Conformal relativity, a theory of mass. II: Special conformal relativity
Tóm tắt
We first review the classical construction of the conformal («angle») geometryM
n+1 out of the elements of the more familiarn-dimensional Euclidean («length») geometryE
n
, guided by Liouville's theorem on the conformal group. After generalizing to indefinite metrics, we specialize to the space-time case. The exposition is didactic with asides on physical reasons and implications. After the inhomogeneous formalism the projective formalism, in which the conformal group acts linearly, is treated. The latter is essential to gauge theories of interaction. Then, on the basis of this geometry, we seek the generalization of the special-relativistic formulap
μ=mdx
μ/dσ, σ≡proper time,m=constant, for the 4-momentum of a classical particle.
Tài liệu tham khảo
From now on the part of the group connected to the identity element will be understood unless stated otherwise. Thus we exclude improper transformations like reflections, which also preserve the line element (1).
The proofs may be found in old-fashioned books on «higher geometry» likeF. Klein:Vorlesungen über höhere Geometrie, 3rd ed. (Berlin, 1926). Theorem (6) is a combination of several theorems, only one of which is historically called Liouville's theorem.
For at least some kinds of force.
See ref. (16) SF II, sect.3, of ref. (13) of part I.
AllX a withX n+2=0 are lumped into a «point at infinity» in the projective sense. Note, however, that both spheres (x μ, λ) and (x μ, −λ) are mapped into the same projective point. Ignore this doubling for the moment.
SeeF. Klein:Vorlesungen über höhere Geometrie, 3rd ed. (Berlin, 1926); orW. F. Osgood:Functions of a Complex Variable, Chap. 3.6 (New York, N. Y., 1938).
See, for example,R. L. Ingraham:Nuovo Cimeéto,12, 825 (1954), eq. (2.12).
The theory of classical motion and the self-force in the conformal theory appears inR. L. Ingraham:Nuovo Cimento,27 B, 293 (1975);34 B, 189 (1976). Refer to these as SF I and II. See alsoR. L. Ingraham:Nuovo Cimento,39 B, 331 (1977), Errata for SF II. Equation (30) is nothing more or less than the equation of ageodesic in (a curved)M 5, as will be shown in a later publication. The trajectory-dependent (i.e. particle-dependent) constant ζ′ is a first integral which always exists.
See SF I, sect.5, of ref. (13). This should be the self-force for at least some kinds of particles, probably all electrically charged ones. In that case, the theory interpretsL as the classical radiusR c (more precisely,L=3/2R c). On the basis of conformal QFT we expect that the SF strengthL will depend on the particle couplings. For example, the neutron should haveL≪classical radius of the proton, because it can radiate only massive quanta. However, we have not yet been able to recover the classical potential (42) from the (in principle exact) QFT in the sense that the Coulomb 1/r is recovered by «inverting» the electron-electron one-photon exchange graph in quantum electrodynamics.
SF II, sect.3, of ref. (13).
SF I, sect.7, of ref. (13). See also clarifying remarks in SF II, sect.6.