Deriving the Dirac Delta Function Equation in Field Theory

• touqra
In summary, this equation was found in a book on particle physics and it is apparently gibberish. The equation states that the amplitude of a Feynman diagram containing four delta functions is the square of the amplitude of the delta function itself.
touqra
I found this equation in a field theory book, which I can't figure how it was derived:

$$\delta(x-a) \delta(x-a) = \delta(0) \delta(x-a)$$

touqra said:
I found this equation in a field theory book, which I can't figure how it was derived:

$$\delta(x-a) \delta(x-a) = \delta(0) \delta(x-a)$$

From the definition of the delta function

$$\int_{ - \infty }^\infty {f(x)\delta (x - a)dx = f(a)}$$

you get

$$\int_{ - \infty }^\infty {\delta (x - a)\delta (x - a)dx = \delta (a - a)} = \delta (0)$$

I expect this is part of the answer.

Ugh. If that's supposed to be the dirac delta distribution, then both sides of that equation are gibberish. What is the context in which you saw it?

From what I know (very slightly!), products of distributions like the delta "function" cannot be defined in a consistent manner.

Hurkyl said:
Ugh. If that's supposed to be the dirac delta distribution, then both sides of that equation are gibberish. What is the context in which you saw it?

I was reading a section dealing with cross sections and scattering. He calculated some amplitude, A (first order) for a Feynman diagram which contains four 4-momentum delta functions. And with that amplitude, we need to get this invariant amplitude, iM which is the square of A.
Squaring A yields us 8 delta functions.
He states that 8 delta functions is bad news, and basically he gave a simple example, which was the one I posted in this forum.

It's from Gauge Theories in Particle Physics Volume I by I J R Aitchison, page 152.

Equations such as $$f(x) \delta(x-a) = f(a)$$ are supposed to be read as $$\int_{-\infty}^{\infty} f(x) \delta(x-a) dx= f(a)$$, I think dropping the integral sign is just some sort of convention, not one I'm a fan of though... I think people keep swapping limits and integral signs too, I think things like that should be made more consistent, as you can't do stuff like that in general.

But yeah don't forget the integral sign!

delta(0) can sometimes be taken to mean the volume of space (with appropriate 2pi factors) in relating delta function normalization with box normalization.

What is the Dirac Delta Function Equation?

The Dirac Delta Function Equation is a mathematical tool used in field theory to represent a point source or impulse in space. It is defined as a function that is zero everywhere except at the origin, where it is infinite, and has an area of one under the curve. It is often denoted as δ(x) or δ(x-x0) to represent the location of the impulse.

Why is the Dirac Delta Function Equation important in field theory?

The Dirac Delta Function Equation is important in field theory because it allows us to describe point sources or impulses in a continuous field. This is useful in many areas of physics, including electromagnetism, quantum mechanics, and fluid dynamics. It also simplifies many mathematical calculations and can be used to solve differential equations.

How is the Dirac Delta Function Equation derived in field theory?

The Dirac Delta Function Equation can be derived in field theory using the definition of a delta function as a limit of a sequence of functions. This involves taking a sequence of functions that converge to the delta function and then using the properties of limits to derive the final equation. This process is often referred to as the "sifting property" of the delta function.

What are the applications of the Dirac Delta Function Equation?

The Dirac Delta Function Equation has many applications in physics and engineering. It is commonly used in signal processing, image processing, and control systems. It is also used in quantum mechanics for describing position and momentum operators, and in electromagnetism for calculating electric and magnetic fields around point charges.

Are there any limitations to the Dirac Delta Function Equation?

One limitation of the Dirac Delta Function Equation is that it is not a true function and cannot be evaluated at every point. It is also not continuous, making it difficult to use in some mathematical operations. Additionally, the use of the delta function in field theory assumes that the underlying space is continuous, which may not always be the case in certain physical systems.

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