The Hilbert Space

David Hilbert; via Wikimedia Commons.

As I began planning last week for this entry, I initially thought that I’d follow the “here is what astronomers discovered in [insert month]” precedent I had established the month before. But then I changed my mind; as with any aspiring physicist, I fell into the rabbit hole and found many interesting concepts in astrophysics that I thought I needed to learn. Over the week I spent planning this entry, I discovered at least six new concepts that I thought needed to be discussed; as a result, I chose to pause new releases of astronomers’ monthly discoveries in order to explain the paradigms I discovered over the last week. Some notable examples outside of Hilbert space, which is the topic for today, include supersymmetry (which I actually discovered upon answering a question in my class in which “selectron” was a possible answer), quarks and the Doppler effect. The interesting thing about both this chapter and the subsequent chapter on supersymmetry is that I have absolutely no background knowledge on either of the topics; in previous entries, I had some existing knowledge, although limited, but today I have none. Nevertheless, I am excited to set my (and your) toes in new waters; let us now consider the Hilbert space through the lenses of mathematics and quantum mechanics. Please do cut me some slack, though, for this topic combines abstract algebra with advanced calculus, two very complicated subjects for teenagers. Anyway, let us explore the Hilbert space, the subject of one of the most complicated entries I will ever write.

A short history of the Hilbert space

(As a disclaimer, let me just say that even the history of the Hilbert space is difficult to understand; I have always been able to understand quantum mechanics with effort, but with generalizations of Euclidean Space, I’ve blanked. The history is rich, but it requires a significant amount of background knowledge, all of which we will discuss in this entry.)

When the Greek philosopher Euclid first envisioned the Euclidean space in his Elements, he argued that it is the fundamental space of geometry, characterized by a three-dimensional space with x, y and z coordinates. It is now understood, however, that such a space can have any finite number of dimensions: it is an nth-dimensional coordinate system in which a point contains a value for every dimension; in the Cartesian coordinate system (a Euclidean space over a real-numbered plane, ℝ2), for example, a function such as y(x) = x2 obtains values of y(1) = 1, or y(40) = 1600 (any point defined by the function has real number values for every dimension in the space). 

Vector applications in the ℝn Euclidean space, such as gradient or slope fields, are known as vector spaces.* These spaces, which are fundamental to calculus, are equally fundamental to the development of the Hilbert space. The first generalizations of the vector space began to gain traction in the late nineteenth century, which led to subsequent developments of vector spaces and, soon, the Hilbert space; when two mathematicians, David Hilbert and Erhard Schmidt, were studying integral equations, they discovered that two square-integrable (that is, functions whose absolute value squared can be integrated over any interval without reaching infinity) and real-valued functions on some interval have a dot product–which means that the square-integrable functions can be modeled as vectors.

As the two mathematicians discovered that square integrable functions can be considered vectors, they modeled those vectors in a vector space. The space eventually gained in 1929 when John von Neumann, the legendary mathematician, coined the term “Hilbert space,” after David Hilbert.

The discovery of Hilbert space led to many applications, including functional analysis–the study of mathematical analysis dealing with functionals. Hilbert spaces are the vector spaces for most physical phenomena: velocity, acceleration, wave functions, everything. Most operations in calculus-based physics revolve around Hilbert space. Many of the entries already published revolve around the Hilbert space–even the properties of black holes (such as angular momentum or angular velocity), for example, can be explained with vectors in the Hilbert space. Beyond classical mechanics, the Hilbert space provides the most robust mathematical formulation of quantum mechanics, and is responsible for much of the developments in quantum mechanics itself.

Understanding the Hilbert space itself

So, we understand the history of the Hilbert space, but just as important as that history is the concept itself. Let us now seek to understand the Hilbert space.

As Hilbert space is a mathematical concept that is not as easily explained in English as it is explained in math, so we will be forced to integrate some calculus and linear algebra.**

The Hilbert space essentially generalizes ideas in linear algebra and calculus (which are typically modeled in finite-dimensional Euclidean spaces) to infinite-dimensional spaces. To understand this in greater depth, let us consider two vectors in any Cartesian plane (a two-dimensional Euclidean space of any set of coordinates). Imagine two vectors, the first with the coordinates <x1, y1>, and the second with <x2, y2>: with this information, one can find the dot product either by multiplying the x and y components of each vector and then adding the answers, or by multiplying the magnitudes of the two vectors and the cosine of the angle between them. One important distinction, however, is that these are Cartesian vectors–they have only two dimensions. Now, if you were to expand these Cartesian vectors to the Hilbert space, the vectors would instead exist in an nth (n ∈ ℝ if n > 0) dimensional space. As such, a Hilbert space can simply be defined as a finite- or infinite-dimensional vector space consisting of square-integrable functions that can be modeled as dot products of one another; such spaces can include typical Euclidean spaces, such as three-dimensional (n = 3) Euclidean vector spaces.

 The Hilbert space, as stated earlier, can generalize the Euclidean to an infinite number of dimensions. In the vectors we shared earlier, a Euclidean vector would have coordinates with n (again, n ∈ ℝ if n > 0) components, yet instead of <d1, d2> in a two-dimensional Euclidean space, Hilbert spaces model that vector as <d1, d2, … d>. An interesting characteristic of the Hilbert space is that it contains infinite sequences of real numbers: think of a vector in three-dimensional Euclidean space, whose coordinates are <x1, y1, z1>; as we generalize the vectors in the Euclidean space, these vectors can be modeled as infinite series of complex numbers, so long as they converge to a finite value. Although wacky and confusing, the Hilbert space is certainly quite entertaining.

Applications in quantum mechanics and mathematics

The Hilbert space is literally everywhere in physics, from the Schrodinger equation to wave functions to virtually every physical value which is modeled as a vector in any environment. 

Wave functions

Let us first consider wave functions; consider this example of a wave function equation

$$\int_{-\infty}^{\infty} \| f(x) \|^2 dx =1$$

Let’s break this up to explain this long function: |f(x)²| is the probability density function, a function whose value at any given point provides a probability that the value of the random variable would equal the point. When considering this wave function, or any other wave function into the Schrodinger Wave Equation, we can either express it as a sine or cosine graph; after solving the Schrodinger Wave Equation for that particular wave function, we may determine the behavior of a quantum system in a force field (a force field is a vector field in which a force acts upon a particle or an object in space–a Hilbert space!).

Graphical models of physical quantities in space

As the Hilbert space is omnipresent in quantum mechanics, so is it in the rest of physics, simply because vector spaces are used in the rest of physics (I even worked with Hilbert spaces just last week as part of my physics II class). In physics, there are scalars (e.g. speed, volume, mass) which have magnitudes, and there are vectors (e.g. velocity, force, field strength) which have magnitude and direction. Hilbert spaces essentially arise naturally in any situation in which an infinite-dimensional function space may exist, and indeed, vector spaces generated by vectors in space lead to the existence of a Hilbert space.

Wrapping it up

Obviously, Hilbert space is not an easy topic; I would say that this was the most difficult entry I have ever written.*** Anyway, the Hilbert space is an infinite-dimensional vector space that was discovered as a generalization of Euclidean space, the fundamental geometric space that many of us understand (at least to a limited extent) already. The Hilbert space is analogous with an infinite-dimensional Euclidean space where the summation of all the coordinates for a particular vector always converges to a finite number. Hilbert spaces have many applications in mathematics and physics; in quantum mechanics, it is responsible for the wave function and the rise of quantum mechanics–it can be said that quantum mechanics is known as well as it is now because of the development of the Hilbert space. And, considering quantum mechanics is indirectly responsible for 25% of the global economy, this is quite significant. The Hilbert space is heavy on advanced calculus and other abstract mathematical applications that I have yet to understand, so, of course, I suggest that you dive into the links in the description so that you can better understand this topic–perhaps even better than I do now. 

I have always had a deep interest in the Hilbert space; over the last five or six months, I have been watching Sean Carroll–the well-known academic proponent of the many-worlds interpretation–and his lectures at Caltech. In these lectures, almost everything Carroll discussed was expressed through the Hilbert space, so I had trouble understanding most of what he said; I mainly tuned into the streams to get exposed to the English of quantum mechanics, as I knew not the math. The study I did on Hilbert space has proven to me that there is yet so much more to learn about quantum mechanics, and even as I arrogantly propagate my knowledge on quantum mechanics to my peers, my knowledge is merely a tiny bulb of light under a radiant star of physical knowledge. Quantum mechanics combines advanced calculus, trigonometry, algebra, and many other mathematical concepts, and I have covered only algebra, calculus, and an introduction to proof-based algebra. There is yet a long way to go in my pursuit of knowledge on quantum mechanics, but I am nevertheless positive; I am excited for what I do not know now, and what I will discover as the years progress. As always, take care and stay curious, everyone.

* They are also known as vector fields, which are specific cases of vector spaces in which each position in the space is accompanied by a vector. Vector functions, which occupy vector spaces but are not vector fields, only have vectors at every point in which they are defined; vector fields, such as the electric field, have vectors for every coordinate in the entire space.

** In the original entry, written in February 2021, I wrote that “I will do my best to explain it in terms of English, though, because I have not even begun calculus, outside of doing some of AP Calc AB online (and Hilbert space relies heavily on integrals, differential equations, and multivariable calculus, none of which I have learned in depth).” Now, however, I am much better able to understand much of the Hilbert space (excluding some of the linear algebra), as I have completed calculus I, II, and III, and I am presently taking linear algebra.

*** Just wait until abstract algebra… this is nothing.

If you have any questions, comments, or corrections, please comment on this post or email with your concerns. Thank you.


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