#### Main text

In this section I will follow some derivations made in the text.

##### Integration by parts of the lagrangian

Separately analyzing the integral of the term:

Changing dummy indices and reordering:

Integrating by parts:

Changing dummy indices:

It’s a contraction between and :

Factoring:

Expressing the bilinear terms of the lagrangian in this form we get:

Putting the whole integral together:

,

matching (1).

##### Propagator

Let be the differential operator:

.

We can define it’s inverse by the condition:

Doing a Fourier transform:

As must be Lorentz invariant, we can write it as:

.

Putting this definition in the previous expression:

This gives us two equations:

Putting all together:

**Why this doesn’t work?**

It’s easy, the derivation is incoherent since the first step:

This term doesn’t make sense.

##### Bypassing Maxwell – Propagator

Let’s start with the general form:

If we multiply both sides by and apply (9), , we get

.

So the equation has the form

.

If we now work in the rest frame of the particle for :

.

So the numerator in the propagator is

,

as we found starting from the Maxwell Equations.

##### I.5.1

Definition of :

Let’s first start writing all terms with four covector indices , , and that are symmetric under , and :

This means that our general expression for the numerator will be:

Eq. 13 for reference:

Now we need to apply two conditions to get (13):

Let’s calculate the result of multiplying by and :

Now let’s analyze the results of products that aren’t completely contracted (OK, I could have started with those 😀 ):

Analyzing the product between and :

Replacing by its definition:

Let’s try the values and , using in a reference frame where it is at rest:

Now substituting and :

Finally, using we get:

Now we know better the form of the numerator:

We cannot learn anything more multiplying , as this terms go to zero for any value of . But we can use :

Then, up to an overall proportionality constant, we have:

Evaluating the components of this equation at rest for and :

.

Finally: