We propose a quantization of linear, volume preserving, maps on the discrete and finite 3-torus T_N^3 represented by elements of the group SL(3,Z_N). These flows can be considered as special motions of the Nambu dynamics (linear Nambu flows) in the three dimensional toroidal phase space and are characterized by invariant vectors, a, of T_N^3. We quantize all such flows which are necessarily restricted on a planar two-dimensional phase space, embedded in the 3-torus, transverse to the vector a . The corresponding maps belong to the little group of the vector a in SL(3,Z_N) which is an SL(2,Z_N) subgroup. The associated linear Nambu maps are generated by a pair of linear and quadratic Hamiltonians (Clebsch-Monge potentials of the flow) and the corresponding quantum maps, realize the metaplectic representation of SL(3,Z_N) on the discrete group of three dimensional magnetic translations i.e. the non-commutative 3-torus with deformation parameter the N-th root of unity.
Other potential applications of our construction are related to the quantization of deterministic chaos in turbulent maps as well as to quantum tomography of three dimensional objects.
The classical and quantum features of Nambu mechanics are analyzed and fundamental issues are resolved. The classical theory is reviewed and developed utilizing varied examples. The quantum theory is discussed in a parallel presentation, and illustrated with detailed specific cases. Quantization is carried out with standard Hilbert space methods. With the proper physical interpretation, obtained by allowing for different time scales on different invariant sectors of a theory, the resulting non-Abelian approach to quantum Nambu mechanics is shown to be fully consistent.
We present briefly the deformation philosophy and indicate, with references, how it was applied to the quantization of Nambu mechanics and to particle physics in anti De Sitter space.
We propose a generalization of cubic matrix mechanics by introducing a canonical triplet and study its relation to Nambu mechanics. The generalized cubic matrix mechanics we consider can be interpreted as a 'quantum' generalization of Nambu mechanics.
Phase Space is the framework best suited for quantizing superintegrable systems, naturally preserving the symmetry algebras of the respective hamiltonian invariants. The power and simplicity of the method is fully illustrated through new applications to nonlinear sigma models, specifically for de Sitter N-spheres and Chiral Models, where the symmetric quantum hamiltonians amount to compact and elegant expressions. Additional power and elegance is provided by the use of Nambu Brackets to incorporate the extra invariants of superintegrable models. Some new classical results are given for these b…
Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectat…
We propose a variant formulation of Hamiltonian systems by the use of variables including redundant degrees of freedom. We show that Hamiltonian systems can be described by extended dynamics whose master equation is the Nambu equation or its generalization. Partition functions associated with the extended dynamics in many degrees of freedom systems are given. Our formulation can also be applied to Hamiltonian systems with first class constraints.
Using the framework of Nambu's generalised mechanics, we obtain a new description of constrained Hamiltonian dynamics, involving the introduction of another degree of freedom in phase space, and the necessity of defining the action integral on a world sheet. We also discuss the problem of quantising Nambu mechanics.
Following the Nambu mechanics framework we demonstrate that the non-dissipative part of the Lorenz system can be generated by the intersection of two quadratic surfaces that form a doublet under the group SL(2,R). All manifolds are classified into four dinstict classes; parabolic, elliptical, cylindrical and hyperbolic. The Lorenz attractor is localized by a specific infinite set of one parameter family of these surfaces. The different classes correspond to different physical systems. The Lorenz system is identified as a charged rigid body in a uniform magnetic field with external torque and t…
The Landau-Lifshitz-Gilbert (LLG) equation describes the dynamics of a damped magnetization vector that can be understood as a generalization of Larmor spin precession. The LLG equation cannot be deduced from the Hamiltonian framework, by introducing a coupling to a usual bath, but requires the introduction of additional constraints. It is shown that these constraints can be formulated elegantly and consistently in the framework of dissipative Nambu mechanics. This has many consequences for both the variational principle and for topological aspects of hidden symmetries that control conserved q…
A general algebraic condition for the functional independence of 2n-1 constants of motion of an n-dimensional maximal superintegrable Hamiltonian system has been proved for an arbitrary finite n. This makes it possible to construct, in a well-defined generic way, a normalized Nambu bracket which produces the correct Hamiltonian time evolution. Existence and explicit forms of pairwise compatible multi-Hamiltonian structures for any maximal superintegrable system have been established. The Calogero-Moser system, motion of a charged particle in a uniform perpendicular magnetic field and Smorodins…
We propose a generalization of the standard geometric formulation of quantum mechanics, based on the classical Nambu dynamics of free Euler tops. This extended quantum mechanics has in lieu of the standard exponential time evolution, a nonlinear temporal evolution given by Jacobi elliptic functions. In the limit where latter's moduli parameters are set to zero, the usual geometric formulation of quantum mechanics, based on the Kahler structure of a complex projective Hilbert space, is recovered. We point out various novel features of this extended quantum mechanics, including its geometric asp…
We develop a Hamilton-Jacobi-like formulation of Nambu mechanics. The Nambu mechanics, originally proposed by Nambu more than four decades ago, provides a remarkable extension of the standard Hamilton equations of motion in even dimensional phase space with a single Hamiltonian to a phase space of three (and more generally, arbitrary) dimensions with two Hamiltonians (n Hamiltonians in the case of (n+1)-dimensional phase space) from the viewpoint of the Liouville theorem. However, it has not been formulated seriously in the spirit of Hamilton-Jacobi theory. The present study is motivated to su…
Phase Space is the framework best suited for quantizing superintegrable systems--systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally, as illustrated on nonlinear $σ$-models, specifically for Chiral Models and de Sitter $N$-spheres. Classically, the dynamics of superintegrable models such as these is automatically also described by Nambu Brackets involving the extra symmetry invariants of them. The phase-space quantization worked out then leads to the quantization of the cor…
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and Rössler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotati…
We study Abelian generalized deformations of the usual product of polynomials introduced in hep-th/9602016. We construct an explicit example for the case of $su/2$ which provides a tentative of a quantum-mechanical description of Nambu Mechanics on $R^3$. By introducing the notions of strong and weak triviality of generalized deformations, we show that the Zariski product is never trivial in either sense, while the example constructed here in a quantum-mechanical context is only strongly non-trivial.
We review some aspects of Nambu mechanics on the basis of the works previously published separately by the present author. Main focuses are on three themes, its various symmetry structures, their possible relevance to string/M theory, and a Hamilton-Jacobi like reformulation. We try to elucidate the basic ideas, most of which were rooted in more or less the same ground, and to explain motivations behind these works from a unified and vantage viewpoint. Various unsolved questions are mentioned. We also include some historical account on the genesis of the Nambu mechanics, and discuss (in the Ap…
Phase Space is the framework best suited for quantizing superintegrable systems--systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally. We illustrate the power and simplicity of the method through new applications to nonlinear sigma-models, specifically for Chiral Models and de Sitter N-spheres, where the symmetric quantum hamiltonians amount to compact and elegant expressions, in accord with the Groenewold-van Hove theorem. Additional power and elegance is provided by the us…
This lecture consists of three parts. In part I, an overview is given on the so-called Matrix theory in the light-front gauge as a proposal for a concrete and non-perturbative formulation of M-theory. I emphasize motivations towards its covariant formulation. Then, in part II, I turn the subject to the so-called Nambu bracket and Nambu mechanics, which were proposed by Nambu in 1973 as a possible extension of the ordinary Hamiltonian mechanics. After reviewing briefly Nambu's original work, it will be explained why his idea may be useful in exploring higher symmetries which would be required f…
In the framework of Nambu Mechanics, we have recently argued that Non-Hamiltonian Chaotic Flows in $ R^{3} $, are dissipation induced deformations, of integrable volume preserving flows, specified by pairs of Intersecting Surfaces in $R^{3}$. In the present work we focus our attention to the Lorenz system with a linear dissipative sector in its phase space dynamics. In this case the Intersecting Surfaces are Quadratic. We parametrize its dissipation strength through a continuous control parameter $ε$, acting homogeneously over the whole 3-dim. phase space. In the extended $ε$-Lorenz system we …
We present several non-trivial examples of the three-dimensional quantum Nambu bracket which involve square matrices or three-index objects. Our examples satisfy two fundamental properties of the classical Nambu bracket: they are skew-symmetric and they obey the Fundamental Identity. We contrast our approach to the existing literature on the quantum deformations of Nambu mechanics. We also discuss possible applications of our results in M-theory.
In analogy to Nambu's generalization of mechanics, we present a generalization of Poisson sigma models to higher dimensional world volumes. We find corresponding generalizations of string sigma models and open-closed string relations for background fields to the case of p-branes. As an application, we discuss a DBI-type effective action for open membranes.
With the deconstruction technique, the geometric information of a torus can be encoded in a sequence of orbifolds. By studying the Matrix Theory on these orbifolds as quiver mechanics, we present a formulation that (de)constructs the torus of {\em generic shape} on which Matrix Theory is ``compactified''. The continuum limit of the quiver mechanics gives rise to a $(1+2)$-dimensional SYM. A hidden (fourth) dimension, that was introduced before in the Matrix Theory literature to argue for the electric-magnetic duality, can be easily identified in our formalism. We construct membrane wrapping st…
Based on our recent work on Quantum Nambu Mechanics $\cite{af2}$, we provide an explicit quantization of the Lorenz chaotic attractor through the introduction of Non-commutative phase space coordinates as Hermitian $ N \times N $ matrices in $ R^{3}$. For the volume preserving part, they satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its vol…
In a recent paper, by exploiting the notion of Morita equivalence for field theories on noncommutative tori and choosing rational values of the noncommutativity parameter $θ$ (in appropriate units), a general one-to-one correspondence between the $m$-reduced conformal field theory (CFT) describing a quantum Hall fluid (QHF) at paired states fillings $ν=% \frac{m}{pm+2}$ and an Abelian noncommutative field theory (NCFT) has been established . That allowed us to add new evidence to the relationship between noncommutativity and quantum Hall fluids\cite% {ncmanybody}. On the other hand, the $m$-re…
We classify the supersymmetric mass deformations of all the super Yang-Mills quantum mechanics, which are obtained by dimensional reductions of minimal super Yang-Mills in spacetime dimensions: ten, six, four, three and two. The resulting actions can be viewed as the matrix descriptions of supermembranes in nontrivial backgrounds of one higher dimensional supergravity theories. We also discuss the utmost generalization of the light-cone formulation of the Nambu-Goto action for a p-brane, including time dependent backgrounds.
We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi-Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov-Shatashvili limit of the refined topological string, but there are non-perturbative corrections determi…
Recently, a novel bootstrap method for numerical calculations in matrix models and quantum mechanical systems is proposed. We apply the method to certain quantum mechanical systems derived from some well-known local toric Calabi-Yau geometries, where the exact quantization conditions have been conjecturally related to topological string theory. We find that the bootstrap method provides a promising alternative for the precision numerical calculations of the energy eigenvalues. An improvement in our approach is to use a larger set of two-dimensional operators instead of one-dimensional ones. We…
We describe the statistical mechanics of a melting crystal in three dimensions and its relation to a diverse range of models arising in combinatorics, algebraic geometry, integrable systems, low-dimensional gauge theories, topological string theory and quantum gravity. Its partition function can be computed by enumerating the contributions from noncommutative instantons to a six-dimensional cohomological gauge theory, which yields a dynamical realization of the crystal as a discretization of spacetime at the Planck scale. We describe analogous relations between a melting crystal model in two d…
We derive asymptotic behaviors of the Nambu-Bethe-Salpeter (NBS) wave function at large space separations for systems with more than 2 particles in quantum field theories. To deal with $n$-particles in the center of mass flame coherently, we introduce the Jacob coordinates of $n$ particles and then combine their $3(n-1)$ coordinates into the one spherical coordinate in $D=3(n-1)$ dimensions. We parametrize on-shell $T$-matrix for $n$-particle system of scalar fields at low energy, using the unitarity constraint of the $S$-matrix. We then express asymptotic behaviors of the NBS wave function fo…
Quantum Mechanics on Lie Groups: II. Path Integrals
CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris（法国国家科学研究中心，巴黎综合理工学院，巴黎理工学院）；Université Claude Bernard Lyon 1, ICJ UMR 5208, CNRS（里昂第一大学，ICJ联合研究实验室，法国国家科学研究中心）
The statistical model of crystal melting represents BPS configurations of D-branes on a toric Calabi-Yau three-fold. Recently it has been noticed that an infinite-dimensional algebra, the quiver Yangian, acts consistently on the crystal-melting configurations. We physically derive the algebra and its action on the BPS states, starting with the effective supersymmetric quiver quantum mechanics on the D-brane worldvolume. This leads to remarkable combinatorial identities involving equivariant integrations on the moduli space of the quantum mechanics, which can be checked by numerical computation…
We perform a quantization of 4-dimensional Nambu-Goto theory of open string in light cone gauge, related in Lorentz-invariant way with the world sheet. This allows to obtain a quantum theory without anomalies in Lorentz group. We consider a special type of gauge fixing conditions, imposed in oscillator sector of the theory, which lead to a relatively simple Hamiltonian mechanics, convenient for canonical quantization. We discuss the algebraic and geometric properties of this mechanics and determine its mass spectrum for the states of spin singlet S=0.
We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle--point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of ``loop quantum mechanics'', we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string conden…
We study static fermion bags in the 1+1 dimensional Gross-Neveu and Nambu-Jona-Lasinio models. It has been known, from the work of Dashen, Hasslacher and Neveu (DHN), followed by Shei's work, in the 1970's, that the self-consistent static fermion bags in these models are reflectionless. The works of DHN and of Shei were based on inverse scattering theory. Several years ago, we offered an alternative argument to establish the reflectionless nature of these fermion bags, which was based on analysis of the spatial asymptotic behavior of the resolvent of the Dirac operator in the background of a s…
We construct explicitly the quantization of classical linear maps of $SL(2, R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of $SL(2, Z)$ to the subgroup $SL(2, Z)/Γ_l$, $Γ_l$ being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, $O_{l}(2)\subset SL(2,l)$, for arbitrary values of l are determin…
This note relates topics in statistical mechanics, graph theory and combinatorics, lattice quantum field theory, super quantum mechanics and string theory. We give a precise relation between the dimer model on a graph embedded on a torus and the massless free Majorana fermion living on the same lattice. A loop expansion of the fermion determinant is performed, where the loops turn out to be compositions of two perfect matchings. These loop states are sorted into co-chain groups using categorification techniques similar to the ones used for categorifying knot polynomials. The Euler characterist…
In this article we present a self contained review of the principles of Matrix Theory including the basics of light cone quantization, the formulation of 11 dimensional M-Theory in terms of supersymmetric quantum mechanics, the origin of membranes and the rules of compactification on 1,2 and 3 tori. We emphasize the unusual origins of space time and gravitation which are very different than in conventional approaches to quantum gravity. Finally we discuss application of Matrix Theory to the quantum mechanics of Schwarzschild black holes. This work is based on lectures given by the second autho…
