Why Quantum Mechanics formulas use complex numbers instead of biquaternions?

Using biquaternions (or dual real numbers) instead of complex numbers in quantum mechanics could potentially lead to interesting mathematical explorations. However, such a substitution could fundamentally change the nature and predictions of quantum theory. In a Cartesian space, each axis are measured by real numbers, like biquaternions, they are orthogonal, means no way to their changes impact each other. But complex space, the imaginary axis has a property that if squared, it moves the angle and eventually changes the number on the real axis. This is a famous work of Euler, and later became the fundamental concept of the Schrodinger equation of quantum mechanics.

Although we can never measure a complex number directly, we do see evidence of complex numbers, such as the Noble prize winning theory of Jefferson effect, there are endless supercurrents without any voltage when two superconductors are put together with a weak link. It is caused by phase differences of complex numbers of two electrons.

Biquaternions differ significantly from complex numbers mathematically, and these differences would have profound effects on the theory.

1. Algebraic Structure Differences: The algebraic structure of biquaternions is different from that of complex numbers. Notably, biquaternions are not algebraically closed, meaning not all polynomial equations have solutions in the realm of biquaternions. This could result in some fundamental equations in quantum theory lacking solutions.

2. Challenges in Physical Interpretation: Many concepts in quantum mechanics, such as the phase and probability amplitude of wave functions, are deeply rooted in the properties of complex numbers. Using biquaternions might require a complete rethinking of these concepts, leading to drastically different physical interpretations.

3. Issues with Probability Interpretation: In quantum mechanics, an important step is to calculate the square of the modulus of the wave function to determine the probability of finding a particle at a specific location. Due to the different structure of biquaternions compared to complex numbers, this probability interpretation might no longer be applicable or would require fundamental modifications.

4. Potential for New Theories: Nonetheless, exploring the application of biquaternions in quantum mechanics could lead to new theoretical frameworks or models. Such exploration might provide novel insights in certain specific contexts, even though these theories could be vastly different from current quantum mechanics.

In summary, while using biquaternions instead of complex numbers in quantum mechanics might offer some interesting theoretical exploration, such a replacement could lead to a completely different physical theory, possibly necessitating a fundamental reworking of concepts and formulations. The current framework of quantum mechanics deeply relies on the mathematical properties of complex numbers, properties which are either absent or manifest differently in biquaternions. Therefore, any new theory based on biquaternions would require rigorous mathematical and physical validation.

What is the reason that I was think of using biquaternions? It is symmetry. The biquaternions are more perfect than complex numbers. Symmetry is beautiful and often the source of most fundamental theories. Unfortunately, the world we live in is not perfect.