Background, Introduction, and Overview

of Quantum Field Theory

Background

Branch of QFT Addressed Herein

There are two main branches to (ways to do) quantum field theory called

· the canonical quantization approach, and

· the path integral approach

This web site, at present, addresses only the first of these. It is the easiest way (see reviewer comments below) to be introduced to the subject.

Recommended Text

The best text for beginners is Quantum Field Theory by Mandl and Shaw, and it deals only with the canonical quantization approach. This web site is patterned to some degree after this book, though it is augmented by other pedagogic material. Quotes on Mandl and Shaw from one reviewer:

“Let me put it this way: I consider a serious mistake for any student NOT to use this book as their first book in Quantum Field Theory. This book is the absolute must for any beginner before he or she moves on into the "fancier" books of the field.

I consider it a CRIME for a physics student to start learning Quantum Field Theory with the path integral approach. You loose immediately the physical picture and the particle content of the theory because you are confronted right from the start with mathematical structures that you have never seen and handling them correctly takes away the physics content of the subject. Believe me I have been there! With Mandl and Shaw you will always be close to the quantum of the field, which is the particle, you will see it right in front of your eyes being created, propagated and then annihilated, and you will have a clear picture of what is really going on (quantum theory permitting of course).

I consider Mandl to be one of the most prominent pedagogists in the physics field and I have the utmost respect for him. The reason for this is that I have gained a very strong background in both Quantum Field Theory and Statistical Mechanics just by reading his books.

The serious student of particle physics will eventually have to move on to the path integral approach, renormalization of the electro-weak theory, renormalization group, QCD etc. BUT without having a solid background in the topics included in Mandl's and Shaw’s book this effort will be fruitless and frustrating. Take my word for it.”

For mathematicians, I recommend Quantum Field Theory for Mathematicians, by Robin Ticciati.

Introduction

Why Quantum Field Theory?

The quantum mechanics (QM) courses students take prior to quantum field theory (QFT) generally treat a single particle such as an electron in a potential (e.g., square well, harmonic oscillator, etc.), and the particle retains its integrity (e.g., an electron remains an electron throughout the interaction.) There is no general way to treat an interaction where a particle and its antiparticle annihilate one another to yield neutral particles such as photons (e.g., e + e⁺ → 2γ.) Nor is there any way to describe the decay of an elementary particle such as a muon into other particles (e.g. μ → e + ν + ).

Here is where QFT comes to the rescue. It provides a means whereby particles can be annihilated, created, and transmigrated from one type to another. In so doing, its utility surpasses that provided by ordinary QM.

How Quantum Field Theory?

As an example, consider the interaction between an electron and a positron known as Bhabba scattering shown in Figure 1. At x₂, the electron and positron annihilate one another to produce a photon. At x₁, this photon is transmutated back into an electron and a positron. Antiparticles like positrons are represented by lines with arrows pointing opposite their direction of travel.

Note that we can think of this interaction as an annihilation (destruction) of the electron and the positron at x₂ accompanied with creation of a photon, and then followed by the destruction of the photon accompanied by creation of an electron and positron at x₁. Unlike the electrons and positrons in this example, the photon here is not a “real” particle, but transitory, short-lived, and undetectable, and is called a “virtual” particle (which mediates the interaction between real particles.)

What we seek and what, as students eventually see, QFT delivers, is a mathematical relationship, called a “transition amplitude”, describing a transition (i.e., interaction) of the sort shown pictorially via the “Feynman diagram” of Figure 1. It turns out that the square of the absolute value of the transition amplitude equals the probability of the interaction occurring. This is similar to the square of the absolute value of the wave function in QM equaling the probability of finding the particle.

QFT employs creation and destruction operators acting on states (i.e., kets), and these creation/destruction operators are part of the transition amplitude. We illustrate the general idea with the following grossly oversimplified transition amplitude, reflecting the interaction process of Figure 1. Be cautioned that we have omitted a few more formal, and ultimately essential, ingredients in (1), in order to make it simpler and easier to grasp the fundamental concept.

(1)

In (1), the ket represents the incoming electron and positron at x₂. The bra represents the outgoing electron and positron at x₁. ψ _d is an operator that destroys an electron (at x₂); an operator that destroys a positron (at x₂); ψ_c creates a positron (at x₁); and creates an electron (at x₁.) The comprise photon operators that create (at x₂) and destroy (at x₁) a virtual photon, with the lines underneath indicating that the photon is virtual and propagates from x₂ to x₁. The mathematical procedure and symbolism representing this virtual process is called a “contraction”. When the virtual particle (photon here) is represented as a mathematical function, it is known as the “Feynman propagator” or simply, “the propagator”.

Note what happens to the transition amplitude as we proceed, step-by-step, through the interaction process. At x₂, the incoming particles (in the ket) are destroyed by the destruction operators, so at an intermediate point, we have

, (2)

where as it turns out, the destruction operators leave the vacuum ket with a factor K₂in front of it. The value of this factor is determined by the formal mathematics of QFT.

In (2), the incoming particles have been destroyed and, but for the virtual photon, only the vacuum remains. The virtual photon propagator has a factor associated with it as well, designated below as K_p. The photon then propagates from x₂ to x₁, where it is annihilated, and the creation operators create an electron and positron (ket with factor K₁ in front.) Thus, at x₁ and thereafter, we have

, (3)

where we note that in QFT the bracket of a multiparticle state (inner product of multiparticle state with itself) such as that shown in (3) always equals unity.

Finally, the probability of the interaction occurring turns out to be

. (4)

The quantity K₁K_pK₂ = S_Bhabba arising in (3) depends on particle momenta, spins, and rest masses, as well as the inherent strength of the electromagnetic interaction, all of which one would rightly expect to play a role in the probability of an interaction taking place.

From the interaction probability, scattering cross sections can be calculated.

From Whence Creation and Destruction Operators?

In ordinary QM, the solutions to the relevant wave equation, the Schroedinger equation, are states (particles or kets.) Surprisingly, the solutions to the relevant wave equations in QFT are not states (not particles.) In QFT, it turns out that these solutions are actually operators that create and destroy states. Different solutions exist that create or destroy every type of particle and antiparticle. In this unexpected (and, for students, often strange at first) twist lies the power of QFT.

Overview: The Structure of Physics and QFT’s Place Therein

Students are often confused over the difference (and whether or not there is a difference) between relativistic quantum mechanics (RQM) and quantum field theory (QFT.) The following discussion, summarized in Unifying Chart 1, should help to distinguish them.

In using Unifying Chart 1, the reader should be aware that, depending on context, the term “quantum mechanics” can mean i) only non-relativistic (“ordinary”) quantum mechanics (NRQM), or ii) the entire realm of quantum theories including NRQM, relativistic quantum mechanics (RQM), and QFT. In the chart, we employ the second of these.

Classical mechanics has both a non-relativistic and a relativistic side, and each contains a theory of particles (localized entities, typically pointlike objects) and a theory of fields (entities extended over space and time). All of these are represented in the first row of Unifying Chart 1. Properties (dynamic variables) of entities in classical particle theories are “total” values, such as object mass, charge, energy, momentum, etc. Properties in classical field theories are density values, such as mass density, charge density, field amplitude at a point, etc.

Quantum theories (bottom row in the table) parallel their classical siblings, and can be considered deduced (to some degree) from those siblings via a procedure known as quantization (more formally, “canonical quantization”.) “First quantization” is concerned with particle theories.

First Quantization (Particle Theories)

1) Assume the quantum particle Hamiltonian has the same form as the classical particle Hamiltonian.

2) Replace the classical Poisson brackets with commutator brackets (divided by ih),e.g.,

{x_i, p_j} = δ_ij [x_i, p_j] = ihδ_ij (5)

In so doing, the classical properties (dynamic variables) of position and its conjugate 3-momentum become quantum non-commuting operators.

Second quantization is concerned with field theories.

Second Quantization (Field Theories)

1) Assume the quantum field Hamiltonian density has the same form as the classical field Hamiltonian density.

2) Replace the classical Poisson brackets for conjugate property densities with commutator brackets (divided by ih), e.g.

(6)

where π_s is the conjugate momentum density of the field _s, and different values for r and s mean different fields. In so doing, these classical field dynamic variables become quantum field non-commuting operators (and this has major ramifications for QFT.)

Unifying Chart 1. The Structure of Physics

	Non-relativistic		Relativistic
	Particle	Field	Particle	Field
Classical (non-quantum) mechanics	Newtonian particle theory	Newtonian field theory (continuum mechanics + gravity)	Relativistic macro particle theory	Relativistic macro field theory (continuum mechanics + e/m + gravity)
Properties (Dynamic Variables) Operators	1^st quantization		1^st quantization	2^nd quantization
Quantum mechanics	NRQM (or just “QM”)	No theory generally taught.	RQM	QFT (not gravity)

Note that because quantum field applications almost invariably involve photons or other relativistic particles, there is little real world application for a non-relativistic quantum field theory. Further, a relativistic theory will always suffice for any non-relativistic application. Hence, the QFT we restrict our study to is specifically relativistic.

As an aside, quantum theories of gravity such as superstring theory are not included in the above chart, as QFT in its standard model form cannot accommodate gravity. Thus, the relativity in QFT is special, but not general, relativity.

Conclusions: RQM is similar to NRQM in that both are particle theories. They differ in that RQM is relativistic. RQM and QFT are similar in that both are relativistic theories. They differ in that QFT is a field theory and RQM is a particle theory.

Ramifications: NRQM employs the (non-relativistic) Schroedinger equation, whereas RQM and QFT must employ relativistic counterparts sometimes called “relativistic Schroedinger equations”. Students of QFT soon learn that each spin type (spin 0, spin , and spin 1) has a different relativistic Schroedinger equation. For a given spin type, that equation is the same in RQM and in QFT, and hence both theories have the same form for the solutions to those equations.

The difference between RQM and QFT is in the meaning of those solutions. In RQM, the solutions are interpreted as states (particles, such as an electron), just as in NRQM. In QFT, though it may be initially disorienting to students previously acclimated to NRQM, the solutions turn out not to be states, but rather operators that create and destroy states. It further turns out that while RQM (and NRQM) are amenable primarily to single particle states, QFT can readily accommodate multi-particle states. Further, QFT can handle transmutation of particles from one kind into another (e.g., muons into electrons), whereas NRQM and RQM can not.

Caveats: In spite of the above, there are some contexts in which RQM and QFT may be considered more or less the same theory, in the sense that QFT encompasses RQM. By way of analogy, classical relativistic particle theory is inherent within classical relativistic field theory. For example, one could consider an extended continuum of matter which is very small spatially, integrate the mass density to get total mass, the force/unit mass to get total force, etc., resulting in an analysis of particle dynamics. The field theory contains within it, the particle theory.

In a somewhat similar way, QFT deals with relativistic states (kets), which are essentially the same states dealt with in RQM. QFT, however, is a more extensive theory and can be considered to encompass RQM within its structure. In QFT, for example, one can create and destroy states, which cannot be done in RQM, and which is necessary for analyzing transmigration of particles from one kind into another (by destroying the original particle(s) and creating the final particle(s).)

And in both RQM and QFT (as well as NRQM), operators act on states in similar fashion. For example, the expected energy measurement is determined the same way in both theories, i.e.,

(7)

with similar relations for other observables.

These similarities and differences, as well as others, are summarized in Unifying Chart 2.

Unifying Chart 2. Comparison of Three Theories

	NRQM	RQM	QFT
Wave equation	Schroedinger	Klein-Gordon (spin 0) Dirac (spin ) Proça (spin 1) Special case of Proça: Maxwell (spin 1 massless)	Same as RQM at left
Solutions to wave equation	States	States	Operators that create and destroy states
Particles per state	Single^*	Single^*	Multi-particle
Expection values		As at left, but relativistic.	As at left in RQM.
Phenomena:
1. bound states	Yes, non-relativistic	Yes, relativistic	Yes, but usually not studied in introductory courses
2. scattering
a. elastic	a. Yes	a. Yes	a. Yes
b. inelastic (transmutation)	b. No	b. Yes and no. (i.e., cumbersome, but can be done, though only for particle/antiparticle creation & destruction.)	b. Yes
3. decay
a. composite particles	a. Yes (tunneling)	a. Yes	a. Yes
b. elementary particles	b. No	b. No	b. Yes
4. vacuum fluctuations	No	Yes and no. (Only for simple particle/antiparticle loops.)	Yes
Coordinates
1. Cartesian (plane waves)	Free, 1D potentials, particles in “boxes”	As at left	Used primarily for free particles, particles in “boxes”, and scattering.
2. Spherical (spherical waves)	Bound states and scattering.	As at left.	Not usually used in elementary courses.
Interactions:
1. e/m	No, though can pseudo model for non relativistic cases	As at left	Yes
2. weak	No	No	Yes
3. strong	No^*	No^*	Yes
4. gravity	No	No	Not at present.

*Some caveats exist for this chart. For example, NRQM and RQM can handle certain multiparticle states (e.g. hydrogen atom), but QFT generally does it more easily and more extensively. And the strong force can be modeled in NRQM and RQM by assuming a Yukawa potential, though a truly meaningful handling of the interaction can only be achieved via QFT.

QFT/Quantum Electrodynamics Overview

Unifying Chart 3 below is an outline summary of QFT primarily as applied to quantum electrodynamics (QED), the quantum study of the electromagnetic field and its interactions with matter. QED is the simplest form of QFT and the starting point for the study of QFT.

In my “hoped-to-one-day-complete” text, several chapters of explanation will accompany Unifying Chart 3, though for the time being, none of that is presented here. The chart should help considerably as a holistic nutshell overview of the theory and its development for students using Mandl and Shaw, though it should be helpful to those using other texts as well.

Specific advantages of Unifying Chart 3 include the following.

· The general development of the theory is patently manifest in summary form and not hidden under many pages of unwieldy derivations. This is facilitated by each block being, as a rule, either derived from previous blocks or used to derive subsequent blocks.

· Parallel development of the three spin types allows easy comparison of similarities and differences between scalars (spin 0), fermions (spin ), and vectors (spin 1).

· Review for exams should be more efficient with this summary overview.

· Future referencing to find a particular relationship is rapid much more so than looking it up in texts or old notes.

· The block diagram format has proven pedagogic advantage in computer science, and we are merely applying that concept to physics. There is a world of difference between trying to understand a program by reading it one line at a time or starting with a block diagram overview showing which subsections of the program perform which tasks. In learning the program, continual reference to the block diagram helps immensely. Similar logic applies in physics. One can read the text one line at a time or one can use a block diagram showing central concepts and milestones as one proceeds through the text.

Unifying Chart 3 has 4 parts to it (of which only the first two are completed and shown herein). These are

Part 1. From Field Equations to Propagators and Observables (see comments below)

Part 2. From Operators and Observables to Feynman Rules

Part 3. Scattering and Decay (not completed)

Part 4. Renormalization (not completed)

Note that you can enlarge an equation for easier viewing by clicking once on it. Click again to return it to normal size. Also note that a colon (:) before operators represents normal ordering (destruction operators placed on right hand side and operate first), and that units are chosen such that h = c = 1 (see your text for definition of “natural” units.)

Comment #1 on Part 1 of Unifying Chart 3

Perhaps the most fundamental and far-reaching principle in QFT is step #2 of the 2^nd quantization postulates (see (6)) in which the field and its conjugate momentum do not commute. It is because of this that the solutions to the field equations in QFT contain operator factors that create and destroy particle states. Why this is so is not obvious at this point, so we simply note here that it turns up as the theory is developed.

Comment #2 on Part 1 of Unifying Chart 3

Using 2^nd quantization (see above), we can start by assuming the classical Hamiltonian density, deduce the Lagrangian density, and from that, the field equation (equation of motion of the field.) For example, for a free (no interactions), complex scalar field in classical relativistic theory (with superscript “0” = scalar and subscript “0” = free), the Hamiltonian density is (see Gravitation by Misner, Thorne, and Wheeler (21.34) p. 504 for real scalar field example)

(8)

and from this, the Lagrangian density is

(9)

Thus, from the 4-dimensional Euler-Lagrange equation (with below),

(10)

we obtain the field equation

(11)

In essence then, for step #1 of 2^nd quantization we could begin with any one of the relations (8), (9), or (11), as each is just a different way of saying the same thing. In Unifying Chart 3, we start with the field equation in the first block, rather than the Hamiltonian density, because that is the usual starting point in most texts.

Step #2 of 2^nd quantization, the field commutation relations, is introduced in the chart at the bottom of the first page (or thereabouts depending on your printer.)

Click here to go to Part 1. From Field Equations to Propagators and Observables. It has many equations and can take a little time to download.