ISREAL MATHEMATICAL CONFERENCE PROCEEDINGS Vol. 12, 1999
PRESENTED
AT THE SYMPOSIUM ON
THE HERITAGE OF EMMY
NOETHER IN ALGEBRA, GEOMETRY, AND PHYSICS
Bar Ilan University, Tel
Aviv, Israel, December 2-3, 1996

E. Noether's Discovery of the Deep Connection Between Symmetries and
Conservation Laws

Nina Byers

Physics Department
University of California at Los Angeles
Los
Angeles, CA. 90095

Abstract

Emmy Noether proved two deep theorems, and their converses, on the
connection between symmetries and conservation laws. Because these theorems
are not in the mainstream of her scholarly work, which was the development
of modern abstract algebra, it is of some historical interest to examine how
she came to make these discoveries. The present paper is an historical
account of the circumstances in which she discovered and proved these
theorems which physicists refer to collectively as Noether's Theorem. The
work was done soon after Hilbert's discovery of the variational principle
which gives the field equations of general relativity. The failure of local
energy conservation in the general theory was a problem that concerned
people at that time, among them David Hilbert, Felix Klein, and Albert
Einstein. Noether's theorems solved this problem. With her
characteristically deep insight and thorough analysis, in solving that
problem she discovered very general theorems that have profoundly influenced
modern physics.

Introduction

Emmy Noether's paper "Invariante Varlationsprobleme" [1],
hereafter referred to as I.V.,  profoundly influenced 20th
century physics. It was presented to the July 16, 1918 meeting of the
Könighche Gesellschaft der Wissenschaften zu Göttingen by Felix Klein
presumably because Noether was not a member of the Gesellschaft1.
One wonders if she was even present when the paper was read. The paper proved
two theorems and their converses which revealed the general connection between
symmetries and conservation laws in physics. They led to a deeper
understanding of laws such as the principles of conservation of energy,
angular momentum, etc., and also were instrumental in the great discoveries of
gauge field symmetries of the 20th century. [2]

Remarkably these theorems are something of a departure from the main line
of Noether's mathematical research, which was the development of modern
abstract algebra. An examination of the historical circumstances in which she
made these discoveries sheds light on how she came to do this work. The work
followed upon David Hilbert's discovery of the variational principle from
which he derived the field equations of general relativity. At this time David
Hilbert, Felix Klein and others in Göttingen were intensely interested in the
recently completed general theory of relativity. There were unresolved issues
regarding the question of energy conservation in the theory. Published
correspondence [3]
between Hilbert and Klein regarding this indicates that Noether's help was
requested by Hilbert to clarify these issues, and that she then did the work
whose results are published a few years later in I.V.. The discussion
and proofs of the two theorems in I.V. clarified and resolved the
issues regarding energy conservation, as Hilbert acknowledged in his 1924
Grundlagen der Physik article [4].
From her Collected Works [5],
it would seem that after this and some additional work on differential
invariants in general relativity [6],
Noether returned to the main line of her mathematical research.

Here I will present first a chronology of events leading up to Noether's
publication of the I.V. paper; then a discussion of some rather
difficult issues regarding energy conservation in general relativity which her
work resolved; and finally a description and discussion of the two theorems in
the I.V. paper.

Chronology of the Events Leading to the Discovery

In 1915, Emmy Noether was invited to join the team of mathematicians
assembled in Göttingen by David Hilbert. Hermann Weyl reports [7]:
"To both Hilbert and Klein, Emmy was welcome as she was able to help them with
invariant-theoretic knowledge." She was thirty-three at that time, having
received a doctorate in mathematics from the University of Erlangen seven
years earlier, and written the eleven papers listed in Appendix A. The first
in the list is her thesis which was done under the supervision of Paul Gordan.
For her thesis she calculated all the 331 invariants of ternary bi-quadratic
forms! Shortly thereafter she took the abstract approach to algebra following
Hilbert's 1888 basis theory paper. She worked unpaid in Erlangen supervising
students and sometimes lecturing for her ailing father. After her father died,
she joined Hilbert and his team in Göttingen. [8]

In June-July 1915, shortly after Noether arrived, Albert
Einstein gave six lectures in Gottingen on the general theory of relativity.
At that time the theory was not yet finished; he had not yet found the
complete field equations. However, the basic ideas were clear and his audience
found them compelling. After giving the lectures, Einstein said [9]:
"To my great joy, I completely succeeded in convincing Hilbert and Klein." He
had been working to generalize the special theory of relativity to include
gravity since 1905. In 1907 he discovered the importance of the equality of
gravitational and inertial mass and formulated the equivalence principle, but
it took another eight years to complete the theory. Finally in November 1915,
having found the complete field equation, he submitted the famous paper [10]
that gives the theory in its final form. Remarkably, in the same month,
Hilbert submitted a manuscript [11]
in which the same field equations are obtained as the solution to a
variational problem. Hilbert and Einstein had independently found the field
equations at about the same time [9].
In November 1915, Emmy Noether wrote to Ernst Fischer: "Hilbert plans to
lecture next week about his ideas on Einstein's differential invariants,and so
.. [we] had better be ready" [8].
It would seem, therefore, that she began to study relativity theory then. Out
of that study came two papers which Hermann Weyl characterized as giving "the
genuine and universal mathematical formulation of two of the most significant
aspects of general relativity theory: first, the reduction of the problem of
differential invariants to a purely algebraic one by use of normal
coordinates; and second, the identities between the left sides of Euler's
equations of a problem of variation ..." [7].
Regarding the first paper referred to by Weyl [6],
Einstein wrote to Hilbert: "Yesterday I received from Miss Noether a very
interesting paper on invariant forms. I am impressed that one can comprehend
thesematters from so general a viewpoint. It would not have done the old guard
at Göttingen any harm had they picked up a thing or two from her. ..." [12].
The second paper referred to by Weyl is the I.V. paper we will discuss
at length.

Hilbert had been interested in the fundamental laws of
physics for many years [13].
The paper in which he published his derivation of the field equations for
general relativity is entitled Grundlagen der Physik [11].
It was an early effort to formulate a unified field theory of gravity,
electromagnetism, and matter. In this was unsuccessful, and Hilbert omitted
this paper from his collected works [14].
However, when specialized to gravity, Hilbert's derivation of the field
equations is an original and important contribution to the general theory. The
Lagrangian he introduced is known as the Hilbert-Einstein Lagrangian, and his
formulation of the theory is in widespread use today. Pais writes: "Hilbert
was not the first to apply the principle to gravitation. Lorentz had done so
before him. So had Einstein, a few weeks earlier. Hilbert was the first,
however, to state this principle correctly." As regards Hilbert's 1915 paper,
Hermann Weyl [13]
wrote: "Einstein's more sober procedure ... [has] proved the more fertile.
Hilbert's endeavors must be looked upon as a forerunner of a unified field
theory of gravitation and electromagnetism. However, there was still much too
much arbitrariness involved in Hilbert's Hamiltonian function; subsequent
attempts (by Weyl, Eddington, Einstein himself, and others) aimed to reduce
it. .. But the problem of a unified field theory stands to this day as an
unsolved problem."

Though the general theory of relativity was completed in
1915, there remained unresolved problems. In particular, the principle of
local energy conservation was a vexing issue. In the general theory, energy is
not conserved locally as it is in classical field theories - Newtonian
gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the
general theory has been perplexing many people for decades. In the early days,
Hilbert wrote about this problem as 'the failure of the energy theorem '. In a
correspondence with Klein [3],
he asserted that this 'failure' is a characteristic feature of the general
theory, and that instead of 'proper energy theorems' one had 'improper energy
theorems' in such a theory. This conjecture was clarified, quantified and
proved correct by Emmy Noether. In the note to Klein he reports that had
requested that Emmy Noether help clarify the matter. In the next section this
problem will be described in more detail and an explanation given of how
Noether clarified, quantified, and proved Hilbert's assertion. One might say
it is a lemma of her Theorem II.

Klein was working on this problem of energy conservation, or
as he termed it Hilbert's energy vector, in 1916. He wrote in this
correspondence with Hilbert that "Frl. Noether advised me continually
throughout my work, and it was really only through her that I was led to the
material presented in this letter. When I spoke recently with Frl. Noether
about my result concerning your energy vector, she told me that she had
derived the same thing from your Note2
a year ago and written it up in a manuscript (which I examined). " Klein
presented a paper with his results on Hilbert's energy vector to the July 19
meeting of the Gesellschaft. In this paper [14],
he acknowledges helpful contributions from Noether. However, in the annotation
to this paper his collected works he makes it clear that his result is a
special case of her 'far-reaching' theorem. He reports that he presented her
paper to the Gesellschaft the following week, and says that she also proved
and generalized some of Hilbert's ideas. An English translation of that
interesting annotation is given in Appendix B.

In his famous 1924 Grundlagen der Physik paper3,
David Hilbert credits Emmy Noether with having solved the problem of an energy
theorem in the general theory, and refers to I.V.. In the concluding
section of her paper, she refers to Hilbert's having said that 'the failure of
the energy theorem' is a characteristic feature of the general theory. The
last section is entitled "A HILBERTIAN ASSERTION", and says in part [16]:
"From the foregoing ... [we] obtain the proof of an assertion of Hilbert
concerning the connection between the failure of proper energy conservation
laws and general relativity, and indeed in a general group- theoretic
setting." It seems to me this remark is a notable understatement. Her powerful
results are deep and general; far more than simply a verification of Hilbert's
assertion.

In 1919, Noether chose I.V. for her Habilitation
thesis and presented it to the University of Gttingen along with the twelve
previously published papers and two additional manuscripts, one of which [17]
contained a number of important ideas which had a significant impact on the
development of modern abstract algebra. Years before she had been refused
Habilitation the right to lecture under University auspices). Hermann Weyl
records [17]
"during the war Hilbert tried to push through Emmy Noether's Habilitation in
the Philosophical Faculty in Gttingen. He failed due to the resistance of the
philologists and historians. It is a well-known anecdote that Hilbert
.[declared] at the faculty meeting; 'I do not see that the sexof a candidate
is an argument against her admission as Privatdocent. After all, we are a
university not a bathing establishment.' "It was only after the liberalization
of Germany in the post-WWI period that Habilitation was granted, and she was
finally able to lecture officially at the University. She had given lectures
in the preceeding years, but the posted notices stated that they were offered
by Herr Professor David Hilbert, given by Frl. E.Noether, with no tuition
required. These lectures were becoming famous and drawing the attendance of
mathematicians from all over Europe. Without Habilitation she was working
unpaid.

It is quite clear from the application papers written for
Habilitation that Noether knew the scope and importance of her results for
physics. She wrote about the I.V. paper [8]"...
The general results contain, as special cases, theorems on first integrals
[conservation laws] ... in mechanics; furthermore, conservation theorems and
interdependences among the field equations in the theory of relativity... "
She remarks that this work was "an outgrowth of my assistance to Klein and
Hilbert in their work on Einstein's general theory of relativity."

`Proper' and `Improper' Conservation Laws

In contemporary terminology the general theory of relativity
is a gauge theory. The symmetry group of the theory, is a gauge group. It is
the group of all continuous coordinate transformations with continuous
derivatives, often called the group of general coordinate transformations. It
is a Lie group that has a continuously infinite number of independent
infinitesimal generators. In Noether's terminology such a group is an infinite
continuous group. The symmetry group of special relativity, the Poincare'
group4
is a Lie subgroup of the group of general coordinate transformations. It
has a finite number (7) of independent infinitesimal generators. Noether
refers to such a group as a finite continuous group. This distinction between
a Lie group with a finite (or countably infinite) number of independent
infinitesimal generators and an infinite continuous group is what
distinguishes Noether's theorem I and theorem II in I.V.. Theorem I
applies when one has a finite continuous group of symmetries, and theorem II
when there is an infinite continuous group of symmetries. Field theories with
a finite continuous symmetry group have what Hilbert called `proper energy
theorems'. Physically in such theories one has a localized, conserved energy
density; and one can prove that in any arbitrary volume the net outflow of
energy across the boundary is equal to the time rate of decrease of energy
within the volume. As will be shown below, this follows from the fact that the
energy-momentum tensor of the theory is divergence free. In general
relativity, on the other hand, it has no meaning to speak of a definite
localization of energy. One may define a quantity which is divergence free
analogous to the energy-momentum density tensor of special relativity, but it
is gauge dependent: i.e., it is not covariant under general coordinate
transformations. Consequently the fact that it is divergence free does not
yield a meaningful law of local energy conservation. Thus one has, as Hilbert
saw it, in such theories `improper energy theorems.'

A key feature for physics of Noether's I.V. paper is
the clarity her theorems brought to our understanding of the principle of
energy conservation. As Feza Gursey wrote [18]:
"Before Noether's Theorem the principle of conservation of energy was shrouded
in mystery, leading to the obscure physical systems of Mach and Ostwald.
Noether's simple and profound mathematical formulation did much to demystify
physics." Noether showed in her theorem I that the principle of energy
conservation follows from symmetry under time translations. This applies to
theories having a finite continuous symmetry group; theories that are Galilean
or Poincar invariant, for example. In general relativity, on the other hand,
energy conservation takes a different form as will be shown below. Noether's
theorem 11 applies in the case of general relativity and one sees that she has
proved Hilbert's assertion that in this case one has `improper energy
theorems', and that this is a characteristic feature of the theory. It is
owing to the fact that the theory is a gauge theory; i.e., that it has an
infinite continuous group of symmetries of which time translations are a
subgroup. Indeed generally she defines as `improper' divergence relationships,
which vanish when the field equations are satisfied, which correspond to a
finite continuous subgroup of an infinite continuous group. Generally they do
not have the required invariance or covariance properties under the larger
group. For example, in general relativity a divergence free energy-momentum
(pseudo) tensor can be constructed but it is gauge dependent (see below).
Because it is not covariant under general coordinate transformations, it is
more properly called a pseudotensor. Such pseudotensors are covariant with
respect to the linear transformations of the Poincar group and may be used in
asymptotic spacetime regions far from gravitating sources to derive a
principle of energy conservation. [19]

To illucidate these matters further, we discuss in some
detail field theories of matter, gravity, electromagnetism, etc. in both
special and general relativity. In special relativity these theories have a
`proper energy theorem' in the sense of Hilbert and we will show how `proper
energy theorems' give a principle of local energy conservation. In general
relativity, on the other hand, the proper energy theorem becomes improper in
that the energy-momentum tensor for which the theorem holds is gauge
dependent. As will be shown below, there is transfer of energy to and from the
gravitational field and it has not meaning to speak of a definite localization
of the energy of the gravitational field in space5.
Consequently we do not have a principle of local energy conservation in
spacetime regions in which there exist gravitational fields.

The theories we discuss here are field theories which may be
formulated in terms of a variational principle; i.e., their field equations
may be obtained from Hamilton's principle. This was Hilbert's great
contribution to the general theory of relativity. He showed the correct field
equations can be derived from Hamilton's principle, and Noether's work
followed upon this. In this formulation, the theory is defined by a Lagrangian
L. The field equations are derived from Hamilton's principle
that the action

(1)

is an extremum. The Lagrangian depends on the field
quantities and their derivatives. Symbolically let phi represent the
field quantities and, for simplicity, assume L depends only on
phi and its first derivatives. Then Hamilton's Principle gives
the field equations: viz.,

(2)

Here we discuss in some detail laws of energy conservation in
special and general relativity where Noether's theorem I applies in the first
case and theorem II in the second.6
When the action is invariant with respect to transformations of the Poincar
group, Noether's theorem I gives a divergence-free energy-momentum tensor
T mu   nu(x) which, symbolically in the notation
of (2), for scalar fields is given by 7

(3)

More generally one may write the energy-momentum tensor as 8

(4)

Theorem I shows invariance of the action under spacetime
translations implies that the divergence of T mu   nu
is proportional to the left side of (2) 9.
Consequently when the field equations are satified, the divergence vanishes
and we have 10

(5)

This implies local energy conservation. The energy density at
the spacetime point x is given by the T00(x)
and the T0i(x) components give the energy crossing a
unit area per unit time in the direction of the normal i. The time-like
equation (5), i.e., with mu = 0, follows from the invariance of the
theory under time translations. To show that this gives local energy
conservation, it is convenient to use a 3-vector F for the
T0i components; then (5) reads

(6)

Integration of (6) over any spatial volume V shows, by
Gauss's theorem, that the rate of change of energy in V equals the net energy
flow across the bounding surface per unit time. This holds for arbitrary and,
therefore, arbitrarily small volumes. Thus (5) gives, in the language of
contemporary physicists, local energy conservation. This is a `proper energy
theorem' in the sense of Hilbert.

In general relativity, on the other hand, the components of
the metric tensor gmu nu are the gravitational field
variables and generally spacetime dependent. They are included as field
variables in the Lagrangian of the theory with the addition of the
Hilbert-Einstein term R; L has in it, as before, the electromagnetic,
matter, etc. field variables. The action is

(7)

where g is the determinant of g mu nu is a
coupling constant and R is the Riemannian curvature scalar. Following Hilbert,
one may derive the field equations of general relativity from the condition
that (7) is an extremum with respect to variations of g mu nu
. One obtains

(8)

where Rmu nu is the Ricci tensor
constructed by contraction from the Riemann curvature tensor, and R is the
Ricci or curvature scalar. The Riemann curvature tensor is constructed from
the metric tensor g mu nu and its first and second
derivatives. The source term on the right side of (8) is the energy-momentum
tensor Tmu nu given by (4).[19]
It contains contributions from the electromagnetic fields, matter fields, etc.
but not from gravitational fields. Since gravitational fields carry energy and
momentum, T00 is not the total energy density. As will be shown
below, Tmu nu is not divergence-free. It cannot and does not
play the same role as regards energy-momentum conservation as in special
relativity. This surely was immediately clear to Hilbert in 1915. What was
apparently not clear to Hilbert and Klein in the early days was how the
principle of conservation of energy is satisfied in this theory. This will be
discussed below.

For theories whose symmetry group is an infinite continuous
group, the main results of theorem II are that there are certain identities,
or "dependencies" as she called them [1],
between Lagrange functions of the theory and their derivatives. This is quite
different from the results of theorem I that for each infinitesimal generator
of the finite continuous group there is a quantity whose divergence vanishes
when the Lagrange functions vanish (field equations are satisfied). In general
relativity, the "dependencies" give the Bianchi identities for components of
the curvature tensor. These results are covariant under the full group. In
particular, one has the contracted Bianchi identity

(9)

This equation states that the covariant divergence of the
left side of (8) vanishes. The covariant divergence differs substantially from
the ordinary divergence which we have been discussing up to now. For a second
rank tensor, e.g. Tmu nu , the covariant divergence is given
by

(10)

where the Christoffel symbols (the gamma symbols)
depend upon the metric tensor gmu nu and its derivatives and
are non-vanishing when the Riemann curvature is different from zero. The
covariant divergence is an invariant; i.e., the covariant divergence of a
second rank tensor gives a first rank tensor. Thus (9) implies

(11)

This is often referred to as the energy-momentum conservation
law for general relativity. The presence of the metric tensor and its
derivatives in (10) shows the energy-momentum transfer contained in (11)
between the gravitational fields and the source fields in T mu
nu .

From (10) it is clear the ordinary divergence T mu
nu, nu   is in general different from zero in
spacetime regions with gravitational fields. It is interesting to note,
however, that at any given spacetime point one may choose a set of coordinates
for which the gravitational fields vanish (gmu nu reduces to
the flat spacetime Minkowski metric and the Christoffel symbols vanish). This
is guaranteed by the equivalence principle which states that one can always
choose a coordinate system such that spacetime in the neighborhood of a given
point is Minkowski (flat). Thus one may see why it is not meaningful to speak
of a localized energy density for gravitational fields.

The physics of these relations is somewhat complicated. In
regions of spacetime near gravitating sources, where the Riemann curvature is
non-vanishing, there is failure of a principle of local energy conservation.
The energy balance locally cannot be discussed independently of the
coordinates one uses to calculate it, and consequently different results are
obtained in various different coordinate frames — some being artifacts of the
calculation itself. An amusing example of this is the case of an accelerating
mass. For simplicity, let us take the mass and acceleration small, so that the
deviations from Newtonian physics are small and may be treated perturbatively
- the so-called post-Newtonian approximation. In this approximation the theory
is similar to electromagnetic theory. In electromagnetic theory, an
accelerating charge radiates and there is the radiation reaction force which
tends to decelerate it; there is energy loss to the radiation field. In
Einstein gravity, the accel erating mass is a source of gravitational
radiation and one may calculate the radiation reaction force as one does in
electromagnetic theory. A naive calculation along these lines yields a
radiation reaction force term (å term in the equation of motion with a the
acceleration) that has the opposite sign to the electromagnetic case; the
particle appears to speed up gaining energy as it radiates! This is an
unphysical radiation reaction force. The calculation is not gauge invariant
and depends on the choice of coordinates used to describe the motion of the
particle.

However, though local energy conservation fails, there is a
large scale principle of energy-momentum conservation in the general theory.
The problem in principle was solved by Noether. She pointed out in I.V.
that it is possible to construct a divergence-free quantity now generally
referred to as the energy-momentum pseudotensor T mu nu
eff.     Jackiw et al. [21]
have recently constructed such a pseudotensor using the methods of Noether
introduced in her theorem II. Other constructions have been made by Weinberg
[22]
and Landau and Lifshitz [20].
Such pseudotensors have the form

(12)

and satisfy

(13)

The tmu nu are not unique, and they are
gauge dependent. Owing to this gauge dependence, Tmu
nueff is called a pseudotensor because it does not
transform as a second rank tensor under the full group of general coordinate
transformations. However, these constructions are covariant under the linear
transformations of the Poincaré group and may be used to calculate energy and
momentum in gravitationally field free regions and asymptotically field free
regions. Using such pseudotensors, one can formulate a principle of energy
conservation by integrating over a large, asymptotically flat spacelike
hypersurface.11
In so doing one obtains a total energy-momentum four-vector Palpha
which includes the energy and momentum carried by gravitational
radiation and is constant in time. The tmu nu contributions
take gravitational radiation into account, and the calculations are done over
distances large compared to the wavelength of the radiation. This total
energy-momentum vector behaves as a special relativistic four-vector for any
transformation which is a Lorentz transformation far from sources and the
vector is independent of the choice of coordinates used in the calculation. [19]
Such a calculation is applied to the observed decrease of the orbital period
of the binary pulsar PSR 1913+1916, and used to estimate the associated flux
of gravitational radiation.12

The Theorems

The main results ofI.V.. are two theorems which are
proved along with their converses. They will be described here. Felix Klein's
statements [16]
of the theorems are given in Appendix B.

Though they apply more generally, we shall describe the
theorems as they apply to field theories. Consider an integral I
over independent variables x of a functional L depending
on functions (field variables) of these x variables phi(x) and
their derivatives; viz.,

(14)

The functional L may also depend directly on
the x. The integral I in physics is the action in the
sense of Hamilton's Principle, and the functional L is the
Langrangian of the theory. The theorems apply to systems for which
I is an invariant of a Lie group of transformations which
transform, x -> y, phi -> chi ,... ; i.e.,
when one has

(15)

integrated over an arbitrary region of x space and the
corresponding y space.

The two cases of a finite and an infinite Lie group are
separately considered. In Noether's notation, a finite Lie group is denoted by
Grho; it is a continuous group of transformations whose most
general transformation depends analytically on a finite (or countably
infinite) number rho of independent parameters. An infinite Lie group,
on the other hand, is denoted by Ginfinity-rho. It is a
continuous group whose most general transformation depends on rho
independent, arbitrary functions and their derivatives. The results of
theorem I apply when the action I is an invariant of a finite Lie
group, and the results of theorem II when it is an invariant of an infinite
Lie group. The results are different in the two cases.

Theorem I for Grho gives the result that
there is a divergence for each of the rho independent generators of the group
which is proportional to a Lagrange function. In her terminology the Lagrange
functions of the theory are the left hand side of equations of motion (field
equations) of which the right hand side vanishes. Thus when the equations of
motion are satisfied, the divergences vanish. For the spacetime displacements
of the Poincaré group, the four divergences are four tensor component
divergences. The vanishing of these divergences give energy-momentum
conservation. The U(1) Lie group of global gauge transformations in
electrodynamics is another example. Corresponding to the generator of this
group, one has a divergence free current and from this one proves charge
conservation.13

Theorem II applies when the action is an invariant of an
infinite Lie group. The statement of the theorem for this case is very general
and, aside from its application to general relativity, it applies in a wide
variety of other cases. For example, quantum chromodynamics and other gauge
field theories are theories to which it applies. From theorem II, one has
identities beteen Lagrange functions and their derivatives. See Appendix B.
These identities Noether calls ‘dependencies’. Generally they are covariant
with respect to the group. For example, the Bianchi identities in the general
theory of relativity are examples of such ‘dependencies’.

It is important to point out that in I.V. proofs are
given for the converses of theorems I and II as well. For example, theorem I
states that if the action I is an invariant of a finite Lie
group, there are rho divergences which vanish when the equations of
motion are satisfied. The converse also is proved I.V..; namely that
the rho vanishing divergences imply that the action is an invariant of.
This result was important in the history of discovery of symmetries in
elementary particle physics. [2]

I conclude this talk with a quotation from I.V. where
Noether says that Theorem II is "proof of a Hilbertian assertion about the
connection of the failure of proper laws of conservation of energy with
general relativity ... in a generalized group theory version." One of the
reasons why her theorems have been very important for 20th century physics is
because they are in ‘a generalized group theory version‘.

FOOTNOTES:

The Royal Society of London, founded in 1662, elected its first female
member in 1945; the Academie des Sciences, Paris, was founded in 1666 and
elected its first female member in 1962.

Ref.[11]

This is included in his Collected Works [4].
It has the same title as the 1915 paper which was omitted from the Collected
Works.

The group of Lorentz transformations and spacetime translations.

A clear exposition regarding gravitational field energy is given by
Landau and Lifshitz [20].

In the next section, Noether's theorems will be described more
generally.

We use Minkowski notation for the components of the spacetime point
x; viz., xmu with mu = 0, 1, 2, 3 with xO
= t and xi for three orthogonal space directions
(i = 1,2,3); gmu nu is the Minkowski metric tensor
whose diagonal components are -1, 1, 1, 1 . The notation is
used also.

This form is useful later when we generalize to general relativity.

This is called by Noether a Lagrange function.

Repeated indices are summed. Relativists' notation for this divergence
is Tmu nu, nu=0.

This is done in Refs. [19]
and [20].

A detailed discussion of this and references to the literature can be
found in Ref.[19].

If, on the other hand, the action is invariant with respect to local
gauge transformations, the symmetry is of an infinite Lie group
corresponding to the possibility of independently transforming the gauge at
every spacetime point. Theorem II applies in this case. As in theorem I
case, one gets electric current conservation from theorem II. However, if
non-Abelian gauge transformations are considered, the result is more
complicated.

Acknowledgements

I would like to thank Mina Teicher and her colleagues for
organizing this very interesting and rewarding Symposium; and my colleague
Basil Gordon for the help he afforded me in understanding and appreciating the
workings of the minds of these great mathematicians, and for his English
translations of their papers, and for his close reading of this manuscript in
its various drafts. I would also like to thank my colleague Aaron Grant for a
critical reading of the final draft.

Appendix A

Über die Bildung des Formensystems der ternären biquadratischen Form.
Sitz. Ber. d. Physikal.-mediz. Sozietät in Erlangen 39 (1907), pp. 176-179.

Über die Bildung des Formensystems der ternären biquadratischen Form.
Journal f. d. reine u. angew. Math. 134 (1908), pp. 23-90.

Zur Invariantentheorie der Formen von n Variabeln. J. Ber. d. DMV 19
(1910), pp. 101-104.

Zur Invariantentheorie der Formen von n Variabeln. Journal f. d. reine
u. Angew. Math. 139 (1911), pp. 118-154.

Rationale Funktionenkörper. J. Ber. d. DMV 22 (1913). pp. 316-319.

Körper und Systeme rationaler Funktionen. Math. Ann. 76 (1915), pp.
161-196.

Der Endlichkeitssatz der Invarianten endlicher Gruppen. Math. Ann. 77
(1916), pp. 89-92.

Über ganze rationale Darstellung der Invarianten eines Systems von
beliebig vielen Grundformen. Math. Ann. 77 (1916), pp. 93-102.

Die allgemeinsten Bereiche aus ganzen transzendenten Zahlen. Math. Ann.
77 (1916), pp. 103-128.

Die Funktionalgleichungen der isomorphen Abbildung. Math. Ann. 77
(1916), pp. 536-545.

Gleichungen mit vorgeschriebener Gruppe. Math. Ann 78 (1918), pp.
221-229.

Appendix B

The following quotation is from Felix Klein's annotation to his paper
"Über die Differentialgesetze für die Erhaltung von Impuls und Energie in
die Einsteinschen Gravitationstheories" [14]
which can be found in his annotataed collected works Gesammelte
mathematische Abhandlungen, Julius Springer, Berlin 1921; Ch. XXXI
(erster band, p. 559). In this quotation "The main theorem given In section
2 above" refers to his paper which he is annotating.

"I presented the main theorems of Frl. Noether at the
July 26 meeting ... The main theorem given in section 2 above is a
special case of the following far-reaching theorem of Frl. Noether:

If an integral I is invariant under a continuous group
Grho with rho parameters, then rho
linearly independent combinations of the Lagrangian expressions arc
divergences.

Concerning in particular the claim made by Hilbert [that
the failure of 'proper' energy theorems was characteristic of the theory],
the following is an exact formulation according to Frl. Noether:

If an integral I is invariant under the translation group, the energy
relations are improper if and only if I is invariant under an
infinite-dimensional group which contains the translation group as a
subgroup.

Incidentally, the theorem of Hilbert that there are four
relations between the field equations [Bianchi identities] also finds its
generalization in Frl. Noether's work. Her theorem is as follows:

If an integral I is invariant under a group with p
arbitrary functions in it with their derivatives up to the s'th order, then there are rho identities
between the Lagrangian expressions and their derivatives of order up to
s."

English translation courtesy of Basil Gordon.

References

E. Noether, "Invariante
Varlationsprobleme", Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen,
Math-phys. Klasse (1918), 235-257; English translation M.
A. Travel, Transport Theory and Statistical Physics 1(3) 1971,183-207.

Nina Byers, History of Original Ideas and Basics Discoveries in
Particle Physics, H. B. Newman and T. Ypsilantis ed., Plenum Press, New
York (1996).

Correspondence between D. Hilbert and F. Klein published by F. Klein in
Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math.-phys. Klasse (1918)
as Zu Hilberts erster Note über die Grundlagen der Physik;
included in his annotated collected works Ref. [15]; see Ch. XXXI
(erster band, p. 560). English translation courtesy of Basil Gordon.

D.Hilbert, Gesammelte Abhandlungen, Springer-Verlag, New
York, 1970.

Emmy Noether, Collected Papers, Springer-Verlag 1983.

E. Noether, "Invarianten beliebiger Differentialausdrücke,",
Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse
(1918) 37 - 44.

H. Weyl, Scripta Matheinatica III. 3 (1935) 201-220; an English
translation of this memorial lecture is given in Ref. [8].

Auguste Dick, Emmy Noether (1882 -1935), Birkhauser 1981;
English translation by H. I. Blocher.

A. Pais, Subtle is the Lord, Oxford University Press 1982.

A. Einstein, Sitzungsberichte, Preussische Akademie der
Wissenschaften, 1915, p. 844.

D. Hilbert, "Die Grundlagen der Physik," Göttinger Nachrichten,
Math.-phys. Klasse (1915), S. 395-407.

Clark Kimberling in Emmy Noether, A Tribute to Her Life and Work;
James W. Brewer and Martha K. Smith, ed.; Marcel Dekker, Inc.

Constance Reid, Hilbert, Springer-Verlag, New York (1970).

F. Klein, "Über die Differentialgesetze für die Erhaltung von
Impuls und Energie in die Einsteinschen Gravitationstheories.", Nachr.
d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse (1918).

Felix Klein, Gesammelte Mathematische Abhandlungen, Julius Springer,
Berlin 1921.

English translation courtesy of Basil Gordon.

E. Noether and W. Schmeidler, "Moduln in nichtkommutativen Bereichen,
insbesondere aus Differential- und Differen-zenaus-drücken", Math. Zs. 8
(1920), pp. 1-35.

Feza Gursey as quoted by Nathan Jacobson in his lengthy Introduction to
Emmy Noether, Collected Works, Ref. [5].

See, e.g., Ignazio Ciufolini and John Archibald Wheeler, Gravitation and
Inertia, Princeton University Press, Princeton, N.J. 1995.

L. Landau and E. Lifshitz, The Classical Theory of Fields, English
translation by M. Hamermesh, Addison-Wesley Publishing Co., Inc., Reading,
MA 1951.

D. Bak, D. Cangemi, and R. Jackiw, Phys. Rev.D 49: 5173 (1994).

Steven Weinberg, Gravitation and Cosmology, John Wiley & Sons, Inc.,
New York 1971.