## 概覽

Faddeev-Popov ghosts嘅需要，源自響用路徑積分定義(en:path integral formulation)量子場論時，啲路徑積分唔應重覆咁數有規範變換(en:gauge symmetry) lung 住嘅場型(? en:field configuation)，因為渠地對應住同一物理態。所以路徑積分嘅測度加咗一個因子，令到我哋唔可以用以前嘅做法（好似用費曼圖）來直接由作用去攞一啲結果。但通常嘅做法亦有可能，喺改過條作用之後，行得通。 所以我地成日要加多幾個場先，呢啲場就叫＂鬼場＂喇. 呢個過程就叫Faddeev-Popov 方法 （睇埋BRST 量子化）。啲鬼場係種計算工具，唔代表真正外界既粒子：佢地＂只係＂可以喺費曼圖中出現，作為虛粒子(virtual particles).

The exact form or formulation of ghosts is dependent on the particular gauge chosen. The Feynman-'t Hooft gauge is usually the simplest gauge, and is assumed for the rest of this article.

Faddeev-Popov 鬼違反自旋－統計定理。例如，楊振寧－米爾斯理論中，(such as quantum chromodynamics) the ghosts are complex scalar fields (spin 0), but they anticommute (like fermions). In general, anticommuting ghosts are associated with bosonic symmetries, while commuting ghosts are associated with fermionic symmetries. Every gauge field has an associated ghost, and where the gauge field acquires a mass via the Higgs mechanism, the associated ghost field acquires the same mass (in the Feynman-'t Hooft gauge only, not true for other gauges).

In Feynman diagrams the ghosts appear as closed loops, attached to the rest of the diagram via a gauge particle at each vertex. Their contribution to the S-matrix is exactly cancelled by a contribution from a similar loop of gauge particles with only 3-vertex couplings or gauge attachments to the rest of the diagram. (The loop of gauge particles makes other contributions that survive the ghost cancellation.) The opposite sign of the contribution of the ghost and gauge loops is due to them having opposite fermionic/bosonic natures. (Closed fermion loops have an extra -1 associated with them; bosonic loops don't.)

The Lagrangian for the ghost fields $c^a(x)\,$ in Yang-Mills theories (where $a$ is an index in the adjoint representation of the gauge group) is given by

$\mathcal{L}_\mathrm{ghost} = \partial_\mu \overline{c}^a\partial^\mu c^a + g f^{abc}(\partial^\mu\overline{c}^a) A_\mu^b c^c$

The first term is a kinetic term like for regular complex scalar fields, and the second term describes the interaction with the gauge fields. Note that in abelian gauge theories (such as quantum electrodynamics) the ghosts do not have any effect since $f^{abc} = 0$ and, consequently, the ghost particles do not interact with the gauge fields.