Shogo Tanimura

Relativistic generalization and extension to the non-Abelian gauge theory of Feynman's proof of the Maxwell equations

Abstract R. P. Feynman showed F. J. Dyson a proof of the Lorentz force law and the homogeneous Maxwell equations, which the obtained starting from Newtons law of motion and the commutation relations between position and velocity for a single nonrelativistic particle. We formulate both a special relativistic and a general relativistic versions of Feynmans derivation. Especially in the general relativistic version we prove that the only possible fields that can consistently act on a quantum mechanical particle are scalar, gauge, and gravitational fields. We also extend Feynmans scheme to the case of non-Abelian gauge theory in the special relativistic context.

Reduction of quantum systems on Riemannian manifolds with symmetry and application to molecular mechanics

This paper deals with a general method for the reduction of quantum systems with symmetry. For a Riemannian manifold M admitting a compact Lie group G as an isometry group, the quotient space Q=M/G is not a smooth manifold in general but stratified into a collection of smooth manifolds of various dimensions. If the action of the compact group G is free, M is made into a principal fiber bundle with structure group G. In this case, reduced quantum systems are set up as quantum systems on the associated vector bundles over Q=M/G. This idea of reduction fails, if the action of G on M is not free. However, the Peter–Weyl theorem works well for reducing quantum systems on M. When applied to the space of wave functions on M, the Peter–Weyl theorem provides the decomposition of the space of wave functions into spaces of equivariant functions on M, which are interpreted as Hilbert spaces for reduced quantum systems on Q. The concept of connection on a principal fiber bundle is generalized to be defined well on the...

Gauge field, parity and uncertainty relation of quantum mechanics on S1

We consider the uncertainty relation between position and momentum of a particle on S 1 (a circle). Since S 1 is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate treatment. Recently Ohnuki and Kitakado have formulated quantum mechanics on S D (a D-dimensional sphere). Armed with their formulation, we examine this subject. We also consider parity and find a phenomenon similar to the spontaneous symmetry breaking. We discuss problems which we encounter when we attempt to formulate uantum mechanics on a general manifold

An extension of Fourier analysis for the n-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian

We solved the Schrodinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus Tn=Rn/Λ is defined as a quotient of the Euclidean space Rn by an arbitrary n-dimensional lattice Λ. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schrodinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian....

Exact solutions of holonomic quantum computation

Holonomic quantum computation is analyzed from geometrical viewpoint. We develop an optimization scheme in which an arbitrary unitary gate is implemented with a small circle in a complex projective space. Exact solutions for the Hadamard, CNOT and 2-qubit discrete Fourier transformation gates are explicitly constructed.

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.

Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation

The isoholonomic problem in a homogeneous bundle is formulated and solved exactly. The problem takes a form of a boundary value problem of a variational equation. The solution is applied to the optimal control problem in holonomic quantum computer. We provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a k-dimensional unitary gate which operates on an N-dimensional Hilbert space with N⩾2k. Our construction is applied to several important unitary gates such as the Hadamard gate, the CNOT gate, and the two-qubit discrete Fourier transformation gate. Controllers for these gates are explicitly constructed.

Spontaneous breaking of the rotational symmetry induced by monopoles in extra dimensions

Abstract We propose a field theoretical model that exhibits spontaneous breaking of the rotational symmetry. The model has a two-dimensional sphere as extra dimensions of the space–time and consists of a complex scalar field and a background gauge field. The Dirac monopole, which is invariant under the rotations of the sphere, is taken as the background field. We show that when the radius of the sphere is larger than a certain critical radius, the vacuum expectation value of the scalar field develops vortices, which pin down the rotational symmetry to lower symmetries. We evaluate the critical radius and calculate configurations of the vortices by the lowest approximation. The original model has a U (1)× SU (2) symmetry and it is broken to U (1), U (1), D 3 for each case of the monopole number q =1/2,1,3/2, respectively, where D 3 is the symmetry group of a regular triangle. Moreover, we show that the vortex configurations are stable against higher corrections of the perturbative approximation.

Inequivalent Quantizations and Holonomy Factor from the Path-Integral Approach ☆

Abstract A path-integral quantization on a homogeneous spaceG/His proposed, based on the guiding principle “first lift toGand then project toG/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 we present a unified viewpoint to discuss the basic features of quantizing onG/Hobtained in various approaches so far.

Quantum mechanically induced Hopf term in the O(3) nonlinear sigma model

Abstract The Hopf term in the (2+ 1)-dimensional O (3) nonlinear sigma model, which is known to be responsible for fractional spin and statistics, is re-examined from the viewpoint of quantization ambiguity. It is confirmed that the Hopf term can be understood as a term induced quantum mechanically, in precisely the same manner as the θ-term in QCD. We present a detailed analysis of the topological aspect of the model based on the adjoint orbit parametrization of the spin vectors, which is not only very useful in handling topological (soliton and/or Hopf) numbers, but also plays a crucial role in defining the Hopf term for configurations of nonvanishing soliton numbers. The Hopf term is seen to arise explicitly as a quantum effect which emerges when quantizing an S 1 degree of freedom hidden in the configuration space.