Four-Dimensional Vector in Special Relativity

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SUMMARY

The discussion focuses on the momentum-energy vector and four-dimensional wave vector in the context of special relativity, specifically represented as \(\vec{P}=(\vec{p},i\frac{E}{c})\) and \(\vec{K}=(\vec{k},i\frac{\omega}{c})\). It establishes the relationship \(\vec{P}=\frac{h}{2\pi}\vec{K}\) due to the equation \(E=\frac{h}{2\pi}\omega\). The conversation highlights the shift from using an imaginary basis for time in four-vectors to a mixed signature metric, represented as \(ds^{2}=x^{2}+y^{2}+z^{2}-c^{2}t^{2}\), emphasizing the unique nature of time in four-dimensional space-time.

PREREQUISITES
  • Understanding of four-vectors in physics
  • Familiarity with special relativity concepts
  • Knowledge of wave vectors and their applications
  • Basic grasp of quantum mechanics, particularly Planck's constant
NEXT STEPS
  • Study the implications of mixed signature metrics in relativity
  • Explore the relationship between energy and frequency in quantum mechanics
  • Learn about the applications of four-dimensional vectors in modern physics
  • Investigate the historical context and evolution of four-vector notation
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Physicists, students of theoretical physics, and anyone interested in the mathematical foundations of special relativity and quantum mechanics.

zhangyang
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About momentum-energy vector ,we have :

[itex]\vec{P}[/itex]=([itex]\vec{p}[/itex],i[itex]\frac{E}{c}[/itex])

and four dimentianal wave vector :

[itex]\vec{K}[/itex]=([itex]\vec{k}[/itex],i[itex]\frac{\omega}{c}[/itex])

They also satisfy the ralation :

[itex]\vec{P}[/itex]=[itex]\frac{h}{2\pi}[/itex][itex]\vec{K}[/itex],

because E=[itex]\frac{h}{2\pi}[/itex][itex]\omega[/itex].

It is interesting.
 
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FYI: very few people use the imaginary basis for time anymore, when speaking of 4-vectors. Instead of a (+,+,+,+) metric with one imaginary basis, a mixed signature metric is used.
 
ds[itex]^{2}[/itex]=x[itex]^{2}[/itex]+y[itex]^{2}[/itex]+z[itex]^{2}[/itex]-c[itex]^{2}[/itex]t[itex]^{2}[/itex]

In the four dimensional space-time vector,the concept of time has been bent,because time has the meaning of evolution and irreversibility.So it can't convert into space freely.
 

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