Is the Quotient of Two Operators AB-1 or B-1A?

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Discussion Overview

The discussion centers around the interpretation of the quotient of two operators, A and B, specifically whether A/B should be expressed as AB-1 or B-1A. Participants explore the implications of this notation within the context of abstract algebra and operator theory.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants assert that A/B is conventionally interpreted as AB-1, but they express a lack of clarity on the reasoning behind this notation.
  • One participant argues that the notation A/B is ambiguous and not commonly used in non-commutative algebra, suggesting that it should be avoided.
  • Another participant emphasizes that the expression A/B could potentially mean different things, such as B-1A or other forms, depending on the context.
  • A reference to a specific textbook on quantum mechanics is mentioned, indicating that the notation is used in that context, but this does not resolve the ambiguity discussed.
  • There is a side discussion about cultural perspectives on writing direction, with references to Arabic and Hebrew scripts, and how this might influence interpretations of notation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of A/B, with multiple competing views presented regarding its meaning and usage. The discussion remains unresolved regarding the appropriate notation in non-commutative contexts.

Contextual Notes

Limitations include the ambiguity of the notation A/B in non-commutative algebra and the lack of clarity on its accepted meaning among participants. The discussion also highlights cultural differences in writing direction that may affect understanding.

wasi-uz-zaman
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HI, Suppose there are two operators A and B , We have to find A /B - Will it equal to AB-1
OR B-1 A , Because i have read that it equals to AB-1 , BUT i could not find reason for that.
thanks
 
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wasi-uz-zaman said:
HI, Suppose there are two operators A and B , We have to find A /B - Will it equal to AB-1
OR B-1 A , Because i have read that it equals to AB-1 , BUT i could not find reason for that.

There is no reason - its just A/B is read left to right so you tend to write it as AB^-1.

Thanks
Bill
 
I think that arab physicists will not be convinced by this answer!
 
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wasi-uz-zaman said:
HI, Suppose there are two operators A and B , We have to find A /B
Who asks you this? I have never seen this notation and I think it isn't used precisely because it is ambiguous.

Generally in abstract algebra, \frac{x}{y} is a shorthand notation for xy^{-1} in the case that the algebraic structure is commutative. If the structure is not commutative, it simply isn't used.
 
I'd like to second kith's opinion. Do not use \frac{x}{y} unless you are dealing with commutative objects. At best, it would be confusing, in other cases, it would be simply wrong. Note, for example,that xy^{-1} and y^{-1}x are not even the only things this could possibly mean. Who says it should not be y^{-1/2}xy^{-1/2} or something entirely different?
 
hello , i have read this in BOOK " QUANTUM MECHANICS CONCEPT AND APPLICATION" (SECOND EDITION) BY Zettili , on problem 2.12 on page 147.
 
naima said:
I think that arab physicists will not be convinced by this answer!
how do you know i am arab
 
bhobba said:
There is no reason - its just A/B is read left to right so you tend to write it as AB^-1.

Thanks"
Bill
hello i have read this in book " quantum mechanics concepts and applicatiotion" by zettlii page 147 2nd edition
 
wasi-uz-zaman said:
how do you know i am arab

He doesn't. Naima was pointing out that Bhobba's answer wouldn't be particularly helpful to someone whose native language is written right-to-left, and Arabic is the first example that came to mind.
 
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You are right.
Israeli also write from right to left and up to down.
It becomes more complicated with traditional japonese.
Things can be read from up to down and then from right to left so ## \frac {x}{y}## has no asymmetry.
 

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