An Extension of Feynman-Bunge-Corben's Relation regarding the Position-Operator of Dirac Electron to Arbitrary Operators. II

Abstract

In the preceding note 1 l (which is hereafter referred to as I), the mathematical formulas concerning Q and dQ/ dt are proved and such significant quantities as r, 'ii, ii, are exhibited as examples. The physical meanings are discussed in this note. i) r: Pryce · proposed previously three kinds of mass-center in his relativistic theory: 2 l (1) mass-center q, (2) proper mass-center X, and (3) point q satisfying canonical relations. Among these examples regarding Dirac free electron, Pryce's q (1) corresponds just exactly to 'gerade' part of r,2). 3) and Pryce's q (3) to Foldy-Wouthuysen's mean-position operator. 4 l Then a comparison of Pryce's proper X (2)with 'the covariant r' can be considered.Really X agrees with r up to the order of (h/mc) ( p/mc), and as p--0, not only X, but q and q all approach r in different orders. Therefore, r is simple and, in some sense, can be regarded as' 'one of masscenters'.In connection with this, the author can mention two characteristics of Q as well as r, which are distinct from those of 'gerade'part of Q and of Foldy-Wouthuysen's mean-operator for Q: 1. Q is covariant (at least 'weakly'); 2. even if there exists general electromagnetic field, the calculations of Q and dQ!dt can be carried out easily. ii) a and fla, etc.,: The 'covariant spin op~rators' that Calogero 5 ) and others treated have a peculiarity of conserved quantity when external field is absent. 'The spin operators (j and { §a' are also covariant and they play precessional motions of so-called Kramers's type when external fields exist. Therefore, under no external field, ii, fJa, etc., are also conserved. It stands to reason that 'ii and fJa, etc., coincide with the 'covariant spin operators'. iii) 1C: As mentioned in I, Bunge, Cor ben and Fradkin-Good discussed the motion of the ele.ctron starting from r and fJa, respectively. The author doubts such a procedure as to adhere to one Q. This 'ii' is also to be discussed at the same time, and in dn/dt (1. 11), the effect of magnetic moment on the motion is manifest unlike in dn/dt (1. 9) depending on velocity a. In general, the time derivatives of r, ii .• and 1!, etc., have a peculiarity that it is independent of the velocity operator a, and consequently of the so-called 'Zitterbewegung', in contrast with those of r, a, and n, etc. If any quantity Q includes a high frequency term due to 'Zitterbewegung' as U exp(i2mT), where U is independent of the proper time T, the adoption of new (l for Q defined by (l=Q+ (i/2m)dQ/dT, results in excluding that high frequency term of Q. It is supposed the Q has a role similar to this (l. At all events, it should be noted that in (1.3) and (1.5), da/dt-[aXH], dfJa/dt-[fJaXH],