LE~T]~R]~ AL ~UOVO CIMENTO VOL. 9, N. 4 26 Gennaio 1974
Muon Decay and Equivalence Principle.
Department o/ Physics, Teehnion.Israel Institute o] Technology - Hai]a
(rieevuto il 30 Ottobre 1973)
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
(1) v = f (g~dx~ axa) 89 + f e A = a~ ,
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
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.
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
(2) A~,-+ A~, + ~;~/Ox ~ ,
where 2 is an arbitrary function of the co-ordinates. The proper time is then altered by the amount eyd)~.
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.
(1) YA. S~tORODINSKII: Zurn. Eksp. Teor. Fiz. Pis. Red., 16, 499 (1972) (English translation: J ETP Let&, 16, 356 (1972)).