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Quantum Field Theory in a Nutshell

Princeton University Press

Chapter One

Who needs quantum field theory?

Quantum field theory arose out of our need to describe the ephemeral nature of life.

No, seriously, quantum field theory is needed when we confront simultaneously the two great physics innovations of the last century of the previous millennium: special relativity and quantum mechanics. Consider a fast moving rocket ship close to light speed. You need special relativity but not quantum mechanics to study its motion. On the other hand, to study a slow moving electron scattering on a proton, you must invoke quantum mechanics, but you don't have to know a thing about special relativity.

It is in the peculiar confluence of special relativity and quantum mechanics that a new set of phenomena arises: Particles can be born and particles can die. It is this matter of birth, life, and death that requires the development of a new subject in physics, that of quantum field theory.

Let me give a heuristic discussion. In quantum mechanics the uncertainty principle tells us that the energy can fluctuate wildly over a small interval of time. According to special relativity, energy can be converted into mass and vice versa. With quantum mechanics and special relativity, the wildly fluctuating energy can metamorphose into mass, that is, into new particles not previously present.

Write down the Schrödinger equation for an electron scattering off a proton. The equation describes the wave function of one electron, and no matter how you shake and bake the mathematics of the partial differential equation, the electron you follow will remain one electron. But special relativity tells us that energy can be converted to matter: If the electron is energetic enough, an electron and a positron ("the antielectron") can be produced. The Schrödinger equation is simply incapable of describing such a phenomenon. Nonrelativistic quantum mechanics must break down.

You saw the need for quantum field theory at another point in your education. Toward the end of a good course on nonrelativistic quantum mechanics the interaction between radiation and atoms is often discussed. You would recall that the electromagnetic field is treated as a field; well, it is a field. Its Fourier components are quantized as a collection of harmonic oscillators, leading to creation and annihilation operators for photons. So there, the electromagnetic field is a quantum field. Meanwhile, the electron is treated as a poor cousin, with a wave function [PSI](x) governed by the good old Schrödinger equation. Photons can be created or annihilated, but not electrons. Quite aside from the experimental fact that electrons and positrons could be created in pairs, it would be intellectually more satisfying to treat electrons and photons, as they are both elementary particles, on the same footing.

So, I was more or less right: Quantum field theory is a response to the ephemeral nature of life.

All of this is rather vague, and one of the purposes of this book is to make these remarks more precise. For the moment, to make these thoughts somewhat more concrete, let us ask where in classical physics we might have encountered something vaguely resembling the birth and death of particles. Think of a mattress, which we idealize as a 2-dimensional lattice of point masses connected to each other by springs (Fig. I.1.1) For simplicity, let us focus on the vertical displacement [which we denote by [q.sub.a](t)] of the point masses and neglect the small horizontal movement. The index a simply tells us which mass we are talking about. The Lagrangian is then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Keeping only the terms quadratic in q (the "harmonic approximation") we have the equations of motion [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking the q's as oscillating with frequency [omega] we have [[SIGMA].sub.b] [k.sub.ab][q.sub.b] = m[[omega].sup.2[q.sub.a]. The eigenfrequencies and eigenmodes are determined, respectively, by the eigenvalues and eigenvectors of the matrix k. As usual, we can form wave packets by superposing eigenmodes. When we quantize the theory, these wave packets behave like particles, in the same way that electromagnetic wave packets when quantized behave like particles called photons.

Since the theory is linear, two wave packets pass right through each other. But once we include the nonlinear terms, namely the terms cubic, quartic, and so forth in the q's in (1), the theory becomes anharmonic. Eigenmodes now couple to each other. A wave packet might decay into two wave packets. When two wave packets come near each other, they scatter and perhaps produce more wave packets. This naturally suggests that the physics of particles can be described in these terms.

Quantum field theory grew out of essentially these sorts of physical ideas.

It struck me as limiting that even after some 75 years, the whole subject of quantum field theory remains rooted in this harmonic paradigm, to use a dreadfully pretentious word. We have not been able to get away from the basic notions of oscillations and wave packets. Indeed, string theory, the heir to quantum field theory, is still firmly founded on this harmonic paradigm. Surely, a brilliant young physicist, perhaps a reader of this book, will take us beyond.

Condensed matter physics

In this book I will focus mainly on relativistic field theory, but let me mention here that one of the great advances in theoretical physics in the last 30 years or so is the increasingly sophisticated use of quantum field theory in condensed matter physics. At first sight this seems rather surprising. After all, a piece of "condensed matter" consists of an enormous swarm of electrons moving nonrelativistically, knocking about among various atomic ions and interacting via the electromagnetic force. Why can't we simply write down a gigantic wave function [PSI] ([x.sub.1], [x.sub.2], ..., [x.sub.N), where [x.sub.j] denotes the position of the jth electron and N is a large but finite number? Okay, [PSI] is a function of many variables but it is still governed by a nonrelativistic Schrödinger equation.

The answer is yes, we can, and indeed that was how solid state physics was first studied in its heroic early days, (and still is in many of its subbranches.)

Why then does a condensed matter theorist need quantum field theory? Again, let us first go for a heuristic discussion, giving an overall impression rather than all the details. In a typical solid, the ions vibrate around their equilibrium lattice positions. This vibrational dynamics is best described by so-called phonons, which correspond more or less to the wave packets in the mattress model described above.

This much you can read about in any standard text on solid state physics. Furthermore, if you have had a course on solid state physics, you would recall that the energy levels available to electrons form bands. When an electron is kicked (by a phonon field say) from a filled band to an empty band, a hole is left behind in the previously filled band. This hole can move about with its own identity as a particle, enjoying a perfectly comfortable existence until another electron comes into the band and annihilates it. Indeed, it was with a picture of this kind that Dirac first conceived of a hole in the "electron sea" as the antiparticle of the electron, the positron.

We will flesh out this heuristic discussion in subsequent chapters.

Marriages

To summarize, quantum field theory was born of the necessity of dealing with the marriage of special relativity and quantum mechanics, just as the new science of string theory is being born of the necessity of dealing with the marriage of general relativity and quantum mechanics.

Chapter Two

The professor's nightmare: a wise guy in the class

As I noted in the preface, I know perfectly well that you are eager to dive into quantum field theory, but first we have to review the path integral formalism of quantum mechanics. This formalism is not universally taught in introductory courses on quantum mechanics, but even if you have been exposed to it, this chapter will serve as a useful review. The reason I start with the path integral formalism is that it offers a particularly convenient way of going from quantum mechanics to quantum field theory. I will first give a heuristic discussion, to be followed by a more formal mathematical treatment.

Perhaps the best way to introduce the path integral formalism is by telling a story, certainly apocryphal as many physics stories are. Long ago, in a quantum mechanics class, the professor droned on and on about the double-slit experiment, giving the standard treatment. A particle emitted from a source S (Fig. I.2.1) at time t = 0 passes through one or the other of two holes, [A.sub.1] and [A.sub.2], drilled in a screen and is detected at time t = T by a detector located at O. The amplitude for detection is given by a fundamental postulate of quantum mechanics, the superposition principle, as the sum of the amplitude for the particle to propagate from the source S through the hole [A.sub.1] and then onward to the point O and the amplitude for the particle to propagate from the source S through the hole [A.sub.2] and then onward to the point O.

Suddenly, a very bright student, let us call him Feynman, asked, "Professor, what if we drill a third hole in the screen?" The professor replied, "Clearly, the amplitude for the particle to be detected at the point O is now given by the sum of three amplitudes, the amplitude for the particle to propagate from the source S through the hole [A.sub.1] and then onward to the point O, the amplitude for the particle to propagate from the source S through the hole [A.sub.2] and then onward to the point O, and the amplitude for the particle to propagate from the source S through the hole [A.sub.3] and then onward to the point O."

The professor was just about ready to continue when Feynman interjected again, "What if I drill a fourth and a fifth hole in the screen?" Now the professor is visibly losing his patience: "All right, wise guy, I think it is obvious to the whole class that we just sum over all the holes."

To make what the professor said precise, denote the amplitude for the particle to propagate from the source S through the hole [A.sub.i] and then onward to the point O as A(S [right arrow] [A.sub.i] [right arrow] O). Then the amplitude for the particle to be detected at the point O is

A(detected at O)= [summation over (i)] A(S [right arrow] [A.sub.i] [right arrow] O) (1)

But Feynman persisted, "What if we now add another screen (Fig. I.2.2) with some holes drilled in it?" The professor was really losing his patience: "Look, can't you see that you just take the amplitude to go from the source S to the hole [A.sub.i] in the first screen, then to the hole [B.sub.j] in the second screen, then to the detector at O, and then sum over all i and j?"

Feynman continued to pester, "What if I put in a third screen, a fourth screen, eh? What if I put in a screen and drill an infinite number of holes in it so that the screen is no longer there?" The professor sighed, "Let's move on; there is a lot of material to cover in this course."

But dear reader, surely you see what that wise guy Feynman was driving at. I especially enjoy his observation that if you put in a screen and drill an infinite number of holes in it, then that screen is not really there. Very Zen! What Feynman showed is that even if there were just empty space between the source and the detector, the amplitude for the particle to propagate from the source to the detector is the sum of the amplitudes for the particle to go through each one of the holes in each one of the (nonexistent) screens. In other words, we have to sum over the amplitude for the particle to propagate from the source to the detector following all possible paths between the source and the detector (Fig. I.2.3).

A(particle to go from S to O in time T) =

[summation over ((paths))] A(particle to go from S to O in time T following a particular path) (2)

Now the mathematically rigorous will surely get anxious over how [[SIGMA].sub.(paths)] is to be defined. Feynman followed Newton and Leibniz: Take a path (Fig. I.2.4), approximate it by straight line segments, and let the segments go to zero. You can see that this is just like filling up a space with screens spaced infinitesimally close to each other, with an infinite number of holes drilled in each screen.

Fine, but how to construct the amplitude A(particle to go from S to O in time T following a particular path)? Well, we can use the unitarity of quantum mechanics: If we know the amplitude for each infinitesimal segment, then we just multiply them together to get the amplitude of the whole path.

In quantum mechanics, the amplitude to propagate from a point [q.sub.I] to a point [q.sub.F] in time T is governed by the unitary operator [e.sup.-iHT], where H is the Hamiltonian. More precisely, denoting by |q> the state in which the particle is at q, the amplitude in question is just <[q.sub.F]|[e.sup.iHT]|[q.sub.I]>. Here we are using the Dirac bra and ket notation. Of course, philosophically, you can argue that to say the amplitude is <[q.sub.F]|[e.sup.-iHT]|[q.sub.I]> amounts to a postulate and a definition of H. It is then up to experimentalists to discover that H is hermitean, has the form of the classical Hamiltonian, et cetera.

Indeed, the whole path integral formalism could be written down mathematically starting with the quantity <[q.sub.F]|e.sup.-iHT]|[q.sub.I]>, without any of Feynman's jive about screens with an infinite number of holes. Many physicists would prefer a mathematical treatment without the talk. As a matter of fact, the path integral formalism was invented by Dirac precisely in this way, long before Feynman.

A necessary word about notation even though it interrupts the narrative flow: We denote the coordinates transverse to the axis connecting the source to the detector by q, rather than x, for a reason which will emerge in a later chapter. For notational simplicity, we will think of q as 1-dimensional and suppress the coordinate along the axis connecting the source to the detector.

Continues...

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Excerpted from Quantum Field Theory in a Nutshell by A. Zee Copyright © 2003 by Princeton University Press. Excerpted by permission.
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