<https://www.physicsforums.com/threads/question-about-coefficients-of-massless-quantum-fields.949541/>
<http://schema.org/articleBody>
"From Chapter 5.9 Weinberg's QFT Vol 1, massless fields are defined as:  \n \\psi_l(x)=(2\\pi)^{-3/2}\\int d^{3}p\\sum_{\\sigma}[k a(p,\\sigma)u_l(p,\\sigma)e^{ipx}+\\lambda a^{c\\dagger}(p,\\sigma)v_l(p,\\sigma)e^{-ipx}]\nWith coefficients defined by the conditions:  \nu_{\\bar{l}}(p,\\sigma) =\\sqrt{|k|/p^0} \\sum_{l}D_{\\bar{l}l}(L(p))u_l(k,\\sigma)\nv_{\\bar{l}}(p,\\sigma) =\\sqrt{|k|/p^0} \\sum_{l}D_{\\bar{l}l}(L(p))v_l(k,\\sigma)\nu_{\\bar{l}}(p,\\sigma) exp(i\\sigma \\theta(k,W) =\\sqrt{|k|/p^0} \\sum_{l}D_{\\bar{l}l}(W)u_l(k,\\sigma)\nv_{\\bar{l}}(p,\\sigma) exp(-i\\sigma \\theta(k,W) =\\sqrt{|k|/p^0} \\sum_{l}D_{\\bar{l}l}(W)v_l(k,\\sigma)\nWhere D_{\\bar{l}l}(L(p)) is a general, irreducible representation of the homogenous Lorentz group restricted to standard boosts, L(p) that take the standard momentum k = (0,0,k) into arbitrary momentum p and D_{\\bar{l}l}(W) is the Lorentz representation restricted to the little group for massless particles. Now Weinberg says that the equations for v are just the complex conjugates of the equations for u so that we can adjust the constants k and \\lambda so that\nv_l(p,\\sigma)=u_l(p,\\sigma)^*\nHowever, taking the complex conjugates of the equations of u:\nu_{\\bar{l}}(p,\\sigma)^* =\\sqrt{|k|/p^0} \\sum_{l}D_{\\bar{l}l}(L(p))^*u_l(k,\\sigma)^*\nu_{\\bar{l}}(p,\\sigma)^* exp(-i\\sigma \\theta(k,W) =\\sqrt{|k|/p^0} \\sum_{l}D_{\\bar{l}l}(W)^*u_l(k,\\sigma)^*\n\nThis is where I get stuck. The above will be true if D_{\\bar{l}l}(L(p))^*=D_{\\bar{l}l}(L(p)) and D_{\\bar{l}l}(W)^*=D_{\\bar{l}l}(W). However, this does seem to necessarily be true. Is there another way to prove Weinberg's claim?"
<https://www.physicsforums.com/threads/question-about-coefficients-of-massless-quantum-fields.949541/>