Canonical Nambu mechanics: relevance to string/M-theory and approaches to quantization
Tamiaki Yoneya
备注：
Written version of an invited talk in the workshop "Space-time topology behind formation of micro-macro magneto-vortical structure by Nambu mechanics", Osaka City University, Sept. 28--Oct.1, 2020, corrected typos, added sentences, and added references, version to be published in a special section of the journal PTEP
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 Appendix) some parallelism of various ideas behind the Nambu's paper with Dirac's old works which are related to the description of vortical flows in terms of gauge potentials.
We find a topologically non-trivial structure of the Nambu monopole in two Higgs doublet model (2HDM), which is a magnetic monopole attached by two topologically stable $Z$ strings ($Z$ flux tubes) from two opposite sides. The structure is in sharp contrast to the topological triviality of the Nambu monopole in the standard model (SM), which is attached by a single non-topological $Z$ string. It is found that the Nambu monopole in 2HDM possesses the same fiber bundle structure with those of the `t Hooft-Polyakov monopole and the Wu-Yang description of the Dirac monopole, as a result of the fac…
We present an explicit matrix algebra quantization of the algebra of volume-preserving diffeomorphisms of the $n$-torus. That is, we approximate the corresponding classical Nambu brackets using $\mathfrak{sl}(N^{\lceil\frac{n}{2}\rceil},\mathbb{C})$-matrices equipped with the finite bracket given by the completely anti-symmetrized matrix product, such that the classical brackets are retrieved in the $N\rightarrow \infty$ limit. We then apply this approximation to the super $4$-brane in $9$ dimensions and give a regularized action in analogy with the matrix quantization of the supermembrane. Th…
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…
We show that a Nambu-Goto string has a nontrivial zero length limit which corresponds to a massless particle with extrinsic curvature. The system has the set of six first class constraints, which restrict the phase space variables so that the spin vanishes. Upon quantization, we obtain six conditions on the state, which can be represented as a wave function of position coordinates, $x^μ$, and velocities, $q^μ$. We have found a wave function $ψ(x,q)$ that turns out to be a general solution of the corresponding system of six differential equations, if the dimensionality of spacetime is eight. Th…
基于我们最近在量子Nambu力学$\cite{af2}$上的工作，通过在$ R^{3}$中引入非对易相空间坐标作为Hermitian $ N \times N $矩阵，给出了Lorenz混沌吸引子的一个显式量子化.对于体积保持部分，它们满足由两个Nambu哈密顿量之一引起的对易关系，第二个产生唯一的时间演化。耗散以自洽的方式以量子力学的方式并入，具有正确的经典极限，而不引入外部自由度。由于其体积相空间收缩，它违反了量子对易关系。我们证明了矩阵Lorenz系统的Heisenberg-Nambu演化方程发展到N个独立Lorenz吸引子的快速退相干。另一方面，存在弱耗散状态，其中体积保持非耗散扇区的量子力学性质存活很长时间。
On the basis of the intimate relation between Nambu dynamics and the hydrodynamics, the hydrodynamics on a non-commutative space (obtained by the quantization of space), proposed by Nambu in his last work, is formulated as ``hydrodynamics of granular material''. In Part 1, the quantization of space is done by Moyal product, and the hydrodynamic simulation is performed for the so obtained two dimensional fluid, which flows inside a canal with an obstacle. The obtained results differ between two cases in which the size of a fluid particle is zero and finite. The difference seems to come from the…
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…
It is shown that the Dirac-nambu-Goto brane can be described as a point particle in an infinite dimensional brane space with a particular metric. This suggests a generalization to brane spaces with arbitrary metric, including the "flat" metric. Then quantization of such a system is straightforward: it is just like quantization of a bunch of non interacting particles. This leads us to a system of a continuous set of scalar fields. For a particular choice of the metric in the space of fields we find that the classical Dirac-Nambu-Goto brane theory arises as an effective theory of such an underly…
It is shown how a string living in a higher dimensional space can be approximated as a point particle with squared extrinsic curvature. We consider a generalized Howe-Tucker action for such a "rigid particle" and consider its classical equations of motion and constraints. We find that the algebra of the Dirac brackets between the dynamical variables associated with velocity and acceleration contains the spin tensor. After quantization, the corresponding operators can be represented by the Dirac matrices, projected onto the hypersurface that is orthogonal to the direction of momentum. A conditi…
We study classical solutions to the Nambu-Goto string on $AdS_2 \times S^1$ and $AdS_3 \times S^1$ corresponding to strings stretched between wrapped branes extending in $AdS$. The solutions are obtained by analytic continuation of giant magnon solutions, cut at the position of the branes. These solutions carry one or two $SO(2,4)$ charges and a single $SO(6)$ charge. We compute their energies and show their relation to $\frac{1}{2}$-BPS geometries. Their relevance to the $SL(2)$ sector of $\mathcal{N}=4$ SYM is also discussed.
In this Letter, we study black hole area quantization in the context of gravitational wave physics. It was recently argued that black hole area quantization could be a mechanism to produce so-called echoes as well as characteristic absorption lines in gravitational wave observations of merging black holes. One can match the spontaneous decay of these quantum black holes to Hawking radiation calculations. Using some assumptions, one can then estimate the natural widths of these states. As can be seen from a classical paper by Bekenstein and Mukhanov, the ratio between width and spacing of nonsp…
We formulate the quantum field theory description of neutron-antineutron oscillations in the framework of canonical quantization, in analogy with the Bardeen--Cooper--Schrieffer (BCS) theory and the Nambu--Jona-Lasinio model. The physical vacuum of the theory is a condensate of pairs of {\it would-be neutrons and antineutrons} in the absence of the baryon-number violating interaction. The quantization procedure defines uniquely the mixing of massive Bogoliubov quasiparticle states which represent the neutron. In spite of not being mass eigenstates, neutron and antineutron states are defined on…
We study the Fock quantization of scalar fields with a time dependent mass in cosmological scenarios with flat compact spatial sections. This framework describes physically interesting situations like, e.g., cosmological perturbations in flat Friedmann-Robertson-Walker spacetimes, generally including a suitable scaling of them by a background function. We prove that the requirements of vacuum invariance under the spatial isometries and of a unitary quantum dynamics select (a) a unique canonical pair of field variables among all those related by time dependent canonical transformations which sc…
This paper provides a systematic study of gauge symmetries in the dynamical fuzzy spectral triple models for quantum gravity that have been proposed by Barrett and collaborators. We develop both the classical and the perturbative quantum BV formalism for these models, which in particular leads to an explicit homological construction of the perturbative quantum correlation functions. We show that the relevance of ghost and antifield contributions to such correlation functions depends strongly on the background Dirac operator $D_0$ around which one perturbs, and in particular on the amount of ga…
