Study of non-equilibrium dynamics become more and more important in many areas of physics such as heavy ion collisions, cold atoms, early universe and so on. Also from theoretical point of view, formalism of non-equilibrium steady state has not been established yet. Non-equilibrium dynamics appears, for example in heavy ion collision, in perception of glasma and to fix a QCD critical end point. Study of QCD at finite temperature and chemical potential has revealed nontrivial phase structure which contains rich physics. When we study phase structure, the critical point plays a very important role. It is well known that when the system in local equilibrium approaches to a critical point, relaxation to equilibrium state takes infinitely long time. This situation is quite different from physics in heavy ion collision, in which relaxation becomes much faster than we expected from perturbation. However, we still consider that understanding of dynamics near critical points is important.

When we theoretically deal with critical phenomena, scaling hypothesis plays a central role. This idea is as follows: Since the correlation length and the relaxation time of the system diverge, the system looks almost the same under scale transformation of the system. Thus physics in a critical region do not depend on details of systems. This is a basic idea of universality class, which was found in experiment.

In critical dynamics, macroscopic effective theory, so called mode coupling theory, has often been used to describe critical dynamics and achieved successful results. However, the dynamic universality class of QCD seems still controversial, when mode coupling theory is applied to the critical end point of QCD phase diagram. Also, we desire to understand critical dynamics of QCD from the dynamics of quarks and gluons. Thus, taking a simple field theory, we attempt to deal with dynamic critical phenomena by microscopically. Unfortunately it is almost desperate to describe dynamics in local equilibrium regions by microscopic perturbation theory, even away from equilibrium. The reason is that ordinary perturbations are expansions about number of collisions, but local equilibrium is realized by collisions of infinitely many times. Therefore some resummation is also needed to describe local equilibrium and to obtain transport coeσcients. This is why we employ 2PI formalism. 2PI formalism resums higher order diagrams in order that conservation laws are automatically satisfied and leads to hydrodynamic equations, and it helpful to derive transport coeσcients.

The goal of this thesis is two-fold. One is to evaluate an off-critical exponent v in the 2PI formalism. The other is to describe critical dynamics from microscopic theory. We employ 2PI formalism, since it systematically resums higher order contributions of 1PI formalism and automatically satisfies hydrodynamic equations in approximation level. The former property improves a value of v, and the latter property will be essential for critical dynamics. We take O(N) ψ4 model, which corresponds to, for example, Ising model (N = 1), 4He superuid transition (N = 2) and QCD chiral phase transition (N = 4). We employ 1=N expansion in NLO of ψ4 model.

In static critical phenomena, we can explicitly see diagrams which are resummed in 1PI and 2PI formalisms and confirm that 2PI takes higher order terms in 1PI diagrams. From the scaling relations, if we obtain two exponents (except for two exponents at a critical point, which are related to each others), we can completely determine all exponents of the system. Therefore, evaluations of off-critical exponent v is important. It is also known that 2PI formalism improves the evaluation of η in 1/N NLO comparing the 1PI result. Therefore we apply 2PI formalism to improve an evaluation of another (and from scaling relation, the other) exponent v. In this calculation, we derive a self-consistent equation of a three point vertex function Γ(2,1)(x; y; z) ~<ψ(x)ψ(y)ψ2(z) > (we show the strict definition of Γ(2,1)(x, y, z) in Sec.3), which is related to v. This relation between Γ(2,1) and v is shown by Ma. Using Γ(2;1), we can estimate v at a critical point, although v is an exponent near a critical point.

In dynamic critical phenomena, we attempt to extract dissipative modes, which control dynamics in a critical region and their dispersions are w~ iDp(2) (D is a transport coeσcient), by calculating a two-point correlation function. Then we evaluate a dynamic critical exponent ^z. In 1PI formalism, we fail to extract dissipative modes and there appears propagating modes, whose dispersions are w ~ ±csp + iΓ (cs is a sound velocity and Γ is a decay rate). This stems from perturbation that expands correlation functions around a free theory. In contrast, we expect that a correlation function from 2PI effective action has dissipative modes, since they satisfy hydrodynamic equations in approximation level. Hence, we assume that a two-point correlation function in critical region has a dissipative pole,Solving a self-consistent equation, we will obtain ^z from microscopic theory. However, this calculation is solely tentative and we have not evaluated ^z correctly yet.

This thesis is organized as follows. In Sec.2 we show the role of resummation which resolves a secularity problem. And we introduce 2PI effective action comparing ordinary 1PI effective action. Then we show the 1=N NLO diagrams in 2PI which we use through this thesis. In Sec.3, we evaluate static critical exponents η and v, which corresponds to a two-point correlation function on and near a critical point. We review the evaluation of η in 1/N expansion by Ma (1PI) and by Alford, Berges and Cheyne (2PI), then we compare two results and confirm the validity of resummation. Next, we show 1PI estimation of v (Ma) and calculate by 2PI formalism, then we compare the 1PI and 2PI results. We find that the differences come from diagrams of higher order by comparing resummed diagrams in both 1PI and 2PI formalism. In Sec.4, we proceed to dynamic critical phenomena and try to estimate a dynamic critical exponent ^z from microscopic Hamiltonian. From linear response theory, we can deal with critical dynamics by correlation functions of equilibrium state. Firstly, we review the conventional estimation of ^z; the mode coupling theory, and we introduce an action from which we can derive Langevin equations and show perturbation of mode coupling theory. Secondly, we attempt to evaluate a dynamic critical exponent from 1PI effective action in microscopic dynamics. Thirdly, we propose a tentative idea to apply 2PI formalism to critical dynamics. Finally, we discuss evaluations obtained in this section. In Sec.5, we summarize this thesis. Details of some complicated calculations are shown in Appendices.

本論文は5章から成り、第1章では緒言と研究の背景が述べられている。

第2章では、本論文で用いられる理論形式の基礎である2粒子既約有効作用理論の物理的および数学的背景が詳述されている。 特に、この章では、摂動級数展開の破綻とその総和法による解決について、永年項を含む常微分方程式を例にとった解説がなされたのち、場の量子論における摂動級数の総和を可能にする手法としての2粒子既約有効作用と1/N展開法を組み合わせた理論形式が、真空および有限温度の場合に論じられている。

第3章と第4章は、本論文の核心部分となっている。

第3章では、2次相転移点近傍における静的臨界指数について、2章で導入した2粒子既約有効作用と1/N展開を組み合わせた理論形式を用いた計算が行われている。まず、臨界点上での相関関数の距離依存性と関係する臨界指数η、臨界点付近での相関長の温度依存性と関係する臨界指数νについて、従来の1粒子既約有効作用を用いた計算法を紹介し、つぎに2粒子既約有効作用を用いた場合に、どのようなファインマングラフが総和されるかを考察している。さらに、臨界指数νを求めるため、2粒子既約有効作用を出発点に、複合演算子を含む頂点関数についての新しい自己無撞着方程式が導出されている。 この方程式は、2点関数に関するカダノフ-ベイム方程式の自然な拡張になっている。最後に、 O(N)φ4模型を例にとり、頂点関数に関するこの自己無撞着方程式の積分核に1/N展開を適用することで、νのN依存性に関する高次項を総和した新しい表式を得ている。この結果は、1粒子既約有効作用に対する1/N展開に比べてN<10で有意な違いを与え、第一原理計算によるνの値とも矛盾しない結果であることが示されている。

第4章では、2粒子既約有効作用の方法の、動的臨界現象への応用が試みられている。まず、臨界点近傍での動的緩和過程を特徴付ける臨界指数zについて、従来のモード結合理論の結果が議論されている。つぎに、1粒子既約有効作用、2粒子既約有効作用の両方を、O(N)φ4模型における時間依存2点相関関数に適用し、zに関する解析的な表式を与えている。得られた結果は、zが1の周りでの展開となっており、zが2の周りでの展開を与えるモード結合理論とは異なる表現となっていることを示し、その違いの物理的理由について考察している。

第5章では、本論文で得られた結果のまとめと、今後の展望が記述されている。

本論文では、有限温度の場の量子論における2次相転移点近傍の臨界指数に対して、従来の繰り込み群法やモード結合理論とは異なる2粒子既約有効作用の方法を適用し、自己無撞着方程式を基礎に、積分核に関する1/N展開による系統的な計算が可能であることが示されている。この成果は、場の量子論における臨界現象を扱う上での新しい端緒を開くものとして大きな意義を持つ。

なお、本論文の主要部である第3章と第4章の内容は、森松治氏、板倉数記氏、藤井宏次氏との共同研究であるが、論文提出者が主体となって理論的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。

以上の観点から、申請者に博士(理学)の学位を授与できると認める。