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Right below eq. (2.5.37), it is written that [tex]A[/tex] and [tex]B[/tex] can be simultaneously diagonalized by [tex]\Psi_{k,a,b}[/tex]. From the content, I inferred that [tex]\Psi_{k,a,b}[/tex] is also the eigenstate of the energy-momentum operator [tex]P^\mu[/tex] with eigenvalue [tex]k=(0,0,1,1)[/tex]. But, since [tex][A,P^\mu]\neq 0[/tex] and [tex][B,P^\mu]\neq 0[/tex], there should not be such simultaneous eigenstate for [tex]A,B,P^\mu[/tex].

Please help me on this. Thanks in advance.