The vacuum structure of light-frontφ 1+1 4 -theory

Abstract

We discuss the vacuum structure of φ 4 -theory in 1+1 dimensions quantised on the light-front x + = 0. To this end, one has to solve a non-linear, operator-valued constraint equation. It expresses that mode of the field operator having longitudinal light-front momentum equal to zero, as a function of all the other modes in the theory. We analyse whether this zero mode can lead to a non-vanishing vacuum expectation value of the field φ and thus to spontaneous symmetry breaking. In perturbation theory, we get no symmetry breaking. If we solve the constraint, however, non-perturbatively, within a mean-field type Fock ansatz, the situation changes: while the vacuum state itself remains trivial, we find a non-vanishing vacuum expectation value above a critical coupling. Exactly the same result is obtained within a light-front Tamm-Dancoff approximation, if the renormalisation is done in the correct way.

Statistical physics and light-front quantization

“…The simplicity of the light-front vacuum, usually considered an advantage, seems to bear problems as far as phase transitions are concerned. Clearly, thermal field theory on the lightfront would be rendered useless if it turned out that the zero-mode problem [23,24] would have to be solved before any statements about phase transitions could be made. However, the fact that the statistical weight of a configuration is maximized for minimal equal-time energy rather than for minimal light-front energy has important consequences for the formation of condensates at low temperatures.…”

“…This is possible, even in a gauge theory, because the gauge link between the fermion fields at r 1 and r 2 can be absorbed into a redefinition of the field operators, see Eq. (22,23). Note that at T = 0, G α,β (k) = G R α,β (k) for positive energy solutions and G α,β (k) = G A α,β (k) for negative energy solutions.…”

Critical Coupling for Two-dimensional $ϕ^4$ Theory in Discretized Light-Cone Quantization

“…When the constraint equation is solved perturbatively, it leads to additional interaction terms in the Hamiltonian. For some processes studied, contributions of these new terms, however were found to vanish [32,33] in the infinite volume limit (L → ∞). We also note that since the zero mode is a constrained field in two-dimensional φ 4 theory in the symmetric phase, it may be omitted when one solves the quantum theory for non-zero mass eigenstates.…”

Manifestation of a Nontrivial Vacuum in Discrete Light-Cone Quantization

“…It has been believed that the true vacuum is trivial and only the zero mode is responsible for spontaneous symmetry breaking (SSB) in scalar field theories [5,8]. There are some studies that rely on a combination of constrained zero modes and trivial vacuum [9]. The vacuum triviality, however, results from an assumption that normal-ordered Hamiltonians are positive semidefinite.…”