Quantum Inequalities and Particle Creation in the Presence of an External, Time-Dependent Mamaev-Trunov Potential

1 United States Military Academy at West Point, West Point, New York 10996-1790, USA a) Corresponding author: kalistaschauer@gmail.com b) michael.pfenning@westpoint.edu c) jared.cochrane.mil@mail.mil Abstract. In 2011, Mr. Dan Solomon proposed a model of a quantized scalar field interacting with a time-dependent Mamaev-Trunov potential in two-dimensional Minkowski spacetime. This model is governed by the Klein-Gordon wave equation with a time-dependent potential. Mr. Solomon claims that this model violates both the classical energy conditions of special relativity and the quantum energy conditions of quantum field theory in curved spacetime. Every classical energy condition can be violated, and their natural replacements are known as quantum inequalities. Mr. Solomon attempted to prove violations of the spatial and temporal quantum inequalities, and he correctly assumed that the negative energy splits into two fluxes at the Cauchy surface, where the potential is turned off. Unfortunately, Solomon neglects the contribution to the energy density due to particle creation when the potential is turned off at time t = 0 . In this project, we calculate the contribution to the stress energy tensor due to particle creation. We show that while the classical energy conditions are violated, the quantum energy inequalities hold, contrary to Mr. Solomon’s statements. SCIENTIFIC BACKGROUND Mathematical Background The mathematical foundation of quantum mechanics consists of wave functions and operators. Wave functions express the state of a system while operators represent observables. Linear algebra is the underlying mathematics of quantum mechanics, where abstract vectors represent wave functions and observables are performed as linear transformations [1]. Quantum mechanics uses Dirac notation to represent a vector as a ‘ket’, shown as . The dual a⟩ ∣ vector for a ket is a ‘bra’, with the inner product ‘bra-ket’ written as . a∣b〉 〈 An inner product space is a vector space over the real or complex numbers containing inner products or dot products. The vector spaces in which wavefunctions exist are called Hilbert spaces. Hilbert spaces are finite-dimensional and span the complex numbers [2]. A Hilbert space is a Banach space where the norm, or mapping, is an inner product. Hilbert spaces are mathematically easier to handle than general Banach spaces due to orthogonality. A Hilbert space is a complete inner product space, an example of which is the collection of square integrable functions,