mmr The quantity a2x(a; Ah) is a monotonic increasing function of a and attains its 4irp

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1 VOL. 46, 1960 PHYSICS: J. SCHWINGER 257 The quntity 2X(; Ah) is monotonic incresing function of ttins its minimum vlue t = 0. Therefore, if we let A(z) = limit 2 (40) - 0 2X(; A) the minimum field strength, Hmin, required to stbilize the Wdverse flow, Q = l[1-(1- p)fl, yu < 1, (41) is given by mmr = A(y) [2U12 (1 - ) Red]. (42) 4irp 10. A further fct my be noted. If Q = constnt, the chrcteristic vlues of ( = p + mg) cn be determined redily in terms of the corresponding chrcteristic vlues,5 o, when H = 0. Thus 0f = [ 0 4 V/(o2 + 4Q2A2)]. (43) 2 * The reserch reported in this pper hs in prt been supported by the Geophysics Reserch Directorte of the Air Force Cmbridge Reserch Center, Air Reserch Development Comm, under Contrct AF 19(604)-2046 with the University of Chicgo. I Chrsekhr, S., "The Stbility of Viscous Flow Between Rotting Cylinders in the Presence of Mgnetic Field," Proc. Roy. Soc. (London), A (1953); lso, Niblett, E. R., "The Stbility of Couette Flow in n Axil Mgnetic Field," Cndin J. Phys., 36, (1958). These ppers del with the cse of two cylinders rotted in the sme direction; the cse when they rotte in opposite directions hs recently been investigted will be published in due course. 2 Velikhov, E. P., "Stbility of n Idelly Conducting Liquid Flowing Between Cylinders Rotting in Mgnetic Field," J. Exptl. Theoret. Phys. (U.S.S.R.), 36, (1959). 3 For the significnce of this discriminnt see Chrsekhr, S., "The Hydrodynmic Stbility of Inviscid Flow Between Coxil Cylinders," these PROCEEDINGS, 46, 137 (1960). 4 I m grteful to Dr. H. W. Reid for communicting to me his unpublished results. I These modes hve been discussed by Lord Kelvin, "Vibrtions of Columnr Vortex," Mthemticl Physicl Ppers, IV Hydrodynmics Generl Dynmics (Cmbridge: The University Press, 1910) pp ; for more recent ccount, see Bjerknes, V., J. Bjerknes, H. Solberg, T. Bergeron, Physiklische Hydrodynmik (Berlin: Springer, 1933), chp. 11. THE GEOMETRY OF QUANTUM STATES By JULIAN SCHWINGER HARVARD UNIVERSITY Communicted December 9, 1959 An erlier note' contins the initil stges in n evolution of the mthemticl structure of quntum mechnics s the symbolic expression of the lws of microscopic mesurement. The development is continued here. The entire discussion remins restricted to the relm of quntum sttics which, in its lck of explicit reference to time, is concerned either with idelized systems such tht ll properties re unchnged in time or with mesurements performed t common time.

2 258 PHYSICS: J. SCHWINGER PROC. N. A. S. The uncontrollble disturbnce ttendnt upon mesurement implies tht the ct of mesurement is indivisible. Tht is to sy, ny ttempt to trce the history of system during mesurement process usully chnges the nture of the mesurement tht is being performed. Hence to conceive of given selective mesurement M(, b') s compound mesurement is without physicl impliction. It is only of significnce tht the first stge selects systems in the stte b', tht the lst one produces them in the stte ; the interposed sttes re without mening for the mesurement s whole. Indeed, we cn even invent nonphysicl stte to serve s the intermediry. We shll cll this mentl construct the null stte 0, write M(, b') = M(, 0)2M1(0, b') (1) The mesurement process tht selects system in the stte b' produces it in the null stte, M(0, b') = 4D(b% con be described s the nnihiltion of system in the stte b'; the production of system in the stte following its selection from the null stte M(, 0) = (), cn be chrcterized s the cretion of system in the stte. Thus the content of (1) is the indiscernibility of M(, b') from the compound process of the nnihiltion of system in the stte b' followed by the cretion of system in the stte, M(, b') = 'F()iD(b'). (2) The extension of the mesurement lgebr to include the null stte supplies the properties of the T D symbols. Thus T()t = -1(), -(b')t= T(b') T()T(b') = 4()>(b') = 0, M(, b')i(c') = 'I()M(b', c') = 0, (3) wheres M(, b') T(c') = <b' Ic' > TI(), c1()m(b', c') = < b' > 4(c'), (4) () (b') = < Ib' > 11(0). The fundmentl rbitrriness of mesurement symbols expressed by the substitution M (, b') --- i' () M (, b')e',(b') (5) implies the ccompnying substitution '() -* eip() 'J'(), 4(b') -* ei<(b') 4(b'), (6) in which we hve effectively removed p(0) by expressing ll other phses reltive to it.

3 VOL. 46, 1960 PHYSICS: J. SCHWINGER 259 The chrcteristics of the mesurement opertors M(, 6') cn now be derived from those of the I (D symbols. Thus M(, b')t = 4D(b')t()t = T(b')4.() = M(b', ), tr M(b', ) = tr c()ti(b') = < Ib' >, while M(, b')m(c', d') -M(, b') *I(c')4.(d') = <b' Ic' > TI() 1 (d') = <b' Ic'> M(, d'). In ddition, the substitution (6) trnsforms the mesurement opertors in ccordnce with (5). The vrious equivlent sttements contined in (3) show tht the only significnt products-those not identiclly zero-re of the form Ad(4, bm, XNI, (DX, in ddition to X Y. where the ltin symbols re opertors, elements of the physicl mesurement lgebr. According to the mesurement opertor construction (2), ll opertors re liner combintions of products AdP, X = Z,Ii() < IXfb' >4(b') b' the evlution of the products XI, tx, X Y reduces to the ones contined in (4), *()((b')=(c')=t() < VC' b' >, 4() T(b') ci(c') = < Ib' > '1(c'). Hence, in ny mnipultion of opertors leding to product CM', the ltter is effectively equl to number, P()F(b') = < b' >, in prticulr 4()'() = 5(, ). (7) It should lso be observed tht, in ny ppliction of 1 s n opertor we hve, in effect 1 = EM() = 2T()D(). Accordingly, which shows tht The brcket symbols X = 'i b ()D()X'(b')4(b'), 4'() XN1(b') = < IX b' >. < I= c(), Ib' > = TI(b')

4 260 PHYSICS: J. SCHWINGER PROC. N. A. S. re designed to mke this result n utomtic consequence of the nottion (Dirc). In the brcket nottion vrious theorems, such s the lw of mtrix multipliction, or the generl formul for chnge of mtrix representtion, pper s simple pplictions of the expression for the unit opertor 1 = S > < t. / We hve ssocited ' 4D symbol with ech of the N physicl sttes of description. Now the symbols of one description re linerly relted to those of nother description, (b') = ET()-c()I(b') = ZT() < Ib' >, (8) c1() = i< jb' > 4(b'), (9) which lso implies the liner reltion between mesurement opertors of vrious types. Arbitrry numericl multiples of T or 4D symbols thus form the elements of two mutully djoint lgebrs of dimensionlity N, which re vector lgebrs since there is no significnt multipliction of elements within ech lgebr. We re thereby presented with n N-dimensionl geometry-the geometry of sttesfrom which the mesurement lgebr cn be derived, with its properties chrcterized in geometricl lnguge. This geometry is metricl since the number bm' defines sclr product. According to (7), the vectors -t() I() of the - description provide n orthonorml vector bsis or coordinte system, thus the vector trnsformtion equtions (8) (9) describe chnge in coordinte system. The product of n opertor with vector expresses mpping upon nother vector in the sme spce, XT(b') = 'T()4)()XT(b') = ZT() < IX lb' >, 4D()X = < X lb'> ci(b'). The effect on the vectors of the -coordinte system of the opertor symbolizing property A, A = is givuen by A*() = T(), ci()a = D(), which chrcterizes I() 4'() s the right left eigenvectors, respectively, of the complete set of commuting opertors A, with the eigenvlues. Associted with ech vector lgebr there is dul lgebr in which ll numbers re replced by their complex conjugtes. The eigenvectors of given description provide bsis for the representtion of n rbitrry vector by N numbers. The bstrct properties of vectors re relized by these sets of numbers, which re kownn s wve functions. We write E l>< = > *(). 'T

5 VOL. 46, 1960 PHYSICS: J. SCHWINGER 261 =p () < q5() =D > If t ' re in djoint reltion, cj connected by = 4t, the corresponding wve functions re +() = ()* The sclr product of two vectors is, in prticulr, D1 2 Z > < 142 = EZ() IP2() t*= >()*#(t). 0 t which chrcterizes the geometry of sttes s unitry geometry. I1cI)2 is represented by the mtrix < 1w*42 Jb' > = (b wve functions tht represent XT (DX re < jxt = E < IX Ib' > #p(b') DX lb' > = E () < IX b' >. The opertor On plcing X = 1, we obtin the reltion between the wve functions of given vector in two different representtions, = << b' > IP(b') 001) E O(/) < lb' >. From the viewpoint of the extended mesurement lgebr, 4b,6 wve functions re mtrices with but single row, or column, respectively. It is convenient fiction to ssert tht every Hermitin opertor symbolizes physicl quntity, tht every unit vector symbolizes stte. Then the expecttion vlue of property X in the stte ' is given by X>~ = *tx = Z jp()*< X l" > qt(). # In prticulr, the probbility of observing the vlues in n A-mesurement performed on systems in the stte 'E is p(, I) = <M() >w = t l> < = #j() 12. The geometry of sttes provides the elements of the mesurement lgebr with the geometricl interprettion of opertors on vector spce. But opertors con-

6 262 PHYSICS: J. SCHWINGER PROC. N. A. S. sidered in themselves lso form vector spce, for the totlity of opertors is closed under ddition under multipliction by numbers. The dimensionlity of this opertor spce is N2 ccording to the number of linerly independent mesurement symbols of ny given type. A unitry sclr product is defined in the opertor spce by the number <X Y> = tr(xty) _ <YtIXt> which hs the properties <x Y>* = <Y X <x Ix> > o. The trce evlution tr M(b', )MI(", b") = 6(, ")b(b', b") chrcterizes the M(W', b') bsis s orthonorml. <M(, b') M(", b") > = S(u'b', b"), the generl liner reltion between mesurement symbols, M(c', d') = E< c'><d' b'> M(, b'), b' cn now be viewed s the trnsformtion connecting two orthonorml bses. chnge of bsis is described by the trnsformtion function <b' c'd'> = <M(b') M(c', d') = < c'> <d' b'>, which is such tht <b' c'd'> = <d'c b'>. This <b' c'd'>* = <c'd' lb'> = <b' Id'c'>. One cn lso verify the composition property of trnsformtion functions, A, <olb' c'd'> <c'd' c'd' e'f> = <b' e'f'>. The probbility relting two sttes ppers s prticulr type of opertor spce trnsformtion function, p(, b') = < b'> <b' > = < Ib'b'>. Let X (), = 1.. N2, be the elements of n rbitrry orthonorml bsis, <X() JX()> -(, = ). The connection with the M(, b') bsis is described by the trnsformtion function <b' > = tr M(b', )X c) = <IX() Ib'>.

7 VOL. 46, 1960 PHYSICS: J. SCHWINGER 263 We lso hve <lb'> = <b' I>* = <b' IX()t l> the trnsformtion function property E <b' I> < lbb> = 6(b', Wb") cquires the mtrix form i, < IX() lb'> <b' IX()t Jw > = 5(, `)5b', by). If we multiply the ltter by the b-mtrix of n rbitrry opertor Y, the summtion with respect to b' b' yields the -mtrix representtion of the opertor eqution E X()YX()t= 1 tr Y, the vlidity of which for rbitrry Y is equivlent to the completeness of the opertor bsis X(). Since the opertor set X()t lso forms n ortbonorml bsis we must hve the prticulr choice Y = 1/N gives 1 EZX()tYX() = 1 tr Y, OX()X()t = - EX ()It X () = 1. The expression of n rbitrry opertor reltive to the orthonorml bsis X (), X = E X()x(), defines the components x() = <X() IX> <IX> t For the bsis M(, b'), the components re x(b') = tr M(b', )X = </ X lb>, the elements of the b-mtrix representtion of X. The sclr product in opertor spce is evluted s <X = IY> Ex()*y() <x Ix> = E Ix() 12 > O. On ltering the bsis the components of given opertor chnge in ccordnce with X() = E < 1 lx(fm)i For mesurement symbol bses this becomes the lw of mtrix trnsformtion.

8 264 PHYSICS: J. SCHWINGER PROC. N. A. S. There re two spects of the opertor spce tht hve no counterprt in the stte spces-the djoint opertion the multipliction of elements re defined in the sme spce. Thus X() = E (3)X(3)t (#3) (h) = tr X() X (), where X()X (A) = E (f3y)x ()t -y - X(,y) <y l# > (i y) = (do ) = (-yf) - tr X ()X (,)X (y) <yl/> = tr X(y)tX ()X( 3). Some consequences re < IX>* = Z (4) < A lxt> <-IXY> = E <y f> <IX> <3IY>, which generlize the djoint multipliction properties of mtrices. The elements of the opertor spce pper in the dul role of opertor oper on defining mtrices by < jx l"> = < XX(")> - Z < I"> < IX>. The mesurement symbol bses re distinguished in this context by the complete reducibility of such mtrices, in the sense of <b' X I"b" > = < IX l"> 6(b', Otherwise expressed, the set of N mesurement symbols M(, b'), for fixed b', or fixed, re left right idels, respectively, of the opertor spce. The possibility of introducing Hermitin orthonorml opertor bses is illustrted by the set For ny such bsis $ ": 2-l/2[M(, ") + M(", )] b"). 2 i [M(, ") -M(, )] M() <,'> = (C) = 6(, ) < IX>* = < lxt>,

9 VOL. 46, 1960 PHYSICS: J. SCHWINGER 265 which implies tht Hermitin opertor X hs rel components reltive to Hermitin bsis, therefore <XiX> = X()2 >0. Thus the subspce of Hermitin opertors is governed by Eucliden geometry, chnge of bsis is rel orthogonl trnsformtion, X() = E (,3)X(fl). When the unit opertor (multiplied by N- 1/2) is chosen s member of such bses it defines n invrint subspce, the freedom of orthogonl trnsformtion refers to the N2-1 bsis opertors of zero trce. Importnt exmples of orthonorml opertor bses re obtined through the study of unitry opertors. 1 Schwinger, J., these PROCEEDINGS, 45, 1542 (1959).