1. Notation: indices μ, ν,ξ,...,=1,..., 4 label Cartesian co-ordinates in space-time withx 4≡t≡ time. Indices α, β,...,=1,..., 5 withx 5≡λ. dx 2≡g μνdx μdx ν withg μν=diag(+++-)1. A space-time vector notation is often used:v·a≡v μ a μ≡g μν v μ a ν v 2≡v·v, etc. Do not confuse 4th components,e.g. R 4, with scalars,e.g. (R 2)2≡(R·R)2! The symbolx stands forx μ. Units:c=1.V 5 is the space of space-time spheres labelled byx α=(x 1,...,x 4, λ), a 5- dimensional Riemannian space with line element (1).
2. R. L. Ingraham:Nuovo Cimento,27 B, 293 (1975); inLectures in Theoretica Physics, Vol.13 (Boulder, Colo, 1971);phys. Rev.,106, 1324 (1957);Nuovo Cimento,16, 104 (1960); see also various work byY. Murai andL. Castell on conformal physics in theV 5 formalism.
3. Equations (13) through (17) of ref. (2). We have restricted our attention to external fields of classical type:F extμν no function of λ,F ext5ν=0. Moreover, in this paper letF extμν depend only onx(τ).
4. This is amplified in Sect.6.
5. dϑ>0 conventionally. Signk is chosen to make $$\dot x^4 > 0$$ . Then Sgne is determined by eq. (8).