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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 witheare associated with an incoming particle and terms with^{i(p⋅x - Ept)}eare 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 thateis a complex expression of an incoming (going right) wave since^{i(p⋅x - Ept)}Re[ewhich 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 ofe(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}t

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?

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# A Wave of an outgoing anti-particle in quantum field theory?

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