In lattice regularization of relativistic quantum field theories, which plays a central role in the analyses of the theories on a non-perturbative level, Euclidean formulation makes it nontrivial whether or not the lattice scheme indeed defines a quantum theory which permits the probability interpretation. Because taking the continuum limit is too difficult a mathematical problem, it is more desirable that a quantum theory can already be reconstructed before taking the continuum limit. Therefore, it is an important issue to study the possibility of reconstructing quantum theories from lattice field theories, and the reflection positivity condition is known to ensure that a Hilbert space of state vectors and a Hamiltonian operator are reconstructed from a lattice model.

Besides the reconstruction problem, it was considered to be impossible to directly deal with the exact chiral symmetry on the lattice, since Nielsen-Ninomiya's no-go theorem was proved. It was the discovery of the overlap Dirac operator which opened up a new way to treat the exact chiral symmetry on the lattice based on the Ginsparg-Wilson relation.

The Ginsparg-Wilson relation expresses the chiral symmetry on the lattice, but other fundamental properties of lattice models, such as locality and possibility of reconstructing a quantum theory, have to be additionally established. The overlap Dirac operator was proved to be local in the sense that it has exponentially decaying tails, but the possibility of reconstructing a corresponding quantum theory was not completely understood.

In this thesis, we will prove that the overlap Dirac fermion satisfies the reflection positivity condition and therefore if the infinite volume limit is assumed the lattice overlap fermion model indeed defines a quantum mechanical system with a Hilbert space of state vectors and self-adjoint energy-momentum operators in it. Furthermore, it will be found that the joint spectrum of energy-momentum operators is supported on [0,∞)×[-π, π]d-1, where d is space-time dimension. This result shows that the Ginsparg-Wilson relation, based on which the exact lattice chiral symmetry is formulated, is compatible with these quantum theoretical properties. It will be also proved that the overlap boson model violates the reflection positivity condition. This implies that the lattice Wess-Zumino model formulated by using overlap boson violates the reflection positivity condition, and one can not be indifferent toward the violation of quantum theoretical unitarity condition when one uses this formulation of the lattice Wess-Zumino model in practical applications. These are main conclusions of this thesis.

In chapter 1, the motivation of studies in this thesis will be presented.

In chapter 2, a mathematical scheme of lattice field theories which contain scalar, spinor, and gauge fields will be given, and Wilson's lattice QCD will be introduced.

In chapter 3, generic lattice models satisfying the basic assumptions (A1)-(A4) will be considered and it will be proved that these models are ensured to indeed define physically acceptable quantum theoretical systems with strongly commuting Hamiltonian and momentum operators (H, P) supported on [0,∞) × [-π, π]d-1. It will be also proved that such theories permit the Umezawa-Kamefuchi-Kallen-Lehmann spectral representation of the two point correlation functions, with positive spectraldensity functions. This tells us that positivity of the spectral density is necessary for a model to satisfy the conditions (A1)-(A4).

In chapter 4, the overlap Dirac operator will be introduced and basic properties will be proved. It will be shown that, if gauge fields are admissible, then the overlap Dirac operator has exponentially decaying tails.

In chapter 5, the main chapter of this thesis, it will be proved that the overlap Dirac fermion satisfies the reflection positivity condition, and that the overlap boson violates the reflection positivity condition. As an application, the lattice Wess-Zumino model will be discussed in the viewpoint of the refection positivity. It will be pointed out that the gauge interacting case is still an open problem. But the method we will present here will not seem to be used straightforwardly in the gauge interacting case. This will need some new idea.

In chapter 6, we will summarize the thesis.

本論文は6章からなる。第1章は、イントロダクションであり、本論文の研究対象となっている、格子場の理論、ユークリッド時空上の場の理論とミンコフスキー空間上の場の理論の関係、Wilson の格子QCD理論、overlap Dirac operatorを用いた格子場理論、などについて、歴史的背景および動機が書かれている。また1.1節では本論文の構成、および主な結果がまとめられている。

第2章は格子場の理論について一般的な定式化が記述されている。Wilson の格子QCD 理論もここで導入され、ゲージ不変性が示されている。

第3章では、格子場の理論から量子論を再構成する手続きが詳細に記述されている。具体的には、与えられた格子場の理論が(A1)~(A4)の4つの仮定が満たせば、そこから量子論が構成され、ハミルトニアン及び運動量演算子が構成される事が示されている。また、3.4節では、格子場の理論が(A1)~(A4)の仮定を満たせば、その2点相関関数が正のspectral densityを持つKallen-Lehmann表示で表される事が証明されている。またここから逆に、正のspectral densityを持つ事が、格子場の理論が(A1)~(A4)を満たすための必要条件である事が結論づけられている。なお3.4節の内容は論文提出者の単著論文(雑誌投稿準備中)に基づいている。

第4章では、overlap Dirac operatorを用いた格子場理論が導入され、それが厳密なchiral symmetryを保っている事が記述されている。

第5章が本論文の主要部分である。第4章で導入されたoverlap Dirac operatorを用いた格子場理論は厳密なchiral symmetryを満たすという点で非常に魅力的であるが、第3章で説明された4つの条件(A1)~(A4)のうち、reflection positivityと呼ばれる条件(A3)を満たすかどうか、きわめて非自明であった。5.1節では、overlap Dirac operatorを用いた格子場理論がreflection positivityを満たす事が証明されている。また5.2節では、overlap scalar boson の格子場理論の場合にはreflection positivityの条件を破っている事が示されている。5.3節ではWess-Zumino模型への応用が議論されている。

第6章は結論にあてられている。またAppendixにおいては、本論文で用いられた様々な定理がまとめられている。

なお、本論文第5章の一部は菊川氏との共同研究に基づいているが、論文提出者が主体となって証明・解析を行ったもので、論文提出者の寄与が十分であると判断する。

したがって、博士(理学)の学位を授与できると認める。