields we generalize the single-component scalar field introduced above. In particular, the field
variables $Q_k\equiv\p_k\ra Q_{k,\a}\equiv\p_{k,\a}$ as well as $P_k\ra P_{k,\a}$ defined so that
\bn [Q_{k,\a},P_{l,\beta}]=i\hbar\s a^{-s}\s\delta_{k,l}\s\delta_{\a,\beta}\;. \en
We also introduce $D_{k,\a}=\half (P_{k,\a}\s Q_{k,\a}+Q_{k,\a}\s P_{k,\a})$ (no summation)
which has the commutation properties
\bn [Q_{k,\a},D_{l,\beta}]=i\hbar a^{-s}\s\delta_{k,l}\s\delta_{\a,\beta}\s Q_{k,\a}\;.\en
For the coherent states we adopt
\bn |p,q\>\equiv e^{\t-i\Sigma'_{k,\a}\s p_{k,\a}\s Q^2_{k,\a}/(2q_{k,\a}\hbar)\s\s a^s}
\,e^{\t i\Sigma'_{k,\a}\ln (|q_{k,\a}|/\ell)\,D_{k,\a}\s a^s/\hbar}\,|\eta\>\;. \en
Hereafter, the study of the multi-component field case follows rather closely that of the
single-component case.
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\end{document}
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\equiv e^{\t ip\s Q^2/2\s q\s\hbar}\,e^{\t i\ln(|q|/\ell)D/\hbar}\s|\eta\>\;;\en
here $\ell>0$ is a fixed factor to cancel the dimensions of $q$, and the fiducial unit vector $|\eta\>$ is
chosen to be symmetric in the sense $\<-x|\eta\>=\

=\ell^2$; hereafter, without loss of
generality, we shall assume units are chosen so that $\ell=1$. Observe
that in this formulation the coherent states $|-p,-q\>=|p,q\>$. We note further that
\bn e^{\t -i\ln(|q|)D/\hbar}\s Q\s e^{\t i\ln(|q|)D/\hbar}\hskip-1.3em&&=|q|\s Q\;, \no\\
e^{\t -i\ln(|q|)D/\hbar}\s P\s e^{\t i\ln(|q|)D/\hbar}\hskip-1.3em&&= P/|q|\;, \no\\
e^{\t -ip\s Q^2/2q\hbar}\s D\s e^{\t ip\s Q^2/2q\hbar}\hskip-1.3em&&= D+pQ^2/q \no\\
e^{\t -ip\s Q^2/2q\hbar}\s P\s e^{\t ip\s Q^2/2q\hbar}\hskip-1.3em&&= P+pQ/q \;. \en
and thus it follows that
\bn \

\s q^{2j}+\half m^{-1}\

= c_j+O(\hbar)$. In summary, it follows that
\bn I_{Q restricted}=\tint[\half(p{\dot q}-q{\dot p})-\half {m}^{-1}p^2-
\Sigma_{j=0}^J\s c'_j\s q^{2j}-\half m^{-1}{\tilde c}_{-1}\s\hbar^2\s q^{-2}\s]\,dt\;, \label{e62}\en
where we have also set ${\tilde c}_{-1}\equiv \