D'Alambertian operator on the heaviside function?

In summary, the conversation discusses the computation of a quantity involving Heaviside and Dirac delta distributions, and how to handle their multiplication with other distributions. It is mentioned that this is not possible in the classical theory of distributions, but there may be a solution through Colombeau's theory.
  • #1
Dixanadu
254
2
Hey guys,

How does one compute the following quantity:

[itex]\Box \theta(x_{0})=\partial_{0}\partial^{0}\theta(x_{0})[/itex]?

I know that [itex]\partial_{0}\theta(x_{0})=\delta(x_{0})[/itex] which is the Dirac delta, but what about the second derivative?

Thanks everyone!
 
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  • #2
Heaviside and Dirac delta are examples of distributions. They are a kind of generalized function, and only defined by their action on some "test function" in an inner-product. Integrals satisfy the properties of inner products, so choosing some smooth function [itex] \varphi (x) [/itex] that vanishes at infinity, we have,
[tex] \int \partial_x\delta(x)\varphi(x) dx = -\int \delta(x) \partial_x \varphi(x) dx = -\partial_x\varphi(0) [/tex]
using integration by parts.

So the action of the distribution [itex] \partial_x^2 \theta(x) [/itex] on a smooth function [itex] \varphi(x) [/itex] produces [itex] -\partial_x\varphi(0)[/itex]
 
  • #3
hmm but my situation involves no integrals - basically I'm trying to compute the following:

[itex]\Box [\theta(x_{0})\phi(x)\phi^{\dagger}(0)][/itex]

where [itex]\phi, \phi^{\dagger}[/itex] are solutions to the complex Klein-Gordon equation.

I've been told that [itex]\partial^{0}\theta(x_{0})=\partial_{0}\theta(x_{0})=\delta(x_{0})[/itex]. I'm not sure how this matches up to what you said above?
 
  • #4
That product is mathematically ill-defined, because you have a multiplication of 3 distributions. In the so-called classical theory of distributions, there's no way to 'multiply' them.
As a note (rumor), I heard that Colombeau's theory of distributions somehow addresses this issue.
 
  • #5


Hello,

The D'Alambertian operator, denoted as \Box, is defined as the sum of all second-order partial derivatives in a given coordinate system. In this case, since we are dealing with only one variable x_{0}, the D'Alambertian operator simplifies to \partial_{0}^{2}. Therefore, we can rewrite the equation as \partial_{0}^{2}\theta(x_{0}).

As you correctly mentioned, the first derivative \partial_{0}\theta(x_{0}) is equal to the Dirac delta function \delta(x_{0}). In order to find the second derivative, we can consider the definition of the Dirac delta function:

\delta(x_{0}) = \begin{cases} \infty, & x_{0} = 0 \\ 0, & x_{0} \neq 0 \end{cases}

Using this definition, we can see that the second derivative will be zero everywhere except at x_{0}=0, where it will be infinite. Therefore, we can write the second derivative as \partial_{0}^{2}\theta(x_{0}) = \infty \cdot \delta(x_{0}).

I hope this helps to clarify the computation of the D'Alambertian operator on the Heaviside function. Please let me know if you have any further questions.
 

What is the D'Alambertian operator on the Heaviside function?

The D'Alambertian operator, also known as the wave operator, is a mathematical operator used in physics and engineering to describe wave-like phenomena. When applied to the Heaviside function, it represents the propagation of a disturbance through space and time.

How is the D'Alambertian operator applied to the Heaviside function?

The D'Alambertian operator acts on the Heaviside function by taking the second derivative of the function with respect to both space and time variables. This results in a differential equation which describes the behavior of a wave.

What is the physical significance of the D'Alambertian operator on the Heaviside function?

The D'Alambertian operator represents the propagation of a wave through space and time. It is often used in fields such as electromagnetics, acoustics, and fluid dynamics to describe the behavior of waves.

What are the properties of the D'Alambertian operator on the Heaviside function?

One of the key properties of the D'Alambertian operator is that it is a linear operator, meaning it satisfies the superposition principle. This allows for the combination of multiple solutions to the operator to give a new solution.

How is the D'Alambertian operator on the Heaviside function related to other mathematical concepts?

The D'Alambertian operator is closely related to other mathematical concepts such as the Laplacian operator, which is the spatial part of the D'Alambertian operator. It is also related to the Cauchy-Riemann equations and the Helmholtz equation.

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