Hello. I'm studying a course of the Quantum Field Theory and I got a question in a canonical quantization of a scalar field. I don't write a full expression of the field quantization here but the textbook said terms with ei(p⋅x - Ept) are associated with an incoming particle and terms with ei(-p⋅x + Ept) are for an outgoing anti-particle (p is positive). Here, natural units are used so 3-dimensional momentum vector p is equal to the wavenumber. Since my undergraduate student years, I have always agreed that ei(p⋅x - Ept) is a complex expression of an incoming (going right) wave since Re[ei(p⋅x - Ept)] = cos(p⋅x-Ept), which is obviously right-going wave. I thought If the wave of the particle (ex: an eigenfunction of the particle in the Quantum mechanics) propagates to the right, the particle itself really goes to the right. However, If I take real part of ei(-p⋅x + Ept) (it is a phase factor of terms for the outgoing anti-particle) to see how a wave of the outgoing anti-particle propagates, Re[ei(-p⋅x + Ept)] = cos(-p⋅x+Ept)=cos(p⋅x-Ept), which is same to that of the incoming particle! I think I have some misunderstood concepts in my mind but I don't know what it is. Could you please tell me what was I wrong?