Commutation Relations: Relativistic Quantum Mechanics

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SUMMARY

The discussion confirms that the usual commutation relations, such as those between position and momentum, remain valid in one-particle relativistic quantum mechanics (RQM) exemplified by the Dirac equation. However, in quantum field theory (QFT), the situation is more complex due to the absence of a time operator, which leads to the demotion of position to a parameter and the promotion of the wave equation to an operator. This distinction is crucial for understanding the foundational differences between RQM and QFT.

PREREQUISITES
  • Understanding of the Dirac equation in relativistic quantum mechanics
  • Familiarity with quantum field theory (QFT) concepts
  • Knowledge of commutation relations in quantum mechanics
  • Basic grasp of operators in quantum mechanics
NEXT STEPS
  • Research the implications of the Dirac equation on particle behavior in RQM
  • Explore the concept of operators in quantum field theory
  • Study the role and significance of time in quantum mechanics
  • Investigate the differences between classical and quantum interpretations of position and momentum
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Physicists, quantum mechanics students, and researchers interested in the foundational aspects of relativistic quantum mechanics and quantum field theory.

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Does the usual commutation relations, e.g. between position and momentum, remains valid in relativistic quantum mechanics?
 
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Yes in one particle RQM like the Dirac equation.
But things get more complicated in QFT.
 
No, because there's no such thing as time operator in quantum mechanics.

Daniel.
 
As far as I'm aware, there's no such thing as a time operator in field theory. My understanding was that, for field theory, they demoted position to a parameter like time, and promoted the wave equation to the status of an operator.
 

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