Um modelo para a superfície líquida no estudo da dinâmica do espalhamento de Xe e Ne pelo esqualano

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  Um modelo para a superfície líquida no estudo da dinâmica do espalhamento de Xe e Ne pelo esqualano
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  MA THEMA TICS: J. M. COOK or mk   1, is aneven entire function of order 1 and type p <ir, which can be shown to be O(z-') as z - through real values; it is real on the real axis, and changes sign at all integers not of the form mk or mk   1. By a theorem of Paley and Wiener,6 F(z) = fo cos zu h(u) du, where h(u) belongs to L2. Write g(u) = h(u) cos '/2u; then F(n   1/2) =  P g(u) cos (n   '/2)U sec1/2u du - (_ 1)n Jtp g(7r - u) sin (n   1/2)u csc '/2u du. Thus (- 1)'F(n   1/2) is the nth partial sum of the Fourier seriesof the evenfunction which is -irg(7r - u) for7r - p < u <7r and zero for 0 _ u <r - p.Furthermore, (- 1) IF(n   1/2) > 0 when n is not a multiple of k, and so for a sequence of integers of density arbitrarily close to 1,if k is large enough. I  On saitfort peu de choses sur I'approximation oriente6 dans les espaces fonctionnels r'ticules : Favard, J.,  Remarques sur I'approximation des fonctionscontinues, Acta Sci. Math., Szeged, 12, part A,101-104 (1950). 2 Fejer, L.,  Gestaltliches uber die Partialsummen und ihrer Mittelwerte bei derFourierreihe und der Potenzreihe, Z. angew. Math. u. Mech., 13, 80-88 (1933). 3 Levinson, N., Gap and DensityTheorems, New York, 1940, chap. II. Levinson, N., op. cit., chap. VII. 5 Duffin, R. J., and Schaeffer, A. C.,  Power Series with Bounded Coefficients, Am. J. Math., 67, 141-154 (1945). 6 Paley, R. E. A. C., and Wiener, N., Fourier Transforms in the ComplexDomain, New York, 1934, p.13. THE MATHEMATICS OF SECOND QUANTIZATION* BY J. M. COOK DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO Communicated by M. H. Stone, May 16, 1951 In order to make a beginning on the problem of constructinga mathe- matically rigorous foundation for quantum field theory, we define the annihilation and creation operators and the particle- and field-observables as transformations on Hilbert space, and investigate their domains, adjoints, commutation relations, normality and other properties. The resulting formalism is given a physical interpretation which is illustrated by applications. The state of an elementary particle is represented by a point in the Hilbert space T and an observable by theoperator A on 9S. Then the VOL. 37, 1951 417  MA THEMA TICS: J. M. COOK stateof a system of n such particles is represented by apoint in the tensor product  9 = 9 ... ® and theobservable by (ZA8('   ... ®A6(i n)) -, where T- is the closure' of T. In field theory, j   Ej e(n) n =0 co n is the state space2 and Q(A) = ED (A'( ',) . (.A. A1i n)) the cor- n = 0 i = 1 responding observable. Q(A) exists when A is densely defined, closed and linear. It is defined to be 0 on T('), the one-dimensional space of no- particle states. Q preserves commutation, order and adjoint relations, and normality. If His self-adjoint, thenexp (iQl(H))Q2(A) exp (-iQ(H)) = Q(exp (iH)A exp (-iH)).Under certain conditions (e.g., when A and B are bounded), Q(aA   bB) = (aQ(A)   bQ(B))- and Q [A B]) = [Q(A), Q(B)]f (where [A, B] = AB   BA). If P is a projection, the eigenvalues of Q(P) are the occupation numbers m of P9. In the cor- responding eigenstates, exactly m of the particles have theproperty P. With the spectral theorem, this gives us the standardenergy expressions hk2kckNk except that there is no infinite null-point energy I2kwk/2 to be subtracted.3 A similar result holds for the null-point momentum, no artificial summation to zero being necessary. To every permutation 7r in the symmetric group HIn corresponds the unitary operator UT n 9?(n) defined by UTl ...--- n = (1) )... 0 T(n)- These operators U,T generate a ring 9,, isomorphic to the groupalgebra of H.. For Gn e 9. and e 9E, we construct the densely defined, closed,linear transformations co(+) = (E EIGn)(4)®), w*( ) = ((O )*(E DGn*)n n O n== 0 on a, where (+X®) maps (n) into 9?(n + 1) by (4)0)4)o1 ... 04)n = 4)41 04 ).. The mapping w obeys the rules wc(+) = w*(*)*, ((4)) = o(aO   b )   (aw(4))   b ()) ), co*(a)   bo) = (a*w *(4)   b*o*( )), exp (-iQ(H))w(O) exp (iQ(H)) = w(exp (-iH)O), exp (-iQ(H))w*(4) exp (iQ (H))   w *(exp (-iH) )), [Q[A], co(4)]-   co(A 4), and [Q2(A), Co*(4))] = - W*(A*4)). The center of ,, is spanned by a set of orthogonal projections P =   nT(7r) U/fn indexed by the characters r of H,n. The alternating and symmetric characters an and Sn give us the subspaces W =   Pan) a n co and 5 = (E P,) of antisymmetric and symmetric wave-functions. n = 0 Setting G,, equal to ViPan or -IP,,, we getthe creation operators '0a(4)) or wc(4)) and annihilation operators Wa*(4)) or w o*(), on or e (to which they must be restricted), for the Fermi-Dirac or Bose-Einstein cases, respectively. Both Wa(4) and Oa*(4)) are bounded: jJW,a(9b)|| = |Wa*(4b))H = ||qb||. The domains of co(4) and ws*(4) are the same as that 418 PROC. N.A. S.  MA THEMA TICS: J. Al. COOK4 of VQ(P[,1) (on i), where P[,] is the projection of 51 on the subspace [4)] spanned by 4. If 01, 2 e) ?, we define 412* to be the transformation on 9) such that ( 102*)O = (4), 2) 1. Then Wa(4))a*( ) = Q(4)*), a*(#)w(4) = (4), /   Q(4)*), and (w(4))w*(#)) = 2(4#*), (ws*( )wS(4)) = (4), )I   Q(00*)i from which it follows, if [A, B]+ = AB   BA, that [Wia()) Wa*(4')]+ = [ws()), ws*(O)] - =  4,O  )I, and [Wa(4)), Wa( )]+ = [wh(), Ws(4')]P = [Wa*(4)), Wa*( )]+ = [cs*(4)), Ws*( )]   0. If {4)} is an orthonormal basis of 9?, then   wCa(0i) } is irreducible on 2l and { W°(s is irreducible on ei. By purely formalmanipulations it can be seen that Q(A) corresponds to the expression2  2m, nC(4m)(An), Om)W*()n) if o is Wa or ws. The operators i(ws(4)) - ws*(4)))/V2 and (co5(4)   cos*(4)))/X/0 are essentiallyself-adjoint,' so their closures p(q5) and q(4) exist and are self- adjoint. The commutation relations [q(4)), q(4,)]- = [p(o), p( )] = ((4,, 4) - (4), 4))I/2 and [q(o), p(4 )]   = i(( ,4)   (4),  ))I/2 reduce to the standard ones when 4 and 6 are elements of an orthonormal basis. Time-dependent commutation relations enable us to avoidthe singular Dirac 5-function and the Jordan-Pauli invariantD-function.Field quantities have physical meaning only as averages over a region. Point- dependence introduces divergencies into the mathematics, so p and q here depend onelements of Hilbertspace (asdistributions with respect to which theaverages are taken) rather than points of Euclidean 3-space E3. The formalism is illustrated by a derivation of the Yukawa-potential, and by the following completely rigorizable, relativistically invariant, divergence-free (as far as it goes) derivation of Maxwell's equations: A photon is represented by 6 in 5) = 5?2(E3) ®l4, where S4 is a 4-dimensional Hilbert space. If k, = -ih(bl/x), etc., and k = V/k 2   ky2   k *2 then the Hamiltonian is   = ck @e. The Lorentz group acts on time- dependent elements of 5) by having exp (- (itc/h)k)4), for 4 e 62(E3), transform like a scalar, and the orthonormal basis P1, P2, P3, P4 of S4 trans- form contragrediently to x, y, , ct. The four-vector wave-functions come inpairs,k and  , as co- and contravariant components for the same par- ticle. Expectation values are written (A , ,). We restrict photons to besuch that their covariant wave-functions must be in the subspace P of all; = 41ip1   420P2   4)3Op3   4)40P4(4i E V2(E3)) for which (b/1x)4i   (6/by)42   (6/6z)43 - c-1(6/bt)q64 = 0. This eliminates thephysical influence of longitudinal and scalar componentsfrom expectation values, and leaves only two effective polarization states. They are perpendicular to the direction of motion of the photon and have the desired spin proper- ties. Now let P(4,) = i(w5(#)-Xw*(#))7V and Q( ) = (wo( )   w*(^))  /V2. Then if 4 e 22(E3) is areal-valuedfunction on E3 for taking 419 OL. 37, 1951  MATHEMATICS: FUGLEDE AND KADISON PROC.N. A. S. field-value averages, thecovariant four-potential operators are Ai(o) = tQ(V/H-1'Spi), i = 1, 2, 3, and t(o) = kQ(VHiY®p4). Averages over mutually space-likeregions commute. The total energy of the field is Q2(H)   c(Al(si)   A2(s2)   Aa(s3)   4(s4)), where s1, S2,S3, s4 is any real, square-integrable, contravariant four-current density. Expectation values of (V ..A -. ci(b/8t)c)(4) are always zero (on photons), and (02A ) (4) - -c-'(si, 4))I, i = 1, 2, 3, (Q24)(4)   c-1(S4, 0)I, so Maxwell's equations are satisfied. A photon is polarized parallel to its electric vector and per- pendicular to its magnetic vector-both perpendicular to its momentum. Its energy satisfies Planck's relation E = hp, where v is the frequency of the induced field. *This note summarizes a longer paper submitted for publication elsewhere. It was written, with the continuing advice of Prof. I. E. Segal, for presentation to the Depart- ment of Mathematics of the University of Chicago in partialfulfillmentof requirements for the Ph.D. Most of the work was done while under contract with the Office of Naval Research. 1 Stone, M. H., Linear Transformations in Hilbert Space, Am. Math. Soc. Coll. Publ., XV, New York, 1932. 2 Fock, V., Zeits. f. Phys., 75, 622-647 (1932). 3Wentzel, G., Einfuhrung in die Quantentheorie der Wellenfelder, FranzDeuticke, Vienna, 1943. ON A CONJECTURE OF MURRAY AND VON NEUMANN By BENT FUGLEDE AND RICHARD V. KADISON* THE INSTITUTE FOR ADVANCED STUDY Communicated by John von Neumann, April 7, 1951 1. Introduction.-In this note the authorspresent a proof of a conjec- ture of F. J. Murray and J. v. Neumann' concerning normalcy offactors. A ring of operators2 CR is said to be normal if each subring 3 of CR coincides with the set of operators in CR each of which commutes withevery operator in W'M, where W' is the ring of operators in CR each of which commutes with every operator in S. In symbols, normalcy requires that ('aY)a'R = 8 for each subring 3 of CR. The center of a normal ring (R consistsof the operators a I, a complex (put 3 = {a I}); i.e., MR is a factor. J. v. Neu- mann- proved8 thatthe factor (B of all bounded operators is normal. The question of which factors are normal was raised by F. J. Murray and J. v. Neumann (R.O. I, p. 185). They showed that all factors in case I (the discrete case) are normaland exhibited examples of non-normal factorsin case II (the continuous case). Their later results establish the non- normalcy of each member of a restricted class of factors in case II, viz., 420
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