The Fourier-Minkowski transform?

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SUMMARY

The Fourier-Minkowski transform is essential for preserving the invariance of the spacetime interval in theoretical physics. The necessity of having opposite signs for the time and space components in the exponentials is crucial for ensuring that the 4-momentum operator can be accurately represented as i∂μ during the Fourier transformation of wave functions. This convention is not arbitrary; it serves a fundamental role in maintaining the mathematical consistency of relativistic quantum mechanics. Further exploration of this topic can clarify its implications in various physical theories.

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  • Understanding of Fourier transforms in quantum mechanics
  • Familiarity with Minkowski spacetime concepts
  • Knowledge of 4-momentum operators in physics
  • Basic principles of invariance in relativistic theories
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The discussion is beneficial for theoretical physicists, students of quantum mechanics, and researchers interested in the mathematical foundations of relativity and field theories.

jason12345
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Can anyone suggest any references explaining the motive behind it's definition?

I'm unfortunately too thick to see the necessity of the sign of the time part of the exponentials being opposite to that of the space part. It seems that the transform must preserve some property of the invariance of the space time interval, but i don't see what.

Thanks
 
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Isn't it just a convention to ensure that the 4-momentum operator can be represented as [itex]i\partial_\mu[/itex] (when we Fourier transform a wave function)?

This question has been asked a couple of times recently, but I don't think I read the answers. Try searching for it.
 

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