CPT 205 ORO-3992-161
The RiRBPil Hilbert Spacc* and Quantum Mechanics
by
A. Bohm
I------NOTICE------I Thii report prepared j s *>f vwrV ■ sponsoreJ b> the l imed Siaw% < • >scf!uv •Supported in part by the U. S. Atomic Energy Commission, Contract At(*»0-1) 3992. DISTRIBUTION PREFACE Thr purpose of this paper is not to pr<.-3t»nt now results, hut to make a little-known arc.i in the mathematical founda tion:; of quantum mecnanins accessable to a wider audience.* Therefore the discussion of the physics and mathemat tc:; will be limited to the bare essentials. The whole subio^t is developed in terms of the simple, well-known example of the harmonic oscillator, and the mathematical notions have been introduced in the least general form which is compatible with the purpose of this paper (Unfortunately, the mathematics had to be abbreviated to an extent which, in the second part, precludes the possibility of proving the mathematical statement;;, bu* -ftieh, hopefully, still allow: one to under stand the content of these mathematical nta tomon ts). The general mathematical background has been treated in revoral 1 2 monographs * written by the pioneers of this area, to which we refer the interested reader. Remarks (ending with the symbol o) have been inserted throughout the paper to mention generalizations and additional related material. One can argue that the subject described here is very useful for physics, because it makes the Dirac formalism rigorous and therewith gives a mathematical iustification for all the mathematically undefined operations which *It is based on a series of lectures given to mathemati cians and physicists. "The Rigged Hilbert Space and Quantum Mechanics" The fourth and fifth lines of Page 9, which read That is, X.^, i = l,2,..n are the generators of A, then the operator ® 2 & = £ X. is essentially self-adjoint i=l 1 (e.s . a . ) . should be replaced by the following: ^ 2 Also, the operator A = J X. , where i=l 1 X^, i = l,2,..n are the generators of A, is essentially self-adjoint 2 physicists have h«*cn usirtf; for . It .tlr.o c.iws ,t ;* li v e l y «.}i f t ir*j«w m V for quantum jw»ch.»nir;’. *htn ihe von Neumann axioms. But all this appears tc me negligible comj'.»!'t»d to the wain feature of this mathematical structure, its he.mty. The ftiggei Hilbert Space haa been one of the most beautiful pieces of 4rt I r»ave come .icrosr. ,*»nd h**i» giv»*:t “ <* *cy whenever I !ooke ! ar it . J hope that th«* ?;is»- p 1 i : i ’ picture which is %hown in thi:- p »pvr :;t i H convey:: enoufch .*>f th: • N ’.iutv to please even thc>‘.*» pUyvici.: ts wht- tvout new ».\thejn t? icr* beciurte they ;»o m-tih.-a.kt ic,* S r ff ’tr in phy.:»c-il ealculat i*'nK. !t» th.* !'.«!» 'winr. : r. t j-nluftory sccti«'n .i Jjm.w- work fv'r the forrul-*? i w <*f quantum acch.micc in the Figged Hilbert Spac* it ?i/en. Though modifications cf this frame - work which incorporate the same mathematical structure are possible, it is within this framework that the subject is developed in the succeeding sections. In section I the algebraic representation space for the algebra generated by the momentum an«i position operator is constructed in a way very familiar to physicists, the essential assur.pt ion is that there exists at least one eigenvector of the energy operator. In section II this algebraic space is equipped with two different topologies, the usual Hilbert space topology 3 and a stronger, nuclear topology. This nuclear topology has it purpor.e 3. the elements of the .jlpe?ra arc repre- ^•nu*t! by continuous operator* and 2. there exist eigen vectors a! the monw-ntum and position operators . <-as explained in the following sectionr.-} neither of which io the case for the u.-vual Hilbert space formulation. In section III the upaces of anti linear functionals are described and the Rigged (filbert Space ir. constructed. faction IV gives the definition of generalized eigenvec tor.. and a statement of the .*«»»«'lear Spectr.t] Theorem, which *M».i«Mnrees the existence oi .% ,-»»»r.plete . . t g e n e n l i .v d eigenvector*?! «*? momon turn u»d position op« r-1 11 »*•:-.. Keali7,a- tiois:; of the Kigged Hilbert Space by spai.*---- of functions and distributions are described and in Appendix* V a simple derivation of the Schrodinger realization is given. ACKNOWLEDGEMENT: These lecture notes have grown under the influence of discussions with the participants of the Mathematical Physics Seminar at the Mathematics and Physics Departments of the University of Texas. I am particularly grateful to F.. C. G. Sudarshan who read different versions of the manuscript. R. 8 . Teese has been of great help in the preparation of this paper. 0. INTRODUCTION To obtain a physical theory Mans to obtain a nathe- statical image of a physical system. The Mathematical image for the doswin of quantum phenomena i s , according to von Mtiwinn^, the set of operators in a Hilbert space. For the case that no superselection rules are present (irreducible systems) the physical observables are identified with the set of all sel^adjoint operators in the Hilbert space H and the ("pure") physical states are identified with the set of ell unit rays, or equivalently, the one dimensional projection operators in H. This one-to-one correspondences between the physical objects of a system and the operators in «t Hilbert space has two deficien cies: 1) It does not contain all the information one has in quentuB mechanics 2) It contains more information than one can ever obtain in physics. To elaborate on 1.): It is well known that all (separable infinite dimensional) Hilbert spaces are isomorphic and so are — roughly speaking — their algebras of operators. This ■m m that all physical systems Mould be equivalent, which 5 irt obviously n«*t t\ u ' imsp (un]«':??** one Inis only one physical uy:itcm, which ttu»n would have to be the microphysical worlu> In every practical case one has to consider an isolated physical system which — though it has to be large enough so that it doe* not become trivial — must be small enough that one may command a view of its mathematical image. The various microphysical systems in nature have quite distinct properties and must therefore have quite distinct mathematical images. Or, in other words, for a particular system some observables are more physical than others, and for a certain isolated physical system some observables are completely unphysical; e.g. if the system consists of the bound states of the hydrogen atom, then for this physical system one can never prepare a state in which the electron position is in a narrow domain of space, because a measure ment of the observable position will destroy this physical system. Thus for a particular physical system not all mathe matical observables are physical observables. Further, for a particular physical system sons observables appear to be more basic than others,and which observables are more basic depends upon the particular physical system. Therefore it appears appropriate to make a distinction 6 between observables and operators. An observable is a physical quantity and th definition of an observable consists of either a prescription for measuring the quantity itself or a definite expression for it in terms of other Measurable quantities. If not all operators are observables then also not all vectors can be physical states, because the preparation of a physical state constitutes the measurement of one or more obr.ervablcs, and a pure state that results from this measure ment is an *»*p,eiu:iate of the oh;;crvubles. Thus eigcnr.tatcr. of oponators which are not observables cannot be prepared. Turther—-and this is connected with 7.— there are vectors which are connected with operators that are observables but which are not physically realizable. These are the vectors that lie outside the domain of definition of an observable thai in represented by an unbounded operator, p.g. the vector which would correspond to the stat® of infinite energy. With regard to 2 .): The Hilbert space is infinite dimen sional and it is infinite dimensional in a very particular sense, namely it is coaplete with respect to a very particular topology, i.e. with respect to a very particular meaning of convergence of infinite sequences. Since an infinite nunber of states oan never be prepared, physical measurements 7 cannot tell ur. any tin up. about infinite scquencon, but can at roost give us information about arbitrarily large but finite sequences. Therefore physics cannot give us suffi cient information to tell how to take the limit to infinity, i.e ., how to choose the topology. On the other hand, we are also not permitted to choose a particular finite dimension N for the space of physical s+ates, because after one has made N experiments one can always perform an (N+l)**1 experiment. Therefore the dimen sion of the space must bf larger than any given finite num ber. I believe that the choice of the topology will always be »i mathematical generalization, but then one should choose the topology such that it is most convenient. At the time when quantum mechanics was developed the Hilbert space was the only linear topological space that had structures which were required by quantum mechanics. That is, it is linear to fulfill the superposition principle, and it could accommodate infinite dimensional matrices and differential operators. Therefore it was natural that von Neumann chose the Hilbert space when he wanted to give a mathematically rigrrous formulation of quantum mechanics. With this choice, though, he could not accommodate all the 8 it) futures of Dirac's fomulation of quantum mechanics, which von Neumann says i:> "«carcely to be surpassed in brevity and elegance" but "in no way satisfies the requirements of mathematical rigour". Since that time* stimulated by the Dirac formalism, a new branch of mathematics, the theory of distributions, has be*n developed. The abstraction of this theory provided the mathematical tool for a mathematically rigorous formu lation of quantui mechanics, which not only overcomes the Above mentioned shortcomings of von Neumann's axioms but also has all the niceties of the Dirac formalism. This mathematical tool is the Rigged Hilbert Space, which was introduced around 1960 by Gelfand** and collaborators 2) and Haurin in connection with the spectral theory of self- adjoint operators. Its use for the mathematical formulation of Quantum Mechanics was suggested around 1965 by J. E. Roberts5* 6) and A. Bohm ; the suggestion to choose the xopology such that the operators are continuous was made (for the canonical commutation relation) by Kristensen, Heljbo and Time I'oulncn* He formulate the general framework by replacing the von Neumann axiom which states the one-to-one correspondence between observables and sel&adjoint operators in the Hilbert 7) •pace by the following basic assumption A physical observable is represented by a linear operator in a linear spaoe (spaoe of states). 9 The mathematical image of a physical system is an algebra A of linear operators in a linear scalar product space. The linear space is equipped with the weakest nuclear topology that makes this algebra an algebra of continuous operators. That is, if X^, i = l, 2 ,..n are the generators of A, then the n 2 operator A = J X. is essentially self-adjoint (e.s.a.). i=l 1 We emphasize again: An observable is the physical quan tity which is defined by the prescription for its measurement. The algebra is the mathematical structure which is defined by the algebraic relations between its generators which repre sent some basic physical quantities and by other mathematical conditions, like the space in which the elements of the algebra act as operators. The connection between the quantities calculated in the mathematical image, such as the matrix elements C^|>, Aij») or Tr(AW) and the quantities measured in the physical world, such as the expectation value (average value of a measurement) MCA), is given by M(A) = (ty,A$) or in general MCA) = TrCAW). These connections are stated by further basic assumptions which we will not discuss in this paper. Within the framework of the above stated basic assumption the actual physical question is how to find the algebra. In non-relativistic quantum mechanics one can usually obtain 10 this, algebra JYom corrvsj'omlence with tho oorivsponding classical system. When no corresponding classical system exists one has to conjecture this algebra from the experi- mental data. 11 I. Tht* Alfii'br.iic ULruetutv oT the Space. ol oMU's We shall treat in detail one of the simplest physical models, the one-dimensional harmonic oscillator, in the framework of the basic assumption stated in the preceeding section. We will give a formulation which easily generalizes to more complicated physical models, and shall remark for 8 ) which algebras these generalizations are already known. The algebra 9 ) of operators for the one-dimensional harmonic oscillator is generated by the operators H representing the observable energy ( 1) P representing the observable momentum Q representing th#» observable position and the defining algebraic relations are ( 1 ) PQ - QP = 'j- 1 H " 2in P + ~ Q A, m, w are constants, universal or characteristics of the system. A is an algebra of linear operators in a linear space, with scalar product^, (*,*)(but ♦ is not a Hilbert space) Further: P,Q,H are symmetric operators, i.e. ( 2) (P*,*) = <$,P*) for all $,i|» e ( 1) and ( 2) do not specify the mathematical structure 12 completely. There are many such linear spaces in which * is an algebra of operators (whether the requirement that its topology be nuclcar strictions is unknown). Therefore we have to make one further assumption to specify the algebra of operators, i.e. the representation of (1) and (2). This requirement can be formulated in two equivalent ways: (3a)H is essentially self adjoint^(e.s.a.) or (3b)There exists at least one eigenvector of H. We w ill use this requirement in the form (3 b ), as it is this form in which it is always used by physicists. Remark; We remark that either of these requirements leads to a repre sentation of (1) which integrates to a representation of the group generated by P,Q,1 (Weyl group); 3a) by a theorem of Nelsor?^ and 3b)because it leads to the well known ladder representation and ladder representations are always integrable^tin this parti cular case one can easily see that H in the ladder repre sentation will be e.s.a.). The generalization of (3) is the requirement that i 2 i : I X. be e.s.a. for the case that * is the enve loping algebra of a Lie group, this will always lead 12 ) to an integrable representstion (Nelson theorem) , i.e. a representation that is connected with a representation of .i f.roup (oven though no symmetry may be involved), o The procedure to find the ladder representation is we Known and will be sketched only very briefly: One defines a = j < / % Q ♦ / = P> CO a* = h Q - P> ^2 J /maiH (i)l = a, a : j l - j l These operators clearly fulfill ( $ , ai|») = (a%,»|>) for ev*»ry ♦,!!> e * and <$, N$) = (N as a consequence of ( 2). As a consequence of (1) a and a+ fulfill ( 6)aa+ -a+a=l (7a) As a consequence of (3a) N is e.s.a. ( 7b) or as consequence of <3b) there exist such that 1** (7b) M#x = \4X From the c .r . ( 6) tlu*n follows ♦ ♦ v N(.i4^) = a a a - (a a - 1) a $ x = a(a a - 1) #x = ( 8) = a(N- l)4A = aCX-l)^A = (A-l)(a*x) i.e. a#x is an eigenvector with eigenvalue (X-l) or a$x = 0 Further from c.r. ( 6) follow | | a % J |2 = (* x,a a % A) = <#x,a+a #x> ♦ ($x, l*x> = I |a#xH 2 ♦ I U xl I2 * 0 as i 0 i .e. (9) t 0 always. Further follows from ( 6) ( 10) N (a % x> = ( X*1) a % i.e. *s always an eigenvector of N with eigenvalue (*♦1). Starting with #x , which was assumed to exist, one can obtain eigenvectors ♦ ^ = a ^ * s 0, 1,2,... with eigenvalue (A-a) of N ■ ♦»- ■ a - ) *x- After a finite number of steps m , _ n Proof : *♦1 m»N ^ = (X-jr.)( there fore I u * . j i 2 (11) (A-m) = --- ^=2-- > 0 i.e. there exists a Q. such that A-in * ♦ » - « = 0 We call 6 the normalized vector for which a C ) N* = 12 o 0 Then one defines the states 16 which have the property ( mj X>l ! • n I I = 1 2>MVn = n n*n 3)(*n»^m) n m " * nra u) For every # there exists a + t 0. n n*i The linear space spanned by we call f ,i.e . T is the set of all vectors * * ? “ Cn) *n n=0,l ... (n> where a ' c t and a is a natural number which is arbitrarily large but finite. We call R. the space spanned by 4^ , i.e. R. = (a+^laef>,(R^ .ire the energy el gcnup-iccs). Using R^ instead of we have a formulation which immediately applies when R^ is not one dimensional. Then * is th® algebraic direct sum of the R^ ¥ = I R. alg.i In other words f is the set of all sequences ( 6 >000...) with 4. e R. O 1 1 I 1 1 where the alget aic operations of the linear space are defined by ♦ * ♦ = (*o * ♦<>• * 1 * »1 * " • 1 » <«♦_. ) o 1 17 The scalar product ia m (+,*) = E ($. ,i|> ) • I 1 1=0 1 And the norm is ii*ii2 = ” i = 0 1 As the sum goes only up to an arbitrarily large but finite number m the question of convergence of the sum of numbers (♦i»^i> does not arise. f is a linear space without any topological structure. We now equip f with a topology, i.e. we construct a linear topological space. 18 I I . The Topological Structure of the Space of States A s«M • is .1 I in*'.it' ti>pologi«Ml spurv ( in which lh«* fir'3t .ixiOiii of <-oun1 al'i 1 i t v is I u 1 fi I led) i I f I) * is a linear space II) For a sequence of elements of • the notion of a limit element is defined (the limit is unique and a subsequence of a convergent sequence converges to the same limit point) I I I ) The algebraic operations of the linear space are continuous i.e. l)a) If ♦ » ♦ -* ♦ e ♦ then also a+ -*■ af for act n n b) If € * a ♦ att then also a $ ♦ a# for every n n 2) If ♦ and * -* i> then $ * if for n n n n $ c 4} n « l ,2 ...... n n This is not the most general definition of a linear topological space but it is sufficient for our present purpose (we w ill later on also use spaces in which the first axiom of countability is not fulfill?*?) In a given linear space the convergence can be defined in various ways leading to various linear topological spaces. All the topological notions like continuity, denseness, bound edness, closure, completeness... depend then on the definition of convergence. We shall introduce into y three different topologies. 19 As a first example we introduce into ¥ the well known Hilbert space topology (one can check that this really fulfills the conditions la) lb) and 2) of the d efin itio n .) Def : 1U) (1) - $ for 7 - » !1*y - for To understand the relation between ¥ and the Hilbert space H let us introduce the following notion: Def: A sequence txn) of elements in a space with a norm (Normed Space) is called a Cauchy sequence with respect to T#if for every e > 0 there exists (?) an N such that I I* - x | I < e for every m,n > N. ' 1 n m1 1 J Def: A linear topological space R is called complete if every Cauchy sequence (xn) has a limit element x which is an element of the space R. If a space is not complete it can be completed by adjoining all limit elements of Cauchy sequences to it . It is easy to see that * is not ^-complete: Let us consider the infinite sequence (3) h = (h,, h„, h„ ... h. ...) with h. e R. 1 2 3 x x i h. i 0 for all i x 2 and £ I I h -1 I < * i =0 1 20 Fro® this follows I !^n I I ■* 0 for n • and «•» £ I I h . | | ^ ♦ 0 for n ■* • i*n»l 1 h is not an « learnt of V, because it is an infinite sequence. Let us consider the sequence Sl» S2* S 3 --- Wltn Si c ¥ where fc) S “ (h. i h* i ••• h f 0^ 0 •*••) n 1 2 n AS Sn “ S* * <0»°» °» hnn * ...... hm* 00 ...... > tM have " Sn-S. " = " W W J I * 0 ,o r n ~ i.e. (Sn) is a Cauchy sequence. The T„-liait element of S. is h: ft n because ||S - h ||2 * I | |h. ||* -*0 for n-*« i*n*l i But h 4 f . If we adjoin to ? all the liait elements of Cauchy sequences, 21 i.e. take ir. addition to the elements t = ( s. ,*2 , . . . .* n 0 0 0 n arbitrarily large, the elements h of the kind defined in (3 ), then we obtain a com plete norced space in which the norm is given by the scalar product; t-.is is the Hilbert space H. Thus, the Hilbert IS) space K is the completion off with respect to the topology defin^.J ; y ( j). I W . A suSspace S of a c«>»pl«'tc topological space X is called den:;»* if S u (all liait points of S) = X. Thus T is a 7^-dense subspace of H. The spacc H is the space of all h * (h^th^ ...), lu c Rj, which fulfill (9); this space is written as m H * £ I R. i«o and is called the Hilbertian direct sum or orthogonal direct suit of the R^. Thus the Hilbertian direct sum is the completion of the algebraic direct sum with respect to x 91 We now introduce into V a new topology which we call t*. is defined in the following way: Me t«lce the original scalar product < * * • I and define 22 the following quantities ( 6) (♦,♦) = (♦,(N*l)Pf> p = 0,1,.’, ... P -and 11 ♦ I Ip " . Remark: Before proceeding we note some properties of the operator N: N is essentially self-adjoint. This is easily proved using one of the criteria of essentially sclf-adjointness: Lemma: An operator A is e.s.a. if (A^l)’1 is continuous and has a dense domain in H. The spectrum of (N+l)~* is consequently it is con tinuous. Its domain is f , which is (tense in H. Conse- qurntly N is e.s.a.. As a oonsequence N*1 is c.u.a. Further (N*1)P for every p is o.s.a.*6* From the spectrum of CN+1)P one sees immediately that (K*X)P is positive definite. From this it is easy to see that for every p fulfills the condition of a scalar product and <6a> I If I IG1 Ilf I lx< Ilf ! I2i ••• Remark: Further, these norms are compatible. I.e .. if a seqn»nre converges with respect to one norm and is a Cauchy sequence with respect to another, then it also converges with respect to this other norm, o A space in which a countable number of norms (scalar- 23 products)are defined is called a count ably normed (countably ••al^r product) r.f>ace. Vt" now «te»l'iru* tf by: IVI : (?) + + + {!# - ♦!| - 0 for every p. Y V P From this definition one immediately sees T* XH { 8) From - 0 •* follows 0 but not vice versa. Therefore is called stronger (finer) than and is c.Olft! weaker (c*»urs«»*‘) than t*. Def: A sequence {x ) is called a t .-Cauchy sequence if for everv Y 9 p and for every t > 0 there exists an M * N(c,p) such that < 11 *Y ** * 11 < e *°** H ,Y > H . WP There are more Cauchy sequences than -Cauchy se quences because for - Cauchy sequences (9 ) needs to be fulfilled only for p * 0. We completed with respect to i.e. adjoin to ¥ the limits points of all t^-Cauchy sequences. The linear topologi cal space that we obtain by completing * with respect to we call ♦. Then T C ♦ , T is T^-dense in ♦. 2*4 • is called a countably Hilbert space. As there are more t^-Cauchy sequences than t^-Cauchy sequences it must be that CIO) ‘T C § C M. Thus f is -r^-dense in ♦ f is t^-dense in H and • is t^- dense in H, as already f is dense in H . To get a feeling for # let us see which of the infinite se quences (3) that are elements of H are also elements of ♦ . In order that h = Ch ,h_ e K be a limit point of a t -Cauchy sequence i.e. in order that: S -*> h n one must have 11S - h|| **■ 0 for every p , i.e. " P (11) CCS -h) ,(!!♦ D PCS -h)) ♦O for every p. n n i . e . , one wist have • • _ * Chj,CK*l)p h.) = E Ci*l)p| |h.| | ♦0 for every p. i»n*l i=n*l Therefore, only those h c N are -limit points of -Cauchy se quences of elements of f -—that means elements of *— for which £ • is called the t.-direct sun of R and is written f n 12) • = r t R n-o To see what the use of the various spaces might mean for physics we recall that R^ are the energy eigenspaoes with energy value (13) E = /« (n ♦ J-) n * Thus T describes states with arbitrarily high but finite energy. H contains the infinite energy" state , and • contains states which have an Infinitely small admixture of infinite energy states Clearly physics can only tell us something about T. In fact for most real physical systems, e.g. diatomic mole cules, whose idealization is the harmonic oscillator, only the very lowest energy levels are relevant* for higher energy the diatomic molecule is no longer a harmonic oscillator and finally not even an oscillator. # and • are mathematical idealisations, though • appears "closer" to reality. The question there is, why do we want these mathematical idealisations and why do we have a preference for one over the other. The advantage that • has over * is matheme tioal convenience , 26 which can be vaguely suMarized by saying ♦ adaits the Dirac formalism. Two aspects of this formalism are: 1) All algebraic operations with the operators are allowed and no questions with respect to the doaain of definition arise. ?) For every e.s.a. operator there exists a complete system of generalized eigen vectors. 1) follows from the fact that all eleaents of the algebra A are continuous operators with respect to and 17) therefore uniquely defined on the whole space ♦ . To show this it is sufficient to show that P and Q or a and a* are continuous because of the Theorea: The product and sua of continuous operators are continuous operators. Titus in frncTal an algebra is an algebra of continuous opera torn iff the generators are continuous operators. We recall that (in spaces in which the topology can be defined by the convergence of sequences (i.e . spaces with the first axioa of cowtability)) an operator A is continuous if for all sequences {# } with 4 *0 it follows that A# ♦0. Y Y Y To prove that a (and a ) is x continuous let us consider ♦ _ a and a on t . We use the leama (proved in Appendix Ik Tor every ♦ c f (in) (♦, a (K * l )P a4#) < k(#, (Itn)**1#) where k is some constant, k < • . 27 Proof: Let ^ ♦ 0 «* y ♦ ' *"* 11#y1 0 for every p (15) *-+ (# ,(lin)p # ) * 0 for every p. Y Y To show that ♦ '♦ _ a # ♦ 0 we have to show that v (16) ||a+ 0 for every q. i.e. that ( a % * ( 4y i a(«4l)^ a % y) •* 0 for every q. By (It): (# , a(N*l'q a4# ) < k(# .CIM)**1# ) y y - y y but by (IS) the r.h.s. of this inequality ■» 0, consequently also the t.h .s. which proves (76). We reaark that the convergence of ||a+ # || ♦ 0 for y ♦ • for a fixed q follows froa the con- v q verqence of ||# || * 0. Therefore in the case of a finite Yq^l nuaber of noras a4 will not be a continuous operator. As the topology in the case of a finite nuabcr of noras is equiva lent to the topology given by the highest nor* (i.e. sequences that converge with respect to one converge also with respect to the other and vice versa) this fact iaplies that a 4 cannot 28 be a continuous operator in the Hilbert space topology which is well known. The proof of the continuity of a is analogous. Therewith we have shown that a and a* are continuous operators on the dense subspace V C * and can therefore be uniquely II) extended to operators in the whole apace ♦. We denote the operators a»a*, P*Q. .extended to the whole space + again by a v a4 , P,Q... a and a*and therewith the whole algebra are defined on the whole space + and domain questions do not arise in the algebraic operations. The t $ -continuous operators ?«Q»H considered as operators 19) in M are not closed operators (they are tf-continuous and consequently -closed but not t--closed), because if i et T * M and f ♦f but f4#,then Af e# for every y but A is not defined T on f. If Af then we define Af by Af * * . Ha can do this for all ftff which are ^-liait points of sose sequences f e# and for which Af t -converges. The operator t y n A defined in this way, i.e. whose doatain D(A) is the set of all these fdf,is the closure of A in the completion of ♦ to K. Thus in correspondence to the relation (10) between the spaces: t C H we have the relation { i n A C S between the operators. 29 (however in general 0(A) $ U K So far we know of the operator* P,Q,H(or N) that they are t ^-continuous sywsetric operator® on I. Therefore P*D F (i.e. D(P*> D & ¥ ) ) (IS) and Q* D Q . HCand h) is e.s.a.** and therefore (19) H* s if iU* * »T) However as a consequence of (19) it follows that also P,Q, are e.s.a^i.e.: (20) P* * P D(P*) * IXP) Q**P Remark: K*I is, except for soae constant factors, the Nelson operator, which is because of (19) e.s.a.. Therefore --as mentioned before— f,Q,l are the group generators o f a group of transformations (Weyl Group). As a consequence of the fact that P,Q are elements of tiie l«ie algebra of this group it follows then that P and Q are also e .s .a ., by a theorem of Nelson and Stinespring?*20*© He now turn to the 2ndaspect of the Dirac formalism, the existence of a complete set of generalised eigenvectors, which follows from the fact that for the o p e r a t o r s in + the nuclear spectral theorem applies. The explanation of this requires »DM further mathematical preparations. Either as a consequence of assumption (3b) as grown in Rmark on p. 12 or by assumption (3a). 30 DejF: A bounded self-adjoint operator B is Hilbert-Schmidt iff R = EXj^Pj^ where the projections P^ project on finite riimen- 2 sional spaces H and JX|X | dim H ) < • K K IC Instead of giving the original definition of a nuclear space^*^ we shall use a theorem of Roberts^?) which gives a necessary and sufficient condition for * to be nuclear. Def; t is a nuclear space iff there exists an e.s.a. t - conti nuous operator A e A, whose inverse is Hilbert Schmidt. It is now very easy to see that our ♦ is nuclear, because N is e.s.a., the spectrum of is j~, n = 1,2,3,... and R^ 1 2 xs one dimensional, thus £(~) < •• Thus N is the operator that fulfills the condition of the above definition. Summarizing, the space ♦ constructed above is a linear topological nuclear space in which all elements of the algebra are continuous operators. Remark concerning generalizations: The construction of a nuclear space described above is iamediately generalized to a more general algebra. The analogue c f the Lemma (1*0 IV) (#, x (A *1 )P X ♦) < k( 2 where X is one of the generators X^ and 6- EX^ (Nelson operator) holds for all enveloping algebras (Lemma by Nelson). Therewith the continuity of the algebra in a linear topological space in which the topology is defined by the countable number of scalar 31 products (♦,#>p = <«MA+1)P*) follows immediately. Further if A is e.s.a, then all the symmetric*1^ generators are also e.s.a. (Theorems by Nelson and Nelson and Stinespring). ( 1***) is much stronger than required for the proof of the continuity of the generators. The continuity of the gene rators (and therewith the wholo algebra) can already be proved if instead of p ♦ 1 on the r.h.s. of (1*»*) one has P+n, where n is any finite integer. Therefore it appears that the continuity of the generators cam already be proved for any finitely generated associative algebra. The nuelearity is a much harder property to establish. It has been proven for the cases that the algebra is the enveloping algebra E(G) of the following groups G: G nilpotent (because then E(G) is isoaorphic to the enveloping algebra generated by P^, Qo, a s l , 2 ,...m with [Pa, 0,3 * I ‘a. 1 for some m (Theorem by Kirillov^?^and we have just an m dimen sional generalization of the above described case) 24) G semi-simple (A. BOhm) G = A semidirect product, with A abelian and K compact, ( b . Nagel) G Poincare group for some of the representations (B. Nagel 32 III Conjugate Space of * An antilinear functional F on a linear space y is a function, the values F($) = <♦ |F> of which are complex numbers, which satisfies: (1) F(a$+B$) = a F (f ) + 8F ( * ) c f ;a ,0 e t or in the other notation ( 1> The functional F is called t -continuous i f f from T t <2) ■* $,y * follows F($y> -*• F($) 1 i where -*• means convergence of complex numbers (for every c > 0 there exists an Cq,i)(e) such that |F($){ < e for all $ J withI M I q<4 >* The set of antilinear functionals on a space f may be added and multiplied by numbers according to <4|aF1+eF2> = a<^|F1>+$<^|F2 > (3) (aF^er^ <*) = aF1(^)+0F2(^) fI The functional aFj+0F2 defined by (3) is again a linear functional. Thus the set of antilinear functionals on a linear space * forms a linear space. This space is called the algebraic V dual or algebraic conjugate space and denoted V . 33 Continuous antilinear functionals have not only to fu lfill ( 1) but also ( 2), therefore the set of continuous antilinear functionals, which also forms a linear space, is smaller than the set V and depends--like all topological notions — on the particular topology that has been chosen. We shall denote the set of all continuous functionals X X by ♦ and the set of all ^-continuous functionals by K . T* XH As every sequence with #Y 0 has the property ^ -*• 0 i.e. there are more convergent sequences than convergent sequences and F($v> F(0>) = 0 must be fu lfille d for a ll (ff) X -convergent sequences if F = F e H , and for all t^-con vergent sequences if F * p ^ ^ e the set of F^^ e will ( ♦) X bo smaller than the set of F e ♦ (because there are more X) X stringent conditions on elements of H . Therefore: H X is the smallest set and ¥ is the largest set of the set * MX .X vx " 9 • » r • In a scalar product space each vector feY defines an antilinear functional by (S )< * | f >=F(*> ( * , f ) It is easy to see that F defined by the vector f as given in (5) fulfills the condition (1) if (♦ , f) fulfills the con ditions for a scalar product. 3U Further if 4> -► <> then for every f e y (even for every f e H) (#^,f) ♦ ( converges also weakly). Thus the functional defined by (5) fulfills F($y> -*• F(4>), i.e. it is a continuous antilinear functional. Thus the scalar product defines a continuous antilinear functional. In the Hilbert space the converse is also true, i .e . any antilinear ^-continuous functional can be written as a scalar product according to the Frechet-Riesz theorem: C H) For every antilinear "^-continuous functional F there exists a unique vector f e H such that ( 6) «*|F(H)>=F(H)( Therefore wo can identify the Hilbert space H ami its conju- X X gate H by equating F e H with the f e H given by ( 6). Then with (1 1 ,9 ) we have the situation: (7) f C * C H = HX ( H) For ^-continuous functionals F the symbols <| > and < >) ( tf) are the same, after th* identification F = f: , (9) <*|f> = (*,f) However for the larger class of x^-continuous functionals F the symbol <$|F> is defined, whereas (♦ jF) is not unless X X F e * is already e H in which case we identify: (10) <*(F> = <*|F(W)> = ($,F) 35 In ♦ one can introduce various topologies and there with various meanings of convergence of countable sequences. An example is the weak convergence (in analogy to the weak convergence in H): X Def: A sequence of functionals {Fy}C# converges (weakly) to a functional F if f (11) < On«i can show that ♦ is already complete with respect to X the topology t of this convergence. X Remark: ♦ is a space in which the topology cannot be com pletely described by the description of the passage to the limit of countable sequences (♦ does not satisfy the first axiom of countability and is, therefore* a more general topo logical space than those defined by our definition in II). 26) Therefore we cannot attempt to prove the following statements. © X As $ is a linear topological space we can consider the X X t -continuous antilinear functionals $ on 4> , i.e. the set XX X ♦ of all those functions on 0 which satisfy X ' _ ^ y (1 ) 0>(aF1+8F2 ) = a ♦ (F ^ + 8 and X - T (?X) $(Fy) - MF) tor every Ty - f XX ♦ is a linear topological space if addition and multiplica tion are defined by: and (weak)convergence by: XX - (11X) *Y < FU > -*■ 2 7) One can prove that there corresponds an antilinear X continuous functional $ on ♦ to each element ♦ e ♦ defined by the equation (12) ♦(F) = fT or (12) convergence with respect to Thus, with the identification 4 = 4, given by ( 12), ♦xx = . (♦is reflexive). The Hilbert space W is certainly reflexive (because already H = H) and as the functionals in H are given by the scalar product the relation corresponding to ( 12) is the property 37 of the scalar product: ( 1.0 lor all t.-continuous functionals F cn ♦ which are also $ -continuous functionals f on ♦ we identify f with F or we have the situation ( 1**) WX C *X which with (7) gives (15) « C H C *X This triplet is called Rigged Hilbert space or Olfand Triplot. 1'or every continuous linear operator A in ♦ one can X X define the adjoint operator A in ♦ by: (16) <$|AX |F> s F(A If A is a continuous operator in 9 and F is a continuous X antilinear functional then A is a continuous operator in *x • 9 i.e., it has in particular the property: (17) AXF -► AXF for all F -*■ F. y y x x x In particular the adjoint operators P ,Q ,H of the X X operators P»Q,H are t -continuous operators in $ . Remark: We have defined a continuous operator only by the convergence of sequences. This is possible only in spaces with the first axiom of countability. Though we have not 38 given the general definition of continuous and bounded operators wo shall remark on the connection botwoen them as theso two notions are used interchangeably. In spaces with the first axiom of countability every continuous operator is bounded and every bounded operator is continuous. The Hilbert space and the space ♦ are such spaces. In general, e.g. in the space X ♦ , continuous operators are not bounded, and the condition (17) % is only necessary for A to be a continuous map.o For every r^-continuous symmetric operator A we have, in correspondence to the relation (8) between the spaces 15) * C H L — + Y 18) the relation between the operators A C. A C A C A , and A C A = A+ C A* i f A is e . s . a . 39 IV Generalized Eigenvectors and Nuclear Spectral Theorem Before wc define generalized eigenvectors of an operator A in the space 4> let us recall the situation in the Hilbert space H . A vector h c H is called an eigenvector of an operator A on the Hilbert space iff Ah = ah where a is a number. In general, it is not possible to find for every operator on W an eigenvector. In fact it is well known that the operators r and Q do not have any eigenvector in H. Yet physicists, following Dir/ic, always work with "eigenvectors" of P and Q P|p> = p|p> and use the assumption that these eigenvectors form a "complete" system in the sense that / |P> We will now give a definition and a theorem that provides a mathematical justification of this procedure. Def: Let A be an operator in A generalized eigenvector of the operator A corresponding to the generalized eigenvalue X X is an antilinear functional, F c ♦ , such that (1 ) F(A*) = holds for every ♦ e ♦, which may be written in the form: 1 ’ ) AX |F> = *X|F> <+0 We shall use this del inition only for (he case that A is e.s.a. (ofter. one defines the generalised eigenvector by F(A*$) = AF($) where A+ is the adjoint operator of A). Let us assume A has generalized eigenvectors in the Hilbert space i.e. F in (1) is an element of H . Then (1) reads 2) for every 3) A+f-I"f and i f A is e.s.a. A+f = Af = Xf = Xf Thus a generalized eigenvector which is an element of the Hilbert space is an ordinary eigenvector corresponding to the same eigenvalue (for the case that A is not e.s.a. + it is an eigenvector of A ). Before we formulate the nuclear spectral theorem let us start with a few remarks. Every self-adjoint operator A in a finite dimensional Hilbert space has a complete system of eigenvectors. I.e. there exists an orthonormal set of eigenvectors h,. V=|X.) e H ( X . ) ' i X.) = >>=ai i i'i such that for every h e H n 5) h = I |X.)(X.|h) n = dim H i = l 1 1 The set of all eigenvalues A = {\ ,\ , ,.X } is called the i. ^ n spectrum of A. For an infinite dimensional Hilbert space this statement is no more true, in fact we know that there are operators that have no eigenvector in H. (e.g.Q). The spectrum of an operator A in H is defined as the set A of all these X for which A-Xl has no inverse. The spectrum of a self-adjoint 2 8) operator' is a subset of t h < ? real axis . If the sell adioint operator A in the infinite dimensional Hilbert space H has only a discrete spectrum,i.e. if A = {X ,X2 »..} then the above state ment carries over to the infinite dimensional case. To avoid complications which are inessential for the principal problems and inapplicable for the special problem of the one-dimensional harmonic oscillator we restrict ourselves to cyclic operators. Def: A operator A is cyclic if there is a vector f e H such k that A f, k = 0 ,1 ,2 ,... span the entire Hilbert space. It is clear that the operator P and Q are cyclic because Q • = (a + a ) ♦ , k = 0 ,1 ,2 ..., any n span the entire H. The generalization of the above statement to the case of a cyclic self adjoint operator with discrete spectrum is given by the following spectral theorem: There exists an orthonormal set of eigenvectors 42 Ah. = A. h. i A. e A, (h. , h. ) = 5. . A. X A . 1 A* A* A * A « i i l 3 1 * niK'h that lot' »'v*'i'v h r (/ h = £ h (h ,h) = Z| A.)(A.|h) 1=1* • A 1* 1A « 1 • X 1 ( 8) = / d)i(X) J X) ( A I h) A The measure y is concentrated on the eigenvalue X^ e A and one point sets have measure one, i.e. = 1 The operator H is an example of such an operator. We remark that all eigenvectors (7) of A appear in the decom position (8). If A has continuous spectrum, then there are no such eigenvectors in H. However a generalization of the above statement holds in the form of the following Wuclear Spectral 29 ) Theorem: X Let 4 C He t be a rigged Hilbert space and A a cyclic e.s.a. x^-continuous operator. Then there exists a set of generalized eigenvectors (9) Fx = |A> e *X CIO) AX |X> = X| A> , AeA i.e. for every ♦ e $ .XX such that for every c ♦ ) and some uniquely de fined positive measure y on A which is written formally ( 12) If we set ^ In (Jl) then we see that from the vanish ing of all components $ corresponding to the operator A, it follows that ||$|| = 0 i.e. that $ = 0. Because of this property the set of generalized eigenvectors | A> occurring in (11) or (12) is called complete (in analogy to the completeness of the system of ordinary eigenvectors in ( 8)). This spectral theorem assures the existence of a com plete set of generalized eigenvectors of the operator Q (and P) , i.e. of generalized vectors that fulfill: (13) QX |x> = xjx> pX |p> = p|p> and with the help of which every $ e * can be written X spectrum P where X =spectrum Q. The fact that there is a complete set of eigenvectors does not tell us what this spectrum is. 4U It is widely known that the spectrum of Q and P is the real line. However it ip not so widely known that the .leri- 30 ) v.ition of this is far from trivial. The reason for this is that the problem of the physicist was the reverse of the problem described here, namely to find the defining assumptions (1,1) and (1,3) from the spectrum of Q which was assumed to be the real line. (cf. remark on p. i+g) . We will in the following only describe the results , the derivation of which will be given in the Appendix V. The set of generalized eigenvalues of Q and P is the complex plane. The spectrum of Q and P is the following subset of the set of generalized eigenvalues, spectrum Q - {x| -a> (15 i spectrum P - {p | -® In general, an e.s.a. operator has more generalized eigenvectors (and more generalized eigenvectors that are continuous functionals) than appear in the spectral decom position (11), (12). (17) The measure in (1U) is the ordinary Lebesgue measure so that after normalization we have (16) ♦ = / dx|x> livery vov'ti'r <(> c # i ; fully i'h.i!MOtc! i;i'.l bv it;; v-oj«- ponents vectors |x> (in the same way as every vector x in the 3-dimen sional space can be equivalently described by its components 3 i . -*■ ■* i ■* x with respect to 3 basis vectors e. according tc x = E x e. 1 i = l 1 or a vector h t H by its components (h ^,h ) with respect to a complete basis system i = l,2,...). The function is called a realization of the vector $ and the set of func tions space ♦. The realization of is the space S of ull infinitely differentiable functions $(x) = their derivatives tend to zero for |xj^* more rapidly than any power of |~j (Schwartz Space). In this realization of ♦ by S the operator Q is realized by the multiplication with x and the operator F is realized by the differential operator in S: The elements of the algebra A are then realized by polynomials in x and ^ . We can, according to (111*12) consider * in (14) or *♦6 X X (11) as the functional on the space * at the element F e ♦ : ♦ (F) = and write (18) The l.h.s. of (is) is well defined and, consequently, sc is the r.h.s. However one factor in the product of the inte grand, (18), In this realization the space H is realized by the space 2 of square integrable functions L (x). In distinction to the Hilbertspace formulation, in which the momentum operator P is realized by the differential operator on the subset of all 2 d' 2 differentiable functions h(x)eL (x) for which also ~ h(x)eL (x) , dx the momentum operator P in the present formulation is realized by the differential operator on S. We remark, that in the derivation of these results (Appendix V) one has to use the defining assumption (1,3). This is a natural assumption for the harmonic oscillator, because H is the energy operator of the harmonic oscillator. For other physical, even if one assumes that its algebra be generated by P^ , fulfilling the canonical commulation ? 2 relation (1,1), the requirement that the operator be e.s.a. need not be such a natural assumption and therefore33® (17) not the most natural possibility. Also it may well be that for other physical systems the algebra of observables may bo more naturally defined by other operators than the I'. and Q . . l l Remark concerning generalization For the case of a more general algebra of linear operators the e.s.a. operators are not cyclic (more than one quantum number is needpd to characterize the pure statos). One then m*eds a complete system of commuting operators to obtain a complete set of generalized eigenvectors. {A^}, k = 1 ,2 ,...N is a system of commuting operators iff 1.) CA. , ^ 1 = 0 for all i i k N 2 2 .) I A is e.s.a, k=l K A^ is a complete commuting system i f f there exists a vector ♦ * such that (A$f A g algebra generated by fA^}} spans the space ♦(or H ) . An antilinear functional F on t is a generalized eigenvector for the system A^ i f for any k = 1,...N <*e Tho set of numbers A = ( A ^ \ X ^2 \ .. . A^^ ) are called gene ralized eigenvalues corresponding to the generalized eigen vector F. = |A^^ A^^ . A^^ >. There exists a complete system A of generalized eigenvectors according to the general form of the Nuclear Spectral Theorem: Let k = 1,2,..N be a complete system of commuting e.s.a. t —continuous operators in the rigged Hilbert space $ ♦ CH C *X Then there exists a set of generalized eigenvectors I v v e *x A£|X(1 )..A(N)> - A(k)|A(1 )..A(N)> A*k* G A*k* = spectrum such that for every $ e * and some uniquely defined measure . . ( 1) A2) .(H) |i on A - A x A . . A I - t A M xk(1) *(N>^,,(1) ,(N)|~ $ - j dw( A) I X ..A > This theorem gives the precise formulation of the famous Dirac conjecture if the starting point is a precisely defined a lg e b r a . The mathematical task that has to be solved if one starts with a well defined algebra is to find a complete com- *»s muting system and the spectrum of this complete commuting system. The problem to dote mine when a system is complete ir; far from trivial. Already for the simplest cases of enveloping algebras of group representations the number N is not inde pendent of the particular commuting system, i.e. if is one particular complete commuting system then another commuting system containing an operator may require N+1 32) operators B^ , ®2’ **®N+1 ln order 'to be comPlete* The problem of a physicist is usually the reverse of this mathematical problem. From the experimental date, he finds out how many quantum numbers are required, and what are the possible values of these quantum numbers. This gives him the complete commuting system {A } and its spectrum. He Jv then conjectures the total algebra A by adjoining to a minimum number of other operators such that the matrix elements of elements of A calculated from the properties of this algebra agree with the experimental values of the corres ponding observables, o Summarizing: Using the rigged Hilbert space we have reproduced the main features of the Dirac formalism. For the particular case of the harmonic oscillator we have in fact reproduced everything of Diracs formalism. The difference between this formulation and the usual Hilbert space formu lation appears to be minor from a physicists point of view but is essential from a mathematical point of view and leads to tremendous mathematical simplification; in fact it justi fies the mathematically undefined operations that the physi cists have been accustomed to in their calculations. 51 APPENDIX I LINEAR SPACES, LINEAR OPERATORS A linear senLarproiiurt space * is (-not necessarily a spaoe of function;; but-) a net of mathematical (imagined) objects for which three operations are defined. I Addition of elements of * , which has the following properties: (i) For h»f,g e t f ♦ g t ♦ and f ♦ g = g ♦ f» (f ♦ g) +h= f + (g + h) = f ♦ g ♦ h ( i i ) There exists an element 0 e ♦ such that f + 0 = f def (iii) h ♦ f = g has a unique solution h * g - f II Multiplication by complex numbers, which has the following properties o(f ♦ g) = af ♦ ag, for f»ge #, a e t = complex numbers (a ♦ 0)f = of ♦ 0f a , 8 e t o(6f) = (o0)fi Of = 0 III Scalar product of two elements e ♦ i .e . with every pair of elements e ♦ there iu associated a complex number (♦»♦> with the properties (♦,*) = (?:♦) (iii) (♦,♦) ^ 0 with (♦,♦) = 0 only for $ = 0 An linear operator A is (-not necesuarily a differen tial operation but-) a map from (a subset of) ♦ into ♦ which has the property: A((*i ^ + o2 f2 > = a 1 Af^ + a2 A f? a l ,Cl2 £ fl ,f2 £ * APPENDIX II ALGEBRA A set A is an (associative) algebra with unit element i f f (i) A is a linear space over (Z. (ii) For every pair A,B e A, there is defined a product AB e A such that (AB)C = A(BC) A(B + C) = AB + AC (A + B)C = AC ♦ BC (aA)B = A(oB) = a AB (iii) There exists an element 1 e A such that 1A = A for a ll A e A. A set K of elements of A is called a system of generators iff the smallest closed subalgebra with unit which contains K coincides with A. The element 1 is not to be called a generator. Let -die elements of K be called X^ i = 1,2,. .n. Then each element of A can be written 54 We restrict ourselves to the case that n is finite and we w ill not discuss topologies of A , i .e . we assume that the above sums for every A are arbitrarily large but finite. Defining algebraic relations are relations among the genera tors (2) P(X±) = 0 where PCx^.) is a polynomial with complex coefficients of n variables x . . An element B e A l (3) B : b° ♦ Z b 1 X. + E b1^ X. X. + ____ b1 . . . e C is equal to the element A i f f (3) can be brought into the same form Cl) with the same coefficients C°, C1 , C1 .... by the use of the defining relations C2). The best known examples of associative algebras are the enveloping algebras eCG) of Lie groups G. An enveloping algebra is the associative algebra gene rated by X^ , X^ .. X^ in which the multiplication is defined by the commutation relations n k (4) PCX.) = X. X. - X. X. - I C.. X. = 0 i 1 3 3 1 k=l 13 C . . are called the structure constants of the Lie group G. 13 The comutation relations (4) do in general not suffice to 55 fully define an enveloping algebra of linear operators in given linear topological spaces. I.e ., there are several algebras of linear operators in linear topological spaces that fulfill a given commutation relation. In order to specify a particular algebra of operators one has to require additional algebraic relations and other properties (such as £ X.2 be e .s .a .) . l 56 APPENDIX III LINEAR OPERATORS IN HILBERT SPACES We recall here a few of the definitions and important theorems from the theory of linear operators in Hilbert space. Let A :K ■* K be a linear but not necessarily bounded operator on a Hilbert space K and let D(A) be dense in K. Then A*, the adjoint of A, is defined in K by (A*f ,h) = (f,Ah). The domain D(A*) is the set of all vectors f €• K such that (f,Ah) = (z,h) holds for all h €E D(A); the vector z is then uniquely defined and z = A*f. An operator A on K is called symmetric if D(A) is dense in K and A C A* (i.e ., D(A) C D(A*) and Af = A*f for every f € D(A))j it is called self-adjoint if D(A) is dense in K and A = A*. An operator A on K is called closed if the relations lim f = f, lint Af = g, f e D(A) n n ° n n * n -*■ » imply f 6 D(A) and Af = g. Closedness is a weaker condition than continuity since, if an operator A on K is continuous, then lim f = f for f 6 D(A) n n n + «• 57 implies that the sequence (Afn) converges, while if it only closed, then the convergence of the sequence {f } for f € D(A) does n n not imply the convergence of the sequence {Af^}. However, if A is closed then, in particular, it has the property that two ; sequences {Af^} and cannot converge to different limits if the corresponding sequences ffn ) and {gn > converge to the same limit. A* is always closed. I f A is not closed, it is sometimes possible to find an extension of A which is closed. An operator A admits of a closure A i f f the relations f € D(A), f' € D(A), lim f = f, lim f» = f, n n n n n -*■ eo n -*■ «* lim Af s g, lim Af' = g' n n imply g = g1. In this case D(A) consists of all f € K for which there exists a sequence tfn) E D(A) which satisfies the conditions: i) lim f = f and ii) {Af } converges. Then by definition n n Af = lim Af . n An operator A on K is called essentially self-adjoint if A is self-adjoint. Physical observable are assumed to be represented by essentially self-adjoint operators. An operator A on K is called Hermitian if A is bounded and self-adjoint. An operator A on K is called unitary 58 i f IjAfjJ - }Jf|) for all f 6 K. A unitary operator satisfies the relations A*A = AA* = 1. Let M be a closed subspace of a Hilbert space 1C. An operator P which associates with each f G K its projection f^ on M, P:K •* U is called a projection operaror on M. Project tion operators are linear, Hermitian (i.e ., P*=P, D(P)=K) and idempotent (i.e ., P 2 = P ); and every linear, Hermitian, idempotent operator with D(P) = K is a projection operator. Two projection operators P^ and P2 are called orthogonal iff P^?2 s 0 ; in this case P^IC and P^K are orthogonal, i.e ., for every h^ E P^K and every h2 E P^JC, we have (l^,!^) = 0 . 59 APPENDIX IV: PROOF OF LEMMA (1i+) SECTION II BY INDUCTION (Al) From the c.r. fellows: (N+l)a+ = a*(N+2) a(N+l) = (N+2)a Ci^O is true for p = 1 because ( $ , a (N + l )a % ) = ($ ,aa+(N+2)$) = ( = (<)>,(N+1)2^) + ( because (^(N+]>f>) < ( ^ ( N + l ) 2 Assume (14) is true for p = q: i.e. (A2) U,a(N*l)qa%) < k(^ ,(N+l)q41^) for every $ and calculate ($,a(N+l)q+1a%) = U,a(N+l)(N+l)q_1(N+l)a%) = because of < k((N+2) $» (Ii+l)q“ 1(N+2)4>) because (A2) is valid for every in particular for < kK(N+l)*,(N+l)q“ 1(N«-l)$) ♦ (<|>,(N+l)q(>) + ((N+l)4>,(N+l)q' 14>) ♦ (♦, < k'k (♦,(N+l)q+S> because of ( 6a). Consequently (1*0 has been shown to be also fulfilled for p = q+1 and therefore it is generally true. 6 0 APPENDIX V: We will show in this Appendix that the defining assump tions, i . e . ( 1 , 1 ) ( 1 ,2 ) ( 1 ,3) and the T^-continuity of P and Q, w ill lead to the Schrodinger representation in the Schwartz* space Sc . 31> 33) We choose one of the energy eigenstates • , n = 1 ,2 ,... and calculate (using a$^ = «/n ♦ ^, a+# - /r:^l * .. ) : n n*l for n = 0 one obtains ( 1 > /r ^ <*1|1<> ( 1) is a recurrence relation for <*n lx>» which can be brought into a well known form by introducing J £ ~ • - <♦ | ( 2 ) y * — »fn(y) = ^ ”n! TTTxT Assuming <*0 I*> * 0 ) mu o' ( 1) is then written ( 1 ’) fn+1(y) = 2yfn(y) - 2n fn„ 2 U') f ^ y ) = 2y f Q <1’ ) are the well known recurrence relations for the Hermite polynomials and have solutions for any complex y. Thus we conclude that for any complex value x there is 61 an antilinear (not necessarily continuous) functional |x> = I’ which is a generalized eigenvector, x The functionals <*n ix> arP given by (3 ) <♦ I x> = /--- <♦ | x> H ( -- ) ’ x e t X " V n ! ° " /mu As the operator Q is e.s.a. (as a consequence of (1,3)) o 8) the opectrum of Q must be real" . Thus the spectrum is only a subset of generalized eigenvalues and in the spectral de~ composition (*+ ) ♦ = /d u (x ) lx> If we consider ♦ (or any # e ♦) as functional at the n generalized eigenvector F , = |x*> e $ ,x' e spectrum Q, then, according to (III, 12) , we obtain from (5 ) ♦ (F ,) = Thus dv>(x) (6 ) dii(x) We consider now the generalized eigenvectors of the operator P: PX |p> = PX F = p F P P or lfiSTw ♦ Using P = v f y (a-a ) one can proceed in complete analogy to tho procedure for Q following eq. (1). One obtains that the set of generalized eigenvalues of P is the complex plane and 1 p (3 ) <* |p> = in ,--- <♦ |p> H ( £■— ) p n /_n , o n /hm In analogy to the case for Q, also in the spectral de composition of with respect to the operator F: (Up) = /dy(p) IpxpIV only those (5 p) ( 6p) dy(p) The analogy between the p- and x- spectral decomposi tion as expressed e.g. by the analogy between (3 ) and (3 ) x p 63 we should have expected as the assumptions we started with, (1,1) and (1 ,3 ) are symmetric in Q and P. We now calculate the scalar product of 4>n and $ using (U ) with (3 ) and (4 ) with (3p): X x p r (7) 6 = $ ) r /du(x) <♦ |x> = — — /dy(x)|<4> | x> | 2 H ( * r r ~ ) H ( >- *— ■) j —— /^r~T ' 'o ' 1 n , m »2®;nj 2 n : ^ /mu /mu ~ /dy(p) <9 |p> . /.n .(spectrum P) /ftmw /fimui 2 m! 2 n! Comparing this with the orthogonality relations for the Hermite polynomials 1 +flB 2 ( 8) --J dy e"y H (y) H (y) = 6 n !2 /n JL m n mn and taking into account that the Hermite polynomials are only orthogonal polynomials if associated with the interval 2 _• < y < +oo and the weight e ^ (one can define H^(y) by ( 8) and derive ( 1!) for real y) we conclude; o i mu> x2 (9x) dii(x)|<»o|x>|2 = f S Z . e -JK72 dx (10^) spectrum Q = X = { x }-® anti 2 ______—P. - . . . | . 2 / I ^ -molTi • 2 ( y du(p)|<*0lp>l - e dp (10 ) spectrum P = {p|-°° If we agree to normalize the generalized eigenvectors such that (11 ) (11 ) ^ ’ Ip* = P then according to (6 ) and (6 ) x p (12 ) dyCx) = dx x (12 ) dp(p) = dp P and mu 2 - J tT x (13 ) <♦„!*> * /ITT 7 7 ~ Hn( / r ~ > e * ¥ 2 n! /ir /mw “ V ' « r ’ • ' Thus for we know the matrix elements of Q in the basis of generalized eigenvectors of Q ( 10^) and the matrix elements of P i n ' the basis of generalized eigenvectors of P ( 10p) We now want to calculate the matrix elements of P in the basis of generalized eigenvectors of Q and the matrix elements of Q in the basis of generalized eigenvectors of P. In order to do this, we will consider t as a functional n X at the generalized eigenvector z $ , p e spectrum P , and use the spectral decomposition (14 ) X F^ e # , x e Spectrum Q, and use the spectral decomposition (4 ) : P (14 ) The Hermite polynomials have the following property: 2, +°° i£n 2 n (IS) in e '" H Ct\) = JdC - e < n HnC5> n -00 / 2 tt Inserting (13 / and (13 ) into this relation it follows P x +00 / i t \ <♦ Jp>=/ dxg~ ■ < * |x> (1V n 1 n 66 or taking the complex conjugate ____ (16 ) “ V 0 7 ) <*Ip > V5^f'elXPM (17 ) and (17 ) together give x P (18> We emphasize that (18 ) does not follow from the "hermiti- city property of the "scalar product" particular property of the operators P and Q (and~as a conse- qucnce thereof-—of the Tourier transformation). For the general case of two arbitrary systems of gene- X — ralized eigenvectors |X>; A |X> = A(A> and j 8>; BX|B> = 0|B> it is not always true that though ( 111, 12) holds always. It is now simple to calculate the matrix element of P in the basis of generalized eigenvectors of Q, using (17^) 67 > = / dp f (19 ) In the same way using d ? ^ ) one obtains (19p) It is now easy to see what the realization of $ by the functions the operators P and 0 are defined on 4>, the functions (♦*Q%) = /dx xn| ($,QnP% ) = (r-)m /dx As all algebraic expressions in P and Q are continuous operators in 4 the sequence $v(x) = realization of the convergent sequence ^ converges to ♦ (x) = which the convergence is defined in the above described way 68 is the space of test functions S (Schwartz space). Concluding this appendix we emphasize that in deriving (19) and (10) we have made use of (1,3), because by (1,3) we were led to (1) and therewith to the Hermite polynomials, whose properties have been used extensively to establish (19) and ( 1 0 ) . 69 REFERENCES AND FOOTNOTES 1) I. M. Gelfaml, G. K. Shilov, Generalized Functions, Vol. II, in particular, Ch. I ., Academic Press, New York 1967, I. M. Gelfand, N. J. Vilenkin, Generalized Functions, Vol. IV ,in particular Ch. I. 2) K. Maurin, General Eigenfunction Expansions and Unitary Representations of Topological Groups. Workawa 196 8. 3) J. von Neumann, Mathematical Foundations of Quantum Mechanics; Springer Berlin 19 32, Princeton University Press 1955. 4) P. A. M. Dirac, The Principles of Quantum Mechanics, Clarendon Press Oxford 1958. 5) J. E. Roberts, Journal Math. Phys. ]7 , 1097 (1966). Also J. P. Antoine J. Math. Phys. 1£, 53, 2276(1969). 6) A. Bohm, Boulder Lectures in Theoretical Physics, Vol. 9A, 25 5(19 66 ). 7) The explanation of the content of this basic assumption is the subject of this paper. A ll the unknown mathematical notions will be defined or described in the following sec tions .... Appendices KLinear Space), II(Algebra) and III(Operators in Hilbert Space) contain further definitions which are not given in the paper. ! 70 8) The formulation given here is a special case of the construction of Rigged Hilbert Spaces for enveloping algebras of Lie groups which has been suggested in A. Bohm Joum. Math. Phys. 1557(1966), Appendix; and independently! i by B. Nagel, College de France Lectures (1970). A de 16) tailed treatment of e (S U (l,l)), which is slightly more complicated than the example treated here, has been given in 6 . Lindblad, B. Nagel, Ann. Inst. Poincar£ Section A- (N.S.) 13, 27(1970). 17) 9) See Appendix II 10) See Appendix I 11) See Appendix III 18) 12) E. Nelson, Ann. Math. 70, 572(1959). 13) H. D. Doebner Proceedings of the 1966 Istambul Summer Institute. 1*0 In a scalar product space the norm || || is defined by 19) 11*11 = A # ,* ) . IS) The construction of the completion R of the space R consists of the following: The elements of the space R 20) are all possible Cauchy sequences x ={xn>, x^€ R, where two such sequences x = {xR} and y * {yR > are not considered ^ distinct if I |x - y 11 -*■ 0 and the elements x € R are 1 1 n n' 1 identified in R with the sequence { x , x , x , ...... }. The 71 operations with these sequences are defined by ax = (ax }, x + y = (x + y }, (x,y) = lim (x ,y ) n n n n -*• oo n n One can verify that R is then a complete scalar product space, i.e. a Hilbert space That (N+l)^ is e.n.a. can he proved in many ways. It also follows from the fact that (N+l)^ is an elliptic element in the enveloping algebra of a group representation; see ref.20). It is well known that P, Q and aa+ cannot be continuous operators with respect to and are, therefore, not defined on the whole Hilbert space. T* . rt Let •+• $ i.e. ($v - not defined on As A is a continuous operator A(#v -♦) -*• 0. Therefore we can define A on $ by A# = x.-lim9 A*,. V The definition of closed and closable operators is given in Appendix III. P,Q,H are closable because they are symmetric and a symmetric operator admits a closure. E. Nelson, W. F. Stinespring, Amer. Journ. Math. £1, 547, (1 9 5 9 ). The original definition of nuclearity for countably normed spaces (Grothendieck, Gelfand and Kostyuchenko) is: i is nuclear iff for any m there is an n such that 72 the mayuuug ♦ ♦ is nuclear i.e. has the form ti in <0 ♦n B ♦ J where is the completion k=l of f with respect to the norm f{ ||^ and ,{^)are orthonormal systems in the spaces *n and ♦ respectively, > 0 and E 22) J. E. Roberts Commun. Math. Phys. 3, 98(1966). 23) A. A. Kirillov, Dokl. Akad. Nauk. SSSR 130, 966(1960). 2*0 A. Boha, Appendix B of Joum . Math. Phys. 8 , 1557(1967). 25) B. Nagel, Lecture Notes, College de France (1970). 26) For a proof we refer to Ge If and, Shilov, ref.l.) Vol. II, Ch. I , Sect. 5 , 6 and Ge1fand»Vilenkin, ref.l.) Vol. IV, Ch. I, Sect. 3, 1. 27) This statement holds for countably Hilbert spaces i f V t is the strong topology and in nuclear spaces —where strong and weak topology coincide— also for the weak topologx See ref. 25. 28) For a sirtple proof of this statement sen e.g. Liustemik Sobolev, Clemente of Functional Analysis 531 or Akhiezer, Glazman, Theory of Linear Operators in Hilbert Space §»»3. 29) For a proof of the Nuclear Spectral Theorem see ref. 1.) and ref. 2 .) 30) J. Dixnier, Comp. Hath. 13, 263, (19S8) and references thereof. 73 31) See also, P. Kristensen, L. Meljbo, E. Thue Poulsen, Commun. Mnth. I'hys. 175, (1965). 32) G. Lindblad, B. Nagel, Ann. Inst. Poincar€ Sect. A (N.S.) 13^, 27(1970). 33) For the possibility of weakening assumption (1,3) see: L. C. Mejlbo, Math. Scand. 13^, 129(1963); see also H. G. Tillmann, Act. Sci. Math. 2 2 5 8(196 3); C. Foias, L. Geh£r* B. Sz-Nagy, Act. Sci. Math. 21^, 78(1960).