Green s function, wavefunction and Wigner function of the MIC-Kepler problem

Green s function, wavefunction and Wigner function of the MIC-Kepler problem Tokyo University of Science Graduate School of Science, Department of Mathematics, The Akira Yoshioka Laboratory Tomoyo Kanazawa 1

Outline 1. Hamiltonian description for the MIC-Kepler problem The reduced Hamiltonian system by an S 1 action The 4-dimensional conformal Kepler problem. Green s function of the MIC-Kepler problem The infinite series of its wavefunction 3. Wigner function of the MIC-Kepler problem

1. Hamiltonian description for the MIC-Kepler problem In 1970, McIntosh and Cisneros studied the dynamical system describing the motion of a charged particle under the magnetic force due to Dirac s monopole field of strength µ and the square inverse centrifugal potential force besides the Coulomb s potential force. B µ = µ r 3 x B µ = µ r Ñ Ò Ø ÓÖ Ù ØÓ Ö ³ Ð Ü Þ Ö ¾ ÓÙÐÓÑ ³ ÓÖ Ü Ö ¾ Ü ¾ ÑÖ ÒØÖ Ù Ð ÓÖ Ù ØÓ Ö ³ Ð µ is quantized as µ ħ Z. Ç Ü Ö Ü Ý 3

The Hamiltonian description for the MIC-Kepler problem is given by T. Iwai and Y. Uwano 1986 as follows. The MIC-Kepler problem is the reduced Hamiltonian system of the 4-dimensional conformal Kepler problem by an S 1 action, if the associated momentum mapping ψu, ρ µ 0. Ü Þ Ç Ê Ü Ôµ Ì Ê Ô Üµ Ü Ë ½ Ùµ Ý ½ µ ½ Ù ¼ ¾ ¼ Ù ¼ Ù ¼ ¼ µ ½ Ù ¾ Ù Ì Ê Ù µ Ë ½ Ù Ê ψu, ρ is invariant under the S 1 action, then let ψ 1 µ T uṙ4 be a subset s.t. { ψ 1 µ = u, ρ T uṙ4 ψu, ρ = 1 u ρ 1 + u 1 ρ u 4 ρ 3 + u 3 ρ 4 = µ. } 4

The MIC-Kepler problem is a triple T Ṙ 3, σ µ, H µ where σ µ = dp x dx + dp y dy + dp z dz µ x dy dz + y dz dx + z dx dy, r3 H µ x, p = 1 m p x + p y + p z + µ mr k r. Its energy hyper surface : H µ = E Φx, p r H µ E = 0 is equal to π µ Φ u, ρ = 0 Ü Ôµ Ê Ù µ where π µ ψ 1 µ T Ṙ 3, Ü Ôµ Ì Ê Ù µ Ì Ê ½ µ π µ Φ = u H E, u u 1 + u + u 3 + u 4. 5

The conformal Kepler problem is a triple T Ṙ 4, dρ du, H where dρ du 4 j=1 dρ j du j, Hu, ρ = 1 1 m 4u 4 j=1 ρ j k u. Since u = r > 0 invariant under the S 1 action, π µφ = 0 is equal to H E = 0. The energy hyper surface H = E is equivalent to Ku, ρ= ϵ where Ku, ρ is the Hamiltonian of 4-dimensional harmonic oscillator: Ku, ρ = 1 m 4 j=1 ρ j + 1 mω 4 u j j=1 { m > 0 mass of pendulum ω > 0 angular frequency considering only the case where the real parameter E < 0 and putting both the constant mω 8E and a real parameter ϵ 4k. 6

We solved the harmonic oscillator by means of the Moyal product, which brought the following functions. Moyal propagator -exponential i ui +u ħ t+iy K f, ρ e = cos ωz 4 exp [ i ħω K ui + u f, ρ ] tan ωz where u i = u i 1, ui, ui 3, ui 4 Ṙ4 and u f = u f 1, uf, uf 3, uf 4 Ṙ4 denote initial point and final point of motion respectively. Feynman s propagator K u f, u i ; z = t + iy = m ω [ 1 4π ħ sin ωz exp i mω ħ = n= C n e inωt 1 sin ωz { u i + u f cos ωz u i u f } ] where C n ω π π/ω π/ω K u f, u i ; τ + iy e inωτ dτ. 7

Green s function Gu f, u i ; ϵ = lim y +0 = m ω π ħ i ħ exp 1 0 n= N l 1!l!l 3!l 4! H l 1 C n e inωt [ mω ħ u i + u f ] N=0 mω ħ ui 1 H l3 mω ħ ui 3 e y +iϵ ħ t+iy dt l 1 +l +l 3 +l 4 =N H l1 mω ħ uf 1 H l3 mω ħ uf 3 H l mω 1 ϵ N + ħω ħ ui H l4 mω ħ ui 4 where n N = l 1 + l + l 3 + l 4 l 1, l, l 3, l 4 N {0}, H l X is the Hermite polynomial : dl H l X = 1 l e X dx l e X. H l mω ħ uf H l4 mω ħ uf 4 8

Moreover, we denote by Ψ N u the wave function of 4-dimensional harmonic oscillator on Ṙ 4 Ψ N u = mω πħ satisfying 1 N l 1!l!l 3!l 4! H l1 mω ħ u 1 exp H l mω ħ u ˆK Ψ N u = ϵ Ψ N u where ˆK = ħ [ mω ] ħ u 1 + u + u 3 + u 4 m H l3 mω ħ u 3 4 j=1 u j + 1 mω mω H l4 ħ u 4 4 j=1 u j. Then we verify Gu f, u i ; ϵ = N=0 1 ϵ N + ħω Ψ Nu f Ψ N u i. 9

à À ÖÑÓÒ Ó ÐÐ ØÓÖ Ø Ã Ø Þ ¼ Ø Ý ¼ Ý ¼ ¼µ Þ ¼ à ¼ Ý ¼ ¼ ½ Ñ ¾ Ì Ê ½ µ À Ñ ¼ ÓÒ ÓÖÑ Ð Ã ÔÐ Ö Ù Ù µ Ä Ý ¼ ¼ Ã Ù Ù Þ ¼ µ Ì Ê ¼ Ñ À ÅÁ ¹Ã ÔÐ Ö Ü Ü µ ÓÖ Ü Ü µ 10

. Green s function of the MIC-Kepler problem We suppose E mk ħ N + N = 0, 1,,, then reduce the Green s function of the conformal Kepler problem Gu f, u i ; ϵ 4k assumed mω 8E to the Green s function of the MIC-Kepler problem G + x f, x i ; E or G x f, x i ; E by an S 1 action. G + x f, x i ; E and G x f, x i ; E denote the Green s functions in the following local coordinates τ + and τ respectively. τ + : π 1 U + u πu, φ + u = xr, θ, ϕ, expiν/ U + S 1 x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ, u 1 = r cos θ cos ν + ϕ u 3 = r sin θ cos ν ϕ, u = r cos θ sin ν + ϕ, u 4 = r sin θ sin ν ϕ where U + = { xr, θ, ϕ Ṙ 3 ; r > 0, 0 θ < π, 0 ϕ < π }, 0 ν < 4π. 11

τ : π 1 U u πu, φ u = x r, θ, ϕ, expi ν/ U S 1 x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ, u 1 = r sin θ cos ν + ϕ u 3 = r cos θ cos ν + 3 ϕ, u = r sin θ sin ν + ϕ, u 4 = r cos θ where U = { x r, θ, ϕ Ṙ 3 ; r > 0, 0 θ < π, 0 < ϕ π }, 0 ν < 4π. sin ν + 3 ϕ z θ r = r x ϕ z r = r x x O θ y x ϕ O y 1

Iwai and Uwano also investigated the quantized system 1988 : The quantized conformal Kepler problem is defined as a pair L R 4 ; 4u du, Ĥ where L R 4 ; 4u du : Ĥ = ħ m 1 4u The Hilbert space of square integrable complex-valued functions on R 4, 4 j=1 u j k u : The Hamiltonian operator. They introduce complex line bundles L l l Z on which the linear connection is induced from a connection on the principal fibre bundle π : Ṙ 4 Ṙ 3. By an S 1 action, L R 4 ; 4u du is reduced to the Hilbert space, denoted by Γ l, of square integrable cross sections in L l over Ṙ 3. 13

The quantized MIC-Kepler problem is the reduced quantum system Γ l, Ĥ l where j stands for the covariant derivation with respect to the linear connection whose curvature gives Dirac s monopole field of strength lħ/, Ĥ l = ħ m 3 j=1 j + lħ/ mr k r. Cross section in L l corresponds uniquely to an eigenfunction of the momentum operator ˆN ˆN = iħ u + u 1 u 4 + u 3 u 1 u u 3 u 4 ˆN Ψu = l ħ Ψu Ψu L R 4 ; 4u du, l Z., Accordingly, the introduction of L l is understood as a geometric consequence of the conservation of the angular momentum associated with the U1 S 1 action. 14

We can obtain the wave function of the quantized MIC-Kepler problem, denoted by Ψ N x Γ l, as the following cross section with either of the local coordinates. i x U +, Ψ + N x π e i l ν/ Ψ N, l u ii x U, Ψ N x π e i l ν/ Ψ N, l u where L R 4 ; 4u du Ψ N, l u satisfies Ĥ Ψu = E Ψu ˆK Ψu = ϵ Ψu s.t. ϵ 4k and mω 8E ˆN Ψu = l ħ Ψu. Finally, we can calculate the Green s function of the MIC-Kepler problem by the following infinite series consists of Ψ N x with either of the local coordinates. G x f, x i ; E = mω /8 = N=0 1 4k N + ħω Ψ Nx f Ψ N x i 15

Proposition 1-1. [The Green s function of the MIC-Kepler problem] i When x i, x f U +, G + x f, x i ; E = mω /8 = m ω 4πħ e mω ħ r i+r f 1 mω N=0 4k N + ħω ħ 1 ri r f cos θ i k 1!k!k 3!k 4! cos θ f P r i cos θ i, r i sin θ i P N k1 +k 3 ri r f sin θ i sin θ f r f cos θ f, r f sin θ f k +k 4 e ik 1 k k 3 +k 4 ϕ i ϕ f / where { k1 + k k 1, k, k 3, k 4 N {0} s.t. + k 3 + k 4 = N k 1 + k k 3 k 4 = l Z l = µ/ħ, P X, Y is the following polynomial. P X, Y = k 1 k j=0 s=0 j!s! ħ j+s k mω 1 C j k3 C j k C s k4 C s X j Y s 16

Proposition 1-. [The Green s function of the MIC-Kepler problem] ii When x i, x f U, G x f, x i ; E = mω /8 = m ω 4πħ e mω ħ r i+ r f 1 mω N N=0 4k N + ħω ħ 1 ri r f sin θ i k 1!k!k 3!k 4! sin θ k1 +k 3 f ri r f cos θ i cos θ f P r i sin θ i, r i cos θ i P r f sin θ f, r f cos θ f iii When x i, x f U + U, the following correlation of G with G + is shown by r = r, θ = π θ and ϕ = π ϕ. G x f, x i ; E = G + x f, x i ; E e i l ϕ i ϕ f k +k 4 e ik 1+3k k 3 3k 4 ϕ i ϕ f / Incidentally, we can also find the following proposition. 17

Proposition. [The wave function of the MIC-Kepler problem] i When x U +, Ψ + N mω mω N x = P rcos θ, rsin θ πħ ħ k1!k!k 3!k 4! r θ k1 +k 3 r θ cos sin ii When x U, Ψ mω mω N P rsin θ, rcos θ N x = πħ ħ k1!k!k 3!k 4! k1 +k θ 3 k +k [ θ 4 r sin r cos exp iii When x U + U, Ψ N x = Ψ+ N e mω ħ r k +k 4 [ exp ik 1 k k 3 + k 4 ϕ ]. e mω ħ r ik 1 + 3k k 3 3k 4 ϕ x e i l ϕ ]. where N = 0, 1,,, the combination of non-negative integers k 1, k, k 3, k 4 and the polynomial P are the same as those shown in Proposition 1. 18

3. Wigner function of the MIC-Kepler problem We showed the energy-eigenspace of the MIC-Kepler problem in our proceeding as follows. Theorem [The eigenspace of the MIC-Kepler problem] Its eigenspace associated with the negative energy E N = mk ħ N + N = 0, 1,, is spanned by f N u, ρ = 1N πħ 4 e N+ L na 4b + 3 b 3 L n b 4b + 1 b 1 L n c 4b + b L n d 4b + 4 b 4 where n a, n b, n c, n d N {0}, l Z s.t. n a + n d N + l n b + n c N l i.e. l N N and l are simultaneously even or odd. The dimension is N + l + 1 N l + 1 = N + l + N l + 4. 19

In the above-mentioned theorem, L n X denotes the Laguerre polynomial of degree n s.t. n L n X = 1 α n! α! n α! Xα, L n X ξ n = 1 1 ξ exp ξ 1 ξ X Further, α=0 n=0 4b + 3 b 3 = mω ħ u 1 + u + 1 mħω ρ 1 + ρ + ħ u 1ρ u ρ 1. 4b + 1 b 1 = mω ħ u 1 + u + 1 mħω ρ 1 + ρ ħ u 1ρ u ρ 1 4b + b = mω ħ u 3 + u 4 + 1 mħω ρ 3 + ρ 4 ħ u 3ρ 4 u 4 ρ 3 4b + 4 b 4 = mω ħ u 3 + u 4 + 1 mħω ρ 3 + ρ 4 + ħ u 3ρ 4 u 4 ρ 3. We reduce the eigenfunction f N u, ρ on T Ṙ 4 to that on T Ṙ 3 with the above-mentioned local polar coordinates. 0

Proposition 3-1. [The Wigner function of the MIC-Kepler problem] Suppose x U + U, i f N r, θ, ϕ, p r, p θ, p ϕ = 1N πħ 4 e N+ [ C N + L na A + mk r1 + cos θ [ N + L nc B E + mk r1 cos θ ii f N r, θ, ϕ, p r, p θ, p ϕ = 1N πħ 4 e N+ [ L na N + mk L nc N + mk [ Ã + B + C r1 cos θ Ẽ r1 + cos θ ] ] ] ] N + L nb mk N + L nd mk L nb N + mk L nd N + mk [ [ [ A + B + D r1 + cos θ F r1 cos θ Ã D + r1 cos θ B F + r1 + cos θ where p x dx + p y dy + p z dz = p r dr + p θ dθ + p ϕ dϕ = p r d r + p θ d θ + p ϕ d ϕ. [ 1 ] ] ] ]

Proposition 3-. [The Wigner function of the MIC-Kepler problem] Functions A, B, C, D, E and F on T U + U are as follows. 1 cos θ Ar, θ, ϕ, p r, p θ, p ϕ p r r1 + cos θ p θ r 1 + cos θ Br, θ, ϕ, p r, p θ, p ϕ p r r1 cos θ + p θ { r mk Cr, θ, ϕ, p r, p θ, p ϕ p ϕ + r1 + cos θ ħn + + µ r { mk Dr, θ, ϕ, p r, p θ, p ϕ p ϕ r1 + cos θ ħn + µ r { mk Er, θ, ϕ, p r, p θ, p ϕ p ϕ + r1 cos θ ħn + µ r { mk Fr, θ, ϕ, p r, p θ, p ϕ p ϕ r1 cos θ ħn + + µ r } } } }

Proposition 3-3. [The Wigner function of the MIC-Kepler problem] Functions Ã, B, C, D, Ẽ and F on T U + U are as follows. 1 + cos θ Ã r, θ, ϕ, p r, p θ, p ϕ p r r1 cos θ + p θ r 1 cos θ B r, θ, ϕ, p r, p θ, p ϕ p r r1 + cos θ p θ { r mk C r, θ, ϕ, p r, p θ, p ϕ p ϕ r1 cos θ ħn + + µ r } { mk D r, θ, ϕ, p r, p θ, p ϕ p ϕ + r1 cos θ ħn + µ r } { mk Ẽ r, θ, ϕ, p r, p θ, p ϕ p ϕ r1 + cos θ ħn + µ r } { mk F r, θ, ϕ, p r, p θ, p ϕ p ϕ + r1 + cos θ ħn + + µ r } 3

Proposition 3-4. [The Wigner function of the MIC-Kepler problem] Using the following equivalences : we verify Then we have r = r, cos θ = cos θ, p r = p r, p θ = p θ, p ϕ = p ϕ Ã = A, B = B C = C, D = D, Ẽ = E, F = F. f N r, θ, ϕ, p r, p θ, p ϕ = f N r, θ, ϕ, p r, p θ, p ϕ. References [1] Bayen F., Flato M., Fronsdal C., Lichnerowicz A. and Sternheimer D., Deformation Theory and Quantization. II. Physical Applications, Ann.Phys.111 1978 111 151. 4

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