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

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SUMMARY

The discussion clarifies that the fourth power of Dirac's Delta function refers to a four-dimensional delta function, specifically represented as δ4(x - x') = δ(x - x')δ(y - y')δ(z - z')δ(t - t'). The integral mentioned, ∫hμσ dzμ/dτ dzσ/dτ dτ, is evaluated over one dimension, resulting in three remaining delta functions. This indicates that the energy-momentum tensor Tμν(x) is concentrated along the particle's worldline, as referenced in MTW, page 180.

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hello,
Please attached snapshot. Does the integral 7.9 equal to 0.5 \inthμσ 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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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.
 

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