Functional Analysis of Quantum Systems with Infinite Degree ofFir Freedom

(1)证明了一维量子自旋链纯态中半无限区域的Haag对偶性。作为推论，我们看到一维有限范围哈密顿量的平移不变纯基态具有分裂性质。(2)一维量子自旋链的平移不变纯态包含一个无限纠缠的副本当且仅当态不具有分裂性质时，如果Alice和Bob系统是一维晶格的半无限区域。我们还考虑了当态不是纯的而是因子的情况。在这种情况下，当且仅当贝尔常数达到相对于给定状态的任何正常状态的最大值时，该状态包含无限纠缠的一个副本。特别是，短程相互作用的Gibbs态不能包含一个无限纠缠的副本，因为贝尔常数取最小值1。非平衡定态(Ness)可以提供不纯态的新例子。(3)我们考虑了玻色凝聚系统的Ness的模拟。我们发现，对于相同温度的耦合玻色子，约瑟夫森流是不为零的。(4)在假设系统的时间演化具有一定的混合性(渐近阿贝尔性)的情况下，我们研究了在D.Ruelle意义下Ness的熵产生的涨落性质，得到了大数定律，并研究了类似的熵产生中心极限定理。在某些情况下，限制不是高斯的。

(1) We proved the Haag duality for half-infinite regions in pure states of 1-dim quantum spin chain.As a corollary, we see that translationally invariant pure ground states of 1-dim finite range Hamiltonians have the split property.(2)A translationally invariant pure state of a 1-dim quantum spin chain contains one copy of infinite entanglement if and only if the state does not have the split property provided that Alice and Bob systems are half-infinite regions of the 1-dim lattice.We also considered the case when the state is not pure but factor. In such cases, the state contains one copy of infinite entanglement if and only if the Bell's constant attains its maximal value for any normal states relative to the given one. In particular, the Gibbs states for short range interactions cannot contain one copy of infinite entanglement because the Bell's constant takes its minimum value 1. New examples of states which are not pure can be provided by non-equilibrium steady states(NESS).(3)We consider an analogue-of NESS for Bose systems with condensation. We found that Josephson current is non-vanishing for coupled Bosons with the same temperature.(4)We studied fluctuation property of entropy production of NESS in the sense of D.Ruelle and obtained the law of large number under the assumption that the time evolution of the system has certain mixing property(Asymptotic Abelian).We also investigated an analogue of the central limit theorem for entropy production. For some cases, the limit is not Gaussian.