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Introduction to the Standard Model
Lecture 8: Quantisation and Feynman Rules
Quantisation of Gauge Fields
problem with gauge fields:Given the field equation: M we see that becauseMμν∂ν= 0,Mis not invertible. The problem can be solved by using the fact that not all degrees of freedom forAμare physical (observable). This can be seen by applying a gauge transformation toAμ: A We can now choose Λ such that∂μAμ= 0 which removes one degree of freedom from the vector fieldAμ. The gauge function Λ is not completely determined; there isanother gauge freedom such that?Λ?= 0. Then we have A μ→A??μ=Aμ+∂μΛ +∂μΛ?where∂μΛ can be used to remove∂μAμand the term∂μΛ?can be used to remove another
degree of freedom,e.g.A0= 0 . Thus,Aμnow only has two degrees of freedom; the other two can be "gauged away." The mode expansion for the 4-component gauge field is Aμ(x) =?d3k
(2π)33 r=0? Notice thatAμ(x) =A?μ(x); the gauge fields are real-valued.Appyling the gauge condition:
μAμ= 0 =?kμεrμ= 0
A0= 0 =?εμ= (0,ε)?
=?k·ε= 0 This is thetransversalitycondition; it is consistent with the observation that EM radiation is tranversly polarised. Note:kμεμr= 0 is manifestly covariant whereasA0= 0 is not. 1 By choosing a reference framekμ=ω(1,0,0,1), the polarisation vectors read:1μ= (0,1,0,0)
2μ= (0,0,1,0)?
Linearly polarised
Using a basis change, one obtains the polarisation vectors which correspond to circularly polarised light:±μ=?1
⎷2(εμ1±εμ2) =?1⎷2(0,1,±i,0)μare thehelicity eigenstatesof the photon.
Note:ε1,2,ε±μcorrespond to the two observable degrees of freedom of the gauge fieldAμ.
The quantisation of gauge fields is non-trivial because∂μAμ= 0 cannot be implemented at the operator level due to a contradiction with the canonicalcommutation relations. This issue is solved by the Gupta-Bleuler formalism (see Rel. QFTnotes for more detail): only quantum states which correspond to transverse photons(ε±/ε1,2) are relevant for observables. unphysical degrees of freedom do not contribute in the scattering matrix (S-matrix) elements. The Gupta-Bleuler formalism works for anyU(1) gauge theory but fails for Non-Abelian theories (this was later solved by Fedeev and Popov in 1958 -see Modern QFT).