# Von Neumann's uniqueness theorem (CCR representations)

• A
• Heidi
In summary, the conversation discusses the paper "Redei's theorem on complete continuity and its uniqueness" and the proof that P is a projector. The proof can be found in von Neumann's original article, where he uses the concept of "Kern" (integral kernel) to show that ASA = kA. The conversation also touches on the translation of "Kern" in modern math English and how to derive formulas for A and SA from the kernels mentioned in the paper.
Heidi
Hi Pfs,
Please read this paper (equation 4):
https://ncatlab.org/nla b/files/RedeiCCRRepUniqueness.pdf
It is written: Surprise! P is a projector (has to be proved)...
where can we read the proof?

thanks Demystifier.

Demystifier
Thank you for bringing it up. I will check in von Neumann's original proof or some other source.

The only explicit proof is in von Neumann's original article.

Heidi
thanks, it's a good opportunity to revise my German a bit (I studied English and German when in high school)

Tell me if this is correct:
To prove that A and AS(u,v)A only differ by a numerical factor, Von Neumann
calculates the "Kern" of A then of SA and then of ASA. As these "kerns" differ by a m
multiplicative constant k, then ASA = k A.
I would like to know how to translate the german word "Kern" in modern math english. Is it really integral kernel? or characteristic functional?
How to derive his forulas for A and SA?

Yes, an integral kernel is the modern term. As for the calculations themselves are all made by von Neumann. Later accounts (Putnam for example) are telegraphic, no explicit calculations

vanhees71
In the paper Von Neumann considers three operators containing integrals. For each of them , he gives its kernel .
How to retrieve A , SA and ASA from these kernels?

## 1. What is Von Neumann's uniqueness theorem?

Von Neumann's uniqueness theorem is a mathematical result in the field of quantum mechanics that states that there is only one possible representation of the canonical commutation relations (CCR) on a Hilbert space. This means that the CCR, which describe the fundamental commutation relationships between position and momentum operators, can only be represented in one specific way.

## 2. What are CCR representations?

CCR representations are mathematical representations of the canonical commutation relations (CCR) on a Hilbert space. They describe the fundamental commutation relationships between position and momentum operators in quantum mechanics. These representations are important because they allow us to understand and manipulate the behavior of quantum systems.

## 3. How does Von Neumann's uniqueness theorem relate to quantum mechanics?

Von Neumann's uniqueness theorem is a fundamental result in quantum mechanics because it shows that there is only one possible way to represent the canonical commutation relations (CCR) on a Hilbert space. This is important because it helps us understand the behavior of quantum systems and make predictions about their properties.

## 4. What are the implications of Von Neumann's uniqueness theorem?

The implications of Von Neumann's uniqueness theorem are far-reaching. It means that the CCR, which are fundamental to our understanding of quantum mechanics, can only be represented in one specific way. This has implications for the behavior and properties of quantum systems, and also has applications in fields such as quantum computing and quantum information theory.

## 5. How is Von Neumann's uniqueness theorem proven?

Von Neumann's uniqueness theorem is proven using mathematical techniques such as functional analysis and operator theory. It involves showing that any two representations of the CCR on a Hilbert space are unitarily equivalent, meaning that they can be transformed into each other through a unitary transformation. This ultimately leads to the conclusion that there is only one possible representation of the CCR on a Hilbert space.

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