Iterated Commutators and Functions of Operators

8/3/2019 Iterated Commutators and Functions of Operators

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ITERATED COMMUTATORS ANDFUNCTIONS OF OPERATORSby Howard C. VolkinLewis Resedrch CenterCleuehnd, OhioN A T I O N A L A E R O N A U T I C S A N D S PA CE A D M I N I S T R A T I O N W A S H I N G T O N , D . C. O CT O BER 1 9 6 8


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TECH LIBRARY KAFB, NMI11111111111l111Illt111I1Ill

b ITERATED COMMUTATORS AND FUNCTIONS O F OPERATORS

1/ -By Howard C;.V o l k i p--Lewis Research CenterCleveland, Ohio

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ITERATED COMMUTATORS AND FUNCTIONS OF OPERATORS by H o w a r d C. Volkin

L e w i s R e s e a r ch C e n t e r S U M M A R Y

Certain commutation properties of two operators A and B a r e represe nted by polynomials in A and B called iterated commutators. A method is developed for systematically expres sing functions of A and B in ter ms of the iterated commutators. Themethod is related to the ordere d-opera tor calculus of Feynman. Compact express ionsresult because al l iterate d commutators of th e sam e degree always appear in the s ameline ar combination. The proced ure is designed prima rily fo r polynomials, exponentialfunctions, and analyt ic functions of th e op era to r (A + B). Applications to the import antca se of exponential functions a r e given.

INTRODUCTION

Functions of noncommuting ope rato rs have been studied extensively ove r the ye ar s ,usually within the context of pro blem s ari si ng in mathem atic s and quantum physi cs. Asurvey of res ul ts, along with numerous ref ere nce s, has been given recent ly by Wilcox(ref. 1). One approac h in working with such a function is to express i t as an expansion insucces sively higher commut ators of the ope rato rs involved. The purpose of this rep ortis to present a new method by which explicit fo rm s of the de si re d type ca n be developed ina systematic and efficient w a y fo r cert ain functions of inte res t.

Th is method is rela ted to the ordered -opera tor calculus of Feynman (ref. 2). InFeyman's op erator calculus, an ordering param eter indicates the sequence in which noncommuting quantities appear ing an any product ar e to be taken. An operat or A, whichdoes not othe rwise depend on the orde rin g pa ram ete r s , may be written As, with theord erin g index attached as a subscript . Then it is understood tha t the fo rm of As doesnot depend on s , but its or de r of operation does and is defined by th e convention: Theoperato r with smalle st index operates first, and so fo rth . Th us , AsBs, equa ls BA whens' > s , equals AB when s > s ' , and is undefined when s = s ' . In the more generalcas e of an operato r that is a function of the time, the convention is that t ime is the


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ordering parameter. We can use the notation A(s) to include eith er case .With the orderi ng par am et er convention, any functional of th e ope rato r functions

A(s), B(s), . . . has a unique interpr etatio n, becau se the ord er of ope rations in any situation is automatically specified by the pa ram eter values. In manipulating such a functional, the algebra and calculus of ordinary numbers may b e applied, as though A(s),B(s), . . . we re ordina ry commuting functions. ttDisentangling" is th e name given byFeynman to the proc es s of finding an opera tor expre ssion for th e functional in the usualpositional notation, where the order of operation is rep rese nte d by t he position in whichthe operators are written. Simple examples of disentangling a r e th e following:

exp(A1 + Bo) = exp(A) exp(B) (1 )

An example of a full functional is the well-known Dyson for mu la for t he Neumann-Liouville expansion of th e uni tar y time development trans for mat ion . The transformationis an exponential functional of a time-dependent Hamiltonian ope rato r function. Theordering is usually exhibited explicitly by th e chronological opera tor , which may b eomitte d when the Feynm an notation is adopted.

The Feynman formal ism per mits all the re su lts of ordinary analysis to be used inhandling ope rat or functionals. However, no genera l proced ure for disentangling functionals is available. Moreover, th er e does not appea r to be any di rec t way of exploitingany simple commutation properties that the oper ator s may happen to possess. Themethod presented he re dea ls with this second aspect . Certai n commutation propert ies oftwo ope rato rs A and B are repr ese nte d by polynomials in A and B called ite rate dcommu tators. The proced ure given converts polynomials, exponential functions, andana lyt ic funct ions of (A + B) into equivalent fo rm s in te r m s of the iterat ed commutators.A ll iter ate d comm utators of deg ree (j + 1) in A and B a r e combined in a suitable sumto form a new operator G di1- By working only with the G (j) and their com mutators, thecalculations are simplified and the r es ult s can be given in an ex tremel y compact notation.

In the next section, the iterate d commutators are defined and thei r relevant proper t ies ar e g iven. In th e s ect ion EXPANSION IN ITERATED COMMUTATORS, relationsa r e developed that convert a suitable function of two op er ato rs into the des ire d form.Applications to the important case of exponential ope rato rs a r e given in the sectionEXPONENTIAL FUNCTIONS. As the derivations required a r e generally quite straightforward , complete details a r e given only where deemed necessa ry and a suitable methodis indicated in all other cases.

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SYMBOLS operatoropera tor with order ing index sopera tor which may depend on ordering par ame ter s operator operator in eq. (29) operator in Zassenhaus formula op erator in expansion of C analytic function of xlin ear combination of it era ted commut ators operator in BCH formula operator in expans ion of H A + B F[X(A + B)]-lFIA(A1 + Bo)] ope rat or in expansion of P F [ W + B)] - F [ m 1 + Bo)] ordering p arameter [A, . . . [A, [A, B]] . . .1, n commutations [. . . [[A, B], B] . . .,B], n commutations

ITERATED COMMUTATORSThe simple iterated commutator is

{ (A)", (B) I = [A, . . . [A, [A, B]] . . .]


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which defines a linear operation on B obtained by n su cc es si ve commutations with theoperator A. The for m of equation (3 ) is a polynomial whose te rm s each have n factorsof A and one of B . Simi lar ly ,

The case n = 0 is specifie d by

It is seen that

Clearly, the iterated commutators (eqs. (3 ) and (4)) arise when two operators, oneof which is ra ised to a power, are transposed. The following f our t heor ems are established by induction:

n[An, B] = () @)}An-'

r= l

r=1

A fami liar example in which thes e iterate d commutators occur is the Lie expansion,which is the power series

exp(hA) B exp(-XA) =

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There is no unique way to prescribe a commutator {(A)", (B)" ] that reduces tothe forms in equations (3 ) and (4 ) for the special cases n = 1 and m = 1, respectively.Various gener alizat ions suggest thems elves. The extended definition of iter ate d commuator is chosen to be

In the "L-type" of extended com mutato r (eq. (9)), the simp le commutator (eq. (3)) withthe m-fold iter ation of the op era tor A (the opera tor appearing in the left-hand position)is formed first, followed by the remaining (n-1) iter ations of the other ope ra to r, B. AnR-type commutator, fo r which the prescription is reversed, is easily se en from equation (5) to sat isfy the rela tion

The iterated commutators (eqs. (9) and (10)) consist of a s u m of products each having mfactors of A and n fa ct or s of B; they desc ribe the commutation of a power of oneoperat or with a power of anoth er operator . From equations (6), (7), and (8) followdirectly the reduction formula s

n m

n m

Equations (7a) and (12), for example, convert any polynomial in A and B into a sumof te rms, each of which is a product of i te ra ted commut ators and an or de re d productm nA B .

B A is the differentiation operat or with re spe ct to a par ame ter on which the ope ra tor B depends, then equation (3) becom es the nth der iva tive of B, that is ,

dnBdtn

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and equation (9) de sc rib es the commutation pro pe rt ie s of the deriv ative s with the powers of B

Employing in tur n th e i denti ties [A, BIT = -[A t tB ] and equation (10) giv es th e followingfo r the H ermitian adjoint:

{ (A)m, (B)n}Lt= (-l)m+n-l{ (At)m, (Bt)n}L (15

EXPA NSION IN ITERATED COM MUTATO RSFirst so me useful disentangling exp res sio ns involving the quantities (A1 + Bo)n wil l

be developed. Th es e "order ed powers" of (A + B) are simplyn+ B ~ ) ~A ~ =C () A ~ - ~ B ~

r=O

that is , they a r e obtained as ord ina ry binomial expansions in which A1 and Bo aret reated as though they commuted. When each t e r m is writt en in the us ual positional convention, powers of A appea r always to the left of powers of B. Multiplying both s idesof equation (17) fr om th e le ft by (A + B) gives

n(A2 + B2)(A1 + Bo)n = (A1 + - (r!l)p+l-r,B]Br-l

r = l

where, fo r convenience, both notations have been used on the righ t. After using, in su ccession, equations (7b), ( 6 ) , (7a), and ( 9 ) , the second te rm on the right-hand side ofequation (18) becomes

. r= l t= l p=O6


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The substitution? j = p + t , k = t - 1 and 1 = n + 1 - r - t sep ara te the summation intoindependent sums over 1 and k. The res ult allows equation (18) to be written in the form

wherei-1

k=OThe operator G( j is a homogeneous polynomial of degree ( j + 1) in the opera tors Aand B. The subscript j is brack eted to signify that it has no ordering significance. Itis found, for example, tha t

On the right-hand si de of equation (19), all the iterated co mmutators (eq. (9)) whoseindices m and n add up to the sam e value appea r in the single ope rat or (eq. (20)) forwhich j = m + n - 1 and in none of the oth er s. If only a finit e number of it er at ed commutato rs a r e nonvanishing, then all G with values of j larg er than a certai n value( jwill vanish. As an illustration, consider the case where A is proportional to p2 , B isa polynomial in x of degree I , and [p,x] = c , an ord ina ry numbe r. With the help of theidentity

m[pm,x2] =Ey )Z ( L - 1). . .( L - r + 1)crxL-r pm - r

r= lit is found that

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where qr(x) is a polynomial in x of degree 2 - 2m + r .2 mThe iterated commutatorvanishes if m > 2 . On th e other hand, when m 5 2, { (p ) , (x2 )j-m+l }L vanishesif j 2 2m + 1. Hence, when j 2 22 + 1, each left commutator (eq. (9)) appearing in G(j)vanishes and GU) = 0.

With the help of t he r ec urs io n re lati on (eq. (19)), th e following disentangling form ulacan be verifi ed dir ect ly by induction:

(A1 + Bo)n - (A + B)"n- 1 n-2

= (1") + (r;2) ('f')'A +B)n-2-rG(i)G(j)r ; l ) (~+ B ) " - ~ - ~ Gr= l r=2 i+j=r

+x-3 r+2 h+i+lr= 3 h+i+j=r (rr3) ( j )( i ) ( A B)n-3-rG(h)G(i)G(j)

The number N of summation te rm s on the right equals the larg est in teger that is l e s sthan or equal to (1/2)n. The fin al te rm SN is (n - l)(n - 3 ) . . . (5)(3)[G(11] (1/2)n ,when n is even, and is the sum term for r equal to (1/2) (n f l ) , when n is odd. Adual fo rm of equation (21) holds, with or de re d powers of (A + B) appearing on the rightin place of ordinary powers:

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(A1 + B O P - (A + B)"

n-3+ >: (r:3)r;2)r+i+1) (.I)(Al+Bo)n-3-rG . G . G(h)+. . . + (-l)N-lSN (22)(1)r= 3 h+i+j=rThe form of the dual expression (eq. (22) ) is similar to equation ( 21 ) except that (1) th eor de r of the sub scri pts on the G(j 1 factors is reve rsed and (2) the sign on successivesum ter ms alternate s, beginning with a plus sign on the first t e rm.

The or de r of t he opera tor f ac to rs in each te rm of equation (21) can be r eve rse d bytaking th e Herm itia n adjoint of each side. The Herm itian adjoint of a functional that iswritten in the Feynman notation is gotten by replacing every o pera tor A(s) by A? (s) andexactly reversing the orde ring convention on the p ara met ers of the adjoint operato rs.~-F o r a simple function of ope rato rs, the resu lt is clearly F(A1, Bo)? z F ( A & , BI). Equa

6) s obtained from the form (20) by substituting AT, B t fo r A, Bion (15) shows that G?.respectively, and inserting an overall factor of ( - l ) J .If A and B are Hermitian,then GL ) = (-l$GG).

The application of equat ions ( 21 ) and (22) to an alytic functions of (A + B) is straightforward . Let the analytic function be

F(x) =>: fnxn (23)n=O n!

In the formulaF[IX(A + B)] = FIIX(A1 + Bo)] + R (24)

the ter m R has the series expansion

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Equation (22) give s the o per ato r R(n) direc tly in te r ms of the constant fn , orderedpowe rs of (A + B), and th e Gti ) * An alternative formulation is the disentangling form ulaF [ A ( A ~+ B ~ ) ]= F[A(A + B)]P (26)

where

Equating th e coeff icien ts of li ke powers of h in equation (26), with th e help of equation (21), yields the operator s P(n). The first f ew t e rm s a r e

A dual fo rm of equation (26) can als o be developed, namely,

where P-' is expre ssed in te r m s of o rde red powers of (A + B). Othe r expansionmethods f o r analytic functions of (A + B) and refere nces to earlier work in this a re aa r e given by Kumar (ref. 3).

EXPONENTIAL FUNCTIONSThe opera tors P(4 in equation (27) tak e a particularly simple form when F is theexponential function, in which ca se f n = 1 fo r all valu es of n. Thi s leads to a compact

state ment of the theoremexp(hA)exp(hB) = exph(A + B)exp(C) (29)

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where

If equation (29) is written in the f orm

the coefficient of (An/n! ) in the power s er ie s expansion of t he left-hand s ide is given byequation (21) . The corresponding coefficient on the right is

1

h+i+j+3 i+j+22 >: $(r:m)k?). * ( h+l ) ( i+ l )m = l r=m a+b+. ..+i+j=r

x ( A + B )n-r -m E(a)E (b) - * . E(h)E(i)E(j) (32)In the mth product te rm of the previou s sum (eq. (32)), the re appear m fac to rs of E ( j1and m binomial coefficients. The EU) a r e found by equating coeffici ents of the s a m epower of h on each si de of equation (31). The re su lt s through ninth or de r in h a r e

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Note that each of the t er ms occurring in E (j ) is essent ially predetermined towithin a numeric al coefficient. Consider any single t er m in the sum of terms thatmakes up the complete expression for a given E(j) If the su bscript on every G(i)appearing in the term has its numeri cal value inc rea sed by unity, then the number sso obtained add up to ( j + l ) , he degree of the iterated commutators in G(j ) *The for mula of Zassen haus (ref. 4 )

expb(A + B g = exp(XA)exp(AB)expe;;($3xpt$). . . (33)cor resp ond s to the complete factor izatio n of exp(-C) in the in ve rse for m of equation (29).With the he lp of equat ion (22), the operators C c i ) are obtained by equating coefficientsof l ike powers of A . Again, the res ult s out to the ninth or der in X are

Because of the Jacobi and other identities, the exp res sion s given fo r the oper ato rsE(j) and C (j ) are not alto gether unique. However, they display concisely the iter ated-commutator natu re of the te rm s in the expansions of equations (30) and ( 3 3 ) . No attempthas been made to obtain gen era l exp res sion s. Never theless , fo r certain types of ter msthe gene ral form of the nu meri cal coefficients can be i nfer red fro m the specific valuesalready given. Fo r example, C ( j ) contains the te rm ( - 1 / 2 ) j ( j - l ) [G(l) ,G(j-2) 1.This proced ure may als o be applied to the Baker-Campbell-Hausdorff (BCH) formula

exp(XA)exp(XB) = exp(H), (34)12


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where

H = AHo + ($)H(l) + ($)H(2) + . . . (3 5)H o = A + B

In the case of the BCH form ula, the method offers no gr eat advantage over previousmethods (refs. 4 and 5) that have been developed specifi cally fo r this proble m. However,the H(j) can be computed within th e frame wor k of the proc edu re given her ein withoutintroducing additional el ements (such as operator derivatives o r auxiliary opera tor functions) that the mo re speci aliz ed methods employ. Equating the coefficient of (hn/n!) inthe expansions of [exp(hH) - exp(hHo)l to the exp ressi on given by equation ( 21 ) yields

H(l)G(1)

When express ed in te rm s of the G ( j ) , the H(j) contain considerably more te rm sthan either the EU ) o r the C ( j ) . The additional terms are the ones containing Ho.Moreover, the relations

and

'No such simple relations exist for higher G(j), however.13


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can be used to reduce the expressions f or the H( j ) . For example, the following can bewritten:

There exist any number of inter mediat e for mul as between the BCH for m (eq. (34))and equation (29), all of which can be wri tten as

with equation (29) by n = 1 and equation (33) by the limit n - 03. The operat ors on theright-hand side of equation (35) can be evaluated by the pr oc edur e given her ein . Inparticular, the operat ors in the first exponential on the right a re those of the BCHformula, as follows dir ectly fro m the fact that the righ t-hand-sid e expansions of equations (34) and (36) are the sa me up to the nth power of A.Lewis Researc h Cente r,

National Aeronauti cs and Space Admin istra tion,Cleveland, Ohio, August 21, 1968,

129-02-07-07-22.

REFERENCES1. Wilcox, R. M. : Exponential Operators and Parameter Differentiation in Quantum

Physics. J. Math. P hys ., vol. 8 , no. 4, Ap r. 1967, pp. 962-982.2. Feynman, Richard P. : An Operat or Calcu lus Having Applications i n Quantum Elec

trody namic s. Phys. Rev ., vol. 84, no. 1, Oct. 1, 1951, pp. 108-128.3. Kum ar, Kailash: Expansion of a Function of Noncommuting Operators. J . Math.

Ph ys ., vol. 6 , no. 12 , Dec. 1965, pp. 1923-1927.

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4. Magnus, Wilhelm: On he Exponential Solution of Differential Equations fo r a LinearOperator. Comm. Pur e Appl. Math., vol. 7, no. 4,Nov. 1954, pp. 649-673.

5. Weiss, G. H. ;and Maradudin, A. A. : The Baker-Hausdorff Fo rm ula and a Problemin Crystal Physics. J . Math. Phys., vol. 3, no. 4, July-Aug. 1962, pp. 771-777.

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Iterated Commutators and Functions of Operators

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