14th Internatonal Summer School n Global Analyss and Mathematcal Physcs Satellte Meetng of the XVI Internatonal Congress on Mathematcal Physcs *** Lectures of Yvette Kosmann-Schwarzbach Centre de Mathématques Laurent Schwartz, Ecole Polytechnque, France Olomouc, Czech Republc August 10-14, 2009
Lecture 1 In Noether s Words: The Two Noether Theorems
Emmy Noether 1918 Invarant Varatonal Problems Invarante Varatonsprobleme Göttnger Nachrchten (1918), pp. 235 257. We consder varatonal problems whch are nvarant under a contnuous group (n the sense of Le). [...] What follows thus depends upon a combnaton of the methods of the formal calculus of varatons and of Le s theory of groups.
The two Noether Theorems In what follows we shall examne the followng two theorems: I. If the ntegral I s nvarant under a [group] G ρ, then there are ρ lnearly ndependent combnatons among the Lagrangan expressons whch become dvergences and conversely, that mples the nvarance of I under a group G ρ. The theorem remans vald n the lmtng case of an nfnte number of parameters. II. If the ntegral I s nvarant under a [group] G ρ dependng upon arbtrary functons and ther dervatves up to order σ, then there are ρ denttes among the Lagrangan expressons and ther dervatves up to order σ. Here as well the converse s vald. 1 1 For some trval exceptons, see 2, note 13.
Questons What varatonal problem s Noether consderng? What s the ntegral I? What are the Lagrangan expressons? Is G ρ a Le group of transformatons of dmenson ρ? In what sense s the ntegral I nvarant? What s a G ρ? What s the formal calculus of varatons?
What varatonal problem s Noether consderng? What s the ntegral I? Noether consders an n-dmensonal varatonal problem of order κ for an R µ -valued functon (n, µ and κ arbtrary ntegers) I = f ( x, u, u ) x, 2 u x 2, dx (1) I omt the ndces here, and n the summatons as well whenever t s possble, and I wrte 2 u x 2 for 2 u α, etc. x β x γ I wrte dx for dx 1... dx n for short. x = (x 1,..., x n ) = (x α ) ndependent varables u = (u 1,..., u µ ) = (u ) dependent varables
Calculus of varatons n a nutshell Consder a varaton of (x, u). Compute the varaton of L, hence that of I. Use ntegraton by parts to obtan the Euler Lagrange equaton. Defne the varatonal dervatve of L also called Euler Lagrange dervatve or Euler Lagrange dfferental. Denote t by δl δq or EL. A necessary condton for a map x u(x) wth fxed values on the boundary of the doman of ntegraton to mnmze the ntegral I = s the Euler Lagrange equaton f ( x, u, u ) x, 2 u x 2, dx EL = 0
Example: Elementary case x = t, u = q = (q ) Lagrangan of order 1 (EL) = δl δq = L q d L dt q The Euler Lagrange equaton n ths case s Lagrangan of order k L q d L dt q = 0. (EL) = δl δq = L q d dt L q + d 2 L dt 2 q... + ( 1) k d k dt k The Euler Lagrange equaton n ths case s L q d dt L q + d 2 L dt 2 q... + ( 1) k d k dt k L = 0. q(k) L q (k)
What are the Lagrangan expressons? Fact. When there are µ dependent varables, there are µ scalar Euler-Lagrange equatons. Ther left-hand sdes are the components of the Euler Lagrange dervatve of L. The Lagrangan expressons n Noether s artcle are the left-hand sdes of the Euler Lagrange equatons. In other words, the Lagrangan expressons are the components of the varatonal dervatve of the Lagrangan, denoted by f, wth respect to the dependent varables, denoted by u = (u ), and Noether denotes these Lagrangan expressons by ψ. Recall that Noether consders the very general case of a multple ntegral, u s a map from a doman n n-dmensonal space to a µ-dmensonal space, and the Lagrangan f depends on an arbtrary number of dervatves of u.
Noether s computaton (p. 238) [...] On the other hand, I calculate for an arbtrary ntegral I, that s not necessarly nvarant, the frst varaton δi, and I transform t, accordng to the rules of the calculus of varatons, by ntegraton by parts. Once one assumes that δu and all the dervatves that occur vansh on the boundary, but reman arbtrary elsewhere, one obtans the well known result, δi = δf dx = ( ψ (x, u, u x, )δu ) dx, (2) where the ψ represent the Lagrangan expressons; that s to say, the left-hand sdes of the Lagrangan equatons of the assocated varatonal problem, δi = 0.
Noether s text (contnued) To that ntegral relaton there corresponds an dentty wthout an ntegral n the δu and ther dervatves that one obtans by addng the boundary terms. As an ntegraton by parts shows, these boundary terms are ntegrals of dvergences, that s to say, expressons, Dv A = A 1 x 1 + + A n x n, where A s lnear n δu and ts dervatves. So Noether wrtes the Euler Lagrange equatons: ψ δu = δf + Dv A. (3) In the modern lterature, A s expressed n terms of the Legendre transformaton assocated to L.
Noether s explct computatons In partcular, f f only contans the frst dervatves of u, then, n the case of a smple ntegral, dentty (3) s dentcal to Heun s central Lagrangan equaton, ψ δu = δf d ( ) f dx u δu, whle, for an n-uple ntegral, (3) becomes ( ) ψ δu = δf f x 1 u x 1 δu x n ( u = du ), (4) dx ( ) f u δu x n. (5)
Noether s explct computatons (contnued) For the smple ntegral and κ dervatves of the u, (??) yelds d dx { (( ) 1 1 + d2 dx 2 { (( 2 2 ) f u (1) f u (2) ψ δu = δf (6) δu + δu + + + ( 1) κ dκ dx κ { ( κ κ ( ) 2 1 ( ) 3 2 ) f u (κ) f u (2) f u (3) δu }, δu (1) + + δu (1) + + ( ) κ 1 ( ) κ 2 f u (κ) f u (κ) and there s a correspondng dentty for an n-uple ntegral; n partcular, A contans δu and ts dervatves up to order κ 1. )} δu (κ 1) )} δu (κ 2)
Noether s proof That the Lagrangan expressons ψ are actually defned by (??), (??) and (??) s a result of the fact that, by the combnatons of the rght-hand sdes, all the hgher dervatves of the δu are elmnated, whle, on the other hand, relaton (??), whch one clearly obtans by an ntegraton by parts, s satsfed. Q.E.D Then Noether states her two theorems.
The theorems of Noether Frst theorem If the ntegral I s nvarant under a group G ρ, then there are ρ lnearly ndependent combnatons among the Lagrangan expressons whch become dvergences and conversely, that mples the nvarance of I under a group G ρ. The theorem remans vald n the lmtng case of an nfnte number of parameters. Second theorem If the ntegral I s nvarant under a group G ρ dependng upon arbtrary functons and ther dervatves up to order σ, then there are ρ denttes among the Lagrangan expressons and ther dervatves up to order σ. Here as well the converse s vald.
Is G ρ a Le group of transformatons of dmenson ρ? In some cases, yes, G ρ s a Le group of transformatons of dmenson ρ. Noether consders the nfntesmal transformatons contaned n G ρ, whch she denotes by y λ = x λ + x λ ; v (y) = u + u. In modern terms, she consders the LIE ALGEBRA of the ρ-dmensonal LIE GROUP, G ρ.
In modern notaton The ρ nfntesmal generators of the Le group are lnearly ndependent VECTOR FIELDS X (1),..., X (ρ), each a vector feld on R n R µ of the form X = n α=1 X α (x) x α + + µ =1 Y (x, u) u [nfntesmal automorphsm of the trval vector bundle F M, F = R n R µ, M = R n ] [projectable vector feld on F M]
Generalzed vector felds But G ρ can be much more general. In fact, for Noether, a transformaton s a GENERALIZED VECTOR FIELD: X = n α=1 X α (x) x α + µ =1 ( Y x, u, u ) x, 2 u x 2, u [NOT a vector feld on the vector bundle F M] [ntroduce JET BUNDLES] Generalzed vector felds wll be re-dscovered much, much later under many names: a new type of vector felds, Le-Bäcklund transformatons.
In what sense s the ntegral I nvarant? Noether defnes nvarance of the acton ntegral fdx: An ntegral I s an nvarant of the group f t satsfes the relaton, I = ( ) f x, u, u x, 2 u, dx x 2 = ( ) f y, v, v y, (7) 2 v, dy y 2 ntegrated upon an arbtrary real doman n x, and upon the correspondng doman n y.
Infntesmal nvarance Then she seeks a crteron n terms of the nfntesmal generators of the nvarance group. Now let the ntegral I be nvarant under G, then relaton (??) s satsfed. In partcular, I s also nvarant under the nfntesmal transformatons contaned n G, y = x + x ; v (y) = u + u, and therefore relaton (??) becomes 0 = I = f f ( y, v(y), v ( x, u(x), u x, ) y, ) dx, where the frst ntegral s defned upon a doman n x + x correspondng to the doman n x. dy (8)
The varaton δu But ths ntegraton can be replaced by an ntegraton on the doman n x by means of the transformaton that s vald for nfntesmal x, = f f ( x, v(x), v ) x, dx + ( y, v(y), v ) y, dy (9) Dv(f. x) dx. Noether then ntroduces the varaton (n modern terms, the vertcal generalzed vector feld) δu = v (x) u (x) = u u x λ x λ, so that she obtans the condton 0 = { δf + Dv(f. x)}dx. (10)
In her own words The rght-hand sde s the classcal formula for the smultaneous varaton of the dependent and ndependent varables. Snce relaton (??) s satsfed by ntegraton on an arbtrary doman, the ntegrand must vansh dentcally; Le s dfferental equatons for the nvarance of I thus become the relaton δf + Dv(f. x) = 0. (11) In modern terms, Le s dfferental equatons express the nfntesmal nvarance of the ntegral by means of the Le dervatve of f wth respect to the gven nfntesmal transformaton.
In her own words (contnued) If, usng (??), one expresses δf here n terms of the Lagrangan expressons, one obtans ψ δu = Dv B (B = A f. x), (12) and that relaton thus represents, for each nvarant ntegral I, an dentty n all the arguments whch occur; that s the form of Le s dfferental equatons for I that was sought. Noether s frst theorem s proved n all generalty! The equatons Dv B = 0 are the conservaton laws that are satsfed when the Euler Lagrange equatons ψ = 0 are satsfed.
Conservaton laws Is the condton DvB = 0 a conservaton law n the usual sense? In mechancs, a conservaton law s a quantty that depends upon the confguraton varables and ther dervatves, and whch remans constant durng the moton of the system. In feld theory, a conservaton law s a relaton of the form B 1 t + n λ=2 B λ x λ = 0, where x 1 = t s tme and the x λ, λ = 2,..., n, are the space varables, and B 1,..., B n are functons of the feld varables and ther dervatves, whch relaton s satsfed when the feld equatons are satsfed. If the condtons for the vanshng of the quanttes beng consdered at the boundary of a doman of the space varables, x 2,..., x n, are satsfed, then, by Stokes s theorem, the ntegral of B 1 over ths doman s constant n tme. In physcs a conservaton law s also called a contnuty equaton.
The second theorem. What s a G ρ? Noether assumes the exstence of ρ symmetres of the Lagrangan, each of whch depends lnearly upon an arbtrary functon p (λ) (λ = 1, 2,..., ρ) of the varables x 1, x 2,..., x n, and ts dervatves up to order σ. In Noether s notaton, each symmetry s wrtten a (λ) (x, u,...)p (λ) (x) + b (λ) (x, u,...) p(λ) x + + c (λ) (x, u,...) σ p (λ) x σ. In modern terms, such a symmetry s defned by a vector-valued lnear dfferental operator of order σ actng on the arbtrary functon p (λ).
Towards the proof of the second theorem Noether ntroduces the adjont operator of each of these dfferental operators. But she does not propose a name or a notaton for them. She wrtes Now, by the followng dentty whch s analogous to the formula for ntegraton by parts, ϕ(x, u,...) τ p(x) x τ = ( 1) τ τ ϕ x τ p(x) mod dvergences, the dervatves of the p are replaced by p tself and by dvergences that are lnear n p and ts dervatves. In modern terms, call the operators D (λ), = 1, 2,..., µ, and denote ther adjonts by (D (λ) ). The above dentty mples ψ D (λ) (p (λ) ) = (D (λ) ) (ψ )p (λ) modulo dvergences Dv Γ (λ).
Expressng the nvarance of the Lagrangan Now the precedng equaton ψ δu = Dv B =1 (B = A f. x), s wrtten µ ψ D (λ) (p (λ) ) = Dv B (λ) (λ = 1, 2,..., ρ). These relatons mply µ (D (λ) ) (ψ ) p (λ) = Dv(B (λ) Γ (λ) ), where Γ (λ) = =1 µ =1 µ =1 Γ (λ). Snce the p (λ) are arbtrary, (D (λ) ) (ψ ) = 0, for λ = 1, 2,..., ρ.
Dfferental denttes These are the ρ dfferental relatons among the components ψ of the Euler-Lagrange dervatve of the Lagrangan f that are dentcally satsfed. Noether wrtes these ρ denttes as { (a (λ) ψ ) x (b(λ) ψ ) + + ( 1) σ σ (λ = 1, 2,..., ρ). x σ (c(λ) } ψ ) = 0 These are the denttes that were sought among the Lagrangan expressons and ther dervatves when I s nvarant under G ρ.
Improper conservaton laws Noether observes that her denttes may be wrtten µ =1 a (λ) ψ = Dv χ (λ), where each χ (λ) s defned by a lnear dfferental operator actng upon the Lagrangan expressons ψ. She then deduces that each B (λ) can be consdered as the sum of two terms, B (λ) = C (λ) + D (λ), where the quantty C (λ) and not only ts dvergence vanshes on ψ = 0, the dvergence of D (λ) vanshes dentcally,.e., whether ψ = 0 or not. Noether calls these conservaton laws mproper.
What s the formal calculus of varatons? In the formal calculus of varatons, the am s to determne necessary condtons for a map x u(x) to realze a mnmum of the varatonal ntegral, and the second varaton s not consdered, the boundary condtons are assumed to be such that the ntegrals of functons that dffer by a dvergence are equal. In modern terms, the formal calculus of varatons s an algebrac formulaton of the calculus of varatons where functonals defned by ntegrals are replaced by equvalence classes of functons modulo a total dfferental. See Gelfand-Dckey [1976], Gelfand-Dorfman [1979], Mann [1978].
A queston n the general theory of relatvty Noether s research was prompted by a queston n the general theory of relatvty (Ensten, 1915) concernng the law of energy conservaton n general relatvty. She showed that each nvarance transformaton mples a conservaton law. Invarance under a group of transformatons dependng upon a fnte or denumerable number of parameters mples proper conservaton laws. Invarance under transformatons dependng upon arbtrary functons (a contnuous set of parameters) yelds mproper conservaton laws. Later, physcsts workng n general relatvty called the mproper conservaton laws of the second type strong laws.
Important remarks Noether carefully proves the converse of the frst and the second theorem. Noether s aware of the problem of defnng equvalent nfntesmal nvarance transformatons and equvalent conservaton laws n order to make the correspondence 1-to-1. Symmetres up to dvergence were ntroduced by Erch Bessel-Hagen n 1921, Über de Erhaltungssätze der Elektrodynamk, Mathematsche Annalen, 84 (1921), pp. 258 276. He wrtes that he wll formulate Noether s theorems slghtly more generally than they were formulated n the artcle he ctes, but that he s n debt for that to an oral communcaton by Mss Emmy Noether herself.
Emmy Noether versus Wllam Hamlton Noether s artcle deals wth the Lagrangan formalsm. There s NO Hamltonan formalsm n Noether s work. In the Hamltonan formalsm, t follows mmedately from the skew-symmetry of the Posson bracket that, f X H s a Hamltonan vector feld, then for any Hamltonan vector feld X K that commutes wth X H, the quantty K s conserved under the flow of X H. The name Noether theorem appled to ths result s a msnomer.
Some hstorcal facts: Ensten, Klen, Hlbert and the general theory of relatvty Noether s two theorems n pure mathematcs can hardly be understood outsde ther hstorcal context, the ncepton of the general theory of relatvty n the perod of great ntellectual effervescence n Germany and especally n Göttngen that concded wth the war and the frst years of the Wemar republc. Ensten had publshed hs artcle The Feld equatons of gravtaton n 1915, where he frst wrote the Ensten equatons of general relatvty. She wrote qute explctly n her artcle that questons arsng from the general theory of relatvty were the nspraton for her research, and that her artcle clarfes what should be the nature of the law of conservaton of energy n that new theory.
Noether s frst attempts In ther artcles of 1917 and 1918 on the fundamental prncples of physcs, Felx Klen and Davd Hlbert, who were attemptng to understand Ensten s work, sad clearly that they had solcted Noether s assstance to resolve these questons and that she proved a result whch had been asserted by Hlbert wthout proof. 1915, notes wrtten for Hlbert February 1918, a postcard to Klen The fundamental dentty can be read on the verso of her postcard, and she announced the result of her second theorem, but only for a very specal case: the varaton of the u n the drecton of the coordnate lne x κ.
A letter to Klen March 1918, letter to Felx Klen The Noether Theorems: from Noether to S evera
Noether s letter to Klen, March 1918 In ths letter she formulated the fundamental dea that the lack of a theorem concernng energy n general relatvty s due to the fact that the nvarance groups that are consdered are n fact subgroups of an nfnte group dependng upon arbtrary functons, and therefore lead to denttes that are satsfed by the Lagrangan expressons: By my addtonal research, I have now establshed that the [conservaton] law for energy s not vald n the case of nvarance under any extended group generated by the transformaton nduced by the z s. The end of her letter s a prelmnary formulaton of the concluson of her artcle.
Noether s concluson: the dscusson of Hlbert s asserton Hlbert asserted (wthout proof) n early 1918 that, n the case of general relatvty and n that case only, there are no proper conservaton laws. Here Noether shows that the stuaton s better understood n the more general settng of group theory. She explans (p. 255) the apparent paradox that arses from the consderaton of the fnte-dmensonal subgroups of groups that depend upon arbtrary functons. Gven I nvarant under the group of translatons, then the energy relatons are mproper f and only f I s nvarant under an nfnte group whch contans the group of translatons as a subgroup.
Noether s concluson Hlbert asserts that the lack of a proper law of [conservaton of] energy consttutes a characterstc of the general theory of relatvty. For that asserton to be lterally vald, t s necessary to understand the term general relatvty n a wder sense than s usual, and to extend t to the afore-mentoned groups that depend upon n arbtrary functons. In her fnal footnote, Noether remarks the relevance of Klen s observaton [1910] n the sprt of hs Erlangen program [1872]. In Noether s strkng formulaton, Klen s remark becomes: The term relatvty that s used n physcs should be replaced by nvarance wth respect to a group.
Before The Dawnng of Gauge Theory Noether extrapolates from the problems arsng from (1) the nvarance group of the equatons of mechancs and (2) the nvarance group of the general theory of relatvty, to a general theory of nvarance groups of varatonal problems. She made the essental dstncton between the case (1) of nvarance groups that are fnte-dmensonal Le groups and that (2) of groups of transformatons that depend upon arbtrary functons. Ths latter case would become, n the work of Hermann Weyl and, much later, Chen Nng Yang and Robert L. Mlls, gauge theory. The queston of the geometrc nature of the Noether theorems could not even be formulated n 1918. It remaned open - untl the 1970 s.