Mike is the inventor and architect of Pipi and the founder of Ajabbi.
Time required to format a USB drive
The time it takes to fully format a USB drive depends entirely on the drive's total storage size and the speed of the USB port, not the volume of files currently stored on it. A 64GB USB drive will take the exact same amount of time to fully format whether it is completely empty or completely full. Because a full format writes zeros to every gigabyte of storage, speed is limited by the USB drive's write performance.
16 GB Drive: Takes roughly 2 to 5 minutes.
32 GB Drive: Takes roughly 5 to 10 minutes.
64 GB Drive: Takes roughly 10 to 20 minutes.
128 GB Drive: Takes roughly 20 to 45 minutes.
256 GB+ Drive: Can take over an hour.
Two factors determine where the drive falls on that time estimate:
USB Generation: A USB 3.0 or 3.2 drive plugged into a matching blue or Type-C port will format significantly faster than an older USB 2.0 drive, which is capped at very slow data transfer speeds.
Hardware Quality: Cheap, promotional USB drives use low-grade flash memory with incredibly slow write speeds, meaning they can take two to three times longer to format than high-quality name-brand drives.
What is a BadUSB Attack and How to Prevent It?
BadUSB, as the name suggests, is a crafty cybersecurity attack that acts as a puppeteer, controlling your USB devices at will. A BadUSB attack occurs when a USB device has a built-in firmware vulnerability that lets it disguise itself as a human interface device. Once connected to its target computer, a BadUSB could then discreetly execute harmful commands or inject malicious payloads.
A common type of BadUSB attack involves a MalDuino device. It uses a programmable USB device that mimics a keyboard when plugged into a system. This device can be pre-configured to automatically inject numerous malicious keystrokes into an unsuspecting user’s computer, enabling attackers to execute commands and compromise the system within seconds.
Within organisations, a preventative measure such as USB blocking software is a necessity because BadUSB attacks, if undetected or unstopped, could result in the unauthorised execution of commands that instigate security bypass incidents, privilege escalation, DDoS attacks, or malware infections of the host computers, which could then spread to target entire networks.
A BadUSB attack is incredibly dangerous because the malware does not live in the storage partition where your files are kept; it lives inside the USB controller chip's firmware.
When you perform a standard Windows format (even a full format), the computer only interacts with the flash memory storage blocks. It completely ignores the controller chip that dictates how the USB device talks to the computer.
How a BadUSB Attack Works.
The Disguise: The compromised firmware tricks your computer into thinking the USB drive is not a storage device at all, but rather a USB keyboard or network card.
The Execution: The moment you plug it in, the chip sends rapid, invisible keystrokes to your computer.
The Result: It can open your command prompt, download malware from the internet, and compromise your system in under three seconds—all before you even have a chance to open File Explorer or click "Format."
What are the types of BadUSB?
The different types of BadUSB found commercially available are:
MalDuino
WiFi-enabled BadUsb
BadUsb Cables
MalDuino
MalDuino is an open-source Arduino-based BadUSB that injects malicious payloads into a target computer. Gaining widespread attention recently, MalDuino packs several more features than regular BadUSB devices thanks to its onboard computer.
MalDuino devices commonly found today support Micro SD cards and include a set of DIP switches that let users toggle between stored programs on the card.
WiFi-enabled BadUsb
This type of BadUSB is similar to MalDuino in that an Arduino board serves as the base for the device but is specially designed with WiFi capabilities. Once plugged into a target system, these devices allow attackers to introduce malicious payloads into a victim's computer using the WiFi protocol.
WiFi-enabled BadUSB can take different forms depending on the exact purpose and function they serve. The common iterations of this device used today are as follows.
WiFi-enabled keystroke injectors
WiFi keyloggers
WiFi deauthers
WiFi-enabled keystroke injectors
These are the most common types of WiFi BadUSB found today. When plugged into a target computer, these devices remain dormant until an attacker makes further contact through a smartphone or a neighbouring system. Connecting to this BadUSB is as simple as connecting to a WiFi access point.
Once connected, the attacker can inject keystrokes using a suitable scripting language. These devices often come with their own applications that allow hackers to execute scripts remotely.
WiFi keyloggers
WiFi keyloggers are modern hacker hardware that particularly targets desktop computers. They work as a bridge between the keyboard's USB terminal and the computer itself. This BadUSB intercepts input signals from the keyboard and relays them to the hacker's computer.
WiFi keyloggers capture everything the user types, including sensitive information and passwords, without raising suspicion. On desktop computers, these devices can be completely hidden from plain sight and do not affect keyboard performance while in use.
WiFi deauthers
WiFi Deauthers are malicious devices that leverage flaws present in the WiFi protocol to force all users of a WiFi network to disconnect automatically. Subsequently, WiFi Deauthers prevent users from reconnecting as long as the device is active.
WiFi deauthers differ from other forms of BadUSB in that they do not directly affect the systems they connect to; instead, they use them as a power source to disrupt network connectivity and cause downtime.
BadUSB Cables
Gaining widespread attention recently, BadUSB cables look and function like any other USB cable, but they are secretly malicious devices that inject scripts and malware into a computer without the user's knowledge.
Also known as USB Ninja and USB Harpoon, these generic-looking cables hide a BadUSB within their internal circuitry and are more deceptive than many other variants. A BadUSB cable can support functions such as charging and data transfer while malicious activity happens in the background.
How to Deal with a Suspected BadUSB
If you believe a USB drive has compromised firmware, formatting it from Windows will not make it safe. You have three real options:
Physical Destruction (Safest): Throwing the drive in the trash or physically destroying it is the only 100% reliable fix for everyday users. Flash drives are cheap; your data security is not.
Firmware Flashing: You would need to find the exact manufacturer's tools for that specific controller chip and overwrite the firmware. This is highly technical, risky, and often impossible for generic drives.
Hardware Write-Blockers: IT security professionals use specialised hardware to inspect suspicious drives without allowing data to flow back to the PC.
BadUSB Removal
No software-based open-source security tools can safely "remove" or clean BadUSB malware from a standard flash drive. Because the malicious code is hardcoded directly into the hardware's internal controller chip, a computer's operating system cannot reach, overwrite, or clean it through traditional software tools.
However, the open-source community has developed powerful tools to detect and block these attacks, as well as complex technical frameworks used by hardware reverse engineers to overwrite the controller chip entirely.
Open-Source Tools to Detect & Block Attacks
Rather than fixing the drive, these open-source tools sit on your computer and intercept a BadUSB device the second it tries to emulate a keyboard or inject malicious keystrokes:
Anti-BadUSB (Python-based): A popular open-source Python script hosted on GitHub. It continuously monitors keyboard inputs across your operating system. If a newly inserted USB device begins typing commands at superhuman speeds (keystroke injection), it instantly flags and blocks the input before the script can execute.
USB Auth Guard: A lightweight, open-source Linux security tool. It locks down your system's USB ports by default. When a new device is plugged in, it forces a security authentication prompt (via polkit) before the operating system can interact with the device. This completely stops human interface device (HID) exploits.
Linux Kernel udev Monitoring: Built-in open-source Linux subsystems can be configured to catch BadUSB activity. By opening a terminal and running udevadm monitor, you can see the exact moment a device is plugged in. If a USB storage stick suddenly triggers a kernel event trying to install a "HID Keyboard," you will catch the disguise instantly.
Working notes on finding a way for Pipi to use genetic algorithms with Hilbert Spaces and matrix calculations. A
desktop exercise with Google at the moment. Computational experiments on
CPU to come. Some generated Python and CFMLsample code below to start playing with.
It looks promising; it might help address limitations of other methods for using fitness landscapes. The output curves do look better.
This might void the need to use a Cerebras wafer for these fitness landscape calculations.
The next step is to watch free maths lectures on YouTube, then do a lot of drawings and process diagrams. Then play with code and see what happens. 😎
The maths formulas were in LaTeX format and were converted to
non-LaTeX format using the free online LatexConverter.
Making a start on Hilbert Spaces and matrix calculations using CPU
By: Mike Peters & Google Search AI Mode
On a Sandy Beach: 30/08/2026
Mike Peters: Mike is the inventor and architect of Pipi and the founder
of Ajabbi.
Google Search AI Mode: Free and surprisingly good.
...
Hilbert Spaces
Hilbert spaces use matrix calculations extensively, especially when the
space is finite-dimensional or when dealing with linear operators. Here is
how they connect:
Finite-Dimensional Spaces
Direct Equivalence: An
(n)-dimensional Hilbert space is
isomorphic to (ℂⁿ) (complex
numbers) or (ℝⁿ) (real
numbers).
Vectors:
Elements in the space are represented as column vectors.
Operators:
Linear operators acting on the space are represented as (n × n)
matrices.
Inner Product:
The inner product is calculated using matrix multiplication
(specifically, the conjugate transpose of a vector multiplied by another
vector).
Infinite-Dimensional Spaces
Infinite Matrices: In infinite-dimensional Hilbert spaces (like L²), the space of square-integrable functions, operators can be
represented as infinite matrices.
Basis Dependence: By choosing an orthonormal basis, any linear
operator can be written as a matrix with infinitely many rows and
columns.
Quantum Mechanics (Matrix Mechanics)
State Vectors: Quantum states are vectors in a Hilbert
space.
Observables: Physical properties (like position, momentum, or
spin) are Hermitian operators.
Heisenberg's Formulation: Werner Heisenberg's formulation of
quantum mechanics is literally called "matrix mechanics" because it uses
matrix calculations to solve Hilbert space problems.
In machine learning, Hilbert spaces—specifically
Reproducible Kernel Hilbert Spaces (RKHS)—rely heavily on matrix
calculations to make complex, non-linear data linearly separable.
Instead of explicitly mapping data points to an infinite-dimensional space,
ML algorithms use the Kernel Trick to compute everything via an
N × N matrix, where N is the number of
data samples.
The Gram (Kernel) Matrix
The core matrix calculation in RKHS is the Gram Matrix (or Kernel Matrix), denoted as K.
Formula:(Kᵢⱼ = k(xᵢ, xⱼ) = ⟨ Φ(xᵢ), Φ(xⱼ) ⟩_{H})
Meaning: Each entry i,j is the inner product of data points
(xᵢ) and
(xⱼ) in the Hilbert space
(H).
Properties: It is always a symmetric and positive semi-definite
(PSD) matrix.
Core Matrix Operations in RKHS
Machine learning algorithms manipulate this kernel matrix using standard
linear algebra:
Matrix Inversion: Used in Kernel Ridge Regression to solve for
weights: (Kv = λ v).
Eigenvalue Decomposition: Used in Kernel PCA to find non-linear
principal components by solving
(Kv = λ v).
Quadratic Programming: Used in Support Vector Machines (SVMs) to
maximise margins using matrix-vector products involving
(K).
Computational Bottlenecks
Because (K) scales with the number of
data points, scaling Hilbert space methods requires specialised matrix
approximations:
Nyström Approximation: Factorises a low-rank sub-matrix to
approximate the massive
N × N matrix.
Random Fourier Features: Avoids the exact matrix calculation
entirely by mapping data into a randomised, lower-dimensional space.
Genetic Algorithms
Combining genetic algorithms (GAs) with Reproducing Kernel Hilbert Spaces
(RKHS) is entirely possible, highly effective, and actively used in machine
learning research.
In this hybrid framework, the fitness landscape is modelled as a
continuous function belonging to an RKHS. This approach addresses a
major limitation of traditional GAs: evaluating an individual's true fitness
can be computationally expensive (e.g., running a physics simulation or a
wet-lab experiment). The RKHS serves as an
intelligent surrogate model.
Core Mathematical Engines At Work
To make this framework highly effective, three core components work in
tandem:
The Representer Theorem: This is the mathematical backbone of your
RKHS surrogate. It guarantees that your continuous fitness approximation
(̂{f}(x)) can be written as a finite linear combination of kernel
evaluations centred at your sampled points:
(ᶠ(x)=∑ ᵢ₌₁ᴺα ᵢK(x,xᵢ))
This scales the search space completely independently of its true
dimensionality, reducing prediction cost to a simple
(O(N)) vector dot product.
Informed Exploration vs. Exploitation: Instead of letting the GA
evaluate individuals blindly on the surrogate, you can use the RKHS
variance (uncertainty) to construct an Acquisition Function (like Expected
Improvement or Upper Confidence Bound). The GA then maximises this
acquisition function rather than the raw estimated fitness, forcing the
algorithm to intelligently search unmapped areas of the landscape.
Gram Matrix Regularisation: As new true data points are evaluated,
they enter the Gram Matrix. To prevent numerical instability or
overfitting as (N) grows, a small
regularisation ridge ((λ I)) is
maintained, smoothly adjusting the landscape's rigidity without rebuilding
the framework from scratch.
How the Combination Works
[GA Population] ---> [Evaluate on RKHS Surrogate] ---> [Select &
Crossover] ^ | |__________________ [Update RKHS with True Data] _________|
The RKHS as the Fitness Landscape: You treat the unknown fitness
landscape (f(x)) as a
smooth function within an RKHS. By evaluating a small set of initial
points, you use Kernel Ridge Regression or Kriging (Gaussian Process
Regression) to build a continuous, analytical approximation of the
landscape.
Genetic Search on the Surrogate: The GA searches this approximated
RKHS landscape. Because evaluating the RKHS kernel matrix is mathematically
cheap (O(N)) for a new prediction, the GA can evolve through thousands of
generations in seconds.
Adaptive Sampling (Bayesian Optimisation): The best candidates
found by the GA are evaluated using the true, expensive fitness function.
These new data points are added back into the Gram matrix, updating the
RKHS landscape to make it more accurate where it matters most.
Key Advantages of This Hybrid Approach
Global Search in Infinite Dimensions: Traditional GAs struggle
with high-dimensional continuous spaces. By mapping the search space into
an RKHS via a kernel (like the Gaussian RBF kernel), the GA
implicitly searches a highly expressive, infinite-dimensional feature
space.
Guaranteed Smoothness: Functions in an RKHS are mathematically
smooth and bounded [6]. This eliminates extreme, discontinuous noise in
the fitness landscape, allowing the GA to find gradient-like paths toward
global optima without getting trapped in microscopic local minima.
Exact Kernel Alignment: You can use a GA to optimise the
hyperparameters of the RKHS kernel itself (e.g., finding the optimal
length-scale (σ ) of an RBF kernel)
to perfectly align the geometry of the Hilbert space with your actual
problem landscape.
Real-World Use Cases
Protein Design & Biochemistry: Designing new molecules where
the "fitness" is binding affinity. The RKHS models the chemical landscape
based on known proteins, and the GA mutates amino acid sequences to find
optimal binders.
Hyperparameter Optimisation: Optimising complex neural network architectures where the
evaluation landscape is highly non-linear and expensive to
compute.
Robotics Control: Evolving locomotion parameters for robotics
where physical testing is slow or dangerous.
Implementation Challenges
The Matrix Bottleneck Returns: As your GA discovers more points
and updates the RKHS landscape, your Gram matrix
(K) grows. Inverting this matrix to
update the landscape scales at O(N³). You will need to use Nyström approximations or sparse
online kernel methods if your total evaluation points exceed 10,000.
Exploration vs. Exploitation: GAs are aggressive exploiters. If
your RKHS surrogate model is inaccurate in an unexplored region, the GA
might get stuck optimising a "fake" peak. You must include an uncertainty
metric (like the variance in a Gaussian Process) to force the GA to
explore unknown areas.
Kernel Ridge Regression or Kriging (Gaussian Process Regression)
Kernel Ridge Regression (KRR) and
Kriging (Gaussian Process Regression, or GPR) are both kernel-based
methods for learning non-linear functions from data, but they differ
fundamentally in their underlying statistical framework, objective
functions, and outputs.
Key Differences
Core Approach: KRR minimises a regularised
mean-squared error loss function in a Reproducing Kernel Hilbert Space (RKHS). GPR uses a probabilistic (Bayesian) approach, defining a Gaussian
process prior over functions and updating it with a likelihood function
based on observed data.
Uncertainty Estimation: KRR outputs point predictions only. GPR
(and Kriging) naturally quantifies uncertainty, providing full posterior
distributions, variance estimates, and confidence intervals.
Hyperparameter Optimisation: KRR typically
optimises kernel parameters using grid search with cross-validation on a
loss function. GPR optimises hyperparameters via gradient ascent on
the marginal likelihood.
Terminology & Origin: Kriging originated in geostatistics
(mining) to find the Best Linear Unbiased Predictor (BLUP), whereas GPR
stems from machine learning and stochastic processes. Mathematically,
standard Kriging is equivalent to GPR under matching covariance and prior
assumptions
When to Use Which
Use Kernel Ridge Regression when: You need a fast, deterministic,
regularised non-linear regressor and do not require predictive uncertainty
bounds.
Use Kriging / Gaussian Process Regression when: You need confidence intervals for predictions, want to optimise
hyperparameters via marginal likelihood, or are modelling
spatial/geostatistical data.
Python Code (not tested)
Here is a clean, modular Python template using scikit-learn to stitch a Genetic Algorithm loop to a Gaussian Process (RKHS surrogate) framework.
import numpy as np
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import Matern
print(f"Current Best Score Found: {np.max(y_train):.4f}\n")
print("Optimization Complete.")
Enhancements
Regularisation (Ridge): The alpha=1e-6 in GaussianProcessRegressor serves as the λI identity-matrix offset. This prevents numerical inversion crashes if the GA proposes points structurally close to existing data.
Normalisation Engine: normalize_y=True transforms unstable fitness functions to a standard normal distribution mapping, stabilising kernel hyperparameter tuning during .fit().
Scalability Hand-off:run_ga_on_surrogate can run for massive populations or generations; it never interacts with true_expensive_fitness, protecting your heavy computation budget.
Script-Only Architecture: Written purely within <cfscript> blocks to remove legacy tag overhead and maximise processing throughput for vector arithmetic loops.
Native Box-Muller Generator: Because ColdFusion’s native rand() produces a flat uniform distribution, a standard Box-Muller transformation was written directly into the mutate() method to correctly inject normal (Gaussian) distribution mutations.
Surrogate Mapping Array Matrix: Replaces Python arrays with native array structs. The predictSurrogateFitness method models continuous field behaviours natively via distances weighted dynamically across the current known points.
This one got me thinking. John von Neumann's 5 laws might be useful inside Pipi for determining any state. I need to do a lot more reading.
The 5 Quantum Commandments of John von Neumann
An arrow in Hilbert space shall represent the quantum state of any object.
Altering the object shall make the arrow rotate smoothly through Hilbert space to represent a new state.
Distinct axes in Hilbert space shall reflect different possible sets of properties of the object.
The shadow the arrow casts onto an axis shall encode the odds of that possibility being realised when a measurement takes place.
Upon measurement, the arrow shall randomly and instantaneously jump to align with the axis representing the observed outcome, and the object shall acquire a fully determined property.
- Quanta Magazine
The original article has some SVG dynamic diagrams which are missing here.
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Last Updated
02/09/2026
In Hilbert Space, All Things Are Quantumly Possible
By: Charlie Wood
Quanta Magazine: 26/08/2026
Charlie Wood: Staff Writer, Quanta Magazine.
...
To explore quantum phenomena, we must leave the familiar world and enter the abstract realm of Hilbert space.
...
At the heart of quantum mechanics lie a few sacred rules for how to use the theory. First and foremost is, roughly, that thou shalt not think about ordinary objects presently whizzing through ordinary space. Rather, quantum mechanics predicts — in exquisite detail — all the possible ways that an object might turn out to be in the future. Exploring those possible futures requires tracking an entirely different mathematical object — an arrow known as a vector, one oriented in an expansive, alien domain.
These arrows aren’t pointing at locations. “It’s a much more abstract space than that,” said Lucien Hardy, a physicist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada. They’re “really pointing in a direction in a possibility space.”
This possibility space is called Hilbert space, and it acts as the primary arena for quantum physics.
The early quantum pioneers didn’t realize — at first — that the arcane math that strikingly captured the conduct of atoms had left the real world behind. It took a visionary mathematical physicist, John von Neumann, to recognize and define the quantum world as a Hilbert space. Once he did, exploring the ins and outs of Hilbert space would lead physicists to a deeper, more unified understanding of quantum physics.
Here’s how von Neumann’s first commandment of quantum physics came to be, and how to understand it.
What Is Hilbert Space?
Von Neumann’s commandments, or axioms, were his way of making sense of the two distinct forms of quantum mechanics developed back-to-back in the 1920s. First came Werner Heisenberg’s “matrix mechanics” in 1925. It used inscrutable tables and, in later formulations, interminable towers of numbers to calculate the odds that an electron circling an atom would jump to a higher or lower orbit. The next year, Erwin Schrödinger introduced his “wave mechanics.” It used waves to track, for instance, the probability of a particle being found at a certain location in space. While the pictures evoked by these two physicists looked completely distinct, they yielded identical predictions. Heisenberg and Schrödinger had come up with two radically different incarnations of one theory. But what was that theory?
The question fascinated David Hilbert, a renowned mathematician who had devoted much of his life to rebuilding physics on a sturdy foundation of crisp axioms. He got von Neumann thinking about the problem in the mid-1920s. In 1927 the 23-year-old prodigy — building on insights from Paul Dirac — solved it in a single-author trilogy of papers.
A bearded man sits in a wicker chair.
The mathematician David Hilbert sought a mathematical structure that would unify the different forms of quantum mechanics.
MacTutor
“The ideas specifically were von Neumann’s, but the inspiration — why do you axiomatize and what for — this is something that he took from Hilbert,” said Leo Corry, a historian of mathematics.
These papers laid out the rules for quantum mechanics, carefully defining the theory’s central objects and how they behaved. Von Neumann showed that Heisenberg’s towers and Schrödinger’s waves were reflections of the same entity, just as 0.5 and ½ indicate the same point on the number line. They both represented the main character in quantum mechanics: the quantum state.
Everything has a state. A coin can read heads or tails. A grandfather clock’s bob can take on any number of positions as it swings. Most of physics amounts to capturing an object’s state and predicting how it will change.
Von Neumann’s quantum rules complicate the notion of a state. Before you observe a quantum object, it does not have a fixed set of properties, such as a specific position. Instead it has a combination of possible properties unique to quantum mechanics — a “quantum superposition.” A superposition combines, for instance, all possible places the particle might end up being. Those possibilities can be precise and informative; perhaps there is a 99% chance you’ll find your particle to your left and a 1% chance you’ll find it to your right. You know how to place your bets, but you can’t know for sure if you’ve won until you check.
Von Neumann rendered the quantum state as a mathematical arrow called a vector. This arrow points in some direction through a space capturing all the possible futures of any quantum object — a Hilbert space.
Imagine a quantum traffic light with three possible states — red, yellow, or green. Its arrow exists in a three-dimensional Hilbert space, where the three axes represent the three possible future colors. Until the moment the light is observed, it doesn’t have a color, but rather a mixture of possible colors. So its arrow points into the space’s central region. The more closely the arrow aligns with, say, the red axis, the more likely the light is to shine red.
Diagram
Mark Belan/Quanta Magazine
The state of any object, from an electron to a galaxy, can be captured by such a vector, pointing in some direction through such a Hilbert space. This is von Neumann’s first rule of quantum mechanics.
What Happens in Hilbert Space?
An arrow moves through Hilbert space in one of two ways. Von Neumann’s other commandments specify how.
The first possibility corresponds to what happens before an observation. As the world influences the object, changing its state, the arrow turns smoothly through Hilbert space. It might get closer to the green axis, which would make our traffic light more likely to be measured as green, or to red or yellow. The point is that all this happens smoothly and predictably.
Then, if you actually observe the system, the vector will instantly and randomly snap onto either the red, yellow, or green axis. The more aligned it is with one axis, the more likely it is to snap to that axis instead of the others, but its fate is ultimately unpredictable. Let’s say it goes green. You’ll observe a green light, and there is now a 100% chance that it will still be green in subsequent measurements, because the arrow is fully aligned with the green axis. The quantum superposition is no more.
Diagram
What Properties Define Hilbert Space?
The more possible futures an object has, the bigger its Hilbert space. A coinlike particle with two possible futures is a “qubit,” the computational building block of quantum computers. It has a two-dimensional Hilbert space. Our three-color traffic light has a three-dimensional Hilbert space. But that’s just the beginning. A freely floating particle could be found in any location in the universe, so its Hilbert space must span an infinite number of dimensions.
This size — whether it’s two dimensions or an infinite number — is the only fundamental feature of a Hilbert space, according to von Neumann’s rules. The axes are arbitrary and imagined by us; they aren’t intrinsic to the space.
Consider an electron. It has one state, one arrow, pointing in a vast Hilbert space. Its Hilbert space spans all possible measurements — energy, position, momentum, etc. If you are curious where the electron might be, you can mark the space with the axes that represent possible positions. If you are wondering where the particle might be going, you apply a different set of axes, those representing possible momenta. No matter which measurement you intend to make, the underlying Hilbert space remains the same.
This freedom to carve up Hilbert space as we see fit is what allowed Heisenberg and Schrödinger to come up with two distinct versions of the same theory. Heisenberg’s picture essentially put in axes and let them rotate around the vector, while Schrödinger’s picture did the opposite: It put in a fixed set of axes and let the vector rotate relative to them. They were two completely different mathematical perspectives on the same arrows, in the same Hilbert spaces.
A man sits wearing a suit and tie.
John von Neumann developed an underlying structure for quantum mechanics that involves arrows moving inside an abstract Hilbert space.
US Department of Energy
In the first of his 1927 papers, von Neumann laid out two mathematical criteria that defined such a space. First, it had to be “complete.” It couldn’t be missing any regions or points. And second, you had to be able to calculate the alignment between a state and an axis, which you can visualize by imagining a light shining straight down onto an arrow so it casts a shadow on an axis. (The longer the shadow, the more aligned with that axis the arrow is.) The space had to allow for this operation, known as an inner product. Any space with these two features, no matter its size or origin, was a Hilbert space.
“This was a major step in creating what we call Hilbert space quantum mechanics,” said Miklós Rédei, a philosopher of physics at the London School of Economics. “It’s a beautiful example of how mathematical generalization or abstraction takes place.”
Von Neumann referred to these abstract spaces as Hilbert spaces because his mentor Hilbert had been the first mathematician to explore specific spaces with infinite dimensions in the early 1900s. Hilbert’s work had relied on those spaces being complete and having an inner product, but he didn’t think of them as examples of a more general class of spaces until his protégé grouped them together. The older mathematician may have been surprised to find his name attached to this new mathematical structure. “Dr. von Neumann, I am really curious to know what these Hilbert spaces are, after all,” Hilbert reportedly asked during a 1929 lecture.
Is Hilbert Space Real or Just an Abstraction?
A century after the birth of quantum mechanics, the theory has left physicists in an awkward position. We live in a world where objects change position as they move through three dimensions of physical space. But our most fundamental theory takes place somewhere else, in von Neumann’s vast realm of possibilities. What does that imply about the reality of our world, or that of Hilbert space?
Mathematically Crucial Fine Print
In quantum physics, Hilbert spaces use complex numbers, which involve the imaginary number i, and negative regions. But von Neumann’s fourth commandment guarantees that the odds of any possibility will always come out as a real, positive number.
To Sean Carroll, a philosopher and physicist at Johns Hopkins University, the message is clear. If quantum mechanics is the fundamental theory of nature, then Hilbert space should be considered the fundamental theater of reality, he argued in a 2022 paper(opens a new tab). One of his lines of research seeks to distill our familiar world from the disorienting Hilbert space that encapsulates all the ways the universe could possibly be.
Other physicists take a more pragmatic stance. Jonathan Sorce, a physicist at Princeton University, says that Hilbert space is a handy mathematical construction that is remarkably useful for describing many quantum systems — but not all of them. He belongs to a community of researchers searching for a mathematical construction that can describe the fabric of space and time as a quantum object. Such a theory is a prerequisite for answering big questions such as what goes on at the heart of a black hole.
Physicists asking these questions have recently focused on an even more abstract space that seems especially well suited for their purposes. This kind of space is made up of the things you could do to a Hilbert space, such as slicing it up in different ways or rotating one slice into another. In this arena, they have found that black holes seem a bit less mysterious.
This sort of über-space is known today as a von Neumann algebra. Von Neumann himself helped develop it as a potential remedy for some logical inconsistencies(opens a new tab) with Hilbert space that troubled him. “I would like to make a confession which may seem immoral: I do not believe in Hilbert space anymore,” he wrote in a 1935 letter while exploring the virtues of algebras.
Sorce, for his part, doesn’t share von Neumann’s desire for one space to rule them all. He’s content to use whichever mathematical construction best suits the quantum object he’s studying. Often it’s a Hilbert space. Sometimes it’s a von Neumann algebra. And occasionally it might even be one of the many other spaces mathematicians have cooked up over the last century.
“There’s a whole zoo of these things,” he said.
The 5 Quantum Commandments of John von Neumann
An arrow in Hilbert space shall represent the quantum state of any object.
Altering the object shall make the arrow rotate smoothly through Hilbert space to represent a new state.
Distinct axes in Hilbert space shall reflect different possible sets of properties of the object.
The shadow the arrow casts onto an axis shall encode the odds of that possibility being realised when a measurement takes place.
Upon measurement, the arrow shall randomly and instantaneously jump to align with the axis representing the observed outcome, and the object shall acquire a fully determined property.
