Feynman Path Integral Formulation

222 6 Lattice Regularized Quantum Gravity
To actually show this, it will be useful to introduce an artificial integration ∫ [dΩ]
over the gauge group parameters, by writing for the Euclidean partition function
∫
Z = [dA]e −S(A)
∫
= const. [dΩ][dA]e −S(A)
∫
= const. [dΩ][dA Ω ]e −S(A)
∫
= const. [dΩ][dA]e −S(A Ω −1 ) . (6.213)
To take into account the effect of the Ω variables one first integrates over them, and
then deduces from this procedure an effective action for the A fields,
S eff (A) =S 0 (A)+δS inv (A) . (6.214)
The new effective action is made out of the original invariant contribution S 0 (A),
plus a correction from δS inv (A), which is obtained from
∫
exp{−δS inv (A)} = [dΩ] exp{−S(A Ω −1)} . (6.215)
In general there is no guarantee that the additional contribution δS inv (A) will be
local, i.e. of the form
∫
δS inv (A) = dx δL inv [A(x)] . (6.216)
But the δS inv (A) term will indeed be local provided certain conditions are met, i.e.
that the correlations in the Ω variables will be sufficiently short ranged. Then at
distances large compared to the correlation length ξ Ω of the Ω variables it is possible
to expand δL inv [A(x)] in terms of locally gauge invariant terms of the type
tr(F μν ) 2 ,tr(∇ σ F μν ) 2 , etc. At sufficiently large distances one expects only terms
with the lowest dimensions to be important, the leading one being tr(F μν ) 2 , whose
effect will be just to renormalize the bare gauge coupling.
Clearly the argument for the decoupling of the Ω variables only works if there
are long range correlations in the unperturbed gauge variables A μ (so that the
two correlation lengths can be compared to each other, and thus the notion of
“short range” makes sense) which implies g ≫ 1 in an asymptotically free gauge
theory, or in general that one is close to an ultraviolet fixed point of the gauge
theory.
In summary, the expectation is that if the non-invariant correction is small there
are no significant changes compared to the invariant theory, which is a rather remarkable
result. On the other hand if the non-invariant correction is large then a
phase transition appears and one moves into a qualitatively new phase. The argument
is nice because of its simplicity, but still leaves one major issue open: namely