Worldline formalism for a confined scalar field

Abstract

The worldline formalism is a useful scheme in quantum field theory which has also become a powerful tool for numerical computations. The key ingredient in this formalism is the first quantization of an auxiliary point-particle whose transition amplitudes correspond to the heat-kernel of the operator of quantum fluctuations of the field theory. However, to study a quantum field which is confined within some boundaries one needs to restrict the path integration domain of the auxiliary point-particle to a specific subset of worldlines enclosed by those boundaries. We show how to implement this restriction for the case of a scalar field confined to the D-dimensional ball under Dirichlet and Neumann boundary conditions, and compute the first few heat-kernel coefficients as a verification of our construction. We argue that this approach could admit different generalizations.

Worldline master formulas for the dressed electron propagator. Part I. Off-shell amplitudes

“…Extensions to curved space [54] and quantum gravity [55,56] have also been considered, addressing in particular induced effective actions and graviton self-energies [57][58][59], QED in curved spaces [60], gravitational corrections to the Euler-Heisenberg Lagrangians [61,62] and related amplitudes [63], and studies of one-loop photon-graviton conversion in strong magnetic fields [64,65]. The case of higher spin fields has also been approached using worldlines [66][67][68][69], as has quantum field theory on non-commutative spaces [70][71][72][73] and spaces with boundary [74][75][76].…”

Dynamical metasurfaces: electromagnetic properties and instabilities

“…In a series of works [89][90][91][92][93], Fosco et al regarded the Casimir energy as a functional of surface shapes and performed the derivative expansion [94] to obtain the PFA formula at the lowest order and higher-order corrections within the worldline formalism [95][96][97][98][99][100]. Such expansion is consistent with the studies of the cylinder-plane [101][102][103], sphere-plane [87,88,[104][105][106],…”

New Techniques for Worldline Integration

“…Beyond standard QFT, the worldline approach has shown to be a natural tool for the construction of higher-spin field interactions [16,20], and it lends itself to generalization to noncommutative spaces [4,5,33,63] as well as spaces with boundary [21,22,36].…”