Ian & Stuart's Australian Mac 1993 September/September 93.iso/Archives/Applications/Text/Conversion/RTF->TeX/RTF.tex1

Text File | 1992-11-27 | 7.6 KB | 172 lines | [TEXT/R*ch]

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\Times The two-dimensional Dirac equation for a function {\it
$\psi$\/}DFO2():DFO2(){\GreekAB  }$^2$~ ~{\GreekAB \`{U}}$^2$ is
given by


\parindent=72pt  {\it $_t$\/}{\it $\psi$\/}({\it t\/},{\it x\/})  =
[{\it $\alpha$\/}( {\it $_x$\/}  {\it iA\/}$_1$({\it t\/},{\it
x\/}))  {\it i\/}.{\it m\/}{\it $\beta$\/} + {\it i\/}.{\it
A\/}$_0$({\it t\/},{\it x\/})]{\it $\psi$\/}({\it t\/},{\it x\/}),
{\it t\/} $\in$ {\GreekAB  }, {\it x\/} $\in$ {\GreekAB  },\qquad (XX)


\parindent=0pt in a system of physical units in which the light
velocity{\it  c\/} and Planck's constant h are equal to one. The 2 2
matrices {\it $\alpha$\/} and {\it $\beta$\/} are hermitian with {\it
$\alpha$\/}$^2$ = {\it $\beta$\/}$^2$ = 1 and {\it
$\alpha\beta$\/} + {\it $\beta\alpha$\/} = 0. Both {\it A\/}$_0$
and {\it A\/}$_1$ are real valued functions on {\GreekAB  }$^2$.
The number {\it m\/} represents the rest mass of the particle whose
state is associated with the function {\it $\psi$\/}.

If for the moment we suppose that the functions {\it A\/}$_0$ and
{\it A\/}$_1$ are both identically zero on {\GreekAB  }$^2$, then
equation (XX) becomes


\parindent=72pt  {\it $_t$\/}{\it $\psi$\/}({\it t\/},{\it x\/})  =
[{\it $\alpha$\/} {\it $_x$\/}  {\it i\/}.{\it m\/}{\it
$\beta$\/}]{\it $\psi$\/}({\it t\/},{\it x\/}),DFO24(){\it t\/} $\in$
{\GreekAB  }, {\it x\/} $\in$ {\GreekAB  }.\qquad (XY)


\parindent=0pt On any space {\it L\/}{\it $^p$\/}({\GreekAB
}$^2$,{\GreekAB \`{U}}$^2$), 1 $\le$ {\it p\/} $<$ , the operator
{\it $\alpha$\/} {\it $_x$\/}  is associated with a continuous,
uniformly bounded group {\it S\/}{\it $_p$\/}({\it t\/}), {\it t\/}
$\in$ {\GreekAB  }, of linear transformations on {\it L\/}{\it
$^p$\/}({\GreekAB  }$^2$,{\GreekAB \`{U}}$^2$): there exists a 2
2 unitary matrix {\it U\/} such that {\it U\/}{\it $\alpha$\/}{\it
U\/}SUP6(1) = B(ACO2HS3(1,0,0,1)), so that {\it US\/}{\it
$_p$\/}({\it t\/}){\it U\/}SUP6(1){\it $\phi$\/}({\it x\/}) = ({\it
$\phi$\/}$_1$({\it x\/}+{\it t\/}),{\it $\phi$\/}$_2$({\it
x\/}{\it t\/})) for {\it $\phi$\/} $\in$ {\it L\/}{\it
$^p$\/}({\GreekAB  }$^2$,{\GreekAB \`{U}}$^2$). For {\it p\/} =
, we have continuity for the weak*-topology {\it $\sigma$\/}({\it
L\/}$$({\GreekAB  }$^2$,{\GreekAB \`{U}}$^2$),{\it
L\/}$^1$({\GreekAB  }$^2$,{\GreekAB \`{U}}$^2$)). Each of the
component functions {\it $\phi$\/}$_1$ and {\it $\phi$\/}$_2$ is
transformed according to the action of a translation in {\GreekAB  }$^2$.

It follows from the Trotter product formula [K ] that the operator
{\it $\alpha$\/} {\it $_x$\/}  {\it i\/}.{\it m\/}{\it $\beta$\/} is
also associated with with a uniformly bounded group of operators on
{\it L\/}$$({\GreekAB  }$^2$,{\GreekAB \`{U}}$^2$), so Theorem
XYZ shows that there are countably additive operator valued measures
associated with equation (XY) and solutions to equation (XX) can be
represented by integrals with respect to these measures.

Similarly, the wave equation in two space-time dimensions is


 {\it $_t$\/}$^2${\it $\psi$\/}({\it t\/},{\it x\/})  = {\it
c\/}$^2$  {\it $_x$\/}$^2${\it $\psi$\/}({\it t\/},{\it x\/})  ,
 {\it $\psi$\/}(0,{\it x\/}) = {\it f\/}({\it x\/})  ,  {\it
$\psi$\/}{\it $_t$\/}(0,{\it x\/}) = {\it g\/}({\it x\/}) , {\it x
\/}$\in${\it  \/}{\GreekAB  }, {\it t\/}   0.

\hfil\break
Let {\it v\/}({\it t\/},{\it x\/}) =  {\it $_t$\/}{\it u\/}({\it
t\/},{\it x\/}) - {\it c\/}  {\it $_x$\/}{\it u\/}({\it t\/},{\it
x\/}) ,  {\it $_t$\/}{\it v\/}({\it t\/},{\it x\/}) + {\it c\/}
{\it $_x$\/}{\it v\/}({\it t\/},{\it x\/}) = 0, with


{\it v\/}(0,{\it x\/})  =  {\it g\/}({\it x\/}) - {\it cf\/} '({\it
x\/}), {\it u\/}({\it x\/}) = {\it f\/}({\it x\/}), for all {\it x
\/}$\in${\it  \/}{\GreekAB  }.

\hfil\break
On setting{\it  $\phi=$ \/}B(A({\it u\/},{\it v\/})), the equation
becomes the first order system


 {\it $_t$\/}{\it $\phi$ \/}= {\it A\/}  {\it $_x$\/}{\it
$\phi$\/} + {\it iQ\/}{\it $\phi$ \/} , {\it Q \/}= B(A(0{\it
\/}-{\it i\/},0 0)),  {\it A  \/}= B(A({\it c  \/}0,0 -{\it
c\/})),{\it \/}


with the initial condition   {\it $\phi$\/}(0,{\it x\/}) ={\it
\/}B(A({\it f\/}({\it x\/}),{\it g\/}({\it x\/}) - {\it cf\/} '({\it
x\/}))), {\it x \/}$\in${\it  \/}{\GreekAB  }.

The operator {\it A\/} {\it $_x$\/}{\it $\phi$\/} + {\it iQ\/} is
also associated with with a uniformly bounded group of operators on
{\it L\/}$$({\GreekAB  }$^2$,{\GreekAB \`{U}}$^2$), so
perturbations to the wave eqaution may also be represented in terms of
path integrals.

\parindent=18pt Similar considerations apply to the {\it N\/} {\it
N\/} hyperbolic system of the first order


\parindent=15pt  {\it $_t$\/}{\it $\psi$\/}({\it t\/},{\it x\/})  =
[ISU({\it l\/}=1,{\it d\/}, ){\it P\/}{\it $_l$\/}( {\it
$_x$\/}$_{SDO2(}${\it $_i$\/}$_)$  {\it iA\/}{\it $_l$\/}({\it
t\/},{\it x\/})) + {\it iQ\/} + {\it iV\/}({\it t\/},{\it x\/}) ]{\it
$\psi$\/}({\it t\/},{\it x\/}), 0 $<$ {\it t\/} $<$ {\it T\/}, {\it x
\/}$\in$ {\bf R}{\it $^d$\/},\qquad (XXA)

\hfil\break
\parindent=0pt where 0 $<$ {\it T\/} $<$ , and {\it P\/}{\it
$_l$\/}, 1 $\le$ {\it l\/} $\le$ {\it d\/}, and {\it Q\/} are
constant {\it N\/} {\it N\/}-matrices, and {\it A\/}{\it
$_l$\/}({\it t\/},{\it x\/}), 1 $\le$ {\it l\/} $\le$ {\it d\/}, and
{\it V\/}({\it t\/},{\it x\/}),  0 $\le$ {\it t\/} $\le$ {\it T\/},
{\it x \/}$\in$ {\bf R}{\it $^d$\/} are real-valued functions. The
function {\it $\psi$\/} has values in {\bf C}{\it $^N$\/}. It is
assumed that {\it P\/}{\it $_l$\/} , 1 $\le$ {\it l\/} $\le$ {\it d
\/}have only real eigenvalues, and that they are simultaneously
diagonalizable. The path space measures associated with the first
order hyperbolic system (XXA) were first considered by T. Ichinose
[Ich ] who examined properties of the fundamental solution of the
system (XXA).

The alternate viewpoint using the Trotter product formula outlined
above was formulated in [Jeff  ] as an application of Theorem ZZZ. The
essence of this approach is that there is a collection of dynamical
systems, represented by translations along the {\it x\/}{\it
$_i$\/}-axes, 1 $\le$ {\it i\/} $\le$ {\it d\/} for the equations
(XXA), that act independently on components of the state vector prior
to suffering a {\it mixing\/} of components via a semigroup of
operators, for example, the semigroup generated by the constant matrix
{\it iQ\/} in equation (XXA).

The operator ISUIN({\it l\/}=1,{\it d\/}, ){\it P\/}{\it $_l$\/}
{\it $_x$\/}$_{SDO2(}${\it $_i$\/}$_)$  in (XXA) can be written
more suggestively as\hfil\break
 IIN({\it $_K$\/},, )ISUIN({\it l\/}=1,{\it d\/}, ){\it
$\lambda$\/}{\it $_i$\/} {\it $_x$\/}$_{SDO2(}${\it
$_i$\/}$_)$ d{\it R\/}({\it $\lambda$\/}$_1$,...,{\it
$\lambda$\/}{\it $_d$\/}). The matrices {\it P\/}{\it $_l$\/} , 1
$\le$ {\it l\/} $\le$ {\it d\/} are simultaneously diagonalizable, so
there exists a discrete spectral measure {\it R\/} acting on {\GreekAB
\`{U}}{\it $^N$\/} such that {\it P\/}{\it $_i$\/} =  IIN({\it
$_K$\/},, ){\it $\lambda$\/}{\it $_i$\/} d{\it R\/}({\it
$\lambda$\/}$_1$,...,{\it $\lambda$\/}{\it $_d$\/}), for each 1
$\le$ {\it l\/} $\le$ {\it d\/}; the set {\it K\/} is the joint
spectrum of the system of matrices ({\it P\/}$_1$,...,{\it P\/}{\it
$_d$\/}).

Our aim now is to apply these ideas in the general setting of a direct
sum of dynamical systems over a single measure space ({\it
$\Sigma$\/},{\ZapfChancery E},{\it $\mu$\/}):



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