- 3

- 0

For a real scalar field \phi, assume that H = H_free - \int d^3 x\ J \phi. J(x, t) is just some real number, source, or background field, without second quantization. Now, what is the amplitude \psi(x, t) for finding a particle at time t(before, during, or after source is on/off) at position x? The J(x,t) is nonzero only for finite period of time. And the initial state is vacuum, when t --> -\infty .

This question looks simple. However, I cannot find a solution which satisfies both causality and Lorentz invariance.