Spontaneous symmetry breaking in light front field theory

A semiclassical picture of spontaneous symmetry breaking in light front field theory is formulated. It is based on a finite-volume quantization of self-interacting scalar fields obeying antiperiodic boundary conditions. This choice avoids a necessity to solve the zero-mode constraint and enables one to define unitary operators which shift the scalar field by a constant. The operators simultaneously transform the light front Fock vacuum to coherent states with lower energy than the Fock vacuum and with nonzero expectation value of the scalar field. The new vacuum states are noninvariant under the discrete or continuous symmetry of the Hamiltonian. Spontaneous symmetry breaking is described in this way in the two-dimensional $\lambda {\varphi }^4$ theory and in the three-dimensional $O(2)$-symmetric sigma model. A qualitative treatment of topological kink solutions in the first model and a derivation of the Goldstone theorem in the second one are given. Symmetry breaking in the case of periodic boundary conditions is also briefly discussed.

Figure 1

Regularized delta function ${\delta }_{\Lambda }(x^{-})$ for $L=20$, $\Lambda =3·{10}^5$, and $ϵ={10}^{-4}$.

Figure 2

Unregularized delta function ${\delta }_{\Lambda }(x^{-})$ (the parameter $ϵ=0$) for $L=20$, and $\Lambda =3·{10}^5$. The oscillations are rather strong and do not vanish for increased values of $\Lambda$.

Figure 3

Regularized sign function ${ϵ}_{\Lambda }(x^{-})$ for $L=20$, $\Lambda =3·{10}^5$, and $ϵ={10}^{-4}$ in the neighborhood of $x^{-}=0$. The function is a constant to a very high precision over the whole interval except for a tiny neighborhood of the point $x^{-}=0$ (or of the endpoints $x^{-}=\pm L$ in the case of the shifted function ${ϵ}_{\Lambda }(L-x^{-})$).

Figure 4

Detailed behavior of the unregularized ($ϵ=0$) sign function ${ϵ}_{\Lambda }(x^{-})$ for $L=20$, $\Lambda =3·{10}^5$ around $x^{-}=0$. Although the function is again indistinguishable from a constant inside the finite interval, it oscillates around the point $x^{-}=0$ in contrast to the case with a nonzero regulator $ϵ$.