Transformation properties of spinors Lorentz Transformations of Spinors Bilinear Covariants The Photon Slides from Sobie and Blokland Physics 424 Lecture 15 Page 1 Lorentz Transformations of Spinors Spinors are not four-vectors, therefore they do not transform ¡ via . How do they transform? ¤ ¢ ¢ £ where for motion along the ¥ -axis, §©¨ § ¤ ¦ § §¨ § ¦ ¦ Physics 424 Lecture 15 Page 2 3 Page entz Lor a not is ¢ ¢&% ¢ explicit Spinor ¢ ¢ ! the ¤ a ¤ ¢$# ¤ page), ¢ ith ¤ using "! ¢ W ¢ £ £ evious 15 e pr "! yourself ¢ Lectur ¢ the ¢ Scalar for ¦ on a ¤ ¢ transformation, ¢ of (check entz ¦ ' ¤ Lor ¤ a Making . esentation epr Consider Under Since scalar r 424 Physics 4 Page minus : couple a spinor add / % ¢ ¢&% can well-placed adjoint / we ¢ # , few the ¢$# a yourself), / ¢ scalar Spinor a this need / ¢ ¢ defining 15 e ¦ by make 0! check . to Lectur ¢ Adjoint ¢ ¦ der spinor contractions (again, . or ¢ a . + ¢-, + ¢ in to The ¦ ) )* scalar ¤ ( -vector . signs entz ¤ four (i.e., as Lor minus a of Just Since is signs 424 Physics 5 Page : .) % ) other to 3 Family every ¦ 3 the # with ) happened 5 of 5 . what ; by 4 esentation, ¦ 15 e , 3 3 know epr 5 Sheep r Lectur ¦ anticommutes to 3 -matrix ell 3:9 Dr want )87 and Black and 6 ¦ don't additional 3 The : an 2 Bjorken eally 1 r the ou Define (Y In Note: 424 Physics Another Scalar? + ¢ ¢ We have already seen how is a Lorentz scalar. . 3 . 3 ¤ ¤ Since ¦ (check this too), 3 + ¢< ¢ is also a Lorentz scalar. + + 3 ¢< ¢ ¢ ¢ This gives us 2 Lorentz scalars: and . What's the difference? Physics 424 Lecture 15 Page 6 ¢ 7 ¢ . ¢ Page 3 = 3 3 . . . ¢ . 3 ¢ + ¢ ¢< ¢ = . £ £ £ £ ¢ 3 ¢ + ¢ . pseudoscalar £ a is ¢ ¢ 15 e 3 Parity + ¢ Lectur ¢ ¢ . . = ¢ and . . transformation ¢ ¢ scalar + ¢ ¢ ¢ = ue parity tr £ £ £ £ a a is ¢ ¢ + + ¢ ¢ Under Since 424 Physics Bilinear Covariants ¢ ¢A@ /?> There are 16 possible products of the form . These 16 products can be grouped together into bilinear covariants: + ¢ ¢ Scalar 1 component + 3 ¢ ¢ Pseudoscalar 1 component + ¢ ¢ ) Vector 4 components + 3 ¢ ¢ ) Pseudovector 4 components + ¢ ¢ * ) Antisymmetric tensor 6 components 4 B C ) * * ) , 7 Note that: Physics 424 Lecture 15 Page 8 Why This is Useful 3 3 6 9 DFE D ) ) )* We have a simple basis set 7 7 7 7 for any matrix, therefore we can always simplify more complicated combinations of matrices. The tensorial and parity character of each bilinear is evident. This makes it easy to see why the QED interaction Lagrangian + H ) ¢< ¢ G ) leads to a parity-conserving electromagnetic force mediated by a vector (i.e., spin-1) boson. To describe the parity-violating weak interaction, we could + + 3 ¢ ¢ ¢ ¢ ) ) (and do) mix vector and axial interactions. Physics 424 Lecture 15 Page 9 10 to Page to allowed 5 e M * ¦ ar equations R Q D<L ) ) I H we ¦ 7 ) P I so ¦ ) ) ! H H ) Maxwell M ) M * and I £ D L the ) I * ¦ H and photons ) O condition H 15 H * e K 7 K N entz Lectur determined ¦ and ¦ simplifies Lor H * )* J EM , ) the ) I I transformation ) uniquely I condition ¦ equation not K e entz gauge demand ar a e O Lor can e 7 N wher Maxwell's W The make 424 Physics Another Constraint Even with the Lorentz condition, we can make further gauge H H I R ) ) ) ! transformations of the form £ without H R M K K D 5 ) ) L ¦ disturbing ¦ so long as . As a result, we can impose an additional constraint. H 5 We typically choose to set ¦ and thereby work in the Coulomb gauge: O S$T 5 ¦ Physics 424 Lecture 15 Page 11 Free Photons M ) 5 For a photon in free space ( ¦ ), the potential is given by H K ) 5 ¦ . The plane-wave solution is >VU [ H ) XZY ) W ¥ § G ¦ ) ) 5 [ [ Y ¦ where is the polarization vector and ) . ) Although Y has 4 components, not all are independent. The ) 5 [ Y ¦ Lorentz condition requires that ) Furthermore, the . 5 5 \ Y Y ¦ ¦ Coulomb gauge implies that and T \ Since Y is perpendicular to , the photon is transversely polarized and there are only 2 independent polarization states. Physics 424 Lecture 15 Page 12.