Why Does Quantum Mechanics Require Complex Numbers?

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Discussion Overview

The discussion centers on the role of complex numbers in quantum mechanics, exploring whether their use is merely a mathematical artifact or if it reflects deeper aspects of the nature of quantum theory. The conversation touches on theoretical implications and interpretations of quantum mechanics.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants question whether the use of complex numbers in quantum mechanics is a mathematical artifact or indicative of deeper truths about nature.
  • One participant argues that complex numbers are essential for accurately describing the interference of probabilities, citing the Aharonov-Bohm effect as evidence that the complex phase of the wave function is not just a mathematical tool.
  • Another participant suggests that any phenomenon described by a complex number can also be represented by two real numbers, although this may complicate the notation.
  • A further point is made comparing the representation of rational numbers as pairs of integers, suggesting that while alternative representations exist, they may not capture the essence of reality as effectively as the original form.
  • There is a reiteration of the question regarding the underlying nature of reality and its potential complexity, with a note that physics does not typically address "why" questions.

Areas of Agreement / Disagreement

Participants express differing views on the significance of complex numbers in quantum mechanics, with no consensus reached on whether their use is purely mathematical or indicative of deeper physical truths.

Contextual Notes

Some limitations in the discussion include unresolved assumptions about the nature of complex numbers and their implications, as well as the lack of clarity on the philosophical aspects of the questions posed.

MichPod
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Is the fact that QM uses complex numbers should be considered as a math artefact (as it is the case when complex numbers are used for alternate current circuit analysis), or, alternatively, it has some deep and important relation to the nature (or at least to the nature of the quantum theory)? If the later is true, could you please clarify or bring some links discussing the topic?
 
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Complex numbers are needed in order to correctly describe the interference of probabilities. However, the complex phase of the wave function is not merely a mathematical artifact, as is shown by the Aharonov-Bohm effect.
 
Whatever can be described by one complex number can be described, as well, by two real numbers. Some denotations may look more cumbersome in such a real notation, that's all.
 
Denis said:
Whatever can be described by one complex number can be described, as well, by two real numbers. Some denotations may look more cumbersome in such a real notation, that's all.

Well, a rational number may be described as a pair of integer numbers as well, but we believe (most of us) that the rational numbers give probably more adequate description of the reality than such a pair.
And... even if we pass to two real numbers instead of one complex number, we still will emulate the complex arithmetic with them. The question is - why? Is the nature in its underneath somehow "complex"?
 
MichPod said:
The question is - why? Is the nature in its underneath is somehow "complex"?
Physics doesn't answer "why" questions.

I suggest you read the link I gave, and other threads on PF on the subject (look for "Similar Discussions" below).

Thread closed.
 

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