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%  *******************  Journal refs **********************

\def\aop#1#2#3{{\it Ann. Phys.} {\bf {#1}} (19{#2}) #3}
\def\cjp#1#2#3{{\it Can. J. Phys.} {\bf {#1}} (19{#2}) #3}
\def\cmp#1#2#3{{\it Comm. Math. Phys.} {\bf {#1}} (19{#2}) #3}
\def\cqg#1#2#3{{\it Class. Quant. Grav.} {\bf {#1}} (19{#2}) #3}
\def\jcp#1#2#3{{\it J. Chem. Phys.} {\bf {#1}} (19{#2}) #3}
\def\ijmp#1#2#3{{\it Int. J. Mod. Phys.} {\bf {#1}} (19{#2}) #3}
\def\jgp#1#2#3{{\it J. Geom. and Phys.} {\bf {#1}} (19{#2}) #3}
\def\jmp#1#2#3{{\it J. Math. Phys.} {\bf {#1}} (19{#2}) #3}
\def\jpa#1#2#3{{\it J. Phys.} {\bf A{#1}} (19{#2}) #3}
\def\mplA#1#2#3{{\it Mod. Phys. Lett.} {\bf A{#1}} (19{#2}) #3}
\def\np#1#2#3{{\it Nucl. Phys.} {\bf B{#1}} (19{#2}) #3}
\def\pl#1#2#3{{\it Phys. Lett.} {\bf {#1}} (19{#2}) #3}
\def\phm#1#2#3{{\it Phil.Mag.} {\bf {#1}} ({#2}) #3}
\def\prp#1#2#3{{\it Phys. Rep.} {\bf {#1}} (19{#2}) #3}
\def\pr#1#2#3{{\it Phys. Rev.} {\bf {#1}} (19{#2}) #3}
\def\prA#1#2#3{{\it Phys. Rev.} {\bf A{#1}} (19{#2}) #3}
\def\prB#1#2#3{{\it Phys. Rev.} {\bf B{#1}} (19{#2}) #3}
\def\prD#1#2#3{{\it Phys. Rev.} {\bf D{#1}} (19{#2}) #3}
\def\prl#1#2#3{{\it Phys. Rev. Lett.} {\bf #1} (19{#2}) #3}
\def\ptp#1#2#3{{\it Prog. Theor. Phys.} {\bf #1} (19{#2}) #3}
\def\tmp#1#2#3{{\it Theor. Mat. Phys.} {\bf #1} (19{#2}) #3}
\def\rmp#1#2#3{{\it Rev. Mod. Phys.} {\bf {#1}} (19{#2}) #3}
\def\zfn#1#2#3{{\it Z. f. Naturf.} {\bf {#1}} (19{#2}) #3}
\def\zfp#1#2#3{{\it Z. f. Phys.} {\bf {#1}} (19{#2}) #3}

\def\asens#1#2#3{{\it Ann. Sci. \'Ecole Norm. Sup. (Paris)} {\bf{#1}} 
(#2) #3} 
\def\aihp#1#2#3{{\it Ann. Inst. H. Poincar\'e (Paris)} {\bf{#1}} (#2) #3} 
\def\cras#1#2#3{{\it Comptes Rend. Acad. Sci. (Paris)} {\bf{#1}} (#2) #3} 
\def\prs#1#2#3{{\it Proc. Roy. Soc.} {\bf A{#1}} (19{#2}) #3}
\def\pcps#1#2#3{{\it Proc. Camb. Phil. Soc.} {\bf{#1}} (19{#2}) #3}
\def\mpcps#1#2#3{{\it Math. Proc. Camb. Phil. Soc.} {\bf{#1}} (19{#2}) #3}

\def\amsh#1#2#3{{\it Abh. Math. Sem. Ham.} {\bf {#1}} (19{#2}) #3}
\def\am#1#2#3{{\it Acta Mathematica} {\bf {#1}} (19{#2}) #3}
\def\aim#1#2#3{{\it Adv. in Math.} {\bf {#1}} (19{#2}) #3}
\def\ajm#1#2#3{{\it Am. J. Math.} {\bf {#1}} ({#2}) #3}
\def\amm#1#2#3{{\it Am. Math. Mon.} {\bf {#1}} (19{#2}) #3}
\def\adm#1#2#3{{\it Ann. der  Math.} {\bf {#1}} ({#2}) #3}
\def\aom#1#2#3{{\it Ann. of Math.} {\bf {#1}} (19{#2}) #3}
\def\cjm#1#2#3{{\it Can. J. Math.} {\bf {#1}} (19{#2}) #3}
\def\cpde#1#2#3{{\it Comm. Partial Diff. Equns.} {\bf {#1}} (19{#2}) #3}
\def\cm#1#2#3{{\it Compos. Math.} {\bf {#1}} (19{#2}) #3}
\def\dmj#1#2#3{{\it Duke Math. J.} {\bf {#1}} (19{#2}) #3}
\def\invm#1#2#3{{\it Invent. Math.} {\bf {#1}} (19{#2}) #3}
\def\ijpam#1#2#3{{\it Ind. J. Pure and Appl. Math.} {\bf {#1}} (19{#2}) #3}
\def\jdg#1#2#3{{\it J. Diff. Geom.} {\bf {#1}} (19{#2}) #3}
\def\jfa#1#2#3{{\it J. Func. Anal.} {\bf {#1}} (19{#2}) #3}
\def\jlms#1#2#3{{\it J. Lond. Math. Soc.} {\bf {#1}} (19{#2}) #3}
\def\jmpa#1#2#3{{\it J. Math. Pures. Appl.} {\bf {#1}} ({#2}) #3}
\def\ma#1#2#3{{\it Math. Ann.} {\bf {#1}} ({#2}) #3}
\def\mz#1#2#3{{\it Math. Zeit.} {\bf {#1}} ({#2}) #3}
\def\ojm#1#2#3{{\it Osaka J.Math.} {\bf {#1}} ({#2}) #3}
\def\pams#1#2#3{{\it Proc. Am. Math. Soc.} {\bf {#1}} (19{#2}) #3}
\def\pems#1#2#3{{\it Proc. Edin. Math. Soc.} {\bf {#1}} (19{#2}) #3}
\def\pja#1#2#3{{\it Proc. Jap. Acad.} {\bf {A#1}} (19{#2}) #3}
\def\plms#1#2#3{{\it Proc. Lond. Math. Soc.} {\bf {#1}} (19{#2}) #3}
\def\pgma#1#2#3{{\it Proc. Glasgow Math. Ass.} {\bf {#1}} (19{#2}) #3}
\def\qjm#1#2#3{{\it Quart. J. Math.} {\bf {#1}} (19{#2}) #3}
\def\qjpam#1#2#3{{\it Quart. J. Pure and Appl. Math.} {\bf {#1}} ({#2}) #3}
\def\rcmp#1#2#3{{\it Rend. Circ. Mat. Palermo} {\bf {#1}} (19{#2}) #3}
\def\rms#1#2#3{{\it Russ. Math. Surveys} {\bf {#1}} (19{#2}) #3}
\def\top#1#2#3{{\it Topology} {\bf {#1}} (19{#2}) #3}
\def\tams#1#2#3{{\it Trans. Am. Math. Soc.} {\bf {#1}} (19{#2}) #3}
\def\zfm#1#2#3{{\it Z.f.Math.} {\bf {#1}} ({#2}) #3}
% *******************   Main text *********************
%\begin{ignore}
\begin{title}  
\vglue 1truein
\righttext {MUTP/98/4}
%\righttext{hep-th/96}
\vskip15truept
%\leftline{\today}
%\vskip 30truept
\centertext {\Bigfonts \bf On the relevance of the multiplicative anomaly}
\vskip 20truept 
\centertext{J.S.Dowker\footnote{dowker@a13.ph.man.ac.uk}}
\vskip 7truept
\centertext{\it Department of Theoretical Physics,}
\centertext{The University of Manchester, Manchester, England}
\vskip 20truept
\centertext {Abstract}
\vskip10truept
\begin{narrow}
We shed doubt on a commonly used manipulation in computing the partition
function for a matrix valued operator together with the attendant invocation 
of the multiplicative anomaly.
\end{narrow}
\vskip 5truept
\righttext {March 1998}
\vskip 60truept
%\righttext{Typeset in \jyTeX}
\vfil
\end{title}

\pagenum=0
%\end{ignore}
%\section{\bf Introduction}
%\begin{ignore}

In some recent works, [\pref{EVZ,EFVZ,Evans}], the multiplicative anomaly 
in the \zf\ definition
of the functional determinant has been discussed from a physical point
of view. In these calculations the anomaly arises when the field operator is 
matrix valued.
For example, for two real free scalar fields of different masses,
computing the functional determinant in two ways apparently yields different
answers. The elements of the calculation are outlined in Evans [\pref{Evans}] 
so, for
convenience, let us refer to equations (1) and (2) of this work. The classical
action is written in two ways
$$
S_a={1\over2}\int\big(\phi_1A_1\phi_1
+\phi_2A_2\phi_2\big) dx,
$$ where
$A_i=-\nabla^2+m_i^2$, and in matrix form
$$
S_b={1\over2}\int\widetilde\bPhi A\bPhi dx
$$
where 
$$
\bPhi=\left({\phi_1\atop\phi_2}\right),\quad A=\left(\matrix{
-\nabla^2+m_1^2&0\cr0&-\nabla^2+m_2^2}\right).
$$

Although trivially $S_a=S_b$, when the functional integral for the partition
function is formally
evaluated, two different answers appear. The reason given is that for
$S_a$ one naturally gets (we leave off standard factors and exponents)
$$\Det A_1\times\Det A_2
\eql{det1}$$ 
while $S_b$, gives
$$\Det A=\Det\big(A_1A_2\big)
\eql{det2}$$ 
and these are not the same. 

This last statement is certainly correct, and is a statement of the 
multiplicative anomaly. In this short note we wish to investigate, not
this mathematical anomaly, but the step leading to (\peq{det2}). This
equation appears in [\pref{EVZ,EFVZ}] where it is attributed to Benson 
{\it et al} [\pref{BBD}]  who
state it without comment. Our opinion is that this relation is not
obvious. It says that, when evaluating the functional determinant of $A$, the
finite algebraic determinant of $A$ can be taken first. The
reasons why we find this to be unnatural, and even wrong, are as follows.

Firstly, the most natural, and the most usual, way of implementing the 
\zf\ method in the
vector/matrix valued case is to take the vector index $i$ together with the
space-time coordinate $x$ as a generalised continuous index. (This has been
a standard procedure, employed most extensively by De Witt.) It leads, in
particular, to the split form (\peq{det1}).

Now, the functional integral formula for the determinant 
is an extension to the continuous, functional case of a standard finite
dimensional formula. We can check (\peq{det2}) by considering
a finite dimensional restriction. Thus replace the action by
$$
S_f={1\over2}\sum_{ij\al\be} \phi_{i\al} A_{ij\al\be}\phi_{j\be}
$$
where integers $\al$ and $\be$ play the roles of the arguments, $x$ and $y$,
of the (nonlocal) operator $A$ and have finite ranges. 
The multiple integral over the 
variables $\phi_{i\al}$ will then involve an ordinary determinant of the
matrix $A$ where the matrix indices are the pairs $(i,\al)$ and $(j,\be)$,  
and this is the correct answer.
For example, if the range of $\al$ and $\be$ is 1 to 2, then the determinant 
is a four by four one. 

The argument leading to 
(\peq{det2}) now would give
$$
\det_{\al\be}\big(\det_{ij} A_{ij\al\be}\big)
$$
\ie one takes the determinant on the $ij$ indices first and then that
on the $\al\be$ indices of the resulting expression. It is easily seen that 
these two routes give different answers. In our example, the second gives a 
sum of eight terms, each a product of four $A$ coefficients, while the 
four by four determinant expands to 24 such terms. 

Our conclusion is that if one uses the natural, and in our view correct, 
implementation of the \zf\ approach, it should not be necessary, at least in 
the vector valued case, to invoke the multiplicative anomaly, nor its 
specific expression.


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\end{putreferences}
\bye








