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\begin{document}
\openup 1.5\jot
\centerline{e to the A, in a New Way, Some More to Say}
\vspace{1in}
\centerline{Paul Federbush}
\centerline{Department of Mathematics}
\centerline{University of Michigan}
\centerline{Ann Arbor, MI 48109-1109}
\centerline{(pfed@math.lsa.umich.edu)}
\vspace{1in}
\centerline{\underline{Abstract}}
Expressions are given for the exponential of a hermitian matrix, $A$. Replacing $A$ by $iA$ these are explicit formulas for the Fourier transform of $e^{iA}$. They extend to any size $A$ the previous results for the $2 \times 2, \ 3 \times 3$, and $4 \t
imes 4$ cases.
\vfill\eject
In a previous paper,[1], explicit formulas were obtained for the Fourier transform of $e^{iA}$, making manifest the theorem of E. Nelson [2], on the support of such Fourier transformations. But this was only done for $2\times2, 3 \times 3$ and $4 \times
4$ matrices. We now treat the general case.
Let $A$ be an $r \times r$ hermitian matrix. We write
\be {\rm Det}(1-A) = \sum^r_{j=0} P_j(A) \ee
where $P_j(A)$ is homogeneous of degree $j$ in the entries of $A$. The formulas we obtained for the exponential of $A$ are as follows:
\be
(e^A)_{\al\beta} = \frac 1{\Gamma(r)} \sum^r_{j=0} P_j(A) \frac {d^{r-j}}{ds^{r-j}} \left( \int d\Omega e^{sTr(AW)} W_{\al\beta} s^r\right) \bigg|_{s=1}
\ee
or
\be
(e^A)_{\al\beta} = \frac 1{\Gamma(r)} \sum^r_{j=0} P_j(A) \frac {d^{r-j}}{ds^{r-j}} \left( \int d\Omega e^{s} n_\al\bar{n}_\beta s^r\right) \bigg|_{s=1}
\ee
Here $\vec n$ is a unit vector in $\C^r$, and $W_{ij} = n_i \bar n_j$, a rank one hermitian matrix. $\int d\Omega$ denotes a normalized integral over all such $\vec n$, an integral over the unit sphere in $\C^r$ with unitary-invariant measure.
In fact the formulas in (2) and (3) do not coincide with formulas in [1] when $r=2,3$ or 4, but formulas {\it of such type} are not unique. We do not know the full scope of such non-uniqueness.
We sketch a derivation/proof of formulas (2) and (3), especially emphasizing the {\bf ideas}. We note the relation between gaussian integrals in $d=2r$ real dimensions, and integrals over the corresponding unit sphere $S^{d-1}$.
\begin{eqnarray}
\frac 1 {\cal N} \int dx_ie^{-\Sigma x^2_i} \prod^{2N} x_{\al(i)}
&=& \frac 1 {\cal N} \int r^{d-1} r^{2N} e^{-r^2} dr\int d\Omega' \prod^{2N} n_{\al(i)} \\
&=& \int_{S^{d-1}} d\Omega \prod^{2N}n_{\al(i)} \cdot \frac{\int dre^{-r^2}r^{2N+d-1}}{\int dr e^{-r^2} r^{d-1}} \\
&=& \int_{ S^{d-1}} d\Omega \prod^{2N}n_{\al(i)} \frac{\Gamma\left(\frac{2N+d}{2}\right)} {\Gamma\left( \frac {d}{2} \right)} .
\end{eqnarray}
Here $n_i$ is the unit vector parallel to $x_i$, and $\int d\Omega'$ is integral over the sphere in its usual measure and $\int d \Omega$ the normalized spherical measure. From (6) we see {\it the integral over a unit sphere of a homogeneous polynomial o
f degree $2N$ is ``approximately" $1/N!$ the gaussian integral of the same polynomial.}
We note that multiplying the term in $A^k$ by $\frac 1 {k!}$ induces a transform (formally) as follows
\be 1 + A + A^2 + \cdots = \frac 1{1-A} \longrightarrow e^A \ . \ee
We consider the gaussian integral formula:
\be
\frac{{\rm Det}(1-A)}{{\cal N}} \int dx_i e^{-\Sigma|x^2_i|+} x_\al\bar{x}_\beta = \left( \frac 1{1-A}\right)_{\al \beta}.
\ee
In the expansion of the integrand on the left side of (8) each power of $A$ has associated to it two powers of $x$. Thus converting from a gaussian integral to an integral over a unit sphere approximately multiplies each power of $A^N$ by $\frac 1 {N!}$,
which would convert $\frac 1 {1-A}$ to $e^A$. The {\bf wrong} formula we get putting these ideas together would yield:
\be
" {\rm Det}(1-A) \int d\Omega e^{} n_\al\bar{n}_\beta = (e^A)_{\alpha\beta} "
\ee
It is just a couple of hours work to correct formula (9) (since there are powers of $A$ in Det$(1-A)$, not just in the integrand; and because the transformation to the unit sphere does not yield exactly $1/N!$) and arrive at our formulas (2) and (3).
\underline{Acknowledgment}: I would like to thank Professor Alexander Barvinok for an all important discussion on evaluating integrals over the unit sphere.
\vfill\eject
\centerline{References}
\begin{itemize}
\item[[1]] P. Federbush, ``e to the A, in a New Way", math-ph/9903006, to be published in the {\it Michigan Math. Journal}.
\item[[2]] E. Nelson, {\it Operants: A functional calculus for non-commuting operators}, Functional Analysis and Related Fields, Proceedings for a conference in honor of Professor Marshal Stone (Univ. of Chicago, May 1968)(F.E. Browder, ed.), Springer-Ve
rlag, Berlin, Heidelberg, and New York, 1970, pp. 172-187. MR 54:978.
\end{itemize}
\end{document}