What is complete integrability in quantum mechanics


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1 INTERNATIONAL SOLVAY INSTITUTES FOR PHYSICS AND CHEMISTRY Proceedings of the Symposium Henri Poincaré (Brussels, 89 October 2004) What is complete integrability in quantum mechanics L. D. Faddeev St. Petersburg Department of Steklov, Mathematical Institute, Russian Academy of Sciences
2 When I was invited to speak on Solvay Conference, dedicated to the 150 anniversary of H. Poincare, I faced a noneasy problem. Being mathematical physicist, dealing with problems, stemming from quantum theory, I could not find a suitable subject to speak about. Finally I decided to discuss the notion of complete integrability, central to great works of H. Poincare, in its relation to quantum mechanics. It happened, that in the last 30 years interest to complete integrability was revived due to the development of the Inverse Scattering Method in the theory of solitons. It was realized, that many soliton equations have Hamiltonian formulation and are completely integrable see e. g. [1] and references there. The quantization of these dynamical models led to beautiful mathematical structures as YangBaxter relation and was the base for the formulation of Quantum Groups. Thus it became natural to understand what integrability means in quantum domain. Based on this development (see e. g. [2] and references there) I present here some general considerations and illustrate them by the fundamental example of quantum soliton theory magnetic chain, following the exposition in [2], where one can find more details. Thus though this text is not very original, I hope that it will stimulate attentive reader to think about the exact nature of the complete integrability in quantum theory. The notion of complete integrability assumes central place in classical mechanics. Almost all great names in mechanics are associated with development of this notion and it is vividly discussed in textbooks. It is not so in quantum mechanics. In this essay I shall discuss several reasons for this phenomenon and after that present an instructive example of passage from classical mechanical treatment of a particular completely integrable dynamical model to quantum description retaining the integrability intact. The relation between quantum and classical pictures are that of deformation and contraction, Planck constant playing the role of the deformation parameter (see e. g. [3]). The construction of a quantum variant of a given dynamical model, known in classical formulation (quantization) is not unique. I use to say, that the quantization is rather an art than science. Thus the transfer of the features of classical mechanics into their quantum analogue is not automatic. Moreover, these features could be modified in this passage. Here we shall illustrate this by the notion of complete integrability. For that we should first remind the basic aspects of the formulation of mechanical system. The main objects in this formulation are the algebra of observables and the evolution in time. In classical mechanics we begin with phase space Γ, dim Γ = 2n, endowed by a nondegenerate 2form ω = 2n i,k Ω ik d ξ i d ξ k. Here ξ i, i = 1,... 2n are (local) coordinates on Γ. Real functions f : Γ R constitute the algebra of observables A 0. Any given observable g defines one parameter family of automorphisms of A 0 by means of the equation for an arbitrary observable f df(s) ds = {g, f(s)}, f(0) = f, where the Poisson bracket {, } is constructed by means of matrix Ω ik inverse 2
3 to Ω ik : ik f g {f, g} = Ω ξ i ξ k The Poisson bracket introduces the structure of a Lie algebra into the algebra of observables. A particular observable H, called Hamiltonian, defines in this way the time evolution. The equation df = {H, f} (1) dt is called the equation of motion. In conventional formulation of quantum mechanics algebra of observables A h is that of linear selfadjoint operators A, B,... in a complex Hilbert space H. The role of Lie operation plays the commutator [A, B] = i (AB BA). Here i = 1 and real parameter is called Planck constant. The automorphisms are given by one parameter families of unitary operators. In particular Hamiltonian H defines the evolution operator U(t) = e iht/ so that the solution of the equation of motion is given by d dt A(t) = i [HA(t) A(t)H] h A(t) = U 1 (t)a(0)u(t). One cannot help observing, that this explicit formula is much nicer, than the description of the solution of equation (1) in terms of the generating function. We can now turn to the discussion of complete integrability. I shall remind two conventional definitions of this notion in terms of classical mechanics and then observe if they have a natural quantum counterpart. We shall see, that the direct (and may be too naive) reformulation will not be sensible. So let the classical mechanical system (Γ 2n, ω, H) be given. The first definition of complete integrability requires the existence of n 1 functionally independent observables Q i, i = 1,..., n 1 which Poisson commute with H and among themselves {H, Q i } = 0, {Q i, Q k } = 0. Observables Q i are called commuting conservation laws. In the second more elaborate definition one requires the existence of change of variables V such that V (ξ) = (α, I), where angleaction variables α i, I i, i = 1,... n constitute two commuting subsets and Hamiltonian H becomes function of actions I only H = H(I). Both these definitions almost immediately fail if we try to formulate them in quantum terms. For the first one it is just not clear what is the definition of the number of degrees of freedom n. Indeed all infinite dimensional separable 3
4 Hilbert spaces are unitary equivalent, so there is no place for the quantum definition of number of degrees of freedom if we stick to the definition of algebra of observables given above. In the case of the second definition the difficulty is more subtle. It turns out that nonintegrable classical systems could become integrable if we naively generalize the second definition of integrability. Indeed, in the theory of scattering of a particle by potential we deal with the phase space Γ = R 6 with canonical variables momenta and coordinates p 1, p 2, p 3, q 1, q 2, q 3 (we treat particle in the 3dimensional space). The Hamiltonian H(p, q) = p2 2m + v(q) contains a potential function v(q), which is supposed to vanish rapidly enough when q. In general case when v(q) does not satisfy any symmetry (like spherical symmetry v(q) = v( q )), the classical system is not integrable. On the other hand the quantum mechanical Hamiltonian H = 2 2m + v(x) in L 2 (R 3 ) is unitarily equivalent to the operator H 0, defined in L 2 (R 3 ) C N as H 0 ψ(p) = p2 2m ψ(p), H 0 φ n = κ 2 nφ n, which is a function of action variable momentum p of a free particle and energies κ 2 n of bound states. This equivalence is established by the wave operator of quantum scattering theory. Thus either we are to admit, that quantum mechanics is more regular than the classical mechanics, or restrict the admissible automorphisms so that the wave operator will be excluded. I think, that after this general discussion it become clear, that the definition of the algebra of observables of quantum system given above is not detailed enough. And of course we know, that in dealing with concrete quantum systems we begin with the set of distinguished observables momenta and coordinate operators of a particle, spin operators for magnets etc. Thus the Hilbert space and its topology is not the first basic object. Rather it appears as a carrier of representation for these distinguished variables. For instance the linear phase space Γ 2n leads after quantization to the Hilbert space H = L 2 (R n ), where the Lie algebra of linear symplectic transformations is represented, with explicit indication of the number of degrees of freedom. After this representation is chosen we are to define algebra of observables, generated by basic variables, and admissible automorphisms of this algebra. In particular such definition is to exclude the wave operator. I do not know such definition yet. Instead I shall present an instructive example of a classical integrable model, for which the integrability is retained in a natural sense. Such examples will be indispensable for the future definition of the quantum integrability. The model is called spin chain. The phase space Γ is given by Γ = S 2 S 2... S } {{ } 2. N 4
5 Each 2sphere S 2 corresponds to a dynamical variable called spin and product means, that we deal with N spins distributed along the discrete circle Z N, n + N n. The Poisson structure on Γ is defined in local way: the variables s i n : i = 1, 2, 3, n = 1,... N with constraint (s 1 n) 2 + (s 2 n) 2 + (s 3 n) 2 = s 2 corresponding to different spins commute {s i n, s k m} = 0, n m. The bracket for a given spin n is given by where ɛ ikj is antisymmetric tensor. It is clear, that {s i n, s k n} = ɛ ikj s j n, (2) dim Γ N = 2N, so we deal with the system of N degrees of freedom. Quantization of bracket (2) is given by the representation of the Lie algebra SU(2). These representations D s are labeled by semiinteger and have dimensions s = 0, 1 2, 1,... dim D s = 2s + 1. The variables s i n become matrices with commutation relations [s i n, s k n] = iɛ ikj s j n (3) (we do not write explicitly using units = 1). For instance in the case s = 1/2 the Hilbert space is C 2 and spin variables are given by s i n = 1 2 σ i, where σ 1 = ( ) 0 1, σ = ( ) ( ) 0 i 1 0, σ i 0 3 =. 0 1 The full Hilbert space of the model is given by the tensor product H = D s1 D s2... D sn. In what follows we shall consider the isotropic case where all labels s 1, s 2,... s N are equal. Thus the role of distinguished observables is played by generators of the algebra SU(2) for each component of the phase space. Now we should introduce a relevant Hamiltonian. According to the general technique of the Inverse Scattering Method (see [2]) it is done as follows. Consider an auxiliary linear problem ψ n+1 = L n (λ)ψ n, (4) 5
6 where ( ) ( ) ψ 1 ψ n = n λ + is 3 ψn 2, L n = n is n is + n λ is 3, (5) n λ complex ( spectral ) parameter and s ± n = s 1 n ± is 2 n. The linear system (4) has monodromy around our circle T (λ) = L N (λ)l N 1 (λ)... L 1 (λ). Now the following fundamental property takes place: the invariant F (λ) = tr T (λ) defines a commuting family of observables {F (λ), F (µ)} = 0. (6) We shall not proof this here and just refer to [1]. However more instructive proof of similar property in quantum case will be given below. It is clear from construction, that F (λ) is a polinom in λ N 1 F (λ) = 2λ N + λ k Q k+1 so the observables Q k, k = 1,... N define functionally independent commuting set {Q i, Q k } = 0, i, k = 1,... N. Each of them can be taken as a Hamiltonian of a completely integrable system. However there is another possibility to choose a more beautiful Hamiltonian. Observe, that matrix L n is degenerate when λ 2 + s 2 = 0. Thus L n (is) has rank 1 and so it is easy to calculate the monodromy. In particular we get k=0 ln F (is) 2 = n ln(s 2 + i s i ns i n+1). The RHS is a sum of terms describing the near neighbour interaction along our chain. The Hamiltonian H = n ln(s 2 + s i ns i n+1) (7) defines the classical completely integrable dynamical system, which we already called spin chain. The construction of the quantum counterpart is based on the fact, that there is no problem in quantization of the matrix L k (λ) indeed, it contains the dynamical variables s n linearly and we do not face the problem of factor ordering. We shall prove, that the quantum analogue of the monodromy M(λ), defined exactly by some formula, as the classical one (5), also generate a family of commuting operators. For this we begin by observation that the set of commutation 6
7 relations for the matrix elements of matrix operator L n (x) can be written in an elegant way. Consider L n (x) as a matrix in tensor product D s C 2. We shall call the first factor D s the quantum space and the second C 2 the auxiliary space. Let R(λ) be a matrix in the product of two auxiliary spaces C 2 C 2 R(λ) = λi + ip. (8) Here P is a permutation P (a b) = b a. Then the following relation holds in the space C 2 C 2 D s where R 12 (λ µ)l 1 n(λ)l 2 n(µ) = L 2 n(µ)l 1 n(λ)r 12 (λ µ), (9) R 12 (λ µ) = R(λ µ) I, L 2 n(λ) = I L n (λ) and L 1 n acts nontrivially in the product of the first auxiliary space and D s. The relation (9) can be checked easily by means of the basic relation (3). Now taking into account, that matrix elements of L 1 n(λ) and L 2 m(µ) for n m commute, we find that the monodromy M(λ) = L N (λ)... L 1 (λ) (10) considered as a matrix in H C 2 satisfy the same set of relations as a local factor L n (λ) R 12 (λ µ)m 1 (λ)m 2 (µ) = M 2 (µ)m 1 (λ)r 12 (λ µ) from which it follows immediately, that F (λ) = tr M(λ), where trace is taken over the auxiliary space, defines commuting set of operators [F (λ), F (µ)] = 0. In fact the derivation of the classical commutativity (6) can be derived by the contraction 0. Thus we have the set of N commuting operators Q j, j = 1,... N, being coefficients of the polinom F (λ). But as in the classical case we can find a more beautiful Hamiltonian which is a sum of local terms. Begin with the case s = 1/2, so that D s = C 2 and quantum and auxiliary spaces coincide. It follows from the definition, that L( i 2 ) = ip, where P is permutation matrix. Using this one can show, that F ( i 2 ) is a shift operator in H and d dλ F (λ)f 1 (λ) λ=i/2 = H 1/2 n,n+1, 7
8 where Thus the Hamiltonian H 1/2 n,n+1 = 3 s i ns i n I. i=1 H = n H 1/2 n,n+1 is a sum of terms, describing the interaction of neighbouring spins. This Hamiltonian is well known in the theory of magnets and corresponding dynamical system is called Heisenberg spin chain. The case of higher spins requires more work. We begin by the statement, that the basic relation (9) is a particular case of a more general one R s1,s2 (λ µ)r s1,s3 (λ σ)r s2,s2 (µ σ) = taking place in the space = R s2,s2 (µ σ)r s1,s3 (λ σ)r s1,s2 (λ µ), (11) D 123 = D s1 D s2 D s3, where each factor R s,s acts nontrivially in two of D s, corresponding to its spin labels. The relation (9) corresponds to the case s 1 = 1/2, s 2 = 1/2, s 3 = 1/2, where R 1/2,1/2 is given by (8) and L n (λ) = R 1/2,1/2 (λ i/2) in case of s = 1/2. The formula (11) realizes the concrete representation of the abstract YangBaxter relation R 12 (λ µ)r 13 (λ σ)r 23 (µ σ) = R 23 (µ σr 13 (λ σ)r 12 (λ µ), which plays the fundamental role in the theory of quantum integrable models and quantum groups. Using the relation (11) for s 1 = 1/2, s 2 = s, s 3 = s and taking into account, that we know R 1/2,s we get the linear equation for R s,s (λ). The solution can be written as R s,s (λ) = ip r(j, λ). Here operator J is defined by the equation ( s 1 + s 2 ) 2 = J(J + 1), (12) where s 1 and s 2 are spin vectors in the first and second D s. The explicit formula for r looks as follows Γ(J iλ) r(j, λ) = Γ(J + 1 iλ). Now the local Hamiltonian can be constructed due to the relation R s,s (0) = ip as Hn,n+1 s = i d dλ ln r(j n,n+1, λ) λ=0 = 2ψ(J n,n+1 + 1), where J n,n+1 is constructed by formula analogous to (12) with s 1 = s n and s 2 = s n+1. Taking into account, that J n,n+1 (J n,n+1 + 1) = 2( s n, s n+1 ) + 2s(s + 1) 8
9 we can express the local density H s n,n+1 via the invariant In particular for s = 1 we get σ = ( s n, s n+1 ). H 1 n,n+1 = σ σ 2. Classical contraction corresponds to the case s and from the asymptotics of ψfunction one can easily get the classical limit (7). Thus starting from the construction of the Hilbert space as a carrier of the representation of the basic variables the spin operators s n we were able to quantize the dynamical model with Hamiltonian H, given by (7), retaining the integrability. The construction used heavily the auxiliary object the matrix L n (λ). In other words, it did not follow from some universal receipt, rather it was based on very particular property of the classical system. Of course, the concrete system, considered here is a particular example of a large family of models, which appeared in connection with the Inverse Scattering Method. References [1] L. D. Faddeev, L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, SpringerVerlag 1987 (Springer series in Soviet Mathematics). [2] L. D. Faddeev, How algrbraic Bethe ansatz works for integrable models, proceedings of Les Houches School of Physics: Quantum Symmetries, Les Houches 1995, [3] M. Flato, A Short Survey in Modern Statistical Mechanics, in Istanbul 1970, Studies In Mathematical Physics, Dordrecht 1973,
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