Feynman Path Integral Formulation

6 1 Continuum FormulationIn general one expects a wave ψ to transform as ψ → e ihθ ψ under a rotation by anangle θ about the direction of propagation if it has helicity h.Then the results of Eq. (1.32) implies that the two physical modes t ± (made outof polarizations e 11 and e 12 ) have helicity ±2. On the other hand the two unphysicalmodes v ± (made out of polarizations e 13 and e 23 , and which can be made to vanishby a suitable choice of gauge function ε μ ) have helicity ±2. Finally, the remainingtwo unphysical modes e 33 and e 00 have helicity zero. The fact that only two helicities±2 are physical implies that one is dealing with a particle of zero mass and spin two.One would expect the gravitational field h μν to carry energy and momentum,which would be described by a tensor τ μν (h). As in the case of electromagnetism,where one hasT (em)αβ= F αγ F γβ − 1 4 η αβF γδ F γδ , (1.33)one would also expect such a tensor to be quadratic in the gravitational field h μν .Asuitable candidate for the energy-momentum tensor of the gravitational field isτ μν = 1 ()− 18πG4 h μν ∂ λ ∂ λ h σ σ + ..., (1.34)where the dots indicate 37 possible additional terms, involving schematically, eitherterms of the type h∂ 2 h, or of the type (∂h) 2 . Such a τ μν term would have to beadded on the r.h.s. of the field equations in Eq. (1.10), and would therefore act asan additional source for the gravitational field (see Fig. 1.1). But the resulting fieldequations would then no longer invariant under Eq. (1.11), and one would have tochange therefore the gauge transformation law by suitable terms of order h 2 ,soasto ensure that the new field equations would still satisfy a local gauge invariance.In other words, all these complications arise because the gravitational field carriesenergy and momentum, and therefore gravitates.Ultimately, a complete and satisfactory answer to these recursive attempts at constructinga consistent, locally gauge invariant, theory of the h μν field is found in Einstein’snon-linear General Relativity theory, as shown in (Feynman, 1962; Boulwareand Deser, 1975). The full theory is derived from the Einstein-Hilbert actionI E = 1 ∫dx √ g(x)R(x) , (1.35)16πGwhich generalized Eq. (1.7) beyond the weak field limit. Here √ g is the square rootof the determinant of the metric field g μν (x), with g = −detg μν , and R the scalarcurvature. The latter is related to the Ricci tensor R μν and the Riemann tensor R μνλσbywhere g μν is the matrix inverse of g μν ,R μν = g λσ R λμσνR = g μν g λσ R μλνσ , (1.36)g μλ g λν = δ μ ν . (1.37)