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

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 3.1.Fermi-Dirac Statistics 3.2. Field 4.Mechanics of the Bosonic Field 4.1.-Statistics Theorem 4.2.Bose Einstein Statistics !5.Conclusion ! 1. Introduction

While (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 . Recently substantiated further by the prediction and discovery of the Higgs , 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 (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 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 and , 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 - 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

( 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 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 Generation Generation Generation I II III Q u a r k s charm anti-quark, and when its discovery was Name UP CHARM TOP Spin 1/2 1/2 1/2 F e r m i o n s announced by two independent research Charge 2/3 2/3 2/3 Mass 2.4 MeV 1.27 GeV 171.2 GeV groups - at Brookhaven National Laboratory Name DOWN STRANGE BOTTOM Spin 1/2 1/2 1/2 and at Berkeley - on November 10, 1974 the Charge -1/3 -1/3 -1/3 Mass 4.8 MeV 104 MeV 4.2 GeV charm quark was no longer a mathematical Name MUON TAU L e p t o n s Spin 1/2 1/2 1/2 speculation. This would come to be known as Charge -1 -1 -1 Mass 0.511 MeV 105.7 MeV 1.777 GeV ELECTRON MUON TAU the November revolution amongst the physics Name NEUTRINO NEUTRINO community [1]. Spin 1/2 1/2 1/2 Charge 0 0 0 There are the quarks and their anti- Mass <2.2 eV <0.17 MeV <15.5 MeV Name PHOTON GLUON Z B o s n particles that are categorized by ascending Spin 1 1 1 Charge 0 0 0 mass into three generations known as Mass 0 0 91.2 GeV Name W+ W- HIGGS generation I, II, and III. In generation I there Spin 1 1 0 Charge +1 -1 0 are the lightest quarks - the up and down Mass 80.4 GeV 80.4 GeV ≈126 GeV

!3 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 , muons, tau particles, and their corresponding . 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 ’ 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 , gluon, W+, W- and Z bosons [4], and graviton respectively. Though the graviton is yet to be observed, the recent discovery of the is a good indicator that there is a graviton.1 !

1 The Higgs boson is necessary in the SM to preserve gauge invariance. This is discussed in greater detail in 2.2.

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

t (1) " S = ∫ L dt t0

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

t (2) "δ ∫ L dt = δS = 0 t0

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

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

!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,

1 µν 1 µ 2 LSM = − FµνF + (iψˉγ ∂µψ + h. c. ) − mψˉψ − ∣∂µϕ∣ − V(|ϕ|) (3) " 4 2 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) L" SM = LField + LDirac + LHiggs.

The LField is the field strength and creates a gauge Lagrangian [8] for the massive vector fields,

1 (5) " L = − F F µν . Field 4 µν

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 (see [26]). The LHiggs will add the Higgs

!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.2 Though the SM is a [7] that 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 . This works out because

d the spinors that are attributed to a d-dimensional space-time have " 2 2 components [8]. Given that the free Dirac Lagrangian is

(6) L" Dirac = iψˉ/∂ψ − mψˉψ, 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

1 µν 1 µ LSM = − FµνF + (iψˉγ ∂µψ + h. c. ) − mψˉψ + LHiggs (7) " 4 2 .

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

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

µ (8) /"v := γ vµ.

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

µ µ µ handed spinors the expression i!*γ ∂µ! becomes i!L*γ ∂µ!L + i!R*γ ∂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

2 Fermionic spinor fields are discussed in greater detail in section 3.2.

!7 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 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

1 † µ † µ † † LDirac = iψ σˉ ∂µψ + iψ σ ∂µψ + h. c. − m ψ ψ + ψ ψ (9) " 2 [( L L R R) ] ( L R R L).

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

!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:

2 (10) L" Higgs = −∣∂µϕ∣ − V(|ϕ|)

Let the complex spinless ϕ(x) be

1 ϕ(x) = ρ(x)eiθ(x) (11) " √2 so that the partial derivative of this field with respect to the mass of the particle, µ, becomes

!9 1 iθ(x) ∂µϕ(x) = e (∂µρ + iρ∂µθ) (12) " √2 where ρ is just the unitary gauge condition, i.e. re-parameterization of the scalar field, such that

ρ=ϕ√2. Defining the as

1 iθ(x) −1 Dµϕ = e ∂µρ − iqρ Aµ − q ∂µθ = ∂µϕ − iqAµϕ (13) " √2 ( ( )) 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:

1 µν 1 † µ † µ † † 2 LSM = − FµνF + iψ σˉ ∂µψ + iψ σ ∂µψ + h. c. − m ψ ψ + ψ ψ − ∣∂µϕ∣ − V(|ϕ|) (14) " 4 2 [( L L R R) ] ( L R R L) !

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 " A and " A . If the gauge is changed, then these scalar and vector quantities become, respectively,

∂ψ A ⇒ A′ = A + (1) " ∂t . ! ! ! ! A ⇒ A′ = A − ∇ψ

If the E-field and the B-field are defined as ! ! ! ∂A E = −∇A − (2) " ∂t , ! ! ! B = ∇ × A 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].

!10 ! ! ! ∂A ! ⎛ ∂ψ ⎞ ∂ ! ! E = −∇A − = −∇⎜ A + ⎟ − (A − ∇ψ ) (3) " ∂t ⎝ ∂t ⎠ ∂t ! ! ! ! ! ! B = ∇ × A = ∇ × A − ∇ψ ( )

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,

′ iqξ(x) (4) ϕ" (x) → ϕ (x) = e ϕ(x)

′ A" µ(x) → Aµ(x) = Aµ(x) + ∂µξ(x) 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 of M and the conjugate 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

2 T unitary matrix would be M = |i |, as the transpose of the matrix M = |aij| is simply M = |aji| such

!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

iqξ(x) (5) M" ϕ(x) = e ϕ(x).

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

!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 [E(s)−µN(s)] (1) e" kT 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

!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

1 − n (ϵ−µ) e kT = P (2) "Z .

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

ϵ µ ϵ µ −s ( − ) − ( − ) Z = e kT = 1 + e kT ∑ (3) " s .

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,

1 − 1 (ϵ−µ) e kT 1 nP = 1 = 1 ∑ − kT (ϵ−µ) kT (ϵ−µ) (4) "n=0 1 + e 1 + e , and is more frequently known as the Fermi-Dirac distribution [13].

!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 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 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 coordinates3 the boson can change its place in space-time and maintain constant properties. As

Penrose explains, “the function ψ = ψ(u,u; v,v) should be symmetric under the interchange of the particles” [16]:

ψ(u,u; v,v)= ψ(v,v; u,u) (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

ψ(u,u; v,v)= -ψ(v,v; u,u) (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

3 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 group of spinor or tensor indices for each particle.

!16 position in space4 with another fermion, unlike the boson which may maintain its state under 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

4 Penrose and I only discuss spatial coordinates, but there is no reason the same argument cannot extend to temporal coordinates of the wavefunction as well. This is to say, then, that the fermion must change state when it changes position in space-time…

!17 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 to become a complete metric space [20].

Hilbert space is a vector space, which means it is a closed set “under finite vector 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 −−−− |aˆ| = ⟨aˆ, aˆ⟩ (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]

0 = lim d(ni, nj) (2) " min(i,j)→∞ .

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

!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-momenta5 pµ, then they will also hold for momenta that do not satisfy the 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 , 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 ) 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

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

0 2 2 1/2 momenta such that p ≠ Ep = (p + M ) . Secondly, when those sums are expressed as off shell polynomials for particles of integral spin it is necessarily true that these spin sums are

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

5 Here µ is used as in the Einstein notation, where pµ corresponds to the momentum component in each of the four dimensions: d = 3+1.

!19 µ µ true that -FAB (p ) = HAB (-p ).

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

† 0∣ϕˆ (x)ϕˆ (y)∣0 = F (i∂ )D(x − y) (3) ⟨" ∣ A B ∣ ⟩ AB x , where D(x-y) is just a substitution for an integral of Ep with respect to p [19]. Similarly,

Kaplunovsky derives

† 0∣ϕˆ (y)ϕˆ (x)∣0 = H (−i∂ )D(y − x) (4) ⟨" ∣ B A ∣ ⟩ AB x .

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).

† † 0∣ϕˆ (x)ϕˆ (y)∣0 = 0∣ϕˆ (y)ϕˆ (x)∣0 (5) ⟨" ∣ A B ∣ ⟩ ⟨ ∣ B A ∣ ⟩

† † 0∣ϕˆ (x)ϕˆ (y)∣0 = − 0∣ϕˆ (y)ϕˆ (x)∣0 (6) ⟨" ∣ A B ∣ ⟩ ⟨ ∣ B A ∣ ⟩

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

ˆ ˆ† ˆ† ˆ (7) ϕ" A(x)ϕ B(y) = ϕ B(y)ϕ A(x), and for fermionic fields the relation must be true,

ˆ ˆ† ˆ† ˆ (8) ϕ" A(x)ϕ B(y) = −ϕ B(y)ϕ A(x).

Equations (5) through (8) demonstrate that all particles with integral and half-integral spin must be bosons and fermions respectively [19].

!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 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

ϵ µ ϵ µ ϵ µ ϵ µ −s ( − ) − ( − ) −2 ( − ) −3 ( − ) Z = e kT = 1 + e kT + e kT + e kT +. . . ∑ (1) " s .

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

(ϵ−µ) 1 −s kT Z = e = ϵ µ ∑ − ( − ) (2) " s 1 − e kT .

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

!21 particular state will be occupied by n number of bosons is

1 − n (ϵ−µ) e kT = P (3) "Z .

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:

−n ϵ−µ ∞ ∞ e kT 1 ϵ−µ 1 −n kT ( ) n e = n 1 = ϵ−µ kT ∑ Z ∑ ϵ−µ e − 1 n=0 n=0 − 1−e kT (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 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

!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, , 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