arXiv 1007.45922010-07hep-th

作为正则变换的全息重整化

Holographic renormalization as a canonical transformation

Ioannis Papadimitriou

备注：: 37 pages; v2 minor corrections in section 2, 2 references and a footnote on Palatini gravity added. Version to appear in JHEP
发表信息：: JHEP 1011:014,2010
DOI：: 10.1007/JHEP11(2010)014

摘要：

规范/弦对偶引起了人们对一类无穷远边界上的变分问题的注意，除非在经典作用量中加入一定的边界项，否则这些问题是不好定义的。在渐近AdS空间中的超引力背景下，这些问题被系统地用全息重整化方法解决。我们认为，这类先验定义不佳的变分问题远远超出了全息对偶的领域。正如我们所展示的，在非渐近AdS空间中的引力中，在具有某些无界自下势的点粒子中，甚至在平坦或AdS背景中的基本弦中，都出现了完全相同的问题。我们表明，在所有这种情况下的变分问题可以很好地定义由以下程序，这是内在的系统的问题，并不依赖于存在的全息对偶理论：（i）第一步是建设的最一般的渐近解的经典运动方程，继承了一个定义良好的辛形式从相空间。一个定义良好的辛形式的要求是必不可少的，往往导致必要的重新包装的自由度。(ii)一旦渐近解的空间已经被构造在正确的自由度方面，然后存在作为Hamilton-Jacobi方程的某个解而获得的边界项，该边界项同时使变分问题被很好地定义并保持辛形式。这个过程与渐近AdS引力情形下的全息重整化完全相同，但它适用于任何哈密顿系统。

英文摘要：

The gauge/string dualities have drawn attention to a class of variational problems on a boundary at infinity, which are not well defined unless a certain boundary term is added to the classical action. In the context of supergravity in asymptotically AdS spaces these problems are systematically addressed by the method of holographic renormalization. We argue that this class of a priori ill defined variational problems extends far beyond the realm of holographic dualities. As we show, exactly the same issues arise in gravity in non asymptotically AdS spaces, in point particles with certain unbounded from below potentials, and even fundamental strings in flat or AdS backgrounds. We show that the variational problem in all such cases can be made well defined by the following procedure, which is intrinsic to the system in question and does not rely on the existence of a holographically dual theory: (i) The first step is the construction of the space of the most general asymptotic solutions of the classical equations of motion that inherits a well defined symplectic form from that on phase space. The requirement of a well defined symplectic form is essential and often leads to a necessary repackaging of the degrees of freedom. (ii) Once the space of asymptotic solutions has been constructed in terms of the correct degrees of freedom, then there exists a boundary term that is obtained as a certain solution of the Hamilton-Jacobi equation which simultaneously makes the variational problem well defined and preserves the symplectic form. This procedure is identical to holographic renormalization in the case of asymptotically AdS gravity, but it is applicable to any Hamiltonian system.

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