CC BY
21
ISSN 2686-9667. Вестник российских университетов. Математика
Том 24, № 127 2019
© Yoshioka A., 2019
DOI 10.20310/2686-9667-2019-24-127-281-292 УДК 517.9
Star product and star function
Akira YOSHIOKA
Tokyo University of Science 162-8601, Japan, Tokyo, Kagurazaka, Shinjuku, 1-3 e-mail: yoshioka@rs.kagu.tus.ac.jp
Звездочное умножение и звездочные функции
Акира ЙОШИОКА
Токийский научный университет 1-3 Кагуразака, Синдзюку, Токио 162-8601, Япония e-mail: yoshioka@rs.kagu.tus.ac.jp
Abstract. We give a brief review on star products and star functions [8,9]. We introduce a star product on polynomials. Extending the product to functions on complex space, we introduce exponential element in the star product algebra. By means of the star exponential functions we can define several functions called star functions in the algebra. We show certain examples.
Keywords: Moyal product; star product; star product algebra; star exponential functions Acknowledgements: The work is partially supported by Grant-in-Aid for JSPS № 24540097.
For citation: Yoshioka A. Star product and star function. Vestnik rossiyskih universitetov. Matematika - Russian Universities Reports. Mathematics, 2019, vol. 24, no. 127, pp. 281292. DOI 10.20310/2686-9667-2019-24-127-281-292.
Аннотация. Мы даем короткий обзор звездочных умножений и звездочных функций, см. [8,9]. Сначала мы вводим звездочное умножение для многочленов. Затем, распространяя произведение на функции, заданные на комплексном пространстве, мы вводим экспоненты в алгебрах с звездочным умножением. С помощью звездочно показательных функций мы можем определить некоторые функции в этих алгебрах, называемые звездочными функциями. Мы также указываем некоторые примеры. Ключевые слова: умножение Мойяла; звездочное умножение; алгебры со звездочным умножением; звездочно показательные функции
Благодарности: Работа частично поддержана грантом Grant-in-Aid for JSPS № 24540097.
Для цитирования: Йошиока А. Звездочное умножение и звездочные функции // Вестник российских университетов. Математика. 2019. Т. 24. № 127. С. 281-292. DOI 10.20310/2686-9667-2019-24-127-281-292. (In Engl., Abstr. in Russian)
1. Star product on polynomials 1.1. Moyal product
The Moyal product is a well-known example of star product [2,3].
For polynomials f, g of the variables (ui,..., um, vm) , the Moyal product f *O g
is given by the power series of the bidifferential operators - • du -- • dv such that
1 (thV t -
f *o g = f exP Y •dU,- — •dty g = f¿ki (f) u-- •d?) g
k=0 ' ^ '
= fg + th f (- •dl- — •dí) g + ^ ^y^ f (- •di- — •dí^ g
+ ••• + f)k f(- • dUU --u •ty g + ••• (1.1.1)
2! V 2
ih ~2/
where h is a positive number and the overleft arrow d means that the partial derivative is acting on the polynomial on the left and the overright arrow similar, for example
____m
/( d -dl -dty g = Y1 (dVjf duj g d 9uj f dVj g) .
j=i
Although the Moyal product is defined as a formal power series of bidifferential operators, this becomes a finite sum on polynomials.
Proposition 1.1.1. The Moyal product is well-defined on polynomials, and associative.
Other typical star products are normal product *N , anti-normal product *A given similarly by
f *n g = f exp ih( dd -fin) g, f *a g = f exp{ dih(K-<i)} g.
These are also well-defined on polynomials and associative. By direct calculation we see easily
Proposition 1.1.2.
(i) For these star products, the generators (ui,...,um,vi,...,vm) satisfy the canonical commutation relations
[uk,vi]„L = -ihbki, [uk,ui]„L = [vk,vi]„L = 0, (k,l = 1, 2,...,m)
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 Weyl algebra.
Actually the algebra isomorphism
IO :(C[u,v], *0) ^ (C[u,v], *N) is given explicitly by the power series of the differential operator such as
i («V,»^
l=0
And other isomorphisms are given in the similar form.
IN (f) = exp (-|Oud^j (f) = ¿0 ^(du3v)l (f). (1.1.2
Remark 1.1.1. We remark here that these facts are well-known as ordering problem in physics [1].
1.2. Star product
Now we define a star product on complex domain by generalizing the previous products. Let A be an arbitrary n x n complex matrix. We consider a bidifferential operator
_____ ____n__
K Adl = (b^-1, ••• , K )A{dZ, ••• , dWl) = AkiK- (1.2.3)
k,i=i
where (w\, • • • ,wn) is a generators of polynomials. Now we define a star product similar to (1) by
Definition 1.2.1.
f *A g = f exp (KAS) g. (1.2.4)
Remark 1.2.1. [9]
(i) The star product *A is a generalization of the previous products. Actually
• if we put A = ( 0 1m j then we have the Moyal product
\ 1m 0 /
• if A = 2 ( 0 0 ] , then we have the normal product
\ 1m 0 /
• if A = 2 ^ 0 then we have the anti-normal product
(ii) If A is a symmetric matrix, the star product *A is commutative. Then similarly as before we see easily
Theorem 1.2.1. For an arbitrary A, the .star product *A is well-defined on polynomials, and associative.
1.3. Equivalence and geometric picture of Weyl algebra
In this section, we take A as a special class of matrices in order to represent Weyl algebra, cf. [4,7]. We consider the following complex matrices A:
A = J + K,
where K is an arbitrary 2m x 2m complex symmetric matrix and
J = ( 0 -1m
J = 1m 0
Since A is determined by K, we denote the star product by *K instead of *A .
We consider polynomials in variables (wl, ••• ,w2m) = (ul, ••• ,um,vl, ••• ,vm). By a easy calculation one obtains
Proposition 1.3.1.
(i) For a star product *K , the generators (ul,...,um ,vl,...,vm) satisfy the canonical commutation relations
[uk ,vi]*K = -ihbki, [uk ,ui ]*K = [vk ,vi]*K = 0, (k,l = 1, 2,...,m).
(ii) Then the algebra (C[u,v], *K) is isomorphic to the Weyl algebra, and the algebra is regarded as a polynomial representation of the Weyl algebra.
Equialence As in the case of typical star products, we have algebra isomorphisms as follows.
Proposition 1.3.2. For arbitrary star product algebras (C[u,v], *K ) and (C[u,v], *K2) we have an algebra isomorphism IK : (C[u,v], *Ki) d (C[u, v], *K2) given by the power series of the differential operator dw(K2 — Kl)dw such that
IK2(f)=exp(| dw(K2 — Kl)dw) (f),
where dw(K — Kl)dw = Y^ki(K2 — Kl)kidwkdwl . Remark 1.3.1.
1. By the previous proposition we see the algebras (C[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 equivalences are also possible to make for star products *A for arbitrary A's with a common skew symmetric part.
By a direct calculation we have
Theorem 1.3.1. Isomorphisms satisfy the following chain rule: 1■ IK1K iKl = Id, vkuK2,K3
2. {Ik:)-1 = I Ki, ^KuK2
According to the previous theorem, we introduce an infinite dimensional bundle and a connection over it and using parallel sections of this bundle we have a geometric picture for the family of the star product algebras {(C[u, v], *K)}K .
Algebra bundle We set S = {K} the space of all 2m x 2m symmetric complex matrices. We consider a trivial bundle over S whose fibers are the star product algebras
n : E = H (C[u,v], *k) ^ S, n-1(K) = (C[u, v], *k).
K
Then the previous proposition shows that fibers (C[u,v], *K) are mutually isomorphic
and are isomorphic to the Weyl algebra, and the isomorphisms IKl give an isomorphism between fibers.
Connection and parallel sections For a curve C : K = K(t) in the base space S, starting from K(0) = K , we define a parallel translation of a polynomial f E (C[u, v], *K) by
i h
f (t) = exp 4 6W(K(t) - K)dw(f).
It is easy to see f (0) = f .By differentiating the parallel translation we have a connection of this bundle such that
d ih
Vxf(K) = Jtf(t)lt=°(K) = J dwXdw f(K)|t=0, X = K(t)\t=0:
where f (K) is a smooth section of the bundle E .
We set P the space of all parallel sections of this bundle. Since IK are algebra isomorphisms
IK:(f(Ki) *K1 g(Ki)) = {IK: (f(Ki)) *k2 {IK:(g(Ki)),
we have a star product on the space of parallel sections f,g eP by
f * g(K) = f (K) *K g(K).
Then we have
Theorem 1.3.2.
(i) The space of the parallel sections P consists of the sections such that
i h
Vx f = j dw Xdw f = 0, VX.
(ii) The space P is canonically equipped with the star product * , and the associative algebra (P, *) is isomorphic to the Weyl algebra.
Remark 1.3.2. The algebra (P, *) is regarded as a geometric realization of the Weyl algebra.
2. Extension to functions
We want to extend the star products *A for an arbitrary complex matrix A from polynomials to functions, cf. [6].
2.1. Star product on certain holomorphic function space
We want to transfer the star products *A from polynomials to functions. However, the product is not necessarily convergent for ordinary smooth functions, hence we need to restrict the product to certain subset of smooth functions.
There may be many such spaces. In this note we consider the following space of certain entire functions.
Semi-norm Let f (w) be a holomorphic function on Cn. For a positive number p, we consider a family of semi-norms { • |ps}s>0 given by
\f lP,s = sup If (w)l exp(—s|w|p), Iwl = .
wecn
Space We put
Ep = {f : entire \ \ f |ps < , Vs > 0}.
With the semi-norms the space Ep becomes a Frechet space. As to the star products, we have for any matrix A .
Theorem 2.1.1.
(i) For 0 < p ^ 2, (Ep, *A) is a Frechet algebra. That is, the product converges for any elements, and the product is continuous with respect to this topology.
(ii) Moreover, for any A' with the common skew symmetric part with A, the map
I? = j dw(A1 - A)dw}j
is a well-defined algebra isomorphism from (Ep, *A) to (Ep, ) . That is, the expansion convergies for every element, and the operator is continuous with respect to this topology.
(iii) For p > 2, the multiplication *a : £p 'xEq ^ Ep is a ujell-defined for q such that (l/p) + (1/q) = 2 , and (Ep, *a) is a Eq -bimodule.
3. Star exponentials
Since we have a complete topological algebra, we can consider exponential elements in the star product algebra (Ep, *a) , cf. [9].
3.1. Definition
For a polynomial H*, we want to define a star exponential exp* (tH*/ih). However, except special cases, the expansion
+n / TT \ n
t / H*
^ n! V ih
n
is not convergent, so we define a star exponential by means of a differential equation.
Definition 3.1.1. The star exponential exp* (tH*/ih) is given as a solution of the following differential equation
d
-Ft = H* *a Ft, F° = 1. (3.1.1)
dt
3.2. Examples
We are interested in the star exponentials of linear, and quadratic polynomials. For these, we can solve the differential equation and obtain explicit form. For simplicity, we take A as above: A = K + J where K is a complex symmetric matrix.
First we remark the following. For a linear polynomial l = 2= ajWj , we see directly that an ordinary exponential function el satisfies
el / Ei, E Ei+e, We> 0.
Then put a Frechet space
Ep+ ^q>pEq ■
Linear case
Proposition 3.2.1. For l = ■ aj Wj =< a, w > , aj E C , we have
l 2 aK a l
exp t— = exp t • exp t— E Ei+.
ih 4ih ih
Quadratic case
Proposition 3.2.2. For Q* = (wA, w)* where A is a 2mx2m complex symmetric matrix,
2m 11 J - e-2taJ exp t(QJih) = , exp — ( w-—-w
VdetM «M M
where M = I - KJ + e-2ta(I + KJ) and a = AJ
Remark 3.2.1. The star exponentials of linear functions are belonging to £1+ then the star products are convergent and continuous. But it is easy to see
exp^ t(Q*/ih) E £2+, E £2
and hence star exponentials exp^ t(Q*/ih) are difficult to treat. Some anomalous phenomena happen, cf. [5].
4. Star functions
There are many applications of star exponential functions, cf. [8]. In this note we show examples using a linear star exponentials.
In what follows, we consider the star product for the simple case where
A = ( 00 ) • P E C.
Then we see easily that the star product is commutative and explicitly given by
Pi *A P2 = Pi exp ^ ^ ^ dw'ij P2.
This means that the algebra is essentially reduced to the space of functions of one variable w1. Thus, we consider functions f (w) , g(w) of one variable w E C and we consider a commutative star product *T with complex parameter t such that
Recall the identity
f (w) *T g(w) = f (w) exp {2w} g(w).
4.1. Star Hermite function
expf V2tw — 2tA = ^ Hn(w)
r ) n(w)n
n=0
where Hn(w) is an Hermite polynomial. We remark here that
1
exp ^V2tw — ^t2^ = exp^(V2tw*)T=-i.
tn
Since expA^/Ztw*) = Y)(^=0(V2twif)n — we have
' n!
Hn (w) = (V2tw*)
n |
T = -1.
We define * -Hermite function by
Hn(w,T ) = (V2tw*)n, (n = 0,1, 2, •••),
with respect to *T product. Then we have
X fn
exp„(V2tw*) = ^ Hn(w,T) —.
n!
n=°
d
Trivial identity — exp^(v2tw* ) = V2w *exp^(v2tw*) yields at every t E C the identity dt
-=H'n(w, t) + V2wHn(w, t) = Hn+i(w, t), (n = 0,1, 2, •••).
The exponential law exp*(\^2sw*)*exp*(v^2tw*) = exp*(v^2(s + t)w*) yields at every t E C the identity
x—"v n!
Hk(w,T) *t Hi(w,T) = Hn(w,T).
k+l=n
4.2. Star theta function
In this note we consider the Jacobi's theta functions by using star exponentials as an application.
A direct calculation gives
exp*T i tw = exp(i tw — (T/4)t2).
Hence for Re t > 0, the star exponential exp*T ni w = exp(ni w — (t/4)v2) is rapidly decreasing with respect to integer n and then we can consider summations for T satisfying Re t> 0
y^ exp*T 2ni w = exp (2ni w — Tn2) = ^ qn2 e2m w, (q = e-T). This is Jacobi's theta function 93(w,T). Then we have expression of theta functions as
1 x
0i*T (w) = T^ (—1)n exp*^ (2n +1)iw,
n=-X <X
&2*T (w) = ^ exp*T (2n + 1)i w,
n=—x
<X
@3*T (w) = ^ exp*T 2niw,
n=—x
<X
d4*T (w) = ^ (—1)n exp*T 2niw.
n=—cc
Remark that Qk*T (w) is the Jacobi's theta function Qk(w,T), k = 1, 2, 3, 4 , respectively. It is obvious by the exponential law
exp*t 2i w *T dk*T (w) = dk*T (w) (k = 2, 3), exp* 2iw *T 9k*T (w) = —9k*T (w) (k = 1, 4).
Then using exp*T 2i w = e Te2i w and the product formula directly we have
e2i w-Tdk*T (w + iT) = dk*T (w) (k = 2, 3), e2i w-T9k*T (w + iT) = —dk*T (w) (k = 1, 4).
4.3. * -delta functions
Since the *T -exponential exp*(itw*) = exp(itw — (T/4)t2) is rapidly decreasing with respect to t when Re t > 0, then the integral of *T -exponential
/ exp*(it(w — a)*) dt = exp*(it(w — a)*)dt = exp(it(w — a) — (T/4)t2)dt
J —^C J —^C J — tt
converges for any a G C. We put a star 5 -function
5*(w — a) = exp(it(w — a)*)dt,
which has a meaning at t with Re t > 0. It is easy to see that for any element p*(w) £ V* (C) , we have
p*(w) * 5*(w — a) = p(a)5*(w — a), w* * 5*(w) = 0. Using the Fourier transform we have Proposition 4.3.1.
Mw) = 2 J] (—1)n5*(w + n + nn)
n=—x
Mw) = - £ (—1)n5*(w + nn)
n=—x
oo
Mw) = - £ 5*(w + nn)
n=—x tx
04*(w) = - ^ 5*(w + 2 + nn).
n=
Now, we consider the т with the condition Re т > 0 . Then we calculate the integral and
obtain ,
S- ! \ -Vn i 1 , n2 o* (w — a) = —ехЫ — (w — a) тт
ОС
ОС
ОС
Then we have
дз^,т) = 1 ô*(w + nn)= ^Г exp ^—(w + nn)'
2
n
у/т
exp
Л œ ( 1 1 2 1
— I exp ( —2n—w--n т
n I 1\ „ ,2nw n2„
= ^ exp — M—, —).
y'T \ T J iT T
We also have similar identities for other * -theta functions by the similar way.
The author is grateful to V. F. Molchanov and S. Berceanu for valuable discussions and also grateful to the organizers for warm hospitality.
n=
n=
n=
References
[1] G. S. Agarwal, E. Wolf, "Calculus for functions of noncommuting operators and general phasespace methods in quantum mechanics I. Mapping Theorems and ordering of functions of noncommuting operators", Physical Review D, 2:10 (1970), 2161-2186.
[2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, "Deformation theory and quantization. I. Deformations of symplectic structures", Annals of Physics, 111:1 (1978), 61110.
[3] J.E. Moyal, "Quantum mechanics as a statistical theory", Proceedings of the Cambridge Philosophical Society, 45 (1949), 99-124.
[4] H. Omori, "Toward geometric quantum theory", Progress in Mathematics. V. 252: From Geometry to Quantum Mechanics, eds. Y. Maeda, T. Ochiai, P. Michor, A. Yoshioka, Birkhauser, Boston, 2007, 213-251.
[5] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, "Strange phenomena related to ordering problems in quantizations", Journal Lie Theory, 13:2 (2003), 481-510.
[6] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, "Orderings and non-formal deformation quantization", Letters in Mathematical Physics, 82 (2007), 153-175.
[7] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, "Geometric objects in an approach to quantum geometry", Progress in Mathematics. V. 252: From Geometry to Quantum Mechanics, eds. Y. Maeda, T. Ochiai, P. Michor, A. Yoshioka, Birkhauser, Boston, 2007, 303-324.
[8] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, "Deformation Expression for Elements of Algebra", arXiv: math.ph/1104.1708v1.
[9] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, "Deformation Expression for Elements of Algebras (II)", arXiv: math.ph/1105.1218v2.
Список литературы
[1] G. S. Agarwal, E. Wolf, "Calculus for functions of noncommuting operators and general phasespace methods in quantum mechanics I. Mapping Theorems and ordering of functions of noncommuting operators", Physical Review D, 2:10 (1970), 2161-2186.
[2] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, "Deformation theory and quantization. I. Deformations of symplectic structures", Annals of Physics, 111:1 (1978), 61110.
[3] J.E. Moyal, "Quantum mechanics as a statistical theory", Proceedings of the Cambridge Philosophical Society, 45 (1949), 99-124.
[4] H. Omori, "Toward geometric quantum theory", Progress in Mathematics. V. 252: From Geometry to Quantum Mechanics, eds. Y. Maeda, T. Ochiai, P. Michor, A. Yoshioka, Birkhauser, Boston, 2007, 213-251.
[5] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, "Strange phenomena related to ordering problems in quantizations", Journal Lie Theory, 13:2 (2003), 481-510.
[6] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, "Orderings and non-formal deformation quantization", Letters in Mathematical Physics, 82 (2007), 153-175.
[7] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, "Geometric objects in an approach to quantum geometry", Progress in Mathematics. V. 252: From Geometry to Quantum Mechanics, eds. Y. Maeda, T. Ochiai, P. Michor, A. Yoshioka, Birkhäuser, Boston, 2007, 303-324.
[8] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, "Deformation Expression for Elements of Algebra", arXiv: math.ph/1104.1708v1.
[9] H. Omori, Y. Maeda, N. Miyazaki, A. Yoshioka, "Deformation Expression for Elements of Algebras (II)", arXiv: math.ph/1105.1218v2.
Information about the author
Akira Yoshioka, Doctor of Physics
and Mathematics, Professor. Tokyo University of Science, Tokyo, Japan. E-mail: yoshioka@rs.kagu.tus.ac.jp
Received 14 June 2019 Reviewed 25 July 2019 Accepted for press 23 August 2019
Информация об авторе
Йошиока Акира, доктор физико-математических наук, профессор. Токийский научный университет, г. Токио, Япония. E-mail: yoshioka@rs.kagu.tus.ac.jp
Поступила в редакцию 14 июня 2019 г. Поступила после рецензирования 25 июля 2019 г. Принята к публикации 23 августа 2019 г.
