Transformation Properties of Spinors
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 Making a Scalar With a Spinor
Consider "! "!
¢ ¢ ¢ ¢ ¢$# ¢&% ! ¦
Under a Lorentz transformation, ¤ ¤ ¢ ¢ ¢ ¢ £ ¤ ¤ ¢ ¢ £ ' ¤ ¤ Since ¦ (check for yourself using the explicit ¤ representation of on the previous page), ¢ ¢ is not a Lorentz scalar.
Physics 424 Lecture 15 Page 3 The Adjoint Spinor
Just as four-vector contractions need a few well-placed minus )* signs (i.e., ( ) in order to make a scalar, we can add a couple of minus signs to a spinor by defining the adjoint spinor: . + / / / / ¢ ¢-, ¢ ¢ ¢ ¢ ¦ %
# . . ¤ ¤
Since ¦ (again, check this yourself), 0! +
¢ ¢ ¢$# ¢&% ¢ ¢ ¦
is a Lorentz scalar.
Physics 424 Lecture 15 Page 4 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 Parity
Under a parity transformation . ¢ ¢ £
Since . 3 . 3 + + ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ = = = = £ £ . . . . 3 . . ¢ ¢ ¢ ¢ £ £ . . 3 ¢ ¢ ¢ ¢ £ £ 3 + + ¢ ¢ ¢< ¢ £ £ + + 3 ¢ ¢ ¢ ¢ is a true scalar and is a pseudoscalar.
Physics 424 Lecture 15 Page 7 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 D DFE ) ) )*
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 EM and photons
Maxwell’s equation I )* * H * I I ) H * J M K D 7 are not uniquely determined and so we are allowed to H H I R ) ) ) ! make a gauge transformation £ I H ) 5 ¦ We can demand the Lorentz condition ) The Lorentz condition simplifies the Maxwell equations to ) ) H M D K L ¦ Physics 424 Lecture 15 Page 10 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