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Bring two electrons close together, and they repel each other. Nineteenth-century theories explained such effects in terms of fields, which are often represented as lines of force that emanate from an electron and extend throughout space. The field produced by each particle repels other particles that have the same electric charge and attracts those with the opposite charge.

QED is a quantum field theory, and it takes a different view of the forces between charged particles. In QED electrons interact by emitting and absorbing photons, which are the quanta, or carriers, of the electromagnetic field. It is the exchange of photons that accounts for attractive and repulsive forces.

Subject Index

Thus all those ethereal fields permeating the universe are replaced by localized events—namely the emission or absorption of a photon at a specific place and time. The theory allows for some wilder events as well. Field theories describe forces between particles. In classical theories lines of force emanate from one particle and terminate on the other. In electromagnetism upper left the lines spread throughout space; the force between quarks upper right is confined to a flux tube.

Quantum field theories reinterpret these observations: Forces arise from the exchange of carrier particles, or quanta, represented here as wavy lines. In quantum electrodynamics or QED lower left the carriers are photons; in quantum chromodynamics, QCD, lower right they are gluons. Gluons can even emit and absorb other gluons. QCD is also a quantum field theory; it describes the same kinds of events, but with a different cast of characters.

Where QED is a theory of electrically charge particles, QCD applies to particles that have a property called color charge hence the name chromo dynamics. And forces in QCD are transmitted not by photons but by particles known as gluons, the quanta of the color field. The lattice version of a quantum field theory places electrons or quarks at the nodes of a rectilinear grid and confines force carriers to the links between nodes. Lattice QCD gave the first theoretical evidence that quarks cannot escape their confinement. There are at least three major differences between the theories.

First, the electric charges of QED come in just two polarities positive and negative , but there are three varieties of color charge usually labeled red, green and blue. Second, the photons that carry the electromagnetic force are themselves electrically neutral; gluons not only carry the color force but also have color of their own. As a result, gluons respond to the very force they carry. Finally, the color force is intrinsically stronger than electromagnetism. Electromagnetism follows an inverse-square law: The force between electrically charged particles falls off rapidly with increasing distance.

In contrast, the force between color-charged quarks and gluons remains constant at long distances. Furthermore, it's quite a strong force, equal to about 14 tons. A constant force means the energy needed to separate two quarks grows without limit as you pull them apart.

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For this reason we never see a quark in isolation; quarks are confined to the interior of protons and neutrons and the other composite particles known as hadrons. A theory in physics is supposed to be more than just a qualitative description; you ought to be able to use it to make predictive calculations. For example, Newton's theory of gravitation predicts the positions of planets in the sky. Likewise QED allows for predictive calculations in its realm of electrons and photons. Suppose you want to know the probability that a photon will travel from one point to another.

For calculations of this kind Richard Feynman introduced a scheme known as the sum-over-paths method. The idea is to consider every possible path the photon might take and then add up contributions from each of the alternatives. This is rather like booking an airplane trip from Boston to Seattle. You could take a direct flight, or you might stop over in Chicago or Minneapolis—or maybe even Buenos Aires.

In QED, each such path is associated with a number called an amplitude; the overall probability of getting from Boston to Seattle is found by summing all the amplitudes, then squaring the result and taking the absolute value. The trick here is that the amplitudes are complex numbers—with real and imaginary parts—which means that in the summing process some amplitudes cancel others.

Another complication is that a photon has infinitely many paths to choose from, but there are mathematical tools for handling those infinities. A more elaborate application of QED is calculating the interaction between two electrons: You need to sum up all the ways that the electrons could emit and absorb photons. The simplest possibility is the exchange of a single photon, but events involving two or more photons can't be ruled out.

Indeed, the variety of interaction mechanisms is limitless. Nevertheless, QED can calculate the interaction probability to very high accuracy. Because these terms are very small, the one-photon exchange dominates the interaction. This style of calculation—summing a series of progressively smaller terms—is known as a perturbative method. In principle, the same scheme can be applied in QCD to predict the behavior of quarks and gluons; in practice, it doesn't work out quite so smoothly.

One problem comes from the color charge of the gluons.

Whereas a photon cannot emit or absorb another photon, a gluon, being charged, can emit and absorb gluons. This self-interaction multiplies the number of possible pathways. Because this number is close to 1, all possible gluon exchanges make roughly the same contribution to the overall interaction. The single-gluon event can still be taken as the starting point for a calculation, but the subsequent terms are not small corrections; they are just as large as the first term. The series doesn't converge; if you were to try summing the whole thing, the answer would be infinite.

In one respect the situation is not quite as bleak as this analysis suggests. The strength of the coupling varies as a function of distance. The customary unit of distance in this realm is the fermi, equal to 1 femtometer, or 10 —15 meter; a fermi is roughly the diameter of a proton or a neutron. If you measure the color force at distances of less than 0. The "constant" grows rapidly, however, as the distance increases. As a result of this variation in the coupling constant, quarks move around freely when they are close together but begin to exert powerful restraining forces as their separation grows.

This is the underlying mechanism of quark confinement. Because the color coupling gets weaker at short distances, perturbative methods can be made to work at close range. In an experimental setting, probing a particle at close range requires high energy. Thus perturbative QCD can tell us about the behavior of quarks in the most violent environments in the universe—such as the collision zones at the Large Hadron Collider now revving near Geneva. But the perturbative theory fails if we want to know about the quarks in ordinary matter at lower energy. Understanding the low-energy or long-range properties of quark matter is the problem that lattice QCD was invented to address, starting in the mids.

A number of physicists had a hand in developing the technique, but the key figure was Kenneth G. Wilson, now of Ohio State University. It's not an accident that Wilson had been working on problems in solid-state physics and statistical mechanics, where many systems come equipped with a natural lattice, namely that of a crystal. Introducing an artificial lattice of discrete points is a common strategy for simplifying physical problems. For example, models for weather forecasting establish a grid of points in latitude, longitude and altitude where variables such as temperature and wind direction are evaluated.

In QCD the lattice is four-dimensional: Each node represents both a point in space and an instant in time. Thus a particle standing still in space hops along the lattice parallel to the time axis. It needs to be emphasized that the lattice in QCD is an artificial construct, just as it is in a weather model.

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No one is suggesting that spacetime really has such a rectilinear gridlike structure. To get rigorous results from lattice studies, you have to consider the limiting behavior as the lattice spacing a goes to zero. But there are many interesting approximate results that do not require taking the limit. One obvious advantage of a lattice is that it helps to tame infinities. In continuous spacetime, quarks and gluons can roam anywhere; even with a finite number of particles, the system has infinitely many possible states. If a lattice has a finite number of nodes and links, the number of quark-and-gluon configurations has a definite bound.

In principle, you can enumerate all states. As it turns out, however, the finite number of configurations is not the biggest benefit of introducing a lattice. More important is enforcing a minimum dimension—namely the lattice spacing a. By eliminating all interactions at distances less than a, the lattice tames a different and more pernicious type of infinity, one where the energy of individual interactions grows without bound. The most celebrated result of lattice QCD came at the very beginning. The mathematical framework of QCD itself without the lattice was formulated in about ; this work included the idea that quarks become "asymptotically free" at close range and suggested the hypothesis of confinement at longer range.

Just a year later Wilson published evidence of confinement based on a lattice model. What he showed was that color fields on the lattice do not spread out in the way that electromagnetic fields do. As quarks are pulled apart, the color field between them is concentrated in a narrow "flux tube" that maintains a constant cross section. The energy of the flux tube is proportional to its length. Long before the tube reaches macroscopic length, there is enough energy to create a new quark-antiquark pair.

The result is that isolated quarks are never seen in the wild; only collections of quarks that are color-neutral can be detected. Other approaches to particle physics require mastery of some very challenging mathematics, but the lattice methods looked like something I could get a grip on—something discrete and finite, where computing the state of a quantum system would be a matter of filling in columns and rows of numbers.

Those early hopes ended in disappointment. I soon learned that lattice QCD does not bring all of quantum field theory down to the level of spreadsheet arithmetic. There is still heavy-duty mathematics to be done, along with a great deal of heavy-duty computing. Nevertheless, I continue to believe that the lattice version of the weird quantum world is easier to grasp than any other.

My conviction has been reinforced by the discovery of an article, "Lattice QCD for Novices," published 10 years ago by G. Peter Lepage of Cornell University. Lepage doesn't offer lattice QCD in an Excel spreadsheet, but he does present an implementation written in the Python programming language. The entire program fits in a page or two. Lepage's lattice model for novices has just one space dimension as well as a time dimension; in other words, it describes particles moving back and forth along a line segment.

And what the program simulates isn't really a quantum field theory; there are no operators for the creation and annihilation of particles. All the same, reading the source code for the program gives an inside view of how a lattice model works, even if the model is only a toy. At the lowest level is a routine to generate thousands of random paths, or configurations, in the lattice, weighted according to their likelihood under the particular rule that governs the physical evolution of the system.

General Physics (so even mathematicians can understand it!)

The ideas of Lagrangians, Hamiltonians, state spaces, operators and Feynman path integrals are demonstrated to be the mathematical underpinning of quantum field theory, and which are employed to formulate a comprehensive mathematical theory of asset pricing as well as of interest rates, which are validated by empirical evidence. Numerical algorithms and simulations are applied to the study of asset pricing models as well as of nonlinear interest rates.

A range of economic and financial topics are shown to have quantum mechanical formulations, including options, coupon bonds, nonlinear interest rates, risky bonds and the microeconomic action functional. This is an invaluable resource for experts in quantitative finance and in mathematics who have no specialist knowledge of quantum field theory. Categories: New book proposals.

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In this book, the history of the concepts critical to the discovery and development of aluminum, its alloys and the anodizing process are reviewed to provide a foundation for the challenges, achievements, and understanding of the complex relationship between the aluminum alloy and the reactions that occur during anodic oxidation. Empirical knowledge that has long sustained industrial anodizing is clarified by viewing the process as corrosion science, addressing each element of the anodizing circuit in terms of the Tafel Equation. This innovative approach enables a new level of understanding and engineering control for the mechanisms that occur as the oxide nucleates and grows, developing its characteristic highly ordered structure, which impact the practical function of the anodic aluminum oxide.

This book presents and discusses the construction of mathematical models that describe phenomena of flow and transport in porous media as encountered in civil and environmental engineering, petroleum and agricultural engineering, as well as chemical and geothermal engineering. The phenomena of transport of extensive quantities, like mass of fluid phases, mass of chemical species dissolved in fluid phases, momentum and energy of the solid matrix and of fluid phases occupying the void space of porous medium domains are encountered in all these disciplines. The book, which can also serve as a text for courses on modeling in these disciplines, starts from first principles and focuses on the construction of well-posed mathematical models that describe all these transport phenomena.

An exploration of the ethics of practical engineering through analyses of eighteen rich case studies The Ethical Engineer explores ethical issues that arise in engineering practice, from technology transfer to privacy protection to whistle-blowing. After examining the extent to which the actions of engineers in the cases align with the FEREs, McGinn recapitulates key ideas used in analyzing the cases and spells out the main lessons they suggest.

He identifies technical, social, and personal factors that induce or press engineers to engage in misconduct and discusses organizational, legal, and individual resources available to those interested in ethically responsible engineering practice. Combining probing analysis and nuanced ethical evaluation of engineering conduct in its social and technical contexts, The Ethical Engineer will be invaluable to engineering students and professionals. Meets the need for engineering-related ethics studyElaborates four fundamental ethical responsibilities of engineersDiscusses diverse, global cases of ethical issues in established and emerging engineering fieldsIdentifies resources and options for ethically responsible engineering practiceProvides discussion questions for each case.

The internationally bestselling author of The Hidden Life of Trees shows how we can decipher nature's secret signs by studying the weather. In this first-ever English translation of The Weather Detective, Peter Wohlleben uses his long experience and deep love of nature to help decipher the weather and our local environments in a completely new and compelling way. Analyzing the explanations for everyday questions and mysteries surrounding weather and natural phenomena, he delves into a new and intriguing world of scientific investigation.

At what temperature do bees stay home? Why do southerly winds in winter often bring storms?

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