Extension of the Barut-Girardello Coherent State and Path Integral

Extension of the Barut-Girardello Coherent State and Path Integral Kazuyuki FUJII and Kunio FUNAHASHI Department of Mathematics, Yokohama City University, Yokohama 36, Japan April, 997 Abstract We extend the Barut-Girardello coherent state for the representation of SU, to the coherent state for a representation of UN, and construct the measure. We also construct a path integral formula for some Hamiltonian. e-mail address : fujii@yokohama-cu.ac.jp e-mail address : funahasi@yokohama-cu.ac.jp

Introduction Harmonic oscillator is a fundamental physical system and one of the few which is solved exactly. So it has been well studied not only for the system itself but also for application to other physical systems. The coherent state is defined as the eigenstate of the annihilation operator []. It is a useful tool for study of the harmonic oscillator. The properties of the coherent state are also well studied. Extension of the coherent state to various systems has been made. As for SU, group, the Perelomov s generalized coherent state is well-known []. The treatment is very easy. Extension of the generalized coherent state to a representation of UN, has been made [3]. Making use of the coherent state, we have shown the WKB-exactness [4, 3], which means that the WKB approximation gives the exact result. Also we have extended the periodic coherent state [5, 6] to the multi-periodic coherent state of a representation of UN, and have shown the WKB-exactness [7, 8]. There is another coherent state of SU, which is known as the Barut-Girardello BG coherent state [9]. The BG coherent state is defined as the eigenstate of the lowering operator. The BG coherent state has a charasteristic feature. The eigenvalue K of the Casimir operator is valid for K> [], in sipite of the range of the representation of SU, is K / []. Extension of the BG coherent state has been made to some groups []. In this paper we extend the BG coherent state to a representation of UN, to examine the WKBexactness and et al. The contents of this paper are as follows. In Section we compare the coherent states of the harmonic oscillator, the Perelomov s coherent state and the BG coherent state. We extend the BG coherent state and construct the measure in Section 3. In Section 4 we construct a path integral formula in terms of the coherent state constructed in Section 3. The last section is devoted to the discussions.

Coherent States The coherent state of the harmonic oscillator is defined as the eigenstate of the annihilation operator such that a z = z z, z C,. where aa is the annihilation creation operator. Alternative expression of the coherent state is z = e za.. In the harmonic oscillator, both are equivalent. The explicit form is z = n= z n n! n,.3 in the standard notation. The conditions of the coherent state in the Klauder s sense [3] are as follows:. continuity: the vector l is a strongly continuous function of the label l,. completeness resolution of unity: I = l l dl..4 Now consider the extension to the representation of SU,. su, algebra is [K 3,K ± ]=±K ±, [K,K + ]=K 3, K ± =±K ±ik,.5 and the representation is { K, m m =,,, },K,K is an eigenvalue of the Casimir operator,.6 which satisfies K 3 K, m =K+m K, m, K + K, m = m + K + m K, m +, K K, m = mk + m K, m..7 3

The extension of. is known as the Perelomov s generalized coherent state, whose form is where ξ e ξk + K, = m= / K + m ξ m K, m,ξ D,,.8 m D, = {ξ C ξ < }..9 The inner product is and the resolution of unity is ξ ξ = ξ ξ K,. K π D, dξ dξ ξ K+ ξ ξ = K,. where dξ dξ dreξdimξ.. On the other hand, the extension of. is known as the Barut-Girardello coherent state which satisfies The explicit form apart from the normalization factor is where The inner product is z = n= K z = z z..3 z n n!kn K, n,z C,.4 a n a a + a+n..5 z z =ΓKz z K+ I K z z,.6 where Γp is the Gamma function: Γp = dte t t p,.7 4

and I ν z is the first kind modified Bessel function: z ν I ν z = z/ n n!γν + n +..8 The resolution of unity is n= dµz, z z z = K, dµz, z = K K z z K dz dz..9 πγk In contrast with the coherent state of the harmonic oscillator,.8 and.4 are not equivalent. Especially it is remarkable that.9 holds for K>, while. holds for K /. 3 Extension of Barut-Girardello Coherent State In this section we construct the BG type coherent state for some UN, representation and its measure. un, algebra is defined by [E αβ,e γδ ]=η βγ E αδ η δα E γβ, η αβ =diag,,,, α, β, γ, δ =,,N +, 3. with a subsidiary condition E αα + E N+,N+ = K,K=N,N +,. 3. α= We identify these generators with creation and annihilation operators of harmonic oscillators: where a, a satisfy E αβ = a αa β, E α,n+ = a αa N+, E N+,α = a N+ a α, E N+,N+ = a N+ a N+ +, 3.3 [a α,a β ]=, [a α,a β ]=[a α,a β ]=, α, β =,,,N +. 3.4 5

The Fock space is n,,n N+ { n,,n N+ n,n,,n N+ =,,, }, n a nn+ N+,,,, n! n N+! On the representation space it is K n,,n N,K + {n}= where an abbreviation has been used. a a α,,, =, 3.5 {n}= n α n,,n N,K + n α, 3.6 α= α= n = n = We put the form of the coherent state as n N =, 3.7 z C n n N zz n z n N N n,,n N,K + n α, 3.8 {n}= α= to determine the coefficients C n n N z s so as to satisfy the condition: By noting E N+,α z = z α z, α =,...,N. 3.9 a α n,...,n α,...,n N+ = n α n,...,n α,...,n N+, 3. the explicit form of the left-hand side of 3.9 is α =,...,N +, E N+,α z = C n n N zz n znα α zn N N n = n α= n N = n α K + n β n,...,n α,...,n N,K + n β β= β= = z α C n...n α+...n N z n α +K+ n β {n}= β= z n...znα α...zn N N n,...,n α,...,n N,K + 6 n β, 3. β=

where a shift n α n α has been made in the second equality. Thus 3.9 leads to a recursion relation: C n...n α+...n N z n α +K+ n β = C n...n N z. 3. This is easily solved to become K+ Nβ= n β n α! C n...n α...n N z = n α! K + N β= n β C n...n N z. 3.3! Application of 3.3 to all α s α =,...,Nleadsto K! C n...n N z = n!...n N! K+ N β= n β! C...z. 3.4 Putting 3.4 into 3.8, we obtain the form of the coherent state: K! z = Cz n!...n N! K+ N β= n β! zn...zn N N n,...,n N,K + {n}= β= n α, α= 3.5 wherewehavewrittenc... z ascz. Cz is a normalization factor and affects no physical quantity, so hereafter we simply put Cz =. Especially, in the N =case with K K, 3.5 becomes z = K! n= n!k + n! zn K, n, 3.6 which coincides with.4, where n, K +n has been identified with K, n in the representation of SU,,.6. The inner product of the coherent states is z z = where {n}= K! n...n N! K+ N α= n α! z z n z Nz N n N =F N K;z αz α, 3.7 F N K; z F N K; z,...,z N K! n!...n N! K+ N α= n α! z n...z N n N. 3.8 {n}= 7

In the N = case 3.8 becomes F K; z = F K; z =ΓKz K+ IK z, 3.9 where F K; z is the hypergeometric function and I ν z is the first kind modified Bessel function. Now we construct the measure so as to satisfy the resolution of unity: dµ z, z z z = K. 3. The explicit form of the left-hand side of 3. is dµz, z K! z z = {n}= {m}= n!...n N! K+ N α= n α! K! m!...m N! K+ N α= m α! {n} {m} dµ z, z z n zn N N z m zn m N, 3. where dµ z, z σ z, z [dz dz], N [dz dz] drezdimz, 3. α= and an abbreviation has been used. We put {n} n,,n N,K + n α, 3.3 α= z α = r α e iθα, α =,...,N, 3.4 and assume that σ depends only on the radial parts: dµ z, z N = σr,...,r N dr dθ dr N dθ N, N dz dz = dr dθ dr N dθ N. 3.5 8

Then 3. becomes K! 3. = {n}= {m}= n! n N! K+ N α= n α! K! m! m N! K+ N α= m α! {n} {m} N π π dr dr N σr,,r N dθ dθ N r n +m r nn+mn e in m θ e in N m N θ N =π N {n}= N K! n! n N! K+ N α= n α! {n} {m} dr Thus σr,,r N must satisfy π N ΓK dr =Γn + Γn N +Γ dr N σr,,r n r n r n N N. 3.6 dr N σr,,r N r n rn N N K+ n α α= After some considerations, we found the following formula. dr r s =Γs + Γs N +Γ. 3.7 dr N r s N N r + +r N K N K K N r + +r N K+ s α. 3.8 α= See appendix A for the derivation of the formula. This is verified as follows. In the left-hand side of 3.8, we put r = ξ ξ, r = ξ ξ ξ 3,. r N = ξ ξ ξ N ξ N, r N = ξ ξ ξ N, r,,r N ξ,,ξ N =ξn ξ N ξn ξ N 9, 3.9

to obtain l.h.s. = = dξ ξ N +s + +s N + K N K K N dξ ξ N +s + +s N ξ s dξ N ξ +s N +s N N ξ N s N dξ ξ K+N +s + +s N K K N ξ dξ 3 ξ N 3+s 3+ +s N 3 ξ 3 s ξ dξ N ξ N s N ξ s N N Bs +,N +s + +s N Bs N +,s N +, 3.3 where we have used the integral expression of the Beta function: Bp, q = dtt p t q. 3.3 By putting ξ = u, 3.3 the ξ -integral in 3.3 becomes the ξ -integral part = duu K+N+s + +s N K K N u = 4 ΓK + s + +s N ΓN + s + +s N, 3.33 where the formula dxx µ K ν ax = µ µ+ν µ ν Γ Γ, a>, Reµ > Reν, 3.34 4 a has been used see appendix B. By noting the relation 3.3 finally becomes Bp, q = ΓpΓq Γp + q, 3.35 3.3 = Γs + Γs N + ΓK + s + +s N, 3.36 which is just the right-hand side of 3.8. Comparing 3.7 with 3.8, we find that σr,,r N = π N ΓK r + +r N K N K K N r + +r N. 3.37

Therefore we obtain the measure: where In the N = case, 3.38 K K is dµ z, z = z K N K K N z [dz dz], 3.38 π N ΓK z z z. 3.39 dµz, z = z K K K z [dz dz], 3.4 πγk which is just the measure of BG coherent state.9. The explicit form of 3. is z K N K K N z [dz dz] π ΓK z n zn N N n,,n N,K + n α {n}= α= z m zn m N m,,m N,K + m α = K. 3.4 {m}= α= It is remarkable that 3.4 holds for K>. 4 Construction of Path Integral Formula We construct a path integral formula with a Hamiltonian N+ Ĥ c α E αα = µ α E αα + Kc N+ K, µ α c α + c N+. 4. α= α= We have shown the WKB-exactness of the trace formula with this Hamiltonian in terms of the generalized coherent state [3] and the multi-periodic coherent state [7]. In this section we write down the trace formula in terms of our BG coherent state. The matrix element of the Hamiltonian is z Ĥ z = Kc N+ F N K;zαz α + µ α zαz αf N K +;zαz α, 4. α= where F N K; z is given in 3.8.

The Feynman kernel is defined by Kz F, z I ; T z F e iĥt z I = lim M z F i tĥ M zi, t T/M, 4.3 where z I z F is the initial final state and T is time interval. The explicit form of the kernel is M Kz F, z I ; T = lim dµ zi, z i M zj i tĥ zj M i= j= M = lim dµ zi, z i M { F N K;zα jz αj M i= j= [ { i t Kc N+ + µ α zα jz αj F NK +;z }] αjz α j } α= F N K;zα jz αj M = lim dµ zi, z i M { } F N K;zα jz αj M [ exp i= i t M k= { Kc N+ j= zm=z F z=z I + µ α zα kz αk F NK +;z }] αkz α k, 4.4 α= F N K;zαkz α k where the resolution of unity 3. has been inserted in the first equality and O t terms, which finally vanish in M limit, have been omitted in the last equality. The trace formula is defined by Z dµ z, z z e iĥt z = dµ z, z Kz, z; T. 4.5 The explicit form is Z = lim M [ exp + α= M i= i t dµ ziz i M M k= { Kc N+ j= { } F N K;zαjz α j µ α z αkz α k F NK +;z αkz α k F N K;z αkz α k = lim M e ikc N+T exp [ i t M i= M k= α= dµ zi, z i M j= } ] zm=z { } F N K;zαjz α j µ α z α kz αk F NK +;z αkz α k F N K;z α kz αk ]. 4.6

In the N = case, 4.6 is M Z = lim M e ihkt i= π K K zi I K z izi [dz idzi] [ M exp ih t z kzk I K z kzk ] k= I K z kzk, 4.7 where h c + c. Even in the N = case, it seems to be complicated not only to make the WKB approximation but also to calculate exactly. 5 Discussion We have extended the BG coherent state for the representation of SU, to some representation of UN, and constructed the measure. The eigenvalue of the Casimir operator I = N+ α= E αα, K, can be enlarged to K>. We have also constructed the path integral formula with the same Hamiltonian with which the WKB-exactness is examined in terms of some coherent states. However, in this case, the form of the coherent state is so complicated that the WKB approximation as well as the exact calculation seems not to be easy. Showing the WKB-exactness is now under consideration. Appendix A Derivation of the Formula 3.8 The definition of the Gamma function immediately leads to the relation: dr α e aαrα r sα α = a sα+ α Γs α +. A. 3

Multiplying equations A. from α =tonand putting a α =/x for all α=,,n, we obtain dr r s dr N r s N N e R/x = x s + +s N +N Γs + Γs N +, A. where R r + +r N. A.3 In both sides, by multiplying e x x K N and by integrating over x, the right-hand side becomes r.h.s. = Γs + Γs N +Γ K+ s α. A.4 Then the left-hand side is α= l.h.s = = dr r s dr N r s N N dxx K N e R/x x N R K N dr r s dr N r s N duu K+N e u R/u, where a change of variable, u = R/x, has been made in the second equality. A.5 By the integral expression of the modified Bessel function: K ν z = z ν dtt ν e t z /4t, A.6 the u-integral is written by Thus A.5 becomes A.5 = duu K+N e u R/u =R K N K K N R. A.7 dr r s Finally, by A.4 and A.8, we obtain dr r s =Γs + Γs N +Γ dr N r s N N R K N K K N R. A.8 dr N r s N N r + +r N K N K K N r + +r N K+ s α. A.9 α= 4

B The Proof of the Formula 3.34 By the integral representation of the modified Bessel function: K ν z = π z ν dye zy y ν, for ν> /, B. ν! the left-hand side of the formula 3.34 becomes dxx µ K ν ax = π dy y { ν ax dxx µ ν! νe axy}. B. Changing a variable x to t such that axy = t, B.3 leads the x-integral to dxx µ ax νe axy = dte t t µ+ν = a µ y µ+ν ν Γµ + ν. B.4 a µ ν y µ+ν Thus B. becomes B. = π Γµ + ν ν! a µ ν dy y ν. B.5 y µ+ν Further we make a change of a variable such that y =/ t, B.6 to obtain B.5 = = = π ν π ν Γµ + ν dtt µ ν! a µ ν t ν+! Γµ + ν a µ ν+ π µ ν a µ Γ ν+ Γ B µ ν µ + ν, µ + ν Γ µ+ν Γµ + ν Γ µ+ν + where we have used the integral expression of the Beta function:, B.7 Bp, q = dtt p t q, B.8 5

and the relation: By the formula: Bp, q = ΓpΓq Γp + q. B.9 z! z +!= z πz +!, B. the product of the Gamma functions is written as µ + ν µ + ν Γ Γ + = µ+ν+ πγµ + ν. B. Thus B.7 becomes B.7 = µ µ ν µ + ν Γ Γ, B. 4 a which is just the right-hand side of the formula. References [] R. J. Glauber, Phys. Rev. A3 963 766, [] A. Perelomov, Generalized Coherent States and Their Applications Springer-Verlag, Berlin, 986. [3] K. Funahashi, T. Kashiwa, S. Sakoda and K. Fujii, J. Math. Phys. 36 995 459. [4] K. Funahashi, T. Kashiwa, S. Sakoda and K. Fujii, J. Math. Phys. 36 995 33. [5] T.Kashiwa,Int.J.Mod.Phys.A5 99 375. [6] K. Funahashi, T. Kashiwa, S. Nima and S. Sakoda, Nucl. Phys. B453 995 58. [7] K. Fujii and K. Funahashi, to be published in J. Math. Phys. [8] K. Fujii and K. Funahashi, J. Math. Phys. 37 996 5987. [9] A. O. Barut and L. Girardello, Commun. Math. Phys. 97 4. [] C. Brif, A. Vourdas and A. Mann, quant-ph/967. 6

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