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UDC 530.1; 539.1
A METHOD OF INTEGRATION FOR CLASSICAL AND QUANTUM EQUATIONS BASED ON THE CONNECTION BETWEEN CANONICAL TRANSFORMATIONS AND IRREDUCIBLE
REPRESENTATIONS OF LIE GROUPS
A. A. Magazev, I. V. Shirokov
Omsk State Technical University, Mira str., 11, 644050, Omsk, Russia. E-mail: magazev@gmail.com, iv_shirokov@mail.ru
We propose a method for integrating the right-invariant geodesic flows on Lie groups based on the use of a special canonical transformation in the cotangent bundle of group. We also describe an original method of constructing exact solutions for the Klein-Gordon equation on unimodular Lie groups. Finally, we formulate a theorem which establishes a connection between the special canonical transformation and irreducible representations of Lie group. This connection allows us to consider the proposed methods of integrating for classical and quantum equations in the framework of a unified approach.
Keywords: geodesic Bow, the Klein-Gordon equation, canonical transformation, irreducible representation, integrabilitv.
1 Introduction
Numerous researchers in the field of theoretical and mathematical physics often face the problem of integration for differential equations describing various classical and quantum systems. At the same time traditional approaches to solving the classical equations is essentially different from methods of integrating the quantum equations. For example, the general method of integrating the geodesic flows on pseudo-Riemannian manifolds is based on the procedure of the symplectic reduction [1], while the problem of finding exact solutions for relativistic wave equations (Klein-Gordon equation, Dirac equation, etc.) is commonly decided by the technique of separation of variables [2,3].
In this work we discuss the problem of integrating classical and quantum equations on manifolds of Lie groups. Namely, we shall consider Hamilton's equations for the right-invariant geodesic flows on Lie groups and the corresponding Klein-Gordon equation.
The integrable geodesic flows on Lie groups were investigated by many prominent mathematicians such as V. Arnold [4], S. Manakov [5], A. Mishenko and A. Fomenko [6] and others. Here we describe a constructive method for integration of right-invariant geodesic flows in quadratures based on the ideas from the work [7]. Hereafter, following the noncommutative integration method of linear differential equations [8], we develop a technique for constructing the general solution of the Klein-Gordon equation on unimodular Lie groups. In conclusion, we establish a profound connection between this methods which allows us to solve the problems of integrating the classical and quantum equations on Lie group in the framework of a unified approach.
2 Symmetries of classical and quantum equations on pseudo-Riemannian manifolds
Let (M, g) be a pseudo-Riemannian n-dimensional manifold. We denote by U C Ma coordinate chart that trivializes the cotangent bundle T*M, i.e. T*M~ UxK"; the corresponding local coordinates are labelled as (x1,..., x" ,p1,... ,P").
The geodesic flow on (M, g) is defined by the Hamilton's equations
dHcl dpi '
Pi = --
dHcl dxi
(1)
with the Hamiltonian Hcl (x,p) = 1 gij (x)pipj (we assume Einstein summation convention). Here gij is the inverse of the metric tensor gif gikgkj = Sj.
A simplest quantum analogue of Hamiltonian system (1) is the Klein-Gordon equation
H^ := (gijViVj + m2 + (R) ^ = 0.
(2)
Here Vi is the covariant derivative in the direction of the coordinate vector field di := 3/3x\ m is a positive real parameter, R is the scalar curvature of the pseudo-Riemannian manifolds (M, g). Parameter Z is a dimensionless coupling constant.
Let G be the group of motions of the pseudo-Riemannian manifold (M, g). The Lie algebra g of the Lie group G is generated % Killing vectors = (x)di with the commutation relations
[£a,£b] = CCab £c.
(3)
Here CCb are the structure constants of the Lie algebra g.
The Killing vectors allows us to construct constants of motion for Hamiltonian system (1) that
are linear on the fibers of the cotangent bundle T*M:
£'(x,p) = £ (x)p. (4)
We shall call these functions Killing constants of motion. Clearly the span of the Killing constants of motion forms a Lie algebra with respect to the canonical Poisson bracket {•, •}; this Lie algebra is isomorphic to the Lie algebra g:
{scl ,scl}
r^c ¿cl
Cab Sc •
(5)
Furthermore, the vector fields £a considered as differential operators in the functional space CTO(M) are symmetry operators of Eq. (2) since the equality
[H,£,]=0,
holds for all a = 1,..., dimg.
Thus, two algebras are naturally associated with the group of motions of the pseudo-Riemannian manifold (M,g). The first one is the Lie algebra of the Killing constants of motion defined by formula (4); the second one is the symmetry algebra for the KleinGordon equation which is generated by the first-order differential operators £a. Both of these algebras are isomorphic to the Lie algebra g of the group G.
3 Simple transitive actions of Lie groups
G
( M, g)
transitive. This means that for any xi,x2 G M there exists precisely one a G G such that x2 = xia
G
action). It follows that there IS cl smooth one-toM
G
in mind, we represent Eqs. (1), (2) in terms of invariant
G
Let e1,..., en be a basis in the Lie algebra g. Denote by na(x) := (Rx-1 )*ea the right-invariant vector field
G ea
to prove that any right-invariant pseudo-Riemannian gG
g(Va,Vb) = G
ab?
(6)
where G = (Gab) is a constant symmetric non-degenerate n-by-n matrix. In the general case only the left-invariant vector fields £a(x) = (Lx)*ea are the Killing vectors for this metric.
From Eq. (6) it follows that the Hamiltonian of the geodesic flow for the right-invariant metric has the form
Hcl(x,p) = i GabnCl(x,p)nCl(x,p), where nCl(x,p) := tfa(x)Pu GabGbc
(7)
Sa. Since the
Hcl
action of G on T*G, we shall call the corresponding geodesic flow the right-invariant geodesic flow.
It is easily shown that the Klein-Gordon equation for metric (6) can be represent as
H^ = [(Ga6na + Ca) nb + m2 + CR] ^ = 0, (8)
where Ca := Cbab- Note that the scalar curvature for a
G
4 Integration method for the right-invariant geodesic flows
In this section we discuss the problem of integration of arbitrary right-invariant geodesic flows on Lie groups in quadratures. In particular, we describe the method of reduction for the corresponding Hamiltonian system based on using the special canonical transformation in cotangent space T*G.
Denote by g* the dual space to the Lie algebra g of the group G. Let e1,..., en be the basis of g* that is the dual for basis e1,..., en of g: e®(ej) = Sj, i, j = 1,..., n. We define the right momentum mapping Mr : T*G ^ g* by the formula
(Mr(x,p), X) := XaSCl(x,p),
where X = X aea G g. This mapping satisfies the condition o y>, o = o for all y, ^ G
0 denotes the Lie - Poisson bracket
g*
^}0(/)= Cab /c—f/--f = e . W
Similarly, we define the left momentum mapping : T*G ^ g*:
(Ml(x,p),X) := Xancl(x,p),
which satisfies the analogous condition: {^loy, =
Ml o{y,V}fl, y,^ G C~(g*).
It can be proved that the mappings Mr and Ml equivariant with respect to the co-adjoint action Ad* of G
is defined by
(Ad/,X) := (/, Adx-i X),
for all x G G, X G g mid / G g*. It can easily be checked that right and left momentum mappings are connected by the transformation
Ml (x,p) = AdX Mr (x,p).
It is known that any Poisson manifold can be split into a collection of symplectic leaves. A. Kirillov showed that the symplectic leaves of the Lie-Poisson bracket coincide with the co-adjoint orbits of the group
G [9]. Thus, any co-adjoint orbit is a homogeneous symplectic manifold.
Let Ox be the co-adjoint orbit passing through the element A G g*. We denote by wx the symplectic 2-Ox
bracket (9) to the orbit Ox. The 2-form wx is called the Kirillov-Konstant form.
It follows from Darboux's theorem that there are local coordinates q = (ql,...,qm) and n = (ni,..., nm) on orb it Ox such that
Let J = (J\,...,Jr), r = codimOx, be local coordinates in an open neighborhood of the orbit space g*/G. Denote by A( J) some local section of the bundle g* ^ g*/G. We shall consider the function Sx(J) (x; q,n') as a genera,ting function of canonical transformation in the cotangent bundle T*G. This canonical transformation is implicitly defined by Eqs. (15) and the additional equalities
Pi
wx = dna A dqa, a = 1,
, m = 1 dimOx. (10)
dSx(J)(x; q, n') dxi '
dSx(J)(x; q, n')
dJu.
(16)
These coordinates are called canonical coordinates. Denote by fa(q,n; A) the functions which define the transition to the canonical coordinates on the orbit Ox. We restrict attention now to the case of the n-linear transition to canonical coordinates
fa(q,n; A) = X'ü(q)na + xa(q; A).
(11)
where ^ = 1,...,r. Thus, we have the smooth one-to-one coordinate transformation (x,p) ^
(q, n, q', n', J, t) which preserves the canonical T* G
dpi A dx1 = dna A dqa + dn'a A dq'a + dJ^ A dT
After the canonical transformation (15), (16) the Hamiltonian (7) of the right-invariant geodesic flow is converted to the function
The existence of such transition is possible if and only
if there exists a subalgebrap C gc such that (see [10]): HH cl(q',n'; J) = Gabfa(q' ,n'; A(J ))fb(q',n'; A(J ))/2.
The correspondence Hamiltonian system has the form
dim p = dim g — 1 dim Ox,
(A, [p, p]> =0.
(12)
A subalgebra p satisfying the conditions (12) is called a
polarization of element A G g*. Note that polarizations g*
q
formula (11) have the following interpretation. Let P
p
Then q = (qa) are the local coordinates on the left coset Q = G/P. In case of real P, the manifold Q is a Lagrangian submanifold of the symplectic manifold Ox G Q
Ox
q'a =
dH cl(q'
n'; J)
On'
qa = na = J^ = o,
_ Silcl(q',n'; J)
: = oq"- :
dHcl(q',n'; J)
J .
(17)
(18)
AdXf (q,n; A) = f (q',n'; A) ^ q' = xq. Now we introduce the function
(13)
Clearly, the integrability of system (17), (18) is equivalent to the integrability of its subsystem (17).
Recall that an index ind g of the Lie algebra g is defined as the codimension of a regular co-adjoint orbit in g*. The next theorem gives the integrability criterion for an arbitrary right-invariant geodesic flow on the Lie group G.
Theorem 1 An arbitrary right-invariant geodesic flow on T*G is integrable in quadratures if and only if
Sx(x; q,n') = j fa(xq,n'; A)
aa(x),
(14) 1 (dim g — ind g) < 1.
(19)
where fa(q,n; A) are defined by (11) and aa(x) are the right-invariant 1-forms on Lie group such that °a (nb) = a,b = 1,..., dim g. The function (14) is well-defined because the 1-form under the integral is closed.
Lemma 1 The function Sx(x; q,n') satisfies the following relations
dSx(x; q, n') dqa
/a_ dSx(x; q,n')
q = S^L ,
(15)
where the canonical variables (qa, na) and (q'a, n'a) are connected by transformation (IS).
5 Integration of the Klein-Gordon equation on Lie groups
Now we consider the problem of integration for the Klein-Gordon equation on unimodular Lie groups. The most efficient way for constructing exact solutions of the Klein-Gordon equation is the noncommutative integration method of linear differential equations suggested by A. Shapovalov and I. Shirokov [8]. In contrast with the well-known method of separation of variables the noncommutative integration method uses to the maximal extent possible the first-order symmetry algebra of the Klein-Gordon equation.
■-U _
na =
A basic element of the noncommutative integration method is the so-called A-representation of Lie algebra g. We describe the procedure of the construction of this representation.
Let p be a polarization of the regular element A G g*, and let {eA} be a basis in p; A = 1,..., dim p. We define the functional subspace L(G, p) of solutions of the system
(nA(x) - iAA)V(x)=0, ^ G C~(G).
There is a local isomorphism i : L(G, p) ^ L(Q, p) where L(Q, p) is the space of complex-valued functions on the mixed manifolds Q (see [9]). Because the functional subspace L(G, p) is invariant with respect
G
linear operators
4
I ° Ca
d dt
,n (Texp(íec
t=0
4(q; A).
By definition
(^¥>) (q) = ,
DA (x)^(q')dM(q'),
(Ca(x)+ ^a(q; A)) D^, (x) = 0, na(x)+ ¿a(q'; A)) D gq/ (x) = 0.
Moreover, the matrix elements DAq, (x) obey the following the completeness and orthogonality-conditions:
DL(x) Dqq,(x) dM(x) = ¿(q, q)¿(q', q')¿(A, A), (23)
Dgq/(x)D^q,(x) d^(q)d^(q')d^(A) = ¿(x, x).
(24)
Here d^(x) is the invariant Haar measure on group G, and d^(A) is the spectral measure of the Casimir operators of the A-representation. By ¿(x, x), ¿(q, q) and ¿(A,q) we denote the delta-functions for the measures d^(x), d^(q) mid d^(A) respectively.
The relations (23) and (24) allows us to define the direct and inverse Fourier transformations as follows:
^^0= / Dgq/(x)^(x)dM(x)
^(x) = Daq, MV'Afe q')dM(q)dM(q')dM(A).
(25)
(26)
Note that the actions of the operators and after transformation (25) are mapping to actions of the A
£n(x)^(x) ^ in(q; A)^A(q,q'),
(2°) na(xMx) ^ 4(q'; A)^(q,q').
are well-defined. The operators (20) realize an
g
A
coordinates on the manifold Q the operators 4 are ihhomogeneous first-order operators depending on dim Q = (dimg — ind g)/2 independent variables q =
(qa)
Let TA be the lift of the A-representation to a local
G
where DAq, (x) are the matrix elements of the representation TA. Here d^(q) is a measure on manifold Q. We choose the measure d^(q) in such a way that the A-representation operators are anti-Hermitian: it = —ia. In this case the representation T A is a unitary
G
It is easy to verify that the matrix elements of T
Let ^(x) be an arbitrary solution of the KleinGordon equation (8). Using the decompositions (25) and (26) we obtain the equation for an unknown function V>A(q, q'):
[Ga64(q'; A)4(q'; A) + m2 + ZR ^(q, q') = 0.
Note that the variables q = (qa) enter to this equation as parameters.
We say that Eq. (8) is integrable if the problem of finding its general solution is reduced to calculation of quadratures and to solving ODE's. Keeping in mind this definition, we get the next result.
Theorem 2 The Klein-Gordon equation (8) on the unimodular Lie group G is integrable with respect to an arbitrary right-invariant metric if and only if the inequation (19) is satisfied.
6 The connection between the generating
A
representation
We define a measure by the condition
(21) (22)
J ei(q-q/)ndM(n) = ¿(q, q').
The following theorem gives the relation between the generating function SA (x; q, n') of the special canonical transformation in T*G and the matrix elements DAq, (x) of the irreducible represe ntation TA.
Theorem 3 The matrix elements of the irreducible T G
Dgq/ (x) =
d(xq)
dq
-1/2
i(q/n-S (x;q,'
■}}dM(n). (27)
1
o I
e
The Theorem 3 is of fundamental importance in the solving of problem of integrating classical and quantum equations on Lie groups. Indeed, formula (27) gives not only the rule for construction of the matrix elements V^q, (x), but also allows us to consider the methods of integration, proposed in sections 4 and 5, in the framework of a unified approach. In particular, Theorem 3 is explained the fact that the integrability criterion for right-invariant geodesic flows G
corresponding Klein-Gordon equations.
7 Conclusion
In this work we have suggested an integration method for the right-invariant geodesic flows on Lie groups which, in contrast to the symplectic reduction technique, is based on the use of the special canonical
transformation in the cotangent bundle of group.
We also have described an original method of constructing exact solutions for the Klein-Gordon equation on unimodular Lie groups. According to this method, the procedure of integrating the Klein-Gordon equation involves constructing of irreducible representations of Lie group and using the decomposition of the solution space into a sum of irreducible components.
The main result of our work is Theorem 3 which establishes a fundamental connection between the proposed integration methods.
Acknowledgement
This research has been supported by Russian Foundation for Basic Research (grant 14-07-00272-a).
References
[1] Abraham R. et al. 1978 Foundations of mechanics (Addison-Wesley Publishing Company, Inc.).
[2] Miller Jr W. 1977 Symmetry and separation of variables (Addison-Wesley, Reading, Massachusetts).
[3] Bagrov V. G. and Gitman D. 1990 Exact solutions wave equations (Springer).
[4] Arnold V. I. 1966 Annales de I'institut Fourier 16(1) 319.
[5] Manakov S. 1976 Functional Analysis and Its Applications 10(4) 328.
[6] Mishchenko A. S. and Fomenko A. T. 1978 Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 42(2) 396.
[7] Magazev A. A. and Shirokov I. V. Theoretical and Mathematical Physics 136(3) 1212.
[8] Shapovalov A. V. and Shirokov I. V. Theoretical and Mathematical Physics 104(2) 921.
[9] Kirillov A. A. 1976 Elements of the Theory of Representations (Berlin: Springer-Verlag).
[10] Shirokov I. V. 2000 Theoretical and Mathematical Physics 123(3) 754.
[11] Dixmier J. 1977 Enveloping algebras (Newnes).
Received 04.11.2014
А. А. Магазев, И. В. Широков
МЕТОД ИНТЕГРИРОВАНИЯ КЛАССИЧЕСКИХ И КВАНТОВЫХ УРАВНЕНИЙ, ОСНОВАННЫЙ НА СВЯЗИ МЕЖДУ КАНОНИЧЕСКИМИ ПРЕОБРАЗОВАНИЯМИ И НЕПРИВОДИМЫМИ ПРЕДСТАВЛЕНИЯМИ ГРУПП ЛИ
Предложен метод интегрирования иравоинвариантных геодезических потоков на группах Ли, основанный на использовании специального канонического преобразования в кокасательном пространстве группы. Описан также оригинальный метод построения точных решений уравнения Клейна^Гордона на унимодулярных группах Ли. В заключение работы сформулирована теорема, устанавливающая связь между специальным каноническим преобразованием и неприводимыми представлениями группы Ли. Данная связь позволяет рассматривать предложенные методы интегрирования классических и квантовых уравнений в рамках единого подхода.
Ключевые слова: геодезический поток, уравнение Клейна-Гордона, каноническое преобразование, неприводимое представление, интегрируемость.
................ _
Магазев А. А., кандидат физико-математических наук, доцент. Омский государственный технический университет. Пр. Мира, 11, 644050 Омск, Россия. E-mail: magazev@gmail.com
Широков И. В., доктор физико-математических наук, профессор. Омский государственный технический университет. Пр. Мира, 11, 644050 Омск, Россия. E-mail: iv shirokov@mail.ru
