The Mechanics of the Fermionic and Bosonic Fields: An Introduction to the Standard Model and Particle Physics

Evan McCarthy

Phys. 460: Seminar in Physics, Spring 2014 Aug. 27, 2014 !

1.Introduction 2.The Standard Model of Particle Physics

2.1.The Standard Model Lagrangian 2.2.Gauge Invariance

3.Mechanics of the Fermionic Field 3.1.Fermi-Dirac Statistics 3.2.Fermion Spinor Field

4.Mechanics of the Bosonic Field 4.1.Spin-Statistics Theorem 4.2.Bose Einstein Statistics

5.Conclusion !!1. Introduction

While Quantum Field Theory (QFT) is a remarkably successful tool of quantum particle

physics, it is not used as a strictly predictive model. Rather, it is used as a framework within

which predictive models - such as the Standard Model of particle physics (SM) - may operate.

The overarching success of QFT lends it the ability to mathematically unify three of the four

forces of nature, namely, the strong and weak nuclear forces, and electromagnetism. Recently

substantiated further by the prediction and discovery of the Higgs boson, the SM has proven to

be an extraordinarily proficient predictive model for all the subatomic particles and forces. The

question remains, what is to be done with gravity - the fourth force of nature? Within the

framework of QFT theoreticians have predicted the existence of yet another boson called the

graviton. For this reason QFT has a very attractive allure, despite its limitations. According to

!1

QFT the gravitational force is attributed to the interaction between two gravitons, however when

applying the equations of General Relativity (GR) the force between two gravitons becomes

infinite! Results like this are nonsensical and must be resolved for the theory to stand.

Unfortunately no means to purge these anomalies from the theory have yet been developed. For

this reason many physicists have turned to String theory as a means of resolving these anomalies,

as it successfully describes how two gravitons interact without these unwanted infinities.

In this paper I present independent discussions on six of the foundational topics of

modern and particle physics, which are then related and presented as a consequence of each other

in the conclusion. I focus on the structure of the standard model Lagrangian and present a

concise overview of how it is derived and constructed, followed by a discussion on gauge

invariance and how it presented a small crisis for the standard model. Then there will be

discussions on the the statistical distribution of fermions and bosons, and the nature of there

interactions; namely, exploring the concept of spinor fields and the spin-statistics theorem using

these fields.

!2. The Standard Model of Particle Physics

The Standard Model (SM) is a model of particle physics designed to explain how the

particles of matter - known as fermions - and the force carrying particles - known as bosons -

interact according to the four forces of nature. In creating this model, physicists have divided all

the particles of the SM into a few categories to sort them by mass, spin, and type. Fermions

(quarks and leptons) all have a fractional spin and quarks take part in the strong nuclear

interaction so they exist in the nucleus of the atom and thus have not been directly observed.

!2

Experiments done in places like the Large Hadron Collider (LHC) at CERN, FermiLab in

Chicago, and other accelerator laboratories around the world have repeatedly confirmed the

existence of quarks from the theoretical predictions of the SM. The most significant of these

experiments was the discovery of the J/ψ particle. Before 1974 the quark model had remarkable

success as a mathematical theory, and demonstrating the existence of the charm quark would

confirm the validity of the theory. In particle physics there are two sorts of composite particles

called mesons and baryons; all baryons are fermions and all mesons are bosons. Baryons are

composed of three quarks and mesons are composed of two quarks. The J/ψ particle is a meson

and is composed of a charm quark and a

charm anti-quark, and when its discovery was

announced by two independent research

groups - at Brookhaven National Laboratory

and at Berkeley - on November 10, 1974 the

charm quark was no longer a mathematical

speculation. This would come to be known as

the November revolution amongst the physics

community [1].

There are the quarks and their anti-

particles that are categorized by ascending

mass into three generations known as

generation I, II, and III. In generation I there

are the lightest quarks - the up and down

!3

Generation I

Generation II

Generation III

Name UP CHARM TOPSpin 1/2 1/2 1/2

Charge 2/3 2/3 2/3Mass 2.4 MeV 1.27 GeV 171.2 GeVName DOWN STRANGE BOTTOMSpin 1/2 1/2 1/2

Charge -1/3 -1/3 -1/3Mass 4.8 MeV 104 MeV 4.2 GeVName ELECTRON MUON TAUSpin 1/2 1/2 1/2

Charge -1 -1 -1Mass 0.511 MeV 105.7 MeV 1.777 GeV

Name ELECTRON NEUTRINO

MUON NEUTRINO

TAU NEUTRINO

Spin 1/2 1/2 1/2Charge 0 0 0Mass <2.2 eV <0.17 MeV <15.5 MeVName PHOTON GLUON ZSpin 1 1 1

Charge 0 0 0Mass 0 0 91.2 GeVName W+ W- HIGGSSpin 1 1 0

Charge +1 -1 0Mass 80.4 GeV 80.4 GeV ≈126 GeV

F e r m i o n s

Q u a r k s

L e p t o n sB

o s o n s

quarks, in generation II there are the charm and strange quarks, and in generation III the heaviest

quarks - the top and bottom quarks. The up, charm, and top quarks all have a positive 2/3 charge,

and the down, strange, and bottom quarks all have a -1/3 charge [2].

There are also leptons which are particles that do not have any part of the strong nuclear

interaction and so have been able to be directly observed. Most people are to some extent

familiar with leptons, as they are electrons, muons, tau particles, and their corresponding

neutrinos. Leptons are categorized into the same three generations of ascending mass; in

generation I there is the electron an electron neutrino, in generation II the muon and muon

neutrino, and in generation III the tau and tau neutrino. The electron, muon, and tau all have a -1

charge, and the electron neutrino, muon neutrino, and tau neutrino all have a 0 charge, and as all

leptons are fermions they all have a fractional spin.

Forces are experienced between particles of matter, like the gravitational force between

two massive bodies. The SM accounts for this supposed ‘action at a distance’ by assigning a

boson to each force. When two matter particles called fermions interact in this model they

exchange force carrier particles called bosons. All fermions “transfer discrete amounts of energy

by exchanging bosons with each other” and this is perceived as a force acting on those fermionic

particles [3]. The fundamental forces of electromagnetism, the strong nuclear force, the weak

nuclear force, and the gravitational force are theorized to be carried by the photon, gluon, W+, W-

and Z bosons [4], and graviton respectively. Though the graviton is yet to be observed, the recent

discovery of the Higgs boson is a good indicator that there is a graviton. 1

!

!4

The Higgs boson is necessary in the SM to preserve gauge invariance. This is discussed in greater detail in 2.2.1

2.1. The Standard Model Lagrangian

The Lagrangian of a system is effectively the difference between the system’s kinetic and

potential energy. This term is used to describe the mechanics of a system in a very succinct and

efficient mathematical expression. The Lagrangian may be used to trace the path of the system

over some period of time with Hamilton’s variational principle, expressed as the action,

(1) "

As the system evolves over time the sum of all the changes in potential to kinetic energy

represent the totality of the different paths the system could have taken, and the maximized

solution is the path actually taken by the system’s evolution [5], such that

(2) "

The SM Lagrangian, LSM, is known for it’s accuracy in predicting the properties and behavior of

all the fundamental particles. The Standard Model Lagrangian comes out of the unification of the

electromagnetic and weak nuclear forces, known as the electroweak theory. While it is true that

the unitary group U(1) defines the electromagnetic gauge field, it is not accurate to say the SU(2)

group defines the weak nuclear bosons, because the weak nuclear force exhibits particularly

massive gauge bosons. Through a method known as spontaneous symmetry breaking the scalar

Higgs field was introduced to give the W± and Z bosons their large mass [6,8] as an attempt to

rectify these necessarily massive gauge bosons with quantum theory, and thereby maintain gauge

invariance. In this new electroweak theory these gauge bosons are naturally massless and are

S = Ldtt0

t

∫

δ Ldtt0

t

∫ = δS = 0

!5

only given mass by the Higgs field. The electromagnetic and weak nuclear forces are naturally

intertwined in a single SU(2) × U(1) gauge symmetry, and when this interacts with the scalar

Higgs field the result is the electromagnetic and the massive, short range, weak nuclear bosons

that manifest as the photon and weak nuclear observables. The electroweak theory has allowed

the long accepted SM Lagrangian to have been such a tremendous success because it has all

three generations of quarks and leptons, one Higgs boson, and it describes the SU(3) × SU(2) ×

U(1) gauge group [7] which corresponds with the SU(3) unitary group the defines the eight

massless gauge fields known as gluons, and the SU(2) × U(1) unitary group that defines the three

massless gauge fields known as the W± and Z bosons and the one massless gauge field known as

the photon [8].

The SM Lagrangian is simplified to its most general form,

(3) "

and is the mathematical representation of all the particles in the SM. It is effectively the sum of

all the smaller Lagrangians that describe the various parts of the theory. So it could be expressed

in terms of these Lagrangians as,

(4) " .

The LField is the field strength tensor and creates a gauge invariant Lagrangian [8] for the massive

vector fields,

(5) " .

Using standard tensor notation Fµν = ∂µ Aν - ∂ν Aµ . In other texts this term is often expressed so to

also include the gluon and boson field strength tensors (see [26]). The LHiggs will add the Higgs

= − + (i ψ + h. c. ) − m ψ − − V(|ϕ|) LSM14

FµνFµν 12

ψγµ∂µ ψ ϕ∣∣∂µ ∣∣2

= + + LSM LField LDirac LHiggs

LField = − 14FµνF

µν

!6

potential and preserve the gauge symmetry through spontaneously breaking the symmetries that

prevent the the gauge W± and Z bosons from having mass.

The LDirac describes the mechanics of the Dirac field which is a spinor field that generates

fermions, and is known as a fermionic spinor field. Though the SM is a gauge theory [7] that 2

describes the dynamics of particles in a four dimensional space-time, the spinor fields that

comprise the theory merely happen to be also four component spinors. This works out because

the spinors that are attributed to a d-dimensional space-time have " components [8]. Given that

the free Dirac Lagrangian is

(6) " ,

then after expanding the Dirac slash and evaluating the projection operators for the left and right

handed Dirac spinors, the SM Lagrangian may be expressed as

(7) " .

The Dirac slash is a notation developed to simplify the use of gamma matrices, also known as

Dirac matrices, that is for any 4-vector, v, defined as

(8) " .

The 1/2[…] + h.c. term comes from the introduction of left and right handed spinors and

the Hermitian nature of the Pauli matrices σµ. The use of ! here is used much like the µ in the

Einstein notation. In more advanced texts that begin to expand the Lagrangian with left and right

handed spinors the expression i!*γµ ∂µ! becomes i!L*γµ ∂µ!L + i!R*γµ ∂µ!R. The field ! that

generates the fermion is known as a Dirac spinor, and these spinors may be operated on by

projection operators that produce that produce the left and right handed components that

2d2

= i ∂ψ − m ψ LDirac ψ/ ψ

= − + (i ψ + h. c. ) − m ψ + LSM14

FµνFµν 12

ψγµ∂µ ψ LHiggs

v := / γµvµ

!7

Fermionic spinor fields are discussed in greater detail in section 3.2.2

manifest in many areas of physics, particularly the weak interaction [27] between fermions that

produces the weak nuclear force. For the left and right handed components the operators are

respectively defined as 1/2(I-γ5) and 1/2(I+γ5), where I is the identity matrix and γ5 is the matrix

defined as a function of the 4x4 Dirac matrices: γ5 = iγ0γ1γ2γ3 [27]. Hamilton’s variational

principle requires the Lagrangian to be real, and the imaginary part of the Lagrangian eventually

produces “irrelevant end-point contributions to the action” [27] according to Cottingham and

Greenwood. These two conditions allow us to ignore that imaginary part of the Lagrangian via

the introduction of the Hermitian conjugate of the previous terms, h.c., so that we finally have

the Dirac Lagrangian density

(9) " .

This is the form presented in Cottingham and Greenwood. This uses the 2x2 Pauli matrices, σ,

and the assumed definition of the dagger as !* = !†γ0.

In QFT and in the current conception of the SM the process known as spontaneous

symmetry breaking allows there to be massive gauge bosons in a theory that predicts an

otherwise intrinsically massless set of gauge bosons. A spontaneously broken symmetry entails

that the system is rotationally invariant. Remember that the SM and QFT are both just playing

with topology, gauge, and group theories, so particles like the bosons are represented by

matrices. As these matrices are rotated they typically generate different properties of the

corresponding particles. However if a particle’s matrix is rotated and generates the same result as

it would prior to that rotation, then it is known as a rotationally invariant system. In de Wit’s

1995 lecture series [8] he gives ferromagnets as an example of this rotational invariance.

Consider a non-ferromagnetic material in which the atomic spins are all randomly oriented. This

= [(i + i ) + h. c.] − m( + ) LDirac12

ψ†Lσµ∂µψL ψ†

Rσµ∂µψR ψ†LψR ψ†

RψL

!8

system is in a “rotationally symmetric ground state” because while there is no energy added to

the system (that would align the atomic spins) any axis of symmetry may be established, around

which, the system may be rotated and still possess the same properties. That is to say, regardless

of which direction you point the material it will never exhibit a greater than zero magnetization.

So for a ferromagnet in the ground state the atomic spins are all aligned along a single axis

(north-south) and thus there is a net magnetization greater than zero. This magnetization is the

result of spontaneous symmetry breaking [24] - that is, the rotational symmetry of the non-

ferromagnetic material was broken to establish only one definite axis of symmetry along which

the material is magnetized. For a more rigorous mathematical discussion of spontaneous

symmetry breaking I recommend reading Itzykson and Zuber’s Quantum Field Theory [24].

In the case of the gauge bosons the SM predicted they must be massless, however

experiments demonstrate that they are in fact very massive. Like the non-ferromagnetic material,

the predicted massless gauge bosons were represented by a rotationally invariant system. They

had a property known as gauge invariance. Like magnetizing the non-ferromagnetic material

would induce spontaneous symmetry breaking, mass would break the rotational symmetry of the

gauge bosons. The Higgs field was introduced to preserve this gauge invariance and allow them

to have mass. This was done by introducing the complex spinless field ϕ to the Lagrangian:

(10) "

Let the complex spinless scalar field ϕ(x) be

(11) "

so that the partial derivative of this field with respect to the mass of the particle, µ, becomes

= − − V(|ϕ|) LHiggs ϕ∣∣∂µ ∣∣2

ϕ(x) = ρ(x) 12√

eiθ(x)

!9

(12) "

where ρ is just the unitary gauge condition, i.e. re-parameterization of the scalar field, such that

ρ=ϕ√2. Defining the covariant derivative as

(13) "

permits the expression of the Lagrangian in terms of the covariant derivative and in accordance

with the gauge transformations shown in (2.2.4) [8]. This then takes the final form:

(14) "

!2.2. Gauge Invariance

Gauge invariance is demonstrated when there is some species of field that always

produces the same observable field when operated on [9]. Consider the gauge of the

electromagnetic potential, A(x), with scalars and vectors " and " . If the gauge is changed, then

these scalar and vector quantities become, respectively,

(1) " .

If the E-field and the B-field are defined as

(2) " ,

then under the gauge transformations defined in (2.2.1) these fields are gauge invariant because,

expressed as a function of the arbitrary field ψ, the observables are the same [10].

ϕ(x) = ( ρ + iρ θ) ∂µ12√

eiθ(x) ∂µ ∂µ

ϕ = ( ρ − iqρ( − θ)) = ϕ − iq ϕ Dµ12√

eiθ(x) ∂µ Aµ q−1∂µ ∂µ Aµ

= − + [(i + i ) + h. c.] − m( + ) − − V(|ϕ|) LSM14

FµνFµν 12

ψ†Lσµ∂µψL ψ†

Rσµ∂µψR ψ†LψR ψ†

RψL ϕ∣∣∂µ ∣∣2

A !A

A⇒ ′A = A + ∂ψ∂t

!A⇒ ′

!A =

!A −!∇ψ

!E = −

!∇A − ∂

!A∂t

!B =!∇×!A

!10

(3) "

While this example of gauge invariance is unique to electromagnetism, it illustrates the formal

definition that is used in many aspects of physics and mathematics. A system is considered gauge

invariant when there is some genus of scalar and vector potential that always produces the same

observable field when operated on by some field, ψ.

In the context of the SM the aforementioned electromagnetic potential, A(x), becomes an

abelian gauge field typically notated as Aµ. The arbitrary field, ψ, becomes an arbitrary

parameter, ξ(x). Then as seen in [8] the gauge transformations are defined as,

(4) "

"

where q is a parameter that describes the strength of the phase transformations, and ϕ(x) is a

complex spinless field (for example, the spinor field that correlates to the photon or the Higgs

boson: both are spinless and complex particles). These gauge transformations are invariant under

U(1) transformations; the unitary group known as U(1) is just the group of 1×1 unitary matrices.

This means that these gauge transformations are invariant when the fields are operated on by a

1×1 unitary matrix, where a unitary matrix is defined as a complex square matrix, M, such that

the product of M and the conjugate transpose M* is equal to the identity matrix. The conjugate

transpose of a matrix is the matrix obtained by flipping the indices of all the elements, and then

replacing those elements with their complex conjugate. So the most simplistic example of a 1×1

unitary matrix would be M = |i2|, as the transpose of the matrix M = |aij| is simply MT = |aji| such

!E = −

!∇A − ∂

!A∂t

= −!∇ A + ∂ψ

∂t⎛⎝⎜

⎞⎠⎟ −

∂∂t!A −!∇ψ( )

!B =!∇×!A =!∇×

!A −!∇ψ( )

ϕ(x) → (x) = ϕ(x) ϕ ′ eiqξ(x)

(x) → (x) = (x) + ξ(x) Aµ A ′µ Aµ ∂µ

!11

that MT = |i2|, and the complex conjugate of MT = |i2| happens to be equal to itself because the

complex conjugate of -1 is -1. The product of |i2||i2| is equal to |1|, which is the identity matrix.

Therefore the matrix M is unitary. If this M were to operate on ϕ(x), then ϕ’(x) would become

equal to the product Mϕ(x). In this example, given the gauge transformations from (2.2.4), the

field ϕ(x) would transform to exp[iqξ(x)] ϕ(x), producing the relation

(5) " .

This entails that the arbitrary parameter ξ(x), must allow M = exp[iqξ(x)].

Like magnetizing the non-ferromagnetic material in section (2.1) would induce

spontaneous symmetry breaking, intrinsically massive bosons would break the rotational

symmetry of the gauge bosons so the Higgs field was introduced to preserve this gauge

invariance and allow them to have mass. This was so important because the bosons have been

observed to be very massive and so it really saved and even resurrected the Standard Model.

!3. Mechanics of the Fermionic Field

The fermionic field is a mathematical conception designed to generate the fermionic

particles observed in the SM. This is not to say there is a physical field that imparts some

physical force on particles, rather, the field is more like a metaphysical entity that enables

fermions to possess certain properties. The mechanics that govern these properties are

themselves dictated by the dynamics of the field. The field is a spinor field who’s quanta are the

fermionic particle spinors. A spinor is a mathematical entity that, when rotated through 2π

radians, transforms into its negative [11]. It at first seems absurd to assert this property belongs

Mϕ(x) = ϕ(x) eiqξ(x)

!12

to real objects because in the classical realm of physics there is no such object. If I invert my pen

through π radians, then it may be said to be in the negative state, i.e. ‘down’ instead of ‘up.’

However if I continue the rotation through 2π radians it is restored to its original positive state,

i.e. ‘up.’ However in quantum mechanics and particle physics there are objects Penrose calls

spinorial objects that have this property.

!3.1. Fermi-Dirac Statistics

In 1926 Enrico Fermi and Dirac both independently developed a set of statistics that

describes the energy distribution of a weakly interacting, or non-interacting, gas of identical

particles that obey the Pauli exclusion principle [12]. By definition of this principle - that no two

identical fermions can occupy the same state [13] - then all fermions must behave according to

the Fermi-Dirac statistics. The Fermi-Dirac statistics are the set of statistical equations that

describe the distribution of fermions in thermal equilibrium. There is also the set of Bose-

Einstein statistics that will be discussed in section 4, that describe the distribution of bosons.

Consider a system in which a single particle has energy ". Either a state within the system has is

occupied by the particle or it is not, i.e. this state has energy " or 0. If there are multiple n

particles in this state, then this state has energy n" or 0 [13]. After one has derived the Gibbs

factor,

(1) "

the grand partition function, Z, is simply the sum of the Gibbs factors of all the states, s, in the

system. The grand partition function for a system with n fermions is given by substituting the

e− [E(s)−µN(s)]1kT

!13

N(s) with n and the total energy of the state E(s) with n" and determining the sum of these terms

over all the states. Given the probability that a state will be occupied by n fermions is the ratio of

the Gibbs factor for n fermions over all the possible states, then the probability that a state will

be occupied by n fermions is expressed as

(2) " .

From the Pauli exclusion principle it is necessarily true that n may be only 0 or 1, as there cannot

be more than one fermion in the same state in any given system. Under these conditions the

grand partition function for fermions can be calculated as the sum

(3) " .

From (3) it is evident that if n is 0 and there are no fermions in the system then there cannot be

any number of states, s, with energy greater than 0, as there are no particles to energize those

states. So the sum goes from 0 to 0 and Z = 1. From equations (2) and (3) P = 1, which is to say

that if there are no particles in a system, then the probability that the system is filled with n = 0

particles is 1.

Voila! The fundamentals of Fermi-Dirac statistics. Given a system in thermal equilibrium

with constant chemical potential, the particle distribution of n number of fermions within said

system can be determined using equations (2) and (3). Typically this section is summarized by

the expression known as occupancy and is the average number of particles in a given state,

(4) " ,

and is more frequently known as the Fermi-Dirac distribution [13].

= P 1Z

e− (ϵ−µ)nkT

Z = = 1 + ∑s

e−s(ϵ−µ)

kT e− (ϵ−µ)kT

nP = = ∑n=0

1 e− (ϵ−µ)1kT

1 + e− (ϵ−µ)1kT

11 + e (ϵ−µ)1

kT

!14

!3.2. Fermion Spinor Field

As mentioned in section 3, there are objects in physics known as spinorial objects. When

students begin to study quantum physics they learn about the wavefunction of the electron and

how to perform calculations, make simple predictions, and begin to see why the quantum model

of physics is so unique, and frankly, bizarre. Not all students realize is that these wavefunctions

are fermionic wavefunctions that describe a fermion and are themselves consequently spinorial

objects [14].

The order of a matrix is simply a numeration of its rows and columns. A square 4x4

matrix is a matrix of order 4, and a 4x5 matrix would have order 4x5. In physics, the fields like

those that depict fermions and bosons are expressed as matrices. A particle with any spin s may

be depicted by a spinor field of order 2s [15], and as such a spinor field is capable of describing

any particles with half-integer spin, i.e. fermions. Fermions will be depicted by a spinor field of

either an odd or even order. Any particle with spin s = 1/2, 2/2, 3/2 … n/2 will generate a spinor

of order 2s = 1, 2, 3 … n respectively. However, as an order 4s spinor is sufficient to express the

same information expressed with an order 2s tensor [15] we see that, as bosons only have integer

spin, a boson with spin s = 1, 2, 3 … m will generate a tensor of order 2s = 2, 4, 6 … m which

has the same amount of information as the fermionic spinor of order 4s = 2, 4, 6 … n. From this

comparison it is clear that bosons of integer spin may be expressed by either a tensor field or a

spinor field of an order twice as large, but fermions of half - integer spin may only be expressed

by a spinor field, as it is not possible to have a tensor of fractional order.

For an introductory paper, I don’t think it is entirely necessary or even appropriate to

!15

enter into a discussion of spinor and tensor calculus that is used to quantize spinor fields and

generate particles. However I recommend Corson’s Introduction to Tensors, Spinors, and

Relativistic Wave-Equations [25].

!4. Mechanics of the Bosonic Field

The fermionic wavefunctions are spinorial objects, and therein lies the fundamental

distinction between fermions and bosons: the wavefunctions of fermions exhibit this spinor

property of rotational symmetry, whereas bosonic wavefunctions do not. The order of variables

in the wavefunction of the boson is not a sufficient condition to change the value of the

wavefunction. This basically means that as the wavefunction is a function of spatial and temporal

coordinates the boson can change its place in space-time and maintain constant properties. As 3

Penrose explains, “the function ψ = ψ(u,u; v,v) should be symmetric under the interchange of the

particles” [16]:

(1) " .

This is different from the anti-symmetric interchange of two fermions which - as previously

described in section 3 - do not maintain exhibit rotational symmetry, and is shown as

(2) " .

Like the fermionic field, the bosonic field dictates the properties of it’s quanta; namely,

bosons. We see from these expressions that the fermion must change state when it changes

ψ(u,u; v,v)= ψ(v,v; u,u)

ψ(u,u; v,v)= -ψ(v,v; u,u)

!16

In Penrose’s book he uses u and v to denote the points in space and u and v to denote parameters that define the 3

group of spinor or tensor indices for each particle.

position in space with another fermion, unlike the boson which may maintain its state under 4

such an interchange.

!4.1. Spin-Statistics Theorem

The Spin-Statistics theorem demonstrates the connection between the spin of a particle

and its statistics describing its position. Unlike bosons, the wavefunctions of fermions exhibit the

spinor property of rotational symmetry. This is to say that according to the theorem the

wavefunction of the system of two identical particles with integer spin will maintain its value

when the position of the particles are interchanged, i.e. the fields of bosons which have a

symmetric wavefunction and integer spin commute [17]. Whereas, for the antisymmetric

wavefunction of the fermion, the fields are not commutative and the value of the system’s

wavefunction will change when the spin-1/2 particles are interchanged under those same

conditions as seen by equation (17) in [18].

In the modern era of physics ever more advanced experiments are being developed to

probe the quantum realm. With the advent of Bose-Einstein condensates, a super cooled state of

matter in which a gas of bosons is chilled to near absolute zero, physicists are able to observe

further evidence for the validity of the spin-statistics theorem. Not only are there now proofs for

the theorem, but there is physics evidence for it. In light of this wealth of knowledge a proposed

field theory that either ignores or eliminates the necessity of the spin-statistics theorem would

likely fail. So it seems necessary to develop a field theory to which the spin-statistics theorem

!17

Penrose and I only discuss spatial coordinates, but there is no reason the same argument cannot extend to temporal 4

coordinates of the wavefunction as well. This is to say, then, that the fermion must change state when it changes position in space-time…

may apply, namely, it must satisfy three requirements [19]: (1) It must have Lorentz invariance

and relativistic causality such that quantum fields at two points in space x and y are separated in

space by (x-y)2 < 0. (2) All particles defined by the field theory must have positive energies. (3)

The field theory must have a Hilbert space with states which all have positive norms.

It is worth noting at this point that by definition if all the states are physical states, then

they all have positive norms. In Hilbert space the norm is defined as the root of the inner product

of some vector with itself, and allows the vector space to become a complete metric space [20].

Hilbert space is a vector space, which means it is a closed set “under finite vector addition and

scalar multiplication” [21]. This simply means that while operating within this space there are

only a finite number of ways to add any two vectors and to multiply any two scalars. Bearing this

in mind, the Hilbert space possess the norm

(1) " .

This norm must also transform the vector space into a complete metric space, which only means

“every Cauchy sequence is convergent” [22]. For the unfamiliar reader, a Cauchy sequence is

any sequence of points n1, n2, n3, … in a space with a metric (a nonnegative function that acts as

a measure of distance between two points in a given set) that satisfies the equality [23]

(2) " .

Therefore a Hilbert space is a set of vectors such that every member of that set conforms

to equation (1) and because of that conformity any two members of the set conform to the

equality (2).

The remarkable consequence of the spin-statistics theorem is the profound connection

| | = a ⟨ , ⟩a a− −−−√

0 = d( , ) limmin(i,j)→∞

ni nj

!18

between spin and quantum state. As far as physicists can tell, it is a fact that a quantum particle’s

spin plays a vital role in determining its state. That this is how the world operates on very small

scales is mechanically fairly well understood, but why this connection exists is still baffling. For

the more advanced and inquisitive student, I have cited a proof of the spin-statistics theorem [19]

worked out by Dr. Vadim Kaplunovsky of the University of Texas for his students. I will only

provide a summary and walkthrough of the proof.

There are two lemmas which Kaplunovsky demonstrates in more detail that I will simply

take for granted: Firstly, when the two spin sums, FAB (p) and HAB (p) are expressed as

polynomials in the four-momenta pµ, then they will also hold for momenta that do not satisfy the 5

classical equations of motion (known as off shell momenta, as opposed to on shell momenta

which would satisfy the classical equations of motion). Remember the first criterion the the field

must have Lorentz invariance and relativistic causality, so just like beginning physics students

learn about treating momentum in three classical dimensions, this 4-momentum raises that same

concept of multi-dimensional momentum into four space-time dimensions (the traditional three

spatial dimensions and now the added temporal dimension) and this 4-momentum becomes a 4-

component Lorentz vector. Using this notation makes the mathematics easier to compute and it

keeps notation relatively simple. Given the particles with mass M, the energy-momentum

relationship which classically has p0 = Ep for on shell momenta is now expressed for off shell

momenta such that p0 ≠ Ep = (p2 + M 2)1/2. Secondly, when those sums are expressed as off shell

polynomials for particles of integral spin it is necessarily true that these spin sums are

symmetrical such that FAB (pµ) = HAB (-pµ), and for particles of half-integral spin it is necessarily

!19

Here µ is used as in the Einstein notation, where pµ corresponds to the momentum component in each of the four 5

dimensions: d = 3+1.

true that -FAB (pµ) = HAB (-pµ).

With these two lemmas in mind Kaplunovsky may now use annihilation and creation

operators to express the relation between any two free quantum fields, ϕA and ϕB as

(3) " ,

where D(x-y) is just a substitution for an integral of Ep with respect to p [19]. Similarly,

Kaplunovsky derives

(4) " .

Upon further examination of the D(x-y) integral, it may be demonstrated that D(y-x) = D(x-y).

From the second lemma it is clear that while (x-y)2 < 0, equation (3) is equivalent to equation (4)

for particles with an integral spin as seen in equation (5), and for particles with half-integral spin

equation (3) is equivalent to equation (4) but with opposite sign, as seen in equation (6).

(5) "

(6) "

It follows from these relations that when we require (x-y)2 < 0, for bosonic fields the relation

must be true,

(7) " ,

and for fermionic fields the relation must be true,

(8) " .

Equations (5) through (8) demonstrate that all particles with integral and half-integral spin must

be bosons and fermions respectively [19].

⟨0 (x) (y) 0⟩ = (i )D(x − y) ∣∣ϕA ϕ

†B

∣∣ FAB ∂x

⟨0 (y) (x) 0⟩ = (−i )D(y − x) ∣∣ϕ

†B ϕA

∣∣ HAB ∂x

⟨0 (x) (y) 0⟩ = ⟨0 (y) (x) 0⟩ ∣∣ϕA ϕ

†B

∣∣

∣∣ϕ

†B ϕA

∣∣

⟨0 (x) (y) 0⟩ = −⟨0 (y) (x) 0⟩ ∣∣ϕA ϕ

†B

∣∣

∣∣ϕ

†B ϕA

∣∣

(x) (y) = (y) (x) ϕA ϕ†B ϕ

†B ϕA

(x) (y) = − (y) (x) ϕA ϕ†B ϕ

†B ϕA

!20

!4.2. Bose-Einstein Statistics

Section 3.1 introduced the derivation of the Fermi-Dirac statistics, and in much the same

way we may derive the Bose-Einstein distribution. We know from the Pauli exclusion principle

that no two fermions may occupy the same quantum state, but this does not hold for bosons. This

is a principle most of us take for granted everyday. The fact that photons are bosons (carrier

particles for the electromagnetic force) allows them to be aligned in lasers into the same quantum

state to produce a uniform and directed monochromatic beam, and this would not be possible if

they were fermions. This principle is necessary to derive the Bose-Einstein distribution for

bosons. Remember the grand partition function from equation 3.1.3. This was derived for

fermions which can only occupy a particular quantum state with one or zero number of particles

(so s = 1 or s = 0), so the sum ended after 2 terms. However, as any number of bosons can

occupy the same quantum state, the same partition function will carry on ad infinitum. So for

bosons the grand partition function becomes

(1) " .

As seen also in equation (7.25) in Schroeder [13] this sum nicely reduces to

(2) " .

To determine the average number of bosons in particular state equation 3.1.4 will still hold true

for bosons, but the sum will have an infinite upper bound, not 1 as for fermions. Similarly,

equation 3.1.2 will also still hold for bosons. As such, we determine the probability that the

Z = = 1 + + + +. . . ∑s

e−s(ϵ−µ)

kT e− (ϵ−µ)kT e−2 (ϵ−µ)

kT e−3 (ϵ−µ)kT

Z = = ∑s

e−s(ϵ−µ)

kT1

1 − e− (ϵ−µ)kT

!21

particular state will be occupied by n number of bosons is

(3) " .

As already computed for the fermions in section 3.1, we evaluate the occupancy to determine the

average distribution of bosons in the system, and consequently this will be the Bose-Einstein

distribution:

(4) " .

From equations 3.1.4 and 4.2.4 we see that the distribution of fermions and bosons in a system is

determined by the energy of the system in the forms of temperature of the system, T, the

chemical potential, µ, and the individual energy of a single particle, ".

!5. Conclusion

The quantum theory of fields has proven itself to be a reliable and powerful framework

for modern physics. QFT establishes a rigorous mathematical basis that allows mathematicians

and physicists to approach the traditional questions of matter and energy from a modern and

refreshed perspective. Since the advent of quantum mechanics in the early 20th century many

great minds have made countless attempts to rationalize the bizarre nature of modern physics.

One of the more well known attempts was Einstein’s failure to unify gravity and

electromagnetism before his death. However, throughout the decades mathematicians and

physicists have developed tools and methods to approach that problem of unification. QFT came

= P 1Z

e− (ϵ−µ)nkT

n = n = ∑n=0

∞ 1Z

e−nϵ−µ

kT ∑n=0

∞ ( )e−nϵ−µ

kT

1

(1− )e−

ϵ−µ

kT

1− 1e

ϵ−µ

kT

!22

to unify three of the four known forces of nature, namely, electromagnetism, the strong and the

weak nuclear forces. It achieved this unification by approaching the problem from a new

perspective, from which all the forces of nature interacted via particles that were only a little

different from particles such as the electron. Electrons were categorized under the new term as

fermions - particles of matter. These particles carried material from point to point in spacetime

and interacted with each other to create the immediately observable world. But forces interacted

with the world through the exchange of bosons - force carrying particles. These particles would

very literally facilitate the exchange of force between two fermions. The photon is an example of

a boson, as it is responsible for the exchange of electromagnetic force and energy.

From this newfound perspective of field theory all the forces except gravity came

together. Through the use of spin — tensors, more commonly known as spinors [25], even the

particles came to be represented as fields that interact with other fields. The standard model

evolved out of this methodology, although the SM Lagrangian is typically regarded as an ugly

and piecemeal work of clever mathematics designed to give physicists a usable and testable

model for standard particle physics. The SM Lagrangian derived in section 2.1 is able to generate

testable predictions for all the particles in physics, and furthermore it describes the properties of

all the particles. Its formulation sparked a minor crisis over preserving gauge invariance for the

massive bosons which generated the newest revelation in physics, namely, the prediction of the

Higgs field and the recent discovery of the consequently necessary Higgs boson. I expect the

next generation of physicists will begin to tackle the old issue of unifying gravity with the other

forces, but they will not be able to do it without the SM (and/or its replacement) and the methods

of QFT.

!23

While the SM has allowed physicists to grasp a descent understanding of the properties of

particles, field theory has allowed physicists to gain a better understanding of particle

interactions. The use of field theory has produced many more vital components to the cosmic

puzzle of physics including the spin-statistics theorem, Fermi-Dirac statistics, Bose-Einstein

statistics, and more than are discussed in this essay. The scientific community has reached an

impressive level of understanding the universe thorough an understanding of the topics I have

discussed. The mechanics described by the SM Lagrangian, the importance and consequences of

gauge invariance, and the properties and statistical behavior of fermions and bosons has finally

yielded one more attempt to develop a unified field theory of gravitation and quantum

mechanics. QFT and the SM have come together to predict the existence of the graviton - a new

boson responsible for carrying the force of gravity.

!24

References ![1] A. Khare, Curr. Sci., 77 (09), 1210 (1999) [2] B. Tatischeff, I. Brissaud, arXiv:1005.0238 [physics.gen-ph] (2010) [3] CERN, The Standard Model, WWW Document, (http://home.web.cern.ch/about/physics/ standard-model) [4] M. Tegmark, Our Mathematical Universe, (Alfred A. Knopf, New York, NY, 2014), pp. 161-162 [5] G. R. Fowles and G. L. Cassiday, Analytical Mechanics, 7th Ed. (Brooks/Cole, Boston, MA, 2005), pp. 419 [6] T. Gherghetta, B. von Harling, A.D. Medinaa, M.A. Schmidta, JHEP, 2013 (02), 32, 2013 [7] H. Davoudiasl, R. Kitano, T. Li, H. Murayama, Phys. Lett. B, 609, 117 (2005) [8] B. de Wit, Introduction to Gauge Theories and the Standard Model, CERN Academic Training Lecture Series (1995) [9] R. Penrose, The Road to Reality, (Alfred A. Knopf, New York, NY, 2005), pp. 451 [10] A. Messiah, Quantum Mechanics Two Volumes Bound as One, (Dover Publications, Inc., Mineola, NY, 1999), pp. 918 [11] R. Penrose, The Road to Reality, (Alfred A. Knopf, New York, NY, 2005), pp. 204 [12] S. Chaturvedi, S. Biswas, Resonance, 19 (01), 45 (2014) [13] D. V. Schroeder, An Introduction to Thermal Physics, (Addison Wesley Longman, San Francisco, CA, 2000), pp. 263-267 [14] R. Penrose, The Road to Reality, (Alfred A. Knopf, New York, NY, 2005), pp. 594 [15] Said, Salem. "Spinor Field." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/SpinorField.html [16] R. Penrose, The Road to Reality, (Alfred A. Knopf, New York, NY, 2005), pp. 596-597 [17] S. Weinberg, The Quantum Theory of Fields Volume 1, (Cambridge University Press, New York, NY, 1995), pp. 201-202 [18] I. Duck, E.C.G. Sudarshan, Am. J. Phys., 66, 284 (1998) [19] V. Kaplunovsky, Spin-Statistics Theorem, University of Texas Class Handout (Unpublished) (http://bolvan.ph.utexas.edu/~vadim/classes/2008f.homeworks/spinstat.pdf) [20] Weisstein, Eric W. "Hilbert Space." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HilbertSpace.html [21] —— "Vector Space." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/VectorSpace.html [22] —— "Complete Metric Space." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CompleteMetricSpace.html [23] —— "Cauchy Sequence." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CauchySequence.html [24] C. Itzykson, J.B. Zuber, Quantum Field Theory, (McGraw-Hill, New York, NY, 1980), pp. 163-197, 519 [25] E. M. Corson, Introduction to Tensors, Spinors, and Relativistic Wave-Equations, (Hafner Publishing Company, New York, NY, 1953), pp. 16 [26] W.N. Cottingham, D.A. Greenwood, An Introduction to the Standard Model of Particle Physics, (Cambridge University Press, Cambridge, 1998), pp. 37-46 [27] —— An Introduction to the Standard Model of Particle Physics, (Cambridge University Press, Cambridge, 1998), pp. 53-55