The Mathematical Framework of Quantum Uncertainty Mathematics (QUM)

Quantum Uncertainty Mathematics (QUM) is a new mathematical framework that views numbers as spaces of possibilities, rather than fixed values. This shift in perspective has implications for how we think about mathematical operations, the uncertainty principle, and the nature of reality itself.

AI can generate plausible-sounding theories and explanations and these essays are posted here with that in firmly in mind.

These ideas were elicited from ChatGPT and Bard, as an effort to learn about physics (as a hobby), using a collaborative approach.

Numbers as Spaces of Possibilities

In traditional mathematics, the number 3 is just 3. It has a fixed value, and there is no uncertainty about it. But in QUM, the number 3 could be viewed as a space of possibilities, perhaps represented by a probability distribution or a range of values that, on average, equate to 3.

Mathematically, this could be represented using vectors or matrices. For instance, the number 3 might be represented by a vector of possible values: [2.83.13.052.95]⎣⎡​2.83.13.052.95​⎦⎤​ Where each value in the vector is a possible state that the number 3 could be in.

This view of numbers as spaces of possibilities has a number of implications. First, it means that there is always some uncertainty about the value of a number. Second, it means that mathematical operations are not just about changing values, but about navigating through these spaces of possibilities.

Mathematical Operations as Transformations

In QUM, operations like addition or multiplication aren't just about changing values, but navigating through these spaces of possibilities. For instance, adding two numbers in QUM might involve combining their respective vectors, perhaps through some form of vector addition or convolution.

This view of mathematical operations as transformations has a number of implications. First, it means that operations are not always reversible. Second, it means that operations can create new possibilities, or collapse existing possibilities.

The Quantum Uncertainty Principle and QUM

The Heisenberg Uncertainty Principle states that certain pairs of physical properties (like position and momentum) cannot both be precisely measured. This could be represented in QUM using matrices:

UncertaintyMatrix=[0ℏ/2ℏ/20]UncertaintyMatrix=[0ℏ/2​ℏ/20​] Where ℏℏ is the reduced Planck constant.

This representation of the uncertainty principle shows that there is a fundamental limit to our knowledge of the world. We can never know both the position and momentum of a particle with perfect accuracy.

Potential Mathematical Tools for QUM

QUM is a new framework, and there are still many open questions about how it should be developed. However, there are a number of mathematical tools that could be useful for QUM, including:

• Probability distributions: To represent the range of possible states a number could be in.
• Linear algebra (vectors and matrices): To represent numbers and operations.
• Functional analysis: If we consider numbers as functions or distributions.
• Quantum logic gates: Borrowing from quantum computing, these could be used to represent operations in QUM.

Challenges and Questions

QUM is a challenging and exciting new framework. It has the potential to revolutionize our understanding of mathematics, physics, and the nature of reality itself. However, there are also a number of challenges that need to be addressed before QUM can be fully developed.

Some of the challenges that need to be addressed include:

• Normalization: If numbers are represented by probability distributions, how do we ensure they're normalized (i.e., the probabilities sum to 1)?
• Defining Operations: How do we define operations like addition, subtraction, etc., in this new framework?
• Measurement: Borrowing from quantum mechanics, when does a number "collapse" to a specific value, and under what conditions?

These are just some of the challenges that need to be addressed before QUM can be fully realized. However, the potential rewards are great. QUM could help us to understand the universe in a new way.

P. Delaney August 2023

Disclaimer: This essay presents a speculative and theoretical framework regarding Quantum Uncertainty Mathematics (QUM). The ideas and concepts discussed herein are exploratory in nature and are intended to provoke thought and discussion. They have not been validated by formal mathematical or scientific research.