Physicists Prove Quantum Mechanics Cannot Eliminate Complex Numbers
Sabine Hossenfelder
Real-number quantum models cannot reproduce multi-source entanglement correlations unless physicists accept forms of non-locality even more extreme than standard quantum theory.
AI-assisted summary of Do Complex Numbers Exist? — original source. Check the original for full context.
Source published:
Key ideas
- A new preprint demonstrates that standard complex quantum mechanics predicts specific entanglement correlations from independent sources that cannot be reproduced by a real-valued quantum framework without introducing extreme non-locality. (6:58)
- Outside of quantum mechanics, complex numbers are purely a calculational convenience and not physically indispensable, as classical equations can always be rewritten strictly using real numbers. (3:44)
- The paper outlines a concrete experimental protocol that could empirically test whether nature fundamentally requires complex numbers or potentially falsify standard quantum mechanics. (8:10)
Detailed notes
## Background This video was published on YouTube by Sabine Hossenfelder on March 6, 2021. According to the page description, it addresses the question of whether complex numbers genuinely exist or are merely convenient tools in science, focusing on a recent preprint claiming that quantum mechanics cannot be formulated without them.
## TL;DR Sabine Hossenfelder examines an academic dispute triggered by an arXiv preprint asserting that quantum physics strictly requires complex numbers. While classical physics routinely uses imaginary numbers merely as calculational shortcuts for wave mechanics, the Schrödinger equation explicitly incorporates them into quantum mechanics. Although single-particle systems can be decoupled into real-valued equations, the new paper demonstrates that entangled particles from independent sources produce correlations that real-valued quantum mechanics cannot reproduce without invoking extreme non-locality. As a result, physicists who reject imaginary numbers are forced to either accept their necessity or embrace a reality even more non-local than standard quantum theory.
## Atmosphere & style The presentation carries Sabine Hossenfelder's trademark dry, irreverent humor and analytical clarity, framing a high-level academic controversy as an amusing refuge from everyday chaos. It balances straightforward mathematical pedagogy with witty observations about theoretical physicists quarreling over ontological minutiae. The tone shifts from educational and crisp during the foundational math review to engagingly provocative as she unpacks how the new preprint corners skeptics, leaving the viewer with an intellectually stimulating and slightly mischievous sense of scientific debate.
## Bulletin - [06:58](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=418s) A new preprint demonstrates that standard complex quantum mechanics predicts specific entanglement correlations from independent sources that cannot be reproduced by a real-valued quantum framework without introducing extreme non-locality. - [03:44](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=224s) Outside of quantum mechanics, complex numbers are purely a calculational convenience and not physically indispensable, as classical equations can always be rewritten strictly using real numbers. - [08:10](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=490s) The paper outlines a concrete experimental protocol that could empirically test whether nature fundamentally requires complex numbers or potentially falsify standard quantum mechanics.
## Key takeaways - [00:00](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=0s) A hyper-niche academic debate emerged over whether complex numbers physically exist or are merely a calculational tool in scientific theories. - [00:35](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=35s) An arXiv preprint titled 'Quantum physics needs complex numbers' claimed a mathematical proof that physics cannot proceed without imaginary numbers. - [01:15](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=75s) Complex numbers arise by defining the imaginary unit $i$ such that $i^2 = -1$, enabling solutions to all algebraic polynomial equations. - [02:44](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=164s) Unlike standard two-dimensional vector spaces, complex numbers form a mathematical field governed by consistent rules for both addition and multiplication. - [03:44](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=224s) In classical physics, complex numbers simplify wave mechanics via Euler's formula, but they can always be eliminated in favor of real trigonometric expressions. - [04:58](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=298s) Under an empirical scientific criterion, a mathematical structure only 'exists' if it is strictly necessary to describe observed physical reality. - [05:48](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=348s) The Schrödinger equation explicitly contains $i$, but single-system equations can easily be decomposed into coupled real-valued differential equations. - [07:15](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=435s) When entangled particles originate from a single source, their statistical correlations can always be reproduced using real-valued quantum mechanics. - [07:50](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=470s) By analyzing network entanglement from two independent sources, the preprint showed real quantum models fail to match complex predictions without extra non-locality. - [08:10](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=490s) The study proposes an experimental setup to empirically verify the predicted correlations, effectively testing the physical necessity of complex numbers. - [09:28](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=568s) Physicists in the pragmatic 'shut-up and calculate' camp view efforts to eliminate complex numbers as pointless and philosophically uninteresting. - [09:55](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=595s) Physicists opposed to unobservable complex quantities are forced to either accept complex numbers or endorse a theory of nature that is even more non-local. - [10:16](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=616s) An experimental failure to detect the predicted complex correlations would result in the falsification of standard quantum mechanics.
## Detailed walkthrough ### [00:00](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=0s) A Niche Nerd Fight Over Complex Numbers
Sabine Hossenfelder opens by explaining that whenever the world feels overwhelmingly chaotic, she seeks comfort in investigating hyper-niche academic disputes: debates where specialists passionately clash over questions most people never realized were open for debate. The focus of this dispute is an ontological question at the intersection of mathematics and physics: are complex numbers genuinely necessary to describe the universe as we observe it, meaning they physically "exist" in a scientific sense, or are they merely a convenient calculational shortcut?
The latest skirmish was sparked by an arXiv preprint bearing the definitive title *"Quantum physics needs complex numbers"*. The authors presented a mathematical proof responding to previous claims that physics could proceed entirely without imaginary numbers. Computer scientist Scott Aaronson highlighted the preprint on his blog, describing the finding as "striking." However, the broader online physics community reacted with sharp skepticism and dismissiveness. As Sabine recounts, “The responses were, well, not very enthusiastic. They ranged from ‘why fuss about it’ to ‘bullshit’ to ‘it’s missing the point.’” To understand why passions flared, she sets out to review the underlying mathematics and physics before analyzing the paper itself.
---
[01:15](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=75s) The Math of Complex Numbers
Sabine reviews the foundations of complex numbers, starting from basic secondary school algebra. Real numbers—which include integers, fractions, and values with infinitely many digits after the decimal point like $\sqrt{2}$ and $\pi$—cannot solve an equation such as $x^2 = -1$, because squaring any real number always yields a non-negative result. Mathematicians bypass this barrier by introducing an imaginary unit, denoted $i$, defined by the property $i^2 = -1$.
Remarkably, introducing this single entity solves not just $x^2 = -1$, but guarantees that every algebraic polynomial equation has solutions within the complex domain. All such numbers can be written in the form $a + ib$, where $a$ is the real part and $b$ is the imaginary part (both $a$ and $b$ being real numbers). As an aside, Sabine drops a mathematical curiosity: this property fails for $p$-adic numbers, which have infinitely many digits *before* the decimal point.
Graphically, complex numbers are plotted on a two-dimensional complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis, placing $i$ in the upper half. Sabine clarifies that this visual representation does not mean complex numbers are simply equivalent to a standard two-dimensional real vector space $\mathbb{R}^2$:
```mermaid flowchart TD accTitle: Algebraic Structure of Complex Numbers accDescr: Diagram contrasting vector addition with complex field multiplication governed by i squared equals minus one.
subgraph AddLaw["Vector-like Operations"] V["Addition: (a + ib) + (c + id) = (a + c) + i(b + d)"] end
subgraph FieldLaw["Field Operations"] F["Multiplication: (a + ib)(c + id) = (ac - bd) + i(ad + bc)"] I["Governed by: i² = -1"] F --> I end
AddLaw --- FieldLaw ```
While complex numbers add component-wise just like two-dimensional vectors, their multiplication rule sets them apart. Because complex numbers possess consistent internal laws for both addition and multiplication, mathematics classifies them as a "field," making them structurally distinct from a generic coordinate grid.
---
[03:44](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=224s) The Physics of Complex Numbers
In practical non-quantum physics, complex numbers are everywhere because they simplify the mathematics of oscillations. By Euler's formula, multiplying a real number $\alpha$ by $i$ inside an exponential function produces $e^{i\alpha}$, a point that traverses the unit circle centered at zero in the complex plane. Decomposed into its real and imaginary projections, $e^{i\alpha} = \cos(\alpha) + i\sin(\alpha)$, tracing out harmonic oscillations.
Multiplying two wave solutions in exponential form requires only adding their exponents, such as $e^{i\alpha} \cdot e^{i\beta} = e^{i(\alpha+\beta)}$. Doing the same calculation with raw sines and cosines requires cumbersome trigonometric identities. Consequently, physicists routinely perform classical calculations—especially when analyzing electromagnetic radiation—in complex notation, discarding the imaginary component at the very end to keep only the physical real part.
Sabine emphasizes that outside quantum mechanics, this convenience never amounts to physical necessity. “With the exception of quantum mechanics, to which we’ll get in a moment, the complex numbers are not necessary.” Under her scientific criterion, a mathematical construct can only be said to "exist" in a scientifically meaningful sense if it is indispensable for describing empirical observations. Classical physics fails that test; it can always be reformulated purely with real numbers at the cost of algebraic convenience.
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[05:48](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=348s) Complex Numbers in Quantum Mechanics
Quantum mechanics appears fundamentally different at first glance. Its central dynamical law, the Schrödinger equation:
$$i\hbar \frac{\partial}{\partial t}\psi = \hat{H}\psi$$
explicitly features the imaginary unit $i$, forcing the wave function $\psi$ to be complex-valued. However, standard calculational procedures complicate the claim that $i$ is physically real. When solving the equation numerically, physicists routinely separate $\psi$ and the equation into two coupled real-valued differential equations for the real and imaginary components. Furthermore, quantum observables and measurement outcomes are strictly real numbers.
Splitting single-system equations into two coupled real equations shows that single particles do not require an irreducible imaginary unit. As Sabine observes, simply rewriting a single Schrödinger equation into real components is trivial: “What I just said about decomposing the Schrödinger equation is of course correct, but that’s not what they looked at in the paper, that would be rather lame.”
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[06:58](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=418s) The New Paper: Entanglement from Independent Sources
The preprint moves beyond single-particle systems to composite quantum systems involving entanglement. When multiple particles interact, their joint state is represented by a combined complex-valued wave function exhibiting quantum correlations that exceed any classical correlation bounds.
Previous mathematical proofs established that if entangled particles originate from a single common source, their measurement statistics can always be reproduced using an equivalent real-valued quantum formulation (by expanding the system's Hilbert space). The authors of the new preprint examined a more complex network configuration: particles emitted by two completely independent, separate sources.
```mermaid flowchart LR accTitle: Independent Source Entanglement Test accDescr: Particles from two independent sources undergo joint measurement to test real versus complex quantum mechanics.
subgraph Sources["Independent Sources"] S1["Source 1"] S2["Source 2"] end
subgraph Measurement["Measurement & Verification"] M["Joint Measurement / Entanglement Protocol"] C["Check Correlations"] end
S1 --> M S2 --> M M --> C ```
Under this independent-source geometry, the authors proved that standard complex quantum theory predicts specific entanglement correlations that a real-valued quantum theory cannot reproduce—unless the real-valued formulation is granted even stronger non-local behaviors than ordinary quantum theory already exhibits. They also proposed a concrete experimental setup to test these correlations empirically.
Sabine notes that the paper had not yet undergone peer review and could contain an analytical flaw. However, she underlines the stakes if the math holds: “So, this is why it’s cool. They’ve figured out a way to experimentally test if complex numbers exist!” The essential caveat lies in the theoretical ground rules: real numbers can only match the predictions if one is willing to abandon standard locality constraints and make the real-valued model significantly more non-local than standard quantum mechanics.
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[09:28](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=568s) Why the Paper Sparked Controversy
The polarized response to the paper highlights a deep philosophical division among physicists regarding quantum foundations:
- **The "Shut-Up and Calculate" Camp:** Physicists in this group find the debate completely pointless. Because complex numbers work seamlessly within the mathematical formalism, they see no philosophical dilemma needing resolution. For them, spending effort to eliminate imaginary numbers is an exercise in futility. - **The "Real-Numbers-Only" Camp:** Physicists who dislike the complex formalism argue that since physical measurement devices only register real numbers, fundamental physics should not rely on unobservable imaginary values.
Sabine explains that the paper corners this second camp: “If you, on the other hand, are in the camp of people who think there’s something wrong with quantum mechanics because it uses complex numbers that we can never measure, then you are now caught between a rock and a hard place. Either embrace complex numbers, or accept that nature is even more non-local than quantum mechanics.”
A third outcome remains: an actual experiment could fail to find the predicted complex correlations altogether. That result would falsify standard quantum mechanics, which Sabine notes would be the most exciting resolution of all.
---
[10:26](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=626s) Sponsor Message: Brilliant
In the closing segment, Sabine introduces the video sponsor, Brilliant. She recommends the platform's interactive science and mathematics courses, noting specifically that their course on complex numbers offers a hands-on path to mastering the mathematics discussed in the video. She also mentions Brilliant's courses covering the properties and behaviors of quantum objects, where learners test their understanding through built-in exercises. Viewers are invited to register at `brilliant.org/Sabine` for free, with an offer granting 20% off an annual premium subscription to the first 200 subscribers.
## Notable quotes > [09:28](https://www.youtube.com/watch?v=ALc8CBYOfkw&t=568s) Either embrace complex numbers, or accept that nature is even more non-local than quantum mechanics. — Sabine Hossenfelder
## Mentioned Sabine Hossenfelder, Scott Aaronson, Brilliant, Patreon, arXiv, Quantum physics needs complex numbers
## Video description Check out the physics courses that I mentioned (many of which are free!) and support this channel by going to https://brilliant.org/Sabine/ where you can create your Brilliant account. The first 200 will get 20% off the annual premium subscription.
Do complex number exist or are they just a convenient, mathematical tool that we use in science? With the exception of quantum mechanics, it is easy to get rid of complex numbers. But can you do quantum mechanics without complex numbers? A recent paper says no, you can't.
The paper is here: https://arxiv.org/abs/2101.10873
Scott's blogpost is here: https://www.scottaaronson.com/blog/?p=5270
You can support us on Patreon here: https://www.patreon.com/sabine
#science #physics #mathematics
0:00 Intro 01:15 The Math of Complex Numbers 03:44 The Physics of Complex Numbers 05:45 Complex Numbers in Quantum Mechanics 06:58 The New Paper 09:28 Why is it controversial? 10:26 Sponsor Message
[Source](https://www.youtube.com/watch?v=ALc8CBYOfkw)