# First Glance at Infinities¶

This section will show you how the infinities emerge in Quantum Field Theory through a resonated toy of physicists — $$\varphi^4$$ theory. The interaction term of $$\varphi^4$$ theory is

$\mathcal{L}_{\text{int}} = -\frac{1}{4!}g\varphi^4$

Recall that the generating functional of interaction theory is

$W[J] = \exp\{-\mathrm{i}\int\mathcal{L}_{\text{int}}\!\!\left[\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J}\right]\} W_0[J]$

Suppose the interaction is weak enough for us to neglect all terms of order higher than one. Therefore, the generating functional up to the first order is

$W_\uparrow^{(1)}[J] = (1 - \mathrm{i}\int\mathcal{L}_{\text{int}}\!\!\left[\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J}\right]\})W_0[J] = (1 - \frac{\mathrm{i}g}{4!}\int\left[\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J}\right]^4\})W_0[J]$

Notice that in this case the four functional derivatives are at the same point $$x$$. Therefore, a calculation similar to the previous section will show

$\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J(x)}\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J(x)}W_0[J] = \tau(x,x) = \mathrm i\Delta(x;x)\ \ ???$

We see that the right hand side contains the value of Green’s function at zero, which is clearly infinity. With higher order terms which contains even more derivatives, there might be even more infinities.

To deal with this infinities, we use a mathematical technique called renormalization. Looking back to where the infinities come and we find

$\frac{1}{\mathrm{i}}\!\frac{\delta}{\delta J(x)}(-\int\Delta_F(x;y)J(y)\mathrm{d}y)$

makes no sense, since the functional derivative requires that $$\Delta_F(x;y)$$ be a function, but it is essentially a distribution (or generalized function). But things are not so bad. The Green’s function is a distribution which usually has a function correspondence, and for most of the physical equations that we care, it has.

Nonetheless, the function corresponding to $$\Delta(x;y)$$ has no definition on $$x=y$$. Therefore, the above expression hopes to get something undefined. This makes things much easier. If it is undefined, the most straight-forward way is to place an additional definition.

The renormalization procedure does exactly this stuff. Usually, we will impose some conditions (such as define the physical mass as the square-root of minus zero-point of two-point convex function) so that the value $$\Delta(x;x)$$ can be defined.

You are not expected to understand the harangue in the bracket in previous paragraph. The renormalization is a big topic and we will open a new chapter later for a systematically discussion.