Why does the propagator converge?

[geoduck]

The propagator $$\frac{1}{k^2+m^2} $$ diverges in the ultraviolet when integrated over all dkⁿ, with n>=2. However, when you throw in an exponential,

$$\Delta(x)=\int d^nk \frac{e^{ikx}}{k^2+m^2} $$ is convergent for x≠0

Intuitively adding the oscillating exponential decreases the ultraviolet growth by alternating it with + and - that cancel when added.

But consider 1 dimension, and the expression $$ \int_1^\infty \frac{dx}{x^n} $$. This expression is convergent for n>1. Now add an exponential: $$ \int_1^\infty dx \frac{ e^{ix}}{x^n} $$. This improves the convergence: now the expression is convergent for n>0. But for n<0, it doesn't converge.

So intuitively, $$\Delta(x)=\int d^nk \frac{e^{ikx}}{k^2+m^2} $$ should not be convergent for n>=2. But it obviously is, since the propagator is convergent in dimensions greater than 2, I believe.

What exactly is happening with these oscillations at infinity? Also I'm a bit bewildered at why $$\int_{-\infty}^{\infty} \cos(x^2) dx $$ is convergent. I can show it mathematically, but intuitively why does that expression converge and not $$\int_{-\infty}^{\infty} \cos(x) dx $$?

 

[The_Duck]

Also I'm a bit bewildered at why $$\int_{-\infty}^{\infty} \cos(x^2) dx $$ is convergent. I can show it mathematically, but intuitively why does that expression converge and not $$\int_{-\infty}^{\infty} \cos(x) dx $$?

Well, consider the functions $f(M) = \int_{-M}^{M} \cos(x^2) dx$ and $g(M) = \int_{-M}^{M} \cos(x) dx$. Plot the two functions as $M \to \infty$ and it will be clear that $f$ goes to a limit, while $g$ doesn't. Intuitively, the area under each positive and negative oscillation of $\cos(x^2)$ shrinks as $x \to \infty$, so the integral $f$ oscillates less and less. But the oscillations of $\cos(x)$ don't have smaller areas as $x \to \infty$, so the integral $g$ oscillates with a constant amplitude and never settles down to a limit.

 

[USeptim]

The propagator $$\frac{1}{k^2+m^2} $$ diverges in the ultraviolet when integrated over all dkⁿ, with n>=2. However, when you throw in an exponential,

$$\Delta(x)=\int d^nk \frac{e^{ikx}}{k^2+m^2} $$ is convergent for x≠0

Intuitively adding the oscillating exponential decreases the ultraviolet growth by alternating it with + and - that cancel when added.

But consider 1 dimension, and the expression $$ \int_1^\infty \frac{dx}{x^n} $$. This expression is convergent for n>1. Now add an exponential: $$ \int_1^\infty dx \frac{ e^{ix}}{x^n} $$. This improves the convergence: now the expression is convergent for n>0. But for n<0, it doesn't converge.

So intuitively, $$\Delta(x)=\int d^nk \frac{e^{ikx}}{k^2+m^2} $$ should not be convergent for n>=2. But it obviously is, since the propagator is convergent in dimensions greater than 2, I believe.

What exactly is happening with these oscillations at infinity? Also I'm a bit bewildered at why $$\int_{-\infty}^{\infty} \cos(x^2) dx $$ is convergent. I can show it mathematically, but intuitively why does that expression converge and not $$\int_{-\infty}^{\infty} \cos(x) dx $$?

Hi Geoduck,

You are right when you say the Fourier transform to the propagator $$\frac{1}{k^2+m^2} $$ only diverges for x=0 and that's quite logical because the contribution of a position to itself must be a delta function and not only an infinitesimal contribution.

From what I have read, the divergence problem arises when introducing the self-interacting lagrangian that depends on λψ^{4}, this new term modifies the propagator causing a mass renormalization, this mass renormalization turns to be divergent for high energy contributions. The Renormalization Group tries to solve these problems. However, I must recognise this issues related with the interacting lagrangian and its fluctuations scape quite from my comprehension.