Exploring Delta Function in Quantum Mechanics and Quantum Field Theory

In summary, the delta function is a useful mathematical tool that samples the value of a function at a specific point. In the context of quantum mechanics and quantum field theory, it is often used in integrals to simplify equations and make them more manageable. Its usage can be better understood by using the formula \delta[g(x)]=\sum_i\frac{\delta(x-x_i)}{|g'(x_i)|} and referring to resources such as the mathworld website.
  • #1
Ratzinger
291
0
As I read in my quantum mechanics book the delta function is sometimes called the sampling function because it samples the value of the function at one point.
[tex]\int {\delta (x - x')} f(x')dx' = f(x)[/tex]

But then I opened a quantum field book and I found equations like that:
[tex]\phi (x) = \frac{1}{{(2\pi )^{3/2} }}\int {d^4 p\delta (p^2 } - m^2 )A(p)e^{ - ip \cdot x} [/tex]

[tex](\partial _\nu \partial ^\nu + m^2 )\phi (x) = \frac{1}{{(2\pi )^{3/2} }}\int {d^4 p( - p^2 } + m^2 )\delta (p^2 - m^2 )A(p)e^{ - ip \cdot x} [/tex]

Can someone explain me what the delta function does here? What it is sampling? How these equations work?

thank you
 
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  • #2
The delta function does the same thing!

I don't know if "sampling" is a good adjective, but I'll run with it. In the integral

[tex]\int d^4 p \, \delta (p^2 - m^2 )A(p)e^{ - ip \cdot x} [/tex]

you're "sampling" the function [itex]A(p)e^{ - ip \cdot x}[/itex] over the region where [itex]p^2 - m^2 = 0[/itex].

We can reduce this down to the known case, if we like, as follows: we can split this up into an iterated integral, with [itex]p_0[/itex] being the innermost. Then we have:

[tex]
\iiint \int \delta(p_0^2 - \vec{p}\,{}^2 - m^2) A(p) e^{-ip \cdot x} \, dp_0 \, d^3 \vec{p}
[/tex]

so the innermost integral is of the form:

[tex]
\int \delta(x^2 - a^2) f(x) \, dx
[/tex]

Now, if I make the substitution [itex]u = x^2[/itex], we have [itex]du = 2 x dx[/itex] so that [itex]dx = du / 2 \sqrt{u}[/itex], and the integral becomes

(yes, I know I'm being lazy with the bounds -- note that when tracing x over [itex](-\infty, +\infty)[/itex], u traces over [itex][0, +\infty)[/itex] twice! I really should break the integral up into two parts)

[tex]
\int \delta(u - a^2) f(\sqrt{u}) \frac{1}{2 \sqrt{u}} \, du
[/tex]

which becomes (one term for each of the times u traces over [itex]0, +\infty)[/itex]:

[tex]
f(a) \frac{1}{2a} - f(-a) \frac{1}{-2a}
= \frac{f(a) + f(-a)}{2a}
[/tex]

I'll leave it to you to work out the general case when the argument to [itex]\delta[/itex] is an arbitrary function of the dummy variable.

Anyways, if my intuition about these things is anywhere close to accurate, your integral for [itex]\phi(x)[/itex] should reduce to a surface integral over the hypersurface given by the equation [itex]p^2 - m^2 = 0[/itex].
 
Last edited:
  • #3
Ratzinger said:
As I read in my quantum mechanics book the delta function is sometimes called the sampling function because it samples the value of the function at one point.
[tex]\int {\delta (x - x')} f(x')dx' = f(x)[/tex]
But then I opened a quantum field book and I found equations like that:
[tex]\phi (x) = \frac{1}{{(2\pi )^{3/2} }}\int {d^4 p\delta (p^2 } - m^2 )A(p)e^{ - ip \cdot x} [/tex]
[tex](\partial _\nu \partial ^\nu + m^2 )\phi (x) = \frac{1}{{(2\pi )^{3/2} }}\int {d^4 p( - p^2 } + m^2 )\delta (p^2 - m^2 )A(p)e^{ - ip \cdot x} [/tex]
Can someone explain me what the delta function does here? What it is sampling? How these equations work?
thank you
notice here that the delta function has as its argument quadratic functions. It can be a little confusing about what to do with these. Well just use this formula:
[tex]\delta[g(x)]=\sum_i\frac{\delta(x-x_i)}{|g'(x_i)|}[/tex]
by the way, the sum is over the roots of g.
In your equation above, after you use this step everything should be clear- it functions just how you would expect it to.
and check out this page for help with dirac delta functions
http://mathworld.wolfram.com/DeltaFunction.html
 
  • #4
thanks Hurykl, thanks Norman, thanks physicsforums, thanks the Internet

and a great weekend to everybody
 

1. What is a delta function in quantum mechanics and quantum field theory?

A delta function, also known as the Dirac delta function, is a mathematical function used in quantum mechanics and quantum field theory to represent a point-like source or a localized distribution of energy. It is often visualized as an infinitely narrow and tall spike centered at a specific point in space or time.

2. How is the delta function used in quantum mechanics?

In quantum mechanics, the delta function is used to represent the position or momentum of a particle in a state of definite energy. It is also used to describe the probability of finding a particle at a specific location or with a specific momentum.

3. How is the delta function used in quantum field theory?

In quantum field theory, the delta function is used to represent the creation or annihilation of particles at a specific point in space or time. It is also used to calculate scattering amplitudes and to describe the interactions between particles.

4. What are some properties of the delta function?

The delta function has several important properties that make it a useful tool in quantum mechanics and quantum field theory. These include the property of being infinitely narrow and tall, the property of being symmetric, and the property of satisfying the sifting property, which means that it integrates to 1 over its support region.

5. How does the concept of delta function relate to other areas of physics?

The delta function is a fundamental concept in many areas of physics, including classical mechanics, electromagnetism, and signal processing. In these fields, it is used to represent point sources, impulses, and other localized phenomena. It also has applications in engineering, such as in the design of filters and control systems.

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