Quanta and fields, p.1
Quanta and Fields, page 1
ALSO BY SEAN CARROLL
The Biggest Ideas in the Universe
Something Deeply Hidden
The Big Picture
The Particle at the End of the Universe
From Eternity to Here
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Copyright © 2024 by Sean Carroll
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Library of Congress Cataloging-in-Publication Data
Names: Carroll, Sean M., 1966– author.
Title: Quanta and fields: the biggest ideas in the universe / Sean Carroll.
Description: New York : Dutton, [2024] | Includes index.
Identifiers: LCCN 2023052127 (print) | LCCN 2023052128 (ebook) | ISBN 9780593186602 (hardcover) | ISBN 9780593186619 (ebook)
Subjects: LCSH: Quantum theory—Popular works. | Physics—Popular works. | AMS: Quantum theory—Instructional exposition (textbooks, tutorial papers, etc.).
Classification: LCC QC174.123 .C37 2024 (print) | LCC QC174.123 (ebook) | DDC 530.12—dc23/eng/20240223
LC record available at https://lccn.loc.gov/2023052127
LC ebook record available at https://lccn.loc.gov/2023052128
Ebook ISBN 9780593186619
Cover design by Jason Booher
Cover photo of Image of Uranus taken by Voyager 2 in 1986/NASA
book design by tiffany estreicher, adapted for ebook by molly jeszke
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To Ariel
CONTENTS
Introduction
One: Wave Functions
Two: Measurement
Three: Entanglement
Four: Fields
Five: Interactions
Six: Effective Field Theory
Seven: Scale
Eight: Symmetry
Nine: Gauge Theory
Ten: Phases
Eleven: Matter
Twelve: Atoms
Appendix: Fourier Transforms
Index
About the Author
_147003904_
INTRODUCTION
The history of physics has witnessed a number of brilliant, transformative ideas. But when it comes to truly revolutionary shifts—changes of paradigm that upend the way we think about the nature of reality—there have really only been two: classical mechanics in the late seventeenth century, and quantum mechanics in the early twentieth.
Classical mechanics was the theme of The Biggest Ideas in the Universe: Space, Time, and Motion, in which we emphasized the idea of the Newtonian/Laplacian paradigm, all the way up through spacetime and relativity. Now it’s time for us to go quantum.
Quantum mechanics is, according to our best current understanding, the way the world works. The first hints of the need for a change came from work by Max Planck and Albert Einstein that indicated light was not merely a wave, as physicists had previously thought. In the right circumstances, light comes in particles we now call photons. These particles are an example of the quanta of the title—discrete bundles of energy emerging out of the rules of quantum mechanics. But it’s subtler than that. Under different circumstances, things we think of as particles, like electrons and protons and neutrons, exhibit wave-like behavior. Quantum mechanics is going to continually frustrate our desire to put the behavior of physical systems into neat, commonsensical boxes.
Don’t feel bad if the ideas of quantum mechanics seem alien at first. The truth is that physicists themselves don’t agree on what, at rock bottom, quantum mechanics actually says. Physicists are extremely good at using quantum mechanics. We can predict the structure of atoms and molecules or calculate the scattering of particles off each other with exquisite precision. But it’s a bit of a black box. The top quantum physicists in the world don’t agree on what is going on to produce the results they predict and observe so successfully.
This lack of intellectual consensus can be traced to the fact that quantum mechanics seems to attribute special properties to the act of “measuring” or “observing” a physical system. In classical physics, objects have properties like positions and velocities, and you can directly measure them with as much accuracy as you like. Quantum objects seem profoundly different. Measuring the properties of a quantum system tends to dramatically change those properties. In some perfectly reasonable ways of thinking about the theory, a particle such as an electron doesn’t even have properties like “position” or “momentum”—those are possible measurement outcomes, not intrinsic features of the quantum system itself.
For the most part, we’re not going to worry about any of that.[*] Here at the Biggest Ideas in the Universe, our attitude is that of hard-nosed physicists, using well-established ideas to make testable predictions about the world. That will give us more than enough to chew on. The foundational issues are indisputably important; understanding quantum mechanics at a deep level might very well turn out to be crucial to pushing beyond our current theories toward a much more comprehensive picture of reality. But the focus of this book will be on understanding the concepts underlying those current theories, and how they have given us an unprecedentedly accurate picture of the physical world.
The concepts in question include quantum mechanics itself; quantum field theory, which arises naturally when one combines quantum mechanics with the requirements of special relativity; and various deep ideas that have arisen within quantum field theory, including Feynman diagrams, renormalization, gauge theories, symmetry breaking, and the spin-statistics connection.
It’s an enormous amount of material, which I’ve endeavored to boil down to its bare essence. The trick, as with the previous book, is that we are going to include enough mathematical specifics to understand the ideas for real, without reaching a level of detail required to solve problems in the manner of a graduate student studying for their doctorate. You will learn the same ideas that they will, but you won’t have to pull all-nighters doing problem sets.
That requires a slightly different strategy than we employed in Space, Time, and Motion, although the basic aspiration is the same. In that book, I could literally show you all of the equations exactly as a professional would learn them. Here, there is just too much information to make that workable. Quantum field theory is loaded down with nitpicky details and layers of notation, which can get in the way of focusing on the central ideas. And it’s the ideas that matter to us. So there will be times when we will ignore coupling constants, hide indices, treat matrices like numbers. I promise you that this is all in the service of helping you understand what’s really going on, not in obscuring it.
Still, there is going to be math. In Space, Time, and Motion we introduced the basic ideas of calculus, including derivatives (rates of change) and integrals (accumulated amounts of change). In later chapters we dealt with tensors, and the use of Greek letters to denote spacetime indices. All of those are going to be here, in force. As well as the basic physics ideas of mass, energy, and relativity. If you are already familiar with those concepts, this book will be entirely self-contained. If not, Space, Time, and Motion should convey everything you need to know.
It will be a breathtaking ride. At the dawn of the twentieth century, classical mechanics was firmly entrenched. Twenty-five years later, we saw the first complete formulations of quantum mechanics. Twenty-five years after that, quantum electrodynamics was the first well-established quantum field theory. And twenty-five years after that, physicists had put together the Standard Model of particle physics, which remains triumphant to this day. That’s the journey on which we are embarking, and it involves some of the most amazing ideas human beings have ever come across.
* * *
–
I have once again been extraordinarily fortunate to receive detailed feedback on drafts of this book. Enormous gratitude goes to Scott Aaronson, Justin Clarke-Doane, Ira Rothstein, and Matt Strassler, who kept my physics honest and my explanations not as convoluted as they would otherwise have been. My agent, Katinka Matson, has provided sage advice along the way. And huge thanks to my editor, Stephen Morrow, who has been patient and understanding and singularly helpful in shaping this book series into something I hope people will learn from and enjoy.
Patience was especially called for this time around, as the writing process was interrupted by a cross-country move and beginning a position at Johns Hopkins. Thanks to all my new colleagues and students for making everything as smooth as possible and
Most of all, thanks to my wife, Jennifer, who picked up and moved with me, shouldered most of the burden of shaping our new home, and always keeps my writing honest. Looking forward to this new and exciting chapter.
The plot of the Mexican-hat potential in Chapter 10 is adapted from a Mathematica code by Vitaliy Kaurov (https://mathematica.stackexchange.com/questions/19578/how-can-i-make-a-plot-of-the-higgs-potential). The image of the LIGO observatory in Hanford in Chapter 11 is from the LIGO collaboration (https://www.ligo.org/multimedia/gallery/lho-images/Aerial5.jpg). The plot of nuclides in Chapter 12 is based on an example from Learning Scientific Programming with Python by Christian Hill (https://scipython.com/).
ONE
WAVE FUNCTIONS
As the nineteenth century drew to a close, you would have forgiven physicists for hoping that they were on track to understand everything. The universe, according to this tentative picture, was made of particles that were pushed around by fields.
The idea of fields filling space had really taken off over the course of the 1800s. Earlier, Isaac Newton had presented a beautiful and compelling theory of motion and gravity, and Pierre-Simon Laplace had shown how we could reformulate that theory in terms of a gravitational field stretching between every object in the universe. A field is just something that has a value at each point in space. The value could be a simple number, or it could be a vector or something more complicated, but any field exists everywhere through space.
But if all you cared about was gravity, the field seemed optional—a point of view you could choose to take or not, depending on your preferences. It was equally okay to think as Newton did, directly in terms of the force created on one object by the gravitational pull of others without anything stretching between them.
That changed in the nineteenth century, as physicists came to grips with electricity and magnetism. Electrically charged objects exert forces on each other, which is natural to attribute to the existence of an electric field stretching between them. Experiments by Michael Faraday showed that a moving magnet could induce electrical current in a wire without actually touching it, pointing to the existence of a separate magnetic field, and James Clerk Maxwell managed to combine these two kinds of fields into single a theory of electromagnetism, published in 1873. This was an enormous triumph of unification, explaining a diverse set of electrical and magnetic phenomena in a compact theory. “Maxwell’s equations” bedevil undergraduate physics students to this very day.
One of the triumphant implications of Maxwell’s theory was an understanding of the nature of light. Rather than a distinct kind of substance, light is a propagating wave in the electric and magnetic fields, also known as electromagnetic radiation. We think of electromagnetism as a “force,” and it is, but Maxwell taught us that fields carrying forces can vibrate, and in the case of electric and magnetic fields those vibrations are what we perceive as light. The quanta of light are particles called photons, so we will sometimes say “photons carry the electromagnetic force.” But at the moment we’re still thinking classically.
Take a single charged particle, like an electron. Left sitting by itself, it will have an electric field surrounding it, with lines of force pointing toward the electron. The force will fall off as an inverse-square law, just as in Newtonian gravity.[*] If we move the electron, two things happen: First, a charge in motion creates a magnetic field as well as an electric one. Second, the existing electric field will adjust how it is oriented in space, so that it remains pointing toward the particle. And together, these two effects (small magnetic field, small deviation in the existing electric field) ripple outward, like waves from a pebble thrown into a pond. Maxwell found that the speed of these ripples is precisely the speed of light—because it is light. Light, of any wavelength from radio to x-rays and gamma rays, is a propagating vibration in the electric and magnetic fields. Almost all the light you see around you right now has its origin in a charged particle being jiggled somewhere, whether it’s in the filament of a lightbulb or the surface of the sun.
Simultaneously in the nineteenth century, the role of particles was also becoming clear. Chemists, led by John Dalton, championed the idea that matter was made of individual atoms, with one specific kind of atom associated with each chemical element. Physicists belatedly caught on, once they realized that thinking of gases as collections of bouncing atoms could explain things like temperature, pressure, and entropy.
But the term “atom,” borrowed from the ancient Greek idea of an indivisible elementary unit of matter, turned out to be a bit premature. Though they are the building blocks of chemical elements, modern-day atoms are not indivisible. A quick-and-dirty overview, with details to be filled in later: atoms consist of a nucleus made of protons and neutrons, surrounded by orbiting electrons. Protons have positive electrical charge, neutrons have zero charge, and electrons have negative charge. We can make a neutral atom if we have equal numbers of protons and electrons, since their electrical charges will cancel each other out. Protons and neutrons have roughly the same mass, with neutrons being just a bit heavier, but electrons are much lighter, about 1⁄1,800th the mass of a proton. So most of the mass in a person or another macroscopic object comes from the protons and neutrons. The lightweight electrons are more able to move around and are therefore responsible for chemical reactions as well as the flow of electricity. These days we know that protons and neutrons are themselves made of smaller particles called quarks, which are held together by gluons, but there was no hint of that in the early 1900s.
This picture of atoms was put together gradually. Electrons were discovered in 1897 by British physicist J. J. Thompson, who measured their charge and established that they were much lighter than atoms. So somehow there must be two components in an atom: the lightweight, negatively charged electrons, and a heavier, positively charged piece. A few years later Thompson suggested a picture in which tiny electrons floated within a larger, positively charged volume. This came to be called the plum pudding model, with electrons playing the role of the plums.
The plum pudding model didn’t flourish for long. A famous experiment by Ernest Rutherford, Hans Geiger, and Ernest Marsden shot alpha particles (now known to be nuclei of helium atoms) at a thin sheet of gold foil. The expectation was that they would mostly pass right through, with their trajectories slightly deflected if they happened to pass through an atom and interact with the electrons (the plums) or the diffuse positively charged blob (the pudding). Electrons are too light to disturb the alpha particles’ trajectories, and a spread-out positive charge would be too diffuse to have much effect. But what happened was, while most of the particles did indeed zip through unaffected, some bounced off at wild angles, even straight back. That could only happen if there was something heavy and substantial for the particles to carom off of. In 1911 Rutherford correctly explained this result by positing that the positive charge was concentrated in a massive central nucleus. When an incoming alpha particle was lucky enough to score a direct hit on the small but heavy nucleus, it would be deflected at a sharp angle, which is what was observed. In 1920 Rutherford proposed the existence of protons (which were just hydrogen nuclei, so had already been discovered), and in 1921 he theorized the existence of neutrons (which were eventually discovered in 1932).
So far, so good, thinks our imagined fin de siècle physicist. Matter is made of particles, the particles interact via forces, and those forces are carried by fields. The entire mechanism would run according to rules established by the framework of classical physics. For particles this is pretty familiar: we specify the positions and the momenta of all the particles, then use one of our classical techniques (Newton’s laws or their equivalent) to describe their dynamics. Fields work in essentially the same way, except that the “position” of a field is its value at every point in space, and its “momentum” is how fast it’s changing at every point. The overall classical picture applies in either case.
