哈密頓三形式扮演了沿著N向量演化方程的生成子的角色。它決定了哈密頓邊界表示 式，也因而決定了準局域量。能量其意義實為能差，能差的概念總是涉及一個相對的參考值，因此無法唯一定義物理的能量。協變哈密頓法[PRD 72 (2005) 104020]指定了一個適當的邊界表示式，而近期的工作中[PRD 84 (2011) 084047 GRG 44 (2011) 2401]，考慮球對稱時空的情形，我們藉由四維度規在封閉二維面上的適配條件得到令人滿意的結果。本文分析了一般情形的四維度規在封閉二維面的適配條件。我們發現對於一個二維面，滿足等距嵌入到閔氏空間，在度規適配的條件下仍然具有兩個自由度可以決定參考系的選擇。準局域能量的值形成一個集，若 它是這兩個自由函數的泛函，則臨界點為其一階變分的解，而準局域能量則為相應的臨界值。 The Hamiltonian 3-form plays the role of the generator of the evolution w.r.t. the displacement vector. It is uniquely de?ned up to a total di?erential term, the Hamiltonian boundary expression. The latter determines the quasi-local quantities. The meaningful concept of energy involves the di?erence of the dynamical values w.r.t. the reference values, so that we do not have a unique de?nition of the physical energies. For the covariant Hamiltonian approach a suitable boundary expression [PRD 72 (2005) 104020] was identi?ed, and in recent works [PRD 84 (2011) 084047 GRG 44 (2011) 2401] we found satisfactory results obtained from matching the four metrics on a 2-sphere for spherically symmetric spacetimes. Here we analyze the general 4D-metric matching on a closed 2-surface. We ?nd that for a 2-surface which satis?es isometric embedding into Minkowski space there are still two degrees of freedom remaining to determine the choice of reference. The quasi-local energy values form a set, and, if it is a functional of the two free functions, the critical values could be determined by the solution of its variation.
