Diffusion, Quantum Theory, and Radically Elementary Mathematics. (MN-47) (Mathematical Notes, 47) - Softcover

Softcover

ISBN 10: 0691125457 ISBN 13: 9780691125459

Publisher: Princeton University Press, 2006

Synopsis

Diffusive motion--displacement due to the cumulative effect of irregular fluctuations--has been a fundamental concept in mathematics and physics since Einstein's work on Brownian motion. It is also relevant to understanding various aspects of quantum theory. This book explains diffusive motion and its relation to both nonrelativistic quantum theory and quantum field theory. It shows how diffusive motion concepts lead to a radical reexamination of the structure of mathematical analysis. The book's inspiration is Princeton University mathematics professor Edward Nelson's influential work in probability, functional analysis, nonstandard analysis, stochastic mechanics, and logic. The book can be used as a tutorial or reference, or read for pleasure by anyone interested in the role of mathematics in science. Because of the application of diffusive motion to quantum theory, it will interest physicists as well as mathematicians.


The introductory chapter describes the interrelationships between the various themes, many of which were first brought to light by Edward Nelson. In his writing and conversation, Nelson has always emphasized and relished the human aspect of mathematical endeavor. In his intellectual world, there is no sharp boundary between the mathematical, the cultural, and the spiritual. It is fitting that the final chapter provides a mathematical perspective on musical theory, one that reveals an unexpected connection with some of the book's main themes.

About this title

About the Author

William G. Faris is Professor of Mathematics at the University of Arizona.

Excerpt. © Reprinted by permission. All rights reserved.

Diffusion, Quantum Theory, and Radically Elementary Mathematics

Mathematical Notes 47

By William G. Faris

PRINCETON UNIVERSITY PRESS

Copyright © 2006 Princeton University Press
All rights reserved.

ISBN: 978-0-691-12545-9

Contents

Preface, ix, 
Chapter 1. Introduction: Diffusive Motion and Where It Leads William G. Faris, 1, 
Chapter 2. Hypercontractivity, Logarithmic Sobolev Inequalities, and Applications: A Survey of Surveys Leonard Gross, 45, 
Chapter 3. Ed Nelson's Work in Quantum Theory Barry Simon, 75, 
Chapter 4. Symanzik, Nelson, and Self-Avoiding Walk David C. Brydges, 95, 
Chapter 5. Stochastic Mechanics: A Look Back and a Look Ahead Eric Carlen, . 117, 
Chapter 6. Current Trends in Optimal Transportation: A Tribute to Ed Nelson Cédric Villani, 141, 
Chapter 7. Internal Set Theory and Infinitesimal Random Walks Gregory F. Lawler, 157, 
Chapter 8. Nelson's Work on Logic and Foundations and Other Reflections on the Foundations of Mathematics Samuel R. Buss, 183, 
Chapter 9. Some Musical Groups: Selected Applications of Group Theory in Music Julian Hook, 209, 
Chapter 10. Afterword Edward Nelson, 229, 
Appendix A. Publications by Edward Nelson, 233, 
Index, 241, 

CHAPTER 1

Introduction: Diffusive Motion and Where It Leads

William G. Faris

1.1 DIFFUSION

The purpose of this introductory chapter is to point out the unity in the following chapters. At first this might seem a difficult enterprise. The authors of these chapters treat diffusion theory, quantum mechanics, and quantum field theory, as well as stochastic mechanics, a variant of quantum mechanics based on diffusion ideas. The contributions also include an infinitesimal approach to diffusion and related probability topics, an approach that is radically elementary in the sense that it relies only on simple logical principles. There is further discussion of foundational problems, and there is a final essay on the mathematics of music. What could these have in common, other than that they are in some way connected to the work of Edward Nelson?

In fact, there are important links between these topics, with the apparent exception of the chapter on music. However, the chapter on music is so illuminating, at least to those with some acquaintance with classical music, that it alone may attract many people to this collection. In fact, there is an unexpected connection to the other topics, as will become apparent in the following more detailed discussion.

The plan is to begin with diffusion and then see where this leads.

In ordinary free motion distance is proportional to time:

Δx = vΔt (1.1)

This is sometimes called ballistic motion. Another kind of motion is diffusive motion. The characteristic feature of diffusion is that the motion is random, and distance is proportional to the square root of time:

Δx = ±σ [square root of Δt] (1.2)

As a consequence diffusive motion is irregular and inefficient. The mathematics of diffusive motion in explained in sections 1.1–1.3 of this chapter.

There is a close but subtle relation between diffusion and quantum theory. The characteristic indication of quantum phenomena is the occurrence of the Planck constant [??] in the description. This constant has the dimensions M L2/T of angular momentum. The relation to diffusion derives from

σ2 = [??]/m, (1.3)

where m is the mass of the particle in the quantum system. The diffusion constant σ2 has the appropriate dimensions L2/T for a diffusion; that is, it characterizes a kind of motion where distance squared is proportional to time.

In quantum mechanics it is customary to define the dynamics by quantities expressed in energy units, that is, with dimensions M L2/T2. The determination of the time dynamics involves a division by [??], which changes the units to inverse time units 1/T. In the following exposition energy quantities, such as the potential energy function V(x), will be in inverse time units. This should make the comparison with diffusion theory more transparent.

One connection between quantum theory and diffusion is the relationship between real time in one theory and imaginary time in the other theory. This connection is precise and useful, both in the quantum mechanics of nonrelativistic particles and in quantum field theory. This connection is explored in sections 1.4–1.7.

The marriage of quantum theory and the special relativity theory of Einstein and Minkowski is through quantum field theory. In relativity theory a mass m has an associated momentum me and an associated energy mc2 These define in turn a spatial decay rate

mL = mc/[??] (1.4)

and a time decay rate

mT = mc2/[??]. (1.5)

These set the distance and time scales for quantum fluctuations in relativistic field theory. This theory is related to diffusion in an infinite-dimensional space of Euclidean fields. Some features of this story are explained in sections 1.8–1.10 of this introduction and in the later chapters by Leonard Gross, Barry Simon, and David Brydges.

The passage from real time to imaginary time is convenient but artificial. However, in the domain of nonrelativistic quantum mechanics of particles there is a closer connection between diffusion theory and quantum theory. In stochastic mechanics the real time of quantum mechanics is also the real time of diffusion, and in fact quantum mechanics itself is formulated as conservative diffusion. This subject is sketched in sections 1.11–1.12 of this introduction and in the chapters by Eric Carlen and Cédric Villani.

The conceptual importance of diffusion leads naturally to a closer look at mathematical foundations. In the calculus of Newton and Leibniz, motion on short time and distance scales looks like ballistic motion. This is not true for diffusive motion. On short time and distance scales it looks like the Wiener process, that is, like the Einstein model of Brownian motion. In fact, there are two kinds of calculus for the two kinds of motion, the calculus of Newton and Leibniz for ballistic motion and the calculus of Ito for diffusive motion. The calculus of Newton and Leibniz in its modern form makes use of the concept of limit, and the calculus of Ito relies on limits and on the measure theory framework for probability. However, there is another calculus that can describe either kind of motion and is quite elementary. This is the infinitesimal calculus of Abraham Robinson, where one interprets Δt and Δx as infinitesimal real numbers. It may be that this calculus is particularly suitable for diffusive motion. This idea provides the theme in sections 1.13–1.14 of this introduction and leads to the later contributions by Greg Lawler and Sam Buss. The concluding section 1.15 connects earlier themes with a variation on musical composition, presented in the final chapter by Julian Hook.

1.2 THE WIENER WALK

The Wiener walk is a mathematical object that is transitional between random walk and the Wiener process. Here is the construction of the appropriate simple symmetric random walk. Let [xi]1, ..., [xi]n be a finite sequence of independent random variables, each having the values ±1 with equal probability. One way to construct such random variables is to take the set {-1, 1}n of all sequences [xi] of n values ± 1 and give it the uniform probability measure. Then [xi]k is the kth element in the sequence, and the function [xi] [??][xi]k is the corresponding random variable. The random walk is the sequence sk = [xi]1 + ... + [xi]k defined for 0 ≤ k ≤ n. The underlying probability space in this construction is finite, with 2n points.

Here is the construction of the n-step Wiener walk on the time interval [0, T]. Let Δt = T/n be the time step. Fix the diffusion constant σ2 > 0, and let the corresponding space step be

Δx = σ [square root of Δt]. (1.6)

If

tk = kΔt (1.7)

for k = 0,1,2, ... n, define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.8)

Finally, define w(n)(t) for real t with 0 ≤ t ≤ T by linear interpolation. Then w(n) is a random real continuous function defined on [0, T]. This random function is the Wiener walk with time step Δt = T/n. A typical sample path is illustrated in Figure 1.1.

Let C([0, T]) be the metric space of all real continuous functions on the time interval [0, T]. Let µ(n) be the probability measure induced on Borel subsets of C([0, T]) by the random function w(n). That is, the probability of a Borel subset is the probability that the function w(n) is in this subset. This probability measure µ(n) is the distribution of the Wiener walk. It is concentrated on a finite set of 2n piecewise linear continuous paths.

1.3 THE WIENER PROCESS

The Wiener process is a fundamental object in probability theory, describing a particular kind of random path. Another common name for it is Brownian motion, since it is closely related to the Einstein model for Brownian motion of a physical particle. There are other models of the physical process of Brownian motion, so it is clearer to use "Wiener process" for the mathematical object.

The Wiener process may be constructed in a number of ways, but one way to get an intuition for it is to think of it as a limit of the Wiener walk. In this limit the distribution of the Wiener walk, which is given by binomial probabilities, converges to the distribution of the Wiener process, which is Gaussian.

Proposition 1.1 (Construction of Wiener measure)For each n = 1,2,3, ... let µ(n) be the probability measure defined on Borel subsets of C([O, T]) defined by the Wiener walk with time step Δt satisfying nΔt = T. Then there is a probability measure µ defined on the Borel subsets of C([0, T]) such that µ(n) -> µ as n -> ∞ in the sense of weak convergence of probability measures.

This result may be found in texts on probability. The statement about weak convergence means that for each bounded continuous real function F defined on the space C([0, T]) the expectation ∫ F dµ(n) -> F dµ as n -> ∞. This µ is the Wiener measure with diffusion parameter (T2. 1ft is fixed, then the map w [??] w(t) is a function from the probability space C([0, T]) with Wiener measure µ to the real numbers, and hence is a random variable. For each t ≥ 0 this random variable w(t) has mean zero and variance σ2t. Furthermore, the increments of w corresponding to disjoint time intervals are independent. Since the random variable w(t) is the sum of an arbitrarily large number of independent increments, by the central limit theorem it must have a Gaussian distribution. The random continuous function w associated with the Wiener measure is the Wiener process. A typical sample path is sketched in Figure 1.2.

So far the Wiener process has been defined as a random continuous function on a bounded interval [0, T of time. However, it is not difficult to build the unbounded interval [0, +∞) out of a sequence of bounded intervals and thus give a definition of the Wiener process as a random continuous function on this larger time interval. In fact, it is even possible to define the Wiener process for the time interval (-∞, +∞) as a random continuous function satisfying the normalization w (0) = 0. Henceforth the Wiener process will refer to the probability space C((-∞, +∞)) with the probability measure µ defined in this way.

In the following the expectation of a random variable F defined on the space C -∞, +∞)) with respect to the Wiener measure µ is written

µ[F] = ∫ F dµ. (1.9)

That is, the same notation is used for expectation as for probability. For example, the expectation of w(t) (as a function of w) is µ[w(t) = 0, and the variance is µ[w(t)2 = σ2|t|.

Another useful topic is weighted increments of the Wiener process and the corresponding Wiener stochastic integral. Let t1< t2 and consider the corresponding increment w(t2) - w(t1 This is Gaussian with mean zero and variance O'2(t2 - h) = O'2l[tl' t2JI, which is proportional to the length of the interval. Consider two such increments W(t2) - W(tl) and W(t2) - w(tD. The condition of independent increments implies that they have covariance

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.10)

which is proportional to the length of the intersection of the two intervals.

This generalizes to weighted increments. Let f be a real function such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then the Wiener stochastic integral [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a well-defined Gaussian random variable with mean zero. Furthermore, the condition of independent increments implies that the covariance of two such stochastic integrals is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.11)

The independent increment property (1.10) is the special case when f and g are indicator functions of intervals.

Another description of the Wiener process is by a partial differential equation. For t > 0 let ρ](y, t) (as a function of y) be the probability density of the Wiener process at time t, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.12)

Since the density ρ(y, t) is Gaussian with mean zero and variance σ2t it follows that it satisfies the partial differential equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.13)

This is the simplest diffusion equation (or heat equation).

For a Wiener process describing diffusion in finite-dimensional Euclidean space the probability density is jointly Gaussian. There is also a corresponding partial differential equation, in which the second derivative in the space variable is replaced by the Laplace operator.

Later it will appear that in infinite-dimensional space it is preferable to deal with the Ornstein–Uhlenbeck velocity process instead of the Wiener process. The probability distributions are jointly Gaussian, but they have a more complicated time dependence. They still solve a second-order linear partial differential equation. However, this equation involves the sum of the Laplace operator with another operator, the first-order differential along the direction of a linear vector field.

1.4 DIFFUSION, KILLING, AND QUANTUM MECHANICS

The first remarkable discovery connecting quantum mechanics with diffusion theory is that the fundamental equation of quantum mechanics is closely related to an equation describing diffusion with killing. As we shall see, the connection is through the Feynman–Kac formula.

Quantum theory, of course, is the ultimate mystery of modern science. It has many strange features, such as remarkable correlations over long distances. These correlations are experimentally observed, and their peculiar nature takes mathematical shape in the form of a violation of Bell's inequalities. The appendix to [17] gives an account of this subject and its implications.

However strange quantum mechanics may be, there is universal agreement that the wave function ψ for an isolated system satisfies the Schrödinger equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.14)

Here σ2 = h/m and V(x) is the potential energy, here measured in inverse time units.

This is often stated in terms of the Schrödinger operator H defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.15)

Then the Schrodinger equation (1.14) has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.16)

This way of writing the equation differs slightly from the usual quantum mechanical convention. The usual quantum mechanical potential energy and total energy are obtained from the V and H in the present treatment by multiplication by the constant [??]. This converts inverse time units to energy units. In quantum mechanics the dynamics is defined by dividing energy by [??], therefore returning to inverse time units. So in the present notation, with inverse time units for H, the solution of the SchrMinger equation with initial condition ψ[ITL(x) = ψ(x, 0) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.17)

This operator exponential may be interpreted via spectral theory or by the theory of one-parameter semi groups.

(Continues...)

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Edition details

• Publisher: Princeton University Press
• Publication date: 2006
• Language: English
• ISBN 10: 0691125457
• ISBN 13: 9780691125459
• Binding: Paperback
• Number of pages: 256
• Editor: Faris William G.