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

[goodphy]

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 e^{i(p⋅x - E_pt)} are associated with an incoming particle and terms with e^{i(-p⋅x + E_pt)} 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 e^{i(p⋅x - E_pt)} is a complex expression of an incoming (going right) wave since Re[e^{i(p⋅x - E_pt)}] = cos(p⋅x-E_pt), 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 e^{i(-p⋅x + E_pt)} (it is a phase factor of terms for the outgoing anti-particle) to see how a wave of the outgoing anti-particle propagates, Re[e^{i(-p⋅x + E_pt)}] = cos(-p⋅x+E_pt)=cos(p⋅x-E_pt), 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?

 

[A. Neumaier]

The textbook is sloppy and alludes to Dirac's sea interpretation of the negative energy solution in his (or the corresponding scalar) equation. In quantum field theory proper, however, there is no Dirac sea and the antiparticles have their own associated wave functions that behave as intuition (including yours) require.