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C Quantum Field Theory
Moreover, the field metric also defines an inner product between two different
elements of the tangent space or field space:
(δδ 1 , δδ 2 ) = G(()(δδ 1 , δδ 2 ),
(( 1 , , 2 ) = G(()(( 1 , , 2 ).
(C.4)
Remark C.1 (Metric in Component Form) If one has a set of spacetime fields
a (x), then a local norm is defined by
|δδ a |
2
=
dx ρ(x)γ ab
δδ a (x)δδ b (x),
(C.5)
which means that the metric in component form is
G ab (x, y)(() = δ(x − y)ρ(x)γ ab
.
(C.6)
Locality means that all fields are evaluated at the same point. On a curved space, it
is natural to write γ only in terms of the metric g and set ρ(x) =
√
det g(x), such
that the inner product is diffeomorphism invariant.
Since a Gaussian integral is proportional to the square root of the operator
determinant, the integration measure can be determined by considering the Gaussian
integral over the tangent space:
dδδ e
−G(()(δδ,δδ)
=
1
√
det G(()
.
(C.7)
Note that one needs to work on the tangent space because G(() can depend on the
field, which means that the integral
d e
−G(()((,,)
(C.8)
is not Gaussian.
Having constructed the Gaussian measure with respect to the metric G((), it is
now possible to consider the path integral of general functional F of the fields:
d
det G(() F (().
(C.9)
The (effective) action S(() provides a natural metric on the field space by defining
√
det G = e −S , or
S = −
1
2
tr ln G(().
(C.10)