In the present work, projection methods employed to restore the spontaneously broken symmetry in the mean-field approach to many-body problem are reviewed. The symmetry restoration is important to include correlations going beyond the mean-field approximation. Further, in order to cause a direct comparison with the experimental quantities, quantum states must have sharp values of the desired quantum numbers. For Hamiltonian based approaches, projection methods have been developed and applied for restoration of symmetries. Symmetry projected Hartree–Fock–Bogoliubov (HFB) equations have been derived in the Hamiltonian based approach, which have a similar structure to those of bare HFB equations. The explicit expressions for the projected Hartree-Fock (HF) and pairing fields are derived in the present work for a generalized projection operator. Approximations to the projection method, based on the work of Lipkin and Kamlah, are discussed with some technical details. Finally, problems associated in the application of the projection technique to the energy density functional approaches are highlighted.

在目前的工作中，对在多体问题的平均场方法中用于恢复自发破碎对称性的投影方法进行了综述。对称恢复对于包括超出均值场近似的相关性很重要。此外，为了引起与实验量的直接比较，量子态必须具有所需量子数的尖锐值。对于基于哈密顿量的方法，已经开发了投影方法并将其应用于对称性的恢复。对称投影的Hartree–Fock–Bogoliubov（HFB）方程已在基于Hamiltonian的方法中推导，其结构与裸HFB方程的结构相似。在目前的工作中，对于广义投影算子，导出了投影Hartree-Fock（HF）和配对字段的显式表达式。基于Lipkin和Kamlah的工作，对投影方法的近似进行了讨论，并附带了一些技术细节。最后，突出了与将投影技术应用于能量密度函数方法相关的问题。