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VOLUME 2 APRIL 1984 52 Quantum NUMBER 14 Mechanical Path Integrals with Wiener Measures for all Polynomial Hamiltonians John R. Klauder A Tcf T Bell Laboratories, Murray Hill, New Jersey 07974 and Ingrid Daubechies Theoretische Natuurkunde, Vrije Universiteit Brussel, B-1050 Brussels, Belgium (Received 8 December 1983) We construct arbitrary matrix elements of the quantum evolution operator for a wide class of self-adjoint canonical Hamiltonians, including those which are polynomial in the Heisenberg operators, as the limit of well defined path integrals involving Wiener measure on phase space, as the diffusion constant diverges. an arbitrary spin Hamiltonian. PACS numbers: A related construction 03.65.Ca Path integrals for evolution operators of quantum mechanical systems are almost always defined as the limits of expressions involving finitely many integrals. ' Efforts to define them as integrals involving genuine measures on path spaces of continuous paths, or as limits of such integrals, have been largely unrewarding. 2 In our earlier work on this subject we have succeeded in establishing quantum mechanical path integrals with genuine measures for the limited class of quadratic Hamiltonians. In this paper, taking an alternative but closely related approach, we succeed in constructing arbitrary matrix elements of the quantum evolution operator as limits of well defined path integrals involving Wiener measure on phase space as the diffusion constant diverges. Our construction works for any self-adjoint Hamiltonian of a wide but special class (defined below) which includes all Hamiltonians polynomial in the canonical (Heisenberg) operators. leads to an analogous A similar construction description of arbitrary matrix elements of the quantum evolution operator for an arbitrary Hamiltonian composed of spin operators for any fixed As above, spin s 0, extending earlier work. these matrix elements are defined as limits of well defined path integrals involving Wiener measure ) achieves a similar result for defined here on the unit sphere, again as the diffusion constant diverges. Finally we comment on how the spin path-integral expression passes to the canonical one as s We content ourselves here with a statement of our principal results, reserving a precise formulation and detailed proofs to a separate article. For clarity we confine our discussion to a single degree of freedom. The notation is that of our earlier papers. For all . (p, q) E R2 let Canonical case ~. — = exp[i(pQ —qP)](0), q) — [Q, P]=i, denote the canonical coherent states, where i0) is a normalized vector that satisfies (Q+iP) ~0) =0. It follows that a general operator H can be expressed ~p, as H = JIh (p, q) ~p, q) (p, q ~ dp dq/27r, where (pq) = exp[ —(i) /Bp + r)'/r)q )/2] (p qlH~pq). For H the unit operator I, this expression yields h = 1; for H a polynomial in P and Q, it follows that h is a polynomial in p and q. The special class of h Hamiltonians we are able to discuss includes all those for which h (p, q) is polynomially bounded. We suppose hereafter that H denotes the self- 1984 The American Physical Society PHYSICAL REVIEW LETTERS VOLUME 52, NUMBER 14 of a harmonic oscillator of unit frequency; clearly — it follows Ip, q, 0) = Ip, q). For any (8 with lpl adjoint Hamiltonian of interest, and thus that h is real. Now we introduce additional canonical coherent states —qP)]In), n =0, 1, 2, . . . , — Ip, q, n) = exp[i(po where In) denotes the normalized (1 that lp, IPI"I„, I = X q)) — P""lp.q. n) n-0 defines a vector in an associated direct-sum Hilbert space . We define four operators in P'as follows: nth excited state Ep= J Ipq)) ((pqldp dq/27r= X n-0 — Cp= Jtexp[ i eh (p, q)] lp, q) ) ((p, q 2 APR&L 1984 4 A = X — 8 n-0 nl„, Bp= Jth(pq)lpq)) ((pq I dpdq/27r, dp dq/2vr. In arriving at the second form for Ep we have used the basic fact that I & = (m p q n ) (p q m dp dq/2m I n I ) I„=o „I„, where In denotes the unit operator in the nth direct-sum subspace. Observe that E] is the identity operator while P = Eo is the projection onto the 0th subspace P. A real h implies that Bp is a symmetric operain tor in A. Now choose the following parameters: N is a variable positive integer, T a fixed positive time interval, e = T/(N+1), and (8= (1 —ev/2)/(1+ ev/2), v fixed and positive (v is the diffusion constant, as will become apparent). With these identifications, and for polynomially bounded h, we are able to prove5 s ("s" means strong) Lemma 4, I: s-limCg Q~ oo =exp( —v TA —i TBl). under the same conditions, we can prove Lemma Furthermore, s-lim p~ lqt&), ((t& where n)0. lp) 'TH le Iqt&) ((p ), q H. Consequently, is just the self-adjoint Hamiltonian element of the evolution operator is given by PBiP restricted to the 0th subspace E 4, the matrix I(c() I(ci) = lim llm (((t& ICg l(ts) Icq Ip, q"', where (p', q') »"= I = (po, qo) &P2. q2lpl. ), ql) ) = and l(t&), I(C(), expression for J(IIl((ui+s qs+sIlui With the parameters exP' 6V i 2 (Plq2 I qlP2) entries, chosen as above it follows qi» fI{expf —saq (p, , q, &&dp, dqi/2'), 1 (p", q") = (p~+ l, q~+ l). We observe 1+ ev/2 and zero in all remaining respectively, (3). 0 I for (3) ) CA are vectors with 0th entry We now proceed to give a path-integral that & (2) oo Observe that any 2: exp ( —v TA —i TB l ) = P exp ( —i TPB l P) P. — 1 26v that3" [(P2 Pl) + (q2 'ql) ] ' s The form of this expression makes evident the result of the limit N lim ((p",q" ICtl Ip', q') ~ (e 0) as T ) =2me" i Jtexp[i Jt [—, (pq —qp) —h(pq)]dt]dp(v(pq), where p, ~ is a product of two pinned Wiener measures concentrated on continuous paths with a normalized —t2/T) for x =p or q; here the role of v as connected covariance given for tl t2 by (x(tl)x(t2))'= vtl(1 diffusion constant is apparent. Note that f(pq —qp)dt interpreted as f(p dq —q dp) involves well-defined stochastic integrals in any9 (Ito or Stratonovich) sense. Finally, we obtain the following Theorem: ~ s (@Ie 1162 'THI(id) = lim Jt(QIP", q") [2me' t J exp(iS)d p, &v](p', q'I(id) dp" dq" dp' dq' PHYSICAL REVIEW LETTERS VOLUME 52, NUMBER 14 S denotes the "classical action, [ —, (pq —qp) —h (p, q) )dt. where 2 APRIL 1984 " fOT This result achieves our stated goal of a path-integral representation. It is important to compare our results here with those obtained earlier. In Ref. 3 we were able to find an expression for quadratic Hamiltonians which involves the function H(p, q) = (p, q lHlp, q) as the Hamiltonidrift an function in S (rather than h), at the expense of introducing Wiener measures with nonvanishing Thus, at least for the limited set of Hamilterms determined by the usual Hamilton equations of motion. tonians in common, we gain a clearer understanding of the true underlying distinction between two formal but otherwise identical path-integral expressions given by' M)texp(iS) II, dp (t) dq (t), ' 0 where S = [ —, (pq —qp) —h (p, q)]dt or S = [ —, (pq —qp) —H(p, q)]dt. If it H, (p, q), where H, (p, q) denotes the usual classical Hamiltonian. H, (p, q), H(p, q) f f recall that h (p, q) This fact explains how yet another expression, involving Wiener measures without drift and the function H(p, q) representing the Hamiltonian, can be valid insofar as the leading term of the stationary-phase approximation is concerned, (ir ) ]. since in that approximation both H and h are equivalent to H, [neglecting terms The rigorous path-integral definition described in this paper enables variable transformations (e.g. , canoniSuch a poscal transformations) to be examined much more critically than in the usual formal formulation. sibility provides just one motivation for our seeking to define quantum mechanical path integrals in terms of genuine measures on continuous paths. Fix s 0, and for Spin case. With regard to a path integral for spin s we can proceed analogously. S'let e (e, y) " 0 ) — le, = exp( —i QS3) exp( —i eS2) ls, ) 4) — spin-coherent denote normalized 4, E states, where S3ls, ) = s ls, ), and S H =N, Jfh (e, y) le, y) (e, @ld 0, where N, = (2s + 1)/4m, d 0 = sine de dp, represents any operator His expressed' in terms of the usual spherical harmonics YI~ by h(e. $) = X X ' 1't 2 = s (s + 1)I,. in A s It follows that Here the relation between h and (e, P) J"&( (e', P')(e', Q'lHle', P')dQ'. = 1; while for any operator H, h is well defined. Hereafter we asEvidently for H the identity operator sume that H is the self-adjoint spin-operator Hamiltonian of interest, and thus h is real. Next we introduce additional normalized spin-coherent states I„h — &j ) = exp(i —@S3)exp( —i eS2) ) appropriate to spin jand magnetic quantum number le, l ( any P with lPl le, y)) = I 1 it $g j m, where (2/+2s+1)' '+ 2p' '+' ey))(«, @l«& 4/~) 4, . Four operators Ipl""""'"I/„, = X I &p= Jth(e, y)le, ); clearly le, qb, s, ) = le, $). For le @,i+s, ) 0 defines a vector in an associated direct-sum Hilbert space pE=„"I S3lj ) = m lj follows that & = Xe I 0 y))((e, yl(dn/4~), are defined on M, as follows: 'i(i+2s+1)I„„ —, 0 cp= Jf exp[ ieh(e, y)]le,— y))((e, yl(dn/4~) Here we have used the fact that (2/+I) for all j l8, 4j / (H. p j' l(dfl/4m) = B,IJ. j, j ~ m. P = 1 —~p, v We choose Na variable positive integer, Ta fixed positive time integral, e= T/(N+ 1), and fixed and positive. Then again we are able to prove, mutatis mutandis, (1) and (2), with (3) as a 1163 PHYSICAL REVIEW LETTERS VOLUME 52, NUMBER 14 As for the path-integral consequence. where, for 0 & e expression it follows that N «e", p" Icrle', p'11=„f — &,,)1 fIexp& iah&e, , g, )I&dB,/4m), J(II«e, +,, &, +, le, 0 p « 0 (e 1, and up to 2 APRIL 1984 ) terms, p 1 . ((8,, F2~8, , y, ) ) = X(21+2s+ I) [I , —v—i(I+2s+1)](8,, y2, I+s, lH, , y, , I+s, ) I 0 00 = (I+s ev)(82, $2~81, $1) I x x (47r)[l —, e—vl(I+I)]Yt~(82,$2) Yt~(81, $1). I-Om - — I Note here that (82, $2~81, $1) = cos 81 82 cos 42 82+ 41, +i cos ' 81 . 42 sin 2$ &tel (4) 2 while oo (e'" )(82, &t2', 81, $1) = X X exp[ —, tvl(—t+1)]Yt~(82, $2) Yt~(81, &t'1), — I Om I 5 the Laplacian on the unit sphere, is the Markov transition sphere with diffusion constant I . Consequently, with ((8", $" ~CPH', $')) =4me'" lim fOT J exp[i element [s cosHQ —h (8, P)]dt]d p for Brownian motion on the "tvH, Q). (6) Here p, ~ denotes a pinned Wiener measure on the unit sphere with diffusion constant v and weight given by (5) for t = T. To obtain (6) it is necessary to expand (4) to second-order differentials and use an appropriate form of the Ito calculus. Here cos8$ dt = cosH dg represents a well defined stochastic integral (in any sense). Finally we observe that f f (@le with ""ly) = T f [s path-integral S= 11m cosH& jl (ylH", @")(&,e'"Ti2Jte"di ~) «', y'ly) —h (8, g) ]dt, represents the desired expression. In the spin case remarks entirely similar to those of the canonical case apply to an alternative pathintegral definition in which Brownian motion on the sphere in the presence of drift and alternative expressions for the classical Hamiltonians arise. See Ref. 4. " Lastly we remark that if we rescale v in the spin case to v/s, set p = s' cosH, q = s' $ ( —m. ~, then it m), and formally take the limit s follows that the spin path integral becomes the canonical path integral (modulo a trivial phase change). It is a pleasure for us to thank Professor Ludwig fur at the Zentrum Streit for his hospitality Federal Interdisziplinare Bielefeld, Forschung, Republic of Germany, where this work was carried out. One of us (I.D.) acknowledges appointment as a Wetenschappelijk Medewerker at the Interuniversitair Instituut voor Kernwetenschappen, Belgium. ~ «t 1See, e.g. , E. Nelson, J. Math. Phys. (N. Y.) 5, 332 (1964). I. M. Gel'fand and A. M. Yaglom, J. Math. Phys. 1164 I d&" dfI', (N. Y.) 1, 48 (1960); R. H. Cameron, J. Analyse Math. 10, 287 (1962/1963). 'J. R. Klauder and I. Daubechies, Phys. Rev. Lett. 48, 117 (1982). 3 I. Daubechies and J. R. Klauder, J. Math. Phys. (N. Y.) 23, 1806 (1982). 'I. Daubechies and J. R. Klauder, Lett. Math. Phys. 7, 229 (1983). "J. R. Klauder, J. Math. Phys. (N. Y.) 23, 1797 (1982). 5I. Daubechies and J. R. Klauder, to be published. 6J. R. Klauder and E. C. G. Sudarshan Fundamentals of Optics (Benjamin, New York, 1968), Chaps. 7 Quantum and 8. 7The normalization of the vectors ~p, q) ) differs by a factor (1 —~P ) 'i' from that in Ref. 3b. 80ne can prove that vA +iBI, ~here 8]=lim& IBp, generates a strongly continuous contraction semigroup which we denote by exp [ —v TA —i TB1]. 9See, e.g. , K. Ito, in Mathematical Problems in Theoretical Physics, edited by H. Araki (Springer-Verlag, New York, 1975), p. 218. Compare, in this regard, R. Shankar, Phys. Rev. Lett. 45, 1088 (1980). 'J. R. Klauder, in Path Integrals, edited G. J. Papadopoulos and J. T. Devreese (Plenum, New York, 1978), p. 5, and Phys. Rev. D 19, 2349 (1979). ' See, e.g. , F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972). ~