Pure infinitesimal geometry.

Transcription

1 Rene Infntesmalgeomete, Math. Z. 2 (1918), Pue nfntesmal geomety. By Hemann Weyl n Züch 1. Intoducton. On the elatonshp between geomety and physcs. The eal wold n whch ou conscousness s foced to esde s not thee, all at a sngle moment, but happens; t elapses, beng destoyed and bon anew n each moment, a contnuous one-dmensonal sequence of states n tme. The aena of ths tmele happenstance s a thee-dmensonal Eucldan space. Its popetes ae examned by geomety; on the othe hand, t s the poblem of physcs that eal thngs exst n space to be egaded conceptually and to be founded on lastng laws, despte the ephemeal natue of phenomena. Physcs s thus a scence that has geomety at ts foundatons; howeve, the concepts by whch t epesents ealty matte, electcty, foce, enegy, electomagnetc feld, gavtatonal feld, etc. belong to a completely dffeent sphee fom geomety. Ths old nsght egadng the elatonshp between fom and content n ealty, between geomety and physcs, has been ovetuned by Enstenan elatvty theoy 1 ). The specal theoy of elatvty leads to the nowledge that space and tme ae melded nto an ndssoluble unfed entty that we wll call the wold; as a consequence of ths theoy, the wold s a fou-dmensonal Eucldan manfold Eucldan, wth the modfcaton that the quadatc fom that s the bass fo the wold-metc s not postvedefnte, but has an ndex of neta equal to 1. The geneal theoy of elatvty says entely n the spt of moden local acton physcs that ths s vald only nfntesmally, and taes the wold metc to then be the geneal concept that was pesented by Remann n hs Habltaton lectue n whch he clamed that such a measue was based on a quadatc dffeental fom. Hs pncpal nnovaton was the followng nsght: The metc s not a popety of the wold n tself; athe, spacetme as a phenomenon taes the fom of a completely fomless fou-dmensonal contnuum, n the sense of Analyss Stus, but the metc expesses somethng eal that exsts n the wold, that physcal actons ae exeted on matte though centfugal and gavtatonal foces, and convesely, the state of the metc s natually detemned by the dstbuton and popetes of matte. Snce I wanted to lbeate Remannan geomety, whch we wll egad as local geomety, fom one of ts cuently unesolved nconsstences, a last global geometc element emeges that s suggested by ts vey Eucldan past tself, I aved at a wold metc fom whch not only gavtatonal, but also electomagnetc effects, emege, whch, as one may wth good eason assume, thus account fo all 1 ) I efe to the pesentaton n my boo Raum, Zet, Matee, Spnge 1918 (denoted by RZM, n the sequel), and the lteatue that was cted n t.

2 H. Weyl. Pue nfntesmal geomety 2 physcal phenomena 2 ). In ths theoy, all eal events that occu n the wold ae manfestatons of the wold metc; the physcal concepts ae nothng but geometc ones. The sngle dffeence between geomety and physcs conssts of the fact that geomety geneally begns wth a set of axoms that the metc concept essentally embodes 3 ), but physcs must ave at these laws and pusue the consequences n ode to dstngush the eal wold among all possble fou-dmensonal metc spaces 4 ). In ths note, I would le to develop that pue nfntesmal geomety of whch I am convnced the physcal wold s undestood to be a specal case. The constucton of local geomety s popely pefomed n thee steps. At the fst step, one fnds a contnuum, n the sense of Analyss Stus, that s baen of all measuements physcally speang, ths s the vacuum. At the second step one fnds the affnely connected contnuum whch s what I call a manfold n whch the concept of the nfntesmal paallel dsplacement of vectos has meanng; n physcs, the affne connecton appeas n the fom of the gavtatonal feld. Fnally, at the thd step, one fnds the metc contnuum physcally, ths s the ethe, whose states ae manfested by the phenomena of matte and electcty. 2. Topologcal space (vacuum). As a esult of the dffculty nvolved wth gaspng the ntutve chaacte of contnuous connectons though a puely logcal constucton, a completely satsfactoy analyss of the concept of an n-dmensonal manfold s not possble at pesent 5 ). The followng shall suffce: An n-dmensonal manfold may be descbed by n coodnates x 1, x 2,, x n, each of whch tae on a defnte numecal value at each pont of the manfold; dffeent ponts coespond to dffeent systems of values fo the coodnates. If x 1, x 2,, xn s a second system of coodnates then thee ae specfed elatons: x = f ( x 1, x 2,, x n ), ( = 1, 2,, n) between the x and x coodnates of the same abtay pont, n whch the f ae puely logco-athmetcally constucted functons; about them, we assume only that they ae contnuous and that they possess contnuous devatves: f α = x 2 ) A fst communcaton on ths matte appeaed wth the ttle of Gavtaton and Eletztät, n Stzungsbe. d. K. Peuß. Aad. d. Wssenschaften 1918, pp ) Tadtonal geomety mmedately goes feely fom ths patcula poblem to a lesse one, n pncple, by no longe mang the space tself the object of one s nvestgaton, but the specal classes of possble stuctues n space that ae suggested by the space metc. 4 ) I am suffcently audacous as to beleve that the totalty of all physcal phenomena may be deved fom a sngle unvesal law of Natue of the utmost mathematcal smplcty. 5 ) On ths, cf., H. Weyl, Das Kontnuum (Lepzg 1918), n patcula, pp. 77 et seq.

3 H. Weyl. Pue nfntesmal geomety 3 whose detemnant does not vansh. The last condton s necessay and suffcent fo affne geomety to be vald n the nfntesmal lmt and fo the coodnate dffeentals n both systems to be elated by nvetble lnea elatons: (1) dx = αdx. We assume the exstence and contnuty of hghe devatves wheneve t becomes necessay n the couse of ou nvestgaton. Thus, n each case the concept of contnuous and contnuously dffeentable functons of poston, as well as 2, 3, tmes contnuous dffeentablty, has an nvaant sense that s ndependent of the coodnates; the coodnates themselves ae such functons. We shall call an n-dmensonal manfold, about whch we shall consde no othe popetes than the ones that ae ntnsc to n- dmensonal manfolds to use physcal temnology an (n-dmensonal) vacuum. The elatve coodnates dx of one of the nfntesmally close ponts P = (x + dx ) to a pont P = (x ) ae the components of a lne element at P, o an nfntesmal dsplacement PP of P. When we tansfom to anothe coodnate system these components satsfy fomulas (1), n whch α means the values of the appopate devatves at the pont P. In geneal, any n gven numbes ξ ( = 1, 2,, n) n a defnte sequence at a pont P when one establshes a patcula coodnate system fo the neghbohood of P chaacteze a vecto (o a dsplacement) at P; the components ξ ( ξ, esp.) of the same vecto n any two coodnate systems - the unpmed and the pmed systems ae connected by the same lnea tansfomaton fomulas (1): ξ = αξ. One can add vectos at P and multply them wth numbes; they theefoe defne a lnea o affne collecton. Thee ae n unt vectos e at P that ae assocated wth any coodnate system, namely, the ones that possess the components: e e 1 2 n 1, 0, 0,, 0 0, 1, 0,, 0 e 0, 0, 0,, 1 n the chosen coodnate system. Any two (lnealy ndependent) lne elements at P wth the components dx (δx, esp.) span a (two-dmensonal) suface element at P wth the components: dx δx dx δx = x ; any thee (ndependent) lne elements dx, δx, dx at P span a (thee-dmensonal) volume element wth the components:

4 H. Weyl. Pue nfntesmal geomety 4 dx dx dx l δ x δ x δ x l dx dx dx l = x ; etc. A lnea fom at P that depends upon an abtay lne, (suface, volume,, esp.) element at P s called a lnea tenso of an 1 (2, 3,,esp.). By the use of a chosen coodnate system the coeffcents a of these lnea foms: a dx ( 1 a x, 1 al x l,, esp.) 2! 3! l can be nomalzed unquely by the altenaton equement; t says that n the lastdescbed case vz., the ndex tple (l) when the tple s subjected to an even pemutaton one obtans the same coeffcent a l, wheeas the sgn of the coeffcent changes unde an odd pemutaton. Hence: a l = a l = a l = a l = a l = a l. The coeffcents, thus nomalzed, wll be efeed to as the components of the tenso n queston. By dffeentaton, to a scala feld f thee coesponds a lnea tenso feld of an 1 wth the components: f f = x ; to a lnea tenso feld of an 1 f thee coesponds a second an tenso feld: f f f = x x ; to a lnea a second an tenso feld thee coesponds a tenso feld of an 3: f f f f l = + + x x x l l l ; etc. These opeatons ae ndependent of the coodnate system used 6 ). A lnea tenso of an 1 at P may be efeed to as a foce that acts on t. By choosng a patcula coodnate system, such a tenso wll theefoe be chaactezed by n numbes ξ that tansfom contagedently to the dsplacement components unde the tanston to anothe coodnate system: ξ = α ξ. 6 ) RZM, 13.

5 H. Weyl. Pue nfntesmal geomety 5 If η ae the components of an abtay dsplacement at P then ξ η s an nvaant. We wll geneally undestand a tenso at P to mean a lnea fom of one o moe abtay dsplacements and foces at P. Fo example, f one s dealng wth a lnea fom composed of thee abtay dsplacements ξ, η, ζ and two abtay foces ρ, σ : pq l al ξ η ζ ρ pσ q, then we spea of a tenso of an 5 that s covaant n the ndces l and contavaant n the ndces pq of the components of a. A dsplacement s tself a contavaant tenso of an 1 and a foce s a covaant tenso of an 1. The fundamental opeatons of tenso algeba ae 7 ): 1. Addton of tensos and multplcaton by a numbe; 2. Multplcaton of tensos; 3. Contacton. Tenso algeba may thus be establshed n the vacuum t assumes no measuements but, by contast, only the lnea tensos of tenso analyss can be defned. A moton n ou manfold s gven when each value s of a eal paamete s assocated wth a pont n a contnuous manne; by the use of a coodnate system x the moton s expessed by fomulas x = x (s), n whch the x on the ght ae undestood to symbolze functons. If we assume contnuous dffeentablty then we obtan, ndependent of the coodnate system, a vecto at each pont P = P(s) of the moton that has the components: u = dx ds, namely, the velocty. Two motons that tansfom between themselves by means of a contnuous monotone tansfomaton of the paamete s descbe the same cuve. 3. Affnely connected manfolds (wold wth a gavtatonal feld). I. Concept of an affne connecton. If P s nfntely close to the fxed pont P then P s affnely connected wth P when one establshes how each vecto at P goes to a vecto at P by a paallel dsplacement of P to P. It s self-explanatoy that the paallel dsplacement of all the vectos at P to P must theefoe satsfy the followng equement: A. The tansplantaton of all vectos at P to the nfntely close pont P by paallel dsplacement yelds a map of the vectos at P to the vectos at P. 7 ) RZM, 6.

6 H. Weyl. Pue nfntesmal geomety 6 If we use a coodnate system n whch P has the coodnates x, P has the coodnates x + dx, an abtay vecto at P has the components ξ, and the vecto at P that t goes to unde paallel dsplacement has components ξ + dξ then dξ must theefoe depend upon the ξ lnealy: dξ = dγ ξ. the dγ ae nfntesmal quanttes that depend only upon the pont P and the dsplacement PP whose components ae dx, but not on the vecto that ξ that s been subjected to the paallel dsplacement. Fom now on, we consde only affnely connected manfolds; n such a manfold each pont P s affnely connected wth all of ts nfntesmally neghbong ponts. We must place yet anothe equement upon the concept of paallel dsplacement, that of commutatvty: B. If P 1, P 2 ae two ponts that ae nfntely close to P and f the nfntesmal vecto PP 1 goes to P2 P 21 unde a paallel dsplacement of P to P2, but PP 2 goes to PP 1 12 unde a paallel dsplacement to P 1 then P 12 and P 21 must concde. (They defne an nfntely small paallelogam.) If we denote the components of PP 1 by dx and those of PP 2 by δx then ths equement obvously says that: (2) dδx = dγ δx s a symmetc functon of both lne elements d and δ. As a esult, dγ must be a lnea fom n the dffeentals dx : dγ = Γ s dx s, s n whch the coeffcents Γ, whch depend only upon P and ae called the components of the affne connecton, must satsfy the symmety condton: Γ s = Γ s. Due to the manne by whch we fomulated equement B n tems of nfntesmal quanttes, t can be agued that t lacs a pecse meanng. Fo that eason, we would le to establsh explctly by a goous poof that the symmety condton of (2) s ndependent of any coodnate system. To ths end, we consde a (twce contnuously dffeentable) scala feld f. Fom the fomula fo the total dffeental: f df = dx, x

7 H. Weyl. Pue nfntesmal geomety 7 we extact the fact that when ξ ae the components of an abtay vecto at P then: df = f x ξ s an nvaant ths s ndependent of any coodnate system. We defne a change n t by means of a second nfntesmal dsplacement δ, unde whch the vecto ξ wll be dsplaced paallel to t fom P to P 2, and we obtan: δdf = f x x f 2 ξ δ x δγ ξ x. If we agan eplace ξ n ths equaton wth dx and swtch d and δ then ths yelds the nvaant: f = (δd dδ)f = ( dγ δ x δγ dx ) x. The elatons: (δγ δx δγ dx ) = 0 yeld the necessay and suffcent condton fo ths, that any scala feld must satsfy the equaton f = 0. In physcal temnology, an affnely connected contnuum efes to a unvese that s uled by a gavtatonal feld. The quanttes Γ s ae the components of the gavtatonal feld. We wll not need to gve the fomulas by whch these components tansfom unde a tanston to anothe coodnate system hee. Unde lnea tansfomatons, the Γ s behave le the components of a tenso that s covaant n and s and contavaant n, but they lose ths chaacte unde nonlnea tansfomatons. Howeve, the vaatons δγ s that the quanttes Γ expeence when one vaes the affne connecton of the manfold abtaly ae actually the components of a geneally nvaant tenso of the assumed chaacte. What we ae to undestand by the paallel dsplacement of a foce at P to an nfntely close pont P s a esult of the equement that the nvaant poduct of ths foce and an abtay vecto at P emans nvaant unde paallel dsplacement. components of the foce and η those of the dsplacement then fom 8 ): and we deduce the fomula: d(ξ η ) = (dξ η ) + ξ dη = (dξ dγ ξ ) = 0 If ξ ae the 8 ) In the sequel, we employ the Ensten conventon that one always sums ove ndces that appea twce n a tem of a fomula, because wthout t, t would be deemed necessay fo us to place a summaton sgn n font of each one.

8 H. Weyl. Pue nfntesmal geomety 8 dξ = dγ ξ. One can ntoduce a coodnate system x whch I call geodetc at P at each locaton P n such a way that the components Γ s of the affne connecton vansh at the locaton P n such a coodnate system. Next, f x ae abtay coodnates that vansh at P and Γ s mean the components of the affne connecton at the locaton P n ths coodnate system then one obtans a geodetc x by the tansfomaton: 1 (3) x = x 2 Γ sx xs. Namely, f we consde the x to be the ndependent vaables and the dffeentals to be constants then we have, n the Cauchy sense, at the locaton P ( x = 0): s dx hence: dx = dx, d 2 x = Γ sdxdxs ; d 2 x + Γ s dx dx s = 0. Due to ts nvaant natue, the latte equaton eads le: d x + Γ dx dx = 0. 2 s s Howeve, fo abtay constant dx t s satsfed only when all of the Γ s vansh. The gavtatonal feld can theefoe always be made to vansh at a sngle pont fo a cetan choce of coodnate system. By the equement of geodesy at P, coodnates n the neghbohood of P ae detemned up to thd ode when one s gven any lnea tansfomaton;.e., f x, x ae two geodetc coodnate systems at P and the x, as well as the x, vansh at P then by neglectng tems of thd and hghe ode n x one has lnea tansfomaton fomulas x = α x wth constant coeffcents α. II. Tenso analyss. Staght lne. Tenso analyss may fst be completely establshed n an affnely connected space. Fo example, f f ae the components of a tenso feld of an 2 that s covaant n and contavaant n then we mae use of an abtay dsplacement ξ and a foce η, constuct the nvaant: f ξ η and ts vaaton unde an nfntely small shft d of the agument pont P, unde whch ξ and η wll be paallel dsplaced along wth P. One has:

9 H. Weyl. Pue nfntesmal geomety 9 hence: d( f ξ η ) = f x l f l = ξ η dx l f x l Γ f η dγ ξ + f + Γ f l l f ξ dγ η, ae the components of a tenso feld of an 3 that s covaant n l and contavaant n that ases fom the gven second an tenso feld n a manne that s ndependent of the coodnate system. In an affnely connected space the concept of a staght o geodetc lne taes on a pecse meanng. A lne comes about when one consstently dsplaces a vecto paallel to tself n ts own decton and follows the moton of the ntal pont of ths vecto; t can thus be chaactezed as the only cuve that leaves ts decton unchanged. If u ae the components of such a vecto then n the couse of ts moton the equatons: du + Γ αβ u α dx β = 0, dx 1 : dx 2 : : dx n = u 1 : u 2 : : u n ae consstently vald. If we then epesent the cuve n tems of the paamete s then we can nomalze t n such a way that one has: dx ds = u dentcally n s, and the dffeental equatons of the staght lne then ead: w 2 d x dx dxα β = + Γ 2 αβ = 0. ds ds ds Fo any abtay moton x = x (s) the left-hand sdes of these equatons ae the components of a vecto that s nvaantly lned wth the moton at the pont s, the acceleaton. In fact, when ξ s an abtay foce at some pont that s paallel dsplaced by a tanston to the pont s + ds, one has: d( u ξ ) ds = w ξ. A moton whose acceleaton vanshes dentcally s called a tanslaton. A staght lne as one can also undestand fom ou explanaton above s to be undestood as the path of moton of a tanslaton.

10 H. Weyl. Pue nfntesmal geomety 10 III. Cuvatue. If P and Q ae two ponts that ae connected by a cuve wth a gven ntal vecto then one can dsplace t fom P to Q paallel to tself along the cuve. The vecto tanslaton thus obtaned s geneally non-ntegable;.e., the vecto that one obtans at Q s dependent upon the path of dsplacement along whch the tanston too place. Only n the specal case whee ntegablty exsts s thee any sense to speang of the same vecto at two dffeent ponts P and Q; one would then undestand such vectos to the ones that go to each othe unde paallel dsplacement. In that case, one calls the manfold Eucldan. One may ntoduce specal lnea coodnate systems n such a manfold that ae dstngushed by the fact that n such systems equal vectos at dstnct ponts have equal components. Any two such lnea coodnate systems ae connected by lnea tansfomaton fomulas. The components of the gavtatonal feld vansh dentcally n a lnea coodnate system. In the nfntesmal paallelogam that was constucted above ( 3, I., B.) we too an abtay vecto wth the components ξ at a pont P, dsplaced t paallel to tself fst to P 1 and then to P 12, and anothe tme fom P to P 2 and then to P 21. Snce P 12 concdes wth P 21 we can tae the dffeence of these two vectos at that pont and thus obtan a vecto whose components ae obvously: Fom: t follows that: δdξ = ξ = δdξ dδξ. dξ = dγ ξ = Γ l dx l ξ m Γ l dxlδ xmξ Γ lδ dxl ξ + dγ δγ ξ, x and due to the symmety of δdx l : ξ Γ m Γ l = dxlδ xm + ( dγ δγ dγ δγ ) ξ. xl xm We thus obtan: ξ = R ξ, n whch R s the lnea fom of the two shfts d and δ, whch ae ndependent of the dsplaced vecto ξ, o futhemoe, of the suface element that they span, whch has the components: x lm = dx l δx m dx m δx l, namely: (4) R = R lm dx l δx m = 1 2 R lm x lm, (R ml = R lm ), Γ xl Γ x (5) R m l lm = + ( ΓlΓm ΓmΓl ) m.

11 H. Weyl. Pue nfntesmal geomety 11 If the η ae components of an abtay foce at P then η x s an nvaant. It follows that R lm ae the components of a tenso of an 4 at P that s covaant n lm and contavaant n, namely, the cuvatue. The vanshng of the cuvatue dentcally s the necessay and suffcent condton fo the manfold to be Eucldan. Along wth the sew symmety descbed n (4) that the components of the cuvatue satsfy, they also satsfy the cyclc dentty: R lm + R lm + R ml = 0. In essence, the cuvatue at a pont P s a lnea map, o tansfomaton, P that assocates each vecto ξ wth a vecto ξ ; ths tansfomaton tself depends lnealy upon the suface element at P: P = P dx δx = 1 2 P x (P = P ). The cuvatue s theefoe best undestood as a lnea tansfomaton tenso of an 2. In ode to goously pove the nvaance of the cuvatue tenso beyond objectons of the sot that mght pehaps be ased fo the nfntesmal changes descbed above, one employs a foce feld f, defnes the change d(f ξ ) of the nvaant poduct f ξ n such a manne that unde the nfntesmal shft d the vecto ξ wll be dsplaced paallel to tself. If one eplaces the nfntesmal shft dx wth an abtay vecto ρ at P n the esultng expesson then one obtans an nvaant blnea fom n two abtay vectos ξ and ρ at P. Wth t, one defnes the change that ases fom a second nfntesmal shft δ and taes the vectos ξ, ρ along wth t n a paallel fashon, and then eplaces the second shft wth a vecto σ at P. One fnds the fom: δd(f ξ ) = δdf ξ + df δξ + δf dξ + f δdξ. Due to the symmety of δdf, swtchng d and δ and then subtactng yelds the nvaant: and one thus acheves the desed poof. (f ξ ) = f ξ, 4. Metc manfold (the ethe). I. Concept of a metc manfold. A manfold caes a measue at the pont P when the lengths of lne elements at P can be compaed; we thus assume the valdty of the Pythagoean-Eucldan law n the nfntesmal doman. Theefoe, a numbe ξ η shall coespond to any two vectos ξ, η at P, namely, the scala poduct, whch s a symmetc blnea fom n ts dependence upon both of them; ths blnea fom s clealy not absolutely detemned, but only up to an abtay popotonalty facto ths dffeent fom 0. Theefoe, t s not actually the fom ξ η, but the equaton ξ η = 0 that s gven; two vectos that satsfy t wll be called

12 H. Weyl. Pue nfntesmal geomety 12 pependcula. We assume that ths equaton s non-degeneate;.e., that the only vecto at P that s pependcula to all of the vectos at P s the vecto 0. Howeve, we do not assume that the assocated quadatc fom ξ ξ s postve-defnte. If t has an ndex of neta q and one has n q = p then we say befly that the manfold s (p + q)- dmensonal at the pont n queston; due to the abtaness of the popotonalty facto, the two numbes p, q ae defned only up to the odeng. We now assume that ou manfold caes a measue at evey pont. In ode to facltate the goal of the analytcal epesentaton, we magne: 1. a choce of coodnate system has been made and 2. a choce of abtay popotonalty facto n the scala poduct has been made at each locaton; one thus aves at a efeence system 9 ) fo the analytcal epesentaton. If the vecto ξ at the pont P wth the coodnates x has the components ξ, and η has the components η then one wll have: (ξ η) = g ξ η (g = g ), n whch the coeffcents g ae functons of the x. The g shall not only be contnuous, but twce contnuously dffeentable. Snce they ae contnuous and the detemnant g, by assumpton, s nowhee vanshng the quadatc fom (ξ ξ) has the same ndex of neta q at evey locaton; we can theefoe egad the manfold as (p + q)-dmensonal n all of ts aspects. If we eep the same coodnate system, but mae a dffeent choce of the undetemned popotonalty facto then, nstead of g, we ave at the new quanttes: g = λ g fo the coeffcents of the scala poduct, whee λ s a nowhee-vanshng contnuous (and twce contnuously dffeentable) functon of poston. As a esult of the foegong assumptons, the manfold s only endowed wth an angle measue; the geomety that ths alone wll suppot s called confomal geomety. As s well nown, n the ealm of two-dmensonal manfolds ( Remann sphees ), due to ts mpotance n the theoy of complex functons, t has attaned a fa-eachng level of development. If we mae no futhe assumptons, then the ndvdual ponts of the manfold eman completely solated fom each othe n the metc context. A metc connecton fom pont to pont wll then be fst ntoduced n t when one poposes a pncple fo compang the length unt at a pont P wth the ones at nfntely close ponts. Instead of ths, Remann made the vey fa-eachng assumpton that the unt lengths of lne elements could be compaed wth each othe, not only at the same locaton, but also at any two fntely dstant locatons. The possblty of such a global geometcal compason can, howeve, not exst at all n a puely nfntesmal geomety. The Remannan assumpton s also caed ove nto the Enstenan wold geomety of gavtaton. Hee, ths nconsstency shall be emoved. Let P be a fxed pont and let P * be an nfntely close pont that one aves at by means of the dsplacement whose components ae dx. We choose a patcula coodnate 9 ) I thus dstngush between coodnate system and efeence system.

13 H. Weyl. Pue nfntesmal geomety 13 system. In tems of the length unt that was establshed at P (as well as the emanng ponts of the space), the squae of the length of an abtay vecto at P wll be: g ξ ξ. Howeve, the squae of the length of an abtay vecto ξ * at P * wll be, when we tansfom the unt length that was chosen at P to P*, as we assumed would be possble, gven by: (1 + dϕ ) ( g ) + dg ξ* ξ*, whee 1 + dϕ means a popotonalty facto that devates fom 1 by an nfntely small quantty; dϕ must be a homogeneous functon of the dffeentals dx of ode 1. Namely, f we tansplant the chosen length unt at the pont P to a pont along a cuve that goes fom P to the fntely dstant pont Q then, upon establshng the unt length at Q, we obtan the expesson g ξ ξ fo the squae of the length of an abtay vecto at Q, multpled by a popotonalty facto, that one obtans fom the poduct of nfntely many factos of the fom 1 + dϕ, whch wll tae the fom: Π (1 + dϕ ) Π e dϕ = e Σdϕ = e unde the tanston fom one pont of the cuve to the next. In ode fo the ntegal that appeas n the exponent to be meanngful, dϕ must be a functon of the dffeentals of the sot descbed above. If one eplace g wth g = λ g then, nstead of dϕ, anothe quantty dϕ wll appea. If λ s the value of ths facto at the pont P then one must have: whch yelds: Q d P (1 + dϕ )( g + a g ) = λ (1 + dϕ )(g + dg ), (6) dϕ = dϕ dλ λ. Of the next possble assumptons about dϕ, that t s a lnea dffeental fom, the squae oot of a quadatc fom, the cube oot of a cubc fom, etc., as we now see fom (6), only the fst one s meanngful. We have aved at the followng esult: The metc of a manfold s based on a quadatc dffeental fom and a lnea dffeental fom: (7) ds 2 = g dx dx and dϕ = ϕ dx. Convesely, f the metc s not, howeve, absolutely establshed by these foms, but by 2 any pa of foms ds, dϕ that ognates n (7) by way of the equatons: ϕ

14 H. Weyl. Pue nfntesmal geomety 14 (8) 2 ds = λ ds 2, dϕ = dϕ dλ λ, then these ae equvalent to the fome pa n the sense that both of them expess the same metc. In ths, λ s an abtay, nowhee-vanshng, contnuous (moe pecsely: twce contnuously dffeentable) functon of poston. In all quanttes o elatons that analytcally epesent metc phenomena, the functons g, ϕ must theefoe be ntoduced n such a manne that one has nvaance: 1. unde abtay coodnate tansfomatons ( coodnate nvaance ), and 2. unde the eplacement of (7) wth (8) dλ ( scale nvaance ). = d lnλ s a total dffeental. Thus, wheeas an abtay λ popotonalty facto fo the quadatc fom ds 2 emans at evey locaton, thee exsts an ndetemnacy n dϕ of an addtve total dffeental. We gve a physcal expesson to a metc manfold by egadng t as a wold full of ethe. The patcula metc that esdes n the manfold epesents a patcula state of the ethe-flled wold. Ths state s theefoe to be descbed, elatve to a efeence system, by beng gven the (athmetc constucton of the) functons g, ϕ. Fom (6), t follows that the lnea tenso of an 2 wth the components: ϕ ϕ F = x x s unquely detemned by the metc on the manfold; I call t the metc otaton. It s, I beleve, the same thng as what one calls the electomagnetc feld n physcs. It satsfes the fst system of Maxwell equatons: F F F + + x x x l l l l = 0. Its vanshng s the necessay and suffcent condton fo the change n length unt to be ntegable, and theefoe any assumpton that Remann based metc geomety upon to be vald. We thus undestand, as Ensten dd n hs wold geomety by dectng hs mathematcal hndsght to Remann, that only the gavtatonal phenomena, but not the electomagnetc ones, could be accounted fo. II. Affne connecton on a metc manfold. In a metc space, n place of the equement A fo the concept of nfntesmal paallel dsplacement that was posed n 3, I., one has the fa-eachng equement: A*: that the paallel dsplacement of all of the vectos at a pont P to an nfntely close pont P must be not only an affne, but also a conguent tansplantaton of ths collecton of vectos.

15 H. Weyl. Pue nfntesmal geomety 15 By the use of the pevous notatons, ths equement yelds the equaton: (9) (1 + dϕ )(g + dg )(ξ + dξ ) (ξ + dξ ) = g ξ ξ. Fo any quanttes a that cay an uppe ndex (), we defne the loweng of ths ndex by the equaton: a = g a (and the nvese pocess of asng of an ndex by the nvese equatons). Fo (9), we can wte, n tems of these symbols: (g ξ ξ ) dϕ + ξ ξ dg + 2 ξ dξ = 0. The last tem s: = 2 ξ ξ dγ = 2 ξ ξ dγ = 2 ξ ξ (dγ + dγ ) ; one must theefoe have: (10) dγ + dγ = dg + g dϕ. Ths equaton may be solved fo cetan only when dϕ s a lnea dffeental fom; an assumpton that we aleady nssted above was the only easonable one. Fom (10), o: g (10*) Γ, + Γ, = x + g ϕ, t follows, by tang nto account the symmety popety Γ, = Γ, that: g (11) Γ, = 1 g g x x x (g 2 ϕ + g ϕ g ϕ ) (Γ, = g s Γ ). Ths shows that n a metc space the concept of nfntesmal paallel dsplacement of a vecto s unquely establshed by the gven equement 10 ). I consde t to be the fundamental fact of nfntesmal geomety that when not only a metc, but also an affne connecton, s gven on a manfold the pncple of unt length dsplacement, wth nothng else, leads to the dsplacement of dectons, o physcally speang, the state of the ethe detemnes the gavtatonal feld. When the quadatc fom g dx dx s ndefnte, among the geodetc lnes one can dstngush the null lnes, along whch ths fom vanshes. Snce ths depends only upon 10 ) On ths, cf., Hessenbeg, Vectoelle Begündung de Dffeentalgeomete, Math. Ann Bd. 78 (1917), pp , especally pp. 208.

16 H. Weyl. Pue nfntesmal geomety 16 the behavo of the g, but not at all on the ϕ, these ae theefoe confomal geometc stuctues 11 ). We have placed cetan axomatc equements on the concept of paallel dsplacement and showed that n a metc manfold they can be satsfed n one and only one way. It s, howeve, also possble to defne these concepts explctly n a smple way. If P s a pont of ou metc manfold then we would le call a efeence system geodetc at the pont P when the ϕ vansh and the g assume statonay values n such a system: g ϕ = 0, = 0. x D. Thee s a geodetc efeence system at each pont P. If ξ s a gven vecto at P and P s, howeve, an nfntely close pont, then we undestand the paallel dsplacement of ξ to the coespondng vecto at P to mean that vecto at P that possesses the same components as ξ n the geodetc efeence system assocated wth P. Ths defnton s ndependent of the choce of geodetc coodnate system. It s not dffcult to pove the clam that was utteed n ths statement ndependently of the lne of easonng that was followed hee by dect computaton, and n the same way, to show that the pocess of paallel dsplacement so defned wll be descbed n an abtay coodnate system by the equatons: (12) dξ = Γ ξ dx, wth the coeffcents Γ beng taen fom (11) 12 ). Hee, howeve, whee the nvaant meanng of equaton (12) s aleady cetan, we can conclude ths n a smple way. Fom (11), the Γ vansh n a geodetc efeence system, and equatons (12) educe to dξ = 0. The concept of paallel dsplacement that we deduced fom an axomatc equement theefoe agees wth the one that was defned n D. It only emans fo us to pove the exstence of a geodetc efeence system. To ths end, we choose a geodetc coodnate system x at P that has the pont P tself fo ts ogn (x = 0). If the unt length at P and n ts neghbohood s chosen abtaly and ϕ then means the values of these quanttes at P then one needs only to cay out the tanston fom fom (7) to (8) wth: λ = e n ode to ave at the fact that, along wth the Γ s, also the ϕ, vansh at P. Fom ths see (10*) the geodetc natue of the efeence system thus obtaned follows. The coodnates of a geodetc efeence system at P ae defned up to tems of thd ode n the mmedate neghbohood of P when one s feely gven a lnea tansfomaton, but the unt length s gven up to tems of second ode as long as the addton of a constant facto s gven feely. ϕ x 11 ) Wth ths ema, I would le to coect an ovesght on page 183 of my boo Raum, Zet, Matee. 12 ) One can thus follow the path that I too n RZM 14.

17 H. Weyl. Pue nfntesmal geomety 17 III. Computatonally convenent extenson of the concept of a tenso. The quanttes that we ntoduced n 2 as tensos ae dmensonless; the components depend completely upon the choce of coodnate system, but not on the choce of unt length. In metc geomety, an extenson of ths concept poves to be pefeable: by a tenso of weght e, we shall undestand a lnea fom of one o moe dsplacements and foces at a pont that ae ndependent of the coodnate system, but depend on the unt length n such a way that the fom taes on the facto λ e unde the eplacement of (7) wth (8). The g themselves ae the components of a covaant tenso of an 2 and weght 1. Incdentally, we egad ths extended concept of a tenso only as an ad that we ntoduce meely fo the sae of computatonal convenence; we ascbe an objectve meanng only to the tensos of weght 0. Theefoe, n the sequel wheneve we spea of tensos wth no addtonal menton of the weght, the concept s always to be undestood n ts ognal sense. Any computatonal convenence esdes n the followng fact: If we pefom the pocess of asng one o moe ndces n the components a of a covaant tenso of weght e then we obtan the mxed components of a tenso of weght e 1 n the case of a o a, and a contavaant tenso of weght e 2 n the case of a. We cannot decde, as would usually be the case, how to dentfy the esultng tensos wth the ognal ones snce, along wth dependng upon those tensos, they also depend upon the metc the state of the wold ethe and we wll not consde ths to be gven a po n the slghtest, but leave open the possblty of subjectng t to abtay vtual vaatons. IV. Cuvatue n metc spaces. If ξ, η ae two abtay dsplacements at the pont P, but f ae the components of a foce feld, then t follows that: f η = f η ; (f η ) = f η = ( f η ) = f η ; hence: (13) ξ η = ξ η. On the othe hand, when the vectos ae, as always, paallel dsplaced by vtual dsplacements one has: d(ξ η ) + (ξ η ) dϕ = 0, δd(ξ η ) + δ(ξ η ) dϕ + (ξ η ) δdϕ = 0. The mddle tem n the latte equaton s: and the fst one s: = (ξ η ) δϕ dϕ, = η δdξ + δη dξ + dη δ ξ + ξ δdη. If one exchanges d and δ and subtacts then ths yelds:

18 H. Weyl. Pue nfntesmal geomety 18 (η ξ + ξ η ) + (ξ η ) ϕ = 0, o, on account of (13): (η ξ + ξ η ) + (ξ η ) ϕ = 0. Thus, f we set: (14) ξ = ξ 1 ξ 2 ϕ then we have decomposed ξ nto components that ae pependcula to ξ and components that ae paallel to ξ. One has: and we wte: One then has: ξ = R ξ, ϕ = 1 2 F x, (15) R 1 lm = R δ F, lm 2 lm R = 1 2 R lm x. lm δ = 1 ( = ) 0 ( ). If we lowe the ndex then the quanttes ae sew-symmetc, not only n l and m, but also and. In the decomposton (15), we efe to the fst summand as the decton cuvatue and the second one as the length cuvatue. Length cuvatue = metc otaton. By the natue of the coespondng decomposton (14) of ξ, a theoem follows that justfes ou temnology: The tenso R of decton cuvatue vanshes when and only when the paallel dsplacement of a vecto subjected to a change of decton s ntegable; the tenso F of length cuvatue vanshes when and only when the lewse alteed length s ntegable. Hee, we gve the explct expesson fo the decton cuvatue. We ntoduce, as usual, the Chstoffel thee-ndex symbols and the Remannan cuvatue components by the equatons: = 1 g g g +, 2 x x x = gs, s s G lm = x l m l l m m l +, xm and futhe set, fo an abtay quadatc system of numbes a : and defne: whch maes: (g l a m + g m a l g m a l g l a m ) = a lm ϕ l ϕ = Φ, x ϕ ϕ 1 g 2 (ϕ ϕ ) = ϕ,

19 H. Weyl. Pue nfntesmal geomety 19 1 R = G lm Φ + ɶ ϕ. lm lm 2 lm One obseves hee that the ndvdual tems on the ght-hand sde have no ntnsc sgnfcance: they clealy possess coodnate nvaance, but not scale nvaance. Fo the contacted tensos: R m = R m, G m = G m one has: whee: When we set: Φ = Φ = anothe contacton yelds: n 2 R = G (Φ ϕ ) 1 g 2 (Φ 1 ϕ ), 2 1 ( gϕ ), ϕ = g x R = R = R, ϕ = G = G, n 2 2 n 2 R = G (n 1) ( Φ + ϕϕ ) 4. (ϕ ϕ ). One can deve a tenso fom the dectonal cuvatue that depends only upon the g n the followng manne: 1 *R lm = (n 2) Rlm ( glrm + gmrl gmrl glrm ) + ( gl gm gmgl ) R. n 1 These numbes *R lm ae analogous to the *G lm that one defnes by means of G lm ; thus, f one ases the ndex agan then *G lm = *R lm ae the components of an nvaant tenso of confomal geomety. Ths tenso always vanshes fo n = 2 and n = 3, and fst plays a ole fo n 4. Its vanshng s a necessay (but not suffcent) condton fo the manfold to be mapped to a Eucldan one n a manne that peseves angles. 5. Scala and tenso denstes. I. In topologcal space. If W dx I befly wte dx fo the ntegaton element dx 1 dx 2 dx n s an ntegal nvaant then W s a quantty that depends upon the coodnate system n such a way that unde a tanston to anothe coodnate system t s multpled by the absolute value of the functonal detemnant. If we egad ths ntegal as a measue on an ntegaton doman that s flled wth quantum matte then W s ts densty. Fo that eason, a quantty of the sot descbed may be efeed to as a scala densty. Ths s an mpotant concept that stands on a pa wth that of scala and does not educe to t n

20 H. Weyl. Pue nfntesmal geomety 20 the slghtest 13 ). Analogously, we wll call a lnea fom n one o moe dsplacements and foces that depends upon the coodnate system n such a way that t gets multpled by the absolute value of the functonal detemnant unde a coodnate tanston a tenso densty. We ae justfed n thnng of tensos as ntenstes and tenso denstes as quanttes. The covaant and contavaant expessons wll be employed fo tensos. The geneal concept of tenso densty belongs to pue topology. Howeve, n ths type of geomety the bass fo the analyss of tenso denstes may be constucted only to an analogous degee compaed to the analyss of tensos. In 2, we called a tenso lnea when t s covaant and ts components satsfy the equement that they be altenatng. We shall call a tenso densty lnea when t s contavaant and possesses altenatng components. A lnea tenso densty of an 1 can be egaded as a cuent stength. If w s such a tenso densty then: (16) w = w x s a scala densty that s coupled wth t; f w s a lnea densty of an 2 then: (17) w x = w s a lnea tenso densty of an 1, etc. One poves (16) n a well-nown manne by showng that the left-hand sde epesents the souce stength that s assocated wth the f cuent stength. Fom ths, one obtans (17) wth the ad of a foce feld f = x that ases fom a potental f, and defnes the dvegence of w f : etc. ( w f) x = w f, x II. In affnely connected and n metc spaces. In an affnely connected manfold one can not only defne the dvegence of lnea tenso denstes, but also abtay ones. We shall consde a vecto feld ξ at a pont P to be statonay when the vectos ξ n the neghbong ponts P to P go ove to the vecto ξ at P unde paallel dsplacement,.e., when thee ae total dffeental equatons: 13 ) The compason between scalas and scala denstes coesponds completely to that of functons and Abelan ntegals n the theoy of algebac functons.

21 H. Weyl. Pue nfntesmal geomety 21 dξ + Γ s ξ ξ dx s = 0 o + Γ sξ = 0 x s at P. Obvously, thee s a vecto feld that s statonay at P that s assocated wth any abtay gven vecto ξ at the pont P. One can defne an analogous concept fo foce felds. If one now defnes, e.g., the dvegence of a mxed tenso densty w of an 2 then one maes use of a vecto feld ξ that s statonay at P and constucts the dvegence of the tenso densty ξ w : ( ξ w ) = x w = ξ w Γ w +. x ξ +ξ w x x Ths quantty s a scala densty, and theefoe: w x Γ s w s s a tenso densty of an 1 that ases fom w n a manne that s ndependent of the coodnate system. Howeve, one can not only constuct a tenso densty that has a an that s less by one fom such a tenso densty by tang ts dvegence, but also constuct anothe tenso densty that has a an that s hghe by one fom t by dffeentaton. Next, f s means a scala densty, whch we can egad as the densty of a substance that flls the manfold, and f dv = dx 1 dx 2 dx n s an nfntely small volume element then s dv s the quantum of the substance that flls ths element. We now subject dv to the nfntesmal dsplacement δ (wth the components δx ); by ths, we undestand a pocess by whch the ndvdual ponts of dv expeence nfntesmal dsplacements that tae them to new ponts by paallel dsplacement. The dffeence between the matte quanta that fll up dv and those that fll the dsplacement of dv to the suoundng neghbohood amounts to: One thus has that: (18) (δs s Γ dx ) dv = (δs s s x Γ s γ ) dv. ae the components of a covaant tenso densty of an 1 that ases fom the scala densty s n a manne that s ndependent of the coodnate system. Its vanshng at a locaton shows that the substance tself s unfomly dstbuted. Moeove, (18) can also be deved n a moe computatonal fashon as follows: One maes use of a vecto feld ξ that s statonay at P and taes the densty of the cuent stength sξ :

22 H. Weyl. Pue nfntesmal geomety 22 ( sξ ) x = s ξ + s x ξ = x s Γ x s ξ. In ode to facltate the tanston fom the dffeentaton of scala tenso denstes to abtay ones e.g., mxed tenso denstes w of an 2 one taes, n a now famla sot of way, a vecto ξ that s statonay at P and a statonay foce feld η, and dffeentates the scala densty w ξ η. Contactng the tenso densty that esults fom dffeentaton ove the dffeentaton ndex and a contavaant one gves one the dvegence. The analyss of tenso denstes s theefoe aleady accomplshed n affne geomety. What metc geomety now povdes s meely the followng method of geneatng tenso denstes: one multples an abtay tenso of weght 2 n by g, whee g s the detemnant of g. Example: The eal wold s a (3 + 1)-dmensonal manfold; g s, howeve, negatve, and we use the postve g n place of t. Fom the covaant metc otaton tenso F, whch has weght 0, we obtan the contavaant F of weght 2, and fom t, upon multplcaton by g, we obtan: g F = F. These ae theefoe the components of a cetan lnea tenso densty of an 2 that s nvaant of the state of the ethe; we wll efe to t as the metc otaton densty (electomagnetc feld densty). F (19) = s x s thus a cuent stength (lnea densty of an 1). In (19), we have the second system of Maxwell equatons befoe us, whch admttedly fst taes on a defnte meanng when the electcal cuent s s expessed n yet anothe way n tems of the state of the ethe. In any event, fom ou ntepetaton of the electomagnetc feld, t can, howeve, gve anythng le an electomagnetc feld densty and an electcal cuent only n a fou-dmensonal wold. The ntegal of S = 1 4 F F, whch can be taen ove any wold doman, appeas n physcs as the quantty of electomagnetc acton that s contaned n ths doman. Its meanng s based on the fact that the nfntely small change that t expeences unde an nfntesmal vaaton δg, δϕ of the state of the ethe that vanshes on the bounday of the doman s: = (s δϕ S δg ) dx (S = S ),

23 H. Weyl. Pue nfntesmal geomety 23 n whch s ae the components of the cuent stength that ae defned by (19), and the mxed tenso densty of an 2 wth the components: S = Sδ F F epesents the enegy-momentum tenso densty of the electomagnetc feld. The exstence of all of these quanttes s completely lned wth the dmenson numbe 4. In the fst place, the ntepetaton of physcal phenomena that s advocated hee gves us easonable gounds fo ecognzng that the wold s fou-dmensonal. ϕ = F dx δx s the tace of any tansfomaton: P = P dx δx that the cuvatue defnes. Fom the fom of S, we can defne the tansfomaton: 1 4 g P P (n whch multplcaton means concatenaton). The tace of M tself s a scala densty that s unfom nea S. III. The quantty of acton and ts vaaton. We now etun to pue mathematcs. If W s any scala densty that s unquely defned by the state of the ethe (ndependently of the coodnate system) then we shall (fom the example of Maxwellan theoy) efe to the ntegal nvaant W dx as the quantty of acton that s contaned n the doman of ntegaton. Unde an abtay vaaton of the state of the ethe of the type that was descbed, we set: (20) δ W dx = (w δϕ + W δg ) dx (W = W ). The w ae the components of a contavaant tenso densty of an 1 and the W ae those of a mxed tenso densty of an 2. Thee exst n + 1 denttes between these Lagangan devatves of the acton functon W that ase fom the nvaance of the quantty of acton. Fst, one must have nvaance when one eplaces g wth λg and, at the same tme, ϕ wth ϕ 1 λ ; n ths, we tae λ to be a quantty 1 + δλ that devates λ x fom 1 by an nfntely small amount then (20) must vansh fo:

24 H. Weyl. Pue nfntesmal geomety 24 Ths yelds the fst n + 1 denttes: δg = δg δλ, δϕ = ( δλ ). w 1 (21) + 2 W = 0. x Secondly, we employ the nvaance of the quantty of acton unde coodnate tansfomatons by an nfntesmal defomaton of the ethe 14 ). We dsplace the pont P = (x ) of the ethe to the pont P = ( x ). Howeve, the dsplacement PP must vansh on the bounday of the egon n queston n such a way that ths dsplaced egon s stll flled wth the same quantum of ethe. In a second coodnate system, we ascbe the coodnates x to pont P. If we dsplace the ethe wthout changng ts state then the metc at the pont wll be defned by: x g (x) dx dx and ϕ (x) dx afte dsplacement n these coodnates, o, when we tansfom bac to the old coodnates, by: g ( x) dxdx and ϕ ( x) dx ; hence, at the pont P by: g ( x) dxdx and ϕ ( x) dx. Fo the state of the ethe thus obtaned, the quantty of acton, due to ts nvaance, must possess the same values as t ognally dd. If ths defomaton s nfntesmal x = x + δx then ths yelds: ( δ x ) ( δ x ) g δg = g ( x) g (x) = g + g + δ x x x x ( δ x ) ϕ δϕ = ϕ( x) ϕ (x) = ϕ + δ x. x x (20) must vansh fo these vaatons. If one gnoes the devatves of the components δx of the shft by patal ntegaton then one obtans the equatons: W 1 gs s ( w ϕ) ϕ W + w = 0. x 2 x x x If we use (21) then we fnd that the second of the two tems n culy bacets s: 14 ) Weyl, Ann. d. Phys Bd 54 (1917), pp. 117 ( 2); F. Klen, Nach. d. K. Gesellsch. d. Wssensch. zu Göttngen, math.-phys. Kl. Stzung v. 25 Jan

25 H. Weyl. Pue nfntesmal geomety 25 Now, one has: 1 g 2 xl s g s ϕ W s + F w. + gsϕ W s = 1 (Γ 2, s + Γ s, ) W s, due to the symmety of W s : = Γ, s W s = Γ W s. s Thus, the equatons tae the fnal fom, n whch the nvaant chaacte s evdent: W s (22) Γ sw + F w = 0. x IV. Tanston to physcs. In a metc manfold whose ethe s found n a state of extemal acton, such that n any egon of the wold that s subjected to abtay nfntesmal vaatons of ϕ and g that vansh on the bounday one has: (23) δ W dx = 0, one has the Lagangan equatons: (24) w = 0, W = 0. In physcs, the fst equaton s efeed to as the law of electomagnetsm and the second one as the law of gavtaton. As n mechancs, physcs also states a Hamltonan pncple 15 ): The eal wold s such that ts ethe s found n a state of extemal acton. We now the laws of Natue, whch ae summazed by Hamlton s pncple (23), that goven t when we now how the acton densty W depends upon the state of the ethe. Equatons (24) ae ndependent of each othe, but fve (n = 4) denttes (21), (22) exst between them. In fact, the quanttes g, ϕ can be detemned by the law (24) only to the extent that one can feely tansfom fom a efeence system to any anothe abtay system; howeve, such a tanston depends upon fve abtay functons. The vanshng w of the dvegence, whch s defned by the left-hand sde of the electomagnetc x equaton, s theefoe a consequence of the law of gavtaton, and convesely, the vanshng of the dvegence: W s ΓsW x 15 ) On ths, cf., G. Me, Annalen de Phys, Bd. 37, 39, 40 (1912/13), o the epesentaton of Me s theoy n RZM 25; D. Hlbet, De Gundlagen de Phys (1. Mttelung), Nach. d. K. Gesellsch. d. Wssensch. zu Göttngen, Stzung of 20 Nov

26 H. Weyl. Pue nfntesmal geomety 26 s a esult of the law of electomagnetsm. These fve denttes ae closely connected wth the so-called consevaton laws, namely, the (one-component) law of the consevaton of electcty and the (fou-component) law of the enegy-momentum pncple. They teach us: the consevaton law (upon whose valdty mechancs ests) follows n two ways fom the electomagnetc equatons, as well as the gavtatonal equatons; one would thus le to efe to t as the smultaneous valdty of both goups of laws. The only Ansatz fo the acton densty n a (3 + 1)-dmensonal wold that one must easonably consde s the followng one: W = M + αs, n whch α s a numecal constant, and the meanng of M and S s to be taen fom pat II of ths secton. Dependng on the scope of that aena, one sees whch of them s allowed by ou theoy of the laws of the wold. In fact, as a fst appoxmaton, by estctng to the lnea tems, Hamlton s pncple gves the Maxwellan law of the electomagnetc feld and the Newtonan law of gavtaton. Thus, snce the quantty of acton s a pue numbe, thee ases the possblty of a quantum of acton, whose exstence s egaded by the contempoay physcs as the fundamental atomc stuctue of the cosmos. Hee, we shall not go any futhe nto the physcal mplcatons of the theoy, whch only teats the systematc development of pue nfntesmal geomety and ts assocated analyss of tensos and tenso denstes. Once moe, we emphasze the ponts at whch t depats fom the usual theoy. They ae: the step-wse constucton n the thee levels of topology, affne geomety, and metc geomety, the lbeaton of the latte fom one of the global-geometc nconsstences that has stuc to t snce ts Remannan concepton, and the extenson of the theoy of tensos (ntenstes) to ts opposte, the theoy of tenso denstes (o quanttes). (Receved on 8 June 1918.)