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

*e*are associated with an incoming particle and terms with

^{i(p⋅x - Ept)}*e*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.

^{i(-p⋅x + Ept)}Since my undergraduate student years, I have always agreed that

*e*is a complex expression of an incoming (going right) wave since

^{i(p⋅x - Ept)}*Re[e*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.

^{i(p⋅x - Ept)}] = cos(p⋅x-E_{p}t),However, If I take real part of

*e*(it is a phase factor of terms for the outgoing anti-particle) to see how a wave of the outgoing anti-particle propagates, Re[

^{i(-p⋅x + Ept)}*e*] = cos(

^{i(-p⋅x + Ept)}*-p⋅x+E*), which is same to that of the incoming particle!

_{p}t)=cos(*p⋅x-E*_{p}tI 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?