Kazuyuki Fujii

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Abstract The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t0=t is naturally understood where t0 is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinskis theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.

Introduction to Grassmann manifolds and quantum computation

Geometrical aspects of quantum computing are reviewed elementarily for nonexperts and/or graduate students who are interested in both geometry and quantum computation. We show how to treat Grassmann manifolds which are very important examples of manifolds in mathematics and physics. Some of their applications to quantum computation and its efficiency problems are shown. An interesting current topic of holonomic quantum computation is also covered. Also, some related advanced topics are discussed.

Coherent states, Path integral, and Semiclassical approximation

Using the generalized coherent states it is shown that the path integral formulas for SU(2) and SU(1,1) (in the discrete series) are WKB exact, if it is started from the trace of e−iTĤ, where H is given by a linear combination of generators. In this case, the WKB approximation is achieved by taking a large ‘‘spin’’ limit: J,K→∞, under which it is found that each coefficient vanishes except the leading term which indeed gives the exact result. It is further pointed out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression sometimes leads to a wrong result. Therefore great care must be taken when some geometrical action would be adopted, even if it is so beautiful as the starting ingredient of path integral. Discussions on generalized coherent states are also presented both from geometrical and simple oscillator (Schwinger boson) points of view.

Note on coherent states and adiabatic connections, curvatures

We give a possible generalization to the example in the paper of Zanardi and Rasetti [Phys. Lett. A 264, 94 (1999)]. For this, explicit forms of adiabatic connection, curvature, etc., are given. We also discuss the possibility of another generalization of their model.

Coherent states over Grassmann manifolds and the WKB exactness in path integral

U(N) coherent states over Grassmann manifold, GN,n≂U(N)/(U(n)×U(N−n)), are formulated to be able to argue the WKB exactness in the path integral representation of a character formula. The phenomena is the so‐called localization of Duistermaat–Heckman. The exponent in the path integral formula is proportional to an integer k labeling the U(N) representation. Thus, when k→∞ a usual semiclassical approximation, by regarding k∼1/ℏ, can be performed to yield a desired conclusion. The mechanism of the localization is uncovered by the help of the (generalized) Schwinger boson technique. The discussion on the Feynman kernel is also presented.

Mathematical foundations of holonomic quantum computer

Abstract We make a brief review of an (optical) holonomic quantum computer (or computation) proposed by Zanardi and Rasetti (quant-ph/9904011) and Pachos and Chountasis (quant-ph/9912093), and give a mathematical reinforcement to their works.

JARLSKOG'S PARAMETRIZATION OF UNITARY MATRICES AND QUDIT THEORY

In the paper (math-ph/0504049) Jarlskog gave an interesting simple parametrization to unitary matrices, which was essentially the canonical coordinate of the second kind in the Lie group theory (math-ph/0505047). In this paper we apply the method to a quantum computation based on multi-level system (qudit theory). Namely, by considering that the parametrization gives a complete set of modules in qudit theory, we construct the generalized Pauli matrices which play a central role in the theory and also make a comment on the exchange gate of two-qudit systems. Moreover we give an explicit construction to the generalized Walsh-Hadamard matrix in the case of n=3, 4 and 5. For the case of n=5 its calculation is relatively complicated. In general, a calculation to construct it tends to become more and more complicated as n becomes large. To perform a quantum computation the generalized Walsh-Hadamard matrix must be constructed in a quick and clean manner. From our construction it may be possible to say that a qudit theory with

Exactness in the Wentzel–Kramers–Brillouin approximation for some homogeneous spaces

n\geq 5

BASIC PROPERTIES OF COHERENT AND GENERALIZED COHERENT OPERATORS REVISITED

is not realistic. This paper is an introduction towards Quantum Engineering.

Universal Schwinger cocycles of current algebras in (D+1)-dimensions: Geometry and physics

Analysis of the Wentzel–Kramers–Brillouin (WKB) exactness in some homogeneous spaces is attempted. CPN as well as its noncompact counterpart DN,1 is studied. U(N+1) or U(N,1) based on the Schwinger bosons leads us to CPN or DN,1 path integral expression for the quantity tr e−iHT, with the aid of coherent states. The WKB approximation terminates in the leading order and yields the exact result provided that the Hamiltonian is given by a bilinear form of the creation and the annihilation operators. An argument on the WKB exactness to more general cases is also made.