“Why does it have that name?”

I’ve asked in seminars, in lectures, in offices, and at group meetings. I’ve asked about physical conjectures, about theorems, and about mathematical properties.

“I don’t know.” Lecturers have shrugged. “It’s just a name.”


This spring, I asked about master equations. I thought of them as tools used in statistical mechanics, the study of vast numbers of particles. We can’t measure vast numbers of particles, so we can’t learn about stat-mech systems everything one might want to know. The magma beneath Santorini, for example, consists of about 1024 molecules. Good luck measuring every one.


Imagine, as another example, using a quantum computer to solve a problem. We load information by initializing the computer to a certain state: We orient the computer’s particles in certain directions. We run a program, then read out the output.

Suppose the computer sits on a tabletop, exposed to the air like leftover casserole no one wants to save for tomorrow. Air molecules bounce off the computer, becoming entangled with the hardware. This entanglement, or quantum correlation, alters the computer’s state, just as flies alter a casserole.* To understand the computer’s output—which depends on the state, which depends on the air—we must have a description of the air. But we can’t measure all those air molecules, just as we can’t measure all the molecules in Santorini’s magma.


We can package our knowledge about the computer’s state into a mathematical object, called a density operator, labeled by ρ(t). A quantum master equation describes how ρ(t) changes. I had no idea, till this spring, why we call master equations “master equations.” Had someone named “John Master” invented them? Had the inspiration for the Russell Crowe movie Master and Commander? Or the Igor who lisps, “Yeth, mathter” in adaptations of Frankenstein?


Jenia Mozgunov, a fellow student and Preskillite, proposed an answer: Using master equations, we can calculate how averages of observable properties change. Imagine describing a laser, a cavity that spews out light. A master equation reveals how the average number of photons (particles of light) in the cavity changes. We want to predict these averages because experimentalists measure them. Because master equations spawn many predictions—many equations—they merit the label “master.”

Jenia’s hypothesis appealed to me, but I wanted certainty. I wanted Truth. I opened my laptop and navigated to Facebook.

“Does anyone know,” I wrote in my status, “why master equations are called ‘master equations’?”

Ian Durham, a physicist at St. Anselm College, cited Tom Moore’s Six Ideas that Shaped Physics. Most physics problems, Ian wrote, involve “some overarching principle.” Example principles include energy conservation and invariance under discrete translations (the system looks the same after you step in some direction). A master equation encapsulates this principle.

Ian’s explanation sounded sensible. But fewer people “liked” his reply on Facebook than “liked” a quip by a college friend: Master equations deserve their name because “[t]hey didn’t complete all the requirements for the doctorate.”

My advisor, John Preskill, dug through two to three books, one set of lecture notes, one German Wikipedia page, one to two articles, and Google Scholar. He concluded that Nordsieck, Lamb, and Uhlenbeck coined “master equation.” According to a 1940 paper of theirs,** “When the probabilities of the elementary processes are known, one can write down a continuity equation for W [a set of probabilities], from which all other equations can be derived and which we will call therefore the ‘master’ equation.”

“Are you sure you were meant to be a physicist,” I asked John, “rather than a historian?”

“Procrastination is a powerful motivator,” he replied.

Lecturers have shrugged at questions about names. Then they’ve paused, pondered, and begun, “I guess because…” Theorems and identities derive their names from symmetries, proof techniques, geometric illustrations, and applications to problems I’d thought unrelated. A name taught me about uses for master equations. Names reveal physics I wouldn’t learn without asking about names. Names aren’t just names. They’re lamps and guides.

Pity about the origin of “master equation,” though. I wish an Igor had invented them.


*Apologies if I’ve spoiled your appetite.

**A. Nordsieck, W. E. Lamb, and G. E. Uhlenbeck, “On the theory of cosmic-ray showers I,” Physica 7, 344-60 (1940), p. 353.

This entry was posted in Reflections, Theoretical highlights by Nicole Yunger Halpern. Bookmark the permalink.

About Nicole Yunger Halpern

I’m a theoretical physicist at the Joint Center for Quantum Information and Computer Science in Maryland. My research group re-envisions 19th-century thermodynamics for the 21st century, using the mathematical toolkit of quantum information theory. We then apply quantum thermodynamics as a lens through which to view the rest of science. I call this research “quantum steampunk,” after the steampunk genre of art and literature that juxtaposes Victorian settings (à la thermodynamics) with futuristic technologies (à la quantum information). For more information, check out my upcoming book Quantum Steampunk: The Physics of Yesterday’s Tomorrow. I earned my PhD at Caltech under John Preskill’s auspices; one of my life goals is to be the subject of one of his famous (if not Pullitzer-worthy) poems. Follow me on Twitter @nicoleyh11.

8 thoughts on “Hello, my name is QUANTUM MASTER EQUATION

  1. Haha I was also very curious about why master equation earns its name. Great exploration! Yeah, as Nordsieck, Lamb, and Uhlenbeck mentioned, master equation describes the most fundamental law of the transitions between microscopic states, from which arises the macroscopic random (stochastic) process. And, I believe those who fully understand master equations deserve doctorate degrees 🙂

  2. It’s great fun, and mighty thought-provoking too, to have Google’s Ngram Viewer plot simultaneously the rising prevalence of “Fermi’s Golden Rule” and the “Pauli master equation”.

    How did two jocular phrases that prior to WWII were known only to a few dozen specialists became known to tens of thousands of physicists during the post-WWII physics boom?

    Every quantum physicist discovers a Great Truth connected to the Fermi and Pauli Great Rules:

    • ‘Fermi’s Golden Rule’ and the ‘Pauli master equation’ neatly encapsulate physical insights that are essential to practical calculations — that’s why every quantum physicist knows them, and that’s how young quantum physicists learn them.

    • It’s very nearly impossible to explain Fermi’s Golden Rule and the Pauli master equation to professional mathematicians.

    Explanation  Physicists and engineers (and their textbooks) ground their appreciation of Fermi’s Golden Rule and the Pauli master equation in physical intuition and practical examples. Mathematicians prefer to ground their understanding in universal objects and natural transformations … and yet lamentably, there are at present (to my knowledge) no mathematician-friendly introductory textbooks in statistical mechanics and thermodynamics that maximally substitute considerations of universality and naturality for considerations of physicality.

    Here ‘universal’ and ‘natural’ are understood in the not-too-complicated sensu stricto of introductory texts like Steve Awodey’s Category Theory, Bill Lawvere’s Conceptual Mathematics, and Tom Leinster’s Basic Category Theory

    Differing expectations grounded in differing language — the language of physicality versus the language of universality and naturality — explain why great 20th century mathematicians like Saunders Mac Lane and Vladimir Arnol’d so commonly lamented

    “It has taken me over fifty years to understand the derivation of Hamilton’s equations”
      — Saunders Mac Lane

    “Every mathematician knows that it is impossible to understand any elementary course in thermodynamics.”
      — Vladimir Arnol’d

    The good news is, nowadays these inessential 20th century barriers to understanding are dissolving rapidly … as reflected in the Moore’s Law growth of the terms ‘naturality’ and ‘universality’ on the arxiv.

    Conclusion  Young 21st century mathematicians and scientists (and even young engineers) have nothing to lose — and a vast universe to gain in professional and creative opportunities — by systematically transcribing their old-fashioned (let’s face it!) 20th century physical understanding of quantum statistical mechanics and thermodynamics into lively 21st century terms of universality and naturality.

  3. I always dreamed of becoming a theorist, who does not have to worry about obnoxious little details; “The magma beneath Santorini, for example, consists of about 10^24 molecules.” So do 2 grams of hydrogen, or 18 grams of water.

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