Do Complex Numbers Exist?

Summary

The video script delves into the debate over the necessity of complex numbers in describing the world. It begins by mentioning a recent paper arguing for the indispensability of complex numbers in quantum physics, which sparked controversy. The video provides a brief explanation of complex numbers, their mathematical properties, and their utility in physics, particularly in handling oscillations and waves. It then discusses the use of complex numbers in quantum mechanics, specifically in the Schrödinger equation, and raises the question of whether real numbers could suffice, albeit less conveniently. The crux of the debate is a new paper suggesting that certain quantum entanglements cannot be replicated with real numbers alone, proposing an experimental method to test this claim. The video concludes by acknowledging the paper's potential impact on our understanding of quantum mechanics and the existence of complex numbers, highlighting the ongoing controversy and the possibility of future developments in the field.

Takeaways

• 🤓 **Complex Numbers Controversy**: A recent debate has emerged over whether complex numbers are essential for describing the world, or merely a mathematical tool.
• 📜 **Paper on Quantum Physics**: A preprint paper titled 'Quantum physics needs complex numbers' has reignited discussions, providing a proof for the necessity of complex numbers in quantum physics.
• 💻 **Scott Aaronson's Blog Post**: Computer scientist Scott Aaronson called the paper 'striking', but the response was mixed, with some questioning its relevance.
• 🔍 **Understanding Complex Numbers**: Complex numbers are used to solve equations that real numbers cannot, such as x^2 = -1, by introducing the imaginary unit 'i'.
• 📐 **Complex Plane**: Complex numbers are often represented on a complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis.
• 🧲 **Physics and Complex Numbers**: They are particularly useful in physics for describing oscillations and waves, such as in electromagnetic radiation.
• 🚫 **Non-Necessity in Classical Physics**: Outside of quantum mechanics, complex numbers are not strictly necessary for calculations, though they simplify the process.
• 🌐 **Quantum Mechanics and Wave Functions**: In quantum mechanics, the Schrödinger equation involves complex numbers, which are essential for the wave function ψ.
• 🧬 **Entanglement in Quantum Systems**: The paper discusses whether entanglement in quantum mechanics could be achieved without complex numbers, proposing an experimental test for this.
• ⚖️ **Controversy Over Necessity**: The debate is controversial because altering quantum mechanics to avoid complex numbers could lead to a more non-local theory.
• 🔬 **Experimental Test of Existence**: The paper suggests an experiment that could test the existence of complex numbers, which would be a significant advancement in physics.
• 📚 **Learning Complex Numbers**: The video encourages viewers to learn more about complex numbers and their role in quantum mechanics through resources like Brilliant's courses.

Q & A

What is the main topic of the video?

The video discusses the necessity and existence of complex numbers in describing the world, particularly in the context of physics and quantum mechanics.

What sparked the recent controversy on the role of complex numbers?

The controversy was ignited by a preprint paper titled 'Quantum physics needs complex numbers,' which aimed to prove the necessity of complex numbers in quantum physics.

What is the mathematical definition of complex numbers?

Complex numbers are numbers of the form a + i*b, where 'a' and 'b' are real numbers, 'i' is the imaginary unit with the property that i^2 = -1, and 'a' is the real part while 'b' is the imaginary part of the complex number.

Why are complex numbers considered useful in physics?

Complex numbers are useful in physics because they simplify calculations involving oscillations and waves, such as those found in electromagnetic radiation, by allowing for the use of Euler's formula and complex exponentials.

In what way does the video suggest complex numbers might be physically real?

The video suggests that if the proposed experiment confirms the specific entanglement predicted by complex quantum mechanics, then complex numbers might be considered as existing in a scientifically meaningful way.

What is the significance of the paper's claim that real numbers alone cannot reproduce certain quantum entanglements?

The significance is that it implies complex numbers are not just a mathematical convenience but are fundamentally necessary to describe certain quantum phenomena, which would mean they have a physical existence.

What is the Schrödinger equation, and why is it important in quantum mechanics?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It is important because it allows for the calculation of probabilities of different outcomes in quantum events.

Why might some people be skeptical about the necessity of complex numbers in quantum mechanics?

Some people might be skeptical because, in many cases, calculations can be done using only real numbers, albeit more cumbersomely. They might argue that since the results are the same, complex numbers are not necessary.

What is the 'shut-up and calculate' camp mentioned in the video?

The 'shut-up and calculate' camp refers to those who believe that the mathematical formalism of quantum mechanics, including the use of complex numbers, works well for making predictions and should be used without questioning its ontological basis.

What is the role of the imaginary unit 'i' in complex numbers?

The imaginary unit 'i' is defined by the property that i^2 = -1. It is used to extend the real numbers to complex numbers, allowing for the solution of equations that have no real solutions.

How does the complex plane represent complex numbers?

The complex plane is a two-dimensional plane where the horizontal axis represents the real part of a complex number and the vertical axis represents the imaginary part. It provides a geometric representation of complex numbers.

What is the p-adic number system mentioned in the video?

P-adic numbers are a type of number system that extends the real numbers to include numbers with infinitely many digits before the decimal point. They are used in certain areas of advanced mathematics.