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A Coulomb green function from an SO(2,1) coherent state path integral
Volume 142B, number 5,6
PHYSICS LETTERS
2 August 1984
A COULOMB GREEN FUNCTION FROM AN SO(2,1) COHERENT STATE PATH INTEGRAL Christopher C. GERRY
Department of Physics, St. Bonaventure University, St. Bonaventure, N Y 14778, USA Received 9 February 1984 Revised manuscript received 23 March 1984 A Green function for the Coulomb problem is derived via path integration of SO(2,1) coherent states with the new "time" parameter of Duru and Kleinert.
In recent years there have been a number of attempts to evaluate non-gaussian path integrals b y the use o f nonlinear contact transformations and also by redefining the time variable in order to reduce the path integral to gaussian form [ 1 - 4 ] . In particular, Kleinert and Duru [1 ] redefined the time variable in a path integral expression for the Coulomb Green function and then used the Kustaanheimo-Stiefel transformation [5] to reduce the problem to a four-dimensional harmonic oscillator. However, their transformations were applied globally while it has been shown that such transformations must be justified in each short time interval o f the lattice space formulation o f the path integral [6]. That the transformation suggested in ref. [ 1] can be used to calculate the path integral for the Coulomb Green function in two and three dimensions has also been shown by Inomata and Ho [2] and Inomata [3]. In this paper we consider a Green function for the attractive Coulomb problem (H atom) evaluated in terms o f the coherent states of the SO(2,1) spectrum generating algebra. We have previously [7] considered path integrals over coherent states of the SO(2,1) [~ SU(1,1)] group and the details of the methods of our present calculation may be found there. In that work we did consider a functional integral expression of a resolvent for the H atom but it was not of the standard form. In this paper we shall start with the standard form of the resolvent and make use o f the redefinition of time as in refs. [ 1 - 3 ] . For the H atom the SO(2,1) Lie algebra is realized as [8] K 0 = ½ (rp 2 +r),
K 1 =½ (rp 2 - r ) ,
K 2 =r.p-
i.
(la, b, c)
The relevant representations are the unitary irreducible representations ~ + (k) where k = l + 1 and basis states are Ip. k) such that K 0 is diagonal according to
Kolp, k ) = ( p + k ) l p , k),
p = 0 , 1,2 ....
(2)
In fact p may be identified with the radial quantum number n r [8]. According to Perelomov [9], coherent states are given by If, k) = exp(c~K+ - a ' K _ ) 1 0 , k),
(3)
where K+ = K 1 + iK2, c~= - ~ r e -i~° and ~ = - tanh(r/2) e -i~°. These states may be expanded as oo
[ P ( p + 2k)'~l/2 I~, k ) = (1 - I ~ 12)k p~=O~-pi-F-(~)l ~Ptp, k>.
(4)
Unity is resolved as 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
391
PHYSICS LETTERS
Volume 142B, number 5,6
I = f din, (/j) I/j, k> (/j, kl,
2 August 1984
(5a)
where dtak ( / j ) = 2 k - 1 d2/j zr (1 -I/j12) 2'
(5b)
and there exists the reproducing kernel g k ( / j ' ,/j) = (/j', k L/J,k) = (1 -I/j'12) k (1 -I/j12) k (1 - / j , . / j ) - 2 k
(6)
such that
Kg(/J',/j) =fdUk(/j")Kk(/j',/j " ) K k(/j " ,/j)
(7)
In what follows, since k = l + 1 we shall write l/j, l + l) = I/j, l ). Also we define an auxiliary or physical coherent state I/j, l) as opposed to the group coherent state l/j, l) by the relation L/J, l) = e i0K2 l/j, l).
(8)
The angle 0 is to be adjusted later, but we note that eqs. ( 5 ) - ( 7 ) are unaltered by the replacement I/j, I ) ~ I/j, l). To find the Green function we start with the resolvent operator G ( E ) = ( E - H ) -1 where for the H atom II = -5
p2 c,/r.
(9)
-
We write
a(e) =
f e
(I0)
dr,
0
and then define "-~ ir l ,, Gt(/j", /j'; E ) = (/j"l I G ( E ) I/j'l ) = - ~ J Pk (/j ,/j'; r) dr, 0 where ~ l z~'" , g~"., 7") = (/j"l l e - i ( H - E ) r / h t-'E~,g
I/j', l).
(11)
(12)
We may think o f p / ( / j '',/j'; 7") as a propagator for the zero energy hamiltonian ( H - E ) . We proceed by considering the corresponding "evolution" operator (13)
U(r) = exp [ - i (H - E ) r/h], which from eqs. (1) and (9) may be written U(7") = exp [ - (i/ll) (r/r) (1 rp 2 - rE - a)l
(14a)
= exp ( - (i/n) (r/r) [4 (K0 + K1) - E ( K o - K1) - a]}
(14b)
= exp [ - (i/~t) (r/r) ( ~ - a)],
(14c)
where we have set the terms linear in the SO(2,1) generators as 1
(15)
/k = 7 (K0 + K 1) - E ( K 0 - K 1 ) . rl I
j , ~, i t
Now using eq. (14c) we can write ~'Et~ ,/j'; r) as the functional integral
392
Volume 142B, number 5,6
PHYSICS LETTERS
N-1 lc~,, ~, PEI.g , g ; r ) = N~** Jlim/'...fFI.,j=l where r = NAt a n d / / =
+
N IA_t ( A - a ) dt2l(tJ)1"= ( ~ j j / [ e x p \( - fir(t~)
It/-1,/),
(16)
(t/= (t] + tj_ 1)/2. One can expand this into path integral form [7] N-1
p/((',t';r)
2 August 1984
= lim F . . . F F I g ~ **J ,.' ]=1
N [(~t ) (Atl ,At]~ dlal(tj)lI-I_lex p A il~k t/--z%-i'-tJ At] "= (1 - Itj[ 2 )
iAt(tjllAlt]_l l ) ~r(Fj) (tjllt/_l l)
i At ] r(7,--3 '
(17)
where we have
A = e -i°K2 A ei°K2 -7-1 eO(Ko +K1) - Ee-°(Ko - K1)
(18)
from the Baker Hausdorf-Campbell formula. With At(t*,
t) = (t'l IAitl)/((lltl)
(19)
the path integral is written in the continuous limit as
Ere,
r)= fc~lzt(t)exp i r
(-T------(t~*-Itl 2)
~)-r--~-/]IAt(t*,t)
•
We now follow refs. [1,2] and introduce the new time parameter t
t
s(t) =f do(t) =f r(t) dt and write 1 -J, pEl~ ,t';7") =
(21)
/ (6 °r-fr(s)ds )pE~.¢ ~/ - I ¢~-"
(22)
, ¢ ; a ) da,
0 so that Gl( t ,, ,~ , ; E )
= - ~ i/~q je
i~a//~ -I r/-" ~'.
pE~.¢ , ¢, o) do,
where
N- 1
PEI.~ ,t, O):N__,.o f -I .... lim
"
(23)
N
/.__FI 1 d/al(t/.)FI 1 e x p r/ _. ., .i]2ka], "= "= L ( 1 - . it;12)
[t t j TAt1 - t j • At,~_ • a/ ] i1~o/At(t/'t]-l) J'
a n d where oj = At/r(~.) is the new local short time. If we choose exp(0) = ~ In this case we can write eq. (24) as N-1
(24)
then we have A = (-2E)l/2K
O.
N
PE(t-I,,,t'; 0t) = N_~**flim""'fj~-l- dlat (t/) j __FI1 (till exp[-ioj(--2E)l/2Kofl~] Itj_l,l),
(25)
which can be evaluated exactly. From eq. (4) we have (t/llexp
[-iaj (-- 2E) 1/2 Kofli] It/_ 1l >= exp [ - i a j (l + 1) (-2E)1/2fl~] (tj, l I{t/exp [-iaj (-2E)l/2/h] } l). (26)
Then from eqs. (6) and (7) we have 393
Volume 142B, number 5,6 p~(~ ,~ ;o)
PHYSICS LETTERS
2 August 1984
= exp[-io(-ZE) 1/2 (I+ 1)/h] ( 1 - 1~"12)/+1(1 -l~'12)/+1 { 1 - ~ " * ~ ' e x p [ - i o ( - Z E ) l / 2 / l t ] } -2(l+11, (27/
where o = liraN ~ ~ N~V1 o]. Finally from eq. (18) the Green function in integral form is
GI(~ ,,,~, ; E ) = - h i fexp{io[ot_(_2E)l/2(l+l)l/h)(
1 _ i~,, 12)l+1(1 _ i~, 12)/+1
0 X (1 - ~ " * ~ '
exp[-io(-2E)l/2/h])-2(l+l)do.
(28)
We may now find the energy spectrum by taking the trace of the resolvent operator as f do exp(i [ a - ( - 2 E ) 1/2 0
Tra(e)=
(l+~)]ofll)
X {sin [ ( - 2 E ) 1/2 o/2h] )-1.
(29)
Upon expanding the last factor o f eq. (29) as (1/2i) { s i n [ ( - 2 E ) 1/2 a/211]}-1 = exp [ - i
**
(-2E) 1/2o/2h] ~
q=0
exp [-iq(-2E)l/2a/'h],
(30)
we obtain TrG(E)=I/~
~q f d o e x p { i [ o ~ _ ( _ 2 E ) l / 2 ( l + l + q ) ] o / h } = i ~ [ ~ - ( - 2 E ) l / 2 ( l + l + q ) ] - l . 0
l
(311
q
This obviously has poles at E = - a2/[2(q + l + 1)2]. Identifying q with the radial quantum number n r we obtain the usual spectrum E n = -- c~2/2n 2, where n = n r + l + 1. In the above discussion we have considered only the case of the bound state (E < 0) part of the H-atom spectrum. However, the continuum states (E > 0) may also be obtained from eq. (28) by making an analytical continuation through the Sommerfeld-Watson transformation in a manner as discussed by Duru and Kleinert [1,10]. This stems from the fact that the continuous SO(2,11 eigenstates which diagonalize the non-compact operator K 1 are obtained from the discrete eigenstates of K0, [nl), by a rotation followed by an analytic continuation [11 ]. Finally, we would like to point out that one possible application for the path integral given here to a phase integral approximation which we have previously developed, in the spirit of large N, for anharmonic oscillators [12]. This application will be discussed elsewhere. [1] [2] [3] [4] [5 ] [6] [7] [8] [91 [10] [ 11 ] [12] 394
I.H. Duru and H. Kleinert, Phys. Lett. 84B (1979) 185. R. Ho and A. Inomata, Phys. Rev. Lett. 48 (1982) 231. A. Inomata, Phys. Lett. 87A (1982) 387. P.Y. Cai, A. lnomata and R. Wilson, Phys. Lett. 96A (1983) 117. See A.O. Barut, C.K.E. Schneider and R. Wilson, J. Math. Phys. 20 (1979) 2244. J.L. Gervais and H. Jevicki, Nucl. Phys. Bll0 (1976) 53; C.C. Gerry, J. Math. Phys. 24 (1983) 874. C.C. Gerry and S. Silverman, J. Math. Phys. 23 (1982) 1995; C.C. Gerry, Phys. Lett. 119B (1982) 381. A.O. Barut, Dynamical groups and generalized symmetries in quantum theory (Univ. Canterbury Press, Christchurch, New Zealand, 197 I). A.M. Perelomov, Commun. Math. Phys. 40 (1975) 153. I.H. Duru and H. Kleinert, Fortschr. Phys. 30 (1982) 411. H. Kleinert, in: Lectures in theoretical physics, eds. A.O. Barut and W. Britten, Vol. Xb (1968) p. 427. C.C. Gerry, J .B. Togeas and S. Silverman, Phys. Rev. D28 (1983) 1945.
PHYSICS LETTERS
2 August 1984
A COULOMB GREEN FUNCTION FROM AN SO(2,1) COHERENT STATE PATH INTEGRAL Christopher C. GERRY
Department of Physics, St. Bonaventure University, St. Bonaventure, N Y 14778, USA Received 9 February 1984 Revised manuscript received 23 March 1984 A Green function for the Coulomb problem is derived via path integration of SO(2,1) coherent states with the new "time" parameter of Duru and Kleinert.
In recent years there have been a number of attempts to evaluate non-gaussian path integrals b y the use o f nonlinear contact transformations and also by redefining the time variable in order to reduce the path integral to gaussian form [ 1 - 4 ] . In particular, Kleinert and Duru [1 ] redefined the time variable in a path integral expression for the Coulomb Green function and then used the Kustaanheimo-Stiefel transformation [5] to reduce the problem to a four-dimensional harmonic oscillator. However, their transformations were applied globally while it has been shown that such transformations must be justified in each short time interval o f the lattice space formulation o f the path integral [6]. That the transformation suggested in ref. [ 1] can be used to calculate the path integral for the Coulomb Green function in two and three dimensions has also been shown by Inomata and Ho [2] and Inomata [3]. In this paper we consider a Green function for the attractive Coulomb problem (H atom) evaluated in terms o f the coherent states of the SO(2,1) spectrum generating algebra. We have previously [7] considered path integrals over coherent states of the SO(2,1) [~ SU(1,1)] group and the details of the methods of our present calculation may be found there. In that work we did consider a functional integral expression of a resolvent for the H atom but it was not of the standard form. In this paper we shall start with the standard form of the resolvent and make use o f the redefinition of time as in refs. [ 1 - 3 ] . For the H atom the SO(2,1) Lie algebra is realized as [8] K 0 = ½ (rp 2 +r),
K 1 =½ (rp 2 - r ) ,
K 2 =r.p-
i.
(la, b, c)
The relevant representations are the unitary irreducible representations ~ + (k) where k = l + 1 and basis states are Ip. k) such that K 0 is diagonal according to
Kolp, k ) = ( p + k ) l p , k),
p = 0 , 1,2 ....
(2)
In fact p may be identified with the radial quantum number n r [8]. According to Perelomov [9], coherent states are given by If, k) = exp(c~K+ - a ' K _ ) 1 0 , k),
(3)
where K+ = K 1 + iK2, c~= - ~ r e -i~° and ~ = - tanh(r/2) e -i~°. These states may be expanded as oo
[ P ( p + 2k)'~l/2 I~, k ) = (1 - I ~ 12)k p~=O~-pi-F-(~)l ~Ptp, k>.
(4)
Unity is resolved as 0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
391
PHYSICS LETTERS
Volume 142B, number 5,6
I = f din, (/j) I/j, k> (/j, kl,
2 August 1984
(5a)
where dtak ( / j ) = 2 k - 1 d2/j zr (1 -I/j12) 2'
(5b)
and there exists the reproducing kernel g k ( / j ' ,/j) = (/j', k L/J,k) = (1 -I/j'12) k (1 -I/j12) k (1 - / j , . / j ) - 2 k
(6)
such that
Kg(/J',/j) =fdUk(/j")Kk(/j',/j " ) K k(/j " ,/j)
(7)
In what follows, since k = l + 1 we shall write l/j, l + l) = I/j, l ). Also we define an auxiliary or physical coherent state I/j, l) as opposed to the group coherent state l/j, l) by the relation L/J, l) = e i0K2 l/j, l).
(8)
The angle 0 is to be adjusted later, but we note that eqs. ( 5 ) - ( 7 ) are unaltered by the replacement I/j, I ) ~ I/j, l). To find the Green function we start with the resolvent operator G ( E ) = ( E - H ) -1 where for the H atom II = -5
p2 c,/r.
(9)
-
We write
a(e) =
f e
(I0)
dr,
0
and then define "-~ ir l ,, Gt(/j", /j'; E ) = (/j"l I G ( E ) I/j'l ) = - ~ J Pk (/j ,/j'; r) dr, 0 where ~ l z~'" , g~"., 7") = (/j"l l e - i ( H - E ) r / h t-'E~,g
I/j', l).
(11)
(12)
We may think o f p / ( / j '',/j'; 7") as a propagator for the zero energy hamiltonian ( H - E ) . We proceed by considering the corresponding "evolution" operator (13)
U(r) = exp [ - i (H - E ) r/h], which from eqs. (1) and (9) may be written U(7") = exp [ - (i/ll) (r/r) (1 rp 2 - rE - a)l
(14a)
= exp ( - (i/n) (r/r) [4 (K0 + K1) - E ( K o - K1) - a]}
(14b)
= exp [ - (i/~t) (r/r) ( ~ - a)],
(14c)
where we have set the terms linear in the SO(2,1) generators as 1
(15)
/k = 7 (K0 + K 1) - E ( K 0 - K 1 ) . rl I
j , ~, i t
Now using eq. (14c) we can write ~'Et~ ,/j'; r) as the functional integral
392
Volume 142B, number 5,6
PHYSICS LETTERS
N-1 lc~,, ~, PEI.g , g ; r ) = N~** Jlim/'...fFI.,j=l where r = NAt a n d / / =
+
N IA_t ( A - a ) dt2l(tJ)1"= ( ~ j j / [ e x p \( - fir(t~)
It/-1,/),
(16)
(t/= (t] + tj_ 1)/2. One can expand this into path integral form [7] N-1
p/((',t';r)
2 August 1984
= lim F . . . F F I g ~ **J ,.' ]=1
N [(~t ) (Atl ,At]~ dlal(tj)lI-I_lex p A il~k t/--z%-i'-tJ At] "= (1 - Itj[ 2 )
iAt(tjllAlt]_l l ) ~r(Fj) (tjllt/_l l)
i At ] r(7,--3 '
(17)
where we have
A = e -i°K2 A ei°K2 -7-1 eO(Ko +K1) - Ee-°(Ko - K1)
(18)
from the Baker Hausdorf-Campbell formula. With At(t*,
t) = (t'l IAitl)/((lltl)
(19)
the path integral is written in the continuous limit as
Ere,
r)= fc~lzt(t)exp i r
(-T------(t~*-Itl 2)
~)-r--~-/]IAt(t*,t)
•
We now follow refs. [1,2] and introduce the new time parameter t
t
s(t) =f do(t) =f r(t) dt and write 1 -J, pEl~ ,t';7") =
(21)
/ (6 °r-fr(s)ds )pE~.¢ ~/ - I ¢~-"
(22)
, ¢ ; a ) da,
0 so that Gl( t ,, ,~ , ; E )
= - ~ i/~q je
i~a//~ -I r/-" ~'.
pE~.¢ , ¢, o) do,
where
N- 1
PEI.~ ,t, O):N__,.o f -I .... lim
"
(23)
N
/.__FI 1 d/al(t/.)FI 1 e x p r/ _. ., .i]2ka], "= "= L ( 1 - . it;12)
[t t j TAt1 - t j • At,~_ • a/ ] i1~o/At(t/'t]-l) J'
a n d where oj = At/r(~.) is the new local short time. If we choose exp(0) = ~ In this case we can write eq. (24) as N-1
(24)
then we have A = (-2E)l/2K
O.
N
PE(t-I,,,t'; 0t) = N_~**flim""'fj~-l- dlat (t/) j __FI1 (till exp[-ioj(--2E)l/2Kofl~] Itj_l,l),
(25)
which can be evaluated exactly. From eq. (4) we have (t/llexp
[-iaj (-- 2E) 1/2 Kofli] It/_ 1l >= exp [ - i a j (l + 1) (-2E)1/2fl~] (tj, l I{t/exp [-iaj (-2E)l/2/h] } l). (26)
Then from eqs. (6) and (7) we have 393
Volume 142B, number 5,6 p~(~ ,~ ;o)
PHYSICS LETTERS
2 August 1984
= exp[-io(-ZE) 1/2 (I+ 1)/h] ( 1 - 1~"12)/+1(1 -l~'12)/+1 { 1 - ~ " * ~ ' e x p [ - i o ( - Z E ) l / 2 / l t ] } -2(l+11, (27/
where o = liraN ~ ~ N~V1 o]. Finally from eq. (18) the Green function in integral form is
GI(~ ,,,~, ; E ) = - h i fexp{io[ot_(_2E)l/2(l+l)l/h)(
1 _ i~,, 12)l+1(1 _ i~, 12)/+1
0 X (1 - ~ " * ~ '
exp[-io(-2E)l/2/h])-2(l+l)do.
(28)
We may now find the energy spectrum by taking the trace of the resolvent operator as f do exp(i [ a - ( - 2 E ) 1/2 0
Tra(e)=
(l+~)]ofll)
X {sin [ ( - 2 E ) 1/2 o/2h] )-1.
(29)
Upon expanding the last factor o f eq. (29) as (1/2i) { s i n [ ( - 2 E ) 1/2 a/211]}-1 = exp [ - i
**
(-2E) 1/2o/2h] ~
q=0
exp [-iq(-2E)l/2a/'h],
(30)
we obtain TrG(E)=I/~
~q f d o e x p { i [ o ~ _ ( _ 2 E ) l / 2 ( l + l + q ) ] o / h } = i ~ [ ~ - ( - 2 E ) l / 2 ( l + l + q ) ] - l . 0
l
(311
q
This obviously has poles at E = - a2/[2(q + l + 1)2]. Identifying q with the radial quantum number n r we obtain the usual spectrum E n = -- c~2/2n 2, where n = n r + l + 1. In the above discussion we have considered only the case of the bound state (E < 0) part of the H-atom spectrum. However, the continuum states (E > 0) may also be obtained from eq. (28) by making an analytical continuation through the Sommerfeld-Watson transformation in a manner as discussed by Duru and Kleinert [1,10]. This stems from the fact that the continuous SO(2,11 eigenstates which diagonalize the non-compact operator K 1 are obtained from the discrete eigenstates of K0, [nl), by a rotation followed by an analytic continuation [11 ]. Finally, we would like to point out that one possible application for the path integral given here to a phase integral approximation which we have previously developed, in the spirit of large N, for anharmonic oscillators [12]. This application will be discussed elsewhere. [1] [2] [3] [4] [5 ] [6] [7] [8] [91 [10] [ 11 ] [12] 394
I.H. Duru and H. Kleinert, Phys. Lett. 84B (1979) 185. R. Ho and A. Inomata, Phys. Rev. Lett. 48 (1982) 231. A. Inomata, Phys. Lett. 87A (1982) 387. P.Y. Cai, A. lnomata and R. Wilson, Phys. Lett. 96A (1983) 117. See A.O. Barut, C.K.E. Schneider and R. Wilson, J. Math. Phys. 20 (1979) 2244. J.L. Gervais and H. Jevicki, Nucl. Phys. Bll0 (1976) 53; C.C. Gerry, J. Math. Phys. 24 (1983) 874. C.C. Gerry and S. Silverman, J. Math. Phys. 23 (1982) 1995; C.C. Gerry, Phys. Lett. 119B (1982) 381. A.O. Barut, Dynamical groups and generalized symmetries in quantum theory (Univ. Canterbury Press, Christchurch, New Zealand, 197 I). A.M. Perelomov, Commun. Math. Phys. 40 (1975) 153. I.H. Duru and H. Kleinert, Fortschr. Phys. 30 (1982) 411. H. Kleinert, in: Lectures in theoretical physics, eds. A.O. Barut and W. Britten, Vol. Xb (1968) p. 427. C.C. Gerry, J .B. Togeas and S. Silverman, Phys. Rev. D28 (1983) 1945.