Feynman Path Integral Formulation

10 1 Continuum Formulation
1
2 m2 [ h μν h μν − (h μ μ) 2 ] , (1.58)
is added to the gravitational Lagrangian. As in electromagnetism, such a term violates
the local gauge invariance of Eq. (1.11), and in the general theory it spoils
general coordinate invariance and the principle of equivalence. Since a massive
spin-two particle has five polarization states and a massless one two, one has a
clear mismatch in the number of physical degrees of freedom, even as m → 0, as
discussed originally in (van Dam and Veltman, 1970; Zakharov, 1970). In particular
these authors point out the fact that the discontinuity which appears in the classical
theory when the graviton mass goes to zero implies that the bending of light by the
sun for massive gravitons is only 3/4 of the experimentally confirmed general relativistic
effect, thereby ruling out the possibility of a massive graviton, no matter how
small its mass is. Furthermore the discontinuity does not seem to go away when a
cosmological constant term is included (Dilkes et al, 2001). The latter acts in some
way as a mass-like term, but does not increase the number of polarization states
of the graviton since it does not break general covariance, so that a persistence of
the discontinuity would be expected, for fixed cosmological constant. The problem
is not entirely new, as it arises in gauge theories as well, where one finds that the
only way to give the gauge particle a mass without spoiling local gauge invariance
is via the Higgs mechanism. This result would suggest that a smooth limit might be
obtained if a graviton mass is generated spontaneously by some sort of dynamical
mechanism.
In the general theory, the energy-momentum tensor for matter T μν is most suitably
defined in terms of the variation of the matter action I matter ,
∫
δI matter = 1 2
dx √ g δg μν T μν , (1.59)
and is conserved if the matter action is a scalar,
∇ μ T μν = 0 . (1.60)
Variation of the gravitational Einstein-Hilbert action of Eq. (1.35), with the matter
part added, then leads to the field equations
R μν − 1 2 g μνR + λg μν = 8πGT μν . (1.61)
Here we have also added a cosmological constant term, with a scaled cosmological
constant λ = λ 0 /16πG, which follows from adding to the gravitational action a term
λ 0
∫ √ g.
1
1 The present experimental value for Newton’s constant is ¯hG/c 3 =(1.61624(12) × 10 −33 cm) 2 .
Recent observational evidence [reviewed in (Damour, 2006)] suggests a non-vanishing positive
cosmological constant λ, corresponding to a vacuum density ρ vac ≈ (2.3 × 10 −3 eV ) 4 with ρ vac related
to λ by λ = 8πGρ vac /c 4 . As can be seen from the field equations, λ has the same dimensions
as a curvature. One has from observation λ ∼ 1/(10 28 cm) 2 , so this new curvature length scale is
comparable to the size of the visible universe ∼ 4.4 × 10 28 cm.