arXiv subjects

Izumi Tsutsui

Publications and source records attributed to Izumi Tsutsui.

61 records · Page 4Linked to original sources

arXiv1996

Inequivalent Quantizations and Holonomy Factor from the Path-Integral Approach

A path-integral quantization on a homogeneous space G/H is proposed based on the guiding principle `first lift to G and then project to G/H'. It is then shown that this principle gives a simple procedure to obtain the inequivalent quantizations (superselection sectors) along with the holonomy factor (induced gauge field) found earlier by algebraic approaches. We also prove that the resulting matrix-valued path-integral is physically equivalent to the scalar-valued path-integral derived in the Dirac approach, and thereby present a unified viewpoint to discuss the basic features of quantizing on $G/H$ obtained in various approaches so far.

hep-th↗

arXiv1996

Regularization of Toda lattices by Hamiltonian reduction

The Toda lattice defined by the Hamiltonian $H={1\over 2} \sum_{i=1}^n p_i^2 + \sum_{i=1}^{n-1} ν_i e^{q_i-q_{i+1}}$ with $ν_i\in \{ \pm 1\}$, which exhibits singular (blowing up) solutions if some of the $ν_i=-1$, can be viewed as the reduced system following from a symmetry reduction of a subsystem of the free particle moving on the group $G=SL(n,\Real )$. The subsystem is $T^*G_e$, where $G_e=N_+ A N_-$ consists of the determinant one matrices with positive principal minors, and the reduction is based on the maximal nilpotent group $N_+ \times N_-$. Using the Bruhat decomposition we show that the full reduced system obtained from $T^*G$, which is perfectly regular, contains $2^{n-1}$ Toda lattices. More precisely, if $n$ is odd the reduced system contains all the possible Toda lattices having different signs for the $ν_i$. If $n$ is even, there exist two non-isomorphic reduced systems with different constituent Toda lattices. The Toda lattices occupy non-intersecting open submanifolds in the reduced phase space, wherein they are regularized by being glued together. We find a model of the reduced phase space as a hypersurface in ${\Real}^{2n-1}$. If $ν_i=1$ for all $i$, we prove for $n=2,3,4$ that the Toda phase space associated with $T^*G_e$ is a connected component of this hypersurface. The generalization of the construction for the other simple Lie groups is also presented.

hep-th↗

arXiv1996

Quantum mechanical Liouville model with attractive potential

We study the quantum mechanical Liouville model with attractive potential which is obtained by Hamiltonian symmetry reduction from the system of a free particle on $SL(2, \Real)$. The classical reduced system consists of a pair of Liouville subsystems which are `glued together' in such a way that the singularity of the Hamiltonian flow is regularized. It is shown that the quantum theory of this reduced system is labelled by an angle parameter $θ\in [\,0,2π)$ characterizing the self-adjoint extensions of the Hamiltonian and hence the energy spectrum. There exists a probability flow between the two Liouville subsystems, demonstrating that the two subsystems are also `connected' quantum mechanically, even though all the wave functions in the Hilbert space vanish at the junction.

hep-th↗

arXiv1995

Induced Gauge Fields in the Path Integral

The path integral on a homogeneous space $ G/H $ is constructed, based on the guiding principle `first lift to $ G $ and then project to $ G/H $'. It is then shown that this principle admits inequivalent quantizations inducing a gauge field (the canonical connection) on the homogeneous space, and thereby reproduces the result obtained earlier by algebraic approaches.

hep-th↗

arXiv1995

The canonical connection in quantum mechanics

In this paper we investigate the form of induced gauge fields that arises in two types of quantum systems. In the first we consider quantum mechanics on coset spaces G/H, and argue that G-invariance is central to the emergence of the H-connection as induced gauge fields in the different quantum sectors. We then demonstrate why the same connection, now giving rise to the non-abelian generalization of Berry's phase, can also be found in systems which have slow variables taking values in such a coset space.

hep-th↗

arXiv1993

BPST instanton and Spin from inequivalent quantizations

We present a simple alternative to Mackey's account of the (infinite) inequivalent quantizations possible on a coset space G/H. Our reformulation is based on the reduction ${\rm G \rightarrow G/H}$ and employs a generalized form of Dirac's approach to the quantization of constrained systems. When applied to the four-sphere $S^4 \simeq {\rm Spin(5)/Spin(4)}$, the inequivalent quantizations induce relativistic spin and a background BPST instanton; thus they might provide a natural account of both of these physical entities.

hep-th↗

arXiv1993

On the emergence of gauge structures and generalized spin when quantizing on a coset space

It has been known for some time that there are many inequivalent quantizations possible when the configuration space of a system is a coset space G/H. Viewing this classical system as a constrained system on the group G, we show that these inequivalent quantizations can be recovered from a generalization of Dirac's approach to the quantization of such a constrained system within which the classical first class constraints (generating the H-action on G) are allowed to become anomalous (second class) when quantizing. The resulting quantum theories are characterized by the emergence of a Yang-Mills connection, with quantized couplings, and new 'spin' degrees of {}freedom. Various applications of this procedure are presented in detail: including a new account of how spin can be described within a path-integral formalism, and how on S^4 chiral spin degrees of {}freedom emerge, coupled to a BPST instanton.

hep-th↗