12 Scalar Green’s functions in flat spacetime

12.1 Green’s equation for a massive scalar field

To prepare the way for our discussion of Green’s functions in curved spacetime, we consider first the slightly nontrivial case of a massive scalar field Φ (x ) in flat spacetime. This field satisfies the wave equation

(□ − k2)Φ(x) = − 4π μ(x), (12.1 )
where αβ □ = η ∂α∂β is the wave operator, μ (x) a prescribed source, and where the parameter k has a dimension of inverse length. We seek a Green’s function G (x,x′) such that a solution to Eq. (12.1View Equation) can be expressed as
∫ Φ (x) = G (x,x ′)μ(x ′)d4x′, (12.2 )
where the integration is over all of Minkowski spacetime. The relevant wave equation for the Green’s function is
2 ′ ′ (□ − k )G(x,x ) = − 4πδ4(x − x ), (12.3 )
where δ4(x − x′) = δ(t − t′)δ(x − x′)δ(y − y′)δ(z − z′) is a four-dimensional Dirac distribution in flat spacetime. Two types of Green’s functions will be of particular interest: the retarded Green’s function, a solution to Eq. (12.3View Equation) with the property that it vanishes when x is in the past of x′, and the advanced Green’s function, which vanishes when x is in the future of x ′.

To solve Eq. (12.3View Equation) we appeal to Lorentz invariance and the fact that the spacetime is homogeneous to argue that the retarded and advanced Green’s functions must be given by expressions of the form

Gret(x, x′) = 𝜃 (t − t′)g(σ ), Gadv(x, x′) = 𝜃(t′ − t)g(σ), (12.4 )
where σ = 1η (x − x ′)α(x − x ′)β 2 αβ is Synge’s world function in flat spacetime, and where g (σ ) is a function to be determined. For the remainder of this section we set ′ x = 0 without loss of generality.

12.2 Integration over the source

The Dirac functional on the right-hand side of Eq. (12.3View Equation) is a highly singular quantity, and we can avoid dealing with it by integrating the equation over a small four-volume V that contains x′ ≡ 0. This volume is bounded by a closed hypersurface ∂V. After using Gauss’ theorem on the first term of Eq. (12.3View Equation), we obtain ∮ G;αd Σα − k2 ∫ GdV = − 4π ∂V V, where dΣα is a surface element on ∂V. Assuming that the integral of G over V goes to zero in the limit V → 0, we have

∮ ;α Vlim→0 G dΣ α = − 4π. (12.5 ) ∂V
It should be emphasized that the four-volume V must contain the point x′.

To examine Eq. (12.5View Equation) we introduce coordinates (w,χ, 𝜃,ϕ) defined by

t = w cos χ, x = w sinχ sin 𝜃 cosϕ, y = w sinχ sin 𝜃 sin ϕ, z = w sin χ cos𝜃,

and we let ∂V be a surface of constant w. The metric of flat spacetime is given by

ds2 = − cos2χdw2 + 2w sin2χdwd χ + w2 cos2χd χ2 + w2 sin2 χd Ω2

in the new coordinates, where dΩ2 = d𝜃2 + sin2 𝜃dϕ2. Notice that w is a timelike coordinate when cos 2χ > 0, and that χ is then a spacelike coordinate; the roles are reversed when cos2 χ < 0. Straightforward computations reveal that in these coordinates, 1 2 σ = − 2w cos2χ, √ --- 3 2 − g = w sin χ sin 𝜃, ww g = − cos2 χ, wχ −1 g = w sin2 χ, χχ − 2 g = w cos2 χ, and the only nonvanishing component of the surface element is dΣw = w3 sin2χd χdΩ, where dΩ = sin 𝜃d𝜃dϕ. To calculate the gradient of the Green’s function we express it as G = 𝜃(±t )g(σ) = 𝜃(±w cosχ )g (− 1w2cos 2χ) 2, with the upper (lower) sign belonging to the retarded (advanced) Green’s function. Calculation gives ;α 4 2 ′ G dΣ α = 𝜃(± cos χ)w sin χg (σ)dχd Ω, with a prime indicating differentiation with respect to σ; it should be noted that derivatives of the step function do not appear in this expression.

Integration of G;αdΣ α with respect to d Ω is immediate, and we find that Eq. (12.5View Equation) reduces to

∫ π lim 𝜃(± cosχ )w4 sin2 χg′(σ)dχ = − 1. (12.6 ) w→0 0
For the retarded Green’s function, the step function restricts the domain of integration to 0 < χ < π∕2, in which σ increases from − 1w2 2 to 1w2 2. Changing the variable of integration from χ to σ transforms Eq. (12.6View Equation) into
∫ 𝜖 ∘ ------ lim 𝜖 w (σ∕𝜖)g′(σ )dσ = − 1, w(ξ) := 1-+-ξ, (12.7 ) 𝜖→0 −𝜖 1 − ξ
where 1 𝜖 := 2w2. For the advanced Green’s function, the domain of integration is π ∕2 < χ < π, in which σ decreases from 12w2 to − 12w2. Changing the variable of integration from χ to σ also produces Eq. (12.7View Equation).

12.3 Singular part of g(σ )

We have seen that Eq. (12.7View Equation) properly encodes the influence of the singular source term on both the retarded and advanced Green’s function. The function g(σ) that enters into the expressions of Eq. (12.4View Equation) must therefore be such that Eq. (12.7View Equation) is satisfied. It follows immediately that g(σ) must be a singular function, because for a smooth function the integral of Eq. (12.7View Equation) would be of order 𝜖 and the left-hand side of Eq. (12.7View Equation) could never be made equal to − 1. The singularity, however, must be integrable, and this leads us to assume that g′(σ ) must be made out of Dirac δ-functions and derivatives.

We make the ansatz

g(σ) = V (σ)𝜃(− σ) + Aδ(σ ) + B δ′(σ) + Cδ′′(σ ) + ⋅⋅⋅, (12.8 )
where V(σ ) is a smooth function, and A, B, C, …are constants. The first term represents a function supported within the past and future light cones of ′ x ≡ 0; we exclude a term proportional to 𝜃(σ ) for reasons of causality. The other terms are supported on the past and future light cones. It is sufficient to take the coefficients in front of the δ-functions to be constants. To see this we invoke the distributional identities
σ δ(σ) = 0 → σδ ′(σ) + δ(σ) = 0 → σ δ′′(σ) + 2δ′(σ ) = 0 → ⋅⋅⋅ (12.9 )
from which it follows that 2 ′ 3 ′′ σ δ(σ) = σ δ (σ ) = ⋅⋅⋅ = 0. A term like f(σ)δ(σ ) is then distributionally equal to f(0)δ(σ), while a term like ′ f(σ)δ (σ) is distributionally equal to ′ ′ f (0)δ(σ) − f (0)δ(σ), and a term like f(σ)δ′′(σ) is distributionally equal to f(0)δ′′(σ) − 2f′(0)δ′(σ) + 2f ′′(0 )δ(σ ); here f(σ ) is an arbitrary test function. Summing over such terms, we recover an expression of the form of Eq. (12.9View Equation), and there is no need to make A, B, C, …functions of σ.

Differentiation of Eq. (12.8View Equation) and substitution into Eq. (12.7View Equation) yields

∫ 𝜖 [∫ 𝜖 A B C ] 𝜖 w (σ∕𝜖)g′(σ )dσ = 𝜖 V ′(σ )w (σ∕𝜖)dσ − V(0)w (0) − --˙w(0) + -2w¨(0) − -3w(3)(0 ) + ⋅⋅⋅ , −𝜖 −𝜖 𝜖 𝜖 𝜖

where overdots (or a number within brackets) indicate repeated differentiation with respect to ξ := σ∕𝜖. The limit 𝜖 → 0 exists if and only if B = C = ⋅⋅⋅ = 0. In the limit we must then have A w˙(0) = 1, which implies A = 1. We conclude that g(σ ) must have the form of

g(σ) = δ(σ ) + V (σ)𝜃(− σ), (12.10 )
with V (σ) a smooth function that cannot be determined from Eq. (12.7View Equation) alone.

12.4 Smooth part of g (σ)

To determine V (σ) we must go back to the differential equation of Eq. (12.3View Equation). Because the singular structure of the Green’s function is now under control, we can safely set x ⁄= x′ ≡ 0 in the forthcoming operations. This means that the equation to solve is in fact 2 (□ − k )g(σ) = 0, the homogeneous version of Eq. (12.3View Equation). We have ′ ∇ αg = g σα, ′′ ′ ∇ α∇ βg = g σασ β + gσ αβ, ′′ ′ □g = 2σg + 4g, so that Green’s equation reduces to the ordinary differential equation

2σg′′ + 4g′ − k2g = 0. (12.11 )
If we substitute Eq. (12.10View Equation) into this we get
2 ′′ ′ 2 − (2V + k )δ(σ ) + (2σV + 4V − k V )𝜃(− σ) = 0,

where we have used the identities of Eq. (12.9View Equation). The left-hand side will vanish as a distribution if we set

1 2σV ′′ + 4V ′ − k2V = 0, V (0) = − -k2. (12.12 ) 2
These equations determine V (σ) uniquely, even in the absence of a second boundary condition at σ = 0, because the differential equation is singular at σ = 0 while V is known to be smooth.

To solve Eq. (12.12View Equation) we let V = F(z)∕z, with √ ----- z := k − 2σ. This gives rise to Bessel’s equation for the new function F:

z2Fzz + zFz + (z2 − 1)F = 0.

The solution that is well behaved near z = 0 is F = aJ1 (z ), where a is a constant to be determined. We have that J1(z) ∼ 1z 2 for small values of z, and it follows that V ∼ a ∕2. From Eq. (12.12View Equation) we see that a = − k2. So we have found that the only acceptable solution to Eq. (12.12View Equation) is

√ ----- V (σ ) = − √-k---J1(k − 2σ ). (12.13 ) − 2σ

To summarize, the retarded and advanced solutions to Eq. (12.3View Equation) are given by Eq. (12.4View Equation) with g(σ) given by Eq. (12.10View Equation) and V(σ ) given by Eq. (12.13View Equation).

12.5 Advanced distributional methods

The techniques developed previously to find Green’s functions for the scalar wave equation are limited to flat spacetime, and they would not be very useful for curved spacetimes. To pursue this generalization we must introduce more powerful distributional methods. We do so in this subsection, and in the next we shall use them to recover our previous results.

Let 𝜃+(x,Σ ) be a generalized step function, defined to be one when x is in the future of the spacelike hypersurface Σ and zero otherwise. Similarly, define 𝜃− (x, Σ) := 1 − 𝜃+(x,Σ ) to be one when x is in the past of the spacelike hypersurface Σ and zero otherwise. Then define the light-cone step functions

𝜃±(− σ) = 𝜃±(x,Σ )𝜃(− σ), x ′ ∈ Σ, (12.14 )
so that 𝜃+(− σ) is one if x is within + ′ I (x ), the chronological future of ′ x, and zero otherwise, and 𝜃− (− σ ) is one if x is within − ′ I (x ), the chronological past of ′ x, and zero otherwise; the choice of hypersurface is immaterial so long as Σ is spacelike and contains the reference point x ′. Notice that 𝜃+(− σ) + 𝜃− (− σ) = 𝜃(− σ). Define also the light-cone Dirac functionals
δ (σ) = 𝜃 (x,Σ )δ(σ), x ′ ∈ Σ, (12.15 ) ± ±
so that δ+(σ), when viewed as a function of x, is supported on the future light cone of x′, while δ− (σ ) is supported on its past light cone. Notice that δ+ (σ ) + δ− (σ) = δ(σ). In Eqs. (12.14View Equation) and (12.15View Equation), σ is the world function for flat spacetime; it is negative when x and x ′ are timelike related, and positive when they are spacelike related.

The distributions 𝜃± (− σ ) and δ±(σ) are not defined at x = x ′ and they cannot be differentiated there. This pathology can be avoided if we shift σ by a small positive quantity 𝜖. We can therefore use the distributions 𝜃 (− σ − 𝜖) ± and δ (σ + 𝜖) ± in some sensitive computations, and then take the limit + 𝜖 → 0. Notice that the equation σ + 𝜖 = 0 describes a two-branch hyperboloid that is located just within the light cone of the reference point ′ x. The hyperboloid does not include ′ x, and 𝜃+ (x,Σ) is one everywhere on its future branch, while 𝜃− (x,Σ ) is one everywhere on its past branch. These factors, therefore, become invisible to differential operators. For example, 𝜃′ (− σ − 𝜖) = 𝜃 (x, Σ)𝜃′(− σ − 𝜖) = − 𝜃 (x,Σ)δ(σ + 𝜖) = − δ (σ + 𝜖) + + + +. This manipulation shows that after the shift from σ to σ + 𝜖, the distributions of Eqs. (12.14View Equation) and (12.15View Equation) can be straightforwardly differentiated with respect to σ.

In the next paragraphs we shall establish the distributional identities

l𝜖→i0m+ ðœ–δ± (σ + 𝜖) = 0, (12.16 ) ′ l𝜖→i0m+ ðœ–δ± (σ + 𝜖) = 0, (12.17 ) lim 𝜖δ′′(σ + 𝜖) = 2 πδ (x − x′) (12.18 ) 𝜖→0+ ± 4
in four-dimensional flat spacetime. These will be used in the next subsection to recover the Green’s functions for the scalar wave equation, and they will be generalized to curved spacetime in Section 13.

The derivation of Eqs. (12.16View Equation) – (12.18View Equation) relies on a “master” distributional identity, formulated in three-dimensional flat space:

√ ------- lim -𝜖- = 2π-δ3(x), R := r2 + 2𝜖, (12.19 ) 𝜖→0+ R5 3
with ∘ ------------ r := |x| := x2 + y2 + z2. This follows from yet another identity, ∇2r −1 = − 4π δ3(x ), in which we write the left-hand side as lim + ∇2R −1 𝜖→0; since R−1 is nonsingular at x = 0 it can be straightforwardly differentiated, and the result is 2 −1 5 ∇ R = − 6 𝜖∕R, from which Eq. (12.19View Equation) follows.

To prove Eq. (12.16View Equation) we must show that 𝜖δ±(σ + 𝜖) vanishes as a distribution in the limit 𝜖 → 0+. For this we must prove that a functional of the form

∫ 4 A± [f ] = 𝜖li→m0+ ðœ–δ± (σ + 𝜖)f (x)d x,

where f (x) = f(t,x) is a smooth test function, vanishes for all such functions f. Our first task will be to find a more convenient expression for δ±(σ + 𝜖). Once more we set ′ x = 0 (without loss of generality) and we note that 2 2 2(σ + 𝜖) = − t + r + 2 𝜖 = − (t − R)(t + R), where we have used Eq. (12.19View Equation). It follows that

δ(t ∓ R ) δ± (σ + 𝜖) = --------, (12.20 ) R
and from this we find
∫ ∫ ∫ f (±R, x ) 3 𝜖 4 3 2π 4 3 A ±[f] = 𝜖li→m0+ 𝜖----R-----d x = 𝜖li→m0+ R5R f(±R, x)d x = 3-- δ3(x )r f(±r, x)d x = 0,

which establishes Eq. (12.16View Equation).

The validity of Eq. (12.17View Equation) is established by a similar computation. Here we must show that a functional of the form

∫ ′ 4 B ±[f ] = lim + 𝜖δ±(σ + 𝜖)f(x)d x 𝜖→0

vanishes for all test functions f. We have

d ∫ d ∫ f (±R, x ) ∫ ( ˙f f ) B± [f ] = lim 𝜖--- δ± (σ + 𝜖)f (x)d4x = lim 𝜖--- ----------d3x = lim 𝜖 ± --2 − --3 d3x 𝜖→0+ ∫d𝜖 𝜖→0+∫ d𝜖 R 𝜖→0+ R R -𝜖-( 3 ˙ 2 ) 3 2π- ( 3 ˙ 2 ) 3 = 𝜖li→m0+ R5 ±R f − R f d x = 3 δ3(x ) ±r f − r f d x = 0,
and the identity of Eq. (12.17View Equation) is proved. In these manipulations we have let an overdot indicate partial differentiation with respect to t, and we have used ∂R ∕∂𝜖 = 1∕R.

To establish Eq. (12.18View Equation) we consider the functional

∫ ′′ 4 C± [f ] = 𝜖l→im0+ 𝜖δ±(σ + 𝜖)f(x)d x

and show that it evaluates to 2πf (0,0). We have

∫ ∫ -d2 4 -d2 f(±R,-x-)-3 C ±[f] = l𝜖→im0+ 𝜖d 𝜖2 δ±(σ + 𝜖)f(x)d x = 𝜖l→im0+ 𝜖d𝜖2 R d x ∫ ( ¨ ˙ ) ∫ ( ) = lim 𝜖 f--∓ 3 f--+ 3 f-- d3x = 2π δ (x) 1-r2 ¨f ± r ˙f + f d3x 𝜖→0+ R3 R4 R5 3 3 = 2πf(0,0 ),
as required. This proves that Eq. (12.18View Equation) holds as a distributional identity in four-dimensional flat spacetime.

12.6 Alternative computation of the Green’s functions

The retarded and advanced Green’s functions for the scalar wave equation are now defined as the limit of the functions 𝜖 ′ G ±(x,x ) when + 𝜖 → 0. For these we make the ansatz

𝜖 ′ G± (x,x ) = δ±(σ + 𝜖) + V (σ )𝜃± (− σ − 𝜖), (12.21 )
and we shall prove that 𝜖 ′ G ±(x,x ) satisfies Eq. (12.3View Equation) in the limit. We recall that the distributions 𝜃± and δ± were defined in the preceding subsection, and we assume that V (σ ) is a smooth function of σ (x, x′) = 12ηαβ(x − x′)α(x − x′)β; because this function is smooth, it is not necessary to evaluate V at σ + 𝜖 in Eq. (12.21View Equation). We recall also that 𝜃+ and δ+ are nonzero when x is in the future of x ′, while 𝜃− and δ− are nonzero when x is in the past of x′. We will therefore prove that the retarded and advanced Green’s functions are of the form
′ 𝜖 ′ [ ] Gret(x, x) = 𝜖li→m0+ G +(x,x ) = 𝜃+(x,Σ ) δ(σ) + V(σ )𝜃(− σ ) (12.22 )
and
[ ] Gadv(x,x ′) = lim G𝜖− (x,x′) = 𝜃− (x,Σ ) δ(σ) + V (σ )𝜃(− σ) , (12.23 ) 𝜖→0+
where Σ is a spacelike hypersurface that contains x ′. We will also determine the form of the function V (σ).

The functions that appear in Eq. (12.21View Equation) can be straightforwardly differentiated. The manipulations are similar to what was done in Section 12.4, and dropping all labels, we obtain (□ − k2)G = 2 σG ′′ + 4G ′ − k2G, with a prime indicating differentiation with respect to σ. From Eq. (12.21View Equation) we obtain G′ = δ′ − V δ + V ′𝜃 and G ′′ = δ′′ − V δ′ − 2V ′δ + V ′′𝜃. The identities of Eq. (12.9View Equation) can be expressed as ′ (σ + 𝜖)δ(σ + 𝜖) = − δ(σ + 𝜖) and ′′ ′ (σ + 𝜖)δ (σ + 𝜖) = − 2 δ(σ + 𝜖), and combining this with our previous results gives

(□ − k2)G 𝜖(x,x′) = (− 2V − k2)δ (σ + 𝜖) + (2σV ′′ + 4V ′ − k2V )𝜃 (− σ − 𝜖) ± ± ± − 2𝜖δ′′±(σ + 𝜖) + 2V 𝜖δ′± (σ + 𝜖) + 4V ′ðœ–δ± (σ + 𝜖).
According to Eq. (12.16View Equation) – (12.18View Equation), the last two terms on the right-hand side disappear in the limit + 𝜖 → 0, and the third term becomes ′ − 4πδ4(x − x ). Provided that the first two terms vanish also, we recover (□ − k2)G (x, x′) = − 4π δ4(x − x ′) in the limit, as required. Thus, the limit of G 𝜖±(x, x′) when 𝜖 → 0+ will indeed satisfy Green’s equation provided that V (σ) is a solution to
′′ ′ 2 1-2 2σV + 4V − k V = 0, V (0) = − 2k ; (12.24 )
these are the same statements as in Eq. (12.12View Equation). The solution to these equations was produced in Eq. (12.13View Equation):
k ( √ ----) V(σ ) = − √-----J1 k − 2σ , (12.25 ) − 2σ
and this completely determines the Green’s functions of Eqs. (12.22View Equation) and (12.23View Equation).
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