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Supplemental Lecture 14

Introduction to the Foundations of Quantum Field Theory

For Physics Students

I. Particles and Anti-Particles

Abstract

This Essay, “Particles and Anti-Particles” explains how locality, causality and special relativity

imply, in the context of quantum field theory, that each charged particle must be accompanied by

an anti-particle of opposite charge but equal mass. The Essay consists of two parts:

“Background” which presents preliminary ideas on non-relativistic harmonic motion by

introducing creation and annihilation operators, illustrating them with coherent states, and then

presents “second quantization” in the context of non-relativistic many body quantum mechanics.

These ideas set the stage for the next portion of the Essay which presents the primary result, that

relativistic quantum field theory predicts the existence of anti-particles. We will see that current

conservation, locality and causality are the crucial ingredients here.

The prerequisites for these Essays are: 1. An understanding of special relativity at the level of the

textbook, and 2. An undergraduate physics course on quantum mechanics. The fundamentals of

quantum field theory will be developed within these Essays.

This Essay supplements material in the textbook: Special Relativity, Electrodynamics and

General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The

term “textbook” in these Supplemental Lectures will refer to that work.

Keywords: Harmonic Oscillator, Creation and Annihilation Operators, Heisenberg Uncertainty

Relation, Coherent States, Second Quantization, Field Operators, Relativistic Quantum Field

Theory, Charge Conservation, Current Conservation, Causality, Particles and Anti-Particles.

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Contents

One Dimensional Harmonic Oscillator ........................................................................................2

Creation and Annihilation Operators ........................................................................................2

Coherent States ........................................................................................................................7

Second Quantization ................................................................................................................. 13

Relativistic Quantum Fields and Anti-Particles ......................................................................... 20

Relativistic Fields and Operators ........................................................................................... 20

Lorentz Transformations of Fields, States and Operators ....................................................... 27

Charged Relativistic Fields .................................................................................................... 29

References ................................................................................................................................ 32

One Dimensional Harmonic Oscillator

Creation and Annihilation Operators

The basic building block of a local, causal, relativistic quantum field theory is the

creation/annihilation operator. We can illustrate the properties of these operators in the context of

the non-relativistic, one-dimensional harmonic oscillator.

Consider a point particle of mass m attached to a spring with a spring constant k which provides

the potential energy,

𝑉(𝑥) =1

2𝑘𝑥2 1.1a

To simplify the notation, introduce the frequency 𝜔, 𝜔 = √𝑘/𝑚, so that Eq. 1.1a becomes,

𝑉(𝑥) =1

2𝑚𝜔2𝑥2 1.1b

and 𝜔 will prove to be the characteristic frequency of the harmonic motion of the classical

system, such as 𝑥(𝑡) = 𝑥0 sin(𝜔𝑡 + 𝛿0).

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Our task here is to find the spectrum of the Hamiltonian, 𝐻 = −ℏ2

2𝑚

𝑑2

𝑑𝑥2 +1

2𝑚𝜔2𝑥2, using

operator methods. In your quantum mechanics course, you solved the Schrodinger Equation [1],

𝐻Ψ = (−ℏ2

2𝑚

𝑑2

𝑑𝑥2 +1

2𝑚𝜔2𝑥2) Ψ = 𝐸Ψ 1.2a

for stationary states using differential equation techniques. This approach hides the universal,

fundamental features of the problem that will guide us in all our developments in quantum field

theory.

In the operator approach we try to diagonalize the Hamiltonian operator directly,

𝐻 =1

2𝑚[(

ℏ

𝑖

𝑑

𝑑𝑥)

2

+ (𝑚𝜔𝑥)2] =1

2𝑚[𝑝2 + (𝑚𝜔𝑥)2] 1.2b

Classically we can factor 𝐻,

𝑝2 + (𝑚𝜔𝑥)2 → (𝑚𝜔𝑥 − 𝑖𝑝)(𝑚𝜔𝑥 + 𝑖𝑝) Classically 1.3a

But quantum mechanically p and x do not commute. Instead, the basic commutator of quantum

mechanics reads [𝑝, 𝑥] = −𝑖ℏ, so Eq. 1.3a is replaced with,

𝑝2 + (𝑚𝜔𝑥)2 = (𝑚𝜔𝑥 − 𝑖𝑝)(𝑚𝜔𝑥 + 𝑖𝑝) + 𝑖[𝑝, 𝑚𝜔𝑥] 1.3b

So, if we define creation and annihilation operators,

𝑎† =1

√2𝑚𝜔ℏ(𝑚𝜔𝑥 − 𝑖𝑝) 1.4a

𝑎 =1

√2𝑚𝜔ℏ(𝑚𝜔𝑥 + 𝑖𝑝) 1.4b

Then,

𝐻 = ℏ𝜔 (𝑎†𝑎 +1

2) 1.5

The terminology “creation” and “annihilation” will become apparent soon. We can write the

Hermitian observables x and p in terms of 𝑎† and a,

𝑥 = √ℏ

2𝑚𝜔(𝑎 + 𝑎†) 𝑝 =

1

𝑖√

𝑚𝜔ℏ

2(𝑎 − 𝑎†) 1.6

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The basic commutator of a and 𝑎† follows from [𝑝, 𝑥] = −𝑖ℏ,

[𝑎, 𝑎†] = 1 1.7

Our next task is to derive the properties of a and 𝑎†. We will see that the spectrum of H consists

of states labelled by a non-negative integer n, |𝑛 >, each having an energy 𝐸 = ℏ𝜔 (𝑛 +1

2). Of

particular note is the fact that the lowest energy state, |0 >, the “ground” state, has non-zero

energy 𝐸0 =1

2 ℏ𝜔.

In order to discover the spectrum of H, Eq. 1.5, the stationary states, we need the properties of

the “number operator”,

𝑁 = 𝑎†𝑎 1.8

It is convenient to focus on the commutators of N, a and 𝑎†. First, calculate

[𝑁, 𝑎†] = 𝑎† [𝑁, 𝑎] = −𝑎 1.9

The derivation of these relations follows from Eq. 1.7, 𝑎†𝑎 = 𝑎𝑎† − 1,

(𝑎†𝑎)𝑎 = (𝑎𝑎† − 1)𝑎 = 𝑎(𝑎†𝑎) − 𝑎

which gives the second entry in Eq. 1.9. The spectrum of H follows immediately. Label the

eigenstates of N as,

𝑁|𝑛 >= 𝑛|𝑛 > 1.10

where we shall show that n must be a non-negative integer. To see this consider,

𝑛 < 𝑛|𝑛 >=< 𝑛|𝑁|𝑛 >=< 𝑛|𝑎†𝑎|𝑛 >=< 𝜒|𝜒 > ≥ 0 1.11

where we called 𝑎|𝑛 > = |𝜒 > and noted that the norm is positive semi-definite in the last

inequality. The right-hand side vanishes only for 𝑛 = 0 which implies 𝑎|0 > = 0. This is the

ground state for the harmonic oscillator with the energy 𝐸0 =1

2 ℏ𝜔.

To make an excited state of H we can apply 𝑎† to |0 >. The energy of the resultant state is

calculated using Eq. 1.9. Consider,

𝑁(𝑎†|0 >) = (𝑎†𝑁 + 𝑎†)|0 >= (𝑎†|0 >)

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So,

|1 >= 𝑎†|0 > , 𝐸1 = ℏ𝜔 (1 +1

2) 1.12

In general |𝑛 > is proportional to (𝑎†)𝑛|0 > but (𝑎†)𝑛|0 > is not normalized to unit probability.

To see this, let’s calculate the norm of 𝑎†|𝑛 >,

(𝑎†|𝑛 >)†(𝑎†|𝑛 >) =< 𝑛|𝑎𝑎†|𝑛 >=< 𝑛|(𝑎†𝑎 + 1)|𝑛 >= 𝑛 + 1 1.13

where we took |𝑛 > to be normalized < 𝑛|𝑛 > = 1. We learn that,

𝑎†|𝑛 > = √𝑛 + 1 |𝑛 + 1 > 1.14

Similarly,

𝑎|𝑛 > = √𝑛 |𝑛 − 1 > 1.15

Iterating Eq.1.14,

|𝑛 > =1

√𝑛𝑎†| 𝑛 − 1 > =

1

√𝑛(𝑛−1)(𝑎†)2|𝑛 − 2 > =

1

√𝑛!(𝑎†)𝑛|0 > 1.16

Using Eq.1.16 and the definition of the creation operator 𝑎†, Eq. 1.4a, we can obtain explicit

expressions for all the eigenfunctions 𝜓𝑛(𝑥) =< 𝑥|𝑛 >. We first find |0 > from the condition

𝑎|0 > = 0, which reads,

1

√2𝑚𝜔ℏ(𝑚𝜔𝑥 + 𝑖

ℏ

𝑖

𝑑

𝑑𝑥) 𝜓0(𝑥) = 0 1.17a

which produces the differential equation,

(𝑚𝜔

ℏ𝑥 +

𝑑

𝑑𝑥) 𝜓0(𝑥) = 0 1.17b

which is solved by a Gaussian, 𝜓0(𝑥)~𝑒𝑥𝑝(𝑎𝑥2). In detail,

𝜓0(𝑥) = √𝑚𝜔

𝜋ℏ

4 𝑒𝑥𝑝 (−

𝑚𝜔

2ℏ𝑥2) 1.18

where the pre-factor guarantees that ∫[𝜓0(𝑥)]2 𝑑𝑥 = 1.

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Given 𝜓0(𝑥) we can calculate (𝑎†)𝑛𝜓0(𝑥) using Eq. 1.4a again. The result will clearly be a

polynomial of degree n multiplied by the universal Gaussian 𝑒𝑥𝑝 (−𝑚𝜔

2ℏ𝑥2). The polynomials

are the famous Hermite polynomials whose formulas, properties and recursion relations follow

from the algebra of the creation and annihilation operators 𝑎† and a.

Now let’s reconsider the ground state and the fact that its energy is not zero. Instead, it has a

“zero point energy”, 𝐸0 =1

2 ℏ𝜔. We saw that 𝐸0 =

1

2 ℏ𝜔 appears in Eq. 1.5 because the position

and momentum operators do not commute, in other words, they are not simultaneously

diagonalizable. The Heisenberg uncertainty relation is at play here. The potential has its

minimum at 𝑥 = 0, so the wave function of the ground state will be localized in that vicinity.

Suppose that region has a width ∆𝑥. By the Heisenberg uncertainty relation, the spread in the

momentum of the particle in the ground state will be ∆𝑥∆𝑝 ≥ ℏ 2⁄ . Then we can estimate the

energy of the ground state from the Hamiltonian Eq. 1.2b,

𝐻 =1

2𝑚[𝑝2 + (𝑚𝜔𝑥)2]~

1

2𝑚[(

ℏ 2⁄

∆𝑥)

2

+ (𝑚𝜔∆𝑥)2] 1.19

Eq. 1.19 estimates the ground state energy as a function of its spread in x: if ∆𝑥 is too small the

first term dominates and H grows large and if ∆𝑥 is too large the second term dominates and

again H grows large. The minimum is somewhere in between. Eq. 1.19 is minimized where

𝜕𝐻 𝜕∆𝑥⁄ = 0, for ∆𝑥 = √ℏ 2⁄

𝑚𝜔 where the minimum is

1

2ℏ𝜔, the exact ground state energy.

We will see the importance of the non-zero ground state energy 𝐸0 =1

2 ℏ𝜔 in later discussions

of quantum field theory.

Another informative exercise is the comparison of highly excited states and classical physics.

Consider a classical harmonic oscillator and initiate its motion with a given amplitude

𝑥(𝑡 = 0) = 𝑥0 and vanishing velocity, 𝑣0 = 0. Then it oscillates 𝑥(𝑡) = 𝑥0 cos(𝜔𝑡) and its total

energy is 𝐸𝐶𝑙𝑎𝑠 =1

2𝑚𝜔2𝑥0

2,

𝐸𝐶𝑙𝑎𝑠 =1

2𝑚[𝑝2 + (𝑚𝜔𝑥)2] =

1

2𝑚[𝑚2𝜔2𝑥0

2 sin2(𝜔𝑡) + 𝑚2𝜔2𝑥02 cos2(𝜔𝑡)] =

1

2𝑚𝜔2𝑥0

2 1.20

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Compare this to the energy of the quantum oscillator in the state n , 𝐸𝑛 = ℏ𝜔 (𝑛 +1

2). Relating

the two expressions, we see how to translate “amplitude” to “number of excitations”,

𝑥02 = (2𝑛 + 1)

ℏ

𝑚𝜔 1.21

√ℏ

𝑚𝜔 is the characteristic extent, or “size”, of the ground state, |0 >, 𝐸𝑞. 1.18. The dependence

here is 𝑥02~𝑛. When we discuss the quantization of the electromagnetic field, expressions of this

sort will help us identify the energy carried by a wave of definite frequency with the excitation

number n and the amplitude of the wave with a creation operator 𝑎†.

Coherent States

Besides stationary states |𝑛 >, there are other complete sets of states of the harmonic oscillator

that are important in physical applications. For example, there are “coherent states”, of great

importance in the field of optics. These are minimal uncertainty states which oscillate in time but

maintain their shape. Hence the name “coherent”.

We will introduce these states in an unconventional fashion, following our interest in gravitation

and accelerating reference frames. This discussion will lead to later discussions of quantum

fields in accelerating reference frames (Rindler space). So, let’s place the harmonic oscillator

into a gravitational field g so the harmonic potential becomes,

𝑉 =1

2𝑚𝜔2𝑥2 → 𝑉𝑔 = 𝑉 + 𝑔𝑚𝑥 1.22

The classical physics of the model is simple: the spring elongates to a new equilibrium position

displaced form the origin, but the oscillations around the new point are unchanged from the

original problem. A less trivial problem is the relation between the quantum stationary states in

the gravitational field and the original states where the gravitational field vanishes. If we

complete the square in the potential 𝑉𝑔(𝑥) in Eq. 1.22,

𝑉𝑔 =1

2𝑚𝜔2𝑥2 + 𝑔𝑚𝑥 =

1

2𝑚𝜔2 (𝑥 +

𝑔

𝜔2)2

−𝑔2𝑚

2𝜔2 1.23

So, define the shifted variable,

𝑦 = 𝑥 +𝑔

𝜔2 𝑑

𝑑𝑦=

𝑑

𝑑𝑥 1.24

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So,

𝐻 = −ℏ2

2𝑚

𝑑2

𝑑𝑥2 +1

2𝑚𝜔2𝑥2 1.25a

and,

𝐻𝑔 = −ℏ2

2𝑚

𝑑2

𝑑𝑦2 +1

2𝑚𝜔2𝑦2 −

𝑔2𝑚

2𝜔2 = 𝐻 −𝑔2𝑚

2𝜔2 1.25b

In addition, the creation and annihilation operators for the two problems are related by a constant

g-dependent shift,

𝑎 =1

√2𝑚𝜔ℏ(𝑚𝜔𝑥 + 𝑖𝑝) 1.26a

and,

𝑏 =1

√2𝑚𝜔ℏ(𝑚𝜔𝑦 + 𝑖𝑝) = 𝑎 + √

𝑚

2𝜔3ℏ 𝑔 1.26b

In this language,

𝐻 = ℏ𝜔 (𝑎†𝑎 +1

2) |𝑛 > =

1

√𝑛!(𝑎†)𝑛|0 > 1.27a

𝐻𝑔 = ℏ𝜔 (𝑏†𝑏 +1

2) −

𝑔2𝑚

2𝜔2 |𝑛 >𝑔 = 1

√𝑛!(𝑏†)𝑛|0 >𝑔 1.27b

So, the ground state for the original problem satisfies,

𝑎|0 > = 0 1.28a

And in terms of the annihilation operator for the problem in the gravitational field, 𝑏 = 𝑎 +

√𝑚

2𝜔3ℏ 𝑔,

𝑏 |0 > = +√𝑚

2𝜔3ℏ 𝑔| 0 > 1.28b

In summary, the ground state of H is an eigenstate of the annihilation operator “a” with

eigenvalue zero which is also an eigenstate of the annihilation operator “b” with eigenvalue

+√𝑚

2𝜔3ℏ 𝑔. States of this variety are called “coherent”. Here is the definition: ” A coherent state

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|𝛼 > is an eigenstate of an annihilation operator a with eigenvalue 𝛼, 𝑎|𝛼 > = 𝛼|𝛼 > ". We

have already observed that the ground state |0 > is a coherent state with 𝛼 = 0. This state is

special because it is also an eigenstate of H. But in general coherent states are not energy

eigenstates: their mean energy is,

< 𝛼|𝐻|𝛼 > = ℏ𝜔 < 𝛼 |𝑎†𝑎 +1

2| 𝛼 > = ℏ𝜔 (|𝛼|2 +

1

2) 1.29

The first term on the right-hand side reminds us of the classical result that the energy stored in a

spring varies as its displacement squared, or that the energy in an electromagnetic field varies as

the sum of the squares of the electric and magnetic field strengths.

Note that the eigenvalue 𝛼 can be a complex number. This is possible because it is an eigenvalue

of a non-Hermitian operator. In most cases in this Essay 𝛼 will be a real number but treating 𝛼

more generally as complex has applications in dynamics and optics.

Next, our discussion of the harmonic oscillator in a gravitational field suggests that the coherent

state |𝛼 > should be related to the ground state |0 > by a displacement. To see this, expand |𝛼 >

in terns of {|𝑛 >, 𝑛 = 0,1,2, … }, a complete set of orthonormal states,

|𝛼 > = ∑ |𝑛 >< 𝑛|𝛼 >𝑛 1.30a

But,

|𝑛 > = 1

√𝑛!(𝑎†)𝑛|0 > 1.30b

so,

< 𝑛 |𝛼 > = 1

√𝑛!< 0| 𝑎𝑛 |𝛼 > =

𝛼𝑛

√𝑛!< 0| 𝛼 > 1.30c

giving,

|𝛼 > =< 0|𝛼 > ∑𝛼𝑛

√𝑛!|𝑛 >𝑛 1.30d

Finally, the prefactor < 0|𝛼 > is obtained from the normalization condition,

1 = ∑ < 𝛼|𝑛 >< 𝑛|𝛼 > = ⌈< 0|𝛼 >⌉2𝑛 ∑

⌈𝛼⌉2𝑛

𝑛!𝑛 = ⌈< 0|𝛼 >⌉2𝑒|𝛼|2 1.31a

So,

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< 0|𝛼 > = 𝑒−1

2|𝛼|2

1.31b

Collecting everything,

|𝛼 > = 𝑒−1

2|𝛼|2

∑𝛼𝑛

√𝑛!|𝑛 >𝑛 1.32

But we can combine Eq. 1.30b with 1.32 to write an operator identity,

∑𝛼𝑛

√𝑛!|𝑛 >𝑛 = ∑

𝛼𝑛

𝑛!𝑛 (𝑎†)𝑛|0 > = 𝑒𝛼𝑎†|0 > 1.33

which implies,

|𝛼 > = 𝑒−1

2 |𝛼|2+𝛼𝑎†

|0 > 1.34

In order to write Eq. 1.34 in a fully general form, recall the Baker-Hausdorff identity that you

learned in your quantum mechanics course [1],

𝑒𝐴+𝐵 = 𝑒−1

2[𝐴,𝐵]𝑒𝐴𝑒𝐵 1.35

which is true if the commutator [𝐴, 𝐵] is a complex number, rather than an operator, so it

commutes with A and B. This identity allows us to write,

|𝛼 > = 𝑒𝛼𝑎†−𝛼∗𝑎 |0 > = 𝐷(𝛼)|0 > 1.36

where 𝐷(𝛼) is the “displacement” operator we were after. Now we have a convenient way to

write down a coherent state. The displacement operator has several important properties that are

easy to prove. (In most cases they follow from the commutator [𝑎, 𝑎†] = 1, the Baker-Hausdorff

identity and the expansion of exponential operators, 𝑒𝐴 = ∑1

𝑛!𝑛 (𝐴)𝑛.)

1. 𝐷†(𝛼) = 𝐷−1(𝛼) = 𝐷(−𝛼)

2. 𝐷†(𝛼)𝑎 𝐷(𝛼) = 𝑎 + 𝛼

3. 𝐷†(𝛼)𝑎† 𝐷(𝛼) = 𝑎† + 𝛼∗

4. 𝐷(𝛼 + 𝛽) = 𝐷(𝛼)𝐷(𝛽)𝑒−𝑖𝐼𝑚(𝛼𝛽∗) 1.37

Note that properties #1-3 state that D provides a representation of translations. In particular, if 𝛼

is a real number, then

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𝐷(𝛼) = 𝑒𝛼𝑎†−𝛼∗𝑎 = 𝑒𝛼(𝑎†−𝑎) = 𝑒𝑥𝑝 (𝑖√2ℏ

𝑚𝜔 𝛼 ∙ 𝑝/ℏ)

where we used Eq. 1.5 to identify the momentum operator. This is the standard form of the

translation operator, with the momentum acting as its generator, that you learned in quantum

mechanics [1]. Furthermore, in the case that 𝛼 is a real number properties #2 and #3 imply,

𝐷†(𝛼) 𝑥 𝐷(𝛼) = 𝑥 + 2√ℏ

2𝑚𝜔𝛼 𝐷†(𝛼) 𝑝 𝐷(𝛼) = 𝑝

These results underscore the interpretation we have already given to coherent states.

Next, consider the ground state of the harmonic oscillator in the presence of the gravitational

field. Call it |0 >𝑔. It is defined by, using Eq 1.26b,

𝑏|0 >𝑔= 0 = 𝑎|0 >𝑔+ 𝑐|0 >𝑔 1.38

where 𝑐 = √𝑚

2𝜔3ℏ 𝑔. But if | − 𝑐 > is a coherent state, then 𝑎|−𝑐 > = −𝑐| − 𝑐 >, so |0 >𝑔 , the

ground state of the stretched harmonic oscillator, is a coherent state of the original problem,

|0 >𝑔 = | − 𝑐 > = 𝑒−1

2 |𝑐|2−𝑐𝑎†

|0 > 1.39

We learn that the ground state of the stretched harmonic oscillator is a linear superposition of the

excited states of the original harmonic oscillator. The probability that |0 >𝑔 is the nth excited

state of H is given by the Poisson distribution,

𝑃(𝑛) = |< 𝑛|0 >𝑔|2

=𝑐2𝑛

𝑛!𝑒−𝑐2

1.40

It is interesting that the mean excitation number in the state |0 >𝑔 is,

�̅� =< 0𝑔|𝑎†𝑎|0𝑔 > = |𝑐|2 =𝑚

2𝜔3ℏ∙ 𝑔2 =

1

2∙

𝑚𝜔

ℏ∙

𝑔2

𝜔4 1.41

Finally, let’s consider the time evolution of the coherent state. Let’s work directly in x-space.

The nth eigenstate, 𝜓𝑛(𝑥) evolves in time as,

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𝜓𝑛(𝑡, 𝑥) = 𝜓𝑛(𝑥)𝑒−𝑖

ℏ𝐸𝑛𝑡 = 𝜓𝑛(𝑥)𝑒−𝑖𝑛𝜔𝑡𝑒

−𝑖𝜔𝑡

2 1.42

Then using Eq. 1.32, the x-space coherent state wave function is, at time t,

𝜑𝛼(𝑡, 𝑥) = 𝑒−1

2|𝛼|2

𝑒−𝑖𝜔𝑡

2 ∑(𝛼𝑒−𝑖𝜔𝑡)

𝑛

√𝑛!𝑛 𝜓𝑛(𝑥) 1.43a

This formula suggests that we define a time dependent 𝛼(𝑡) = 𝛼𝑒−𝑖𝜔𝑡 . Then Eq. 1.43a can be

written in the suggestive form,

𝜑𝛼(𝑡, 𝑥) = 𝜑𝛼(𝑡)(𝑥)𝑒−𝑖𝜔𝑡

2 1.43b

Since |𝛼 > is a displacement of the ground state |0 >, Eq. 1.43b indicates that the time variation

of a coherent state is particularly simple: since 𝛼(𝑡) has sinusoidal variation with characteristic

frequency 𝜔, the coherent state oscillates with that frequency and maintains the shape of the

ground state (hence the term “coherent”).

Let’s check some of these points. First calculate the expectation value of x in the state Eq. 1.43b,

using Eq. 1.5,

�̅�𝛼(𝑡) =< 𝜑𝛼(𝑡)|𝑥|𝜑𝛼(𝑡) >= 𝑥0 < 𝜑𝛼(𝑡)|𝑎 + 𝑎†|𝜑𝛼(𝑡) >= 𝑥0(𝛼(𝑡) + 𝛼∗(𝑡)) = √2𝑥0|𝛼| cos 𝜔𝑡

where 𝑥0 =√(ℏ/2mω).

Finally, using the same techniques, we can generalize Eq. 1.36 to time t and find,

|𝛼(𝑡) > = 𝑒𝑥𝑝(−𝑖𝜔𝑡/2) 𝐷(𝛼(𝑡))|0 > 1.44a

whose probability density can be written explicitly in x-space,

|𝜑𝛼(𝑡, 𝑥)|2 =1

√𝜋 𝑥0exp (−

(𝑥−�̅�𝛼(𝑡))2

𝑥02 ) 1.44b

which has all the properties discussed above: for all t the coherent state is a minimal uncertainty

Gaussian wave packet: it does not spread in time and it maintains ∆𝑥∆𝑝 = ℏ/2.

These results will serve us well in later Essays on quantum field theory.

13

Second Quantization

Creation and annihilation operators are also very useful for formulating multi-particle, non-

relativistic quantum mechanics. The operators allow us to describe the transitions of particles

between different single particle states. Of course, non-relativistic particles are neither created or

annihilated in non-relativistic interactions, but the resulting formalism is much clearer and

economical than traditional formulations where n-body wave functions must be written down

explicitly. Each particle is conserved by the non-relativistic potential it experiences and the total

number of particles is conserved as a consequence.

Consider an n-particle system and describe it with single particle states |𝜆 >, where 𝜆 is a

complete list of compatible quantum numbers. Suppose there are 𝑛1 particles in state |𝜆′ >, 𝑛2

particles in state |𝜆′′ >, etc., with the total number of particles n fixed, 𝑛 = 𝑛1 + 𝑛2 + ⋯ . A n-

particle state can be labelled with “occupation numbers”, |𝑛1, 𝑛2, … , 𝑛𝑖, … > with 𝑛 = ∑ 𝑛𝑖𝑖 . This

notation means that there are 𝑛𝑖 identical scalar particles ( “Bose” particles) in the single particle

state |𝜆(𝑖) >. The wave function of the single particle state is denoted < �⃗� |𝜆(𝑖) > which might

be an atomic orbital, or a state of definite momentum 𝒩 𝑒𝑖�⃗⃗�(𝑖)∙𝑥 , etc. We introduce creation and

annihilation operators following the harmonic oscillator formalism. So,

𝑎𝑖|𝑛1, 𝑛2, … , 𝑛𝑖 , … > = √𝑛𝑖 |𝑛1, 𝑛2, … , 𝑛𝑖 − 1, … > 2.1a

and

𝑎𝑖†|𝑛1, 𝑛2, … , 𝑛𝑖, … > = √𝑛𝑖 + 1 |𝑛1, 𝑛2, … , 𝑛𝑖 + 1, … > 2.1b

These relations imply the commutation relations of 𝑎𝑖 and 𝑎𝑗

†, just like in the discussion of the

harmonic oscillator. First, clearly,

[𝑎𝑖, 𝑎𝑗] = 0 [𝑎𝑖†, 𝑎𝑗

†] = 0 2.2a

To determine the commutator of 𝑎𝑖 and 𝑎𝑖

†, consider

[𝑎𝑖, 𝑎𝑖†]|𝑛1, … , 𝑛𝑖 , … > = (𝑎𝑖𝑎𝑖

† − 𝑎𝑖†𝑎𝑖)| > =

𝑎𝑖√𝑛𝑖 + 1 | … , 𝑛𝑖 + 1, … > −𝑎𝑖†

√𝑛𝑖 | … , 𝑛𝑖 − 1, … > =

14

((𝑛𝑖 + 1) − 𝑛𝑖)| … , 𝑛𝑖 , … > = |. . , 𝑛𝑖 , . >

So, since the states |. . , 𝑛𝑖, . > are complete, we have the operator identity,

[𝑎𝑖, 𝑎𝑗†] = 𝛿𝑖𝑗 2.2b

where we noted that 𝑎𝑖 and 𝑎𝑗

† commute when 𝑖 ≠ 𝑗 since they operate on different states. It is

also clear that 𝑁𝑖 = 𝑎𝑖†𝑎𝑖 is a Hermitian operator that counts the number of particles in the state i,

𝑁𝑖|. . , 𝑛𝑖, . > = 𝑛𝑖|. . , 𝑛𝑖, . > 2.3a

So 𝑁 = ∑ 𝑁𝑖 is the total number operator. It also follows from Eq. 2.2 that,

[𝑁𝑖, 𝑎𝑖†] = 𝑎𝑖

† [𝑁𝑖, 𝑎𝑖] = −𝑎𝑖 2.3b

Just as for the harmonic oscillator, these commutation relations imply that there is a “vacuum”,

or “ground” state, for which,

𝑎𝑖|0 > = 0 2.4

for all i. Then we also have 𝑁𝑖|0 > = 0. We can then write the state |. . , 𝑛𝑖 , . > as the appropriate

power of 𝑎𝑖† applied to |0 >, as for the harmonic oscillator, Eq. 1.30b.

Our next task is to invent “field” operators that create and annihilate particles at a point �⃗�. These

are given by the generalized “Fourier transforms”,

𝜓(�⃗�) = ∑ < �⃗�𝑖 | 𝜆(𝑖) > 𝑎𝑖 𝜓†(�⃗�) = ∑ <𝑖 𝜆(𝑖) | �⃗� > 𝑎𝑖

† 2.5

The fundamental properties of the field operator follow from their commutators which are

derived from the commutators of the creation and annihilation operators,

[𝜓(�⃗�), 𝜓(�⃗�)] = [ 𝜓†(�⃗�), 𝜓†(�⃗�)] = 0 2.6a

And, more interesting,

[𝜓(�⃗�), 𝜓†(�⃗�)] = ∑ < �⃗�𝑖𝑗

| 𝜆(𝑖) >< 𝜆(𝑗) |�⃗� > [𝑎𝑖 , 𝑎𝑗†]

= ∑ < �⃗�𝑖 | 𝜆(𝑖) >< 𝜆(𝑖) |�⃗� > = < �⃗�|�⃗� > = 𝛿(�⃗� − �⃗�) 2.6b

15

where we used the completeness of the one-particle states,

1 = ∑ |𝜆(𝑖) >< 𝜆(𝑖)|𝑖 2.7

It is easy to see that the basic commutators, Eq. 2.6a and b, are independent of the choice of

single particle states |𝜆(𝑖) > and are general results.

We can write other familiar operators and states in terms of 𝜓(�⃗�) and 𝜓†(�⃗�). For example,

using

< 𝜆(𝑖)|𝜆(𝑗) > = 𝛿𝑖𝑗 = ∫ < 𝜆(𝑖)|�⃗� > 𝑑�⃗� < �⃗�|𝜆(𝑗) > 2.8a

we have,

𝑁 = ∑ 𝑎𝑖†𝑎𝑖𝑖 = ∑ 𝛿𝑖𝑗𝑖𝑗 𝑎𝑗

†𝑎𝑖 = ∑ ∫ < 𝜆(𝑖)|�⃗� > 𝑑�⃗� < �⃗�|𝜆(𝑗) >𝑖𝑗 𝑎𝑗†𝑎𝑖 = ∫ 𝑑�⃗� 𝜓†(�⃗�)𝜓(�⃗�) 2.8b

We learn that 𝜓†(�⃗�)𝜓(�⃗�) is the particle density operator.

Further, we can create a fully symmetric state of n particles with one at �⃗�1, another at �⃗�2, etc.,

1

√𝑛! 𝜓†(�⃗�1) 𝜓†(�⃗�2) … 𝜓†(�⃗�𝑛)|0 > 2.9

The fact that the 𝜓†′𝑠 commute insures that the state is fully symmetric. These particles are

Bose particles, identical spin-zero, pointlike particles. In your quantum mechanics course, you

studied Fermions, like electrons, which satisfy the Pauli Exclusion Principle and are described by

anti-commuting field operators. We won’t delve into that topic here since our goal concerns anti-

particles in relativistic field theory which can be explained more simply for scalar particles and

fields. We will return to Fermions in a later Essay on the Spin-Statistics Theorem of quantum

field theory.

Consider the state 𝜓†(�⃗�)|0 >. Let’s check that it describes a particle at position �⃗�. First,

[𝑁, 𝜓†(�⃗�)] = ∫ 𝑑�⃗�′ ( 𝜓†(�⃗�′)𝜓(�⃗�′) 𝜓†(�⃗�) − 𝜓†(�⃗�) 𝜓†(�⃗�′)𝜓(�⃗�′))

= ∫ 𝑑�⃗�′ 𝜓†(�⃗�′)𝛿(�⃗�′ − �⃗�) = 𝜓†(�⃗�) 2.10a

Therefore,

𝑁(𝜓†(�⃗�)|0 >) = [𝑁, 𝜓†(�⃗�)]|0 > = 𝜓†(�⃗�)|0 > 2.10b

16

So, 𝜓†(�⃗�)|0 > is an eigenstate of N with eigenvalue unity. Next, the probability amplitude for

finding the particle in the state 𝜓†(�⃗�)|0 > at the position �⃗�1 is,

< �⃗�1| 𝜓†(�⃗�)|0 >= < 0|𝜓(�⃗�1) 𝜓†(�⃗�)|0 > = 𝛿(�⃗� − �⃗�1) < 0|0 > = 𝛿(�⃗� − �⃗�1) 2.11

Similarly, we can construct one particle states in a wave function 𝜒(�⃗�) as,

|𝜒 > = ∫ 𝑑�⃗� 𝜒(�⃗�) 𝜓†(�⃗�) |0 > 2.12

which has the norm < 𝜒|𝜒 > = = ∫ 𝑑�⃗� |𝜒(�⃗�)|2.

Now let’s write some familiar operators in this language. Begin with an operator that does not

change particle number,

< �⃗�1′ , �⃗�2

′ , … , �⃗�𝑛′ |𝐹|�⃗�1, �⃗�2, … , �⃗�𝑛 > = 𝐹(�⃗�1, �⃗�2, … , �⃗�𝑛)𝛿(�⃗�1 − �⃗�1

′ ) ∙∙∙ 𝛿(�⃗�𝑛 − �⃗�𝑛′ ) 2.13

Now use the completeness of the states |�⃗�1, �⃗�2, … , �⃗�𝑛 > 1

√𝑛! 𝜓†(�⃗�1) 𝜓†(�⃗�2) … 𝜓†(�⃗�𝑛)|0 > to

write,

< 𝑛1′ , 𝑛2

′ , … |𝐹|𝑛1, 𝑛2, … > =

∫ 𝑑𝑥1 … 𝑑𝑥𝑛 ∫ 𝑑𝑥1′ … 𝑑𝑥𝑛

′ < 𝑛1′ , 𝑛2

′ , . . |�⃗�1′ , �⃗�2

′ , . . >< �⃗�1′ , �⃗�2

′ , . . |𝐹|�⃗�1, �⃗�2, . . >< �⃗�1, �⃗�2, . . |𝑛1, 𝑛2, . . >

= ∫ 𝑑𝑥1 … 𝑑𝑥𝑛 < 𝑛1′ , 𝑛2

′ , . . |�⃗�1, �⃗�2, . . > 𝐹(�⃗�1, �⃗�2, . . ) < �⃗�1, �⃗�2, . . |𝑛1, 𝑛2, . >

=1

𝑛!∫ 𝑑𝑥1. . 𝑑𝑥𝑛 < 𝑛1

′ , 𝑛2′ , . . | 𝜓†(�⃗�𝑛). . 𝜓†(�⃗�1)|0 > 𝐹(�⃗�1, �⃗�2, . . ) < 0|𝜓(�⃗�1). . 𝜓(�⃗�𝑛)|𝑛1, 𝑛2, . >

2.14

Note that since 𝜓(�⃗�1). . 𝜓(�⃗�𝑛)|𝑛1, 𝑛2, . > is a zero particle state, |0 >< 0| can be replaced by

unity in Eq. 2.14. But the states |𝑛1, 𝑛2, . > are complete, so we have an operator identity,

𝐹 =1

𝑛!∫ 𝑑𝑥1. . 𝑑𝑥𝑛 𝜓†(�⃗�𝑛). . 𝜓†(�⃗�1) 𝐹(�⃗�1, �⃗�2, . . )𝜓(�⃗�1). . 𝜓(�⃗�𝑛) 2.15

A useful case occurs when F is the sum of one particle operators,

𝐹(�⃗�1, �⃗�2, . . ) = ∑ 𝑓(�⃗�𝑖)𝑖 2.16

17

where f might be a function like a common external potential 𝑉(�⃗�𝑖) which effects each particle

equally or it might be an operator like momentum 𝑝(𝑖) =ℏ

𝑖∇⃗⃗⃗(𝑖) or kinetic energy 𝑇(𝑖) =

−ℏ2

2𝑚∇⃗⃗⃗2(𝑖). Then Eq. 2.15 simplifies. Consider the 𝑓(�⃗�𝑛) term first,

1

𝑛!∫ 𝑑𝑥1. . 𝑑𝑥𝑛 < 𝑛1

′ , 𝑛2′ , . . |𝜓†(�⃗�𝑛). . 𝜓†(�⃗�1) 𝑓(�⃗�𝑛)𝜓(�⃗�1). . 𝜓(�⃗�𝑛)|𝑛1, 𝑛2, . > 2.17

Note that ∫ 𝑑 �⃗�1𝜓†(�⃗�1)𝜓(�⃗�1) = 𝑁 which operates on a a one particle state, produces a factor of

1in Eq. 2.17. Then identify, ∫ 𝑑 �⃗�2𝜓†(�⃗�2)𝜓(�⃗�2) = 𝑁 again which acts on a two body state

producing a factor of 2. Continuing, we arrive at,

1

𝑛!(𝑛 − 1)! ∫ 𝑑𝑥𝑛 < 𝑛1

′ , 𝑛2′ , . . |𝜓†(�⃗�𝑛) 𝑓(�⃗�𝑛) 𝜓(�⃗�𝑛)|𝑛1, 𝑛2, . > 2.18

The other 𝑓(�⃗�𝑖) contributions are analyzed similarly. Collecting everything and accounting for

the factors of n, we find the elegant result,

𝐹 = ∫ 𝑑�⃗� 𝜓†(�⃗�) 𝑓(�⃗�) 𝜓(�⃗�) 2.19

This result is an economical way to write expressions for n-particle systems. Compare it to Eq.

2.16.

Next consider a two-particle operator,

𝐹(�⃗�1, �⃗�2, . . ) = ∑ 𝑉(�⃗�𝑖, �⃗�𝑗)𝑖<𝑗 =1

2∑ 𝑉(�⃗�𝑖, �⃗�𝑗)𝑖𝑗,𝑖≠𝑗 2.20

that could describe 2-body interactions acting within the ensemble of n particles. Now we can

repeat the analysis that led from Eq. 2.16 to 2.19 for a single particle operator and show that Eq.

2.20 can be written,

𝐹 =1

2∫ 𝑑�⃗� 𝑑�⃗�′ 𝜓†(�⃗�′) 𝜓†(�⃗�)𝑉(�⃗�, �⃗�′)𝜓(�⃗�)𝜓(�⃗�′) 2.21

The operator ordering is important here: it guarantees that a term with 𝑉(�⃗�𝑖, �⃗�𝑖) does not occur,

in accord with Eq. 2.20, and it makes the Hermiticity of F manifest.

Now we can write the Hamiltonian of any n-body system where there are internal 2-body forces,

𝐻 = 𝐻0 + 𝐻𝐼 2.22a

18

with

𝐻0 = −ℏ2

2𝑚∫ 𝑑�⃗� 𝜓†(�⃗�) ∇⃗⃗⃗2𝜓(�⃗�) + ∫ 𝑑�⃗� 𝜓†(�⃗�) 𝑉𝑒𝑥𝑡(�⃗�) 𝜓(�⃗�) 2.22b

and

𝐻𝐼 =1

2∫ 𝑑�⃗� 𝑑�⃗�′ 𝜓†(�⃗�′) 𝜓†(�⃗�)𝑉(�⃗�, �⃗�′)𝜓(�⃗�)𝜓(�⃗�′) 2.22c

where 𝑉𝑒𝑥𝑡 is a 1-body external potential energy.

In this formulations of quantum mechanics, the states have time dependence but the operators are

generally time-independent. This is called the “Schrodinger” picture. State vectors satisfy the

Schrodinger equation,

𝑖ℏ𝜕

𝜕𝑡|𝜒𝑆(𝑡) > = 𝐻|𝜒𝑆(𝑡) > 2.23

Frequently, especially in relativistic quantum field theory, it is convenient to make the state

vectors time independent and the operators time dependent: the local operators then depend on �⃗�

and t and their space time properties are clearer. This is the “Heisenberg” picture [1], defined

through the unitary transformation,

|𝜒𝐻 > = 𝑒𝑖𝐻𝑡 ℏ⁄ |𝜒𝑆(𝑡) > 𝜓𝐻(�⃗�, 𝑡) = 𝑒𝑖𝐻𝑡 ℏ⁄ 𝜓(�⃗�) 𝑒−𝑖𝐻𝑡 ℏ⁄ 2.24

It is conventional to drop the subscript “H” on operators because the time dependence will label

them clearly. Note that the basic commutation relations of the Schrodinger field operators

become equal-time commutators in the Heisenberg picture,

[𝜓(�⃗�, 𝑡), 𝜓†(�⃗�, 𝑡)] = 𝑒𝑖𝐻𝑡 ℏ⁄ [𝜓(�⃗�), 𝜓†(�⃗�)]𝑒−𝑖𝐻𝑡 ℏ⁄ = 𝛿(�⃗� − �⃗�) 2.25a

And,

[𝜓(�⃗�, 𝑡), 𝜓(�⃗�, 𝑡)] = [ 𝜓†(�⃗�, 𝑡), 𝜓†(�⃗�, 𝑡)] = 0 2.25b

We will discuss these equal-time commutators further when we turn to relativistic quantum field

theory.

Let’s end this discussion with some observations about non-interacting systems. The

Hamiltonian describing the propagation of non-relativistic particles of mass m is,

19

𝐻0 = −ℏ2

2𝑚∫ 𝑑�⃗� 𝜓†(�⃗�, 𝑡) ∇⃗⃗⃗2𝜓(�⃗�, 𝑡) 2.26

following the prescription of Eq. 2.19. Similarly, the total momentum operator is,

�⃗⃗� =ℏ

𝑖∫ 𝑑�⃗� 𝜓†(�⃗�, 𝑡) ∇⃗⃗⃗𝜓(�⃗�, 𝑡) 2.27a

This result is usually written in the manifestly Hermitian form,

�⃗⃗� =ℏ

2𝑖∫ 𝑑�⃗� [ 𝜓†(�⃗�, 𝑡) ∇⃗⃗⃗𝜓(�⃗�, 𝑡) − ∇⃗⃗⃗𝜓†(�⃗�, 𝑡) 𝜓(�⃗�, 𝑡)] 2.27b

Since [𝐻, �⃗⃗�] = 0 we can simultaneously diagonalize both operators. The eigenstates are plane

waves characterized by a wave vector �⃗⃗�. Then the field operator can be written in this basis,

𝜓(�⃗�, 𝑡) =1

√𝑉∑ 𝑎�⃗⃗��⃗⃗� 𝑒𝑖(�⃗⃗�∙𝑥− 𝜔

�⃗⃗⃗� 𝑡)

2.28

with 𝜔�⃗⃗� = ℏ�⃗⃗�2 2𝑚⁄ . The field operator then satisfies the equation of motion,

𝑖ℏ𝜕

𝜕𝑡 𝜓(�⃗�, 𝑡) = [𝜓(�⃗�, 𝑡), 𝐻] = −

ℏ2

2𝑚 ∇⃗⃗⃗2𝜓(�⃗�, 𝑡) 2.29

which follows from Eq. 2.26 and the equal-time commutators Eq. 2.25. It is conventional to

quantize the system by putting it into a 3-dimensional box of volume 𝑉 = 𝐿3 with periodic

boundary conditions. Then the cartesian components of the continuous wave number become

discrete variables 𝑘𝑖 = 𝑛𝑖 ∙2𝜋

𝐿 with 𝑛𝑖 = 0, ±1, ±2, …. When 𝑉 → ∞, the sum over the discrete �⃗⃗�

values is replaced by an integral over �⃗⃗� with,

1

𝑉∑ →

1

(2𝜋)3 ∫ 𝑑�⃗⃗� �⃗⃗� 2.30

The commutation relations for the creation and annihilation operators associated with the

modes 1

√𝑉𝑒𝑖�⃗⃗�∙𝑥 in the 𝑉 → ∞ limit become,

[𝑎�⃗⃗� , 𝑎†�⃗⃗�′ ] = 𝛿(�⃗⃗� − �⃗⃗�′) 2.31a

[𝑎�⃗⃗� , 𝑎�⃗⃗�′] = [𝑎†�⃗⃗�′ , 𝑎

�⃗⃗�

†] = 0 2.31b

20

We can calculate the equal-time commutator of 𝜓 and 𝜓† using Eq. 2.28 and 2.31, and taking

𝑉 → ∞,

[𝜓(�⃗�, 𝑡), 𝜓†(�⃗�′, 𝑡)] =1

𝑉∑ [𝑎�⃗⃗� , 𝑎†

�⃗⃗�′]�⃗⃗�,�⃗⃗�′ 𝑒𝑖(�⃗⃗�∙𝑥− 𝜔�⃗⃗⃗�

𝑡)𝑒𝑖(�⃗⃗�′∙𝑥′− 𝜔�⃗⃗⃗�′ 𝑡) =

1

𝑉∑ 𝑒𝑖�⃗⃗�∙(𝑥−𝑥′)

�⃗⃗� = 𝛿(�⃗� − �⃗�′)

2.32

And finally we can write H, �⃗⃗� and N in terms of 𝑎�⃗⃗� and 𝑎�⃗⃗�

† using Eq. 2.26, 2.27 and 2.28,

𝐻 = ∑ℏ�⃗⃗�2

2𝑚�⃗⃗�

𝑎�⃗⃗�

†𝑎�⃗⃗�

�⃗⃗� = ∑ ℏ�⃗⃗��⃗⃗� 𝑎�⃗⃗�

†𝑎�⃗⃗� 2.33

𝑁 = ∑ 𝑎�⃗⃗�

†𝑎�⃗⃗�

�⃗⃗�

These results will guide us when we turn to relativistic quantum fields.

Relativistic Quantum Fields and Anti-Particles

Relativistic Fields and Operators

In Eq. 2.22 we considered non-relativistic many-body quantum systems with interactions. These

involved instantaneous potentials, like the Coulomb potential between two charged particles.

Now we turn to relativistic theories described by local quantum fields. When we discussed

classical electromagnetism we developed the field strength tensor 𝐹𝜇𝜈 and wrote it in terms of the

vector potential field 𝐴𝜇(�⃗�, 𝑡) in the Lorenz gauge 𝜕𝜇𝐴𝜇 = 0, 𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇. The “source”

for the field 𝐹𝜇𝜈 was the electric current 𝑗𝜇. Maxwell’s equations had the form 𝜕𝜇𝐹𝜇𝜈 = 𝑗𝜈. In

particular, the 𝜈 = 0 equation reduces to Gauss’ Law and the 𝜈 = 1,2,3 components reproduce

Ampere-Maxwell’s Law. In the language of Hamiltonians, the interaction between charges and

the electromagnetic field arises from the interaction term,

ℋ(�⃗�, 𝑡) = 𝑒𝑗𝜇(�⃗�, 𝑡)𝐴𝜇(�⃗�, 𝑡) 3.1

where ℋ(�⃗�, 𝑡) is the Hamiltonian density, 𝐻𝐼 = ∫ 𝑑�⃗� ℋ(�⃗�, 𝑡). In a later Essay we will derive Eq.

3.1 from the principles of electrodynamics, but here we just want to consider some generic

21

features of these formulas. First, the interactions between fields and particles are expressed as

products of fields. In addition, in relativistic theories ℋ(�⃗�, 𝑡) must be a Lorentz scalar, as in Eq.

3.1, constructed from the product of fields at the same space-time point. This feature will allow

the theory to be causal and relativistic. In addition, it will admit local conservation laws that are

fully relativistic. Recall from the textbook that local conservation laws could be relativistic while

non-local conservation laws which occur in Newtonian mechanics were not. Our original

illustration of these ideas was Newton’s Third Law of action-reaction. We considered the earth-

moon system and observed that the law of action-reaction implies that the earth pushes on the

moon with an equal but opposite force that the moon pushes on the earth. The sum of the two

forces is zero and this fact leads to a global conservation law: the sum of the momenta of the

earth and the moon is a constant even as they interact. Note that the conservation is not local: the

two forces are applied a distance 128,000 miles apart. This “makes sense” in Newton’s world

where there is no speed limit and information travels instantaneously. But non-local conservation

laws are impossible in relativity. Let’s review the reasoning here. If such a non-local

conservation law were true in one reference frame, where one event occurs at (𝑐𝑡, �⃗�1) and the

second at (𝑐𝑡, �⃗�2), then we can transform to other frames where the spatially separated events

occur at different times. Therefore, if there were non-local conservation of momentum for all

times in the first reference frame, then the conservation law is not true at all times in the second

frame. This observation led us to reject the idea of instantaneous potentials and led to their

replacement by dynamical fields. We were then able to replace global conservation laws with

local conservation laws and create a system of dynamics consistent with relativity and causality.

This led to the idea that Coulomb’s Law has to be replaced by the electromagnetic field which

introduced a new set of dynamical variables, the electromagnetic field, equal in stature to

propagating charges, the sources of the electromagnetic field. This point found fruition in the

textbook’s derivation of Maxwell’s equations and the wave equation for the electromagnetic

field which predicted that special relativity’s speed limit is the speed of light.

In quantum field theory we can construct local Hermitian observables 𝐴(𝑥) an 𝐵(𝑥) where x

stands for the four vector 𝑥 = (𝑐𝑡, �⃗�). Then causality implies that 𝐴(𝑥) an 𝐵(𝑥) must be

independent variables as long as x and x’ cannot be connected by a light ray, i.e. as long as

(𝑥 − 𝑥′)2 = 𝑐2(𝑡 − 𝑡′)2 − (�⃗� − �⃗�′)2 < 0 is space-like. We write,

22

[𝐴(𝑥), 𝐵(𝑥′)] = 0 3.2

if (𝑥 − 𝑥′)2 < 0. This observation applies, of course, to the interaction Hamiltonian density,

[ℋ(𝑥), ℋ(𝑥′)] = 0 if (𝑥 − 𝑥′)2 < 0. This statement becomes very predictive when we write

ℋ(𝑥) in terms of a product of local, causal fields, like ℋ(𝑥) = 𝑒𝑗𝜇(�⃗�, 𝑡)𝐴𝜇(�⃗�, 𝑡) for

electrodynamics or ℋ(𝑥) = 𝜑4(𝑥) for a neutral, scalar field theory.

Let’s begin with scalar fields for chargeless, non-interacting particles: we need to understand free

neutral scalar fields before considering interactions, and before considering fields that describe

particles that carry a conserved charge. We will follow our non-relativistic discussion but will

have to make adjustments for relativistic kinematics. In particular, single particle states of

definite wave number �⃗⃗� and angular frequency 𝜔 are given by plane waves,

𝑒𝑖(𝜔 𝑡−�⃗⃗�∙𝑥) 3.3a

with ℏ𝜔 = √ℏ2 �⃗⃗�2𝑐2 + 𝑚2𝑐4. So, a scalar field should have the form,

𝜑(�⃗�, 𝑡) = ∫𝑑�⃗⃗�

(2𝜋)32𝜔[𝑎(𝑘)𝑒𝑖(𝜔 𝑡−�⃗⃗�∙𝑥) + ⋯ ] 3.3b

Eq. 3.3b is actually incomplete because there is another criterion 𝜑(�⃗�, 𝑡) must satisfy: it should

be Hermitian so that it describes an observable, quantum quantity. This is certainly the case for

the electromagnetic field 𝐴𝜇(�⃗�, 𝑡) that in the classical theory produces real electric and magnetic

fields. In order to make Eq. 3.3b Hermitian, we must add in the Hermitian adjoint of the term

already recorded. So,

𝜑(�⃗�, 𝑡) = ∫𝑑�⃗⃗�

(2𝜋)32𝜔[𝑎(𝑘)𝑒𝑖(𝜔 𝑡−�⃗⃗�∙𝑥) + 𝑎†(𝑘)𝑒−𝑖(𝜔 𝑡−�⃗⃗�∙𝑥)] 3.3c

Let’s write the phase factors in relativistic form 𝑘 ∙ 𝑥 = 𝑘𝜇𝑥𝜇 = 𝜔 𝑡 − �⃗⃗� ∙ �⃗� with 𝑘𝜇 = (𝜔

𝑐, �⃗⃗�)

and 𝑥𝜇 = (𝑐𝑡, �⃗�). So,

𝜑(𝑥) = ∫𝑑�⃗⃗�

(2𝜋)32𝜔[𝑎(𝑘)𝑒𝑖𝑘∙𝑥 + 𝑎†(𝑘)𝑒−𝑖𝑘∙𝑥] 3.3d

where 𝑘2 = 𝑘𝜇𝑘𝜇 =𝜔2

𝑐2 − �⃗⃗�2 =𝑚2𝑐4

ℏ2 . Finally, we choose the integration measure to be 𝑑�⃗⃗�

(2𝜋)32𝜔.

There are two reasons for this: 1. The combination 𝑑�⃗⃗� 2𝜔⁄ is Lorentz invariant, and 2. The factor

23

(2𝜋)3 provides a convenient normalization for the creation /annihilation operators 𝑎(𝑘) and

𝑎†(𝑘). In order to establish property #1, note that 𝑑4𝑘 𝛿 (𝑘2 −𝑚2𝑐4

ℏ2 ) 𝜃(𝑘0) is invariant under

ordinary Lorentz transformations. Next recall the property of the delta-function,

∫ 𝑑𝑥 𝛿(𝑔(𝑥)) = ∑1

|𝑔′(𝑥𝑖)|𝑖 3.4a

where 𝑥𝑖 are the zeros of the smooth function 𝑔(𝑥). Therefore,

∫ 𝑑𝑘0 𝛿 (𝑘2 −𝑚2𝑐4

ℏ2 ) 𝜃(𝑘0) =1

2𝜔 3.4b

So,

∫ 𝑑4𝑘 𝛿 (𝑘2 −𝑚2𝑐4

ℏ2 ) 𝜃(𝑘0) = ∫𝑑�⃗⃗�

2𝜔 3.4c

which establishes the point. We have seen these identities before in Supplementary Lecture #7

on relativistic kinematics, rapidity, light-cone variables and multi-particle production in high

energy collisions. Relativistic invariance is an essential(!) ingredient in the discussion here so the

factor of 1

2𝜔 in the integration measure represents an important matter of principle!

Next, we can project the creation and annihilation operators out of the expression for 𝜑(𝑥) by

Fourier transform,

∫ 𝑑�⃗� 𝑒−𝑖𝑘∙𝑥𝜑(𝑥) =1

2𝜔𝑎(�⃗⃗�) +

1

2𝜔𝑒2𝑖𝜔𝑡𝑎†(−�⃗⃗�) 3.5a

In order to isolate 𝑎(�⃗⃗�) we need another expression. We choose to Fourier transform 𝜕0𝜑(𝑥)

starting with Eq. 3.3c,

∫ 𝑑�⃗� 𝑒−𝑖𝑘∙𝑥𝜕0𝜑(𝑥) = −𝑖

2𝑎(�⃗⃗�) +

𝑖

2𝑒2𝑖𝜔𝑡𝑎†(−�⃗⃗�) 3.5b

From these relations we can find 𝑎(�⃗⃗�) alone,

𝑎(�⃗⃗�) = ∫ 𝑑�⃗� 𝑒−𝑖𝑘∙𝑥[𝑖𝜕0𝜑(𝑥) + 𝜔𝜑(𝑥)] = 𝑖 ∫ 𝑑�⃗� 𝑒−𝑖𝑘∙𝑥𝜕0𝜑(𝑥) 3.5c

where we define the symbol 𝜕 as 𝑓𝜕𝑔 = 𝑓𝜕𝑔 − 𝜕𝑓𝑔.

24

Next, we need the commutation relations of 𝑎†(�⃗⃗�), 𝑎(�⃗⃗�), and the field operators 𝜑(𝑥), 𝜑†(𝑥),

etc. Following our exercises in the harmonic oscillator and second quantization for non-

relativistic quantum mechanics, the free Hamiltonian should have the form,

𝐻 = ∫𝑑�⃗⃗�

(2𝜋)32𝜔(ℏ𝜔) 𝑎†(�⃗⃗�)𝑎(�⃗⃗�) 3.6a

and similarly, the total momentum operator,

�⃗⃗� = ∫𝑑�⃗⃗�

(2𝜋)32𝜔(ℏ�⃗⃗�) 𝑎†(�⃗⃗�)𝑎(�⃗⃗�) 3.6b

and the total number operator,

𝑁 = ∫𝑑�⃗⃗�

(2𝜋)32𝜔 𝑎†(�⃗⃗�)𝑎(�⃗⃗�) 3.6c

Then we should have the basic commutation relations,

[𝐻, 𝑎(�⃗⃗�)] = − ℏ𝜔 𝑎(�⃗⃗�) [𝐻, 𝑎†(�⃗⃗�)] = ℏ𝜔𝑎†(�⃗⃗�) 3.6d

[�⃗⃗�, 𝑎(�⃗⃗�)] = − ℏ�⃗⃗�𝑎(�⃗⃗�) [�⃗⃗�, 𝑎†(�⃗⃗�)] = ℏ�⃗⃗�𝑎†(�⃗⃗�) 3.6e

[𝑁, 𝑎(�⃗⃗�)] = − 𝑎(�⃗⃗�) [𝑁, 𝑎†(�⃗⃗�)] = 𝑎†(�⃗⃗�) 3.6f

which express the fact that 𝑎†(�⃗⃗�) is a creation operator for a particle that carries energy ℏ𝜔 and

momentum ℏ�⃗⃗�. Let’s check Eq. 3.6d. Let |𝐸 > be an eigenstate of H with eigenvalue E. Let’s

check that 𝑎(�⃗⃗�)|𝐸 > is an eigenstate of H with eigenvalue 𝐸 − ℏ𝜔. From Eq. 3.6d,

𝐻𝑎(�⃗⃗�)|𝐸 > = 𝑎(�⃗⃗�)𝐻|𝐸 > +[𝐻, 𝑎(�⃗⃗�)]|𝐸 > = (𝐸 − ℏ𝜔)𝑎(�⃗⃗�)|𝐸 > 3.7

The considerations to motivate Eq. 3.6e and 3.6f are similar.

Next we can use Eq. 3.6a and d to find the basic commutation relations of the

creation/annihilation operators. Straight-forward algebra analogous to that above produce the

results,

25

[𝑎(�⃗⃗�), 𝑎(�⃗⃗�′)] = [𝑎†(�⃗⃗�), 𝑎†(�⃗⃗�′)] = 0 3.8a

[𝑎(�⃗⃗�), 𝑎†(�⃗⃗�′)] = (2𝜋)32𝜔 𝛿(�⃗⃗� − �⃗⃗�′) 3.8b

Note the factor (2𝜋)32𝜔 in Eq. 3.8b, as compared to the non-relativistic commutator. The right-

hand-side of Eq. 3.8b is the Lorentz invariant quantity (2𝜋)32𝜔 𝛿(�⃗⃗� − �⃗⃗�′) which accompanies

the Lorentz invariant measure 𝑑�⃗⃗� (2𝜋)32𝜔⁄ . (We could, of course, have “guessed” these

results Eq. 3.8 on the basis of the non-relativistic second quantization rules, corrected for the

relativistic phase space, 𝑑�⃗⃗� → 𝑑�⃗⃗� (2𝜋)32𝜔⁄ , but it is best to establish them straight-forwardly.)

Now we can turn to the field operators,

𝜑(𝑥) = ∫𝑑�⃗⃗�

(2𝜋)32𝜔[𝑎(𝑘)𝑒𝑖𝑘∙𝑥 + 𝑎†(𝑘)𝑒−𝑖𝑘∙𝑥] 3.3c

We can separate 𝜑 into its annihilation part and its creation part,

𝜑(𝑥) = 𝜑+(𝑥) + 𝜑−(𝑥) 3.9a

with

𝜑+(𝑥) = ∫𝑑�⃗�

(2𝜋)32𝑝0 𝑎(𝑝)𝑒𝑖𝑝∙𝑥 𝜑−(𝑥) = 𝜑+†(𝑥) = ∫𝑑�⃗⃗�

(2𝜋)32𝑝0 𝑎†(𝑝)𝑒−𝑖𝑝∙𝑥 3.9b

The next property we need here is the commutator of 𝜑+(𝑥) and 𝜑−(𝑥). Given these

commutators we can calculate others. We are particularly interested in understanding causality

and locality in this framework. Using Eq. 3.8a and b, we have

[𝜑+(𝑥), 𝜑−(𝑦)] = ∫𝑑𝑝 𝑑𝑝′

(2𝜋)62𝑝02𝑝′0𝑒𝑖𝑝∙𝑥𝑒−𝑖𝑝′∙𝑦(2𝜋)32𝑝0 𝛿(𝑝 − 𝑝′)

= ∫𝑑�⃗�

(2𝜋)32𝑝0 𝑒𝑖𝑝∙(𝑥−𝑦) ≡ ∆+(𝑥 − 𝑦) 3.10

The right-hand-side of Eq. 3.10 is an important special function of quantum field theory,

∆+(𝑥 − 𝑦) = ∫𝑑�⃗�

(2𝜋)32𝑝0 𝑒𝑖𝑝∙(𝑥−𝑦) 3.11

26

Note that it is Lorentz invariant because the measure is invariant and so is the inner product 𝑝 ∙ 𝑥.

Therefore, it only depends on the invariants one can form from (𝑥 − 𝑦) , which is the square

(𝑥 − 𝑦)2 = (𝑥 − 𝑦)𝜇(𝑥 − 𝑦)𝜇. So ∆+(𝑥 − 𝑦) = ∆+(𝑦 − 𝑥) = ∆+((𝑥 − 𝑦)2).

We are particularly interested in ∆+(𝑥) for space-like 𝑥2. In that case we can take 𝑥0 = 0 and

⌈�⃗�⌉ = √−𝑥2 and reduce Eq. 3.11 to a one dimensional integral and identify a familiar function of

mathematical physics,

∆+(𝑥) = ∫𝑑𝑝

(2𝜋)32√�⃗�2 + 𝑚2𝑒𝑖�⃗�∙𝑥 =

4𝜋

(2𝜋)3∫

𝑝2 𝑑𝑝

2√𝑝2 + 𝑚2

sin 𝑝√−𝑥2

𝑝√−𝑥2

∞

0

=𝑚

4𝜋2√−𝑥2 ∫𝑢 𝑑𝑢

√𝑢2+1

∞

0sin(𝑚√−𝑥2 𝑢) 3.12

where we introduced a dimensionless variable 𝑢 = |𝑝| 𝑚 = 𝑝 𝑚⁄⁄ in the last step. We also

suppressed factors of ℏ and c here and will restore them whenever we need them by dimensional

analysis. The integral here is the Hankel function, a relative of the Bessel function,

∆+(𝑥) =𝑚

4𝜋2√−𝑥2 𝐾1(𝑚√−𝑥2) 3.13

Recall some generic properties of 𝐾1. First, it is exponentially small when its argument is large,

𝐾1~𝑒𝑥𝑝(−𝑚√−𝑥2), and, for 𝑚 → 0, 𝐾1(𝑚√−𝑥2) falls as (𝑚√−𝑥2)−1

. We are particularly

interested in massive scalar particles where we see that the range of 𝐾1(𝑚√−𝑥2) is of order

1 𝑚⁄ . Restoring dimensions, we recognize that this fundamental length is the Compton

wavelength of the particle, ℏ/mc.

In order to better understand the significance of this result, consider the same problem for non-

relativistic fields. Then the boost invariant measure in Eq. 3.12 would be replaced by non-

relativistic phase space 𝑑𝑝, and ∆+(𝑥 − 𝑦) becomes local, ∆+(𝑥 − 𝑦) → 𝛿(�⃗� − �⃗�) when 𝑥0 =

𝑦0. Recall that this remark was already encountered in Eq. 2.32 where the commutator of the

non-relativistic field was calculated from first principles. The appearance of the Dirac delta

function 𝛿(�⃗� − �⃗�) was an expression of the fact that in non-relativistic theory, the single particle

states are complete. Note also that non-relativistic physics does not contain the fundamental

parameter of relativity, the speed of light c, and so there is no Compton wavelength to set a

fundamental scale. As we will discuss further below, we need more degrees of freedom to

27

capture the principles of relativity, locality and causality than in non-relativistic quantum

mechanics. This development parallels the textbook’s earlier development of classical field

theory from Newtonian mechanics. Recall that to accommodate the finite speed limit c, it proved

necessary that the instantaneous electric force had to be replaced by a dynamical electromagnetic

field which described propagating and polarizable electric and magnetic fields. We learned that

the number of degrees of freedom of Newtonian mechanics had to be expanded to include

polarizable electric and magnetic fields. In our present discussion, we will learn that in order to

capture relativity, locality and causality as well as local charge conservation, each charged

degree of freedom must be accompanied by an oppositely charged degree of freedom, an anti-

particle. This is a fundamental implication of the unification of relativity with local field theory.

Lorentz Transformations of Fields, States and Operators

Next let’s understand the Lorentz boost properties of the fields 𝜑(𝑥), 𝜑+(𝑥) and 𝜑−(𝑥), and the

other operators and states that have been introduced so far. In quantum mechanics you learned

how to implement translations and rotations in the language of operators and states [1]. The

components of the angular momentum operator 𝐽𝑖 were identified as the generators of rotations

which were represented by unitary operators, 𝑈(𝜃, �⃗⃗�) = 𝑒𝑥𝑝(𝑖𝜃𝐽 ∙ �⃗⃗� ℏ⁄ ) where �⃗⃗� is the direction

of an axis of rotation and 𝜃 is the angle around it. In order to represent Lorentz transformations,

one proceeds similarly. For a Lorentz transformation, 𝑥′𝜇 = 𝐿 𝜌𝜇

𝑥𝜌 there is a unitary operator

𝑈(𝐿). When 𝑈(𝐿) acts on an n-particle state of scalar particles labelled by the momenta of each

particle,

𝑈(𝐿) |𝑝1, … , 𝑝𝑛 > = |𝐿𝑝1, … , 𝐿𝑝𝑛 > 3.14

where 𝐿𝑝 is short-hand for 𝐿 𝜌𝜇

𝑝𝜌. In addition, 𝑈(𝐿) has no effect on the vacuum,

𝑈(𝐿)|0 = |0 > 3.15

𝑈(𝐿) provides a unitary representation of Lorentz boosts,

𝑈†(𝐿) = 𝑈−1(𝐿) = 𝑈(𝐿−1) 3.16

Explicit formulas for 𝑈(𝐿) can be written down in terms of the generator of boosts, following the

material you learned in quantum mechanics for rotations and the angular momentum operator.

We will not need those explicit expressions here, but the references have all the details [1,2].

28

Next, recall from quantum mechanics, that when a state is rotated,

𝑈(𝜃, �⃗⃗�)|𝜒 > = |𝜒′ > 3.17a

then operators Θ are also rotated through similarity transformations,

Θ′ = 𝑈(𝐿)Θ𝑈−1(𝐿) 3.17b

For example, the creation and annihilation operators for particles of momentum 𝑝 transform as,

𝑈(𝐿)a(𝑝)𝑈−1(𝐿) = 𝑎(𝐿𝑝) 3.17c

which accompanies Eq. 3.14.

Let’s consider another illustration. Consider the operator for the total momentum,

𝑃𝜇 = ∫𝑑�⃗�

(2𝜋)32𝑝0 𝑝𝜇 𝑎†(𝑝)𝑎(𝑝) 3.18

of an ensemble of scalar particles. Then 𝑃𝜇 in the boosted frame is,

𝑈(𝐿)𝑃𝜇𝑈−1(𝐿) = ∫𝑑𝑝

(2𝜋)32𝑝0𝑝𝜇 𝑈(𝐿)𝑎†(𝑝)𝑈−1(𝐿)𝑈(𝐿)𝑎(𝑝)𝑈−1(𝐿)

= ∫𝑑�⃗�

(2𝜋)32𝑝0 𝑝𝜇 𝑎†(𝐿𝑝)𝑎(𝐿𝑝) = ∫𝑑�⃗�′

(2𝜋)32𝑝′0(𝐿−1) 𝜌

𝜇 𝑝′𝜌 𝑎†(𝑝′)𝑎(𝑝′) = (𝐿−1) 𝜌

𝜇 𝑃𝜌 3.19

where we used the invariant nature of the relativistic measure in the second line.

Now we can check the important point in this discussion: 𝜑+(𝑥), 𝜑−(𝑥) and the causal field

𝜑(𝑥) are all Lorentz invariant fields. The proof is straight-forward,

𝑈(𝐿) 𝜑+(𝑥)𝑈−1(𝐿) = ∫𝑑𝑝

(2𝜋)32𝑝0𝑈(𝐿) 𝑎(𝑝)𝑈−1(𝐿)𝑒𝑖𝑝∙𝑥

= ∫𝑑𝐿𝑝⃗⃗⃗⃗⃗⃗

(2𝜋)32𝐿𝑝0 𝑎(𝐿𝑝)𝑒𝑖𝐿𝑝∙𝐿𝑥 = ∫𝑑𝑝′⃗⃗⃗⃗⃗

(2𝜋)32𝑝′0 𝑎(𝑝′)𝑒𝑖𝑝′∙𝐿𝑥 = 𝜑+(𝐿𝑥) 3.20

Now that we know how 𝜑+(𝑥), 𝜑−(𝑥) and 𝜑(𝑥) transform under boosts, we can construct

products of fields and operators and can calculate their transformation properties. This will prove

to be crucial in our discussion of anti-particles in local, causal quantum field theory when we

construct Hamiltonian densities that must be Lorentz scalars.

29

Charged Relativistic Fields

Now let’s return to the main theme of this Essay, the necessity for anti-particles in relativistic

quantum field theory. We learned several important lessons from Eq. 3.10: First, since physical

operators must commute for space-like separations, we cannot use just 𝜑+(𝑥) or 𝜑−(𝑥) to

construct a local, causal quantum field theory of scalar particles. For example, the interaction

Hamiltonian density ℋ(𝑥) should commute with itself for space-like separations,

[ℋ(𝑥), ℋ(𝑦)] = 0 for (𝑥 − 𝑦)2 < 0, and this simple fact cannot be implemented with just 𝜑+

(or just 𝜑−) because of Eq. 3.10.

However, there is a way out of this calamity! We have already seen that the linear combination

𝜑 = 𝜑+ + 𝜑− has important physical properties: it is Hermitian and satisfies the Klein-Gordan

wave equation [1,2], (𝜕𝜇𝜕𝜇 − 𝑚2𝑐4)𝜑(𝑥) = 0, in the theory of free, non-interacting particles,.

Let’s calculate its commutator for (𝑥 − 𝑦)2 < 0,

[𝜑(𝑥), 𝜑(𝑦)] = [𝜑+(x) + 𝜑−(𝑥), 𝜑+(y) + 𝜑− (𝑦)] = [𝜑+(x), 𝜑−(𝑦)] + [𝜑−(x), 𝜑+(𝑦)]

= ∆+(𝑥 − 𝑦) − ∆+(𝑦 − 𝑥)

So,

[𝜑(𝑥), 𝜑(𝑦)] = 0 3.21

because ∆+(𝑥 − 𝑦) = ∆+(𝑦 − 𝑥), and we also used the trivial results, [𝜑+(x),𝜑+(y)] = 0 and

[𝜑−(x),𝜑−(y)] = 0.

Now let’s finally(!) consider the charged scalar field. Suppose its charge is +𝑞. There will be

creation and annihilation operators of four momentum p and charge q which we label 𝑎(𝑝, 𝑞) and

𝑎†(𝑝, 𝑞). In addition to the observables H, P and N already introduced, there is a global,

conserved charge operator Q with the property,

[𝑄, 𝑎(𝑝, 𝑞)] = −𝑞 𝑎(𝑝, 𝑞) 3.22a

Taking the Hermitian adjoint and using 𝑄† = 𝑄, we also have,

[𝑄, 𝑎†(𝑝, 𝑞)] = +𝑞 𝑎†(𝑝, 𝑞) 3.22b

30

The Hamiltonian and the Hamiltonian density ℋ(𝑥) must commute with Q in order to guarantee

charge conservation. In fact, Q can be expressed as the integral over a charge density operator,

the 0-th component of the four vector charged current density 𝐽𝜇, 𝑄 = ∫ 𝑑�⃗� 𝐽0(𝑥). In this way

the conservation of charge can be expressed as a local conservation law, as emphasized earlier

and in the textbook. In specific quantum field theories, 𝐽𝜇(𝑥) is constructed from fundamental,

causal fields. In addition, ℋ(𝑥) will be constructed out of the product of fields at x. So, in order

that [ℋ(𝑥), 𝑄] = 0, ℋ(𝑥) must be constructed out of fields that have simple commutation

relations with Q. If we label the fields with a subscript q, charge, we require,

[𝑄, 𝜑𝑞(𝑥)] = −𝑞 𝜑𝑞(𝑥) [𝑄, 𝜑𝑞†(𝑥)] = +𝑞𝜑𝑞

†(𝑥) 3.23a

Suppose that ℋ(𝑥) is constructed out of the products of fields 𝜑𝑞1(𝑥), 𝜑𝑞2

(𝑥) etc. and adjoints

𝜑†�̃�1

(𝑥), 𝜑†�̃�2

(𝑥) etc. Charge conservation, [𝑄, ℋ(𝑥)] = 0, then requires,

𝑞1 + 𝑞2 + ⋯ − �̃�1 − �̃�2 − ⋯ = 0 3.23b

Explicitly, if ℋ(𝑥) = 𝑔𝜑†�̃�1

(𝑥)𝜑†�̃�2

(𝑥) ⋅⋅⋅ 𝜑𝑞1(𝑥)𝜑𝑞2

(𝑥) ⋅⋅⋅. Then,

[𝑄, ℋ(𝑥)] = [𝑄, 𝑔 𝜑†�̃�1

(𝑥)𝜑†�̃�2

(𝑥) ⋅⋅⋅ 𝜑𝑞1(𝑥)𝜑𝑞2

(𝑥) ⋅⋅]

= 𝑔 [𝑄, 𝜑†�̃�1

(𝑥)] 𝜑†�̃�2

(𝑥) ⋅⋅⋅ 𝜑𝑞1(𝑥)𝜑𝑞2

(𝑥) ⋅⋅ +𝑔𝜑†�̃�1

(𝑥) [𝑄, 𝜑†�̃�2

(𝑥)] ⋅⋅⋅ 𝜑𝑞1(𝑥)𝜑𝑞2

(𝑥)+…

= 𝑔 (�̃�1 + �̃�2 +⋅⋅⋅ −𝑞1 − 𝑞2 −⋅⋅⋅)ℋ(𝑥) = 0

To accommodate this conservation law, Eq. 3.23b, we need to generalize Eq. 3.3 to fields that

describe charged particles. We generalize the separation into creation operators and annihilation

operators,

𝜑𝑞+(𝑥) = ∫

𝑑�⃗�

(2𝜋)32𝑝0 𝑎(𝑝, 𝑞) 𝑒𝑖𝑝∙𝑥 3.24a

𝜑�̃�−(𝑥) = ∫

𝑑�⃗⃗�

(2𝜋)32𝑝0 𝑎†(𝑝, �̃�) 𝑒−𝑖𝑝∙𝑥 3.24b

Then a candidate for the causal field 𝜑𝑞(𝑥) is,

𝜑𝑞(𝑥) = 𝜑𝑞+(𝑥) + 𝜑−𝑞

−(𝑥) = ∫𝑑�⃗�

(2𝜋)32𝑝0[𝑎(𝑝, 𝑞) 𝑒𝑖𝑝∙𝑥 + 𝑎†(𝑝, −𝑞) 𝑒−𝑖𝑝∙𝑥] 3.24c

31

Then Eq. 3.23a is satisfied, as desired, as a consequence of Eq. 3.22a and b!

We learn from this exercise that there must be a doubling of particle species carrying a conserved

quantum number: if an annihilation field 𝜑𝑞+(𝑥) annihilates a particle of charge q, then the

creation field 𝜑−𝑞−(𝑥) must create anti-particles which have charge -q!

Note that most textbooks in quantum field theory [2] do not use the notation 𝑎(𝑝, 𝑞) and

𝑎†(𝑝, −𝑞). Rather, they use the notation,

𝑎(𝑝, 𝑞) → 𝑎(𝑝) 𝑎(𝑝, −𝑞) → 𝑎𝑐(𝑝) 3.25a

where “c” denotes “charge conjugate” and they write the causal field, and its creation and

annihilation parts as,

𝜑(𝑥) = 𝜑+(𝑥) + 𝜑𝑐+†(𝑥) = ∫𝑑�⃗�

(2𝜋)32𝑝0[𝑎(𝑝) 𝑒𝑖𝑝∙𝑥 + 𝑎𝑐†(𝑝) 𝑒−𝑖𝑝∙𝑥] 3.25b

Then,

[𝑄, 𝜑+(𝑥)] = −𝑞𝜑+(𝑥), [𝑄, 𝜑+𝑐(𝑥)] = +𝑞𝜑+𝑐(𝑥) 3.26a

So,

[𝑄, 𝜑(𝑥)] = −𝑞𝜑(𝑥) [𝑄, 𝜑†(𝑥)] = +𝑞𝜑†(𝑥) 3.26b

And for space-like separations the commutator of 𝜑(𝑥) and 𝜑†(𝑦) is,

[𝜑(𝑥), 𝜑†(𝑦)] = [𝜑+(𝑥) + 𝜑𝑐+†(𝑥), 𝜑+†(𝑦) + 𝜑𝑐+(𝑦)]

= [𝜑+(𝑥), 𝜑+†(𝑦)] + [𝜑𝑐+†(𝑥), 𝜑𝑐+(𝑦)] = 0 3.27

by the same observations that led to Eq. 3.21. Eq. 3.27 is the reason behind the term “causal

field”. Note that the vanishing of Eq. 3.27 depends on the prediction that the mass of the particle

and its anti-particle are identical. This insures that the two terms in Eq. 3.27 cancel identically.

Finally, note that 𝜑(𝑥) is not Hermitian. But the Hamiltonian density ℋ(𝑥) must be Hermitian.

So, a possible interaction term which is local, Hermitian, and conserves charge could be

ℋ𝐼(𝑥) = 𝑔 (𝜑†(𝑥)𝜑(𝑥))2

. Then [𝑄, ℋ𝐼(𝑥)] = 0. And, in addition, since [𝜑(𝑥), 𝜑†(𝑦)] = 0 for

(𝑥 − 𝑦)2 < 0, and [𝜑(𝑥), 𝜑(𝑦)] = [𝜑†(𝑥), 𝜑†(𝑦)] = 0, straight-forward algebra gives the

desired result [ℋ𝐼(𝑥), ℋ𝐼(𝑦)] = 0 for (𝑥 − 𝑦)2 < 0.

32

So, we see the necessity of anti-particles! Without them we couldn’t construct a local, causal

quantum field theory of charged particles consistent with local charge conservation.

This was one of the first predictions of quantum field theory that was verified in early cosmic ray

experiments in the 1930’s, with the discovery of positrons, and in early accelerator experiments

in the 1950’s, with the discovery of anti-protons.

Let’s end this Essay with some observations about high resolution measurements. We noted

already that quantum fields carry within them a length scale, the Compton wavelength ℏ/𝑚𝑐 of

the particle the field creates and annihilates. We noted that a theory with only particles, and no

anti-particles, would violate causality at these length scales because ∆+(𝑥 − 𝑦) had this extent.

In the causal theory both particles and anti-particles exist and locality and causality are restored.

But what is the character of a measuring experiment with a resolution finer than the Compton

wavelength ℏ/𝑚𝑐? Imagine a “microscope” (an accelerator, actually) using photons of

wavelength 𝜆. To resolve space-time intervals less than ℏ/𝑚𝑐 we need 𝜆 < ℏ/𝑚𝑐. The energy of

such a photon is

ℏ𝜔 = ℏ2𝜋𝑐

𝜆> ℏ ∙ 2𝜋𝑐 ∙

𝑚𝑐

ℏ~2𝜋𝑚𝑐2

So, the energy of the photon is above the threshold for pair production of the particle whose

internal structure we are trying to observe! As a consequence, the measurement itself produces a

multi-body final state consisting of particles and anti-particles. This was the character of deep

inelastic scattering experiments which “looked inside” target protons using virtual photons

produced by electron beams at the Stanford Linear Accelerator Center (SLAC) in the 1960’s.

They resolved space-time intervals much smaller than the Compton wavelength of target protons

and discovered it’s constituents, colored quarks of the relativistic field theory Quantum

Chromodynamics.

References

1. Steven Weinberg, The Quantum Theory of Fields, Volume 1 Foundations, Cambridge

University Press, Cambridge, U.K., 1995

2. The Principles of Quantum Mechanics, P. Dirac, Oxford University Press, Oxford,

U.K. 1930.