Is Dirac's Delta to the fourth power equal to 0.5?

In summary, The conversation discusses the integral 7.9 and its relation to the fourth power of Dirac's Delta. It is clarified that the integral is not a 4th power, but rather a four-dimensional delta function. This means that Tμν(x) is concentrated on the particle's worldline. For further information, MTW pg180 is recommended.
  • #1
zn5252
72
0
hello,
Please attached snapshot. Does the integral 7.9 equal to 0.5 [itex]\int[/itex]hμσ dzμ/dτ dzσ/dτdτ ?
I'm confused as to the fourth power of Dirac's Delta. Then where does the derivative on x go ?
For more on this, see MTW pg180
thanks
 

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  • #2
It's not a 4th power, it means a four-dimensional delta function. I.e. in Minkowski coordinates,

δ4(x - x') = δ(x - x')δ(y - y')δ(z - z')δ(t - t')

Ofttimes you would integrate this over all four dimensions. In this case there is only one integration, leaving three δ's, so it means that Tμν(x) is concentrated on the particle's worldline.
 

FAQ: Is Dirac's Delta to the fourth power equal to 0.5?

What is the Dirac delta to the fourth?

The Dirac delta to the fourth, also known as delta function to the fourth, is a mathematical function that is used to represent the fourth derivative of the Dirac delta function. It is commonly used in physics and engineering to model impulse responses in systems.

How is the Dirac delta to the fourth defined?

The Dirac delta to the fourth is defined as the fourth derivative of the Dirac delta function. It can be represented mathematically as δ(x)'''' = δ(x)^(4), where δ(x) is the Dirac delta function and '''' represents the fourth derivative operator.

What are the properties of the Dirac delta to the fourth?

Just like the Dirac delta function, the Dirac delta to the fourth has several properties that make it useful in mathematical and scientific applications. These include the sifting property, scaling property, and derivative property. It is also an even function, meaning that δ(x)'''' = δ(-x)''''.

How is the Dirac delta to the fourth used in real-world applications?

The Dirac delta to the fourth is commonly used in physics and engineering to model impulse responses in systems. It can also be used to solve differential equations and is a key component in Fourier transforms and Laplace transforms. It has also been used in signal processing, image processing, and quantum mechanics.

What is the relationship between the Dirac delta function and the Dirac delta to the fourth?

The Dirac delta to the fourth is the fourth derivative of the Dirac delta function. This means that it is derived from the Dirac delta function and shares many of its properties. However, the Dirac delta to the fourth is a more specific function and is used for different purposes compared to the Dirac delta function.

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