Maxwell–Bloch equations, C Neumann system and Kaluza–Klein theory

Maxwell-Bloch equations, C. Neumann system, and Kaluza-Klein theory Pavle Saksida Department of Mathematics Faculty of Mathematics and Physics University of Ljubljana June, 2005 Abstract The Maxwell-Bloch equations are represented as the equation of motion for a continuous chain of coupled C. Neumann oscillators on the three-dimensional sphere. This description enables us to find new Hamiltonian and Lagrangian structures of the Maxwell-Bloch equations. The symplectic structure contains a topologically non-trivial magnetic term which is responsible for the coupling. The coupling forces are geometrized by means of an analogue of the KaluzaKlein theory. The conjugate momentum of the additional degree of freedom is precisely the speed of light in the mediun. It can also be thought of as the strength of the coupling. The Lagrangian description has a structure similar to the one of the Wess-Zumino-Witten-Novikov action. We describe two families of solutions of the Maxwell-Bloch which are expressed in terms of the C. Neumann system. One family describes travelling non-linear waves whose constituent oscillators are the C. Neumann oscillators in the same way as the harmonic oscillators are the constituent oscillators of the harmonic waves. The 2π-pulse soliton is a member of this family. 1 Introduction The Maxwell-Bloch equations are a well-known system of partial differential equations used in the non-linear optics. Roughly speaking, these equations are a semi-classical 1 model of the resonant interaction between light and an active optical medium consisting of two level atoms. We will consider the following form of the Maxwell-Bloch equations without pumping or broadening: Et + c Ex = P − αE, Pt = ED − βP, 1 Dt = − (EP + EP ) − γ(D − 1). 2 (1) The independent variables x and t parametrize one spatial dimension and the time, the complex valued functions E(t, x) and P (t, x) describe the slowly varying envelopes of the electric field and the polarization of the medium, respectively, and the real valued function D describes the level inversion. The constant c is the speed of light in the medium, α represents the losses of the electric field, while β is the longitudinal and γ the transverse relaxation rate in the medium. In our paper we shall assume that α = γ = 0. We shall consider the spatialy periodic case of (1). The Maxwell-Bloch equations are an integrable system (see [1], [2] [3], [4]). In particular, they satisfy the zero curvature condition. The other integrable system which figures in this paper is the C. Neumann system. The C. Neumann system describes the motion of a particle on the n-dimensional sphere S n under the influence of the force whose potential is quadratic. This oscillator was first described in the 19th century by Carl Neumann in [5]. More recently, many authors studied its different geometrical aspects. See [6], [7], [8], and many other texts. We will show that there is an interesting relationship between the MaxwellBloch equations and the C. Neumann oscillator. Results of this paper are motivated by this relationship. The Hamiltonian system (T ∗ SU (2), ωc , Hcn ), where the function Hcn : T ∗ SU (2) → R is given by 1 Hcn (q, pq ) = kpq k2 + Tr(σ · Adq (τ )), 2 σ, τ ∈ su(2), (2) describes the C. Neumann oscillator moving on the three-sphere S 3 = SU (2). The force potential is given by a quadratic form on R4 whose 4 × 4 symmetric matrix has two double eigenvalues. Our theorem 1 claims that the Maxwell-Bloch equations describe a continuous chain of interacting C. Neumann oscillators of the above type. The oscillators in the chain are parametrized by the spatial dimension of the Maxwell-Bloch equations and the interaction between the oscillators is of magnetic type. By this we mean that the acceleration of a given oscillator is influenced by the velocity and not by the position of the neighbouring oscillators. More concretely, the Maxwell-Bloch equations (1) are the equations of motion for the Hamiltonian system (T ∗ LSU (2), ωc + c ωm , Hmb ). Here LSU (2) = {g: S 1 → SU (2)} is the loop group of 2 SU (2) and the Hamiltonian function Hmb : T ∗ LSU (2) → R is given by Z   1 Hmb (g, pg ) = kpg (x)k2 + Tr(σ · Adg(x) (τ (x))) dx. S1 2 We see that the Hamiltonian is precisely the total energy of our chain of the C. Neumann oscillators parametrized by x ∈ S 1 . The symplectic form ωc + c ωm is a perturbation of the canonical form ωc . The perturbation term ωm is the natural pull-back of the 2-form Ωm on LSU (2) which is right-invariant on LSU (2) and whose value at the identity e ∈ LSU (2) is given by Z Tr(ξ ′ (x) · η(x)) dx, ξ(x), η(x) ∈ Lsu(2) = Te LSU (2). Ωm (ξ, η) = S1 The term ωm is responsible for the magnetic type interaction among the neighbouring oscillators in our chain. At the level of the equations of motion the relationship between the C. Neumann system and the Maxwell-Bloch equations is reflected in the following. The equation of motion of the C. Neumann system (T ∗ SU (2), ωc , Hcn ) is (gt g −1 )t = [σ, Adg (τ )]; g(t) : I −→ SU (2) while the Maxwell-Bloch equations can be rewritten in the form (gt g −1 )t + c (gt g −1 )x = [σ, Ad( τ (x)]; g(t, x): I −→ SU (2). (3) More precisely, the above equation is equivalent to the system (1), if we impose the constraint Tr(gt g −1 · σ) = const. The rewritting (3) shows clearly that the stationary (time independent) solutions of the Maxwell-Bloch equations are solutions of our C. Neumann equation. In this paper we consider the equation (3) without the constraint. This makes the discussion easier and clearer. In addition, we believe that the equation (3), being a description of a chain of oscillators, is interesting in itself. A more interesting illustration of the relationship between the Maxwell-Bloch equations and the C. Neumann system is provided by the solutions of the former given in the proposition 3. These solutions are the non-linear travelling waves whose constituent oscillator is the C. Neumann oscillator in the same way as the harmonic oscillator is the constituent oscillator of the harmonic waves. More precisely, the constituent oscillator turns out to be the electrically charged spherical pendulum moving in the field of the magnetic monopole which is positioned at the centre of the sphere. For small oscillations of the spherical pendulum our solutions indeed behave similarly as the harmonic waves. (Indeed, the linearization around the stable 3 equilibrium of our equation yields the harmonic waves.) But we show in section 4 that the famous 2π-pulse soliton is a particular case of the solutions given in the proposition 3. This solution occurs when the constituent oscillator becomes the planar gravitational pendulum. In addition, its energy must be the energy of the separatrix of the pendulum’s phase portrait. The difference between the symplectic structure of a Hamiltonian system and the canonical structure is called the magnetic term. The momentum shifing argument (see e.g. [9] or [10]) tells us that the magnetic term is responsible for a force which depends linearly on the momenta. An example is the Lorentz force of a magnetic field acting on a moving charged particle. Geometrization of such forces can be achieved by analogues of the Kaluza-Klein theory. This approach provides the configuration space in which the motion of a charged particle under the influence of the magnetic force is described by the geodesic motion. In Hamiltonian terms this means that the relevant symplectic structure will be canonical. The geometrization is achieved by the introduction of an additional circular degree of freedom. The extended configuration space is thus a U (1)-bundle over the original configuration space. A key role is played by the connection which is given on this bundle and whose curvature is precisely the magnetic term. In symplectic geometry, the procedure of adding degrees of freedom and their conjugate momenta is called the symplectic reconstruction - a process inverse to the symplectic reduction. Symplectic reconstruction was studied e.g. in [11], [12], [9]. In the case of the Lorentz force, the moment conjugate to the (single) additional dimension is precisely the electric charge of the moving particle. Therefore the new momentum is usually called the charge. We shall see that in the case of the MaxwellBloch equations the role of the Kaluza-Klein charge is taken by the speed of light. In our case the magnetic term ωm is not exact. The class [Ωm ] is a non-zero element in the cohomology group H 2 (LSU (2)). In such cases the idea of the KaluzaKlein geometrization has to be used with some care. It can be performed only when the magnetic term is an integral 2-form. This follows from a well-known theorem of A. Weil. Our proposition 5 claims that, in general, whenever the magnetic term σm of a system (T ∗ N, ωc + σm , H) is integral, there exists the extended Hamiltonian e whose configuration space is the total space of a U (1)-bundle system (T ∗ M, Ωc , H) M → N . The extended system is invariant with respect to the natural U (1)-action, 2 (N ) is the and (T ∗ N, ωc + σm , H) is its symplectic quotient. The class [σm ] ∈ HDR Chern class of M → N . Our theorem 2 describes the Kaluza-Klein description of e (2) be the central extension of the loop group the Maxwell-Bloch system. Let LSU e of the system (T ∗ LSU e (2), Ωc , H e mb ) be LSU (2). Let the Hamiltonian function H Z   1 2 e Tr σ · Adg❡(τ (x)) dx H mb (e g , pg❡) = kpg❡k + 2 S1 4 e where kpg❡k is given by the natural metric on the central extension Lsu(2) = Lsu(2) ⊕ R. Then this system is invariant with respect to the natural U (1)-action. Its symplectic quotient at the level c of the momentum map is the Maxwell-Bloch Hamiltonian e (2) → system (T ∗ LSU (2), ωc + c ωm , Hmb ) on LSU (2). We note that S 1 → LSU 2 LSU (2) is a non-trivial U (1)-bundle whose first Chern class is [Ωm ] ∈ H (LSU (2)). e (2), Ωc , H e mb ) of the MaxwellThe charge in the Kaluza-Klein description (T ∗ LSU Bloch system has a clear physical interpretation. It is precisely the speed of light in the medium in question. Alternatively, it can be thought of as the strength of the coupling among the neighbouring C. Neumann oscillators. The situation described above is reminiscent of the following finite-dimensional one. Let (T ∗ SU (2), Ωc , Hcn ) be the C. Neumann system on SU (2) = S 3 , with the Hamiltonian given by (2). This system is invariant with respect to the U (1)- action which arises from the Hopf fibration S 1 ֒→ S 3 → S 2 given by g → Adg (τ ). The symplectic quotient is (T ∗ S 2 , ωc + ωm , Hsp ), where 1 Hsp (q, pq ) = kpq k2 + Tr(σ · q) 2 and ωm is the pull-back of the volume form Ωm on S 2 . This system describes the spherical pendulum in the magnetic field of the Dirac monopole situated at the centre of S 2 . The form [Ωm ] ∈ H 2 (S 2 ) is the first Chern class of the Hopf fibration. This construction is described in more detail in [13] and in greater generality in [14]. An important merit of the Kaluza-Klein approach lies in the fact that it clarifies the otherwise elusive Lagrangian description of the systems with non-trivial magnetic terms. In theorem 3 we give the Lagrangian expression of the Maxwell-Bloch e (2). The proof is a straightforward system on the extended configuration space LSU application of the Legendre transformation. We stress the fact that the Lagrangian description of a solution, which is not periodic in time, is possible only on the extended configuration space. The presence of the topologically non-trivial magnetic term makes the Lagrangian description on the primary configuration space LSU (2) more involved. This description is given in theorem 4. The Lagrangian has a structure similar to that of the Wess-Zumino-Witten-Novikov Lagrangian. In particular, it is well-defined only for those solutions of the Maxwell-Bloch equations which are temporally periodic. We note that the results and proofs of Section 5 hold with only minor notational changes for a general Hamiltonian system with a non-trivial (but integral) magnetic term. Throughout this paper the group SU (2) can be replaced by any compact semisimple Lie group G. Thus our construction yields a family of integrable infinitedimensional systems (T ∗ LG, ωc + c ωm , Hgmb ) which satisfy the zero-curvature con5 dition. All these integrable systems are systems with the non-trivial magnetic term ωm ∈ Ω2 (LG) and with the geometric phase. The rewriting (3) of the Maxwell-Bloch equations is already used in the papers [15] and [16] 1 by Q-Han Park and H. J. Shin. There it is interpreted as an equation of a field theory. The connection between the principal chiral field theories on the one hand and the Maxwell-Bloch equations, or more precisely, the self-induced transparency theory of McCall and Hahn, on the other, was already established by A. I. Maimistov in [17]. The authors of [15] and [16] find the Lagrangian of the Maxwell-Bloch equations by means of field-theoretic considerations. Our WZWNtype Lagrangian from theorem 4 is essentialy the same as the one found by Park and Shin. The only difference is that we consider the unconstrained equation (3), while Park and Shin take the constraint Tr(gt g −1 · σ) = const into account. They very elegantly and ingeniously subsume this constraint into the U (1)-gauging part of the WZWN theory. The rewriting (3) enables Park and Shin to describe many important features of the Maxwell-Bloch equations, including soliton numbers, conserved topological and non-topological charges, as well as certain symmetry issues. In [16] they also show that the above mentioned generalizations of the equation (3) to Lie groups G other than SU (2) are, in some cases, relevant to the theory of the resonant lightmatter interaction. In particular, they show explicitely that various non-degenerate and degenerate two and three-level light-matter systems can be described by the equation (3) with the appropriate choice of the group G and of the constant τ . Certain choices of these two constants give rise to the systems whose configuration spaces are supported on symmetric spaces of the form G/H, where H ⊂ G is a suitable subgroup. In terms of our Hamiltonian description, these systems are precisely the symplectic quotients of (T ∗ LG, ωc + c ωm , Hgmb ) with respect to the natural action of LH. 2 A rewriting of the Maxwell-Bloch system In this section we shall express the Maxwell-Bloch equations in a form which will reveal their connection with the C. Neumann system. Let the functions E(t, x) and P (t, x) be complex valued and let the values of 1 The references [15] and [16] were brought to the author’s attention by the referees after the submission of this paper. The author was previously not aware of the existence of these two important papers. 6 D(t, x) be real. We shall consider the Maxwell-Bloch equations Et + c Ex = P, Pt = ED − βP, 1 Dt = − (EP + EP ) 2 (4) with spatially periodic boundary conditions: E(t, x + 2π) = E(t, x), P (t, x + 2π) = P (t, x), D(t, x + 2π) = D(t, x). (5) The system (4) can be rewritten in a more compact form. Let the Lie algebra-valued maps ρ(t, x): R × S 1 → su(2) and F (t, x): R × S 1 → su(2) be defined as     1 iD(t, x) iP (t, x) iβ E(t, x) ρ(t, x) = , F (t, x) = . (6) −iβ −iP (t, x) −iD(t, x) 2 −E(t, x) In terms of these maps the system (4) acquires the form ρt = [ρ, F ], where 1 σ= 2 Ft + c Fx = [ρ, σ]  i 0 0 −i  (7) . We observe that the equation ρt = [ρ, F ] is of the Lax form. Therefore, we have ρ(t, x) = Adg(t,x) (τ (x)), F (t, x) = −gt (t, x) · g −1 (t, x) (8) where τ (x): S 1 → su(2) and g(t, x): R × S 1 → SU (2) are arbitrary smooth matrixvalued functions. If we insert the above into the second equation of the system (7), we obtain the following second-order partial differential equation for g(t, x): R × S 1 → SU (2): (gt g −1 )t + c (gt g −1 )x = [σ, Adg (τ (x))]. (9) This is the new rewriting of the Maxwell-Bloch equations that we shall use in this paper. The equation (9) is slightly more general than the Maxwell-Bloch equations (4). It is equivalent to (4), if we add the stipulation hgt g −1 , σi = const. = −β. We will consider the equation (9) as an equation of motion for the group-valued loop g(x)(t) = g(t, x) ∈ {S 1 → SU (2)} = LSU (2), where LSU (2) denotes the loop group of unbased SU (2) loops. In other words, a solution of the equation (9) is a path g(t, x): I −→ LSU (2), 7 t 7−→ g(t, x). Then for every choice of the loop τ (x): S 1 → su(2), together with a choice of the initial conditions g(0, x) ∈ LSU (2) and gt (0, x) · g −1 (0, x) ∈ Lsu(2), we expect solutions g(t, x) of (9). By Lsu(2) we denoted the loop algebra Lsu(2) = {τ : S 1 → su(2)} which is, of course, the Lie algebra of LSU (2). We conclude this section by pointing out that our rewriting of the Maxwell-Bloch equation yields a whole family of integrable partial differential equations. Let G be an arbitrary semi-simple Lie group and let g(t, x): I × S 1 → G be a smooth map. Let us put c = 1. A straightforward check gives the proof of the following proposition. Proposition 1 Let σ ∈ g be an arbitrary element and let τ : S 1 → g be a loop in the Lie algebra g. The equation (gt g −1 )t + (gt g −1 )x = [σ, Adg (τ (x))] satisfies the zero-curvature condition: Vt − Ux + [U, V ] = 0 where U = −(−zσ + gt g −1 ) 3 1 and V = −zσ + gt g −1 − Adg (τ ). z Hamiltonian structure with the magnetic term We shall now take a closer look at the equation (gt g −1 )t + c (gt g −1 )x = [σ, Adg (τ (x))]. Consider first those solutions g(t): I → SU (2) of (9) which are constant with respect to the x-variable. Clearly, such solutions will exist only in the case when τ (x) ≡ τ is a constant element in su(2). The Lie group valued function g(t) will then be a solution of the ordinary differential equation gt g −1 = [σ, Adg (τ )]. (10) For α, β ∈ su(2), let hα, βi = − 12 Tr(α · β) denote the Killing form on su(2). Proposition 2 The equation (10) is the equation of motion for the Hamiltonian system (T ∗ SU (2), ωc , Hcn ), where the Hamiltonian is given by 1 Hcn (g, pg ) = kpg k2 + hσ, Adg (τ )i 2 8 (11) and ωc is the canonical symplectic form on the cotangent bundle T ∗ SU (2) = T ∗ S 3 . This system is a special case of the C. Neumann oscillator on the three-sphere. In the suitably chosen cartesian co-ordinates on R4 the potential of Hcn assumes the form hσ, Adg(~q) (τ )i = λ(q12 + q22 ) − λ(q32 + q42 ) where λ is a positive real number. Proof: First we shall prove that Hcn is indeed the Hamiltonian of the equation (10) with respect to the canonical symplectic form. Let G be an arbitrary compact semi-simple Lie group and T ∗ G its cotangent bundle. Let T ∗ G ∼ = G × g∗ be the trivialization by means of the right translations. In this trivialization the canonical simplectic form ωc on T ∗ G is given by the formula (ωc )(g,pg ) ((Xb , Xct ), (Yb , Yct )) = −hXct , Yb i + hYct , Xb i + hpg , [Xb , Yb ]i. (12) Above ha, xi denotes the evaluation of the element a ∈ g∗ on the element x ∈ g. For the proof see [18]. Let (M, ω, H) be a Hamiltonian system on the symplectic manifold (M, ω). A path γ(t): I → M is a solution of the equation of motion for our system, if γ̇(t) = XH (γ(t)), where XH is the Hamiltonian vector field defined by dH = ω(XH , −). For the Hamiltonian given by (11) we have hdHcn , (δb , δct )i = −h[σ, Adg (τ )]at , δb i + hδct , p♯g i. (13) Here at: g → g∗ and ♯: g∗ → g are defined by αat = hα, −i and β = hβ ♯ , −i. Let us denote XHcn = (Xb , Xct ) ∈ Γ(T ∗ SU (2)) = Γ(SU (2) × su(2)∗ ), where we use the trivialization by the right translations. Then (ωc )(g,pg ) ((Xb , Xct ), (δb , δct )) = −hXct , δb i + hδct , Xb i + hpg , [Xb , δb ]i = (14) h−Xct − {Xb , pg }, δb i + hδct , Xb i and {a, α} denotes the ad∗ -action of a ∈ su(2) on α ∈ su(2)∗ . Comparing (13) and (14) we obtain p♯g = Xb , [σ, Adg (τ )]at = Xct + {Xb , pg } and from this Xb = p♯g , Xct = [σ, Adg (τ )]at . 9 Let γ(t) = (g(t), pg (t)): I → T ∗ G be a path, and let γ̇ = (gt g −1 , (pg )t ) be its tangent at (g, pg ) expressed in the right trivialization. Then the above equations and (gt g −1 , (pg )t ) = (Xb , Xct ) give us (gt g −1 )t = [σ, Adg (τ )] which proves the first part of our proposition. The proof of the second part is a matter of simple checking. An element g ∈ SU (2) is a matrix of the form   4 X g1 + ig2 g3 + ig4 , det (g) = g= gi2 = 1. −g3 + ig4 g1 − ig2 i=1 Let τ=  ia b + ic −b + ic −ia  . Then hσ, Adg (τ )i is the quadratic form hσ, Adg (τ )i = −Tr(σgτ g −1 ) = 2a(g12 + g22 − g32 − g42 ) + 4b(−g1 g4 + g2 g3 ) + 4c(g1 g3 + g2 g4 ) on R4 restricted to the sphere SU (2) = S 3 ⊂ R4 . The 4 × 4-matrix of this quadratic form has two double eigenvalues √ √ λ = 2 a2 + b2 + c2 and µ = −λ = −2 a2 + b2 + c2 which concludes the proof of the proposition. ✷ Let us now return to the equation (9): (gt g −1 )t = −c (gt g −1 )x + [σ, Adg (τ (x))]. This can now be thought of as the equation of motion of a continuous chain of C. Neumann oscillators parametrized by x ∈ S 1 . At the time t the position of the x0 -th oscillator is g(t, x0 ) ∈ SU (2) ∼ = S 3 . The above equation can be written in the form  c (gt g −1 )t (x) = − gt g −1 (x − ǫ) − gt g −1 (x + ǫ) |ǫ→0 + [σ, Adg(x) (τ (x)]. ǫ For every x the acceleration of the oscillator g(t, x) is determined by the potential [σ, Adg(x) (τ (x))] and by the velocities gt g −1 (x ± ǫ) of the infinitesimally close oscillators. The interaction of the neighbouring oscillators is of magnetic type. It depends on the velocities of the particles and not on their position. 10 This interpretation of the Maxwell-Bloch equation suggests a Hamiltonian structure. The configuration space is the space of positions of the continuous C. Neumann chains. This is the space of maps g(x): S 1 → SU (2), in other words, the loop group LSU (2). Thus the phase space will be the cotangent bundle T ∗ LSU (2). The natural choice of the Hamiltonian is the total energy of all the oscillators: Z   1 Hmb (g(x), pg (x)) = kpg (x)k2 + hσ, Adg(x) (τ (x))i dx. (15) S1 2 Let ωc now denote the canonical cotangent symplectic structure on T ∗ LSU (2). It is easily seen that the equation of motion of the Hamiltonian system (T ∗ LSU (2), ωc , Hcn ) is simply (gt g −1 )t = [σ, Adg (τ (x))]. Therefore the canonical symplectic form ωc has to be perturbed by a form which will account for the interaction term (gt g −1 )x . Let (Ωm )e be the skew bilinear form on the loop algebra Lsu(2) given by the formula Z Z hξx , ηidx, ξ(x), η(x) ∈ Lsu(2). hηx , ξidx = − (Ωm )e (ξ, η) = S1 S1 This bilinear form is a Lie algebra cocycle. Let Ωm be the right-invariant 2-form on LSU (2) whose value at the identity e ∈ LSU (2) is (Ωm )e . Since (Ωm )e is a cocycle, the form Ωm is closed. Let proj: T ∗ LSU (2) → LSU (2) be the natural projection and denote the pull-back proj ∗ (Ωm ) by ωm . The form ωm is then a closed differential 2-form on T ∗ LSU (2). Theorem 1 Let (T ∗ LSU (2), ωc + c ωm , Hmb ) be the Hamiltonian system, where the Hamiltonian H is given by (15), the form ωc is the canonical cotangent form, and ωm is the form described above. Then the equation of motion is the Maxwell-Bloch equation (gt g −1 )t + c (gt g −1 )x = [σ, Adg (τ (x))]. Proof: Let ξ(x) and η(x) be two arbitrary elements of the loop Lie algebra Lsu(2). The inner product on Lsu(2) defined by the formula Z hhξ(x), η(x)ii = hξ(x), η(x)i dx S1 is nondegenerate and Ad-invariant with respect to the group LSU (2). By the same symbol we shall also denote the evaluation hhα, aii of the element α ∈ Lsu(2)∗ on 11 an element a ∈ Lsu(2), as well as the induced inner product on Lsu(2)∗ . Thus the Hamiltonian (15) can be written in the form 1 Hmb (g, pg ) = kpg k2 + hhσ, Adg (τ )ii 2 where kpg k2 = hhpg , pg ii. The canonical cotangent form on T ∗ LSU (2) has the expression analogous to (12), namely (ωc )(g,pg ) ((Xb , Xct ), (Yb , Yct )) = −hhXct , Yb ii + hhYct , Xb ii + hhpg , [Xb , Yb ]ii (16) where (Xb , Xct ), (Yb , Yct ) is an arbitrary pair of tangent vectors from T(g,pg ) (T ∗ LSU (2)) written in the right trivialization. The expression of the symplectic form ωc + c ωm in this trivialization is (ωc + c ωm )(g,pg ) ((Xb , Xct ), (Yb , Yct )) = −hhXct , Yb ii + hhYct , Xb ii +hhpg , [Xb , Yb ]ii − c hh(Xb )x , Yb ii. Similarly as in the proof of proposition 2, we have hhdHmb , (δb , δct )ii = −hh[σ, Adg (τ )]at , δb ii + hhδct , p♯g ii and (ωc + c ωm )(g,pg ) ((Xb , Xct ), (δb , δct )) = −hhXct , δb ii + hhδct , Xb ii + hhpg , [Xb , δb ]ii− −c hh(Xb )x , δb ii = hh − Xct − c (Xb )at x − {Xb , pg }, δb ii + hhδct , Xb ii. Again, because of the independence of δb and δct , the above two equations give p♯g = Xb , at Xct + c (Xb )at x = [σ, Adg (τ )] . (17) Solutions of the Hamiltonian system (T ∗ LSU (2), ωc + c ωm , H) are the paths γ(t; x) = (g(t; x), pg (t; x)): I −→ T ∗ LSU (2) ∼ = LSU (2) × (Lsu(2))∗ which are the integral curves of the Hamiltonian vector field XH of the Hamiltonian H. The condition (gt g −1 , (pg )t ) = (Xb , Xct ) and the equations (17) finally give (gt g −1 )t + c (gt g −1 )x = [σ, Adg (τ )] which proves our theorem. 12 ✷ It is clear that the above theorem holds if the group SU (2) is replaced by any compact semi-simple Lie group G. Every such G is endowed with the Killing form h−, −i and the cocycle Z ωm (ξ, η) = − hξx , ηi, ξ, η ∈ Lg S1 on the corresponding loop algebra is well-defined. The equation (gt g −1 )t + c (gt g −1 )x = [σ, Adg (τ (x)] for g(t, x): I ×S 1 → G is the equation of motion of the system (T ∗ LG, ωc +c ωm , Hgmb ), where Hgmb and ωm are defined in the same way as above. (By Hgmb we denoted the Hamiltonian of the generalized Maxwell-Bloch system.) This system describes a continuous chain of oscillators on G given by (T ∗ G, ωc , Hrs ), where 1 Hrs (g, pg ) = kpg k2 + hσ, Adg (τ )i. 2 These are the well-known integrable systems described by Reymann and SemenovTian-Shansky in [19] and [20]. Connection of such systems with Nahm’s equations of the Yang-Mills theory is studied in [21]. 4 Two families of solutions In this section we omit the spatial periodicity condition. It will be convenient to work with the symplectic reduction of our C. Neumann system which was already mentioned in the introduction. Let us denote the position variable of th C. Neumann system (T ∗ SU (2), ωc , Hcn ) be h ∈ SU (2). The corresponding equation of motion is (ht h−1 )t = [σ, Adh (τ )]. (18) This system is invariant with respect to the actions of the circle groups Uτ (1) = {Exp(s · τ )} and Uσ (1) = {Exp(s · σ)} in SU (2). The action of Uτ (1) is the cotangent lif of the action (ρτ )u (h) = h · u on SU (2). In [13] we show that the moment map µ: T ∗ SU (2) → u(1)∗ is given by µ(h, ht h−1 ) = hht h−1 , Adh (τ )i. 13 (19) Above we identified the cotangents and tangents by means of the Riemannian metric and we shall continue to do so below. The symplectic quotient of (T ∗ SU (2), ωc , Hcn ) with respect to ρτ at the level m of the moment map µ is the Hamiltonian system (T ∗ Sτ2 , ωc + m ωdm , Hsp ), where 1 Hsp (q, pq ) = kpq k2 + hq, σi. 2 Here q = Adh (τ ) ∈ Sτ2 ⊂ su(2) = R3 . This system describes the charged spherical pendulum moving on the 2-sphere Sτ2 under the influence of the gravitational force potential hσ, qi and the Lorentz force caused by the Dirac magnetic monopole positioned at the centre of Sτ2 . The charge of the pendulum is m. This system is described in more detail in [13]. The differentiation qt = [ht h−1 , Adh (τ )] = [ht h−1 , q], and the fact that the map [−, q] : Tq Sτ2 −→ Tq Sτ2 ; v 7−→ [v, q] is a rotation through π2 , give us the expression ht h−1 = −[qt , q] + hht h−1 , qi q = −[qt , q] + m q. (20) Since hht h−1 , σit = h(ht h−1 )t , σi, it is now clear from (18) that e m = hht h−1 , σi = h−[qt , q] + m q, σi Ω (21) (gt g −1 )t + c (gt g −1 )x = [σ, Adg (τ )] (22) is a conserved quantity of our magnetic pendulum. This integral is a perturbation of the angular momentum h[qt , q], σi of the pendulum with respect to the axis of gravitation. The perturbation term m hq, σi stems from the presence of the magnetic monopole. Let now be the Maxwell-Bloch equation in which τ is a constant element of su(2). Our first family of solutions describes the waves whose constituent oscillators are the charged spherical pendula in the field of a magnetic monopole. Let g(t, x) = h(kx − ωt) = h(s) take values in SU (2). Then (gt g −1 )t + c(gt g −1 )x = (ω 2 − k ω c)(hs h−1 )s . 14 The map g(t, x) solves the Maxwell-Bloch equation (30) if and only if h(s) is a solution of the C. Neumann equation (hs h−1 )s = [( 1 ) σ, Adh (τ )]. ω2 − ω k c (23) It is important to note that the solutions g(t, x) = h(kx − ωt) indeed satisfy the constraint hgt g −1 , σi = const. This is insured by the fact that (21) is a conserved quantity. Let us express the solution g(t, x) = h(kx − ωt) in terms of the original physical quantities of the Maxwell-Bloch equations, namely in terms of the electrical field E, the polarization of the medium P , and the level inversion D. To this end it is better to use an appropriate solution of a magnetic spherical pendulum. If h(s) is a solution of (23), then q(s) = Adh(s) (τ ): I → Sτ ⊂ su(2) is an evolution of our pendulum. Let us denote   iq3 (s) q1 (s) + iq2 (s) = Adh(s) (τ ): I −→ Sτ2 ⊂ su(2) ∼ q(s) = = R3 (24) −q1 (s) + iq2 (s) −iq3 (s) and let Ω1 (s) = q2 (s)q̇3 (s) − q3 (s)q̇2 (s), Ω2 (s) = q3 (s)q̇1 (s) − q1 (s)q̇3 (s) be the components of the angular momentum with respect to the two directions perpendicular to gravity. Formulae (6), (8), (20), (22) and (23) now yield the proof of the following proposition. Proposition 3 Let (q1 (s), q2 (s), q3 (s)) : I −→ Sτ ⊂ su(2) = R3 be a solution of the magnetic spherical pendulum with charge m, and the gravitational potential equal to   1 hσ, qi. V (q) = ω2 − ω k c The functions E(t, x) = (Ω1 − m q1 )(ωt − kx) + i(Ω2 − m q2 )(ωt − kx) P (t, x) = q1 (ωt − kx) + iq2 (ωt − kx)) D(t, x) = q3 (ωt − kx) solve the Maxwell-Bloch equations (30). 15 The above solutions describe a family of non-linear travelling waves. The constituent oscillators of these waves are the magnetic spherical pendula in the same way as the harmonic oscillators are the constituent oscillators of the harmonic waves. The phase velocity ω/k of our waves increases with the increasing gravitational potential V (q). When V (q) approaches the infinity, the velocity of the waves approaches the speed of light c in the medium. Now we shall show that the famous 2π-pulse solution of the theory of self-induced transparency appears as a special case of the family described above. Let us consider the symplectic quotient of our C. Neumann system at the zero value of the moment map µ given by (19). In this case the reduced system is the usual spherical pendulum (T ∗ S 2 , ωc , Hsp ) without the magnetic monopole. The conserved quantities of this e qt ) = h[qt , q], σi with system are the energy Hsp and the angular momentum Ω(q, e qt ) = 0, this system is reduced to respect to the axis of gravitation. If we have Ω(q, the usual planar gravitational pendulum. Without the loss of generality, we can take τ = σ and confine the path q given by (24) to the circle   iq3 (s) iq2 (s) : I −→ S 1 ⊂ Sσ2 ⊂ su(2) ∼ q(s) = = R3 . iq2 (s) −iq3 (s) If we parametrize this circle by the angle 2θ , we get the path   i cos 2θ(s) i sin 2θ(s) : I −→ S 1 . q(θ(s)) = Adh(θ(s)) = i sin 2θ(s) −i cos 2θ(s) (25) In this case the suitable lift h(θ(s)): I → SU (2) is clearly given by   cos θ(s) sin θ(s) : I −→ U (1) ⊂ SU (2) h(s) = − sin θ(s) cos θ(s) and thus −1 hs h (s) =  0 θ′ (s) ′ θ (s) 0  : I −→ u(1). (26) Recall that g(t, x): I ×R → U (1) ⊂ SU (2) is a solution of the Maxwell-Bloch equation, if g(t, x) = h(kx − ωt) and h(s) is a solution of the suitable C. Neumann oscillator. Let θ(s): I → R be a solution of the gravitational pendulum whose potential is equal to   1 V (θ) = −κ2 cos θ = cos θ. ω 2 − ωkc Then E(t, x) = θ′ (ωt − kx) P (t, x) = sin θ(ωt − kx) (27) D(t, x) = cos θ(ωt − kx) 16 is a solution of the Maxwell-Bloch equations. This can be seen from the equations (6), (8), (25), and (26). The gravitational pendulum has a well-known homoclinic solution which corresponds to the energy the pendulum has at the unstable equilibrium (when it is at rest on the top of the circle). In other words, this is the solution that travels along the separatrix in the phase portrait of the pendulum. It is well known and indeed not difficult to see that this solution is given by θ(s) = 4 arctan (eκs ) − π. For the calculation see e.g. [22]. If we now put this solution into (27), we finally get the 2π-pulse solitonic solution E(t, x) = 2κ sech(κ(ωt − kx)) P (t, x) = sin (4 arctan (e2κ(ωt−kx) ) − π) D(t, x) = cos (4 arctan (e2κ(ωt−kx) ) − π). Remark 1 We note that the above construction of the solutions which stems from the planar gravitational pendulum corresponds to the well-known reduction of the MaxwellBloch equations to the sine-Gordon equation. Our second family of solutions is simpler and it is obtained by the ansatz g(t, x) = u(t, x) · h(t): I × S 1 → SU (2) where h(t): I → SU (2) solves the C. Neumann system (T ∗ SU (2), ωc , Hcn ). If we insert this into (22) and if we take into account that h(t) solves (ht h−1 )t = [σ, Adh (τ )], we see that u(t, x) must commute with σ and that is satisfies the equation (ut u−1 )t + c(ut u−1 )x + [ut u−1 + c ux u−1 , Adu (ht h−1 )] = 0. Commutation of u with σ gives u(t, x) = Exp(f (t, x) · σ) for some function f (t, x): I × S 1 → SU (2). From the above equation we get the following one for f : (ftt + cftx ) · σ + (ft + cfx ) · [σ, ht h−1 ] = 0. The elements σ and [σ, ht h−1 ] are orthogonal with respect to the Killing form on su(2), therefore, we simply have ft + cfx = 0. This is the ”outgoing part” of the wave equation and its D’Alambert solutions are of the form f (t, x) = w(ωt − kx), where w: R → R is an arbitrary function of one variable. Thus, the mapping g(t, x) = Exp(w(ωt − kx) · σ) · h(t) : I × S 1 −→ SU (2) 17 (28) is a solution of the equation (22) for arbitrary function w and for every solution h(t) of our C. Neumann system. In g(t, x) the solution h(t) of the C. Neumann system is rotated in the vertical direction of the Hopf fibration S 1 ֒→ SU (2) → Sτ2 , given by the projection h 7→ Adh (τ ). Rotation is caused by a harmonic wave which travels with the speed of light c = ω/k. In the case when hgt g −1 , σi = const., which corresponds to the Maxwell-Bloch equations (4), we simply have w(ωt − kx) = ωt − kx + a (29) where a is a constant. Then the above discussion and the expressions (6), (8), (28), and (29) in which we neglect the inessential phase shift a, give us the following result. Proposition 4 Let Et + c Ex = P, Pt = ED − βP, 1 Dt = − (EP + EP ) 2 (30) be the Maxwell-Bloch equations. The functions   E(t, x) = ei2(ωt−kx) 2 (Ω1 (t) − m q1 (t)) + i(Ω2 (t) − m q2 (t)) P (t, x) = ei2(ωt−kx) (q1 (t) + iq2 (t)) D(t, x) = q3 (t) solve (30) for every solution (q1 (t), q2 (t), q3 (t)) : I −→ Sτ2 ⊂ R3 of the magnetic spherical pendulum with the charge equal to m. For the longitudinal relaxation rate β we have the expression em − c β=Ω e m is the value of the integral (21) along our chosen solution (q1 (t), q2 (t), q3 (t)) where Ω of the magnetic spherical pendulum. 5 Hamiltonian structure with the canonical symplectic form As we stressed above, in the Hamiltonian system (T ∗ LSU (2), ωc + c ωm , Hmb ) the canonical symplectic structure ωc on T ∗ LSU (2) is perturbed by the 2-form ωm . Let 18 (T ∗ N, ωc + σm , H) be a Hamiltonian system, where ωc is the canonical structure on T ∗ N and σm is the pull-back of some 2-form Σm on N . Being closed, the form Σm is locally exact, (Σm )q = dθq . Then (again locally) a path q(t): I → N is a solution of the Hamiltonian system (T ∗ N, ωc + σm , H) if and only if it is a solution of the system (T ∗ N, ωc , Hs ), where the Hamiltonian function Hs : T ∗ N → R is given by the formula Hs (q, pq ) = H(q, pq + θq ). For the proof see [10], page 158. This shows that the magnetic terms are responsible for forces which depend linearly on the momentum. The geometrization of such forces is provided by the Kaluza-Klein theory, as mentioned in the introduction. First we shall describe the Kaluza-Klein geometrization in general. We have to consider the magnetic terms which can be topologically non-trivial, since this is the case in the Maxwell-Bloch system. We recall the statement of Weil’s theorem. Let N be a manifold and let Σm ∈ Ω (N ) be an integral 2-form. This means that for every 2-cycle S in N the value R of the pairing S Σm is an integer. Weil’s theorem then ensures the existence of the circle bundle φ: M → N equipped with the connection θ, such that the curvature Fθ is precisely the 2-form Σm . Proof of Weil’s theorem can be found in many texts about the geometric quantization, e.g. in [23]. 2 Weil’s connection θ on M decomposes the tangent bundle Tq M into the horizontal and the vertical part, Tq M = Horq ⊕ Vertq . This decomposition induces the decomposition of the cotangent space Tq∗ M = Hor∗q ⊕ Vert∗q . (31) Note that Hor∗q = Ann(Vertq ) and Vert∗q = Ann(Horq ), where Ann is the annihila∗ tor. Let φ∗ : Tφ(q) N −→ Hor∗q be the adjoint of the derivative (Dφ)q : Tq M → Tq N restricted to Horq . The map φ∗ is of course an isomorphism. Let us define the lifted e on T ∗ M by the formula Hamiltonian H    2 e pq ) = H φ(q), (φ∗ )−1 (Hor∗ (pq )) + Vert∗ (pq ) . H(q, (32) q The natural U (1)-action on M lifts to the action ρ : U (1) × T ∗ M −→ T ∗ M which is Hamiltonian with respect to the canonical structure ωc on T ∗ M . Let µ : T ∗ M −→ u(1) = iR be the moment map of ρ. Weil’s theorem enables us to state the following claim. Proposition 5 Let (T ∗ N, ωc + σm , H) be a Hamiltonian system and let the magnetic e term Σm be an integral 2-form on N . Then the Hamiltonian system (T ∗ M, Ωc , H) 19 e given by (32), is whose symplectic structure Ωc is canonical and whose Hamiltonian H, −1 invariant with respect to the action ρ. Its symplectic reduction (µ (ia)/U (1), ωsq , Hr ) is the original system (T ∗ N, ωc + a σm , H). e with respect to the action ρ is a direct consequence of Proof: The invariance of H the fact that the connection θ is invariant with respect to ρ. Whenever the action on the cotangent bundle is lifted from the action on the base space, the moment map µ: T ∗ M → iR is given by µ(q, pq )(ξ) = pq (ξN ), where ξN is the infinitesimal action on the base space. In our case, this gives µ(q, pq ) = pVq , V where pVq = Vert∗q (pq ) is the vertical part of the decomposition pq = pH q + pq given e induces the function H + a2 on the symplectic quotient by (31). This shows that H µ−1 (ia)/U (1). This function differs from our original Hamiltonian by an irrelevant constant. Now we have to prove that the symplectic quotient of (T ∗ M, Ωc ) is indeed (T ∗ N, ωc + a σm ). Let ϑ ∈ Ω1 (T ∗ M ) be the tautological 1-form. Then dϑ = Ωc . For every pair of tangent vectors X(q,pq ) , Y(q,pq ) ∈ T(q,pq ) (T ∗ M ) the well-known formula for the derivative of 1-forms gives   (33) (Ωc )(q,pq ) (X(q,pq ) , Y(q,pq ) ) = X̂(ϑ(Ŷ ) − Ŷ (ϑ(X̂) − ϑ([X̂, Ŷ ]) |(q,pq ) where X̂, Ŷ are the arbitrary vector fields in a neighbourhood of (q, pq ) which extend our tangent vectors. Choose a local trivialization of T (T ∗ M ) and denote X(q,pq ) = (Xb , Xct ), where Xb ∈ Tq M and Xct ∈ Tq∗ M . The tautological form is defined by ϑ(q,pq ) (Xb , Xct ) = pq (Xb ). We can decompose it into the horizontal and the vertical part, ϑ = ϑH + ϑV , by putting H ϑH (q,pq ) (Xb , Xct ) = pq (Xb ), ϑV(q,pq ) (Xb , Xct ) = pVq (Xb ) ∗ ∗ V where pH q ∈ Horq and pq ∈ Vertq . Let us choose the extension vector field X̂ of X(q,pq ) = (Xb , Xct ) defined in some neighbourhood of (q, pq ) in the following way: decompose first Xb = XbH + XbV into the horizontal and the vertical parts. Choose a vector field extending (Dφ)q (XbH ) on N and let X̂bH be its unique U (1)-invariant horizontal lift. The stipulation for the extension X̂bV of XbV is the following: the restriction of the function pVq (XqV ) on µ−1 (ia) ⊂ T ∗ M must be constant. Let now X̂b = X̂bH + X̂bV . Define the field X̂ctH analogously to the definition of X̂bH using the isomorphism φ∗ , let XctV be an arbitrary vertical extension of XctV , and let finally X̂ct = X̂ctH + X̂ctV . We construct Ŷ in the same manner as X̂. Then we have [X̂bV , ŶbV ] = 0 and [Ŷb , X̂b ] = [ŶbH , X̂bH ]. 20 (34) The first equation is obvious. For the second, denote by Φ(s) the flow of the vector field YbV and by ϕ(s) the integral curve of YbV beginning at q. Then   d [ŶbV , X̂bH ] = |s=0 (Dϕ(s) (Φ−1 (s)) X̂bH (ϕ(s)) = 0 ds since X̂bH is U (1)-invariant. The second equation of (34) now follows immediately. From our construction of the fields X̂ = X̂ H + X̂ V and Ŷ = Ŷ H + Ŷ V it also follows: H X̂ V (pH X̂ H (pVq (ŶbV )) = 0 on µ−1 (ia). (35) q (Ŷb )) = 0, H The first equation is true because the function pH q (Ŷb ) is invariant with respect to the action ρ and the field X̂ V is colinear with the infinitesimal action of ρ. The second follows from the fact that pVq (ŶbV ) is constant on µ−1 (ia). We can express ϑH and ϑV slightly more explicitely: H H ϑH (q,pq ) (Xb , Xct ) = pq (Xb ), ϑV(q,pq ) (Xb , Xct ) = pVq (XbV ). (36) Define the projection map Ψ: T ∗ M → T ∗ N by   ∗ −1 H Ψ(q, pq ) = φ(q), (φ ) (pq ) where φ∗ is again the adjoint of the derivative Dq φ restricted to Horq ⊂ Tq M . Formulae (33), (34), (35) and (36) now give   ∗ (X , Y ) = Ψ (ω ) dϑH (X(q,pq ) , Y(q,pq ) ) c (q,p ) (q,p ) q q (q,pq ) (q,pq ) and     i∗ dϑV(q,pq ) (X(q,pq ) , Y(q,pq ) ) = (i pVq ) · (Ψ∗ )(σm ) (q,pq ) (X(q,pq ) , Y(q,pq ) ) where i: µ−1 (ia) → T ∗ M is the inclusion. Here we have used the fact that Σm is the curvature of the connection θ and is therefore given by (Σm )q (Xq , Yq ) = e Hor(Ye )]q ), where X e and Ye are arbitrary vector fields on M extending Vertq ([Hor(X), ∗ Xq , Yq ∈ Tq M . Note that σm = π (Σm ), and that (i pVq ) = a is a real number. Recall now that Ωc = dϑ = dϑH + dϑV . The above expressions show that for every ia ∈ u(1) the pull-back i∗ (Ωc ) via the inclusion map i: µ(ia)−1 → T ∗ M satisfies the relation i∗ (Ωc ) = Ψ∗ (ωc + a σm ). Finally, we note that the natural projection Π: T ∗ M → T ∗ N of the action ρ is precisely the map Ψ. Therefore the above formula completes the proof of the theorem. 21 ✷ Now we shall describe the Kaluza-Klein expression for the Maxwell-Bloch system. It will be instructive to construct it directly, without referring to the proposition 5. The 2-form Ωm ∈ Ω2 (LSU (2)) plays an important role in the theory of the loop e (2) group LSU (2). It is essentially the cocycle associated to the central extension LSU of LSU (2). The central extension e R −→ Lsu(2) = Lsu(2) ⊕ R −→ Lsu(2) of the Lie algebra Lsu(2) is given by 1 1 [(ξ, λ), (η, µ)] = ([ξ, η], (ωm )e (ξ, η)) = ([ξ, η], − 2π 2π Since the skew form extension 1 (ωm )e 2π Z S1 hξx , ηidx). (37) is an integral cocycle on Lsu(2), it defines the central φ e (2) −→ S 1 −→ LSU LSU (2) (38) e (2) is the U (1) principal on the group level. Geometrically, the central extension LSU bundle over LSU (2), equipped with a right-invariant connection θ whose value at the e (2) is given by identity e ∈ LSU e = θ(X, x) = x, θ(X) e ∈ Te LSU e (2) = Lsu(2) e X = Lsu(2) ⊕ iR. e (2). Alternatively, the connection θ is given by the right-invariant distribution in T LSU e (2), it is given by At the identity e ∈ LSU e (2) = Lsu(2) e Te LSU = Lsu(2) ⊕ R = (Horθ )e ⊕ (Vertθ )e . The curvature of θ is equal to the 2-form i Ωm . e (2) → T ∗ LSU e (2) the cotangent lift of the natural Let us denote by ρ: U (1)×T ∗ LSU U (1)-action. We note that we only need the expression of the infinitesimalization at e (2) of this action. However, the reader can easily find the the identity e ∈ LSU e (2) from the information given in [24]. formula for the entire action on LSU e (2) and Clearly, ρ preserves the canonical symplectic structure Ωc on T ∗ LSU ∗e is therefore Hamiltonian. The moment map µ : T LSU (2) −→ iR is given by µ(e g , pg❡) = pg❡(ξρ ), where the vector field ξρ is the infinitesimal action on the base e (2). Let us trivialize the tangent and the cotangent bundles of LSU e (2) space LSU 22 e (2) ∼ by the right translations. Then for every ge we have Tg❡LSU = Lsu(2) ⊕ iR and ∗e ∗ ∼ Tg❡ LSU (2) = (Lsu(2)⊕iR) . Under this identification we have pg❡ = (pg , ψ), ξρ = (0, 1) and therefore µ(e g , pg❡) = ψ. e (2) with Now we shall decompose the canonical symplectic structure Ωc on T ∗ LSU e (2). We shall apply the respect to the natural connection θ on the circle bundle LSU formula (12) for the canonical form on the cotangent bundle over a Lie group to the e (2). In the right trivialization, case when the Lie group is the central extension LSU ∗ e b, X e ct ) ∈ T(g❡,p ) (T ∗ LSU e (2)) = Lsu(2) e e an element (X × (Lsu(2)) has the form g ❡ e b, X e ct ) = ((Xb , xb ), (Xct , xct )), (X Xb ∈ Lsu(2), Xct ∈ (Lsu(2))∗ , xb , xct ∈ R. Formula (12) and the Lie algebra bracket (37) of the central extension then give (Ωc )(g❡,pg❡) = −hXct , Yb i + hYct , Xb i + hpg , [Xb , Yb ]i −xct yb + yct xb Z 1 −ψ · h(Xb )x , Yb idx 2π S 1 e (2) → where pg❡ = (pg , ψ) ∈ (Lsu(2) ⊕ R)∗ . Let the projection map F : T ∗ LSU ∗ T LSU (2) in the right trivializations be given by F (e g , pg❡) = F (e g , (pg , ψ)) = (φ(g), pg ). The above formulae give (Ωc )(g❡,pg❡) = F ∗ (ωc )(g❡,pg❡) + (ωf ib )(g❡,pg❡) + ψ · F ∗ (ωm )(g❡,pg❡) . (39) Here ωc is the canonical structure on T ∗ LSU (2). The second term ωf ib is the canonical cotangent form on the fibre of the map F . For every (g, pg ) ∈ T ∗ LSU (2), the fibre F −1 (g, pg ) is the cotangent bundle T ∗ S 1 over the circle. Finally, F ∗ (ωm ) is the pulle (2) → LSU (2). Recall that ωm back of the curvature ωm of the connection θ on LSU is also the perturbation form in the Maxwell-Bloch Hamiltonian system. e (2) with respect to the action Consider now the symplectic quotient of T ∗ LSU ρ. Let ωsq denote the induced symplectic structure on the symplectic quotient µ−1 (ψ)/U (1). The decomposition (39) proves the following result. e (2) → R be the moment map of the natural action Proposition 6 Let µ: T ∗ LSU ∗e ∗e ρ: U (1)×T LSU (2) → T LSU (2). Then for the symplectic quotient (µ−1 (ψ)/U (1), ωsq ) e (2), Ωc ) we have of (T ∗ LSU (µ−1 (ψ)/U (1), ωsq ) = (T ∗ LSU (2), ωc + ψ ωm ). 23 The above proposition gives us now the expression of the Maxwell-Bloch Hamiltonian system in terms of a canonical symplectic structure. e (2), Ωc , H) e be the Hamiltonian system on T ∗ LSU e (2), where Theorem 2 Let (T ∗ LSU e mb : T ∗ LSU e (2) −→ Ωc is the canonical cotangent symplectic structure and the function H R is given by the formula 1 e mb (e H g , pg❡) = kpg❡k2 + hhσ, Adg❡(τ )ii 2 with σ = 21 diag(i, −i) ∈ su(2) and τ ∈ Lsu(2) an arbitrary loop. Then the moe (2) → R of the U (1)-action ρ is an integral of the system ment map µ: T ∗ LSU ∗e e mb ). For the reduced Hamiltonian system we have (T LSU (2), Ωc , H (µ−1 (ψ)/U (1), ωsq , Hsq ) = (T ∗ LSU (2), ωc + ψ ωm , Hmb ) where (T ∗ LSU (2), ωc + ψ ωm , Hmb ) is the system whose equation of motion is (gt g −1 )t + ψ(gt g −1 )x = [σ, Adg (τ )]. When ψ = c, this is precisely the Maxwell-Bloch equation. e (2) Remark 2 The Kaluza-Klein charge of the additional degree of freedom in LSU is ψ. We can write the above equation in the form  1 (gt g −1 )t (x) = ψ gt g −1 (x − ǫ) − gt g −1 (x + ǫ) |ǫ→0 + [σ, Adg(x) (τ (x))]. ǫ This shows that the charge ψ is the strength of the magnetic interaction between the neighbouring C. Neumann oscillators in the chain. An even clearer description says that the momentum ψ is equal to the speed of light in the medium. The fact that ψ is e (2), Ωc , H) e coincides with the fundamental an integral of the extended system (T ∗ LSU physical law which says that the speed of light ψ = c in the medium is constant. e is invariant Proof of Theorem 2 : We only have to check that the Hamiltonian H with respect to the U (1)-action ρ. For the kinetic energy we have kpg❡k2 = k(pg , ψ)k2 = kpg k2 + ψ 2 which is clearly invariant. In the potential energy term we have the adjoint action of e (2) on an element from (Lsu(2)). e LSU The adjoint action is given by the formula Z   1 −1 e hg gx , βidx . Adg❡(β) = Adφ(g❡) (β, b) = Adg (β), b − 2π S 1 24 e (2) of LSU (2) is central and This can be seen from the fact that the extension LSU e from the formula (37) for the Lie bracket in Lsu(2). Tha natural inclusion of the e element σ ∈ su(2) into the group Lsu(2) fas the form i(σ) = (σ, 0) ∈ Lsu(2) ⊕ R. e Recall that the inner product on Lsu(2) is given by Z hα, βi dx + a · b. (40) hh(α, a), (β, b)ii = R S1 From this we see hhσ, Adg❡(τ (x))ii = S 1 hσ, Adφ(g❡) (τ (x))i dx. This expression is clearly invariant with respect to the action ρ. (The orbits of ρ are φ−1 (g).) The statement of the theorem now follows directly from proposition 6. ✷ In the paper [26] the authors describe a Hamiltonian structure of the MaxwellBloch equation, but their structure is different from the one constructed above. A quick way to establish the nonequivalence of the two structures is to observe that the symplectic structure in [26] does not include the derivatives of the variables with respect to x co-ordinate, while our symplectic structure does. The fact that the Maxwell-Bloch equations are endowed with two nonequivalent Hamiltonian structures is of course very important. We intend to study this topic in another paper. 6 Lagrangian structure of the Maxwell-Bloch equations In this section we shall investigate the Lagrangian structure of the equation (9). To e (2) is simplify the notation we put c = 1. The fact that the magnetic term ωm ∈ LSU topologically non-trivial will play a crucial role. The Lagrangian expression of systems with non-trivial magnetic terms was studied by Novikov in [29]. Although we focus on the Maxwell-Bloch system, our construction of the Lagrangian formulation works for any Hamiltonian system with an integral non-trivial magnetic term. Our construction is different from the one described in [29]. The essential ingredient in our approach is the Kaluza-Klein extension, which makes the problem quite straightforward. The Lagrangian expression of the Maxwell-Bloch equations on the original, nonextended configuration space LSU (2) is more intricate, if less general. In particular, it works only for the temporally periodic solutions of the Maxwell-Bloch equations. It has essentially the same structure as the WZWN-model which was introduced by Witten in [27] and [28]. Again, our construction could be applied to arbitrary Hamiltonian systems with non-trivial magnetic terms. 25 We shall start by applying the Legendre transform to the Kaluza-Klein expression e mb ) of the Maxwell-Bloch system. Let T LSU e (2) be the tangent (T LSU (2), Ωc , H e (2) by the right transbundle. As before, we will work in the trivialization of T LSU e e (2), we have the inner product given by lations. On the Lie algebra Lsu(2) = Te LSU e (2) of the right-invariant metric on (40). By hh−, −iig❡ we denote the value on Tg❡LSU e (2) whose value at the identity is given by (40). Note that the metric hh−, −ii LSU g ❡ is not bi-invariant, since the inner product (40) is not Ad-invariant. We can use our metric for the identification ∗e e (2) = {pg❡ · (e Tg❡∗ LSU g −1 )∗ = hhe g t ge−1 , −ii, e (2)}. get ge−1 ∈ T LSU e (2) → R be given by Let now the Lagrangian L: T LSU Z 1 L(e g , get ) = hhe hσ, Adφ(g❡) (τ (x))idx. g , ge ii − 2 t t g❡ S1 (41) e (2) −→ T ∗ LSU e (2) In the right trivializations, the Legendre transformation F L : T LSU −1 −1 ∗ −1 is given by F L(e g t ge ) = pg❡ ·(e g ) = hhe g t ge , −ii. This gives us the following theorem. e (2) be a solution of the Theorem 3 Let the path γ(t) = (e g (t), pg❡(t)) : I −→ T ∗ LSU e (2), Ωc , H), e and let proj: T ∗ Lg e C LSU (2) → LSU e (2) be Hamiltonian system (T ∗ LSU the natural projection. Then the path e (2) proj(γ(t)) = ge(t) : I −→ LSU is an extremal of the Lagrangian functional Z L(e g (t)) = L(e g (t), get (t))dt (42) I where the function L is given by (41). We shall now take a closer look at the closed extremals of the Lagrangian functional e (2) for which the (42), that is, we will be interested in the loops ge(t) : S 1 −→ LSU value L(e g (t)) is minimal. We shall see that the closed extremals of (42) can be characterized as the extremals of a Lagrangian functional on the non-extended loop group LSU (2). But this Lagrangian will be of a non-standard kind in a similar way that the WZWN functional is. We will prove the following theorem. 26 Theorem 4 Let g(t) : S 1 −→ LSU (2) be a loop in LSU (2). Let D ⊂ R2 be a disc whose boundary is our circle, ∂D = S 1 , and let ĝ : D −→ R be an extension of g to the disc D. Then Z  Z  1 −1 2 ωm (43) L(g(t)) = kgt g k − hhσ, Adg(t) (τ )ii dt + S1 2 ĝ(D) is a well-defined map L : {Loops in LSU (2)} −→ R/Z = S 1 . Furthermore, a loop g(t): S 1 → LSU (2) is an extremal of L if and only if it is a solution of the Maxwell-Bloch equation (gt g −1 )t + (gt g −1 )x = [σ, Adg (τ )]. Proof: The loop group LSU (2) can be endowed with the structure of a Banach manifold in several different ways. (See [24], [25]). Throughout this paper we assume that LSU (2) is equipped with a suitable Banach manifold structure which makes ωm a smooth 2-form. Let {Uα ; α ∈ A} be an open covering of LSU (2) by contractible α α open sets Uα . Consider the family of Hamiltonian systems (T ∗ Uα , ωcα + ωm , Hmb ), α α ∗ α where ωc + ωm denotes the restriction of ωc + ωm to T Uα , and Hmb is the restriction of the Hamiltonian function Hmb . The form ωm is closed on LSU (2), therefore its restriction to any contractible subset Uα is exact by the Poincaré lema. We have α = dθα . ωm Recall now the momentum shifting argument for the Hamiltonian systems with magnetic terms. Let M be a manifold and let Tθ : T ∗ M → T ∗ M be a map defined by the formula Tθ (q, pq ) = (q, pq − θq ). Let H: T ∗ M → R be a Hamiltonian function and let Hθ (q, pq ) = H(q, pq + θq ). Then Tθ pulls the function Hθ back to H and the canonical form ωc back to the magnetically perturbed form ωc + dθ. It is clear that a path q(t): I → T ∗ M is a solution of the Hamiltonian system (T ∗ M, ωc + dθ, H) if and only if it is also a solution of the Hamiltonian system (T ∗ M, ωc , Hθ ). Thus, α α for every α ∈ A the Hamiltonian system (T ∗ Uα , ωcα + ωm , Hmb ) is equivalent to the ∗ ∗ Hamiltonian system (T Uα , ωc , Hα ), where Hα : T U → R is given by Hα (g, pg ) = (Hmb )/Uα (g, pg + θgα ). By means of the Legendre transformation we can now recast our restricted Hamiltonian systems into the Lagrangian form. We have the following result. A path g(t): I → Uα is a solution of the Hamiltonian system (T ∗ Uα , ωc , Hα ) ∼ = α α (T ∗ Uα , ωcα + ωm , Hmb ) if and only if it is an extremal of the Lagrangian functional Lα : {Paths on Uα } → R given by Z   1 −1 2 α kgt g k + θ (ġ(t)) − hhσ, Adg(t) (τ )ii dt. Lα (g(t)) = I 2 27 We can rewrite this Lagrangian somewhat more invariantly as Z Z   1 −1 2 θα . kgt g k − hhσ, Adg(t) (τ )ii dt + Lα (g(t)) = g(I) I 2 Note that θα is determined only up to a closed 1-form. But on the contractible Uα every closed 1-form is also exact. For every 0-form (i. e. a function) β on Uα , we have Z Z β = β(g(b)) − β(g(a)). dβ = g(I) ∂g(I) Therefore, the Lagrangians Lα corresponding to various possible choices of θα differ only by irrelevant constants when the enpoints of the paths g(I) are fixed, and they do not differ at all when we consider the closed paths g(S 1 ). Now we will show that the family of local Lagrangians Lα : {Paths on Uα } → R gives rise to a global Lagrangian L : {Loops on LSU (2)} −→ R/Z = S 1 . Let g: S 1 → LSU (2) be a loop in LSU (2) and let ĝ: D → LSU (2) be an extension of g on the disc D, bounded by our S 1 . Then ĝ(D) is a two-dimensional submanifold in LSU (2) whose boundary is the loop g(S 1 ). Since ĝ(D) is compact, it is covered by a finite subfamily {Uα ; α ∈ A′ } of the covering {Uα }α∈A . The disc D is two-dimensional, therefore S we can assume that at most three different Uα have non-empty intersection. Let α∈A′ Dα = D be a partition of the disc D into a union of curvilinear polygons Dα , such that for every α ∈ A′ we have ĝ(Dα ) ⊂ S Uα , and such that the interiors of the polygons Dα are disjoint. A suitable partition α∈A′ Dα = D is given by the nerve of the covering {Uα ; α ∈ A′ }. In the group of one-dimensional chains in LSU (2) we then have X (∂ĝ(Dα )). g(S 1 ) = ∂ĝ(D) = α∈A′ R R For every α ∈ A′ the theorem of Stokes gives ∂ĝ(Dα ) θα = ĝ(Dα ) ωm . But unlike θα , the form ωm is globally defined. Therefore, we can define Z  Z  1 1 −1 2 L̆(g(S )) = kgt g k − hhσ, Adg(t) (τ )ii dt + ωm . S1 2 ĝ(D) This functional is of course dependent on the choice of the extension ĝ of the map g: S 1 → LSU (2). Let ǧ: D → LSU (2) be another extension of g. Then the chain ǧ(D) − ĝ(D) is a smooth map ǧ(D) − ĝ(D) = g̊(S 2 ) : S 2 −→ LSU (2) 28 of a two-sphere into LSU (2). Now, LSU (2) is diffeomorphic to SU (2) × ΩSU (2), where ΩSU (2) denotes the group of the based loops in SU (2). Since for the singular homology with integer coefficients we have H3 (SU (2)) = H3 (S 3 ) = Z, we also get H2 (ΩSU (2) ∼ = H3 (SU (2)) = Z, and finally H2 (LSU (2) = H2 (SU (2)) × H2 (ΩSU (2)) = Z. The form ωm is closed, but not exact. Therefore, Z Z Z Z ωm ∈ Z ωm − ωm = ωm = ǧ(Dα ) ĝ(Dα ) ǧ(Dα )−ĝ(Dα ) g̊(S 2 ) and, in general, this integer is different from zero. This shows that for different choices of the extension of the loop g on the disc D the values of the functional L̆ : {Loops on LSU (2)} → R can differ by integers. Therefore, the composition L̆ κ {Loops on LSU (2)} −→ R −→ R/Z = S 1 in which κ: R → R/Z = S 1 is the natural projection, is independent of the choice of the map ĝ extending the loop g. This proves that the Lagrangian functional L = κ◦ L̆ given by the formula Z  Z  1 −1 2 L(g) = ωm kgt g k − hhσ, Adg (τ )ii dt + S1 2 ĝ(D) is a well-defined single-valued map L : {Loops on LSU (2)} −→ R/Z as we have claimed in the statemant of the theorem. Finally, we have to show that the extremals of L are precisely the closed solutions of the Maxwell-Bloch Hamiltonian system (T ∗ LSU (2), ωc + ωm , Hcn ). But this is clear from our construction of L. Inside every Uα we have L/Uα = Lα . Let g(t) be an extremal of L. Then its restriction to Uα is an extremal of Lα . We have shown that the corresponding path (g(t), (gt )at ) in the cotangent bundle T ∗ Uα is an integral path of the Hamiltonian vector field Xα defined by the Hamiltonian system (T ∗ Uα , ωcα + α α ωm , Hcn ). But, recalling that Uα is open in LSU (2), we know the Hamiltonian vector field Xα coincides with the restriction of the Hamiltonian vector field X of our original Hamiltonian system (T ∗ LSU (2), ωc + ωm , Hcn ), which completes the proof of our theorem. ✷ Remark 3 The Lagrangian L: {Paths in T LSU (2)} → S 1 is well-defined only for closed paths, i.e. for temporally periodic solutions. For the Lagrangian description of e (2) must be the general non-periodic solutions the extended configuration space LSU used. The interested reader can compare our construction to the results in [30]. 29 We shall conclude this paper with a comparison between the Maxwell-Bloch system and the Wess-Zumino-Witten-Novikov action. Let X ⊂ R3 be a closed twodimensional orientable surface and let f : X → SU (2) be a smooth map. Denote by B the three-dimensional manifold bounded by the surface X, that is, ∂B = X. The Wess-Zumino-Witten-Novikov action is a two-dimensional conformal field theory given by the Lagrangian Z Z 1 1 −1 Lwzwn (f ) = (∇f )f + fˆ∗ (Θ) 4π X 2π B where fˆ: B → SU (2) is an extension of f : X = ∂B → SU (2), and Θ ∈ Ω3 (SU (2)) is the right-invariant three-form whose value at the identity is given by Θ(ξ1 , ξ2 , ξ3 ) = hξ1 , [ξ2 , ξ3 ]i, ξ1 , ξ2 , ξ3 ∈ Te SU (2) = su(2). In other words, the form Θ is the volume form on SU (2) = S 3 with respect to the natural round metric. One can immediately see that Lwzwn is defined only up to addition of integers. Indeed, for two different choices fˆ and fˇ of extensions, the chain fˇ(B) − fˆ(B) is a representative of a class in the homology group H3 (SU (2)) = H3 (S 3 ) = Z. Since Θ is the volume form, it is closed, but R not exact, and therefore [Θ] 3 is a non-zero element in HDR (SU (2)). Thus we have (fˇ−fˆ)(B) Θ ∈ Z, as claimed. Let now X be a sphere S 2 or a torus T 2 . Both can be parametrized as closed paths of simple loops in obvious ways. We will denote the parameter of the closed path by t ∈ S 1 and the parameter on the simple loops by x ∈ S 1 . The WZWN-action for f (t, x): X → SU (2) can then be written as Z  Z  1 1 −1 2 −1 2 kft f k + kfx f k dt dx + Θ. Lwzwn (f ) = 4π X 2π fˆ(B) We shall now compare the topologically non-trivial terms of the WZWN-action and of the Maxwell-Bloch system. The relation between the forms Θ ∈ Ω3 (SU (2)) and ωm ∈ Ω2 (LSU (2)) is described by the following proposition. (See [30] for proof.) Proposition 7 Let ev: S 1 × LSU (2) → SU (2) be the evaluation R map ev(u, g(x)) = g(u), and let τ : Ω3 (SU (2)) → Ω2 (LSU (2)) be defined by τ (α) = S 1 ev∗ (α) . Then ωm = τ (2πΘ) − dβ where β is the 1-form on LSU (2) given by Z 2π 1 hgx g −1 , Xg g −1 i dx, βg (Xg ) = 4π 0 In particular, [τ (Θ)] = [ωm ] ∈ H 2 (LSU (2)). 30 (44) Xg ∈ Tg LSU (2). If we put the formula (44) into the expression (43) for the Lagrangian of the Maxwell-Bloch system, we get Z  Z Z  1 −1 2 L(g) = β. (45) τ (2πΘ) − kgt g k − hhσ, Adg (τ (x))ii dt dx + S1 2 g(S 1 ) ĝ(D) In the third term above we have used Stokes’ theorem and the fact that ∂ĝ(D) = g(S 1 ). A loop g: S 1 → LSU (2) in the loop group LSU (2) can be thought of as a map f (t, x): X → SU (2), where X is a sphere or a torus. Formula (45), expressed in terms of the maps f rather than of the loops g, has the form Z  Z  1 −1 2 −1 −1 Θ (46) A(f ) = kfx f k + hfx f , ft f i − hσ, Adf (τ (x)i dt dx + fˆ(B) X 2 in which the topologically non-trivial term is the same as in the WZWN-action. 7 Conclusion In this paper a new Hamiltonian structure of the Maxwell-Bloch equations is constructed and some of its properties are studied. Our Hamiltonian structure stems from the representation of the Maxwell-Bloch equations as the equation of motion for a continuous chain of C. Neumann oscillators parametrized by the single spatial variable x. The interaction among the oscillators is of magnetic type. This means that the acceleration of the oscillator on the location x0 is influenced by the momenta rather than the positions of the neighbouring oscillators. Our Hamiltonian structure is of the form (T ∗ LSU (2), ωc + c ωm , Hmb ), where ωm is the pull-back of the form ω em ∗ on the loop group LSU (2) via the natural projection π: T LSU (2) → LSU (2). The magnetic nature of the interaction among the oscillators is reflected in the perturbation c ωm of the canonical symplectic structure ωc . The form ω em is topologically non-trivial, but it is integral. It is in fact a generator of the cohomology group H 2 (LSU (2); Z) ∼ = Z. By Weil’s theorem it is therefore the curvature of a connece → LG. The total tion on the topologically non-trivial principal U (1)-bundle LG e (2) is precisely the central extension of the loop group LSU (2). Therespace LSU fore the system (T ∗ LSU (2), ωc + c ωm , Hmb ) is the symplectic quotient of the system e (2), Ωc , H), e where Ωc is the canonical symplectic form on T ∗ LSU e (2) and H e (T ∗ LSU is the suitable Hamiltonian. The value of the moment map at which the quotient is taken is equal to c, that is, to the speed of light in the medium. In other words, the e (2), Ωc , H) e is the extension of (T ∗ LSU (2), ωc +c ωm , Hmb ) in the sense system (T ∗ LSU of the Kaluza-Klein theory. The interaction force is geometrized on the U (1)-bundle 31 e (2) over LSU (2). This is reflected in the fact that the magnetically perturbed LSU e (2). symplectic structure ωc + c ωm on LSU (2) lifts to the canonical structure on LSU The conserved Kaluza-Klein charge in our case is the speed of light in the medium. The Kaluza-Klein extension yealds an easy way to find the Lagrangian for the Maxwell-Bloch equations. This Lagrangian is defined on the space of paths in the e (2). We then construct the Lagrangian on the original configcentral extension LSU uration space LSU (2). Here the nontrivial topology of the situation plays the crutial role. Namely, the Lagrangian contains the Wes-Zumino-Witten-Novikov term. Therefore it is well defined only for temporally periodic solutions of the Maxwelle (2) is well Bloch equations, while the Lagrangian on the Kaluza-Klein extension LSU defined for arbitrary solutions. We construct two families of solutions of the Maxwell-Bloch equations. One of these families nicely illustrates the relation between the Maxwell-Bloch and the C. Neumann systems. Our solutions are nonlinear travelling waves whose constituent oscillator is the magnetic spherical pendulum in the same way as the harmonic oscillator is the constituent oscillator of the harmonic travelling waves. By the expression ”magnetic sperical pendulum” we call an electrically charged spherical pendulum moving in the field of a magnetic monopole situated at the centre of our sphere. The magnetic spherical pendulum is a symplectic quotient of a particular kind of circularly symmetric C. Neumann system, the kind that figures in this paper. The well-known 2π-soliton occurs as a special case of our family of solutions. In this case the constituent oscillator has to be reduced to the planar gravitational pendulum at the critical energy. Our representation of Maxwell-Bloch equations as a chain of interacting oscillators and the associated Hamiltonian structure offer a starting point for many lines of further investigation. It is easily seen that this Hamiltonian system is invariant with respect to the natural action of the loop group LU (1). More generally, Hamiltonian systems (T ∗ LG, ωc +c ωm , Hgmb ) are invariant with respect to the actions of LH, where H are suitable subgroups of G. These actions yield various symplectic quotients. In a forthcoming paper we intend to study some of these quotients and their properties. This topic is directly connected with the multilevel resonant light-matter interaction studied by Park and Shin in [16]. Another interesting topic are partial discretizations of the Maxwell-Bloch equations. If we discretize them with respect to the spatial variable, we get a discrete system of interacting C. Neumann oscillators. In [32] we construct a large number of conserved quantities of such many-body systems. We intend to address different topics concerning the geometry and dynamics of such discretizations in future papers. 32 Acknowledgement I would like to thank professors Pavel Winternitz, Gregor Kovačič, and John Harnad for interesting and stimulating discussions. I would also like to thank both referees for informing me about the references [15] and [16], and for their valuable suggestions. The research for this paper was supported in part by the Ministry of Science and Technology of the Republic of Slovenia. A part of the research was done at the Centre de recherches mathématiques, Montreal, Canada. The hospitality of CRM and especially of professor Jiřı́ Patera is gratefully acknowledged. 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