Quantization of unstable linear scalar fields in static spacetimes

Abstract

We discuss the quantization of an unstable field through the construction of a "one-particle Hilbert space." The system considered here is a neutral scalar field evolving over a globally hyperbolic static spacetime and subject to a stationary external scalar potential. In order to prove our results we assume spacetimes without horizons and that the theory possess a "mass gap." Our strategy consists in building a complex structure, which arises from a suitable positive bilinear form defined over the space of classical solutions of the field equation. Once the space of states of the quantum field has been set, it is possible to study the effect of the time translation symmetry on it. From the time translation operator we obtain an expression for the Hamiltonian operator associated with the unstable sector of the field. This last result coincides with findings from long ago showing that the unstable degrees of freedom of the field behave as non-relativistic particles in a parabolic potential barrier.

Fundamental asymmetry between space and time in quantum field theory

“…In fact, if we impose equal t canonical commutation relations, then [φ(0, 0), φ(t, x)] vanishes for x 2 > t 2 , meaning that the field is causal only if we interpret t as the 'chronological time', and becomes acausal if, instead, we treat T as time (see equation (30)). Vice versa, if we impose equal T canonical commutation relations, then [φ(0, 0), φ(X, T)] vanishes for X 2 > T 2 , and the field is causal if T is interpreted as the 'chronological time', and acausal if, instead, t is treated as time (see equation (35)).…”

“…The situation, here, is quite different from the previous case, because now the commutators are canonical in the coordinates {T, X}, in which the field is tachyonic. Of course, we may look for the 'tachyon analogue' of equations (37) and (38) (see [35] for a formal construction). However, it is easier to approach the problem by looking directly at the commutation relations.…”

Instability of nonminimally coupled scalar fields in the spacetime of thin charged shells

“…Quantum fluctuations and, consequently, the expectation value of the stress-energy-momentum tensor, grow exponentially in time in the presence of these modes (see also Refs. [17,18] for further discussions on the quantization of unstable linear fields in static globally hyperbolic spacetimes). Now, we shall discuss in detail the radial part of modes v ∓ ωlµ and w ∓ Ωlµ , namely, ψ ∓ ωl (r)/r and ψ ∓ Ωl (r)/r, respectively.…”