text:
\section{Introduction}

As was remarked in Ref. \cite{Gon01}, there exists a natural way of
building meson spectroscopy and relativistic models of mesons which might be
based on the exact solutions of the SU(3)-Yang-Mills equations modelling
quark confinement, the so-called confining solutions, and also on the
corresponding modulo square integrable solutions of the Dirac equation in those
confining SU(3)-fields. The given approach is the direct
consequence of the {\it relativistic} QCD (quantum chromodynamics) Lagrangian
since the mentioned Yang-Mills and Dirac equations are derived just from the
latter one. In Ref. \cite{Gon01} both the types of solutions were obtained and
then in Ref. \cite{Gon03} they were successfully
applied to the description of the charmonium spectrum. In its turn, the mentioned
description pointed out the linear confinement to be (classically) governed by
the magnetic colour field linear in $r$, the distance between quarks.

 The results of Refs. \cite{{Gon01},{Gon03}} suggested the following mechanism
of confinement to occur within the framework of QCD (at any rate, for mesons
and quarkonia). The gluon exchange between quarks is realized in such a way
that at large distances it leads to the confining SU(3)-field which may be
considered classically (the gluon concentration becomes huge and gluons form
the boson condensate -- a classical field) and is
a {\em nonperturbative} solution of the SU(3)-Yang-Mills
equations. Under the circumstances mesons are the {\em relativistic bound
states} described by the corresponding wave functions -- {\em nonperturbative}
modulo square integrable solutions of the Dirac equation in this
confining SU(3)-field. For each meson there
exists its own set of real constants (for more details see Section 2)
$a_j, A_j, b_j, B_j$ parametrizing the mentioned confining gluon
field (the gluon condensate) and the corresponding wave
functions while the latter ones also depend on $\mu_0$, the reduced
mass of the current masses of quarks forming
meson. It is clear that constants $a_j, A_j, b_j, B_j,\mu_0$
should be extracted from experimental data. Further application of the
approach to quarkonia (charmonium and bottomonium) in
Refs. \cite{{GC03},{GB04}} confirmed the mentioned picture of linear
confinement and, in particular, gave possibility to estimate the above gluon
concentrations.

The aim of the present Letter is to specify a number of features in the above
confinement scenario, in particular, to show that the confining solutions
found in Ref. \cite{Gon01} and used in Refs. \cite{{Gon03},{GC03},{GB04}} are
in essence the unique ones.

     Further we shall deal with the metric of
the flat Minkowski spacetime $M$ that we write down
[using the ordinary set of local spherical
($r,\vartheta,\varphi$)
or rectangular (Cartesian) ($x,y,z$) coordinates
for spatial part in the forms
$$ds^2=g_{\mu\nu}dx^\mu\otimes dx^\nu\equiv
dt^2-dr^2-r^2(d\vartheta^2+\sin^2\vartheta d\varphi^2)\>, \eqno(1)$$
or
$$ds^2=g_{\mu\nu}dx^\mu\otimes dx^\nu\equiv
dt^2-dx^2-dy^2-dz^2\>, \eqno(2)$$
so the components $g_{\mu\nu}$ take different values depending on the
choice of coordinates.
Besides we have $\delta=|\det(g_{\mu\nu})|=(r^2\sin\vartheta)^2$
in spherical coordinates and the exterior differential $d=\partial_t dt+
\partial_xdx+\partial_ydy+\partial_zdz$ or $d=\partial_t dt+\partial_r dr+
\partial_\vartheta d\vartheta+\partial_\varphi d\varphi$ in the corresponding
coordinates and we denote 3-dimensional vectors by bold font.

  Throughout the paper we employ the system of units with $\hbar=c=1$,
unless explicitly stated otherwise.

\section{Uniqueness of the confining solutions}
If $A=A_\mu dx^\mu=A^a_\mu \lambda_adx^\mu$ is a SU(3)-connection in the
(trivial) three-dimensional bundle $\xi$ over the Minkowski spacetime,
where $\lambda_c$ are the known Gell-Mann matrices, then
we are interested in the confining solutions $A$ of the SU(3)-Yang-Mills
equations
$$d\ast F= g(\ast F\wedge A - A\wedge\ast F) \>,\eqno(3)$$
while the curvature matrix (field strentgh)
for the $\xi$-bundle is
$F=dA+gA\wedge A= F^a_{\mu\nu}\lambda_adx^\mu\wedge dx^\nu$ and $\ast$
means the Hodge star
operator conforming to metric (1), $g$ is a gauge coupling constant.

The confining solutions were defined in Ref. \cite{Gon01} as the
spherically symmetric solutions of the Yang-Mills
equations (3) containing only the components of the
SU($3$)-field which are Coulomb-like or linear in $r$. Additionally
we shall impose the Lorentz condition on the sought solutions.
The latter condition is necessary for
quantizing the gauge fields consistently within the framework of perturbation
theory (see, e. g. Ref. \cite{Ryd85}), so we should impose the given condition
that can be written
in the form ${\rm div}(A)=0$, where the divergence of the Lie algebra valued
1-form $A=A_\mu dx^\mu=A^a_\mu \lambda_adx^\mu$ is defined by the relation (see, e. g.
Refs. \cite{Bes87})
$${\rm div}(A)=\frac{1}{\sqrt{\delta}}\partial_\mu(\sqrt{\delta}g^{\mu\nu}
A_\nu)\>.\eqno(4)$$
It should be noted the following. If writing down the Yang-Mills equations (3)
in components then we
shall be drowned in a sea of indices which will strongly hamper searching for one
or another ansatz and make it to be practically immense. Using the Hodge star
operator as well as the rules of external calculus makes the problem to be
quite foreseeable and quickly leads to the aim. Let us remind, therefore, the
properties of Hodge star operator we shall need (for more details see
Refs. \cite{Bes87}).
\subsection{Hodge star operator $*$ on Minkowski spacetime
in spherical coordinates}
Let $M$ is a smooth manifold of dimension $n$ so we denote an
algebra of smooth functions on $M$ as $F(M)$. In a standard way the spaces
of smooth differential $p$-forms $\Lambda^p(M)$ ($0\le p\le n$) are defined
over $M$ as modules over $F(M)$.
If a (pseudo)riemannian metric $G=ds^2=g_{\mu\nu}dx^\mu\otimes dx^\nu$ is given
on $M$ in local coordinates
$x=(x^\mu)$ then $G$ can naturally be continued on spaces $\Lambda^p(M)$
by relation
$$G(\alpha,\beta)={\rm det}\{G(\alpha_i,\beta_j)\}    \eqno(5)$$
for $\alpha=\alpha_1\wedge\alpha_2...\wedge\alpha_p$,
$\beta=\beta_1\wedge\beta_2...\wedge\beta_p$, where for 1-forms
$\alpha_i=\alpha_\mu^{(i)}dx^\mu$, $\beta_j=\beta_\nu^{(j)}dx^\nu$ we have
$G(\alpha_i,\beta_j)=g^{\mu\nu}\alpha_\mu^{(i)}\beta_\nu^{(j)}$ with the
Cartan's wedge (external) product $\wedge$. Under the circumstances the Hodge
star operator $*$: $\Lambda^p(M)\to\Lambda^{n-p}(M)$ is defined for any
$\alpha\in\Lambda^p(M)$ by
$$\alpha\wedge(*\alpha)=G(\alpha,\alpha)\omega_g\> \eqno(6)$$
with the volume $n$-form
$\omega_g=\sqrt{|\det(g_{\mu\nu})|}dx^1\wedge...dx^n$.
For example, for 2-forms $F=F_{\mu\nu}dx^\mu\wedge dx^{\nu}$ we have
$$
F\wedge\ast F=(g^{\mu\alpha}g^{\nu\beta}-g^{\mu\beta}g^{\nu\alpha})
F_{\mu\nu}F_{\alpha\beta}
\sqrt{\delta}\,dx^1\wedge dx^2\cdots\wedge dx^n,\,\mu<\nu,\,\alpha<\beta
\>\eqno(7)
$$
with $\delta=|\det(g_{\mu\nu})|$.
If $s$ is the number
of (-1) in a canonical presentation of quadratic form $G$ then two most
important properties of $*$ for us are
$$ *^2=(-1)^{p(n-p)+s}\>,\eqno(8)$$
$$ *(f_1\alpha_1+f_2\alpha_2)=f_1(*\alpha_1)+f_2(*\alpha_2)\> \eqno(9)$$
for any $f_1, f_2 \in F(M)$, $\alpha_1, \alpha_2 \in\Lambda^p(M)$, i. e.,
$*$ is a $F(M)$-linear operator. Due to (9) for description
of $*$-action in local coordinates it is enough to specify $*$-action on
the basis elements of $\Lambda^p(M)$, i. e. on the forms
$dx^{i_1}\wedge dx^{i_2}\wedge...\wedge dx^{i_p}$ with $i_1<i_2<...<i_p$
whose number is equal to $C_n^p=\frac{n!}{(n-p)!p!}$.

The most important case of $M$ in the given paper is the Minkowski spacetime
with local coordinates $t, r, \vartheta, \varphi$, where
$r, \vartheta, \varphi$ stand for spherical coordinates on spatial part of $M$.
The metric is given by (1) and we shall obtain the $*$-action on the basis
differential forms according to (6)
$$\ast dt=r^2\sin{\vartheta}dr\wedge d\vartheta\wedge d\varphi,\>
\ast dr=r^2\sin{\vartheta}dt\wedge d\vartheta\wedge d\varphi,\>$$
$$\ast d\vartheta=-r\sin{\vartheta}dt\wedge dr\wedge d\varphi,\>
\ast d\varphi=rdt\wedge dr\wedge d\vartheta,\>$$
$$\ast(dt\wedge dr)=-r^2\sin\vartheta d\vartheta\wedge d\varphi\>,
\ast(dt\wedge d\vartheta)=\sin\vartheta dr\wedge d\varphi\>,$$
$$\ast(dt\wedge d\varphi)=-\frac{1}{\sin\vartheta}dr\wedge d\vartheta\>,
\ast(dr\wedge d\vartheta)=\sin\vartheta dt\wedge d\varphi\>,$$
$$\ast(dr\wedge d\varphi)=-\frac{1}{\sin\vartheta}dt\wedge d\vartheta\>,
\ast(d\vartheta\wedge d\varphi)=\frac{1}{r^2\sin\vartheta}dt\wedge dr\>,$$
$$\ast(dt\wedge dr\wedge d\vartheta)=\frac{1}{r}d\varphi\>,
\ast(dt\wedge dr\wedge d\varphi)=-\frac{1}{r\sin{\vartheta}}d\vartheta,\>$$
$$\ast(dt\wedge d\vartheta\wedge d\varphi)=\frac{1}{r^2\sin{\vartheta}}dr,\>
\ast(dr\wedge d\vartheta\wedge d\varphi)=
\frac{1}{r^2\sin{\vartheta}}dt,\>
\eqno(10)$$
so that on 2-forms $\ast^2=-1$, as should be in accordance with
(8).

At last it should be noted that all the above is easily over linearity
continued on the matrix-valued differential forms (see, e. g.,
Ref. \cite{Car}), i. e., on the arbitrary
linear combinations of forms
$a_{i_1i_2...i_p}dx^{i_1}\wedge dx^{i_2}\wedge...\wedge dx^{i_p}$,
where coefficients $a_{i_1i_2...i_p}$ belong to some space of matrices $V$,
for example, a SU($3$)-Lie algebra. But now the
Cartan's wedge (external) product $\wedge$ should be understood as
product of matrices with elements consisting of usual (scalar) differential
forms. In the SU(3)-case, if $T_a$ are
matrices of
generators of the SU($3$)-Lie algebra in $3$-dimensional representation, we
continue the above scalar product $G$ on the SU($3$)-Lie algebra valued 1-forms
$A=A^a_\mu T_adx^\mu$ and $B=B^b_\nu T_bdx^\nu$ by the relation
$$G(A,B)=g^{\mu\nu}A^a_\mu B^b_\nu{\rm Tr}(T_aT_b)\,,\eqno(11)$$
where Tr signifies the trace of a matrix,
and, on linearity with the help of (5), $G$ can be continued
over any SU($3$)-Lie algebra valued forms.
\subsection{The confining solutions}
Let us for definiteness put $T_a=\lambda_a$.
Under this situation we can take the general ansatz of form
$$A=r^\mu\mit\Gamma dt+A_rdr+A_\vartheta d\vartheta+
r^\nu\mit\Delta d\varphi \>,\eqno(12)$$
where $A_\vartheta= r^\rho T$ and matrices $\mit\Gamma=\alpha^a\lambda_a$,
$\mit\Delta=\beta^a\lambda_a$, $T=\gamma^a\lambda_a$, $A_r=f^a(r)\lambda_a$
with arbitrary real
constants $\alpha^a, \beta^a, \gamma^a$ and arbitrary real functions $f^a(r)$.
It could seem that
there is a more general ansatz in the form
$A=r^{\mu_a}\alpha^a\lambda_a dt+f^a(r)\lambda_adr+
r^{\rho_a}\gamma^a\lambda_a d\vartheta+
r^{\nu_a}\beta^a\lambda_a d\varphi$ but somewhat more complicated considerations
than the ones below show that all the same we should have
$\mu_a=\mu, \nu_a=\nu, \rho_a=\rho$ for any $a$ so we at once consider this
condition to be fulfilled to avoid unnecessary complications.
Then for the above ansatz (12) the Lorentz condition ${\rm div}(A)=0$
takes the form
$$\partial_r(r^2\sin{\vartheta}g^{rr}A_r)+
\partial_\vartheta(r^2\sin{\vartheta}g^{\vartheta\vartheta}A_\vartheta)=0$$
which can be rewritten as
$$\partial_\vartheta(\sin{\vartheta}A_\vartheta)+\sin{\vartheta}
\partial_r(r^2A_r)=0\>, \eqno(13)$$
wherefrom it follows $\cot{\vartheta}r^\rho T+\partial_r(r^2A_r)=0$ while
the latter entails
$$A_r=\frac{C}{r^2}-\frac{\cot{\vartheta}r^{\rho-1}T}{\rho+1}\>
\eqno(14)$$
with a constant matrix $C$.
Then we can see that it should put $C=T=0$ or else $A_r$ will not be spherically
symmetric and the confining one where only the powers of $r$ equal to $\pm1$
are admissible. As a result we come to the conclusion that one should put
$A_r=A_\vartheta=0$ in (12).
After this we have ($[,]$ signifies matrix commutator)
$F=dA+gA\wedge A=-\mu r^{\mu-1}\mit\Gamma dt\wedge dr+
\nu r^{\nu-1}\mit\Delta dr\wedge d\varphi+gr^{\mu+\nu}
[\mit\Gamma,\mit\Delta]dt\wedge d\varphi$ which entails [with the help of
(10)]
$$\ast F=\mu r^{\mu+1}\sin{\vartheta}\mit\Gamma d\vartheta\wedge d\varphi-
\frac{\nu r^{\nu-1}\mit\Delta}{\sin{\vartheta}} dt\wedge d\vartheta-
\frac{gr^{\mu+\nu}}{\sin{\vartheta}}
[\mit\Gamma,\mit\Delta]dr\wedge d\vartheta$$
and the Yang-Mills equations (3) turn into
$$\mu(\mu+1)r^{\mu}\sin^2{\vartheta}\mit\Gamma=
g^2r^{\mu+2\nu}[\mit\Delta,[\mit\Gamma,\mit\Delta]]\>,$$
$$\nu(\nu-1)r^{\nu-2}\mit\Delta=
g^2r^{2\mu+\nu}[\mit\Gamma,[\mit\Gamma,\mit\Delta]]\>.
\eqno(15)$$
It is now not complicated to enumerate possibilities for obtaining the confining
solutions in accordance with (15), where we should put $\mu=-1$, $\nu=1$.
\begin{enumerate}
\item $\mit\Gamma$=0 or $\mit\Delta$=0. This situation does obviously not
correspond to a confining solution
\item $\mit\Gamma=C_0\mit\Delta$ with some constant $C_0$. This case conforms
to that all the parameters $\alpha^a$ describing electric colour Coulomb field
(see Subsection 2.3) and the ones $\beta^a$ for linear magnetic colour
field are
proportional -- the situation is not quite clear from physical point of view
\item Matrices $\mit\Gamma,\mit\Delta$ are not equal to zero simultaneously and
both matrices belong to Cartan subalgebra of SU(3)-Lie algebra. The parameters
$\alpha^a, \beta^a$ of electric and magnetic colour fields are not connected
and arbitrary, i. e.
they should be chosen from experimental data. The given situation is the most
adequate to the physics in question and the corresponding confining solution
is in essence the same which has been obtained as far back as in
Ref. \cite{Gon01} from other considerations.
\end{enumerate}
One can slightly generalize the starting ansatz (12) taking it in the form
$A=(r^\mu\mit\Gamma +A')dt+(r^\nu\mit\Delta +B')d\varphi$
with matrices $A'=A^a\lambda_a$, $B'=B^a\lambda_a$ and constants $A^a,B^a$.
Then considerations along the same above lines draw the conclusion that
the nontrivial confining solution is described by
$\mit\Gamma,\mit\Delta, A', B'$ belonging to Cartan subalgebra.

Let us remind that, by definition,
a Cartan subalgebra is a maximal abelian subalgebra in
the corresponding Lie algebra, i. e., the commutator for any two matrices of
the Cartan subalgebra is equal to zero (see, e.g., Ref. \cite{Bar}). For
SU(3)-Lie algebra the conforming Cartan subalgebra is generated by the
Gell-Mann matrices $\lambda_3, \lambda_8$ which are
$$\lambda_3=\pmatrix{1&0&0\cr 0&-1&0\cr 0&0&0\cr}\,,
  \lambda_8={1\over\sqrt3}\pmatrix{1&0&0\cr 0&1&0\cr
                   0&0&-2\cr}\,.\eqno(16)$$
Then it is easy to gain the only nontrivial
solution in question in the form (which reflects the fact that for any matrix
${\cal T}$ from SU(3)-Lie algebra we have ${\rm Tr}\,{\cal T}=0$)
 $$ A^3_t+\frac{1}{\sqrt{3}}A^8_t =-\frac{a_1}{r}+A_1 \>,
 -A^3_t+\frac{1}{\sqrt{3}}A^8_t=-\frac{a_2}{r}+A_2\>,
-\frac{2}{\sqrt{3}}A^8_t=\frac{a_1+a_2}{r}-(A_1+A_2)\>, $$
$$ A^3_\varphi+\frac{1}{\sqrt{3}}A^8_\varphi =b_1r+B_1 \>,
 -A^3_\varphi+\frac{1}{\sqrt{3}}A^8_\varphi=b_2r+B_2\>,
-\frac{2}{\sqrt{3}}A^8_\varphi=-(b_1+b_2)r-(B_1+B_2)\>, \eqno(17)$$
where real constants $a_j, A_j, b_j, B_j$
parametrize the solution, and we wrote down
the solution in the combinations that are just
needed to insert into the corresponding Dirac
equation (see Section 3). From here it follows one more form
$$A^3_t = [(a_2-a_1)/r+A_1-A_2]/2,\>
A^8_t =[A_1+A_2-(a_1+a_2)/r]\sqrt{3}/2\>,$$
$$ A^3_\varphi = [(b_1-b_2)r+B_1-B_2]/2,
   A^8_\varphi= [(b_1+b_2)r+B_1+B_2]\sqrt{3}/2\>\eqno(18)$$

Clearly, the obtained results may be extended over all SU($N$)-groups with
$N\ge2$ and even
over all semisimple compact Lie groups since for them the corresponding Lie
algebras possess just the only Cartan subalgebra. Also we can talk about the
compact non-semisimple groups, for example, U($N$). In the latter case
additionally to Cartan subalgebra we have centrum consisting from the matrices
of the form $\alpha I_N$ ($I_N$ is the unit matrix $N\times N$) with arbitrary
constant $\alpha$.
The most relevant physical cases are of course U(1)- and SU(3)-ones
(QED and QCD),
therefore we shall not consider further generalizations of the results
obtained but let us write out the corresponding solution for U(1)-case which
will useful for to interpret above solutions (17)--(18) in the more habitual
physical terms. It should also be noted that the
nontrivial confining solutions obtained exist at any gauge coupling constant
$g$, i. e. they are essentially {\em nonperturbative} ones.
\subsection{\rm{U(1)}-case}
Under this situation the Yang-Mills equations (3) turn into the second pair of
Maxwell equations
$$d\ast F= 0 \eqno(19)$$
with $F=dA$, $A=A_\mu dx^\mu$.
We search for
the solution of (19) in the form $A=A_t(r)dt+A_\varphi(r)d\varphi$.
It is then easy to check that $F=dA=
-\partial_rA_tdt\wedge dr+\partial_rA_\varphi dr\wedge d\varphi$ and
according to (10) we get
$\ast F=r^2\sin\vartheta\partial_rA_td\vartheta\wedge d\varphi-
\frac{1}{\sin\vartheta}\partial_rA_\varphi dt\wedge d\vartheta$. From here
it follows that (19) yields
$$\partial_r(r^2\partial_rA_t)=0,\>\partial^2_rA_\varphi=0\>,\eqno(20)$$
and we write down the solutions of (20) as
$$ A_t =\frac{a}{r}+A \>, A_\varphi=br+B \>\eqno(21)$$
with some constants $a, b, A, B$ parametrizing solutions (further for the
sake of simplicity let us put $a=1, b=1\ {\rm GeV}, A=B=0$).

To interpret solutions (21) in the more habitual physical terms let us pass on
to Cartesian coordinates employing the relations
$$\varphi=\arctan(y/x),\>
d\varphi=\frac{\partial\varphi}{\partial x}dx+
\frac{\partial\varphi}{\partial y}dy\> \eqno(22)$$
which entails
$$A_\varphi d\varphi=-\frac{ry}{x^2+y^2}dx+\frac{rx}{x^2+y^2}dy\>\eqno(23)$$
and we conclude that the solutions of (21) describe the combination
of the electric Coulomb field with potential $\Phi=A_t$ and the constant
magnetic field with vector-potential
$${\bf A}=(A_x,A_y,A_z)=(-\frac{ry}{x^2+y^2},\frac{rx}{x^2+y^2},0)=
(-\frac{\sin\varphi}{\sin\vartheta},\frac{\cos\varphi}{\sin\vartheta},0)\>,
\eqno(24)$$
which is {\it linear} in $r$ in spherical coordinates and the 3-dimensional
divergence ${\rm div}{\bf A}=0$, as can be checked directly. Then Eqs. (19) in
Cartesian coordinates takes the form
$$\Delta\Phi=0,\> {\rm rotrot}{\bf A}= \Delta{\bf A}=0\> \eqno(25)$$
with the Laplace operator $\Delta=\partial_x^2+\partial_y^2+\partial_z^2$.
At last, it is easy to check that the solution under consideration satisfies the
Lorentz condition ${\rm div}(A)=0$.

Practically the same considerations as the above ones in electrodynamics
show that the solutions (17)--(18) describe the configuration of the electric
Coulomb-like colour field (components $A_t$) with potentials $\Phi^3,\Phi^8$
and the constant magnetic colour
field (components $A_\varphi$) with vector-potentials ${\bf A}^3,{\bf A}^8$
which are {\it linear} in $r$ in spherical coordinates with 3-dimensional
divergences ${\rm div}{\bf A^{3,8}}=0$ and
$\Delta\Phi^{3,8}=0,\> {\rm rotrot}{\bf A^{3,8}}= \Delta{\bf A^{3,8}}=0$.

\section{Relationship with QCD}
The previous section leads us to the following problem: how to describe
possible relativistic bound states in the obtained confining SU(3)-Yang-Mills
fields?
The sought description should be obviously based on the QCD-Lagrangian.
Let us write down this Lagrangian (for one flavour) in
arbitrary curvilinear (local) coordinates in Minkowski spacetime
$${\cal L}=\overline{\Psi}{\cal D}\Psi -\mu_0\overline{\Psi}\Psi
-\frac{1}{4}(g^{\mu\alpha}g^{\nu\beta}-g^{\mu\beta}g^{\nu\alpha})
F^a_{\mu\nu}F^a_{\alpha\beta}, \,\mu<\nu,\,\alpha<\beta
\eqno(26)$$
where, if denoting
$S(M)$ and $\xi$, respectively, the standard spinor bundle
and $3$-dimensional vector one (equipped with a SU($3$)-connection with
the corresponding connection and curvature matrices
$A=A_\mu dx^\mu=A^a_\mu \lambda_adx^\mu$,
$F=dA+gA\wedge A=F^a_{\mu\nu}\lambda_adx^\mu\wedge dx^\nu$)
over Minkowski spacetime, we can
construct tensorial product $\Xi=S(M)\otimes\xi$. It is clear that $\Psi$ is
just a section of the latter bundle, i. e.
$\Psi$ can be chosen in the form
$\Psi=(\Psi_1,\Psi_2, \Psi_3)$
with the four-dimensional Dirac spinors $\Psi_j$ representing the $j$-th colour
component while $\overline{\Psi}=\Psi^{\dag}(\gamma^0\otimes I_3)$ is
the adjont spinor,($\dag$) stands for hermitian conjugation,
$\otimes$ means tensorial product of matrices,
$\mu_0$ is a mass parameter, ${\cal D}$ is the Dirac operator
with coefficients in $\xi$ (see below). At last, we have
the condition ${\rm Tr}(\lambda_a\lambda_b)=2\delta_{ab}$ so that the
third addendum in (26) has the form $G(F,F)/8$ with $G$ of (11), where
coefficient $1/8$ is chosen from physical considerations.

From general considerations (see, e. g., Ref. \cite{89}) the explicit form of
the operator ${\cal D}$ in local coordinates $x^\mu$ on Minkowski spacetime
can be written as follows
$${\cal D}=i(\gamma^e\otimes I_3)E_e^\mu\left(\partial_\mu\otimes I_3
-\frac{1}{2}\omega_{\mu ab}\gamma^a\gamma^b\otimes I_3-igA_\mu\right),
\>a < b ,\>\eqno(27)$$
where the forms\ $\omega_{ab}=\omega_{\mu ab}dx^\mu$ obey
the Cartan structure equations
$de^a=\omega^a_{\ b}\wedge e^b$, while the
orthonormal basis $e^a=e^a_\mu dx^\mu$ in cotangent bundle and
dual basis $E_a=E^\mu_a\partial_\mu$ in tangent bundle are connected by the
relations $e^a(E_b)=\delta^a_b$. At last, matrices $\gamma^a$ represent
the Clifford algebra of
the quadratic form $Q_{1,3}=x_0^2-x_1^2-x_2^2-x_3^2$ in ${\tenmsbm\fam\msbmfam C}^{4}$.
It should be noted that Greek indices $\mu,\nu,...$
are raised and lowered with $g_{\mu\nu}$ of (1) or its inverse $g^{\mu\nu}$
and Latin indices $a,b,...$ are raised and lowered by
$\eta_{ab}=\eta^{ab}$= diag(1,-1,-1,-1) except for Latin indices connected
with SU($3$)-Lie algebras,
so that $e^a_\mu e^b_\nu g^{\mu\nu}=\eta^{ab}$,
$E^\mu_aE^\nu_bg_{\mu\nu}=\eta_{ab}$ and so on but $\lambda_a=\lambda^a$.

Under the circumstances we can obtain the following equations according to the
standard prescription of Lagrange approach from (26)
$${\cal D}\Psi=\mu_0\Psi\>,\eqno(28)$$
$$d\ast F= g(\ast F\wedge A - A\wedge\ast F) +gJ\>,\eqno(29)$$
where the source $J$ (a nonabelian SU($3$)-current) is
$$J=j_\mu^a\lambda_a\ast(dx^\mu)=\ast j=\ast(j_\mu^a\lambda_adx^\mu)=
\ast(j^a\lambda_a)\>\eqno(30)$$
with currents
$$j^a=j_\mu^adx^\mu=
\overline{\Psi}(\gamma_\mu\otimes I_3)\lambda^a\Psi\,dx^\mu\>,$$
so summing over
$a=1,...,8$ is implied in (26) and (30).

When using the relation (see, e. g. Refs. \cite{Fin})
$\gamma^cE_e^\mu\omega_{\mu ab}\gamma^a\gamma^b=
\omega_{\mu ab}\gamma^\mu\gamma^a\gamma^b=
-{\rm div}(\gamma)$ with matrix 1-form $\gamma=\gamma_\mu dx^\mu$
[where ${\rm div}$ is defined by relation (4)] and also the
fact that $(\gamma^\mu)^{\dag}\gamma^0=\gamma^0\gamma^\mu$, the
Dirac equation for spinor $\overline{\Psi}$ will be
$$i\partial_\mu\overline{\Psi}(\gamma^\mu\otimes I_3)+
\frac{i}{2}\overline{\Psi}{\rm div}(\gamma)\otimes I_3-
g\overline{\Psi}(\gamma^\mu\otimes I_3)A^a_\mu \lambda_a=-\mu_0\overline{\Psi}
\>.\eqno(28')$$
Then multiplying (28) by $\overline{\Psi}\lambda_a$ from left and ($28'$) by
$\lambda_a\Psi$ from right and adding the obtained equations, we get
${\rm div}(j^a)={\rm div}(j)=0$ if spinor ${\Psi}$ obeys the Dirac equation
(28).

The question now is how to connect the sought relativistic bound states with
the system (28)--(29). To understand it let us apply to the experience
related with QED. In the latter case Lagrangian looks like (26) with changing
group SU($3$)$\to$U(1) so $\Psi$ will be just a four-dimensional Dirac spinor.
Then, as is known (see, e. g. Ref. \cite{LL1}), when passing on to the
nonrelativistic limit the Dirac equation (28) converts into the Pauli equation
and further, if neglecting the particle spin, into the Schr{\"o}dinger
equation, parameter $\mu_0$ becoming the reduced mass of two-body system.
The modulo square integrable solutions of the Schr{\"o}dinger equation just
describe bound states of a particle with mass $\mu_0$, or, that is equivalent,
of the corresponding two-body system. Historically, however, everything was
just vice versa. At first there appeared the Schr{\"o}dinger equation, then
the Pauli and Dirac ones and only then the QED Lagrangian. In its turn,
possibility of writing two-body Schr{\"o}dinger equation on the whole owed
to the fact that the corresponding two-body problem in classical
nonrelativistic (newtonian) mechanics was well posed and actually quantizing
the latter gave two-body Schr{\"o}dinger equation. Another matter was Dirac
equation. Up to now nobody can say what two-body problem in classical
relativistic (einsteinian) mechanics could correspond to Dirac equation. The
fact is that the two-body problem in classical relativistic mechanics has
so far no single-valued statement. Conventionally, therefore, Dirac
equation in QED is treated as the relativistic wave equation describing one
particle with spin one half in an external electromagnetic field.

There is, however, one important exclusion -- the hydrogen atom. When solving
the Dirac equation here one considers mass parameter $\mu_0$ to be equal to the
electron mass and one gets the so-called Sommerfeld formula for hydrogen atom
levels which passes on to the standard Schr{\"o}dinger formula for hydrogen atom
spectrum in nonrelativistic limit (for more details see, e. g.
Ref. \cite{LL1}). But in the Schr{\"o}dinger formula
mass parameter $\mu_0$ is equal to the reduced mass of electron and proton. As
a consequence,
it is tacitly supposed that in Dirac equation the mass parameter should be
equal to the same reduced mass of electron and proton as in Schr{\"o}dinger
equation. Just the mentioned reduced mass is approximately equal to that of
electron but, exactly speaking, it is not the case. We remind that for the
problem under discussion (hydrogen atom) the external field is the Coulomb
electric one between electron and proton, essentially nonrelativistic object
in the sense that it does not vanish in nonrelativistic limit at $c\to\infty$.
If now to place hydrogen atom in a magnetic field then obviously spectrum of
bound states will also depend on parameters decribing the magnetic field.
The latter, however, is essentially relativistic object and vanishes at
$c\to\infty$ because, as is well known, in the world with $c=\infty$ there
exist no magnetic fields (see any elementary textbook on physics, e. g.
Ref. \cite{Sav82}). But it is clear that spectrum should as before depend
on $\mu_0$ as well and we can see that $\mu_0$ is the same reduced mass as
before since in nonrelativistic limit we again should come to the hydrogen
atom spectrum with the reduced mass. So we can draw the conclusion that if
an electromagnetic field is a combination of electric Coulomb field between two
charged elementary particles and some magnetic field (which may be generated
by the particles themselves) then there are certain grounds to consider the
given (quantum) two-body problem to be equivalent to the one of motion for one
particle with usual reduced mass in the mentioned electromagnetic field. As a
result, we can use the Dirac equation for finding possible relativistic
bound states for such a particle implying that this is really some description
of the corresponding two-body problem.

Actually in QED the situaion is just as the described one but magnetic field
is usually weak and one may restrict
oneself to some corrections from this field to the nonrelativistic Coulomb
spectrum (e. g., in the Seemann effect). If the magnetic field is strong then
one should solve just Dirac
equation in a nonperturbative way (see, e.g. Ref. \cite{ST}). The latter
situation seems to be natural in QCD where the corresponding (colour) magnetic
field should be very strong (see Section 4)
because just it provides linear confinement of quarks as we could see above
(see also Refs. \cite{{Gon03},{GC03},{GB04}}) and the given field
also vanishes in nonrelativistic limit (for more details see
Refs. \cite{{Gon03},{GC03}}).

At last, we should make an important point that in QED the mentioned
electromagnetic field is by definition always a solution of the Maxwell
equations so
within QCD we should require the confining SU(3)-field to be a solution of
Yang-Mills equations. Consequently, returning to the system (28)--(29), we
can suggest to decribe relativistic bound states of two quarks (mesons) in QCD
by the compatible solutions of the given system. To be more precise, the meson
wave functions should be the {\em nonperturbative} modulo square integrable
solutions of Dirac equation (28) (with the above reduced mass $\mu_0$) in the
confining SU(3)-Yang-Mills field being a {\em nonperturbative} solution of
(29). In general case, however, the analysis of (29) is difficult because of
availability of the nonabelian current $J$ of (30) in the right-hand side of
(29) but we may use the circumstance that the corresponding modulo square
integrable solutions of Dirac equation (28) might consist from the components
of form $\Psi_j\sim r^{\alpha_j}e^{-\beta_j r}$ with some $\alpha_j>0$,
$\beta_j>0$
which entails all the components of the current $J$ to be modulo $<< 1$ at each
point of Minkowski space. The latter will allow us to put $J\approx0$ and we
shall come to the problem of finding the confining solutions for the Yang-Mills
equations of (29) with $J=0$ whose unique nontrivial form has been described in
previous Section and after inserting the
found solutions into Dirac equation (28) we should require the corresponding
solutions of Dirac equation to have the above necessary behaviour. Under
the circumstances the problem becomes self-consistent and can be analyzable
and this has been done actually in Ref. \cite{Gon01}.

It is clear that all the above considerations can be justified only by
comparison with experimental data but now we obtain some intelligible
programme of further activity which has been partly realized in
Refs. \cite{{Gon03},{GC03},{GB04}}.
\section{Verification of confinement criterion and estimates of colour magnetic
field strength}
To illustrate some of the above let us verify the Wilson confinement criterion
\cite{Wil} for the confining solutions (17)--(18) and also adduce numerical
estimates for strength of colour magnetic field responsible for linear
confinement in the ground state of charmonium.
\subsection{Verification of confinement criterion}
The Wilson confinement criterion is in essence the assertion that the so-called
Wilson loop $W(c)$ should be subject to the area law for the confining gluonic
field configuration. In its turn, the latter law is equivalent to the fact that
energy $E(R)$ of the mentioned configuration (gluon condensate) is linearly
increasing with $R$, a characteristic size of some volume $V$ containing the
condensate. We can easily evaluate $E(R)$ for solutions (17)--(18) using the
$T_{00}$-component (volumetric energy density) of
the energy-momentum tensor for a SU($3$)-Yang-Mills field
$$T_{\mu\nu}={1\over4\pi}\left(-F^a_{\mu\alpha}\,F^a_{\nu\beta}\,g^{\alpha\beta}+
{1\over4}F^a_{\beta\gamma}\,F^a_{\alpha\delta}g^{\alpha\beta}g^{\gamma\delta}
g_{\mu\nu}\right)\>. \eqno(31) $$
It is not complicated to obtain the curvature matrix (field strentgh)
corresponding to the solution (17)--(18)
$$F= F^a_{\mu\nu}\lambda_a dx^\mu\wedge dx^\nu=
-\partial_r(A^a_t\lambda_a)dt\wedge dr
+\partial_r(A^a_\varphi \lambda_a)dr\wedge d\varphi
\>, \eqno(32)$$
which entails the only nonzero components
$$F^3_{tr}=\frac{a_1-a_2}{2r^2},\>
F^8_{tr}=\frac{(a_1+a_2)\sqrt{3}}{2r^2},\>
F^3_{r\varphi}=\frac{b_1-b_2}{2},\>
F^8_{r\varphi}=\frac{(b_1+b_2)\sqrt{3}}{2}\>\eqno(33) $$
and, in its turn,
$$T_{00}\equiv T_{tt}=\frac{1}{4\pi}\left\{\frac{3}{4}\left[(F^3_{tr})^2+
(F^8_{tr})^2\right]
+\frac{1}{4r^2\sin^2{\vartheta}}\left[(F^3_{r\varphi})^2+
(F^8_{r\varphi})^2\right]\right\}= $$
$$\frac{3}{16\pi}\left(\frac{a_1^2+a_1a_2+a_2^2}{r^4}+
\frac{b_1^2+b_1b_2+b_2^2}{3r^2\sin^2{\vartheta}}\right)\equiv
\frac{{\cal A}}{r^4}+
\frac{{\cal B}}{r^2\sin^2{\vartheta}}\>\eqno(34)$$
with ${\cal A}>0$, ${\cal B}>0$.
Let $V$ be the volume between two concentric spheres with radii $R_0<R$. Then
$E(R)=\int_VT_{00}\sqrt{\delta}d^3x=
\int_VT_{00}r^2\sin{\vartheta}dr d\vartheta d\varphi\>$. As was discussed in
Ref.\cite{{GB04}}, the notion of classical gluonic field is applicable only
at distances $>> \lambda_B$, the de Broglie wavelength of quark, so we may
take $R_0$ to be of order of a characteristic size of hadron (meson). Besides it
should be noted that classical $T_{00}$ of (34) has a singularity along
$z$-axis ($\vartheta=0,\pi$) and we have to introduce some angle $\vartheta_0$
whose physical meaning is to be clarified a little below. Under the circumstances, with
employing the relations $\int d\vartheta/\sin{\vartheta}=\ln\tan{\vartheta/2}$,
$\tan{\vartheta/2}=\sin{\vartheta}/(1+ \cos{\vartheta})=
(1-\cos{\vartheta})/\sin{\vartheta}$, we shall have
$$E(R)=\int_{R_0}^R\int^{\pi-\vartheta_0}_{\vartheta_0}\int_{0}^{2\pi}
\left(\frac{{\cal A}}{r^2}+\frac{{\cal B}}{\sin^2{\vartheta}}\right)
\sin{\vartheta}dr d\vartheta d\varphi=E_0-\frac{4\pi{\cal A}}{R}+
{2\pi\cal B}R\ln\frac{1+\cos{\vartheta_0}}{1-\cos{\vartheta_0}} \>\eqno(35)$$
with $E_0=\frac{{4\pi\cal A}}{R_0}-
{2\pi\cal B}R_0\ln\frac{1+\cos{\vartheta_0}}{1-\cos{\vartheta_0}}$. It is clear
that at $R\to\infty$ we get $E(R)\sim R$, i. e. the area law is fulfilled.
\subsection{Estimates of colour magnetic field strength}
To estimate $R_0$ and ${\vartheta_0}$ under real physical situation let us
consider the ground state of charmonium $\eta_c(1S)$ and apply the ideology of
Section 3 for describing the given state. We use the parametrization of
relativistic spectrum of charmonium from Ref. \cite{GB04}. Namely, Table 1
contains necessary parameters of solutions (17)--(18) for the sought estimates.
\begin{table}[htbp]
\caption{Gauge coupling constant, mass parameter $\mu_0$ and
parameters of the confining SU(3)-gluonic field for charmonium.}
\label{t.1}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\small $ g$ & \small $\mu_0$ & \small $a_1$  & \small $a_2$ & \small $b_1$
& \small $b_2$ & \small $B_1$ & \small $B_2$ \\
  & \small (GeV) &  &  & \small (GeV) & \small (GeV) & &  \\
\hline
\small 0.46900 & \small 0.62500 & \small 2.21104 & \small -0.751317
& \small 20.2395 & \small -12.6317 & \small 6.89659 & \small  6.89659 \\
\hline
\end{tabular}
\end{center}
\end{table}
As for parameters $A_{1,2}$ of (17)--(18) then we can put them equal to zero
\cite{GB04}.
One can note that the reduced mass parameter $\mu_0$ is consistent with the
present-day experimental limits \cite{pdg} where the current mass of $c$-quark
($2\mu_0$) is accepted between 1.1 GeV and 1.4 GeV. Then wave function
$\Psi=(\Psi_1,\Psi_2, \Psi_3)$ of the
state under consideration [as a modulo square integrable solution of Dirac
equation (28) in the field (17)--(18)] is given by
the form (with Pauli matrix $\sigma_1$)
$$\Psi_j=e^{i\omega_j t}r^{-1}\pmatrix{F_{j1}(r)\Phi_j(\vartheta,\varphi)\cr\
F_{j2}(r)\sigma_1\Phi_j(\vartheta,\varphi)}\>,j=1,2,3\eqno(36)$$
with the 2D eigenspinor $\Phi_j=\pmatrix{\Phi_{j1}\cr\Phi_{j2}}$ of the
euclidean Dirac operator on the unit sphere ${\tenmsbm\fam\msbmfam S}^2$.
The explicit form of $\Phi_j$ is not needed here and
can be found in Ref. \cite{GB04}. The radial parts are
$$F_{j1}=C_jP_jr^{\alpha_j}e^{-\beta_jr}\left(1-
\frac{gb_j}{\beta_j}\right),F_{j2}=iC_jQ_jr^{\alpha_j}e^{-\beta_jr}\left(1+
\frac{gb_j}{\beta_j}\right)\eqno(37)$$
with $\alpha_j=\sqrt{\Lambda_j^2-g^2a_j^2}$, $\Lambda_j=-1-gB_j$,
$\beta_j=\sqrt{\mu_0^2-\omega_j^2+g^2b_j^2}$, $P_j=gb_j+\beta_j$,
$Q_j=\mu_0+\omega_j$ where $a_3=-(a_1+a_2),b_3=-(b_1+b_2), B_3=-(B_1+B_2)$ and
constants $C_j$ are determined
from the normalization conditions
$\int_0^\infty(|F_{j1}|^2+|F_{j2}|^2)dr=1/3$. Energy of the ground state is
obtained as
$$\epsilon=\sum\limits_{j=1}^3\omega_j\equiv
\frac{-\Lambda_1g^2a_1b_1+\alpha_1|\Lambda_1|\mu_0}{\Lambda_1^2}+
\frac{-\Lambda_2g^2a_2b_2+\alpha_2|\Lambda_2|\mu_0}{\Lambda_2^2}+
\frac{-\Lambda_3g^2a_3b_3-\alpha_3|\Lambda_3|\mu_0}{\Lambda_3^2}=
2.9796\ {\rm GeV}\>.
\eqno(38)$$
Then we can put $R_0=\frac{1}{3}\sum_{j=1}^{3}1/\beta_j
\approx0.0399766\ {\rm fm}$ and, adding the rest energies of quarks $2(2\mu_0)$
to $E_0$ of (35), we consider $\epsilon=E_0+4\mu_0=2.9796\ {\rm GeV}$ which
entails $\vartheta_0\approx0.723388$ and one may consider $\vartheta_0$ to be
the parameter determining some cone $0\le\vartheta\le\vartheta_0$ so the quark
emits gluons outside of the cone. It is evident that the Wilson criterion is
purely classic one: quantum mechanically the stay probability of quarks at
distances $>R_0$ is of order $\sum_{j=1}^{3}(|F_{j1}|^2+|F_{j2}|^2)$ and
is damped exponentially fast at $R>R_0$. Just the colour magnetic field defines
this damping through the coefficients $\beta_j$.

At last, using the Hodge star operator in 3-dimensional euclidean space [where
$ds^2=g_{\mu\nu}dx^\mu\otimes dx^\nu\equiv
dr^2+r^2(d\vartheta^2+\sin^2\vartheta d\varphi^2)$,
$\ast(dr\wedge d\vartheta)=\sin{\vartheta}d\varphi,
\ast(dr\wedge d\varphi)=-d\vartheta/\sin{\vartheta},
\ast(d\vartheta\wedge d\varphi)=dr/(r^2\sin{\vartheta}$],
we can confront components $F^3_{r\varphi},F^8_{r\varphi}$ of (33)
with 3-dimensional 1-forms of the magnetic colour field
$${\bf H}^3=-\frac{b_1-b_2}{2\sin{\vartheta}}d\vartheta\>,
{\bf H}^8=-\frac{(b_1+b_2)\sqrt{3}}{2\sin{\vartheta}}d\vartheta $$
that are modulo equal to
$\sqrt{g^{\mu\nu}{\bf H}^{3,8}_{\mu}{\bf H}^{3,8}_{\nu}}$ or
$$H^3=\frac{|b_1-b_2|}{2r\sin{\vartheta}},\>
H^8=\frac{|b_1+b_2|\sqrt{3}}{2r\sin{\vartheta}}\>,\eqno(39)$$
where we shall put $\sin{\vartheta}=1$ for simplicity.
Table 2 contains estimates of magnetic colour field strength in the ground
state of charmonium whereas the Bohr radius
$a_0=0.529177249\cdot10^{5}\ {\rm fm}$ \cite{pdg}. Also when calculating we
applied the relations $1\ {\rm GeV^{-1}}\approx0.21030893\ {\rm fm}\>$,
$1\ {\rm T}\approx0.692508\times 10^{-15}\ {\rm GeV}^2$.

\begin{table}[htbp]
\caption{Magnetic colour field strengths in the ground
state of charmonium.}
\label{t.2}
\begin{center}
\begin{tabular}{|lll|}
\scriptsize $\eta_c(1S)$: & \scriptsize $R_0= 0.0399766\ {\rm fm}$  & \\
\hline
$r$ &  $H^3$ &  $H^8$ \\
 (fm) &  $({\rm T})$ & $({\rm T})$\\
\hline
 $0.1R_0$ &  $0.124857\cdot10^{19}$  &  $0.500516\cdot10^{18}$ \\
\hline
$R_0$ & $0.124857\cdot10^{18}$  & $0.500516\cdot10^{17}$  \\
\hline
$10R_0$ & $0.124857\cdot10^{17}$  & $0.500516\cdot10^{16}$  \\
\hline
$1.0$ & $0.499137\cdot10^{16}$  & $0.200089\cdot10^{16}$  \\
\hline
$a_0$ & $0.943232\cdot10^{11}$ & $0.378114\cdot10^{11}$  \\
\hline
\end{tabular}
\end{center}
\end{table}
It is seen that at the characteristic scales of
the charmonium ground state the strength of magnetic colour field
responsible for linear confinement reaches huge values of order
$10^{17}$--$10^{18}\ {\rm T}$. For comparison one should notice that the most
strong magnetic fields known at present have been discovered in magnetic
neutron stars, pulsars (see, e. g., Ref. \cite{Bo}) where the corresponding
strengths can be of order $10^{9}$--$10^{10}\ {\rm T}$. So the characterictic
feature of confinement is really very strong magnetic colour field between
quarks. In a certain sense the essence of confinement can be said to be just
in enormous gluon concentrations and magnetic colour field strentghs in space
around quarks.

\section{Concluding remarks}
The aim of this Letter we
pursued was to specify a scenario of linear confinement of quarks, at any rate,
for mesons and quarkonia. As we could see, crucial
step consisted in studying exact solutions of the SU(3)-Yang-Mills
equations modelling confinement. Namely structure of the confining solutions
for SU(3)-Yang-Mills equations prompts to the idea that linear confinement
should be governed by magnetic colour field. But the latter is essentially
relativistic object vanishing in nonrelativistic limit which implies
the relativistic effects to be extremely important for the confinement
mechanism \cite{{Gon03},{GC03},{GB04}}.

  Further specification of the confinement mechanism can consist in
the inquiry answer: what the way of gluon exchange between quarks is?
The matter concerns such a modification of gluon
propagator which might correspond to linear confinement at large distances.
As was remarked in Ref. \cite{Gon03}, the sought modification should be
carried out on the basis of the exact
solutions of the SU(3)-Yang-Mills equations modelling quark confinement.
We hope to consider this modification elsewhere.

There is also an interesting possibility of indirect experimental verification
of the confinement mechanism under discussion. Really solutions (21) point out
the confinement phase could be in electrodynamics as well. Though
there exist no elementary charged particles generating a constant magnetic
field linear in $r$, the distance from particle, after all, if it could generate
this elecromagnetic field configuration in laboratory then one might study
motion of macroscopic charged particles in that field. The confining properties
of the mentioned field should be displayed at classical level too but the exact
behaviour of particles in this field requires certain analysis of the corresponding
classical equations of motion.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

The notion of a projective quantum measurement was introduced by
von Neumann\cite{1} in his treatise on the mathematical foundations
of quantum mechanics. The idea is the following: (i) Every quantum
measurement yielding the data ``yes'' or ``no'' to an experimental question
is described by the projection operators ${\cal P}={\cal P}^\dagger=
{\cal P}^2$ (for ``yes'') or ${\cal Q}={\cal Q}^\dagger={\cal Q}^2$
(for ``no''),
\begin{equation}
{\cal P}+{\cal Q}=1.
\end{equation}
(ii) If the initial state of a quantum object {\em before} a
measurement is $|\Psi_{before}>$, and if the measurement yields the
experimental answer ``yes'' to a question ${\cal P}$, then the state
of the quantum object {\em after} the measurement $|\Psi_{after}>$
is constructed according to the {\em collapse} of the quantum state
rule
\begin{equation}
\big|\Psi_{before}\big>\to \big|\Psi_{after}\big>=\
{{\cal P}\big|\Psi_{before} \big>\over
\ \sqrt{\big<\Psi_{before}\big|{\cal P}\big|\Psi_{before}\big>}\ }\ .
\end{equation}
Eq.(2) is part of a calculation scheme which is {\em not} a unitary
development of the quantum state {\em during a measurement}. The unitary
behavior (implicit in the Schr\"odinger equation for a quantum object)
was thought by von Neumann to hold only {\em between measurements}\cite{2}.

To see how this works, consider a beam of photons which has passed through
an ``upwards'' polarizer as in Figs.1(a) and (b). If a second polarizer
has an axis at an angle of $\theta $ with respect to the upward
direction, then one writes the photon wave function $|\uparrow >$ in the
basis
\begin{equation}
\big|\uparrow \big>=\cos \theta \big|\nwarrow \big>+
\sin \theta \big|\nearrow \big>.
\end{equation}
The probability to pass through the second polarizer is then
\begin{equation}
p(\theta )=\big|\big<\nwarrow \big|\uparrow \big>\big|^2
=\cos^2\theta .
\end{equation}

The cases shown in Fig.1 below work as follows: In Fig.1(a), the second
polarizer is set orthogonal to the first polarizer so that
no photons will pass; i.e.
\begin{equation}
p_a(\pi /2 )=\big|\big<\leftarrow \big|\uparrow \big>\big|^2
=\cos^2 (\pi /2)=0.
\end{equation}
This probability can be increased, as in Fig.1(b) below, by placing an
intermediate polarizer between the first and last orthogonal polarizers.

\begin{figure}[htbp]
\begin{center}
\mbox{\epsfig{file=zfig1.eps,height=70mm}}
\caption{A photon beam polarized ``upwards'' will not pass through a second
orthogonal polarizer as shown in (a). If an intermediate polarizer, as shown
in (b), is placed before the final polarizer, then some photons in
the beam will pass through the final polarizer. The intermediate
measurement is then thought to induce the quantum Zeno effect.}
\label{zfig1}
\end{center}
\end{figure}

In Fig.1(b), a fraction $\cos^2 \theta $ of the $|\uparrow >$
photon beam will pass through the intermediate polarizer and
{\em collapse into the state $|\nwarrow >$}. Then, a fraction
$\cos^2 \big((\pi /2)-\theta \big)$ of the $|\nwarrow >$ photon beam
will pass through the final polarizer and again {\em collapse into the state
$|\leftarrow >$}. The above two {\em collapse of the quantum state
processes} dictate that one multiply probabilities. The final probability
that a photon in the upward polarized beam passes through both the
intermediate and final polarizers is then given by
\begin{equation}
p_b(\theta )=\cos^2 \theta \cos^2 \big((\pi /2)-\theta \big)=
(1/4)\sin^2 (2\theta ).
\end{equation}
That an intermediate measurement can {\em increase} the probability of
passing through a polarizer stack, say $(1/4)=p_b(\pi /4)>p_a(\pi /2)=0$,
is an example of what is thought to be a quantum Zeno effect
\cite{3,4,5,6,7,8,9,10,11,12,13}

A textbook quantum Zeno effect problem in quantum mechanics\cite{14} is
shown in Fig.2.

\begin{figure}[htbp]
\begin{center}
\mbox{\epsfig{file=zfig2.eps,height=70mm}}
\caption{A vertical polarized photon beam enters the stack of N=8 polarizers,
and a horizontally polarized photon beam leaves the stack.}
\label{zfig2}
\end{center}
\end{figure}

A beam of photons in state $|\uparrow >$ is incident on a stack of $N$
polarizers  each of which has a polarization axis rotated
by an angle of $\Delta \theta_N =(\pi /2N)$ relative to the previous
polarizer. According to the von Neumann projection postulate\cite{15},
the quantum state of each photon will {\em collapse} $N$ times, yielding
a probability
\begin{equation}
P_N=\big(\cos^{2}(\Delta \theta_N )\big)^N=
\Big\{\cos^{2}\Big({\pi \over 2N}\Big)\Big\}^N
\end{equation}
of passing through the total stack as pictured in Fig.2. Note, in the
formal limit of an {\em infinite number of quantum state collapse events},
\begin{equation}
\lim_{N\to \infty }P_N =1, \ \ \ {\rm (Quantum\ Zeno\ Effect)}.
\end{equation}
Theoretically {\it every photon passes through the stack}. In real
experiments, there is always some attenuation of a photon beam
passing through many polarizers.

Although the large $N$ quantum Zeno effect does appear to be verified
in the laboratory\cite{16}, we have {\em a very uneasy feeling} about
attributing such observations to the rule of {\em collapsing quantum
states}. After all, long before von Neumann considered projective
measurements, Maxwell had developed a reliable electromagnetic radiation
theory which perfectly well determined how an electromagnetic wave will
(or will not) pass through stacks of polarizers. {\em And not even once}
did Maxwell feel obliged to discuss the collapse of the electromagnetic
radiation field functions. For that matter, neither Bohr nor
Einstein\cite{17} had {\em ever} felt obliged to discuss the so-called wave
function collapse.

Our purpose is to discuss why the projection postulate (collapse of the
quantum state), {\em certainly may and probably should}
be eliminated from ones thinking in quantum mechanics.
The application of this view towards photon polarization measurements
is typical of a more general state of affairs. If one returns
to the original Copenhagen interpretation of the quantum mechanical
formalism, then the quantum state describes an {\em ensemble of experiments}.
Each photon certainly does {\em not} carry its own wave function which
might then undergo a collapse whenever a photon disappears (e.g. whenever
a photon is absorbed in a polarizer). The wave function (in reality)
{\em describes an ensemble} of many photons.

The collapse of the quantum state whenever a photon is absorbed
closely resembles the collapse of a probability table (at the
automobile insurance company) every time a car crashes. Actually,
the probability table at the insurance company stays intact
during a car crash. It is the {\em automobile} and {\em not}
the probability that collapses. This constitutes an altogether
{\em different} pile of scrap metal, even if the cross section
for two automobiles to scatter off one another were computed
with a quantum mechanical amplitudes to include the small
diffraction effects.

In Sec.II, it will be shown how the Schr\"odinger equation for photons in
the vacuum is equivalent to the vacuum Maxwell field equations. In
Sec.III, the  Schr\"odinger equation for photons propagating in
continuous media (e.g. polarizers) will be explored. The photon wave
function follows from the Maxwell electromagnetic theory in continuous media.
However, the electromagnetic wave attenuates if the continuous
media exhibits dissipation in the form of finite conductivity.
The complete quantum system (polarizers plus photons)
obeys {\em a big monster} Schr\"odinger equation with a huge
number of degrees of freedom and
{\it does not at all require the projective measurement}
formalism of Eq.(2). This feature of quantum mechanics
is valid for many systems as will be discussed in Sec.IV.

In Sec.V we compute, {\em for a realistic model of polarizers}, the
Maxwell equation formulation of the Zeno effect. The {\em classical Maxwell
wave} computation (without projection operators) yields the same result
as {\em the quantum photon Schr\"odinger amplitude} (without projection
operators).

In an appropriate regime, the projection postulate does indeed
give an accurate answer, but this may only be proved by solving
the full Maxwell theory without the projection postulate. One might be
led to ask why should one {\em ever} add to the quantum theory a
projection theory of measurements. A hint as to
why the projection postulate is so popular might be connected to the fact
that the Maxwell theory computation (to be presented) required a fast
computer. More than mere speed, we required algorithms sufficient to
maintain $\sim 10^3$ significant figure accuracy during the entire
calculation of the photon transmission probability. Without recently
developed precise numerical algorithms, one might be coerced into
employing the projection postulate.

In the concluding Sec.VI, we question the utility of replacing the wisdom
of experimentalists and engineers who design and build laboratory
equipment with a {\em merely} philosophical view of projective
measurements. It is somewhat unreasonable to represent real
laboratory equipment with mathematical projection operators.

\section{Photon Wave Functions}

For a spin $S$ particle\cite{18} one requires three Hermitian matrices
${\bf S}=(S_1,S_2,S_3)$ obeying
\begin{equation}
[S_i,S_j]=i\epsilon_{ijk}S_k, \ \
S_1^2+S_2^2+S_3^2=S(S+1).
\end{equation}
The photon is a spin one ($S=1$) particle so we may choose the spin
matrices
\begin{equation}
S_1=\pmatrix{ 0 & 0 & 0\cr
0 & 0 & -i\cr
0 & i & 0},\ \
S_2=\pmatrix{ 0 & 0 & i\cr
0 & 0 & 0\cr
-i & 0 & 0},
\end{equation}
and
\begin{equation}
S_3=\pmatrix{ 0 & -i & 0\cr
i & 0 & 0\cr
0 & 0 & 0}.
\end{equation}
With ${\bf p}=-i\hbar {\bf \nabla}$, the vacuum Schr\"odinger equation for
free photons reads
\begin{equation}
i\hbar \Big({\partial \psi ({\bf r},t)\over \partial t}\Big)
=c\big({\bf S\cdot p} \big)\psi ({\bf r},t).
\end{equation}
Writing
\begin{equation}
\psi ({\bf r},t)=\pmatrix{F_1({\bf r},t) \cr
F_2({\bf r},t) \cr F_3({\bf r},t) },
\end{equation}
allows one to cast the photon Schr\"odinger Eq.(12) into the (perhaps)
more simple vector form
\begin{equation}
i\Big({\partial {\bf F} ({\bf r},t)\over \partial t}\Big)=
c\ {\bf curl\  F} ({\bf r},t).
\end{equation}
The vector field ${\bf F}({\bf r},t)$ should be constrained to obey
\begin{equation}
div{\bf F}({\bf r},t)=0.
\end{equation}
Eq.(15) is equivalent to only allowing two independent spin states
(say superpositions of only two transverse polarizations),
thereby {\em eliminating} a third (longitudinal polarized)
``zero energy'' mode. Finally, Eqs.(14) and (15) take on a clearly
recognizable form if one
defines the real and imaginary parts of ${\bf F}({\bf r},t)$ to be,
respectively, the electric field ${\bf E}({\bf r},t)$ and the
magnetic field ${\bf B}({\bf r},t)$\cite{19}\cite{20}\cite{21}; i.e.
\begin{equation}
{\bf F}({\bf r},t)={\bf E}({\bf r},t)+i{\bf B}({\bf r},t).
\end{equation}
The photon Schr\"odinger Eq.(12) then appears equivalent to the vacuum
Maxwell theory; i.e.
\begin{equation}
\Big({\partial {\bf E} ({\bf r},t)\over \partial t}\Big)=
c\ {\bf curl\ B}({\bf r},t),
\end{equation}
and
\begin{equation}
\Big({\partial {\bf B} ({\bf r},t)\over \partial t}\Big)=
-c\ {\bf curl\ E}({\bf r},t),
\end{equation}
with the vacuum Gauss law Eq.(15) constraints
\begin{equation}
div {\bf B}({\bf r},t)=0,\ \ \
div {\bf E}({\bf r},t)=0.
\end{equation}

It is of interest to discuss the relativistic Lorentz symmetry of
the photon Schr\"odinger Eq.(12). The proof of Lorentz symmetry is
actually well known since we have already proved equivalence with the
Maxwell theory. However, the discrete Lorentz symmetries are still
worthy of further discussion\cite{22,23,24,25,26,27,28}. If we
consider the classical Lorentz force on a charge equation
${\bf f}=e({\bf E}+({\bf v\times B})/c)$,
and then allow the charge to change sign $e\to -e$, then the
force ${\bf f}$ will remain the same if we also allow
${\bf E}\to -{\bf E}$ and ${\bf B}\to -{\bf B}$. Thus, the electromagnetic
field Eq.(16) is odd under charge conjugation,
\begin{equation}
\hat{C}{\bf F}=-{\bf F}.
\end{equation}
Under time reversal, ${\bf E}\to {\bf E}$ and ${\bf B}\to -{\bf B}$ so
that
\begin{equation}
\hat{T}{\bf F}={\bf F}^*.
\end{equation}
Finally, under parity ${\bf E}\to -{\bf E}$ and ${\bf B}\to {\bf B}$ so
that
\begin{equation}
\hat{P}{\bf F}=-{\bf F}^*.
\end{equation}
Note
\begin{equation}
\hat{T}\hat{C}\hat{P}=1,
\end{equation}
as would be expected from relativistic wave equations.

\section{Photon Wave Functions in Continuous Media}

The Maxwell theory for an electromagnetic disturbance moving through
continuous media\cite{29,30,31,32,33,34} may be formulated starting
from
\begin{equation}
div{\bf E}({\bf r},t)=4\pi \rho ({\bf r},t),\ \ \
div{\bf B}({\bf r},t)=0,
\end{equation}
and
\begin{equation}
\Big({\partial {\bf E} ({\bf r},t)\over \partial t}\Big)=
c\ {\bf curl\  B} ({\bf r},t)-4\pi {\bf J}({\bf r},t),
\end{equation}
\begin{equation}
\Big({\partial {\bf B} ({\bf r},t)\over \partial t}\Big)=
-c\ {\bf curl\  E} ({\bf r},t).
\end{equation}
It is sufficiently general to supplement Eqs.(24), (25) and (26) by
employing a linear causal relationship between the current and the electric
field, i.e.
\begin{equation}
{\bf J}({\bf r},t)=\int_0^\infty \int
{\bf K}({\bf r},{\bf r}',s){\bf \cdot E}({\bf r}',t-s)
d^3 {\bf r}'ds.
\end{equation}
If, for a complex frequency $\zeta $ where $\Im m\ \zeta >0$,
\begin{equation}
{\bf E} ({\bf r},t)=\Re e \big\{{\bf E}_0 ({\bf r},\zeta)
e^{-i\zeta t}\big\},
\end{equation}
\begin{equation}
{\bf J} ({\bf r},t)=\Re e \big\{{\bf J}_0 ({\bf r},\zeta )
e^{-i\zeta t}\big\},
\end{equation}
Eqs.(27), (28) and (29) serve to define the non-local conductivity
${\bf \sigma }({\bf r},{\bf r}',\zeta )$, i.e.
\begin{equation}
{\bf J}_0 ({\bf r},\zeta )=\int {\bf \sigma }({\bf r},{\bf r}',\zeta )
{\bf \cdot E}_0({\bf r}',\zeta)d^3{\bf r}',
\end{equation}
where
\begin{equation}
{\bf \sigma }({\bf r},{\bf r}',\zeta )=\int_0^\infty e^{i\zeta s}
{\bf K}({\bf r},{\bf r}',s)ds.
\end{equation}
Some workers prefer to complete the Maxwell equations in continuous media
by employing a constitutive equation relating the Maxwell displacement
field ${\bf D}({\bf r},t)$ to the electric field ${\bf E}({\bf r},t)$.
In the frequency domain, the dielectric response
${\bf \epsilon }({\bf r},{\bf r}',\zeta )$ of the continuous media is defined
as
\begin{equation}
{\bf D}_0 ({\bf r},\zeta )=\int {\bf \epsilon }({\bf r},{\bf r}',\zeta )
{\bf \cdot E}_0({\bf r}',\zeta)d^3{\bf r}'.
\end{equation}
Just so long as the various response functions are maintained as non-local
(in space and time), which of the many equivalent response functions are
actually used in a calculation remains a matter of convenience.
For example, Eqs.(30) and (32) are equivalent if we choose
\begin{equation}
{\bf \epsilon }({\bf r},{\bf r}',\zeta )=
{\bf 1}\delta ({\bf r}-{\bf r}')+
\Big({4\pi i {\bf \sigma }({\bf r},{\bf r}',\zeta )\over \zeta }\Big).
\end{equation}

The dielectric, magnetic, or conductivity response functions conventionally
used to describe continuous media now make an appearance into the
Schr\"odinger equation for a photon moving through the continuous media
via the photon ``self-energy'' insertion. To see what is involved,
we may write the Maxwell Eqs.(24) and (25) as
\begin{equation}
div{\bf F} ({\bf r},t)=4\pi \rho ({\bf r},t),
\end{equation}
and
\begin{equation}
i\Big({\partial {\bf F} ({\bf r},t)\over \partial t}\Big)=
c\ {\bf curl\  F} ({\bf r},t)-4\pi i{\bf J}({\bf r},t).
\end{equation}
where Eq.(16) has been invoked. Employing Eqs.(21), (27) and
(35), we find our central result for the photon Schr\"odinger equation
in continuous media
$$
i\Big({\partial {\bf F} ({\bf r},t)\over \partial t}\Big)=
c\ {\bf curl\  F} ({\bf r},t)+
$$
\begin{equation}
\int_0^\infty \int {\bf \Sigma }({\bf r},{\bf r}',s)
{\bf F} ({\bf r}',t-s)d^3 {\bf r}'ds.
\end{equation}
where the photon self-energy part is defined as
\begin{equation}
{\bf \Sigma }({\bf r},{\bf r}',s)=-2\pi i
{\bf K}({\bf r},{\bf r}',s)(1+\hat{T}).
\end{equation}
Using Eqs.(13) and (36), one may write the photon Schr\"odinger
equation in the form
$$
i\hbar\Big({\partial \psi ({\bf r},t)\over \partial t}\Big)=
c\big({\bf S\cdot p}\big) \psi ({\bf r},t)+
$$
\begin{equation}
\hbar\int_0^\infty \int {\bf \Sigma }({\bf r},{\bf r}',s)
\psi ({\bf r}',t-s)d^3 {\bf r}'ds.
\end{equation}

In principle the calculation of the self energy part
${\bf \Sigma }({\bf r},{\bf r}',s)$ is equivalent to the
calculation of the conductivity
${\bf \sigma }({\bf r},{\bf r}',\zeta )$
as is evident from Eqs.(21), (31), and (37). In a practical
calculations involving the photon Schr\"odinger equation
in a polarizer, one should use the conductivity and dielectric
response functions as those of the optical engineers who designed
the device\cite{35,36,37,38,39}. Finally, the non-local form of
Eq.(38) is shared by many other quantum mechanical systems as we
shall now briefly explore.

\section{Non-Local Schr\"odinger Equations}

Although we do not adhere to the non-unitary collapse of the
quantum state Eq.(2) as postulated by von Neumann, we do find the
notion of projection operators (also introduced into quantum mechanics
by von Neumann) to be more than just convenient. Eq.(1) will thereby
be invoked in what follows. For a macroscopic
laboratory quantum system we here insist that a {\em monster}
Schr\"odinger equation with a huge number of degrees of freedom still
holds true,
\begin{equation}
i\hbar \Big({\partial \over \partial t}\Big)|\Psi (t)>
={\cal H}_{tot}|\Psi (t)>.
\end{equation}
We do not contemplate the {\em ultimate monster wave function} of the {\em
whole universe}, not only because we are not smart enough to find it, but
because we do not know how to build an experimental
{\em ensemble of universes} to test a hypothetical quantum state of
{\em our universe}. We leave such considerations to people more wise
than are we.

We are happy to say something meaningful about a very small physical
piece of a laboratory quantum state; e.g. a ``projected piece''
\begin{equation}
\big|\psi (t)\big>={\cal P}\big|\Psi (t)\big>.
\end{equation}
We leave the rest of the monster wave function,
\begin{equation}
\big|\phi (t)\big>={\cal Q}\big|\Psi (t)\big>,
\end{equation}
to the engineers who design the experiment. With the little piece of the wave
function Eq.(40) and the big piece of the wave function Eq.(41), we rewrite
Eq.(39) as
\begin{equation}
i\hbar \Big({\partial \over \partial t}\Big)
\pmatrix{|\psi (t)> \cr |\phi (t)>}=
\pmatrix{ H & V\cr V^\dagger & H'}
\pmatrix{|\psi (t)> \cr |\phi (t)>},
\end{equation}
where $H={\cal P}{\cal H}_{tot}{\cal P}$,
$V={\cal P}{\cal H}_{tot}{\cal Q}$,
$V^\dagger={\cal Q}{\cal H}_{tot}{\cal P}$ and
$H'={\cal Q}{\cal H}_{tot}{\cal Q}$.

We can only hope to find $|\psi (t)>$ which obeys
\begin{equation}
i\hbar \Big({\partial \over \partial t}\Big)\big| \psi (t) \big>
=H\big| \psi (t) \big>+V\big| \phi (t) \big>,
\end{equation}
so we solve for the monster piece
\begin{equation}
i\hbar \Big({\partial \over \partial t}\Big)\big| \phi (t) \big>
=H'\big| \phi (t) \big>+V^\dagger \big| \psi (t) \big>,
\end{equation}
subject to the causal boundary condition
\begin{equation}
\big| \phi (t) \big>=-\Big({i\over \hbar}\Big)\int_0^\infty
e^{-iH's/\hbar}V^\dagger\big|\psi(t-s)\big>ds.
\end{equation}

From Eqs.(43) and (45) we find the general non-local Schr\"odinger
equation for the piece of the wave function that is of interest;
It is
\begin{equation}
i\hbar \Big({\partial \over \partial t}\Big)\big|\psi (t)\big>
=H\big|\psi (t) \big>+
\hbar \int_0^\infty \Sigma (s)\big|\psi(t-s)\big>ds,
\end{equation}
where
\begin{equation}
\Sigma (s)=
-\Big({i\over \hbar^2}\Big)Ve^{-iH's/\hbar }V^\dagger .
\end{equation}

The full monster Schr\"odinger Eq.(39) for complete wave function
$|\Psi (t)>$ reduces to the non-local in time  Schr\"odinger Eq.(46)
for the small subspace piece of the wave function  $|\psi(t)>$.
A particular example of Eq.(46) is the photon Schr\"odinger Eq.(38).
The point is that when a photon propagates inside of a continuous media
device, say a polarizer, the {\em monster wave function}
includes {\em both} the photon {\em and} the polarizer degrees of freedom.
In a one photon projected subspace, the quantum state of the polarizer
is fixed. In addition to the polarizer, there is but
one propagating photon. If the photon is absorbed, then an electronic
excitation is produced which heats up the polarizer. The piece of
the monster quantum state representing this absorbed photon situation
is not contained in the single photon subspace. The single photon
subspace wave function is then attenuated due to photon absorption
in the polarizer. All of this (including the attenuation of the Maxwell
wave in the medium) is described by the one photon Schr\"odinger equation,
{\it without} recourse to the projection postulate. Maxwell was so
lucky to be able to write down the Schr\"odinger equation for the photon
in matter, including attenuation, {\em without} knowing anything about
quantum states (be they monster or otherwise).

It is of interest to look for a fixed energy solution of the subspace
wave function Eq.(46) of the form
\begin{equation}
\big|\psi (t)\big>=\big|\psi_E\big>e^{-iEt/\hbar}.
\end{equation}
Eqs.(46), (47) and (48) imply
\begin{equation}
\Big\{H+\Delta (E)-\Big({i\hbar \over 2}\Big)\Gamma (E)\Big\}
\big|\psi_E\big>=E\big|\psi_E\big>,
\end{equation}
where
\begin{equation}
\Delta (E)-\Big({i\hbar \over 2}\Big)\Gamma (E)=
V\Big({1\over E-H'+i0^+}\Big)V^\dagger .
\end{equation}
That the effective energy dependent Hamiltonian on the left hand side of
Eq.(49)
\begin{equation}
{\cal H}_{eff}(E)=H+\Delta(E)-\Big({i\hbar \over 2}\Big)\Gamma (E)
\end{equation}
is not Hermitian, is due to the ``Fermi platinum rule'' transition rates
to leave the small subspace and wander into the remaining monster subspaces.
The Fermi platinum rule
\begin{equation}
\Gamma (E)=\Big({2\pi \over \hbar}\Big)V\delta (E-H')V^\dagger
\end{equation}
is just a little bit more valuable than the more usual
perturbative Fermi golden rule, since Eq.(52) is rigorously true to
all orders of perturbation theory. The fixed energy $E$ case
corresponds to having a fixed photon frequency $\omega =(E/\hbar)$
for the electromagnetic problem at hand. Let us return to this problem.

\section{Maxwell Theory and Polarizers}

The Maxwell theory of a transverse wave moving through a polarizer may
be formulated following the discussion of Born and Wolf\cite{40}.
(i) If a transverse wave propagates in the $z$-direction, then the wave
may be described by four electromagnetic field components, say
\begin{equation}
\chi(z)=\pmatrix{E_1(z)\cr E_2(z)\cr B_1(z)\cr B_2(z)}.
\end{equation}
If the polarizer is oriented by a rotation angle $\theta $,
as in Fig.1(b), then the dielectric response tensor will have the form
\begin{equation}
\epsilon(\theta )=
\left({\tilde{\epsilon}_1+\tilde{\epsilon}_2\over 2}\right)+
\left({\tilde{\epsilon}_1-\tilde{\epsilon}_2\over 2}\right)
e^{-i\tau_2\theta }\tau_3e^{i\tau_2\theta },
\end{equation}
where we have employed the $2\times 2$ Pauli matrices
\begin{equation}
\tau_1=\pmatrix{0 & 1 \cr 1 & 0},\ \ \
\tau_2=\pmatrix{0 & -i \cr i & \ 0},
\end{equation}
and
\begin{equation}
\tau_3=\pmatrix{1 & \ 0 \cr 0 & -1}.
\end{equation}
The complex principal eigenvalues of the dielectric tensor at frequency
$\omega $ have the form
\begin{equation}
\tilde{\epsilon}_j=\varepsilon_j+
\left(4\pi i\sigma_j\over \omega \right),\ \ \
j=1,2\ .
\end{equation}
The real parts of the dielectric eigenvalues $\varepsilon_{1,2}$ determine
the velocities of the waves polarized in the principal directions. The
conductivities $\sigma_{1,2}$ determine the dissipative attenuation of the
waves polarized in the principle directions. For a well designed polarizer,
a wave with an allowed polarization direction will propagate unattenuated
while a wave with a disallowed polarization will be strongly attenuated.

{\em It is the anisotropic dissipative conductivity that is often modeled
by the quantum projective postulate for the case of photon polarization}.
On the other hand, conductivity is evidently {\it classical} since both
Ohm and Maxwell were dead and gone {\em before} quantum mechanics had been
invented.

For the classical Maxwell calculation, the wave at the beginning
$(z=0)$ of the polarizer is related to the wave at the end $(z=a)$
of the polarizer by a $4\times 4$ transfer matrix
\begin{equation}
\chi (a)={\cal M}(\theta ,\xi )\chi (0), \ \ \
\xi =\Big({\omega a\over c}\Big),
\end{equation}
$\chi(z)$ is defined in Eq.(53), and in partitioned matrix form
$$
{\cal M}(\theta ,\xi )=
$$
\begin{equation}
\pmatrix{ \cos(\xi \sqrt{\epsilon(\theta )}) &
-\big({\sin(\xi \sqrt{\epsilon(\theta )})/
\sqrt{\epsilon(\theta )} }\big)\tau_2 \cr
\tau_2 \sqrt{\epsilon(\theta ) }\sin(\xi \sqrt{\epsilon(\theta )}) &
\tau_2\cos(\xi \sqrt{\epsilon (\theta )})\tau_2 }.
\end{equation}
For a stack of $N$ polarizers, each of width $a$ and oriented at angles
$(\theta_1,...,\theta_N)$, one may repeat the above process going from
the beginning $(z=0)$ of the stack to the end $(z=Na)$ of the stack
\begin{equation}
\chi (Na)={\cal M}_{N}\chi (0), \ \
\xi =\Big({\omega a\over c}\Big).
\end{equation}
One Multiplies the transfer matrices for each polarizer in the stack
\begin{equation}
{\cal M}_{tot,N}={\cal M}(\theta_N,\xi)...
{\cal M}(\theta_2,\xi){\cal M}(\theta_1,\xi ).
\end{equation}

Once the total $4\times 4$ transfer matrix is computed, one may find
the transmission amplitudes $T_j(N)$ and reflection amplitudes $R_j(N)$
for the two polarization directions $(j=1,2)$ by solving the linear
algebra problem
\begin{equation}
{\cal M}_{tot,N}
\pmatrix{1+R_1(N) \cr R_2(N)  \cr R_2(N) \cr 1-R_1(N)}=
\pmatrix{T_1(N) \cr T_2(N)  \cr -T_2(N) \cr T_1(N)}.
\end{equation}
For an incident photon in the allowed $1$-polarization direction,
the probability that the photon exits a stack of $N$ polarizers
(rotated into the allowed $2$-polarization direction) is given by
the absolute square of the transmission amplitude,
\begin{equation}
P_N=|T_2(N)|^2.
\end{equation}
From Eqs.(54), (59), and (61)-(63), with each of the angles in
$(\theta_1,...,\theta_N)$ set equal to $(\pi/2N)$, one may compare
the the projection postulate $P_N$ in Eq.(7) to the Maxwell theory,
or equivalently the photon Schr\"odinger Eq.(63), without the projection
postulate. The matrix multiplications in Eq.(61) and the linear algebra
in Eq.(62) are best left to a computer. The results of such a computation
are shown in Fig.3.

\begin{figure}[htbp]
\begin{center}
\mbox{\epsfig{file=zfig3.eps,height=70mm}}
\caption{The Zeno effect probability $P_N$ for a photon to pass
through $N$ polarizers: (i)The solid curve (evaluated for integer $N$) is
the projection postulate in Eq.(7). (ii) The filled circles represent
the Maxwell theory with zero attenuation for the allowed polarization
direction. (iii) The filled squares represent the Maxwell theory with a
small attenuation even for the allowed polarization direction.}
\label{zfig3}
\end{center}
\end{figure}

The solid curve in Fig.3 is the prediction of the projection postulate
Eq.(7). For the Maxwell theory with a reasonable stack of laboratory
polarizers, there may be $\sim 20$ wavelengths of light in each polarizer
in the stack. Thus we chose the parameter $\xi=(\omega a/c)=100$ for
purpose of illustration. We further choose  $\tilde{\epsilon}_1=1$
to describe the polarization direction for which the polarizer
would be transparent and the complex $\tilde{\epsilon}_1=1+2i$
to describe the polarization direction for which the polarizer
would be strongly attenuated (if not absolutely opaque).
The resulting computed points are the filled circles of Fig.3.
For this case, the projection postulate yields a $P_N$ which is quite close
to the full Maxwell theory.

For some regimes (perhaps more close to the laboratory situation),
the Maxwell theory begins to diverge from from the projection postulate
for large $N$. In the laboratory all polarization directions are subject
to small (but certainly finite) attenuation. To test this situation,
we computed the Maxwell theory prediction when the allowed
polarization has a weak attenuation response
$\tilde{\epsilon}_1=1.1+0.001i$ while the disallowed polarization
has a stronger attenuation response
$\tilde{\epsilon}_2=1.1+0.05i$. The results are the filled squares of
Fig.3. For $N\le 3$, there is a Zeno effect in which $P_N$ increases.
However for $N\ge 6$, the is an attenuation induced decrease in $P_N$,
contrary to the ``Zeno effect expectation''. However, one might regard
this attenuation of the allowed polarization direction to be merely an
indication of poor experimental design.

Some final comments about the calculations producing Fig.3 are
worthy of note. The transfer matrix formalism used to calculate the
transmission probability $P_N$ was tedious, but straight forward. However,
the numerical computations required to produce Fig.3 were not entirely
trivial. The first three numerical commercial software programs that
we employed for the calculation produced complete nonsense\cite{41}.
The physical problem was that the Maxwell waves reflect back and forth
within the polarizers. Hence, there exist {\em exponentially attenuated}
terms as well as {\em exponentially amplified} terms in
the transfer matrix. High accuracy was needed for the matrix
multiplications in Eq.(61). Only with high accuracy at intermediate
stages of the calculation, could Eqs.(62) and (63) be
reliably computed. For realistic laboratory polarizers, the
required accuracy turns out to be very high indeed. Fortunately,
Aberth\cite{42} has developed precise numerical algorithms together
with publicly available programming code which allows for $\sim 10^4$
significant figure accurate computations. Much  of the required code
for the standard functions and standard linear algebra was thus available.
We used this code. However, we required $\sim 10^3$ significant figure
computations to produce the reliable plots shown in Fig.3.

\section{Conclusions}

In discussions of the foundations of quantum mechanics there has been
a tendency to regard all interference of waves as explicitly due
to quantum mechanics. This is only sometimes true, since ``waves'' as well
as ``particles'' also have classical limits. In many (perhaps most)
of the electromagnetic experiments and applications, the Maxwell wave
itself is classical. The {\em distinction} between a classical wave,
a classical particle, and a quantum particle has been discussed simply,
{\it yet clearly}, by Feynman\cite{43}.

(i) A quantum particle is {\em certainly not} a classical particle.
Quantum particle dynamics dictates amplitude superposition,
whereas classical dynamics dictates probability superposition.
(ii) A quantum particle is {\em certainly not} a classical wave.
A classical wave is ``smooth'' while a quantum particle comes in ``lumps''.
Photons are lumpy. Photon counters yield one click whenever a
photon is detected. {\em One} photon can never make {\em two} different
counters each give you {\em one half} of a click. That is much too smooth
for quantum mechanics. In quantum mechanics, each photon counter gives
one click or no clicks. The photon is {\it certainly not} a classical
wave. The photon is {\em certainly not} a classical particle. The photon is
{\em certainly not} a little bit classical wave and a little bit
classical particle. (iii) The photon is {\em neither} a classical wave
{\em nor} a classical particle {\em nor} a little bit of both.

Electromagnetic waves, as described by Maxwell, are classical waves.
If a radio station broadcasts a signal, then the signal is
(for all practical purposes) smooth. Your radio gets
some of the signal, and our radio gets some of the signal.
There is enough smoothly distributed classical wave to go around
to all of the radios. The classical Maxwell wave is smooth. One detector
cannot get a fraction of a photon. Remember, photons come in lumps.
One photon can get destroyed in at most one radio receiver. One
detector can get any fraction of a classical Maxwell wave. Remember,
classical waves are smooth. Yet the Schr\"odinger equation for photon
is the same (in a mathematical formalism) as the Maxwell equations for
classical electromagnetic waves. This is because the photon has no mass
and because the photon is a Boson. When very many photons are Bose
condensed into a single quantum state, the result is a Maxwell wave
which is classical for all practical purposes.

The above picture was not all that clear to Einstein (by his own admission),
and we do not claim to know more about photons than did Einstein.
According to Einstein, a Maxwell wave hits a half silvered mirror
and some of the wave goes to one detector and the rest of the wave
goes to a second detector. If the wave represents one photon, then
only one counter goes click. The other counter gets whacked with a
Maxwell wave. Yet this counter counts nothing. When there is but one
photon, only one counter can speak for her. Some Maxwell wave
parts turn out to have have a photon, and some Maxwell wave parts
turn out not have a photon. Some counters go click and some counters
remain mute. You never know. And photons are lumpy. It is a very strange
shell game according to Einstein\cite{44}.

However, the Zeno effect formally attributed to the quantum mechanical
projection postulate has here been argued to be merely a classical wave
effect. Maxwell would not have had to increase his knowledge of
electromagnetic theory by even one iota in order to totally
understand the polarization version of the Zeno effect. No quantum
mechanics is required, not to even speak of the projection postulate.

\vfill \eject

\medskip
\centerline{\bf Acknowledgement}
\medskip
\par \noindent
This work has benefited from the allocation of time at the Northeastern
University High-Performance Computing Center (NU-HPCC).
\medskip