Weyl’s fundamental idea for generalising Riemannian geometry was to note that, unlike for the comparison of vectors at different points of the manifold, for the comparison of scalars the existence of a connection is not required. Thus, while lengths of vectors at different points can be compared without a connection, directions cannot. This seemed too special an assumption to Weyl for a genuine infinitesimal geometry:
“If we make no further assumption, the points
of a manifold remain totally isolated from each other with regard
to metrical structure. A metrical relationship from point to point
will only then be infused into [the manifold] if a principle for carrying the unit of length
from one point to its infinitesimal
neighbours is given.” 60
In contrast to this, Riemann made the much stronger assumption that line elements may be compared not only at the same place but also at two arbitrary places at a finite distance.
“However, the possibility
of such a comparison ‘at a distance’ in no way can be admitted
in a pure infinitesimal
geometry.” 61
([397
], p. 397)
In order to invent a purely “infinitesimal” geometry, Weyl introduced the 1-dimensional, Abelian group of gauge transformations,
besides the diffeomorphism group (coordinate transformations). At a point, Equation (98Let us now look at what happens to parallel
transport of a length, e.g., the norm
of a tangent vector along a particular curve
with parameter
to a different (but
infinitesimally neighbouring) point:
Thus, in Weyl’s connection (100), both the
gravitational and the electromagnetic fields, represented by the
metrical field
and the vector field
, are intertwined. Perhaps, having in mind Mie’s
ideas of an electromagnetic world view and Hilbert’s approach to
unification, in the first edition of his book, Weyl remained reserved:
“Again physics, now the physics of fields, is
on the way to reduce the whole of natural phenomena to one single
law of nature, a goal to which physics already once seemed close
when the mechanics of mass-points based on Newton’s Principia did
triumph. Yet, also today, the circumstances are such that our trees
do not grow into the sky.” 62
([396
], p. 170;
preface dated “Easter 1918”)
However, a little later, in his paper accepted on 8 June 1918, Weyl boldly claimed:
“I am bold enough to believe that the whole of
physical phenomena may be derived from one single universal
world-law of greatest mathematical simplicity.” 63
([397
], p. 385,
footnote 4)
The adverse circumstances alluded to in the first
quotation might be linked to the difficulties of finding a
satisfactory Lagrangian from which the field equations of Weyl’s theory can be derived. Due
to the additional group of gauge transformations, it is useful to
introduce the new concept of gauge-weight within tensor calculus as in
Section 2.1.564
. As the Lagrangian
must have gauge-weight
, we are looking for a scalar
of gauge-weight
. Weitzenböck has shown that the only
possibilities quadratic in the curvature tensor and the line
curvature are given by the four expressions [391]
Weyl did calculate the curvature
tensor formed from his connection (100) but did not get the
correct result65
; it is given by Schouten ([310],
p. 142) and follows from Equation (51
):
Perhaps Bach (alias Förster) was also
dissatisfied with Weyl’s calculations: He went
through the entire mathematics of Weyl’s theory, curvature tensor,
quadratic Lagrangian field equations and all; he even discussed
exact solutions. His Lagrangian is given by , where the invariants are
defined by
While Weyl’s unification of
electromagnetism and gravitation looked splendid from the
mathematical point of view, its physical consequences were dire: In
general relativity, the line element had been identified
with space- and time intervals measurable by real clocks and real measuring
rods. Now, only the equivalence class
was supposed to have a physical meaning: It was as if clocks and
rulers could be arbitrarily “regauged” in each event, whereas in Einstein’s theory the same clocks and rulers had to be used
everywhere. Einstein, being the first expert
who could keep an eye on Weyl’s theory, immediately
objected, as we infer from his correspondence with Weyl.
In spring 1918, the first edition of Weyl’s famous book on differential
geometry, special and general relativity Raum-Zeit-Materie appeared, based on his
course in Zürich during the summer term of 1917 [396]. Weyl had arranged that the page
proofs be sent to Einstein. In communicating this on
1 March 1918, he also stated that
“As I believe, during these days I succeeded in
deriving electricity and gravitation from the same source. There is
a fully determined action principle, which, in the case of
vanishing electricity, leads to your gravitational equations while,
without gravity, it coincides with Maxwell’s equations in first order. In the most general case,
the equations will be of 4th order, though.” 66
He then asked whether Einstein would be willing to
communicate a paper on this new unified theory to the Berlin
Academy ([320], Volume 8B, Document 472, pp. 663-664). At the end of March,
Weyl visited Einstein in Berlin, and finally,
on 5 April 1918, he mailed his note to him for the Berlin Academy.
Einstein was impressed: In April
1918, he wrote four letters and two postcards to Weyl on his new unified field
theory - with a tone varying between praise and criticism. His
first response of 6 April 1918 on a postcard was enthusiastic:
“Your note has arrived. It is a stroke of genious of first rank.
Nevertheless, up to now I was not able to do away with my objection
concerning the scale.” 67
([320
], Volume 8B, Document 498, 710)
Einstein’s “objection” is formulated in his “Addendum” (“Nachtrag”) to Weyl’s paper in the reports of the Academy, because Nernst had insisted on such a postscript. There, Einstein argued that if light rays would be the only available means for the determination of metrical relations near a point, then Weyl’s gauge would make sense. However, as long as measurements are made with (infinitesimally small) rigid rulers and clocks, there is no indeterminacy in the metric (as Weyl would have it): Proper time can be measured. As a consequence follows: If in nature length and time would depend on the pre-history of the measuring instrument, then no uniquely defined frequencies of the spectral lines of a chemical element could exist, i.e., the frequencies would depend on the location of the emitter. He concluded with the words
“Regrettably, the basic hypothesis of the
theory seems unacceptable to me, [of a theory] the depth and
audacity of which must fill every reader with admiration.” 68
([395
], Addendum,
p. 478)
Einstein’s remark concerning the
path-dependence of the frequencies of spectral lines stems from the
path-dependency of the integral (102) given above. Only for
a vanishing electromagnetic field does this objection not hold.
Weyl answered Einstein’s comment to his paper in
a “reply of the author” affixed to it. He doubted that it had been
shown that a clock, if violently moved around, measures proper time
. Only in a static gravitational field,
and in the absence of electromagnetic fields, does this hold:
“The most plausible assumption that can be made
for a clock resting in a static field is this: that it measure the
integral of the normed in this way [i.e., as in Einstein’s theory]; the task
remains, in my theory as well as in Einstein’s, to derive this fact by
a dynamics carried through explicitly.” 69
([395],
p. 479)
Einstein saw the problem, then unsolved within his general relativity, that Weyl alluded to, i.e., to give a theory of clocks and rulers within general relativity. Presumably, such a theory would have to include microphysics. In a letter to his former student Walter Dällenbach, he wrote (after 15 June 1918):
“[Weyl] would say that clocks and
rulers must appear as solutions; they do not occur in the
foundation of the theory. But I find: If the , as measured by a clock (or a ruler), is something
independent of pre-history, construction and the material, then
this invariant as such must also play a fundamental role in theory.
Yet, if the manner in which nature really behaves would be
otherwise, then spectral lines and well-defined chemical elements
would not exist. [...] In any case, I am as convinced as Weyl that gravitation and
electricity must let themselves be bound together to one and the
same; I only believe that the right union has not yet been found.”
70
([320
], Volume 8B, Document 565, 803)
Another famous theoretician who could not side
with Weyl was H. A. Lorentz;
in a paper on the measurement of lengths and time intervals in
general relativity and its generalisations, he contradicted Weyl’s statement that the
world-lines of light-signals would suffice to determine the
gravitational potentials [210].
However, Weyl still believed in the
physical value of his theory. As further “extraordinarily strong
support for our hypothesis of the essence of electricity” he
considered the fact that he had obtained the conservation of
electric charge from gauge-invariance in the same way as he had
linked with coordinate-invariance earlier, what at the time was
considered to be “conservation of energy and momentum”, where a
non-tensorial object stood in for the energy-momentum density of
the gravitational field ([398], pp. 252-253).
Moreover, Weyl had some doubts about the
general validity of Einstein’s theory which he derived
from the discrepancy in value by 20 orders of magniture of the
classical electron radius and the gravitational radius
corresponding to the electron’s mass ([397],
p. 476; [152]).
There exists an intensive correspondence
between Einstein and Weyl, now completely available in
volume 8 of the Collected Papers of Einstein [320]. We subsume some of
the relevant discussions. Even before Weyl’s note was published by the
Berlin Academy on 6 June 1918, many exchanges had taken place
between him and Einstein.
On a postcard to Weyl on 8 April 1918, Einstein reaffirmed his admiration
for Weyl’s theory, but remained firm
in denying its applicability to nature. Weyl had given an argument for
dimension 4 of space-time that Einstein liked: As the Lagrangian
for the electromagnetic field is of
gauge-weight
and
has
gauge-weight
in an
, the integrand in
the Hamiltonian principle
can
have weight zero only for
: “Apart from the [lacking]
agreement with reality it is in any case a grandiose intellectual
performance” 71
([320
], Vol. 8B, Doc. 499, 711). Weyl did not give in:
“Your rejection of the theory for me is
weighty; [...] But my own brain still keeps believing in it. And as
a mathematician I must by all means hold to [the fact] that my
geometry is the true geometry ‘in the near’, that Riemann happened
to come to the special case is due only to
historical reasons (its origin is the theory of surfaces), not to
such that matter.” 72
([320
], Volume 8B, Document 544, 767)
After Weyl’s next paper on “pure
infinitesimal geometry” had been submitted, Einstein put forward further
arguments against Weyl’s theory. The first was that
Weyl’s theory preserves the
similarity of geometric figures under parallel transport, and that
this would not be the most general situation
(cf. Equation (49)). Einstein then suggested the affine
group as the more general setting for a generalisation of
Riemannian geometry ([320
], Vol. 8B, Doc. 551, 777). He repeated this argument in a
letter to his friend Michele Besso from his vacations at the Baltic
Sea on 20 August 1918, in which he summed up his position with
regard to Weyl’s theory:
“[Weyl’s] theoretical attempt does
not fit to the fact that two originally congruent rigid bodies
remain congruent independent of their respective histories. In
particular, it is unimportant which value of the integral is assigned to their world line. Otherwise, sodium
atoms and electrons of all sizes would exist. But if the relative
size of rigid bodies does not depend on past history, then a
measurable distance between two (neighbouring) world-points exists.
Then, Weyl’s fundamental hypothesis is
incorrect on the molecular level, anyway. As far as I can see,
there is not a single physical reason for it being valid for the
gravitational field. The gravitational field equations will be of
fourth order, against which speaks all experience until now [...].”
73
([327
], p. 133)
Einstein’s remark concerning
“affine geometry” is referring to the affine geometry in the sense
it was introduced by Weyl in the 1st and 2nd edition of
his book [396], i.e., through the affine group and not as a suggestion of an affine connexion.
From Einstein’s viewpoint, in Weyl’s theory the line element
is no longer a measurable quantity - the
electromagnetical 4-potential never had been one. Writing from his
vacations on 18 September 1918, Weyl presented a new argument in
order to circumvent Einstein’s objections. The
quadratic form
is an absolute invariant,
i.e., also with regard to gauge
transformations (gauge weight 0). If this expression would be taken
as the measurable distance in place of
, then
“[...] by the prefixing of this factor, so to
speak, the absolute norming of the unit of length is accomplished
after all” 74
([320
], Volume 8B, Document 619, 877-879)
Einstein was unimpressed:
“But the expression for the
measured length is not at all acceptable in my opinion because
is very dependent on the matter density. A very
small change of the measuring path would strongly influence the
integral of the square root of this quantity.” 75
Einstein’s argument is not very
convincing: itself is influenced by matter through
his field equations; it is only that now
is algebraically connected to the matter
tensor. In view of the more general quadratic Lagrangian needed in
Weyl’s theory, the connection
between
and the matter tensor again might become less
direct. Einstein added:
“Of course I know that the state of the theory
as I presented it is not satisfactory, not to speak of the fact
that matter remains unexplained. The unconnected juxtaposition of
the gravitational terms, the electromagnetic terms, and the -terms undeniably is a result of resignation.[...] In
the end, things must arrange themselves such that action-densities
need not be glued together additively.” 76
([320],
Volume 8B, Document 626, 893-894)
The last remarks are interesting for the way in which Einstein imagined a successful unified field theory.
Sommerfeld seems to have been convinced by Weyl’s theory, as his letter to Weyl on 3 June 1918 shows:
“What you say here is really marvelous. In the
same way in which Mie glued to his consequential electrodynamics a
gravitation which was not organically linked to it, Einstein glued to his
consequential gravitation an electrodynamics (i.e., the usual electrodynamics) which had
not much to do with it. You establish a real unity.” 77
[326]
Schouten, in his attempt in 1919 to replace the presentation of the geometrical objects used in general relativity in local coordinates by a “direct analysis”, also had noticed Weyl’s theory. In his “addendum concerning the newest theory of Weyl”, he came as far as to show that Weyl’s connection is gauge invariant, and to point to the identification of the electromagnetic 4-potential. Understandably, no comments about the physics are given ([295], pp. 89-91).
In the section on Weyl’s theory in his article for
the Encyclopedia of Mathematical
Sciences, Pauli described the basic elements
of the geometry, the loss of the line-element as a physical variable, the convincing derivation of
the conservation law for the electric charge, and the too many
possibilities for a Lagrangian inherent in a homogeneous function
of degree 1 of the invariants (103
). As compared to his
criticism with respect to Eddington’s and Einstein’s later unified field
theories, he is speaking softly, here. Of course, as he noted, no
progress had been made with regard to the explanation of the
constituents of matter; on the one hand because the differential
equations were too complicated to be solved, on the other because
the observed mass difference between the elementary particles with
positive and negative electrical charge remained unexplained. In
his general remarks about this problem at the very end of his
article, Pauli points to a link of the
asymmetry with time-reflection symmetry (see [242
], pp. 774-775;
[244
]). For Einstein, this criticism was not
only directed against Weyl’s theory
“but also against every continuum-theory, also
one which treats the electron as a singularity. Now as before I
believe that one must look for such an overdetermination by
differential equations that the solutions no longer have the character of a
continuum. But how?” ([103], p. 43)
In a letter to Besso on 26 July 1920, Einstein repeated an argument against Weyl’s theory which had been removed by Weyl - if only by a trick to be described below; Einstein thus said:
“One must pass to tensors of fourth order
rather than only to those of second order, which carries with it a
vast indeterminacy, because, first, there exist many more equations
to be taken into account, second, because the solutions contain
more arbitrary constants.” 78
([327
], p. 153)
In his book “Space, Time, and Gravitation”, Eddington gave a non-technical
introduction into Weyl’s “welding together of
electricity and gravitation into one geometry”. The idea of gauging
lengths independently at different events was the central theme. He
pointed out that while the fourfold freedom in the choice of
coordinates had led to the conservation laws for energy and
momentum, “in the new geometry is a fifth arbitrariness, namely
that of the selected gauge-system. This must also give rise to an
identity; and it is found that the new identity expresses the law
of conservation of electric charge.” One natural gauge was formed
by the “radius of curvature of the world”; “the electron could not
know how large it ought to be, unless it had something to measure
itself against” ([57], pp. 174, 173,
177).
As Eddington distinguished natural geometry and actual space from world geometry and conceptual space serving for a graphical representation of relationships among physical observables, he presented Weyl’s theory in his monograph “The mathematical theory of relativity”
“from the wrong end - as its author might
consider; but I trust that my treatment has not unduly obscured the
brilliance of what is unquestionably the greatest advance in the
relativity theory after Einstein’s work.” ([59], p. 198)
Of course, “wrong end” meant that Eddington took Weyl’s theory such
“that his non-Riemannian geometry is not to be
applied to actual space-time; it
refers to a graphical representation of that relation-structure
which is the basis of all physics, and both electromagnetic and
metrical variables appear in it as interrelated.” ([59], p. 197)
Again, Eddington liked Weyl’s natural gauge encountered in Section
4.1.5, which made the
curvature scalar a constant, i.e.,
; it became a consequence of Eddington’s own natural gauge in
his affine theory,
(cf. Section 4.3). For Eddington, Weyl’s theory of
gauge-transformation was a hybrid:
“He admits the physical comparison of length by
optical methods [...]; but he does not recognise physical
comparison of length by material transfer, and consequently he
takes to be a function fixed by arbitrary convention and
not necessarily a constant.” ([59
], pp. 220-221)
In the depth of his heart Weyl must have kept a fondness for
his idea of “gauging” a field all
during the decade between 1918 and 1928. As he had abandoned the
idea of describing matter as a classical field theory since 1920, the linking of the
electromagnetic field via the gauge idea could only be done through
the matter variables. As soon as the
new spinorial wave function (“matter wave”) in Schrödinger’s and
Dirac’s equations emerged, he adapted his idea and linked the
electromagnetic field to the gauging of the quantum mechanical wave
function [407, 408
]. In October 1950,
in the preface for the first American printing of the English
translation of the fourth edition of his book Space, Time, Matter from 1922, Weyl clearly expressed that he had
given up only the particular idea of a link between the electromagnetic field and the local
calibration of length:
“While it was not difficult to adapt also Maxwell’s equations of the electromagnetic field to this principle [of general relativity], it proved insufficient to reach the goal at which classical field physics is aiming: a unified field theory deriving all forces of nature from one common structure of the world and one uniquely determined law of action.[...] My book describes an attempt to attain this goal by a new principle which I called gauge invariance. (Eichinvarianz). This attempt has failed.” ([410], p. V)
Pauli, still a student, and with
his article for the Encyclopedia in front of him, pragmatically
looked into the gravitational effects in the planetary system,
which, as a consequence of Einstein’s field equations, had
helped Einstein to his fame. He showed
that Weyl’s theory had, for the static
case, as a possible solution a constant Ricci scalar; thus it also
admitted the Schwarzschild solution and could reproduce all desired
effects [244, 243].
Weyl himself continued to develop
the dynamics of his theory. In the third edition of his Space-Time-Matter [398], at the
Naturforscherversammlung in Bad Nauheim in 1920 [399
], and in his paper
on “the foundations of the extended relativity theory” in
1921 [402
], he returned to his
new idea of gauging length by setting
(cf. Section 4.1.3); he interpreted
to be the “radius of curvature” of the world. In
1919, Weyl’s Lagrangian originally was
together with the
constraint
with constant
([398
], p. 253). As
an equivalent Lagrangian Weyl gave, up to a divergence79
“Moreover, this theory leads to the
cosmological term in a uniform and forceful manner, [a term] which
in Einstein’s theory was introduced
ad hoc” 80
([402],
p. 474)
Reichenbächer seemingly was unhappy about Weyl’s taking the curvature scalar to be a constant before the variation; in the discussion after Weyl’s talk in 1920, he inquired whether one could not introduce Weyl’s “natural gauge” after the variation of the Lagrangian such that the field equations would show their gauge invariance first ([399], p. 651). Eddington criticised Weyl’s choice of a Lagrangian as speculative:
“At the most we can only regard the assumed form of action [...] as a step towards some more natural combination of electromagnetic and gravitational variables.” ([59], p. 212)
The changes, which Weyl had introduced in the 4th
edition of his book [401], and which,
according to him, were of fundamental importance for the
understanding of relativity theory, were discussed by him in a
further paper [400]. In connection with
the question of whether, in general relativity, a formulation might
be possible such that “matter whose characteristical traits are
charge, mass, and motion generates the field”, a question
which was considered as unanswered by Weyl, he also mentioned a
publication of Reichenbächer [271]. For Weyl, knowledge of the charge and
mass of each particle, and of the extension of their
“world-channels” were insufficient to determine the field uniquely.
Weyl’s hint at a solution remains
dark; nevertheless, for him it meant
“to reconciliate Reichenbächer’s idea: matter causes a ‘deformation’ of the metrical field and Einstein’s idea: inertia and gravitation are one.” ([400], p. 561, footnote)
Although Einstein could not accept Weyl’s theory as a physical
theory, he cherished “its courageous mathematical construction” and
thought intensively about its conceptual foundation: This becomes
clear from his paper “On a complement at hand of the bases of
general relativity” of 1921 [73]. In
it, he raised the question whether it would be possible to generate
a geometry just from the conformal invariance of Equation (9) without use of the
conception “distance”, i.e., without
using rulers and clocks. He then embarked on conformal invariants
and tensors of gauge-weight 0, and gave the one formed from the
square of Weyl’s conformal curvature
tensor (59
), i.e.
His colleague in Vienna, Wirtinger
, had helped him in
this81. Einstein’s conclusion was that, by
writing down a metric with gauge-weight 0, it was possible to form
a theory depending only on the quotient of the metrical components.
If
has gauge-weight
, then
is such a metric. In order to reduce the new theory
to general relativity, in addition only the differential
equation
Eisenhart wished to partially
reinterpret Weyl’s theory: In place of putting
the vector potential equal to Weyl’s gauge vector, he suggested
to identify it with , where
is the electrical 4-current vector (-density) and
the mass density. He referred to Weyl, Eddington’s book, and to Pauli’s article in the
Encyclopedia of Mathematical Sciences [117].
Einstein’s rejection of the
physical value of Weyl’s theory was seconded by
Dienes
, if only with a
not very helpful argument. He demanded that the connection remain
metric-compatible from which, trivially, Weyl’s gauge-vector must vanish.
Dienes applied the same argument
to Eddington’s generalisation of Weyl’s theory [51]. Other
mathematicians took Weyl’s theory at its face value
and drew consequences; thus M. Juvet calculated Frenet’s
formulas for an “-èdre” in Weyl’s geometry by generalising a
result of Blaschke for Riemannian geometry [179
]. More important,
however, for later work was the gauge invariant tensor calculus by
a fellow of St. John’s College in Cambridge,
M. H. A. Newman [236
]. In this calculus,
tensor equations preserve their form both under a change of
coordinates and a change of gauge. Newman applied his scheme to a
variational principle with Lagrangian
and concluded:
“The part independent of the ‘electrical’
vector is found to be
, a
tensor which has been considered by Einstein from time to time in
connection with the theory of gravitation.” ([236],
p. 623)
After the Second World War, research following Weyl’s classical geometrical
approach with his original
1-dimensional Abelian gauge-group was resumed. The more important
development, however, was the extension to non-Abelian gauge-groups and the combination
with Kaluza’s idea. We shall discuss
these topics in Part II of this article. The shift in Weyl’s interpretation of the role
of the gauging from the link between gravitation and
electromagnetism to a link between the quantum mechanical state
function and electromagnetism is touched on in Section 7.