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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
Star product and its application
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan
16 May 2014 Vancouver
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
Abstract
∗ We introduce a star product O on polynomials and show some expamles. ∗ ∗ We define general star products Λ, K . Then we introduce exponential element in the star product algebra. Using the star exponential elements we define several functions called star functions in the algebra. We show certain expamples of star functions.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§1. Introduction: the idea
The canonical commutation relation is a basic identity of quantum mechanics, which is given by a pair of operators such as
[p ˆ, qˆ] = pˆ qˆ − qˆ pˆ = i ~
where pˆ = i ~ ∂q and qˆ is a multiplication operator q× acting on the functions of q, and ~ is the Plank constant. The algebra generated by pˆ and qˆ is called the Weyl algebra which plays a fundamental role in quantum mechanics.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
We have another way to give the same algebra by using only functions, not using operators. ∗ Instead of using operators, we introduce an associative product O into the space of functions of (q, p). The product is different from the usual multiplication of functions, but is given as a deformation of the usual multiplication in the following way. (Cf. Bayen-Flato-Fronsdal-Lichnerowicz-Sternheimer [1], Moyal).
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
∗ The product O For smooth functions f, g on R2, we have the canonical Poisson bracket
2 { f, g}(q, p) = ∂p f ∂qg − ∂q f ∂pg, (q, p) ∈ R
In deformation quantization, we very often use the notation of ←− −→ ←− −→ bidifferential opearator ∂p · ∂q − ∂q · ∂p such as (←− −→ ←− −→) { f, g} = f ∂p · ∂q − ∂q · ∂p g = ∂p f ∂qg − ∂q f ∂pg
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
, ∗ For polynomials f g we consider a product f O g given by the exponential power series of the bidifferential operator ←− −→ ←− −→ ∂p · ∂q − ∂q · ∂p such that
( ) ∞ ( ) ←− −→ ←− −→ ∑ ( ) ←− −→ ←− −→ k ∗ = i~ ∂ · ∂ − ∂ · ∂ = 1 i~ k ∂ · ∂ − ∂ · ∂ f O g f exp 2 p q q p g f k! 2 p q q p g = ( ) k 0 ( ) ←− −→ ←− −→ ( ) ←− −→ ←− −→ 2 = + i~ ∂ · ∂ − ∂ · ∂ + 1 i~ 2 ∂ · ∂ − ∂ · ∂ f g 2 f p q q p g 2! 2 f p q q p g ( ) ( ) ←− −→ ←− −→ k + ··· + 1 i~ k ∂ · ∂ − ∂ · ∂ + ··· k! 2 f p q q p g
where ~ is a positive number. The product is well-defined and associative for polynomials.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
Now we calculate the commutator of the functions p and q. We see
( ) ∑∞ ( ) ( ) ←− −→ ←− −→ 1 k ←− −→ ←− −→ k p ∗ q = p exp i~ ∂ · ∂ − ∂ · ∂ q = p i~ ∂ · ∂ − ∂ · ∂ q O 2 p q q p k! 2 p q q p k=0 (←− −→ ←− −→) = + i~ ∂ · ∂ − ∂ · ∂ = + i~ pq 2 p p q q p q pq 2
Similarly we see ∗ = − i~ q O p pq 2 Then the functions p and q satisfy the canonical commutation ∗ relation under the commutator of the product O , = ∗ − ∗ = ~ [p q]∗ p O q q O p i
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
∗ The product O is associative on polynomials, and then we obtain the Weyl algebra given by the ordinary polynomials with the ∗ C , , ∗ product O , ( [q p] O ). ∗ Using this Weyl algebra of the product O , we can obtain some results of quantum mechanics, and further we can discuss some extensions. In this talk, we give a brief review on this point mainly related our investigation ([4], [8]).
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.1. Eigenvalues of Harmonic Oscillator §2. Star calculation of eigenvalues §2.1. Eigenvalues of Harmonic Oscillator
We can calculate the eigenvalues of the harmonic oscillator by ∗ means of the star product O .
Eigenvalues The Schrdingier¨ operator of the harmonic oscillator is ( ) ˆ = − ~2 ∂ 2 + 1 2. H 2 ∂q 2 q The eigenvalues are
= ~ + 1 , = , , , ··· En (n 2 ) n 0 1 2
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.1. Eigenvalues of Harmonic Oscillator Star product calculation
Parallel to the arguments in quantum mechanics, we can calculate ∗ the eigenvalues En by using the star product O and the functions of p and q in the following way. The classical hamiltonian function is
= 1 2 + 2 . H 2 (p q )
We put functions such as 1 1 a = √ (p + iq), a† = √ (p − iq). 2 ~ 2 ~
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.1. Eigenvalues of Harmonic Oscillator
Then we see easily
† 1 ( ) 1 a ∗ a = p ∗ p + i[p, q]∗ + q ∗ q = (p · p + i · i~ + q · q) O 2 ~ O O 2 ~ † ∗ = 1 2 + 2 − 1 Then we see a O a 2 ~ (p q ) 2 and we have = ~ + 1 , = † ∗ H (N 2 ) (N a O a)
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.1. Eigenvalues of Harmonic Oscillator
The commutator with respect to the star product is easily seen
† † † 1 [a, a ]∗ = a ∗ a − a ∗ a = 2(−i)[p, q]∗ = 1 O O 2 ~ Now we set a function
= 1 − † = 1 − 1 2 + 2 f0 π ~ exp( 2 aa ) π ~ exp( ~ (p q ))
By a direct calculation we see
∗ = ∗ † = a O f0 f0 O a 0
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.1. Eigenvalues of Harmonic Oscillator We set a function 1 f = a† ∗ · · · ∗ a† ∗ f ∗ a ∗ · · · ∗ a n n! | O {z O } O 0 O | O {z O } n n † Remark that [a, a ]∗ = 1 is equivalent to ∗ † = † ∗ + = + a O a a O a 1 N 1 then we have a commutation relation ∗ † = † ∗ ∗ † = † ∗ ∗ † = † ∗ + N O a (a O a) O a a O (a O a ) a O (N 1) ∗ = ∗ = Remark also that a O f0 0 yields N O f0 0. Then for example we calculate as ∗ = ∗ † ∗ ∗ = † ∗ + ∗ ∗ = N O f1 N O (a O f0 O a) a O (N 1) O f0 O a f1 By a similar manner we easily see ∗ = ∗ = N O fk fk O N k fk
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.1. Eigenvalues of Harmonic Oscillator
= ~ + 1 Since H (N 2 ) we have the solutions of the star eigenvalue problem
∗ = ∗ = ~ + 1 = , = , , , ··· H O fn fn O H (n 2 ) fn En fn (n 0 1 2 )
Thus we obtain the eigenvalues of the harmonic oscillator.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem §2.2. MIC-Kepler problem (T. Kanazawa-Y [4])
We apply the method to the MIC-Kepler problem, the Kepler-problem with magnetic monopole. Background McIntosh and Cisneros [7] studied the dynamical system describing the motion of a charged particle under the influence of Dirac’s monopole field besides the Coulomb’s potential. Iwai-Uwano [2] gives the Hamiltonian description for the MIC-Kepler problem.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem Point
Iwai-Uwano showed that the classical system of MIC-Kepler problem is obtained by the geometric method of S 1-reduction, or Marsden-Weinstein reduction for symplectic manifolds. Star product method uses only classical system with deformed product. Then by the star product calculation, we expect to deal with the quantized system of MIC-Kepler problem by means of the geometric method of Marsden Weinstein reduction in natural way. We discuss this in this subsection.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem MIC-Kepler problem
Now we introduce the MIC-Kepler problem. We consider a closed two form on R˙ 3 = R3 − {0} such that
3 Ω = ( q1 dq2 ∧ dq3 + q2 dq3 ∧ dq1 + q3 dq1 ∧ dq2)/r √ = 2 + 2 + 2 ∗R˙ 3 where r q1 q2 q3 . We consider the cotangent bundle T and a symplectic form
σµ = dp1 ∧ dq1 + dp2 ∧ dq2 + dp3 ∧ dq3 + Ωµ
∗ 3 where (q, p) = (q1, q2, q3, p1, p2, p3) ∈ T R˙ and the 2-form Ωµ ≡ µ Ω stands for Dirac’s monopole field of strength µ ∈ R.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem
Then the MIC-Kepler problem is given as the triple
∗ 3 ( T R˙ , σµ , Hµ )
where Hµ is the Hamiltonian function such that
( ) µ2 1 2 2 2 k Hµ (q, p) = p + p + p + − 2 1 2 3 2r2 r and k is a positive constant. When µ = 0 the system is just the Kepler problem.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem S 1-action
The MIC-Kepler problem is obtained by the S 1-reduction from the conformal Kepler problem on T ∗R˙ 4 (Iwai-Uwano [2]) as follows. We denote the points by y ∈ R4 and (y, η) ∈ T ∗R4. ∗ 4 We identify the point of T R ∋ (y1, y2, y3, y4, η1, η2, η3, η4) by
∗ 4 ∗ 2 4 T R ∋ (y1, y2, y3, y4, η1, η2, η3, η4) 7→ (z1, z2, ζ1, ζ2) ∈ T C = C
where
z1 = y1 + i y2, z2 = y3 + i y4, ζ1 = η1 + i η2, ζ2 = η3 + i η4
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem The canonical one form θ on T ∗R4 is written as θ(z, ζ) = Re (ζ¯ · dz). The S 1 action on the cotangent bundle T ∗R˙ 4 is given by it it φt :(z, ζ) 7→ (e z, e ζ), (t ∈ R) which preserves the canonical one form θ and then is an exact symplectic action. The induced vector field v(z, ζ) on T ∗R˙ 4 of the action is v(z, ζ) = (iz, i ζ) and then a moment map ψ of the action is given by
ψ(z, ζ) = ιvθ(z, ζ) = Im ζ · z¯ = (ζ · z¯ − ζ¯ · z)/2 i
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem S 1-reduction
Following the Marsden-Weinstein reduction theroy, we consider a level set of the moment map ψ−1(µ) for µ ∈ R. Then the S 1-bundle − − πµ : ψ 1(µ) → ψ 1(µ)/S 1 has the symplectic structure ωµ such that ∗ ∗ ιµdθ = πµωµ, hence we have a reduced symplectic manifold − − ∗ (ψ 1(µ)/S 1, ωµ), where ιµ : ψ 1(µ) → T R˙ 4 is the inclusion map. Then one can show Proposition (Iwai-Uwano [2]) The reduced phase space is diffeomorphic to the symplectic manifold of the MIC-Kepler problem,
−1 1 ∗ 3 (ψ (µ)/S , ωµ) ≃ (T R˙ , σµ)
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem Conformal Kepler problem on T ∗R˙ 4
Now we consider a harmonic oscillator on T ∗R˙ 4 , ζ = 1 |ζ|2 + 1 ω2 | |2 H(z ) 2 2 z Iwai-Uwano [2] introduces the conformal Kepler problem with the Hamiltonian , ζ = 1 , ζ − − 1 ω2 = 1 |ζ|2 − k HCF(z ) 4|z|2 (H(z ) 4k) 8 8|z|2 |z|2 The MIC-Kepler problem is the reduced hamitonian system of the conformal Kepler problem, i.e., ∗ ∗ πµHµ = ιµHCF
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem
The conformal Kepler problem is related to the harmonic oscillator on T ∗R˙ 4 as ( ) | |2 , ζ + 1 ω2 = , ζ − 4 z HCF(z ) 8 H(z ) 4k
Hence the energy surfaces in T ∗R˙ 4 coincide, i.e.,
= − 1 ω2 ⇐⇒ = . HCF 8 H 4k
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem Star product calculation of the eigenvalues
On 8-dimensional phase space T ∗R˙ 4, we have the canonical Poisson bracket and then by the same way as the previous section, ∗ we have the star product O . We consider functions ( √ ) ω b (z, ζ) = 1 z + √i ζ , b (z, ζ)† = b (z, ζ), 1 2 ( √ ~ 1 ω~ 1) 1 1 1 ω √i † b2(z, ζ) = ~ z2 + ζ2 , b2(z, ζ) = b2(z, ζ), 2 ( √ ω~ ) ω † b (z, ζ) = 1 z¯ + √i ζ¯ , b (z, ζ) = b (z, ζ), 3 2 ( √ ~ 1 ω~ 1) 3 3 , ζ = 1 ω + √i ζ¯ , , ζ † = , ζ . b4(z ) 2 ~ z¯2 ω~ 2 b4(z ) b4(z )
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem
We see the commutators of these functions are
† † † , ∗ = , ∗ = , , ∗ = δ , = , , , . [b j bk] [b j bk] 0 [b j bk] jk ( j k 1 2 3 4)
We set = † ∗ + † ∗ + † ∗ + † ∗ . N b1 O b1 b2 O b2 b3 O b3 b4 O b4 Then we see H = ~ω (N + 2), ψ , ζ , † and the moment map (z ) is written in terms of b j b j as
ψ , ζ = ~ − † ∗ − † ∗ + † ∗ + † ∗ . (z ) 2 ( b1 O b1 b2 O b2 b3 O b3 b4 O b4)
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem We put for j = 1, 2, 3, 4
† 1 −2 b b j 1 † k k , , ζ = j , , , ζ = ∗ , ∗ . f j 0(z ) π~ e f j k(z ) k! (b j )∗ O f j 0 O (b j)∗ We consider −→ −→ = , ∗ , ∗ , ∗ , , = , , , . f n f1 n1 O f2 n2 O f3 n3 O f4 n4 n (n1 n2 n3 n4) Parallel to Iwai-Uwano [3], we can calculate the eigenvalues of the MIC-Kepler problem as follows. Similarly as before we easily see
∗ −→ = ~ω + ∗ −→ = ~ω + + + + −→ H O f n (N 2) O f n (n1 n2 n3 n4 2) f n and ~ ψ ∗ −→ = − − + + −→ O f n 2 ( n1 n2 n3 n4) f n
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem
Hence the energy level
= − 1 ω2 ⇐⇒ = ψ = µ HCF 8 H 4k and
is quantized as
4k = ~ω(n1 + n2 + n3 + n4 + 2)
and µ = ~ − − + + 2 ( n1 n2 n3 n4)
Thus the quantized energy level of HCF is − 1 ω2 = − 2k2 2 2 and the strength of magnetic monopole 8 ~ (n1+n2+n3+n4+2) µ = ~ − − + + is quantized as 2 ( n1 n2 n3 n4).
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§2.2. MIC-Kepler problem Thus we have Theorem The eigenvalues of the MIC-Kepler problem with the strength of ~ m magnetic monople 2 is
2k2 E = − , ( n ≥ |m|, and n ± m ≡ 0 mod 2 ). n ~2(n + 2)2
The multiplicity of the eigenvalue En is (n + m + 2)(n − m + 2) 4
This is the same as the ones in Iwai-Uwano [3].
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.1. Examples: Moyal, normal, anti-normal products §3. Star products (Omori-Maeda-Miyazaki-Y [8]) §3.1. Examples: moyal, normal, anti-normal products
For polynomials f, g of the variables (u1,..., um, v1,..., vm), the Moyal product f ∗ g is given by the power series of the O (←− −→ ←− −→) bidifferential operators ∂v · ∂u − ∂u · ∂v such that
( ) ∞ ( ) ←− −→ ←− −→ ∑ ( ) ←− −→ ←− −→ k ∗ = i~ ∂ · ∂ − ∂ · ∂ = 1 i~ k ∂ · ∂ − ∂ · ∂ f O g f exp 2 v u u v g f 2 v u u v g = k! ( ) k 0 ( ) ←− −→ ←− −→ ( ) ←− −→ ←− −→ 2 = + i~ ∂ · ∂ − ∂ · ∂ + 1 i~ 2 ∂ · ∂ − ∂ · ∂ f g 2 f v u u v g 2! 2 f v u u v g ( ) ( ) ←− −→ ←− −→ k + ··· + 1 i~ k ∂ · ∂ − ∂ · ∂ + ··· k! 2 f v u u v g
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.1. Examples: Moyal, normal, anti-normal products
←− where ~ is a positive number and the overleft arrow ∂ means that the partial derivative is acting on the polynomial on the left and the overright arrow similar, for example
(←− −→ ←− −→) ∑m ( ) ∂ · ∂ − ∂ · ∂ = ∂ ∂ − ∂ ∂ f v u u v g v j f u j g u j f v j g j=1
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.1. Examples: Moyal, normal, anti-normal products
Theorem The Moyal product is well-defined on polynomials, and associative. ∗ Other typical star products are normal product N , anti-nomal ∗ product A given similarly by (←− −→) (←− −→) ∗ = ~ ∂ · ∂ , ∗ = − ~ ∂ · ∂ f N g f exp i v u g f A g f exp i u v g
These are also well-defined on polynomials and associative. By direct calculation we see easily the following.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.1. Examples: Moyal, normal, anti-normal products
Proposition
(i) For these star products, the generators (u1,..., um, v1,..., vm) satisfy the canoical commutation relations
[u , v ]∗ = −i~δ , [u , u ]∗ = [v , v ]∗ = 0, (k, l = 1, 2,..., m) k l L kl k l L k l L ∗ ∗ ∗ ∗ where L stands for O , N and A . (ii) Then the algebras (C[u, v], ∗L)(L = O, N, A) are mutually isomorphic and isomorphic to the Wyel algebra.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.1. Examples: Moyal, normal, anti-normal products Acutally the algebra isomorphsm N C , , ∗ → C , , ∗ IO :( [u v] O ) ( [u v] N) is given explicitly by the power series of the differential operator such as ( ) ∑∞ O = − i~ ∂ ∂ = 1 i~ l ∂ ∂ l IN ( f ) exp 2 u v ( f ) l! ( 2 ) ( u v) ( f ) l=0 And other isomophisms are given in the similar form. Remark We remark here that these facts correspond to the well-known ordering problem in physics.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.2. General star product
Now by generalizing these products, we define a star product. Notice here that we consider on complex domain. Biderivation Let Λ be an arbitrary n × n complex matrix. We consider a bidifferential operator
←− −→ ←−− ←−− −−→ −−→ ∑n ←−−−−→ ∂ Λ∂ = ∂ , ··· , ∂ Λ ∂ , ··· , ∂ = Λ ∂ ∂ w w ( w1 wn ) ( w1 wn ) kl wk wl k,l=1
where (w1, ··· , wn) is a generators of polynomials.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.2. General star product
Now we define a star product similalry by
Definition (←− −→) ∗ = i ~ ∂ Λ∂ f Λ g f exp 2 w w g
Then similary as before we see easily Theorem
For an arbitrary Λ, the star product ∗Λ is a well-defined associative product on complex polynomials.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.2. General star product
Remark
(i) The star product ∗Λ is a genralization of the previous products. Acutally ( ) Λ = 0 1 if we put − then we have the Moyal produt ( ) 1 0 0 2 if Λ = , we have the normal product ( 0 0 ) 0 0 if Λ = then the anti-normal product −2 0
(ii) If Λ is a symmetric matrix, the star product ∗Λ is commutative. Furthermore, if Λ is a zero matirx, then the star product is nothing but a usual commutative product.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.3. Star product representation of the Weyl algebra §3.3. Star product representation of the Weyl algebra
In this section, we fix n = 2m and also fix the antisymmetric part of Λ as J below in order to represent the Weyl algebra. Let K be an arbitrary 2m × 2m complex symmetric matrix. We put a completx matrix Λ = J + K where J is a fixed matrix such that ( ) 0 1 J = −1 0
Since Λ is determined by the complex symmetric matrix K, we ∗ ∗ denote the star product by K instead of Λ . Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.3. Star product representation of the Weyl algebra We consider polynomials of variables (w1, ··· , w2m) = (u1, ··· , um, v1, ··· , vm). By an easy calculation one obtains for an arbitrary K Proposition
∗ ,..., , ,..., (i) For a star product K , the generators (u1 um v1 vm) satisfy the canoical commutation relations
[uk, vl]∗ = −i~δkl, [uk, ul]∗ = [vk, vl]∗ = 0, (k, l = 1, 2,..., m) C , , ∗ (ii) Then the algebra ( [u v] K ) is isomorphic to the Weyl algebra, and the algebra is regarded as a polnomial representation of the Weyl algebra.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.3. Star product representation of the Weyl algebra Equialence
Similarly as before, we have algebra isomorphisms as follows. Proposition For arbitrary star product algebras (C[u, v], ∗ ) and (C[u, v], ∗ ) K1 K2 we have an algebra isomorphism IK2 :(C[u, v], ∗ ) → (C[u, v], ∗ ) K1 K1 K2 given by the power series of the differential operator ∂w(K2 − K1)∂w such that ( ) K2 = i~ ∂ − ∂ IK ( f ) exp 4 w(K2 K1) w ( f ) 1 ∑ ∂ − ∂ = − ∂ ∂ where w(K2 K1) w kl(K2 K1)kl wk wl .
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.3. Star product representation of the Weyl algebra
By a direct calculation we have Theorem Then isomorphisms satisfy the following chain rule: K K K 1 I 1 I 3 I 2 = Id K3 K2 K1 ( )− K 1 K 2 I 2 = I 1 K1 K2
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.3. Star product representation of the Weyl algebra
Remark C , , ∗ 1 By the previous proposition we see the algebras ( [u v] K ) are mutually isomorphic and isomorphic to the Weyl algebra. {Hence we have} a family of star product algebras C , , ∗ ( [u v] K ) K where each element is regarded as a polynomial representation of the Weyl algebra.
2 The above equvalences are also possible to make for star
products ∗Λ for arbitrary Λ’s with a common skew symmetric part.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.4. Star exponentials §3.4. Star exponentials Idea of definition
Using polynomial expressions, we can consider exponential elements of certain polynomials in the Weyl algebra.
For a plynomial H∗ of the Weyl algebra, we want to define a star H∗ t i~ ∑exponential( ) e∗ . However, except special cases, the expansion tn H∗ n n n! i~ is not convergent, so we define a star exponential by means of the differential equation as follows.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.4. Star exponentials
Definition H∗ t i~ The star exponential e∗ is given as a solution of the following differential equation
d = ∗ , = . dt Ft H∗ Λ Ft F0 1
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.4. Star exponentials Examples
We are interested in the star exponentials of linear, and quadratic plonomials. For these, we have explicit solutions. Lienar case ∑ = 2m We denote a linear polynomial by l j=1 a jw j. We see Proposition ∑ = =< , > For l j a jw j a w , the star exponential with respect to the
product ∗Λ is t(l/i~) t2 aKa/4i~ t(l/i~) ∗ = e Λ e e
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.4. Star exponentials Quadratic case
Proposition
For a quadratic polynomial Q∗ = ⟨wA, w⟩∗ where A is a 2m × 2m complex symmetric matrix,
m 1 ⟨ 1 − −2tα , ⟩ t(Q∗/i~) 2 ~ w −κ+ −2tα +κ (I e )J w e∗ = √ e i I e (I ) (1) det(I − κ + e−2tα(I + κ))
where κ = KJ and α = AJ.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions §3.5. Star functions
Using exponentials, we can consider several star functions by the paralell way to the standard method in text book.
There are many application of star exponential functions. Today we show examples using a linear star exponentials. In what follows, we consider the star product for the simplest case where ( ) ρ Λ = 0 0 0 Then we see easily that the star product is commutative and ( ←−−−−→) ∗ = i~ρ ∂ ∂ explicitly given by p1 Λ p2 p1 exp 2 w1 w1 p2.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions
This means that the algebra is essentially reduced to space of functions of one varible w1. Thus, we consider functions f (w), g(w) of one variable w ∈ C and
we consider a commutative star product ∗τ with complex parameter τ such that ←− −→ τ ∂ ∂ 2 w w f (w) ∗τ g(w) = f (w)e g(w)
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions §3.5.1. Star Hermite function
Recall the identity ( √ ) ∑∞ − 1 2 = tn exp 2tw 2 t Hn(w) n! n=0
where Hn(w) is an Hermite polynomial. By applying the explicit t(l/i~) t2 aKa/4i~ t(l/i~) formula of linear case e∗ = e e to l = w, we see K √ ( √ ) = − 1 2 exp∗( 2tw∗)τ=−1 exp 2tw 2 t √ ∑ √ = ∞ n tn Since exp∗( 2tw∗) n=0( 2tw∗) n! we have √ = n Hn(w) ( 2tw∗)τ=−1
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions We define ∗-Hermite function by √ n Hn(w, τ) = ( 2tw∗) , (n = 0, 1, 2, ··· )
with respect to ∗τ product. Then we have √ ∑∞ = , τ tn exp∗( 2tw∗) Hn(w ) n! n=0 Identities √ √ √ d = ∗ Trivial identity dt exp∗( 2tw∗) 2w exp∗( 2tw∗) for the product ∗τ yields the identity √ √τ ′ H (w, τ) + 2wHn(w, τ) = Hn+1(w, τ), (n = 0, 1, 2, ··· ). 2 n for every τ ∈ C.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions
The exponential law √ √ √ exp∗( 2sw∗) ∗ exp∗( 2tw∗) = exp∗( 2(s + t)w∗)
for the product ∗τ yields the identity ∑ n! , τ ∗ , τ = , τ . k!l! Hk(w ) τ Hl(w ) Hn(w ) k+l=n
for every τ ∈ C.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions §3.5.2. Star theta function
We can express the Jacobi’s theta functions by using star exponentials. Using the formula of linear case, a direct calculation gives = − τ/ 2 exp∗τ i tw exp(i tw ( 4)t ) Hence for Re τ > 0, the star exponential = − τ/ 2 exp∗τ ni w exp(ni w ( 4)n ) is rapidly decreasing with respect to integer n and then the summation converges to give ∑∞ ∑∞ ( ) ∑∞ = − τ 2 = n2 2ni w, = −τ exp∗τ 2ni w exp 2ni w n q e (q e ) n=−∞ n=−∞ n=−∞
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions
This is Jacobi’s theta function θ3(w, τ) . Similarly we have expression of theta functions as ∑∞ ∑∞ 1 n θ ∗ = − + , θ ∗ = + 1 τ (w) i ( 1) exp∗τ (2n 1)i w 2 τ (w) exp∗τ (2n 1)i w n=−∞ n=−∞ ∑∞ ∑∞ n θ ∗ = , θ ∗ = − 3 τ (w) exp∗τ 2ni w 4 τ (w) ( 1) exp∗τ 2ni w n=−∞ n=−∞ θ θ , τ = , , , where k∗τ (w) is the Jacobi’s theta function k(w ), k 1 2 3 4 respectively.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions
The exponential law of the star exponential yields trivial identities such that
∗ θ ∗ = θ ∗ = , exp∗τ 2i w τ k τ (w) k τ (w)(k 2 3) ∗ θ ∗ = −θ ∗ = , exp∗τ 2i w τ k τ (w) k τ (w)(k 1 4)
= −τ 2i w Then using exp∗τ 2i w e e and the product formula directly we see the above identities are just
2i w−τθ + τ = θ = , e k∗τ (w i ) k∗τ (w)(k 2 3) 2i w−τθ + τ = −θ = , e k∗τ (w i ) k∗τ (w)(k 1 4)
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions §3.5.3. ∗-delta functions
∗ = − τ 2 Since the τ-exponential exp∗(itw∗) exp(itw 4 t ) is raidly decreasing with respect to t when Re τ > 0. Then the integral of ∗τ-exponential ∫ ∞ ∫ ∞ ∫ ∞ − = − = − − τ 2 exp∗(it(w a)∗) dt exp∗(it(w a)∗)dt exp(it(w a) 4 t )dt −∞ −∞ −∞ converges for any a ∈ C. We put a star δ-function ∫ ∞ δ∗(w − a) = exp∗(it(w − a)∗)dt −∞ which has a meaning at τ with Re τ > 0. It is easy to see for any element p∗(w) ∈ P∗(C),
p∗(w) ∗ δ∗(w − a) = p(a)δ∗(w − a), w∗ ∗ δ∗(w) = 0. Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions Using the Fourier transform we have Proposition
∑∞ θ = 1 − nδ + π + π 1∗(w) 2 ( 1) ∗(w 2 n ) n=−∞ ∑∞ θ = 1 − nδ + π 2∗(w) 2 ( 1) ∗(w n ) n=−∞ ∑∞ θ = 1 δ + π 3∗(w) 2 ∗(w n ) n=−∞ ∑∞ θ = 1 δ + π + π . 4∗(w) 2 ∗(w 2 n ) =−∞ Akira Yoshioka Dept. Math. Tokyo University of Science,n Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
§3.5. Star functions
Now, we consider the τ with the condition Re τ√ > 0. Then( we ) π δ − = 2√ − 1 − 2 calcultate the integral and obtain ∗(w a) τ exp τ (w a) Then we have ∑∞ ∑∞ √ ( ) π θ , τ = 1 δ + π = √ − 1 + π 2 3(w ) 2 ∗(w n ) τ exp τ (w n ) n=−∞ n=−∞ √ ( ) ∑∞ ( ) π = √ − 1 − 1 − 1 2τ2 τ exp τ exp 2n τ w τ n n=−∞ √ ( ) π 2 = √ − 1 θ 2πw , π . τ exp τ 3∗( iτ τ )
We also have similar identities for other ∗-theta functions by the similar way.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
Others
linear case : star special functions, star Eisenstein series. quadratic case: group like object, singularities. etc.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
References References
[1 ] Bayen F., Flato M., Fronsdal C., Lichnerowicz A. and Sternheimer D., Deformation Theory and Quantization I, II, Ann. Phys. 111 (1978) 61–151. [2 ] Iwai T. and Uwano Y., The Four-dimensional Conformal Kepler Problem Reduces to the Three-dimensional Kepler Problem with a Centrifugal Potential and Dirac’s Monopole Field. Classical Theory, J. Math. Phys. 27 (1986) 1523–1529. [3 ] Iwai T. and Uwano Y., The Quantised MIC-Kepler Problem and Its Symmetry Group for Negative Energies, J. Phys. A: Math. Gen. 21 (1988) 4083–4104.
Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application ......
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§1. Introduction: the idea §2. Star calculation of eigenvalues §3. Star products
References [4 ] Kanazawa T. and Yoshioka A., Star Product and Its Application to the MIC-Kepler Problem, J. Geom. Symmetry Phys. 25 (2012) 57–75. [5 ] Kanazawa T., A Specific Illustration of Section Derived from ∗-unitary Evolution Function, In: New Developments in Geometric Mechanics, T. Iwai (Ed), RIMS, Kyoto 2012, pp 172–191. [6 ] Kanazawa T., Green’s Function, Wavefunction and Wigner Function of The MIC-Kepler Problem, J. Geom. Symmetry Phys. 30 (2013) 63–73. [7 ] McIntosh H. and Cisneros A., Degeneracy in the Presence of a Magnetic Monopole, J. Math. Phys. 11 (1970) 896–916. [8 ] ArXiv: Hideki Omori, Yoshiaki Maeda, Naoya Miyazaki Akira Yoshioka Akira Yoshioka Dept. Math. Tokyo University of Science, Japan Star product and its application