Muon decay and equivalence principle

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<ul><li><p>LE~T]~R]~ AL ~UOVO CIMENTO VOL. 9, N. 4 26 Gennaio 1974 </p><p>Muon Decay and Equivalence Principle. </p><p>A. P~s </p><p>Department o/ Physics, Teehnion.Israel Institute o] Technology - Hai]a </p><p>(rieevuto il 30 Ottobre 1973) </p><p>In a recent paper with the same title as above, SMOI~ODINSKII (1) suggested that the proper time for a charged particle in an electromagnetic field is given by </p><p>(1) v = f (g~dx~ axa) 89 + f e A = a~ , </p><p>rather than by just the first term on the right-hand side of (1). A consequence of this formula would be that a muon moving in a circular orbit in </p><p>a cyclotron, thus enclosing a large amount of magnetic flux, would have a longer life- time than a free muon moving with the same speed. </p><p>The purpose of this note is to point out that eq. (1) cannot be correct, because it is not gauge invariant. This is easily seen by performing the gauge transformation </p><p>(2) A~,-+ A~, + ~;~/Ox ~ , </p><p>where 2 is an arbitrary function of the co-ordinates. The proper time is then altered by the amount eyd)~. </p><p>In Smorodinskii's paper (1), this difficulty is obviated by tacitly choosing a gauge where A 0 = 0, and taking a closed integration contour in space. These assumptions, however, are quite arbitrary, and it is clear that eq. (t) cannot hold in generul. </p><p>(1) YA. S~tORODINSKII: Zurn. Eksp. Teor. Fiz. Pis. Red., 16, 499 (1972) (English translation: J ETP Let&amp;, 16, 356 (1972)). </p><p>146 </p></li></ul>