dimension*
*
APPENDIX III
The phase
*From Eddington, Arthur Stanley. Fundamental Theory (pp. 46-47). London: Cambridge University Press, 1946.
The usual equations of wave mechanics postulate flat space. I do not think that there is anything to be gained by trying to extend wave mechanics to curved space. Curvature and wave functions are alternative ways of representing distributions of energy and momentum; and it is probably bad policy to mix them.
We
have introduced the curved space of molar relativity theory as a mode of
representation of the extraordinary fluctuation, and have obtained the
fundamental relation between the microscopic constant 
and
the cosmological constants 
Having got what we want out of
it, space curvature no longer interests us; and we return to flat space to
pursue the specialized development of microscopic theory. That does not
mean that henceforth we neglect curvature; we merely refrain from using
the dodge that introduces it. The scale uncertainty, instead of being
disguised as curvature, will be taken into account openly; so that there
is no loss of rigour.
Accordingly the scale is now treated as an additional variate whose probability distribution is specified along with that of the ordinary momenta and coordinates. The variates of a probability distribution occur in conjugate pairs, and the variate conjugate to the scale will be called the phase. Since we have to provide for cases in which the scale reduces to an eigenvalue, the scale is classed as a momentum and the phase as a coordinate. The phase coordinate is represented as a fifth dimension normal to space-time (which is now flat), so that the scale and phase are invariant for the rotations and Lorentz transformations of special relativity theory.*
*This is not the same as the fifth dimension introduced by curvature. In § 6 the scale was represented by a distance O'P' in the u direction; but distances normal to space-time now represent phase. the scale being a momentum.
The scale uncertainty is primarily a fluctuation of the extraneous standard. But fluctuations of the standard are reflected in the measured characteristics of the system. The scale momentum is the measure of a characteristic which we may call the scale-indicator; it is itself unvarying, but its measure shows these reflected fluctuations. In the ordinary momenta the reflected fluctuations of the standard and the fluctuation of the characteristics themselves are inextricably combined; so that we have to introduce one unvarying characteristic to exhibit the scale fluctuation by itself.
We have employed a comparison particle to embody the extraneous standard, and have 'perfected' the object-system by including the comparison particle within it. The introduction of the scale and phase dimension is an equivalent way of perfecting the object-system; and the scale-indicator is the form taken by the comparison particle when it is brought into the object-system. It is a common practice to use a 6-dimensional space to represent a system of two particles. Here one of the particles is a comparison particle, and we need only to extend the object-space by one dimension. Moreover, since the object-system has always to be considered in conjunction with an extraneous standard, the extra dimension is a permanent feature of its representation.
To
represent the extraordinary fluctuation or cosmical curvature the scale
momentum must be given a Gaussian probability distribution with standard
deviation . For most purposes this would be a pedantic refinement; and
the scale may be regarded as a stabilized characteristic. But now that
each particle or small system has its own scale variate, a new field of
phenomena is opened to theoretical investigation, which is suppressed in
the molar treatment of scale as an averaged characteristic. As remarked
previously, the comparison particle to be introduced into a microscopic
object-system is an individual; and the fluctuation of its energy is of
order 1, in contrast to the mean comparison particle whose fluctuations
are of order 
We have therefore to distinguish two steps: the
substitution of an explicit (5-dimensional) for a concealed (curvature)
representation of the mean scale, and in the explicit representation the
substitution of individual scales for the mean scale. Since the mean scale
is practically a stabilized scale, the second step is described as the
de-stabilization of scale.
For
some purposes it is convenient to take an angular momentum as extraneous
standard, so that the scale momentum is an angular momentum and the
corresponding phase coordinate is an angle.* This facilitates the
stabilization of scale - or rather it facilitates the de-stabilization of
the fixed scale commonly assumed. The feature of an angular coordinate is
that 'infinite uncertainty' corresponds to uniform probability
distribution between 0 and 
. Thus, if
J is an angular momentum and 
the corresponding angle, as the uncertainty of J diminishes
tends to a
uniform distribution over the range ; and we pass without
discontinuity from an almost exact (observed) value to an exact
(stabilized) value of J. Conversely, results which assume an exact scale
are extended to a slightly fluctuating scale by spreading the distribution
uniformly over a thickness
in an extra phase dimension. We call
the widening factor. From the widened distribution we can pass
continuously to distributions in which the variation of scale becomes of
serious importance.
*The complete momentum vector contains both linear and angular momentum, so that there is no incongruity in this choice.
The
widening factor must be taken into account when we compare spherical space
(with stabilized scale) and flat space (with fluctuating scale). When the
scale is stabilized we have a spherical space whose total volume is
Preparatory to de-stabilization this is to be re-ordered as
a volume 
of three-dimensional space having a
thickness
in an extra phase dimension. Comparing it with a flat
sphere of radius 
and volume 
we
have
Since
in natural units is a mass m, this is a relation of the
form
and
is an example of the law connecting masses of different multiplicity.
In 
the scale is still exact and the phase necessarily has uniform
distribution over the thickness ; the representation does not
give
any extra freedom. In 
the scale is de-stabilized and the
constraint is relaxed, so that the number of degrees of freedom is raised
from k = 3 to k = 4. Conversely, starting with the volume
of flat space, we multiply it by a thickness
in the
phase dimension, then multiply by 3/4 to stabilize the scale since the
stabilization reduces the number of degrees of freedom from 4 to 3, and so
obtain the volume 
of scale-stabilized space which is
three-dimensional but curved.
