664 DOCUMENT 472 MARCH
1918
werden
die
Gl.
allerdings
4.
Ordnung.[4]
Darf
ich
Ihnen,
wenn
ich’s
ausgearbeitet
habe,
das
Manuskript
(etwa
10
Seiten) zuschicken,
daß Sie’s vielleicht in der Ber
liner Akademie
vorlegen?
Ende März komme ich nach Berlin und würde mich sehr
freuen,
Sie besuchen
zu
dürfen. Oder kommen Sie in den Ferien wieder in die
Schweiz? Mit besten
Grüßen,
in
größter Verehrung
Ihr
H
Weyl
AKS.
[24
007].
The
verso
is addressed
"Herrn
Professor
Dr.
A. Einstein Berlin W 30 Haberlandstr.
5,"
with return address "Abs. H.
Weyl
Zürich
Schmelzbergstr.
20,"
and
postmarked
"Zürich
8 (Flun
tern)
1.III.18.7."
[1]Weyl
1918c.
[2]Weyl
had
taught
a
threehour
course
at
the ETH
on
"Raum,
Zeit,
Materie"
in
summer
semester
1917 (see
ETH
Programm
1917a,
p.
21), published as
Weyl
1918c
(see
preface
thereto).
[3]This
unification
is
based
on
a generalization
of
Riemannian
geometry.
As is
explained
in the
introduction
of
the
first
exposition
of
the
new theory
(Weyl
1918b,
pp.
466467),
Weyl’s point
of
departure
is the
concept
of
parallel displacement,
which had
recently
been introduced into differential
geometry
(LeviCivita 1917a,
Hessenberg
1917).
Since the
parallel displacement
of
a
vector
will
gen
erally
be
path
dependent,
Riemannian
geometry,
unlike Euclidean
geometry,
does not allow
comparison
of
the direction
of
vectors at
points
that
are
not
infinitesimally
close
to
one
another. It
does, however,
allow
comparison
of
the
length
of
such vectors.
Weyl argued
that in
a "truly
local
geometry" ("wahrhafte
NaheGeometrie,"
Weyl
1918b,
p. 466)
such
global comparison
of
lengths
should be
given
up as
well.
Units
of
length
have to
be defined
locally.
As he
put
it in
a subsequent
paper:
"this task cannot be handed
over
to
a
central
gauge
bureau"
("...
diese
Aufgabe
kann nicht
einem zentralen Eichamt
übertragen
werden."
Weyl
1919c,
pp.
102103). To
implement
this
notion,
Weyl stipulated
that the line element ds2
=
guvdxudxv
be determined
only up
to
a
local
gauge
fac
tor
X,
some arbitrary
continuous function
of
the
spacetime
coordinates. In other words,
only
the
ra
tios
of
the various
components
of
the metric
tensor
gmv have
definite values,
not
these
components
themselves.
Hence,
an
invariance
requirement
is introduced
in
addition
to general covariance, namely
invariance under the transformation
guv
 Aguv. Originally,
this
was
called "measuring
rod
invari
ance"
("MaßstabInvarianz,"
Weyl
1918b,
p.
475,
Weyl
1918d,
p.
398),
but it would
come to
be known
as "gauge
invariance"
("Eichinvarianz,"
Weyl
1919c,
p.
114; in the
EinsteinWeyl
correspondence,
this
term
is used for the first time
in
Doc.
661). Allowing
for
the
transport
of
local
length
standards
from
one
point
to another,
Weyl's
next
step
is
to
introduce
an
affine connection
(Weyl
1918b,
pp.
468
470).
He assumed that the inner
product
of
any
two vectors
after
parallel displacement
from
any one
point to
some
neighboring point
is
only proportional
(not
equal, as
in standard Riemannian
geometry)
to
the inner
product
of
these
vectors
before
parallel displacement. Writing
the
proportionality con
stant
as
1
+
dp,
he showed that
d(p
is
a
linear
differential form:
dtp
=
pidxi.
When
gmv
is
replaced
by
Aguv,
dtp changes
into
dtp +
dlogk.
So,
while
guv
is determined
only up
to
a
factor
X,
tpi
is
determined
only up
to
a
term
dlogX/dxi.
This
suggests
that the
vector
field
pi
can
be
interpreted as
the
electromagnetic fourpotential
and the
gauge
invariant
tensor
Fuv
= tU(pv

dvtp^
as
the electro
magnetic
field
(Weyl
1918b,
p.
471).
In this
way, Weyl’s generalization
of
Riemannian
geometry
leads
to
a
unification
of
gravity
and
electromagnetism.
For critical discussion
of
Weyl’s theory, see
Pauli
1921,
sec.
65
(and
note 22 in the
supplementary
notes
added
to
the translation). For historical discussion,
see
Vizgin
1989, 1994, and
Bergia
1993.
[4]Weyl
wanted
to
derive
the
field
equations
in his
new theory
from
a
variational
principle
(Weyl
1918b,
p.
174). Contrary
to what
is
suggested by Weyl’s
remarks
here,
it
turned out to
be easier
to
recover
Maxwell's
equations
from
the
new
theory
in this
way
than
to recover
the
Einstein
field
equa
tions. The curvature
scalar,
the
Lagrangian
for the metric field in Einstein’s
theory,
cannot be used
as
the
Lagrangian
in
Weyl’s theory
because it does
not
give a gaugeinvariant
action. The
simplest
Lagrangian
compatible
with
gauge
invariance
(see
Doc.
499, note 4)
leads
to
fourthorder field
equa
tions for the metric field. As he later
wrote to Einstein, however,
Weyl
convinced
himself
early on
that,
for
static fields and in first
approximation,
his
new theory
does
reproduce
Einstein’s field
equa
tions,
including
the
cosmological
term (see
Doc.
619, note
11,
for further
discussion).