CHAPTER 2
 GENERAL FORMAL SCHEMES FOR RADIATION REACTION

 Historically, Dirac gave the first formal analysis of the radiation reaction effect

for the electromagnetic field of a particle moving in flat spacetime in 1938 [2]. In

the equation of motion for a moving electron, he was able to obtain the additional

force term, named the "Abraham-Lorentz-Dirac (ALD) damping term," apart from

the Lorentz force due to the external electromagnetic field. But this ALD damping

term eventually turns out to vanish in free fall, leaving the particle's motion in

geodesic, and no radiation damping or "self-force" effect occurs in flat spacetime.

 However, Dirac's pioneering idea was succeeded and generalized to curved

spacetime in similarly formal approaches by the following generations. Dewitt

and Brehme [3] extended Dirac's analysis to curved spacetime. Mino, Sasaki, and

Tanaka [4] developed a similar analysis for the gravitational tensor field. Quinn and

Wald [5] and Quinn [6] worked out similar schemes for the radiation reaction of the

gravitational, electromagnetic, and scalar fields by taking axiomatic approaches.

All these generalized versions of the radiation reaction problem show the obvious

existence of non-vanishing damping terms in addition to the ALD damping term,

which would eventually cause radiation reaction in curved spacetime.

 In this ('C Ilpter we review the two main articles on this subject, one by Dirac

[2] and the other by Dewitt and Brehme [3].

 2.1 Dirac: Radiating Electrons in Flat Spacetime

2.1.1 The Fields Associated with an Electron

 The problem to deal with is a single electron moving in an electromagnetic

field in flat spacetime (following the signature convention (+1, -1, -1, -1) as in

Dirac's original note).