### 4.1 Scalar Green’s functions in flat spacetime

#### 4.1.1 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 in flat spacetime. This field satisfies the wave equation

where is the wave operator, a prescribed source, and where the mass parameter has a dimension of inverse length. We seek a Green’s function such that a solution to Equation (233) can be expressed as
where the integration is over all of Minkowski spacetime. The relevant wave equation for the Green’s function is
where 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 Equation (235) with the property that it vanishes if is in the past of , and the advanced Green’s function, which vanishes when is in the future of .

To solve Equation (235) we use 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

where is Synge’s world function in flat spacetime, and where is a function to be determined. For the remainder of this section we set without loss of generality.

#### 4.1.2 Integration over the source

The Dirac functional on the right-hand side of Equation (235) is a highly singular quantity, and we can avoid dealing with it by integrating the equation over a small four-volume that contains . This volume is bounded by a closed hypersurface . After using Gauss’ theorem on the first term of Equation (235), we obtain , where is a surface element on . Assuming that the integral of over goes to zero in the limit , we have

It should be emphasized that the four-volume must contain the point .

To examine Equation (237) we introduce coordinates defined by

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

in the new coordinates, where . Notice that is a timelike coordinate when , and that is then a spacelike coordinate; the roles are reversed when . Straightforward computations reveal that in these coordinates , , , , , and the only nonvanishing component of the surface element is , where . To calculate the gradient of the Green’s function we express it as , with the upper (lower) sign belonging to the retarded (advanced) Green’s function. Calculation gives , with a prime indicating differentiation with respect to ; it should be noted that derivatives of the step function do not appear in this expression.

Integration of with respect to is immediate, and we find that Equation (237) reduces to

For the retarded Green’s function, the step function restricts the domain of integration to , in which increases from to . Changing the variable of integration from to transforms Equation (238) into
where . For the advanced Green’s function, the domain of integration is , in which decreases from to . Changing the variable of integration from to also produces Equation (239).

#### 4.1.3 Singular part of

We have seen that Equation (239) properly encodes the influence of the singular source term on both the retarded and advanced Green’s function. The function that enters into the expressions of Equation (236) must therefore be such that Equation (239) is satisfied. It follows immediately that must be a singular function, because for a smooth function the integral of Equation (239) would be of order , and the left-hand side of Equation (239) could never be made equal to . The singularity, however, must be integrable, and this leads us to assume that must be made out of Dirac -functions and derivatives.

We make the ansatz

where is a smooth function, and , , , …are constants. The first term represents a function supported within the past and future light cones of ; we exclude a term proportional to 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 -functions to be constants. To see this we invoke the distributional identities
from which it follows that . A term like is then distributionally equal to , while a term like is distributionally equal to , and a term like is distributionally equal to ; here is an arbitrary test function. Summing over such terms, we recover an expression of the form of Equation (241), and there is no need to make , , , …functions of .

Differentiation of Equation (240) and substitution into Equation (239) yields

where overdots (or a number within brackets) indicate repeated differentiation with respect to . The limit exists if and only if . In the limit we must then have , which implies . We conclude that must have the form of

with being a smooth function that cannot be determined from Equation (239) alone.

#### 4.1.4 Smooth part of

To determine we must go back to the differential equation of Equation (235). Because the singular structure of the Green’s function is now under control, we can safely set in the forthcoming operations. This means that the equation to solve is in fact , the homogeneous version of Equation (235). We have , , and , so that Green’s equation reduces to the ordinary differential equation

If we substitute Equation (242) into this we get

where we have used the identities of Equation (241). The left-hand side will vanish as a distribution if we set

These equations determine uniquely, even in the absence of a second boundary condition at , because the differential equation is singular at and is known to be smooth.

To solve Equation (244) we let , with . This gives rise to Bessel’s equation for the new function :

The solution that is well behaved near is , where is a constant to be determined. We have that for small values of , and it follows that . From Equation (244) we see that . So we have found that the only acceptable solution to Equation (244) is

To summarize, the retarded and advanced solutions to Equation (235) are given by Equation (236) with given by Equation (242) and given by Equation (245).

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 Section, and in the next we shall use them to recover our previous results.

Let be a generalized step function, defined to be one if is in the future of the spacelike hypersurface , and defined to be zero otherwise. Similarly, define to be one if is in the past of the spacelike hypersurface , and zero otherwise. Then define the light-cone step functions

so that is one if is an element of , the chronological future of , and zero otherwise, and is one if is an element of , the chronological past of , and zero otherwise; the choice of hypersurface is immaterial so long as is spacelike and contains the reference point . Notice that . Define also the light-cone Dirac functionals
so that , when viewed as a function of , is supported on the future light cone of , while is supported on its past light cone. Notice that . In Equations (246) and (247), is the world function for flat spacetime; it is negative if and are timelike related, and positive if they are spacelike related.

The distributions and are not defined at and they cannot be differentiated there. This pathology can be avoided if we shift by a small positive quantity . We can therefore use the distributions and in some sensitive computations, and then take the limit . Notice that the equation describes a two-branch hyperboloid that is located just within the light cone of the reference point . The hyperboloid does not include , and is one everywhere on its future branch, while is one everywhere on its past branch. These factors, therefore, become invisible to differential operators. For example, . This manipulation shows that after the shift from to , the distributions of Equations (246) and (247) can be straightforwardly differentiated with respect to .

In the next paragraphs we shall establish the distributional identities

in four-dimensional flat spacetime. These will be used in the next Section 4.1.6 to recover the Green’s functions for the scalar wave equation, and they will be generalized to curved spacetime in Section 4.2.

The derivation of Equations (248, 249, 250) relies on a “master” distributional identity, formulated in three-dimensional flat space:

with . This follows from yet another identity, , in which we write the left-hand side as ; since is nonsingular at , it can be straightforwardly differentiated, and the result is , from which Equation (251) follows.

To prove Equation (248) we must show that vanishes as a distribution in the limit . For this we must prove that a functional of the form

where is a smooth test function, vanishes for all such functions . Our first task will be to find a more convenient expression for . Once more we set (without loss of generality) and we note that , where we have used Equation (251). It follows that

and from this we find

which establishes Equation (248).

The validity of Equation (249) is established by a similar computation. Here we must show that a functional of the form

vanishes for all test functions . We have

and the identity of Equation (249) is proved. In these manipulations we have let an overdot indicate partial differentiation with respect to , and we have used .

To establish Equation (250) we consider the functional

and show that it evaluates to . We have

as required. This proves that Equation (250) holds as a distributional identity in four-dimensional flat spacetime.

#### 4.1.6 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 as . For these we make the ansatz

and we shall prove that satisfies Equation (235) in the limit. We recall that the distributions and were defined in the preceding Section 4.1.5, and we assume that is a smooth function of ; because this function is smooth, it is not necessary to evaluate at in Equation (253). We recall also that and are nonzero when is in the future of , while and are nonzero when is in the past of . We will therefore prove that the retarded and advanced Green’s functions are of the form
and
where is a spacelike hypersurface that contains . We will also determine the form of the function .

The functions that appear in Equation (253) can be straightforwardly differentiated. The manipulations are similar to what was done in Section 4.1.4, and dropping all labels, we obtain , with a prime indicating differentiation with respect to . From Equation (253) we obtain and . The identities of Equation (241) can be expressed as and , and combining this with our previous results gives

According to Equation (248, 249, 250), the last two terms on the right-hand side disappear in the limit , and the third term becomes . Provided that the first two terms vanish also, we recover in the limit, as required. Thus, the limit of as will indeed satisfy Green’s equation provided that is a solution to
these are the same statements as in Equation (244). The solution to these equations was produced in Equation (245),
and this completely determines the Green’s functions of Equations (254) and (255).