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\parskip=0pt
\parindent=3em

% START Title section
\topinsert
\vskip2.0truepc
\endinsert
    
{\baselineskip=15truept
\parindent=0pt
{\bf             EXISTENCE OF RADIATIVELY INDUCED FERMION RESONANCE (RFR)  FROM THE STOKES PARAMETERS }
}
\vskip32truept
% END Title section
    
% START.author.affiliation
{\parindent=.8truein
{\bf M. W. Evans$^{1,2}$}
\vskip 12truept
    
{\it
$^{1}$Alpha Foundation and Laboratories
    
   Institute of Physics

   11 Rutafa Street
    
    Budapest, Hungary
    
\vskip 12truept
    
    $^{2}$J.R.F., 1975
    
    Wolfson College
    
    Oxford, Great Britain
    
}
\vone

    Received Spring, 1998
}
% END.author.affiliation
%\baselineskip=24truept    
\vone

\noindent
The classical polarization tensor of light is shown to contain a term proportional to the vector product of the Pauli matrix  $\vecsigma_{3}$       with the conjugate product $\AxAast$         of complex magnetic vector potentials. This simple analysis confirms the existence of a first order interaction energy,
$$
En :=: -  {e^{2} \over 2m} \vecsigma_{3} \cdot
\AxAast  \, ,
$$
between classical, circularly polarized radiation and a fermion or classical charged particle.
\vone

\noindent
Key words:  Radiatively induced fermion resonance, Stokes parameters, $\b3$ field.
\vtwo

\noindent
{\bf1.~INTRODUCTION}
\vone

The classical polarization tensor of light [1] contains a term that shows the ability of circularly or elliptically polarized radiation to induce fermion resonance, a phenomenon which we have named radiatively induced fermion resonance ($RFR$) [2--5]. The resonance frequency induced in this way is proportional to $ I /\omega^{2} $    where  $\omega$     is the angular frequency and $I$ the intensity of the radiation, in units of radians per second and watts per square metre respectively. The theory produces [2--5] electron and proton spin resonance at frequencies approaching the visible range under accessible conditions, thus greatly increasing the instrumental resolution of current magnet based instruments of any kind. The $RFR$ phenomenon can also be used to supplement the permanent magnet [6], producing in principle a greater instrumental resolution and site specific effects by second order perturbation theory [5]. 
\vtwo

\noindent
{\bf2.~THE STOKES PARAMETERS}
\vone

Define the four Stokes parameters [6,7] in terms of the components of the magnetic vector potential
$$
\matrix
{
S_{0} :=  A_{x} A_{x}^{\ast}  +  A_{y}A_{y}^{\ast}\, , \cr
S_{1} :=  A_{x} A_{x}^{\ast}  -  A_{y}A_{y}^{\ast}\, , \cr
S_{2} :=  A_{x} A_{y}^{\ast}  +  A_{y}A_{x}^{\ast}\, , \cr
S_{3} :=  -i (A_{x}A_{y}^{\ast}  -  A_{y}A_{x}^{\ast}) \, .
}
\eqno(1)
$$
In circularly polarized radiation
$$
S_{1} = S_{2} = 0
\eqno(2)
$$
and the general relation,
$$
S_{0}^{2} = S_{1}^{2} + S_{2}^{2} + S_{3}^{2} \, ,
\eqno(3)
$$
reduces to
$$
S_{0} = \pm S_{3} \, .
\eqno(4)
$$
Therefore the existence of 
$$
S_{0} := \mid \vecA \cdot \vecA^{\ast} \mid \, ,
\eqno(5)
$$
implies that of
$$
S_{3} := \mp i \mid \AxAast \mid
\eqno(6)
$$
in circularly polarized radiation. The two signs in Eq.~(6) denote left and right handed circular polarization respectively.        
\vtwo

\noindent
{\bf 3.~THE CLASSICAL LIGHT INTENSITY TENSOR}
\vone

We can choose to describe the light intensity tensor [6--10] in terms of a complex, rank two, tensor with a symmetric real and antisymmetric imaginary part,
$$
I_{ij1} \propto
\pmatrix
{
A_{x} A_{x}^{\ast} & A_{x} A_{y}^{\ast} \cr
A_{y} A_{x}^{\ast}  & A_{y} A_{y}^{\ast}
} \,  ,
\eqno(7)
$$
or as a real Stokes matrix,
$$
I_{ij2} \propto
\pmatrix
{
A_{x} A_{x}^{\ast} & - i A_{x} A_{y}^{\ast} \cr
-i A_{y} A_{x}^{\ast} & A_{y} A_{y}^{\ast}
} \,  .
\eqno(8)
$$
In both cases, the tensor can be expressed in terms of the Stokes parameters, respectively,
$$
I_{ij1} \propto {1 \over 2}
\pmatrix
{
S_{0}  + S_{1} & S_{2}  + iS_{3} \cr
S_{2}  - i S_{3} & S_{0}  - S_{1}
} \, ,
\eqno(9)
$$
and
$$
I_{ij2} \propto {1 \over 2}
\pmatrix
{
S_{0}  + S_{1} & S_{3}  - i S_{2} \cr
- S_{3}  -i S_{2} & S_{0}  - S_{1}
} \, .
\eqno(10)
$$
These matrices can be expressed as combinations of Pauli matrices as follows. Resonance can be induced between the states of the appropriate Pauli matrix as usual.
\vtwo

\noindent
{\bf 4.~THE PAULI MATRICES}
\vone

There are several representations possible for the Pauli matrices, as discussed for example by Sakurai [11], they are defined by the $SU(2)$ group symmetry in any situation in dynamics and electrodynamics. Therefore we express the real and physical Stokes matrix (10  ) as a linear combination of the Pauli matrices
$$
\sigma_{0} = 
\pmatrix
{
1 & 0 \cr
0 & 1
} 
\qquad
\sigma_{1} = 
\pmatrix
{
1 & 0 \cr
0 & -1
}
\;\;\;\; 
$$
$$
\sigma_{3} = 
\pmatrix
{
0 & 1 \cr
1 & 0
} 
\qquad
\sigma_{3} = 
\pmatrix
{
0 & i \cr
-i & 0
} 
\qquad
\eqno(11)
$$
which form the $SU(2)$ relation,
$$
\lbrack {\sigma_{1} \over 2} , {\sigma_{2}\over 2 }\rbrack =
-i { \sigma_{3}\over 2} {\rm ~et~cyclicum}.
\eqno(12)
$$
For circularly polarized radiation the Stokes matrix is the real valued sum of two Pauli matrices,
$$
I_{ij2} \propto
{1 \over 2}
\pmatrix
{
S_{0} & S_{3} \cr
-S_{3} & S_{0}
}
=
{1 \over 2}
( S_{0} \sigma_{0} - i  S_{3}  \sigma_{3}) \, ,
\eqno(13)
$$
and it is clear that resonance can be induced between the states of the real valued matrix,
$$
-i \sigma_{3} = 
\pmatrix
{ 
0 & 1 \cr
-1 & 0
}.
$$  
\vtwo

\noindent
{\bf DISCUSSION}
\vone

The energy of interaction between a classical charged particle and the classical Stokes matrix (13) is dimensionally proportional to,
$$
En \propto {e^{2} \over 2m} A^{(0)2} (\sigma_{0} - i \sigma_{3}),
\eqno(14)
$$
where $e / m$ is the charge to mass ratio of the particle. Assuming that the constant of proportionality is unity we arrive at the same result as derived in a number of different ways in Ref. 3 from the minimal prescription. For an electron, and in $SI$ units, the expected resonance frequency is [3,4], 
$$
\omega_{\rm res} = 1.007 \times 10^{28} {I \over \omega^{2}} \, ,
\eqno(15)
$$
and appears in the far infra red to visible region under accessible conditions [3,4 ]. 

In summary we have derived the phenomenon of radiatively induced fermion resonance in the simplest possible way, by re-expressing the Stokes matrix in terms of a sum of two Pauli matrices. Other derivations of the same result are given in Ref. 3. 
\vtwo

\noindent
{\bf ACKNOWLEDGEMENTS}
\vone

The Alpha Foundation is thanked for a professorship, honoris cause, and a presidium membership. Many colleagues worldwide are thanked for helpful internet discussions.
\vtwo

\noindent
{\bf REFERENCES}
\vone

\item{[1]} L.~D.~Landau and E.~M.~Lifshitz, {\it Electrodynamics of Continuous Media} (Pergamon, New York, 1960). 
\item{[2]} M.~W.~Evans, J.-P.~Vigier, S.~Roy, and S.~Jeffers, {\it The Enigmatic Photon, Volume 3: Theory and Practice of the $\b3$ Field}, Chaps.~1 and 2 (Kluwer Academic, Dordrecht, 1996). 
\item{[3]} M.~W.~Evans, {\it Found.~Phys.~Lett.}, submitted for publication. 
\item{[4]} M.~W.~Evans, J.-P.~Vigier and M.~Meszaros, eds., {\it The Enigmatic Photon, Volume 5}, Chap.~3 (Kluwer Academic, Dordrecht, 1999), in preparation. 
\item{[5]} L.~Li, T.~He, X.~Wang, and F.-C.~Liu, {\it Chem.~Phys.~Lett.~} {\bf 268}, 549 (1997). 
\item{[6]} M.~W.~Evans and S.~Kielich, eds., {\it Modern Nonlinear Optics} Vol.~85(2) of {\it Advances in Chemical Physics}, I.~Prigogine and S.~A.~ Rice, eds., paperback edn.~(Wiley Interscience, New York, 1997).
\item{[7]} M.~Born and E.~Wolf, {\it Principles of Optics}, 6th edn.~(Pergamon, Oxford, 1975). 
\item{[8]} L.~D.~Barron, {\it Molecular Light Scattering and Optical Activity} (Cambridge University Press, Cambridge, 1982). 
\item{[9]} K.~Knast and S.~Kielich, {\it Acta Phys.~Polonica} {\bf 55A}, 319 (1979). 
\item{[10]} S.~Kielich, {\it Acta Phys.~Polonica} {\bf 32}, 405 (1967). 
\item{[11]} J.~J.~Sakurai, {\it Advanced Quantum Mechanics}, 11th printing, Chap.~3 (Addison Wesley, New York, 1967).
\bye
