Relativistic Mean-Field Theory in Quantum Hadrodynamics

Aristotle University of Thessaloniki School of Physics

B.Sc. Thesis

Relativistic Mean-Field Theory In Quantum Hadrodynamics

Author: Theodoros Soultanis Supervisor: Theodoros Gaitanos

September, 2016 1 Abstract

The goal of this Thesis is to introduce the reader to the main ideas regarding relativistic mean-field theory and modeling of nuclear matter. The description of nuclear matter is studied in the framework of the nuclear model called as Quantum Hadrodynamics (QHD). In QHD, we treat protons and neutrons as elementary particles where their interaction is mediated through the exchange of sigma-, omega- and rho- mesons. The prescription follow is simple, firstly we present the most important nuclear properties any nuclear model has to reproduce. In the later chapters we study the basic concepts of classical field theory and then we proceed on the quantization of neutral Klein-Gordon and Dirac fields. Consequently, we compute the energy-momentum tensor for each of the most important relativistic fields. The energy momentum tensor provides the energy density and pressure expressions of the system. Finally, we deploy all the previous knowledge and apply it to QHD. We derive the Euler-Lagrange equations of motion for the QHD-I parameter set and in the next chapter we study more advanced QHD parameter sets. In both cases we apply the mean-field approximation to simplify the complicated equations of motion and derive the equation of state of the system. Finally we compute the theoretical expressions of the symmetry energy coefficient.

2 PerÐlhyh

O stìqoc thc Diplwmatik c ergasÐac eÐnai na eisaggoume ton anagn¸sth stic basikèc idèec ìson afor thn sqetikistik jewrÐa tou mèsou pedÐou kai thn montelopoÐhsh thc purhnik c Ôlhc. H perigraf thc purhnik c Ôlhc epitugqnetai sta plaÐsia tou purhnikoÔ montèlou gnwstì kai wc Kbantik Adrodunamik (QHD). 'Opou antimetwpÐzoume ta prwtìnia kai netrìnia wc stoiqei¸dh swmtia twn opoÐwn h allhlepÐdrash pragmatopoieÐtai me thn antallag twn sÐgma-, wmèga- kai ro- mesìniwn. H dom thc ergasÐac eÐnai apl , arqik parajètoume tic pio shmantikèc idiìthtec thc purhnik c Ôlhc tic opoÐec opoiod pote purhnikì montèlo ofeÐlei na anapargei. Sta epìmena keflaia meletoÔme tic ènnoiec thc klassik c jewrÐac pedÐou kai Ôstera proqwroÔme sthn kbntwsh tou pragmatikoÔ Klein-Gordon kai Dirac pedÐwn. En suneqeÐa, upologÐzoume ton tanust enèrgeiac-orm c gia kje èna apì ta pio shmantik sqetikistik pedÐa. O tanust c enèrgeiac-orm c parèqei tic ekfrseic thc puknìthtac enèrgeiac kai pÐeshc tou sust matoc. En katakleÐdi, epistrateÔoume ìlec tic prohgoÔmenec gn¸seic kai tic efarmìzoume sthn QHD. Exgoume tic Euler-Lagrange exis¸seic kÐnhshc gia to QHD-I set paramètrwn kai sto epìmeno keflaio meletoÔme akìmh pio exeidikeumèna QHD set paramètrwn. Kai stic dÔo peript¸seic efarmìzoume thn prosèggish tou mèsou pedÐou gia na aplopoi soume tic polÔplokec exis¸seic kÐnhshc ¸ste na exgoume thn katastatik exÐswsh tou sust matoc. Tèloc, upologÐzoume tic jewrhtikèc ekfrseic tou suntelest thc enèrgeiac summetrÐac.

3 Contents

1 Introductory Concepts 5 1.1 Notation ...... 5 1.2 Nuclear Matter ...... 6 1.3 Nuclear Properties ...... 8

2 Field Theory 9 2.1 Lagrangian Formalism ...... 9 2.2 Hamiltonian formalism ...... 11 2.3 Noether’s Theorem and Conservation Laws in Field Theory ...... 12 2.3.1 Invariance under translation ...... 12 2.3.2 Internal Symmetries-Noether Charge ...... 13 2.4 Relativistic Fields ...... 14 2.4.1 Klein-Gordon Field ...... 14 2.4.2 Dirac Field ...... 16 2.4.3 Photon Field ...... 19 2.4.4 Massive Vector Boson Field ...... 20

3 Quantum Field Theory 23 3.1 Harmonic Oscillator ...... 23 3.2 Neutral Klein-Gordon Field ...... 24 3.3 Dirac Field ...... 28

4 Energy-Momentum Tensor 35 4.1 Relativity-EMT-Perfect Fluid ...... 35 4.2 EMT- Klein-Gordon Field ...... 36 4.3 EMT- Dirac Field ...... 36 4.4 EMT- Massive Vector Boson Field ...... 37

5 Quantum Hadrodynamics - QHD 39 5.1 QHD-I ...... 40 5.2 Relativistic Mean-Field Approximation ...... 41 5.3 Baryon Field - Expectation values ...... 42 5.4 Parameter Set, Equation of State, Observables ...... 47 5.5 Alternative Method - Expectation Values ...... 49

6 Advanced QHD Parameter Sets 51 6.1 Formalism and Conventions ...... 52 6.2 Relativistic Mean-Field Approximation ...... 54 6.3 Expectation Values ...... 56 6.4 Parameter Sets, Equation of State, Observables ...... 58

7 Results 61

Bibliography 64

4 Chapter 1

Introductory Concepts

1.1 Notation

In this work the 3-dimensional vectors are expressed either with bold font, p, or with the indexes i, j such as, xi, while the 4-dimensional vectors are denoted by the Greek indexes, µ, ν such as xµ. The contravariant position 4-vector is given by xµ = x0, x1, x2, x3 = x0, x . (1.1)

We also employ Einstein’ s double-indice summation convention where two repetitive indexes (superscript, subscript) essentially mean a summation over space-time. µ The length of any 4-vector, A , is given in terms of the metric, gµν , of the ﬂat Minkowski space as

2 µ ν A = gµν A A (1.2)

µν where gµν and its inverse matrix, g is given by

1 0 0 0  1 0 0 0  0 −1 0 0  µν 0 −1 0 0  gµν =   & g =   . (1.3) 0 0 −1 0  0 0 −1 0  0 0 0 −1 0 0 0 −1

µν µν βγ γ Note that gµν and g are symmetric. Since g is the inverse matrix of gµν the relation gαβg = δα holds. µ Given the contravariant position 4-vector, x we extract the covariant one xµ:

ν 0 xµ = gµν x = x , −x . (1.4)

Therefore the product of two 4-vectors A and B is given by

µ 0 0 A · B = A Bµ = A B − A · B . (1.5)

The ﬁeld space-time dependence, φ(xµ) is usually expressed as

φ(x, t) or φ(x) (1.6) however sometimes it is omitted for the sake of simplicity. The covariant and contravariant derivatives with respect to contravariant and covariant position 4-vector respectively are hold as ∂ ∂ ∂ ∂ ∂ ∂ ≡ = , , = ∂ , ∇ (1.7) µ ∂xµ ∂x0 ∂x1 ∂x2 ∂x3 0 µ ∂ ∂ ∂ ∂ ∂ ∂ ≡ = , , = ∂0, −∇ . (1.8) ∂xµ ∂x0 ∂x1 ∂x2 ∂x3 where the properties 1.3 where employed. The choice between the two diﬀerent notations of the partial derivative is usually determined by personal criteria. In addition the time derivative is often expressed as ˙ ∂0φ = φ (1.9)

5 Now, in order to simplify a lot of the expressions associated with physical formulas we adopt the natural unit system. In the latter Plank’ s reduced constant, ~, and the speed of light are considered to be dimensionless constants equal to unity: ~ = c = 1 . (1.10) Under these terms every quantity is measured either to Energy units (MeV) or Length units (fm) with the following expression connecting them:

~ c = 1 = 197.33 MeV fm . (1.11) In natural units the famous relativistic energy-momentum relation is thus given as

E2 = p2c2 + m2c4 → E2 = p2 + m2 . (1.12)

1.2 Nuclear Matter

It is believed that shortly after the Big Bang the universe was filled with an astonishingly hot, dense soup made of particles, such as quarks and gluons, moving with speeds near the speed of light. Quarks are the fundamental bits of matter and are bound together through gluons, the mediators of strong interaction. However in these first moments of extreme temperature they were weakly bound and free to move on their own in what is called a quark-gluon plasma. This state of matter is investigated with high-energy collisions of heavy-ions such as the ones that performed in the STAR and PHENIX experiments at Relativistic Heavy Ion Collider (RHIC) in the US. Further research is done in the ALICE, ATLAS and CMS experiments at CERN’s Large Hadron Collider (LHC). Another topic of special interest for nuclear physicists is the study of the interior of neutron stars. Neutron stars are the smallest and densest stars known to exist. Their radius lies in between 11.0 to 11.5 kilometers and their mass is estimated to be between 1.1 and perhaps up to 3 solar masses while the maximum observed mass of neutron stars is about 2.01 solar masses. In general, compact stars of less than 1.39 solar masses are white dwarfs, whereas compact stars with a mass between 1.4 and 3 solar masses should be neutron stars. Based on the current theory regarding the formation of neutron stars, they are the collapsed core of of a massive star. A normal star is sustained against gravitational collapse by the thermal pressure due to the energy release of thermonuclear fusion in its core. Now, throughout this process the core becomes more iron-rich and at the point the fuel is exhausted the core must be supported by electron degeneracy pressure alone. If the electron degeneracy pressure is not strong enough, the core starts to contract under its own gravity, and thus densities where electrons become relativistic will easily be reached. At that point, protons and electrons combine through electron-capture into neutrons resulting a enormous release of neutrinos. Thus, the core becomes more neutron- rich and when the densities reach nuclear density the the gravitational collapse is halted by neutron-degeneracy pressure. The in-falling matter of the outer atmosphere of the star, due the low pressure of the contracting core, rebounds off the stiffened core thanks to the flux of neutrinos, and thus becoming a Type II or Type Ib supernova. The remnant left is the so called neutron star. If the mass of the remnant is more than five solar masses its fate is predestined and it will become a black hole. As the core of a massive star is compressed and collapses into a neutron star, it conserves most of its angular momentum. However, since the remnant star has only a tiny fraction of its original radius it is attributed with a high rotational speed. Furthermore, highly magnetized rotating neutron stars that emit beams of electromagnetic radiation are called pulsars. These peculiar objects were discovered by Jocelyn Bell when she observed a pulsating radio-source in outer space that had characteristics unlike any other radio-source. On February 11, 2016 the LIGO Scientific Collaboration and VIRGO Collaboration teams announced that they observed for the first time gravitational waves, originated from merging of two black holes. Based on the observed signal LIGO scientists estimate that these black holes had masses about 29 and 36 solar masses, and the event took place 1.3 billion years ago. Furthermore, on June 15, 2016, LIGO announced a second robust gravitational wave-signal from two black holes in their final orbits and then their coalescence into a single black hole. Gravitational waves are ripples in the curvature of space-time caused by massive objects with violent ac- celerations. They propagate as waves outward of their source and transfer energy through the gravitational radiation. Einstein predicted the existence of such waves in his theory of General Relativity, about 100 years ago. Gravitational waves open a great window of new scientific research, such as the emerging field ofgravitational wave-astronomy. Gravitational waves can be used to study systems that invisible or almost impossible to measure with other means. For instance they are best for studying and measuring the properties of black holes. Now, potential gravitational wave sources, in which their signal is strong enough to measure, are extremely

6 Figure 1.1: Representation of different ideas and assumption regarding the inter of a neutron star, [16] massive objects moving in significant fractions of the speed of light. Such objects are neutron stars and the types of gravitational waves that can be studied through them are the continuous gravitational waves and the compact binary inspiral gravitational waves. Continuous gravitational waves are produced by a single spinning massive object such as a neutron star. Spinning neutron stars with imperfections in their spherical shape (axial asymmetry) generate gravitational waves whose frequency and amplitude depends on spin rate of the neutron star. Compact binary inspiral gravitational waves are originated from compact binaries, which are made up of two closely orbiting massive and dense stellar objects, such as white dwarfs, neutron stars and black holes. For these reasons the study of neutron stars is of great importance and it will be fruitful regarding gravitational wave research. The interior of a neutron star is still unknown, however there are a lot of ideas and assumptions as shown in Fig.1.1. Given that the average densities of neutron stars are comparable to that of nuclei, it is assumed that neutron stars consist of baryonic matter (protons, neutrons) and therefore can be studied as giant nuclei. Now, the main difference between a neutron star and a nucleus is that the first one is bound together with gravity while the latter is bound with nuclear forces. Taking into account the assumption of baryonic matter in the interior of neutron star, nuclear models can be incorporated to provide the equation of state of the neutron star. These various nuclear models for the interior of neutron star can be tested by comparing the calculated properties, such as the mass-radius relationship to the observed values. In general nuclear models are complex and thus difficult to solve, therefore approximations have to be made. In this thesis we present the nuclear matter model called Quantum Hadrodynamics (QHD), through the approximation of relativistic mean-field theory. Firstly we introduce the basic concepts regarding field-theory, using relativistic field equations, and then we proceed on the quantization of these fields. Later, the QHD-I formalism is introduced and the approximation of mean-field is applied to simplify the equations of motion. Finally we compute theoretically the symmetry energy coefficient for QHD-I. In the following chapter, we study more advanced QHD parameter sets published by various famous authors and compute again the symmetry energy coefficient. In the final chapter we present a few results taken from code-calculations in the framework of QHD. All the calculated quantities in QHD can be used for the study of the interior of a neutron star.

7 1.3 Nuclear Properties

In this section we introduce the basic properties of nuclei and nuclear matter, which any model for the description of dense matter should be able to reproduce. These are saturation density, binding energy, symmetry energy and compression modulus.

Saturation Density The nucleus consists nucleons (protons, neutrons) which stay bound together mainly due to the strong nuclear interaction. Even though strong interaction is short ranged, it is essentially attractive which is a necessity for a stable nucleus. However it seems to be repulsive and distances less than 0.4 fm. Since the strong interaction is short ranged, the interactions is limited between nearest neighbors and thus at a certain density even if we add more nucleons to the system the short-ranged strong force will not reach them, therefore the the central density of the system will not increase. The density at which this occurs is characterized as saturation density. The pressure of a saturated system is zero and the system given that it is undisturbed it will remain in the same state.

Binding Energy In general the binding energy of a system represents the amount of energy we need to oﬀer in order to break it into its constituents. For instance, in a stable system such as a nucleus the total mass is given as the summation of the number of proton masses and the number of neutron masses minus (or plus) the positive ( or negative) binding energy. As a consequence when a bond between particles or atoms is made energy is released, leaving the system at a lower energy state. In a system at saturation density the binding energy has to be at a maximum (or minimum) since at this density the system is in its most stable state, compared to other densities.

Symmetry Energy Based on the famous Segre diagram stable nuclei with low mass number, A, prefer the number of protons to be equal to the number of neutrons. However, at the region of high mass number, A, stable nuclei have more neutrons than protons. The explanation is simple, as the number of protons rises, Z, the repulsive Coulomb forces also rise and thus the system is less stable. This preference between protons and neutrons is accounted for by the symmetry energy. The symmetry energy coeﬃcient, α4, comes from liquid-drop model of the nucleus, and refers to the contribution made by the isospin asymmetry to the total energy of the nucleus. It is given by:

2 ∂ ρn − ρp α4 = where t ≡ . (1.13) ∂t2 ρ t=0 ρ

is the energy density of the system, ρ is the total number density of the system and ρp, ρn are the number densities of protons and neutron respectively. /ρ refers to the energy per nucleon.

Compression Modulus The compression modulus deﬁnes the curvature of the equation of state at saturation density. It is related to the high density behaviour of the equation of state. If the energy density increases as the pressure increases the equation of state is referred to as stiﬀ. In a soft equation of state the energy density increases more gradually with an increase in the pressure. The compression modulus is given by:

h ∂2 i K = 9 . (1.14) ∂t2 ρ ρ=0

8 Chapter 2

Field Theory

In this chapter we study a few basic concepts of field theory. Firstly we introduce the Lagrange formalism generalized for field theory and secondly we study the most important relativistic equations and their respective fields needed for studying nuclear matter.

2.1 Lagrangian Formalism

We assume the reader is already familiar with the Lagrange formalism and Euler-Lagrange equations associated with systems of point masses characterized by a discrete set of coordinates, where the dynamical variables are the generalized coordinates an their time derivatives. In the framework of field theory the Lagrangian formalism is modified. Firstly, in field theory the dynamical variable of the system is the field. We define a field as a continuous function with a specific value for each point of finite or infinite region in time-space. We denote a field as φ(x, t). Since the dynamical variables are the values of the fields, φ (x, t), in each point of space the system is described by an infinite number of degrees of freedom. The Lagrangian of the theory takes the form of a functional of the field. In general a functional is a mapping from a space of function to real numbers. Notice that Lagrangian depends only on the fields. h i L (t) = L φ (x, t) , φ˙ (x, t) . (2.1)

In order to apply the Hamilton’s principle we ﬁrst have to deﬁne the variation of the functional L[φ, φ˙] as

Z δL δL δL = d3x δφ(x) + δφ˙(x) . (2.2) δφ(x) δφ˙(x)

In the right side of the equation we introduced the functional derivatives δL/δφ, δL /δφ˙ with respect to the field φ and its time derivative φ˙ at the space point of x. These derivatives tell us how the value of the functional is changed when the values of the fields, φ, φ˙ are varied at space point of x. Both sides of the upper equation have a time dependence that we have not marked. ˙ ˙ At this point we define the action W [φ, φ] as the integral of L[φ, φ] over a time interval t1 . . . t2. Notice that the action is also a functional of φ, φ˙. Thus the variation of the action takes the following form:

Z t2 h i δW = δ dt L φ, φ˙ t1 Z t2 h i = dt δL φ, φ˙ t1 Z t2 δL δL = dt d3x δφ + δφ˙ . (2.3) ˙ t1 δφ δφ And since the following relation holds ∂ ∂ ∂ δφ˙ = φ˙0 − φ˙ = φ0 − φ = δφ . (2.4) ∂t ∂t ∂t

9 Integration by parts with respect of time, in combination with the boundary conditions δφ(x, t) = δφ˙(x, t) = 0 leads to the following form of the variation of the action:

t Z t2 δL Z δL 2 Z t2 ∂ δL δW = dt d3x δφ + d3x δφ − dtd3x δφ δφ ˙ ∂t ˙ t1 δφ t1 t1 δφ Z t2 δL ∂ δL = dt d3x − δφ . (2.5) ˙ t1 δφ ∂t δφ Therefore, Hamilton’s principle of stationary action, δW = 0, leads to the Euler-Lagrange equations generalized for ﬁeld theory. δL ∂ δL − = 0 . (2.6) δφ ∂t δφ˙ By returning to the discrete description we can better understand the concept of functional derivatives. We assume that space is divided in small regions or cells of size ∆Vi. The values of ﬁeld in each cell can be substituted by the average values of the function φ (x, t), i.e., Z 1 3 φi (t) = d x φ (x, t) . (2.7) ∆Vi ∆Vi ˙ ˙ ˙ Therefore L depends on a discrete set of coordinates φ1, φ1, φ2, φ2 . . . φi, φi.Hence, the variation of the Lagrangian takes the following form.

X δL δL δL = δφ + φ˙ δφ i ˙ i i i δφi X 1 δL 1 δL = δφ + φ˙ ∆V . (2.8) ∆V δφ i ∆V ˙ i i i i i i δφi

Now since we assume the variations at diﬀerent spatial points are independent, by comparing (2.2) and (2.8) the following relations are easily derived:

δL 1 δL (t) = lim , (2.9) δφ (x, t) ∆Vi→0 ∆Vi δφ(t) δL 1 δL (t) = lim , (2.10) δφ˙ (x, t) ∆Vi→0 ∆Vi δφ˙(t) where the point of space x is located at cell ∆Vi. Therefore the functional derivatives essentially mean diﬀeren- tiation with respect of the value the ﬁeld and its time derivative take at the point of x. At this point we express the Lagrangian as a volume integral over density function, the Lagrange density, L. Z L (t) = d3xL φ (x, t) , φ˙ (x, t) , ∇φ (x, t) . (2.11)

As we can see, the Lagrangian density depends on the field function φ (x, t), its time derivative φ˙ (x, t) and also on the gradient ∇φ (x, t). In principle, a dependence of higher-order field derivatives is allowed. This would need to generalize Euler-Lagrange equations. However, such extensions are not necessary. In the following we will assume Lagrangians depending on fields and their first-order derivatives only. In this study we focus on local field theories where the Lagrangian density at x does not depend on the value of the field at other points of space y 6= x. As we showed at (2.10) the functional derivatives essentially mean differentiations of fields at points of space, therefore we substitute the functional derivatives of L with ordinary partial derivatives of L at point x. Hence, we calculate the variation of L(t) in terms of the Lagrange density L:

Z ∂L ∂L ∂L δL(t) = d3x δφ (x, t) + δφ˙ (x, t) + δ∇φ (x, t) . (2.12) ∂φ (x, t) ∂φ˙ (x, t) ∂ (∇φ (x, t))

Derivation of Euler-Lagrange equations of motion is achieved through Hamilton’s principle on the variation of the action. The action is expressed in terms of Lagrange density and since the relations (2.4) and δ∇φ = ∇δφ

10 are valid, integration by parts can be applied as follows:

Z t2 ∂L ∂L ∂L δW = dt d3x δφ (x, t) + δφ˙ (x, t) + δ∇φ (x, t) ˙ t1 ∂φ (x, t) ∂φ (x, t) ∂ (∇φ (x, t)) Z t2 ∂L ∂ ∂L ∂L = dt d3x − − ∇ δφ (x, t) ˙ t1 ∂φ (x, t) ∂t ∂φ (x, t) ∂ (∇φ (x, t)) t x=+∞ Z ∂L 2 Z t2 ∂L + d3x δφ (x, t) + δφ (x, t) ∂φ˙ (x, t) t ∂ (∇φ (x, t)) t1 1 x=−∞ Z t2 ∂L ∂ ∂L ∂L = d3x − − ∇ δφ (x, t) . (2.13) ˙ t1 ∂φ (x, t) ∂t ∂φ (x, t) ∂ (∇φ (x, t)) In the last step we performed the boundary conditions which we introduced before and also we assumed that the ﬁelds and their derivatives on the surface approach zero very fast. Thus by applying the Hamilton’s principle on (2.13) we ﬁnally derive the Euler-Lagrange equations expressed in terms of Lagrange density L:

∂L ∂ ∂L ∂L − − ∇ = 0 (2.14) ∂φ (x, t) ∂t ∂φ˙ (x, t) ∂ (∇φ (x, t)) or using relativistic notation with xµ = x0, x = (t, x) leads to Euler-Lagrange equations in covariant form:

∂L ∂ ∂L − µ = 0 . (2.15) ∂φ (x, t) ∂x ∂ (∂µφ (x, t))

When the Lagrange function depends on several independent ﬁelds φr, r = 1, 2 ...N, then (2.15) can be generalized as follows: ∂L ∂ ∂L − µ = 0 . (2.16) ∂φr ∂x ∂ (∂µφr)

2.2 Hamiltonian formalism

In classical mechanics, Hamilton formalism is an alternative way to express Lagrange mechanics. In contrast to Lagrangian formalism, Hamiltonian formalism is expressed in terms of the generalized coordinates and their conjugate momenta. In order to apply the Hamiltonian formalism in field theory we first have to to define a canonically conjugate momentum to the field variable. Therefore we define the conjugate field as a functional derivative δL ∂L π (x, t) = = . (2.17) δφ˙ (x, t) ∂φ˙ (x, t) The time derivative of the conjugate field is easily derived via the generalized Euler-Lagrange equations (2.6)

∂L π˙ (x, t) = . (2.18) ∂φ (x, t)

And also we deﬁne the Hamiltonian as: Z H (t) = d3x π (x, t) φ˙ (x, t) − L (t) Z = d3x π (x, t) φ˙ (x, t) − L (x) Z = d3x H (x) . (2.19)

Where in the last step we introduced the Hamilton density H(t) as:

H (x) = π (x, t) φ˙ (x, t) − L (x) . (2.20)

11 2.3 Noether’s Theorem and Conservation Laws in Field Theory

Conservation laws describe quantities which do not change in time and also are independent of the dynamical evolution of the system. Every physical theory has to be able to describe properly these conservation laws. The most fundamental laws in physics are the conservation of energy, momentum and angular momentum. Moreover the physical systems usually are associated with additional conserved quantities such as the charge of a particle. In framework of ﬁeld theory the description of these conservation laws is achieved via Nother’ s theorem. This theorem describes the relation between the conserved quantities and the symmetry properties of the system. When a system which is described by the Lagrange density is symmetric under a certain continuous symmetry transformation then a conserved quantity can be deduced. Let the coordinates to be subject to an inﬁnitesimal transformation of the kind

x0µ = xµ + δxµ (2.21) with the corresponding change of the ﬁeld φr(x) given by

0 0 φr (x ) = φr (x) + δφr (x) . (2.22) Obviously the changes of the coordinates/ﬁelds result a change in Lagrange density of

L0(x0) = L(x) + δL(x) . (2.23)

Now if the action integral of the Lagrange density is invariant under the upper transformations then the following relation will hold ∂ ∂L ∂φr ν µ µ δφr(x) − ν δx + L(x)δx = 0 . (2.24) ∂x ∂ (∂µφr) ∂x It is obvious that this relation is an equation of continuity for the conserved current deﬁned by the terms in the square bracket, i.e. ∂ J µ(x) = 0 (2.25) ∂xµ with µ ∂L(x) ∂L(x) ∂φr µν J (x) = δφr(x) − − g L(x) δxν . (2.26) ∂ (∂µφr) ∂ (∂µφr) ∂xν

Now, we can easily derive the corresponding conservation law from the continuity equation. We integrate (2.25) over the three-dimensional space and then we apply Gauss’ divergence theorem as follows: Z Z Z Z I 3 µ 3 0 3 i 3 0 0 = d x∂µJ (x) = d x∂0J (x) + d ∂iJ (x) = ∂0 d xJ (x) + dS · J(x) . (2.27) V V V V S Since the ﬁelds and their derivatives vanish at the surface the value of the integral over the surface is zero. Therefore : Z G = d3xJ 0(x) (2.28) V is a conserved quantity that has constant value in time. So summarising the Noether’ s theorem : Every continuous symmetry transformation leads to a conserved quantity i.e. conservation law.

2.3.1 Invariance under translation The Lagrange density describes the physical properties of the system. Requiring invariance under temporal and spatial translations the conservation of energy and momentum follow as the consequence of Noether’s Theorem. We ﬁrst consider a change in the coordinate system via a space-time translation:

x0µ = xµ + µ . (2.29)

Since the values of the ﬁelds should not change when the coordinates change, it follows that:

0 0 φr(x ) = φr(x) (2.30)

12 meaning that the variations of the ﬁelds vanish δφr = 0. Therefore following Noether’s theorem there has to be a conserved current (2.26) with the corresponding equation of continuity: ∂ ∂L(x) ∂φr µν µ − g L(x) ν = 0 . (2.31) ∂x ∂ (∂µφr) ∂xν

Since ν are arbitrary we can split these factors. Thus by deﬁning the canonical energy momentum tensor as: ∂L(x) ∂φ T µν = r − gµν L(x) . (2.32) ∂ (∂µφr) ∂xν The following relations hold: ∂ T µν = 0 . (2.33) ∂xµ Notice that since ν = 0,..., 3 it implies four equations of continuity and therefore four conserved quantities. We interpret these conserved quantities as the energy E and the momentum vector P of the ﬁeld. Thus in four dimensional notation 1 Z P ν = (E/c, P) = d3x T 0ν (x) = const. (2.34) c V

2.3.2 Internal Symmetries-Noether Charge Except from symmetries which are related to coordinate transformations, the Lagrange density of a given theory is possible to exhibit further invariance properties. This usually happens when the fields are associated with an internal structure, which is indicated by the presence of several field components φr. Now, an infinitesimal internal symmetry transformation is expressed via a mixing of these components and it can be written as

0 X φr(x) = φr(x) + i λrsφs(x) . (2.35) s Let us study the case of the above inﬁnitesimal internal transformation without changing the coordinate system. This reads as:

µ X δx = 0 , δφr(x) = i λrsφs(x) . (2.36) s If the action integral variable is invariant under these transformations, Noether’ s theorem leads to a conserved current: X ∂L X ∂L J µ = δφ = i λ φ (x) . (2.37) ∂ (∂ φ ) r ∂ (∂ φ ) rs s r µ r r,s µ r Therefore by integration over three-dimensional space the conserved quantity, the so called Noether charge, is obtained. Z 3 X Q = d x πr(x)λrsφs(x). (2.38) r,s Let us now study the complex ﬁeld which is described by a Lagrange density invariant under a phase transformation. This kind of symmetry transformation leads to a conserved charge that is interpreted as the electric charge of the particles represented by the ﬁelds.

φ0 = φ + iφ , φ∗0 = φ∗ − iφ∗. (2.39)

It is more convenient to treat both φ and φ∗ as independent complex ﬁelds with corresponding independent complex variations δφ = iφ and δφ∗ = −iφ∗. This way by comparison of (2.35) and (2.39) the transformation coeﬃcients read as:

λ11 = 1 , λ12 = 0 (2.40)

λ21 = 0 , λ22 = −1 (2.41) (2.42)

Thus the Noether (electric) charge takes the following form: Z Q = d3x [π(x)φ(x) − π∗(x)φ∗(x)] . (2.43)

13 2.4 Relativistic Fields

In this section we will study properties of the most important relativistic equations for describing nuclei, the Klein-Gordon, Dirac, Maxwell and Proca equations. Consequently we will apply the classical field theory we introduced in previous section to these equations. Klein-Gordon field (scalar) describes spinless massive particles 1 and Dirac field (spinor) describes massive spin- 2 particles. Massless (photon) and massive spin-1 particles are described by (vector) fields associated with Maxwell’s equations and Proca equation respectively.

2.4.1 Klein-Gordon Field In this section we will study the Klein-Gordon equation which is used to describe spin-0 particles. The (scalar) field associated with Klein-Gordon equation can be either real or complex. The first describes neutral scalar particles and the second electrical charged particles. For the purposes of this work we will only study the real scalar field. Klein-Gordon equation was one of the first attempts of constructing an equation fitted for describing relativistic systems i.e. an equation that satisfies the relativistic energy momentum relation

E2 = p2 + m2. (2.44)

The motivation of the Klein-Gordon equation follows by applying the canonical quantization rules to the relativistic energy-momentum relation (2.44). Hence we substitute the physical quantities like energy, E and momentum, p, with operators, acting on the wave function φ as follows

Eˆ → i∂t pˆ → −i∇ (2.45) and thus

Eˆ2φ = pˆ2 + m2 φ (2.46) leads to the free Klein-Gordon equation

2 2 2 ∂t − ∇ + m φ = 0 (2.47) (2.48) or written in covariant form using the four-divergence as

µ 2 (∂µ∂ + m )φ = 0 . (2.49) Let us now study the solutions of free Klein-Gordon equation. A possible candidate is the ansatz of plane waves of the form φ(x, t) = ei(p·x−Et). Substitution of φ(x, t) to (2.49) leads to

2 2 2 (∂t − ∇ + m )φ(x, t) = 0 (2.50) (−E2 + p2 + m2)φ(x, t) = 0 (2.51) E2 = p2 + m2. (2.52)

Hence relativistic energy-momentum relation is obtained. As we see Klein-Gordon equation allows plane waves as solutions with corresponding positive and negative energy values. p E = ± p2 + m2 . (2.53)

Since plane waves are solutions, we now construct a general solution of the free Klein-Gordon equation as a linear combination of positive and negative energy plane waves. It reads as Z 3 1 i(p·x−ωpt) 2 i(p·x+ωpt) φ = d p αpe + αpe (2.54)

p 2 2 where ωp = + p + m . Interestingly, we see that imposing the transformation p → −p on the second term does not interfere with generality of the solution. Hence φ(x, t) takes the following form Z Z 3 1 i(p·x−ωpt) 3 2 −i(p·x−ω−pt) φ = d p αpe + d p α−pe (2.55) Z 3 1 i(p·x−ωpt) 2 −i(p·x−ωpt) = d p αpe + α−pe , since ω−p = ωp. (2.56)

14 Consequently the complex conjugate solution ,φ∗(x, t) reads as Z ∗ 3 1 ∗ −i(p·x−ωpt) 2 ∗ i(p·x−ωpt) φ = d p (αp) e + (α−p) e . (2.57)

Now, in the case of neutral particle, φ(x, t) has to be a real function i.e. φ = φ∗. The comparison of (2.56), (2.57) provides us the following relation 2 ∗ 1 ∗ α−p = αp ≡ αp . (2.58) Therefore the general solution of the free Klein-Gordon equation reads as Z 3 i(p·x−ωpt) ∗ −i(p·x−ωpt) φ(x, t) = d p αpe + αpe . (2.59)

Moving on the generalization of Klein-Gordon equation to ﬁeld theory, we introduce the Lagrange density of real (neutral) scalar ﬁeld, φ(x, t), with corresponding mass, m, as follows 1 1 L = ∂µφ ∂ φ − m2φ2 (2.60) 2 µ 2 Since the following relations hold ∂L 1 ∂ = − m2 φ2 = −m2φ ∂φ 2 ∂φ ∂L ∂ 1 ν ∂µ = ∂µ ∂ φ∂ν φ (2.61) ∂ (∂µφ) ∂ (∂µφ) 2

1 ∂ να = ∂µ [g ∂αφ∂ν φ] 2 ∂ (∂µφ) 1 = ∂ [gναδµ∂ φ + gναδµ∂ φ] 2 µ α ν ν α 1 = ∂ [∂µφ + ∂µφ] 2 µ µ = ∂µ∂ φ . (2.62)

Euler-Lagrange equation of motion leads to free Klein-Gordon equation

µ 2 (∂µ∂ + m )φ = 0. (2.63)

Furthermore, the conjugate ﬁeld is deﬁned analogous to the conjugate momentum in the classical Lagrangian formalism (2.17). It reads as π(x) = φ˙(x) . (2.64) Consequently the Hamiltonian density takes the following form

H = πφ˙ − L 1 1 = π2 − ∂µφ∂ φ + m2φ2 2 µ 2 1 1 = π2 + (∇φ)2 + m2φ2 (2.65) 2 2 and thus the Hamiltonian of the real Klein-Gordon ﬁeld reads as Z 1 1 H = d3x π2 + (∇φ)2 + m2φ2 . (2.66) 2 2

Klein-Gordon equation and therefore the corresponding ﬁeld, φ, are associated with a conserved current. In order to derive the form of the conserved current we introduce the Lagrange density of the complex Klein-Gordon ﬁeld:

µ ∗ 2 ∗ L = ∂ φ ∂µφ − m φ φ. (2.67) (2.68)

As we can easily verify the upper Lagrange density is invariant under a global phase transformation of the form

φ → φ0 = φ e+iα φ∗ → φ∗0 = φ∗ e−iα (2.69)

15 it reads as 0 µ ∗ +iα −iα 2 ∗ −iα +iα L = ∂ (φ e )∂µ(φe ) + m φ e φe = L . (2.70) For an inﬁnitesimal global phase transformation, we expand the exponentials (2.69) in McLauren series and therefore the transformations take the following form

φ0 = φ + iαφ & φ∗0 = φ∗ − iαφ∗. (2.71)

Thus the variations of the ﬁelds φ, φ∗ read as

δφ = iαφ & δφ∗ = −iαφ∗. (2.72)

Since the Lagrange density is invariant under the upper transformation Noether’s theorem implies the existence of a conserved current(2.26). By treating the ﬁeld, φ, and the complex ﬁeld, φ∗, as independent degrees of freedom we derive the form of the current µ ∂L ∂L  = iα φ − ∗ ∗ (2.73) ∂(∂µφ) ∂(∂µφ )φ And since the relations of partial derivatives of L hold

∂L µ ∗ ∂L µ = ∂ φ , ∗ = ∂ φ . (2.74) ∂(∂µφ) ∂(∂µφ ) The conserved current reads as e jµ = i (φ ∂µφ∗ − φ∗∂µφ) . (2.75) 2m e µ where we added the factor i 2m for interpretation reasons, since j can be interpreted as electrical current density. Consequently the conserved quantity i.e. the Noether charge is given as Z Q = d3x ρ = const. (2.76) where ρ = φ ∂0φ∗ − φ∗∂0φ is the time component of the conserved current jµ. Since ρ is not a positive definite we cannot interpret it as probability density of the field. However it can be interpreted as electrical-charge density and therefore Q represents the total charge of the field. Hence the electrical-charge density can be either positive, negative or zero. As an example let us examine the neutral Klein-Gordon field, where φ = φ∗. In this case we can easily see that ρ vanishes.

2.4.2 Dirac Field In order to describe nuclear systems, and more general Fermionic systems, we have to be able to describe interactions between fermions (proton, neutron). Therefore in this section we introduce the formalism that 1 is best suited to describe spin- 2 particles, i.e. the Dirac equation and the Dirac field. Following the same procedure as for the derivation of the Klein-Gordon field we first introduce a few basic concepts associated with the Dirac equation, and secondly we will apply them to field theory i.e. Dirac field. Dirac’ s idea was to formulate an equation based on a energy-momentum relationship linear to spatial and temporal derivatives. Hence he proposed a Hamiltonian of the following form:

HD = α · p + βm. (2.77)

The quantities α and β were assumed to be constants but had to be determined later, by imposing particular constraints on the theory. Following the canonical quantization rules i.e. substituting the physical quantities by operators (Eˆ = i∂t, pˆ = −i∇), we motivate the Dirac equation as:

Eψˆ = Hψˆ

i∂tψ = −iα · ∇ψ + βmψ. (2.78)

Since the Dirac equation has to be invariant under spatial rotations, α and β cannot be simply numbers hence α and β are considered to be N × N matrices. Dirac equation has to satisfy the relativistic energy-momentum relationship (2.44), meaning that H ·H = p2 +m2. Therefore the matrices have to be independent of space-time, they have to commute with momentum operators and after some algebra the following properties can also be derived:

16 {αi, β} = 0 i = 1, 2, 3 (2.79)

{αi, αj} = 2δij1 i, j = 1, 2, 3 (2.80)

The hermiticity of the Hamilton operator implies that α and β are hermitian matrices:

ˆ † † † † † ˆ H = (α · p + βm) = pi αi + β m = α · p + βm = H. (2.81) For the purposes of this study we will use the Pauli-Dirac representation where α and β traceless, four dimensional matrices with eigenvalues ±1, given as: 0 σi αi = (2.82) σi 0

1 0 β = 2×2 (2.83) 0 −12×2 where σi are the Pauli matrices: 0 1 σ = , (2.84) 1 1 0

0 −i σ = , (2.85) 2 i 1

1 0 σ = . (2.86) 3 0 −1

Since α, β are 4-dimensional matrices the wave-function of the Dirac equation, ψ, must have dimensions of 4 × 1. These objects are diﬀerent from the usual 4-vectors due to their diﬀerent behavior under Lorentz- transformations. For this reason they are called as ”spinors”.

  ψ1 ψ2 ψ =   . (2.87) ψ3 ψ4

The hermitian adjoint of the Dirac spinor is deﬁned as the complex transpose vector:

† ∗ ∗ ∗ ∗ ψ = (ψ1 , ψ2 , ψ3 , ψ4 ) (2.88)

As a possible solution to the free Dirac equation we choose the ansatz of plane waves of the form:

ψ(x) = ωei(p·x−Et) (2.89) where ω is a 4-component column vector written as:

φ ω = (2.90) χ with φ and χ as 2-component spinors. Substitution of (2.89) to Dirac equation (2.78) leads to:

φ m1 σ · p φ E = 2×2 . (2.91) χ σ · p −m12×2 χ

Hence we obtain the following coupled equations:

Eφ = mφ + σ · pχ (2.92) Eχ = σ · pφ − mχ (2.93)

17 Solving for χ and substituting the column vector, ω, it takes the following form:

φ ω = σ·p . (2.94) E+m φ Moreover insertion of the solution with respect of χ to (2.93) gives:

(E + m)(E − m) φ = (σ · p)2φ (2.95)

Since the following identity of the Pauli matrices holds,

(σ · A)(σ · B) = (A · B)12×2 + iσ · (A × B) (2.96) the relativistic energy-momentum relationship is obtained which implies that plane waves are solutions. Thus Dirac equation allows solutions for the positive and negative energies: p E = ± p2 + m2 . (2.97)

We can also express the Dirac equation in a covariant form using the relativistic notation. Thus we introduce the γ matrices, deﬁned as:

γ0 = β (2.98) γi = βαi = γ0αi (2.99)

The properties of γ matrices can be derived through the properties of α, β and Pauli matrices. Hence γ matrices satisfy the algebra :

{γµ, γν } = γµγν + γν γµ = 2gµν (2.100) i2 γ = −14×4 for i = 1, 2, 3. (2.101) 02 γ = 14×4 (2.102) The adjoint γ matrices are deﬁned as :

γµ† = γ0γµγ0. (2.103)

Thus multiplying Dirac equation (2.78) by β = γ0 from the left the covariant form is obtained:

µ (iγ ∂µ − m) ψ = 0. (2.104) Since (2.104) has to be invariant under Lorentz transformations, we have to replace the hermitian adjoint ψ† with the Dirac adjoint spinor deﬁned as :

ψ¯ = ψ†γ0. (2.105)

We now consider the Dirac wave function as a classical field function in space-time. Consequently we apply the Lagrangian field formalism and derive the corresponding equation of motion of the field i.e. Dirac equation. A possible Lagrange density for the Dirac field reads as:

L = iψ†ψ˙ + iψ†α · ∇ψ − mψ†βψ. (2.106)

We consider the spinors ψ and ψ† to be independent ﬁelds, each having four components. By applying the variation principle with respect the adjoint ﬁeld ψ† Euler-Lagrange equation is obtained ∂ ∂L ∂L ∂L = − ∇ (2.107) ∂t ∂ψ˙ † ∂ψ† ∂(∇ψ†) which after some algebra leads to the Dirac equation (2.78)

i∂tψ + iα · ∇ψ − βmψ = 0. (2.108) Multiplying by γ0 the covariant Dirac equation is obtained

µ (iγ ∂µ − m) ψ = 0. (2.109) Using the relativistic notation we can write the Lagrange density (2.106) covariantly as

¯ µ L = ψ (iγ ∂µ − m) ψ. (2.110)

18 Hamilton’s formalism is expressed through the canonically conjugate ﬁelds πψ and πψ† .

∂L † ∂L πψ = = iψ , πψ† = = 0. (2.111) ∂ψ˙ ∂ψ˙ †

Hence the Dirac ﬁeld is associated with two independent ﬁelds, ψ and ψ†. We can also calculate the Hamilton density (2.20) as:

˙ ˙ † † ˙ † ˙ † † H = πψψ + πψ† ψ − L = iψ ψ − iψ ψ − iψ α · ∇ψ + mψ βψ = ψ† (−iα · ∇ + βm) ψ. (2.112)

Therefore the Hamiltonian of the Dirac ﬁeld is the expectation value of Dirac’s diﬀerential operator HD = α · p + βm: Z H = = d3x ψ† (−iα · ∇ + βm) ψ (2.113)

Let us now study the conserved quantities associated with the Dirac ﬁeld. We impose the Lagrange density under a global phase transformation of the ﬁelds

ψ0 → ψ0 = ψeiα , ψ† → ψ0† = ψ†e−iα (2.114) and thus the Lagrange density (2.110) reads as:

0 0† 0 µ 0 L = ψ γ (iγ ∂µ − m) ψ (2.115) † −iα 0 µ iα = ψ e γ (iγ ∂µ − m) ψe † −iα 0 iα µ = ψ e γ e (iγ ∂µ − m) ψ = L . (2.116)

Since the Lagrange density is invariant under the global phase transformation Noether’s theorem implies the existence of a conserved current. For an inﬁnitesimal transformation (2.114) reduces to

ψ0 = ψ(1 + iα) , ψ0† = ψ(1 − iα) (2.117) and the corresponding ﬁeld variations hold as

δψ = iαψ , δψ† = −iαψ† . (2.118)

Therefore the the conserved current (2.26) takes the following form µ ∂L ∂L † ¯ µ j = −ie ψ − † ψ = eψγ ψ. (2.119) ∂(∂µψ) ∂(∂µψ ) We also included the factor e of the electrical elementary charge since in the framework of quantum ﬁeld theory, Dirac current is interpreted as the electrical current density of the ﬁeld. The corresponding conserved quantity is the total electrical charge and holds as Z Z Q = d3x j0(x) = e d3x ψ†ψ. (2.120)

2.4.3 Photon Field Photon is the massless spin-1 particle that is considered to be the mediator of the electromagnetic interaction between charged particles. The electromagnetic phenomena are described by Maxwell’s equations which are associated with two three-dimensional vector fields i.e. the electric and magnetic field strength, E(x, t) and B(x, t). Maxwell’s equations can be written in covariant form by defining the antisymmetric tensor of rank 2, the so called field-strength tensor F µν as

 0 −E1 −E2 −E3 1 3 2 µν E 0 −B B  F =   . (2.121) E2 B3 0 −B1 E3 −B2 B1 0

19 Thus Maxwell’s equations are expressed as

µν ν ∂µF = j (2.122) ∂λF µν + ∂µF νλ + ∂ν F λµ = 0. (2.123) where jµ = (ρ, j) is four-current density. By taking the four-divergence of (2.123) the equation of continuity of the electric current vector is obtained. ν µν ∂ν j = ∂ν ∂µF = 0 (2.124) where in the last step the term vanished since the whole summation is antisymmetric. It is customary to treat the ﬁeld strength, E and B as derived quantities. Hence we employ the fundamental dynamical variable, the vector potential Aµ = (A0, A) which is connected to the ﬁeld strength throughout the following relations

B = ∇ × A , (2.125) ∂A E = − − ∇A , (2.126) ∂t 0 or written covariantly as

F µν = ∂µAν − ∂ν Aµ. (2.127)

Substituting (2.127) to (2.123) leads to the wave equation of the ﬁeld Aµ

µ ν ν µ ν ∂µ∂ A − ∂ ∂µA = j (2.128) where the conservation of the electric current is a deﬁnite since taking the four-divergence immediately results ν to ∂ν j = 0. However the potential Aµ(x) is not an observable quantity and also it is not uniquely deﬁned. A transformation which adds an arbitrary scalar function Λ(x) to Aµ of the form

A0µ = Aµ + ∂µΛ (2.129) will leave field strengths (2.127) invariant and will preserve the wave equations (2.128). Since photons are massless we can only observe two degrees of freedom (two transversal polarization ori- entations). However a massive vector field has four components and hence is associated with four degrees of freedom. Thus we can use the the property of gauge invariance to reduce the degrees of freedom of the field. We achieve that by imposing additional (gauge) constraints on the field, also referred as gauge conditions. Moving on the description of electromagnetic interactions in the framework of field theory we introduce the Lagrange density of the electromagnetic field which interacts with a charged source jµ(x) 1 L = − F F µν − j Aµ. (2.130) 4 µν µ And thus the conjugate momentum is easily calculated as

µ ∂L µ π = = −∂0A . (2.131) ∂ (∂0Aµ)

2.4.4 Massive Vector Boson Field Spin-1 particles are of great importance in the study of fundamental interactions. They are considered the mediators of the interactions between particles. Electromagnetic interactions are mediated by photons, weak interactions by W ± and Z0 and strong interactions by gluons. We can also study less fundamental interactions like nuclear interactions through the massive spin-1 ω− and ρ− mesons. However, not all the mediators are massless. In fact only photon and gluons are supposed to be massless. W ±, Z0, ω− and ρ are massive spin-1 particles. In the previous section we studied the vector field associated with the photons. Now we study the vector field of massive spin-1 particles in a different approach. Till now we introduced the relativistic equations and then generalized them in field theory using their respective Lagrange densities. In order to describe the vector field of massive spin-1 particles we construct the Lagrange density of the field as a combination of a massless vector field(2.130) with an additional mass term of the Klein-Gordon field(2.60). It reads as:

20 1 1 L = − F F µν + m2A Aµ − j Aµ. (2.132) 4 µν 2 µ µ The upper Lagrangian describes the vector field, Aµ of a massive spin-1 neutral particle, m, which interacts µν with a current jµ. Again F is the field strength tensor defined as: F µν = ∂µAν − ∂ν Aµ. (2.133)

Since the following relations hold: ∂ 1 1 ∂ 1 ∂ A Aµ = m2 (A Aν ) = m2 (g AαAµ) ∂Aµ 2 µ 2 ∂Aµ ν 2 ∂Aµ να 1 ∂Aα ∂Aν = m2g Aν + Aα 2 να ∂Aµ ∂Aµ 1 = m2g (δ αAν + Aαδ ν 2 να µ µ 1 = m2(A + A ) 2 µ µ 2 = m Aµ, (2.134)

∂ αβ ∂ αβ ∂ α β β α µ (FαβF ) = µ (∂αAβ − ∂βAα) F + Fαβ µ ∂ A − ∂ A ∂(∂ν A ) ∂(∂ν A ) ∂(∂ν A ) ∂ γ γ αβ = µ (gβγ ∂αA − gαγ ∂βA ) F ∂(∂ν A ) ∂ αγ β βγ α + µ g ∂γ A − g ∂γ A Fαβ ∂(∂ν A ) ν γ ν γ αβ αγ ν β βγ ν α = gβγ δα δµ − gαγ δβ δµ F + g δγ δµ − g δγ δµ Fαβ ν αβ ν αβ αν βν α = gβµδα F − gαµδβ F + g δµ βFαβ − g δµ Fαβ αν αν = −2gαµF − 2g Fµα ν = −4Fµ (2.135) and also ∂ (j Aα) = j δ α = j (2.136) ∂Aµ α α µ µ Euler-Lagrange equation reads as

µν 2 ν ν ∂µF + m A = j (2.137) which is the Proca equation for massive spin-1 particles. We can also express Proca equation in terms of the vector ﬁeld Aµ as

µ ν ν µ 2 ν ν ∂µ∂ A − ∂ (∂µA ) + m A = j . (2.138) Here lies a major difference between the massless and massive cases. Taking the four-divergence of (2.138), the first two terms cancel each other and thus, since m 6= 0, it reads as 1 ∂ Aν = ∂ jν . (2.139) ν m2 ν ν We either assume the source current is conserved, i.e. ∂ν j = 0 or that there are no sources, therefore the above condition of the field reduces to

ν ∂ν A = 0. (2.140) which is the Lorentz gauge condition. In the electro-magnetic field the current is conserved automatically, however in this case the conservation of the source depends on its nature. Consequently since the the Proca field satisfies the Lorentz condition, (2.138) is simplified as

µ ν 2 ν ∂µ∂ A + m A = 0 (2.141)

21 hence the four components of spin-1 vector ﬁeld satisfy the free Klein-Gordon equation. Moving on the Hamiltonian formalism of the Proca theory we obtain the canonically conjugate ﬁeld for each component as ∂L πµ = ∂(∂0Aµ) 1 ∂ αβ ∂ 0 β β α = (∂0Aβ − ∂βAα) F + ∂ A − ∂ A Fαβ 4 ∂(∂0Aµ) ∂(∂0Aµ) 1 = − δ 0δ µ − δ 0δ µ F αβ 2 α β β α 1 = − F 0µ − F µ0 2 = F µ0 (2.142) or

π0 = 0 and πi = Ei. (2.143)

As we can see the temporal 0-component of the field Aµ corresponds with a vanishing canonically conjugate field π0 = 0. At first this seems to be a problem when one wants to apply quantization rules to Proca field, since it is impossible to postulate a canonical commutation relation for A0. However as we already discussed, massive vector bosons should have three degrees of freedom, two transversal and one longitudinal polarization µ 0 states. Since Lorentz condition is satisfied ∂µA = 0, it follows that A is a dependent variable. The free (no sources) Proca equation for the 0-component reduces to

µ0 2 0 ∂µF + m A = 0. (2.144)

Thus, the dynamical variables of the theory are the three-vector fields A and π = E. Knowing the values of these fields the value of A0 is guaranteed. Substituting in (2.144) and solving with respect of A0, it is expressed in terms of the canonically conjugate field Ei as 1 A0 = − ∇ · E. (2.145) m2

22 Chapter 3

Quantum Field Theory

In this chapter we study the basic concepts regarding the ﬁeld quantization. Firstly we introduce the reader to the quantum harmonic oscillator and consequently we examine the quantization of neutral Klein-Gordon and Dirac ﬁelds.

3.1 Harmonic Oscillator

The Hamiltonian operator of an one-dimensional harmonic oscillator with mass, m, and frequency, ω, is given as pˆ2 1 Hˆ = + mω2qˆ2 (3.1) 2m 2 whereq ˆ andp ˆ are the position and momentum operators, which satisfy the canonical commutation relation

[q, ˆ pˆ ] = i~ . (3.2) We also deﬁne the ladder operators as follows

rmω pˆ αˆ = qˆ+ i , (3.3) 2~ mω rmω pˆ αˆ† = qˆ− i . (3.4) 2~ mω And thus,q ˆ andp ˆ are easily expressed in terms ofα ˆ andα ˆ† as r qˆ = ~ αˆ +α ˆ† , (3.5) 2mω r 1 ω pˆ = ~ αˆ − αˆ† . (3.6) i 2 (3.7)

Now, employing the relations (3.2) we extract the commutation properties ofα ˆ andα ˆ†

α,ˆ αˆ† = 1. (3.8)

Furthermore, the Hamiltonian (3.1) expressed in terms ofα ˆ andα ˆ† is given by the following simple form

1 Hˆ = ω αˆ† αˆ + . (3.9) ~ 2

Consequently, we deﬁne the number operator:

Nˆ ≡ αˆ†αˆ (3.10) which later we show that its eigenvalues give the number of ~ω multiples the total energy of the system has above the lowest possible energy.

23 Using Dirac’ s notation we denote its corresponding eigenstates as |ni and eigenvalues as n satisfying the relation

Nˆ |ni = n |ni . (3.11)

Using the the commutation relations (3.8) we see that the statesα ˆ† |ni andα ˆ |ni are also eigenstates of Nˆ, with shifted eigenvalues:

Nˆαˆ† |ni =α ˆ†αˆαˆ† |ni =α ˆ† αˆ†αˆ + 1 |ni = (n + 1)α ˆ† |ni (3.12) and

Nˆαˆ |ni =α ˆ†αˆαˆ |ni = αˆαˆ† − 1 αˆ |ni = (n − 1)α ˆ |ni . (3.13)

We immediately see that the eigenstates |ni are also energy-eigenstates. This result is expected since the number operator, Nˆ, commutes with the Hamiltonian operator, Hˆ .

1 1 Hˆ |ni = ω Nˆ + |ni = ω n + |ni &[H,ˆ Nˆ] = 0 . (3.14) ~ 2 ~ 2

Thus, the energy eigenvalues are given by

1 E = ω n + (3.15) n ~ 2

† and hence the statesα ˆ |ni andα ˆ |ni correspond to energy eigenvalues En+1 = En + ~ω and En−1 = En − ~ω respectively. Thus the continuous application ofα ˆ produces lower energy eigenstates, down to En = −∞. However, since n is a positive deﬁnite:

αˆ |0i = 0 → Nˆ |0i = 0 |0i . (3.17)

All the other states can be constructed by the repeated application ofα ˆ† on the ground state |0i:

1 n |ni = √ αˆ† |ni (3.18) n! where the additional factor guarantees normalization, hn|ni = 1, given that h0|0i = 1. In conclusion the eigenvalues of Nˆ are positive integers starting from n = 0 and thus the energy eigenvalues read as

1 E = n + ω n = 0, 1, 2... (3.19) n 2 ~ with lowest-point energy 1 E = ω. (3.20) 0 2~

3.2 Neutral Klein-Gordon Field

In this section we quantize the real (neutral) Klein-Gordon field. First let us remind the reader a few basic concepts associated with Klein-Gordon field. The Lagrange density of the real Klein-Gordon field, φ(x, t), reads as 1 1 L = ∂µφ(x) ∂ φ(x) − m2φ(x)2 (3.21) 2 µ 2 and as a consequence the Euler-Lagrange equation of motion leads to Klein-Gordon equation:

µ 2 ∂ ∂µ + m φ(x) = 0 . (3.22)

24 The canonically conjugate field is given by π(x) = φ˙(x) (3.23) and hence the Hamilton density reads as 1 H(x) = π(x)2 + (∇φ(x))2 + m2φ(x)2 . (3.24) 2 In the canonical field quantization the classical fields φ(x, t) and π(x, t) are promoted to field operators:

φ(x, t) → φˆ(x, t) , π(x, t) → πˆ(x, t) . These operators have to satisfy the equal-time commutation relations (ETCR):

[φˆ(x, t), πˆ(x0, t)] = iδ3 (x − x0) , (3.25) [φˆ(x, t), φˆ(x0, t)] = [ˆπ(x, t), πˆ(x0, t)] = 0 (3.26) The choice of the ordinary commutators is to ensure the Klein-Gordon field quanta satisfy Bose-Einstein statistics. Since the field operators are time dependent the evolution of the system is obtained through Heisenberg’ s picture. In the latter, the states |Φi are assumed to be time-independent and the time evolution is carried out by the operators. Thus the field operators φˆ(x, t) andπ ˆ(x, t) have to satisfy the following equation of motion ˙ 1 φˆ(x, t) = [φˆ(x, t), Hˆ ] , (3.27) i 1 πˆ˙ (x, t) = [ˆπ(x, t), Hˆ ] , (3.28) i where we introduced the Hamilton operator of the quantized Klein-Gordon field: Z 1 2 Hˆ = d3x πˆ(x, t)2 + ∇φˆ(x, t) + m2φˆ(x, t) . (3.29) 2 Now, let us compute the upper commutation relations. For this purpose we first derive a few useful relations:

h ˆ ˆ 0 i h ˆ ˆ 0 i φ(x), ∇φ(x ) = ∇x0 φ(x), φ(x ) = 0 (3.30)

ˆ 0 ˆ 0 3 0 [ˆπ(x), ∇φ(x )] = ∇x0 [ˆπ(x), φ(x)] = −∇x0 iδ(x − x) = −∇x0 iδ (x − x ) (3.31)

h i πˆ(x), φˆ(x0) = −iδ3(x − x0) (3.32) where there is also an additional time-dependence which was omitted. Thus, using the equations (3.25), (3.30), (3.31), (3.32) we derive the commutators Z 1 [φˆ(x), Hˆ ] = d3x0 [φˆ(x), πˆ(x0)2] + [φˆ(x), (∇φˆ(x0))2] + m2[φˆ(x), φˆ(x0)2] 2 Z 1 = d3x0 [φˆ(x), πˆ(x0)]ˆπ(x0) +π ˆ(x0)[φˆ(x), πˆ(x0)] 2 Z i = d3x0 δ3(x − x0)ˆπ(x0) +π ˆ(x0)δ3(x − x0) 2 = iπˆ(x) and Z 1 [ˆπ(x), Hˆ ] = d3x0 [ˆπ(x), πˆ(x0)2] + [ˆπ(x), ∇φˆ(x0)2] + m2[ˆπ(x), φˆ(x0)2] 2 Z 3 0 1 3 0 0 0 3 0 = d x (−i)∇ 0 δ (x − x )∇φˆ(x ) + ∇φˆ(x )(−i)∇ 0 δ (x − x ) 2 x x Z 1 + d3x0 (−i)m2δ3(x − x0)φˆ(x0) + φˆ(x0)(−i)m2δ3(x − x0) 2 Z = d3x0 iδ3(x − x0)∇2φˆ(x0) − im2δ3(x − x0)φˆ(x0)

= i ∇2 − m2 φˆ(x) .

25 Summarizing, the time evolution of the ﬁeld operators φˆ(x, t) andπ ˆ(x, t) is given by the equations:

˙ φˆ(x, t) =π ˆ(x, t) (3.33) πˆ˙ (x, t) = ∇2 − m2 φˆ(x, t) . (3.34)

Hence, the ﬁeld operator φˆ(x, t) satisﬁes the free Klein-Gordon equation:

¨ φˆ(x, t) = ∇2 − m2 φˆ(x, t) .

Now, in order to put in use our theory we have to construct an explicit representation for the ﬁeld operator, φˆ(x, t). For this purpose, we choose a complete basis of classical functions. Thus, the ﬁeld operator, φˆ(x, t), can be expressed in terms of generalized Fourier decomposition by expanding it with respect of the basis. Hence, we choose the set of normalized plane waves

1 ip·x up(x) = e . (3.35) p 3 2ωp(2π) )

Thus the ﬁeld operator, φˆ(x, t), takes the following form Z ˆ 3 1 ip·x φ(x, t) = d p αˆp(t)e . (3.36) p 3 2ωp(2π) )

Substitution of (3.36) to Klein-Gordon equation, (3.35), leads to the second order diﬀerential equation:

¨ 2 2 αˆp(t) = − p + m αˆp(t) (3.37) where the most general solution reads as

(1) iωpt (2) −iωpt αˆp =α ˆp e +α ˆp e . (3.38)

(1) (2) The operatorsα ˆp andα ˆp are constant in time and ωp stands for

p 2 2 ωp = + p + m . (3.39)

Now, since the classical Klein-Gordon ﬁeld is real, φ∗ = φ, the corresponding ﬁeld operator, φˆ, has to be † (1) (2) hermitian, φˆ = φˆ. Therefore with similar manipulations to Ch.1 we express one of the operators,α ˆp andα ˆp in terms of the other:

1 † (2) αˆp =α ˆ−p . (3.40)

Following the same procedure as in Ch.1 the ﬁnal form of φˆ reads as Z 1 ˆ 3 i(p·x−ωpt) † −i(p·x−ωpt) φ(x, t) = d p αˆpe +α ˆ e (3.41) p 3 p 2ωp(2π) )

(1) whereα ˆp ≡ αˆp . Consequently, using (3.35), the conjugate ﬁeldπ ˆ(x, t) is easily computed: Z r ωp πˆ(x, t) = d3p (−i) αˆ ei(p·x−ωpt) − αˆ† e−i(p·x−ωpt) . (3.42) 2(2π)3) p p

† The operatorsα ˆp andα ˆp are called creation and annihilation operators respectively and their interpretation will be clear later. Their commutation relations are obtained by evaluating the commutator Z Z h ˆ 0 i 3 3 0 −i(p·x+p0·x0) φ(x, t), πˆ(x , t) = d p d p (−iωp0 ) [ˆαp, αˆp0 ] e

† −i(p·x−p0·x0) † i(p·x−p0·x0) − [ˆαp, αˆp0 ]e + [ˆαp, αˆp0 ]e † † i(p·x+p0·x0) − [ˆαp, αˆp0 ]e (3.43)

0 where the normalization constants were omitted and the relativistic notation p · x = ωpt − p · x, t = t, was introduced.

26 † We expect the operatorsα ˆp andα ˆp follow a speciﬁc algebra for the commutator to lead to delta function. Thus we propose the following relations

3 0 [ˆαp, αˆp0 ] = δ (p − p ) (3.44) † † [ˆαp, αˆp0 ] = [ˆαp, αˆp0 ] = 0 (3.45) and we easily verify their validity by substituting to (3.43): Z Z h ˆ 0 i 3 3 0 3 0 −i(p·x−p0·x0) −i(p·x−p0·x0) φ(x, t), πˆ(x , t) = d p d p (iωp0 )δ (p − p ) e + e Z 3 1 1 i(p·(x−x0) −i(p·(x−x0) = i d p 3 ωp e + e (2π) 2ωp = iδ3(x − x0) . (3.46)

The Hamiltonian can be expressed in terms of the creation and annihilation operators in a straightforward way. We substitute (3.41), (3.42) to (3.29) and after some algebra involving the orthogonality relations of time-dependent plane wave functions, the Hamiltonian operator is obtained 1 Z Hˆ = d3p ω αˆ† αˆ +α ˆ αˆ† . (3.47) 2 p p p p p

However, the Hamiltonian manifests pathologies. Using the relations (3.44), the expectation value of any state, including the vacuum, leads to infinite energy: 1 Z Hˆ = d3p ω 2ˆα† αˆ† + δ3(0) . (3.48) 2 p p p Now, let us approach the problem differently. First, we confine the particles into a sufficiently large box and therefore the momentum states p become discrete instead of continuous. Thus, the commutation relations of † creation and annihilation operators,α ˆp,α ˆp read as

h † i αˆp, αˆp0 = δp,p0 (3.49)

h † † i [ˆαp, αˆp0 ] = αˆp, αˆp0 = 0 (3.50) and therefore the Hamiltonian operator reads as

X 1 Hˆ = ω αˆ† αˆ + . (3.51) p p p 2 p

Since the the commutation relations of creation and annihilation operators, (3.50), are similar with the commutation relations of ladder operators, (3.8), the Hamiltonian is easier to interpret. The momentum states, † p, are occupied by particles whose number is given by the expectation value of the number operator,n ˆp =α ˆpαˆp. Each of the particles contributes in the total energy of the system a quantum of energy equal to ωp. However there is also the additional energy of the vacuum (zero-point energy) which is independent of the occupation number. The number operators for diﬀerent momentum states commute with each other:

† † [ˆnp, nˆp0 ] = [ˆαpαˆp, αˆp0 αˆp0 ] (3.52) † † † † † † = [ˆαp, αˆp0 ]ˆαpαˆp0 +α ˆp[ˆαp, αˆp0 ]ˆαp0 +α ˆp0 [ˆαp, αˆp0 ]ˆαp +α ˆp0 [ˆαp, αˆp0 ] = 0 and thus, they also commute with the Hamiltonian:

X 1 [H,ˆ nˆ ] = ω 0 [ˆn 0 , nˆ ] + [ , nˆ 0 ] = 0 ∀p . (3.53) p p p p 2 p p0

Consequently, we deﬁne the total-number operator whose expectation value gives the total number of particles: X Nˆ = nˆp . (3.54) p

27 Since it commutes with the Hamiltonian the total number of particles is conserved: X [N,ˆ Hˆ ] = ωp[ˆnp, nˆp0 ] = 0 , (3.55) p,p0

˙ Nˆ = −i [N,ˆ Hˆ ] = 0 . (3.56)

The operators H,ˆ N,ˆ nˆp commute with each other. Therefore we construct a common set of eigenstates as

nˆpi |np1, np2...npi...i = npi |np1, np2...npi...i (3.57)

where these state vectors |nˆp1, nˆp1, ...i form a basis in Hilbert space (also called Fock space). In Fock space the scalar product is deﬁned as

0 0 n , n ... n , n , ... = δ 0 δ 0 ... (3.58) p1 p2 p1 p2 np1,np1 np2,np2 We deﬁne the vacuum as the state where there is no occupation for any of the momentum states, p:

nˆp |0i = 0 ∀p (3.59) or more accurately, vacuum is the state which is destroyed when any annihilation operator,α ˆp, acts on it, i.e.

αˆp |0i = 0 ∀ p (3.60)

† All the other state-vectors can be constructed by applying repeatedly the creation operators,α ˆp on vacuum, |0i:

n n 1 † p1 † p2 np1, np2, ... = p αˆp1 αˆp2 · · · |0i . (3.61) np1!np2! ···

As we expected, using (3.25), the states are symmetric under permutations of creation operators:

|np1, . . . npi, npj ...i = |np1, . . . npj , npi ...i . (3.62)

In addition the energy of the vacuum (zero-point energy) is given by

X 1 E = ω (3.63) 0 2 p p and since the summation is over infinite momentum states the energy of the vacuum, E0, is infinite. Fortunately physical observables involve energy differences instead of absolute values, hence the constant energy of the vacuum drops out. We achieve that by defining a new Hamiltonian of the form:

ˆ ˆ X † He = H − E0 = ωp αˆpαˆp . (3.64) p

Finally, switching back to the continuum description, the Hamiltonian reads as Z ˆ 3 † H = d p ωp αˆpαˆp . (3.65)

3.3 Dirac Field

1 After we ﬁnished our study on scalar spin-0 particles, we now focus on spin- 2 particles. Therefore we quantize the Dirac Field. As we stated previously its Lagrange density holds as

¯ µ † ˙ † † L = ψ (iγ ∂µ − m) ψ = iψ ψ + iψ α · ∇ψ − mψ βψ. (3.66)

We treat ψ and ψ† as independent ﬁelds, each of them being a 4-component spinor. It is useful to express the Lagrange density in terms of matrix components: ¯ µ L = ψα(iγαβ∂µ − m1αβ)ψβ. (3.67)

28 Now, applying the variation principle with respect to ψ† and ψ we derive Dirac equation

µ (iγ ∂µ − m) ψ = 0 (3.68) and its hermitean conjugate Dirac equation

← ¯ µ ψ(iγ ∂µ − m) = 0 , (3.69) where the arrow on top of the partial derivative indicates action on the function on the left side. We easily notice that the upper Lagrange density is not a real quantity:

→ ← † ¯ µ † † µ† 0† L = [ψ(iγ ∂µ − m)ψ] = ψ (−iγ ∂µ − m)γ ψ ← ← † 0 µ 0 0 ¯ µ = ψ (−iγ γ γ ∂µ − m)γ ψ = ψ(−iγ ∂µ − m)ψ. (3.70) where the properties of γ-matices were used. We overcome this problem by deﬁning the real Lagrange density as 1 1 ↔ L0 = (L + L†) = ψiγ¯ µ∂ ψ − mψψ¯ (3.71) 2 2 µ

↔ where the double-arrowed partial derivative is deﬁned as α ∂µ β = (∂µα)β − α(∂µβ). The canonically conjugate momenta of the Dirac ﬁeld are given by

∂L ˙† ∂L πψ = = iψ , πψ† = = 0 . (3.72) ∂ψ˙ ∂ψ˙ † and thus, indeed there are two independent ﬁelds. However we will treat the quantization using ψ, ψ† instead of ψ, πψ as independent ﬁelds. As previously calculated the Hamilton density reads as

H = ψ† (−iα · ∇ + βm) ψ . (3.73)

In the canonical quantization of Dirac ﬁeld the spinors are promoted to ﬁeld operators:

ψ(x, t) → ψˆ(x, t) , ψ†(x, t) → ψˆ†(x, t) , (3.74)

where, we postulate they satisfy the equal time anti-commutation relations (ETAR):

n ˆ ˆ† 0 o 3 0 ψα(x, t), ψβ(x , t) = δαβδ (x − x ) , n ˆ ˆ 0 o n ˆ† ˆ† 0 o ψα(x, t), ψβ(x , t) = ψα(x, t), ψβ(x , t) = 0 . (3.75)

The choice of anticommutators instead of normal commutators guarantees the correct spin statistics of Fermi- Dirac particles. Consequently, the Hamiltonian operator expressed in terms of matrix elements is given by Z ˆ 3 ˆ† ˆ H = d xψα(x, t)(−iααβ · ∇ + βαβm) ψβ(x, t) . (3.76)

Thus, the time evolution of the ﬁeld operator ψˆ(x, t) is given by Heisenberg’ s equation of motion

˙ 1 ψˆ(x, t) = [ψˆ(x, t), Hˆ ] . (3.77) i However in order we calculate the upper commutator it is useful to introduce the following identity which connects normal commutators and anti-commutators:

[A,ˆ BˆCˆ] = {A,ˆ Bˆ}Cˆ − Bˆ{A,ˆ Cˆ}. (3.78)

29 hence, Z h ˆ ˆ i 3 0 h ˆ ˆ† 0 ˆ 0 i ψα(x), H = d x ψα(x), ψβ(x )(−iαβτ · ∇x0 + ββτ m)ψτ (x ) Z 3 0 h ˆ ˆ† 0 ˆ 0 i h ˆ ˆ† 0 ˆ 0 i = d x ψα(x), ψβ(x )(−iαβτ · ∇x0 ψτ (x ) + ψα(x), ψβ(x )ββτ mψτ (x ) Z 3 0 n ˆ ˆ† 0 o ˆ 0 = d x (−i) ψα(x), ψβ(x ) αβτ · ∇x0 ψτ (x )

ˆ† 0 n ˆ ˆ 0 o n ˆ ˆ† 0 o ˆ 0 − ψβ(x )αβτ · ∇x0 ψα(x), ψτ (x ) + ββτ m ψα(x), ψβ(x ) ψτ (x ) ˆ† 0 n ˆ ˆ 0 o − ψβ(x ) ψα(x), ψτ (x ) Z 3 0 3 0 ˆ 0 3 0 ˆ 0 = d x (−i) δαβδ (x − x )αβτ · ∇x0 ψτ (x ) + mββτ δαβδ (x − x ) ψτ (x ) ˆ ˆ = −iαατ · ∇xψτ (x) + mβατ ψτ (x) ˆ = (−iαατ · ∇x + mβατ ) ψτ (x) (3.79) where in the process we took advantage of the anti-commutation relations, (3.75). Therefore the field operator ψˆ(x, t) satisfies the Dirac equation: ˆ˙ ˆ iψ(x, t) = (−iα · ∇x + mβ) ψ(x, t) . (3.80) The time evolution of the conjugate field ψˆ†(x, t) can be derived similarly. Now, we have to form an explicit representation for the field operators ψˆ(x, t), ψˆ†(x, t). We do that by expanding them in terms of a complete set of classical wave functions and thus, the natural choice are the plane-wave solutions of the Dirac equation:

1 m 1/2 ψ(r)(x, t) = w (p) eir (p·x−ωpt), ∀p (3.81) p 3/2 r (2π) ωp where the index r represents the four independent solutions for each momentum state, p. For r = 1, 2 the p 2 2 solutions have positive energy eigenvalues E = +ωp = + p + m , in contrast to r = 3, 4 where the solutions p 2 2 have negative energy eigenvalues E = −ωp = − p + m . For this purpose we introduced the sign function r = +1 for r = 1, 2 and r = −1 for r = 3, 4. wr(p) are the so called Dirac unit spinors and they satisfy the algebraic equation µ (γ pµ − rm) wr(p) = 0. (3.82) These spinors posses the following orthogonality and completeness properties:

† ωp w 0 ( 0 p)w ( p) = δ 0 , (3.83) r r r r m rr w¯r0 (p)wr(p) = r δrr0 , (3.84) 4 X ωp w ( p)w† ( p) = δ , (3.85) rα r rβ r m αβ r=1 4 X rwrα(p)w ¯rβ(p) = δαβ . (3.86) r=1

(r) Now, these properties guarantee that the plane-wave solutions, ψp , are orthonormal to delta functions. Their inner product is deﬁned as follows, and after some algebra involving the properties of delta functions and the relations, (3.86), we obtain: Z (r) (r0) 3 (r)† (r0) 3 0 ψp (x), ψp0 (x) = d x ψp (x)ψp0 (x) = δrr0 δ (p − p ). (3.87)

Consequently, the expansion of the ﬁeld operator, ψˆ(x, t), in terms of the orthonormal set of Dirac plane-wave functions is given by 4 Z ˆ X 3 (r) ψ(x, t) = d p αˆ(p, r)ψp (x, t) r=1 4 1/2 X Z d3p m = αˆ(p, r) w (p) e−ir p·x (3.88) (2π)3/2 ω r r=1 p

30 and thus, the hermitean conjugate ﬁeld, ψˆ†(x, t), reads as

4 1/2 X Z d3p m ψˆ†(x, t) = αˆ†(p, r) w†(p) e+ir p·x (3.89) (2π)3/2 ω r r=1 p where the operatorsα ˆ†(p, r),α ˆ(p, r) are the creation and annihilation operators of the Dirac ﬁeld. Notice that each momentum state, p, corresponds to four pairs of operators. Now, in order to derive the anti-commutation relations betweenα ˆ†(p, r) andα ˆ(p, r), we project the ﬁeld operator ψˆ(x, t) on a plane-wave using the orthogonality of the set of functions (3.87): Z (r) ˆ 3 (r)† ˆ αˆ(p, r) = ψp (x, t), ψ(x, t) = d x ψp (x, t)ψ(x, t) 4 Z Z X 3 0 0 0 x (r) (r0) = d p αˆ(p , r ) d ψp (x, t) ψp0 (x, t) r0=1 4 Z X 3 0 0 0 3 0 = d p αˆ(p , r ) δrr0 δ (p − p ) r0=1 =α ˆ(p, r) (3.90) hence,

Z d3x m 1/2 αˆ(p, r) = e+ir p·xw†(p) ψˆ(x, t) , (3.91) 3/2 r (2π) ωp

Similarly the form ofα ˆ†(p, r) is given by Z † ˆ (r) 3 ˆ† (r) αˆ (p, r) = ψ(x, t), ψp (x, t) = d x ψ (x, t)ψp (x, t)

Z d3x m 1/2 = e−ir p·xψˆ†(x, t)w (p) . (3.92) 3/2 r (2π) ωp

Therefore the anti-commutation relations are easily computed with the aid of, (3.75), as Z Z † 0 0 3 3 0 n (r)† ˆ ˆ† 0 (r0) 0 o αˆ(p, r), αˆ (p , r ) = d x d x ψpα (x, t)ψα(x, t), ψβ(x , t)ψp0β (x , t) Z Z 3 3 0 (r)† (r0) 0 n ˆ ˆ† 0 o = d x d x ψpα (x, t)ψp0β (x , t) ψα(x, t), ψβ(x , t) | {z } ( 0 δαβ δ x−x ) Z 3 (r)† (r0) = d x ψpα (x, t)ψp0α(x, t)

(r) (r0) = ψp , ψp0 3 0 = δrr0 δ (p − p ) , (3.93) and also

Z Z 0 † 0 0 3 3 0 (r)† (r ) 0 n ˆ ˆ 0 o {αˆ(p, r), αˆ(p , r )} = d x d x ψpα (x, t)ψp0β (x , t) ψα(x, t), ψβ(x , t) = 0 , (3.94)

Z Z † † 0 0 3 3 0 (r) (r0) 0 n ˆ† ˆ† 0 o αˆ (p, r), αˆ (p , r ) = d x d x ψpα (x, t)ψp0β (x , t) ψα(x, t), ψβ(x , t) = 0 . (3.95)

As we expected the creation and annihilation operators,α ˆ†(p, r),α ˆ(p, r), also satisfy the equal time anti- commutation relations. Furthermore, the Hamiltonian operator, Hˆ , of the quantized Dirac ﬁeld reads as Z Hˆ = d3x ψˆ†(x, t)(−iα · ∇ + βm) ψˆ(x, t) (3.96)

31 and thus, the insertion of the the ﬁeld operators (3.88), leads to Z Z Z ˆ X 3 3 0 † 0 0 3 (r)† (r0) H = d p d p αˆ (p, r)ˆα(p , r ) d x ψp (x, t)(−iα · ∇ + βm) ψp0 (x, t) (3.97) rr0 and since the plane-waves are solutions of Dirac equation, they are also eigenfunctions of the Dirac diﬀerential operator HˆD = (−iα · ∇ + βm), hence ˆ (r) (r) HDψp = rωpψp (3.98)

(r) where rωp is the energy eigenvaule of the plane-wave, ψp . Thus, Z Z Z ˆ X 3 3 0 † 0 0 3 (r)† (r0) H = d p d p αˆ (p, r)ˆα(p , r ) r0 ωp0 d x ψp (x, t)ψp0 (x, t) 0 rr | {z } δrr0δ3(p−p0) Z X 3 † = d p rωp αˆ (p, r)ˆα(p, r) (3.99) r by separating the positive and negative energy contributions we obtain: Z 2 4 ! 3 X † X † Hˆ = d p ωp αˆ (p, r)ˆα(p, r) − ωp αˆ (p, r)ˆα(p, r) . (3.100) r=1 r=3 We again adopt the interpretation of number of particles for each state. For this purpose, we deﬁne the (r) particle-number operator for the state, ψp (x, t), as

† nˆp,r =α ˆ (p, r)ˆα(p, r) . (3.101) As we did in the case of Klein-Gordon field we confine our system in a sufficient large box. Therefore, the momentum states , p, become discrete and the anti-commutation relations are altered to

† 0 0 αˆ(p, r), αˆ (p , r ) = δrr0 δpp0 , (3.102) αˆ(p, r), αˆ(p0, r0) = αˆ†(p, r), αˆ†(p0, r0) = 0 . (3.103)

However, since we study a system of fermions the occupation number, np,r, cannot be higher than one. Indeed, 2 † 2 from the relations, (3.103), we see thatα ˆp,r = (ˆαp,r) = 0. As a consequence:

2 † † † † nˆp,r =α ˆp,r αˆp,rαˆp,r αˆp,r =α ˆp,r 1 − αˆp,rαˆp,r αˆp,r =n ˆp,r (3.104) thus, the eigenvalues ofn ˆp,r are 0,1. Let us now return to the upper Hamiltonian. It is clearly problematic since for r=3,4 the contribution of the field quanta to the total energy is negative. Therefore as the occupation of negative energy states grows the total energy of the system drops to negative values without any bound. Nevertheless, we fix this problem with the aid of Dirac’ s hole picture. According to that, we consider as vacuum the state where all negative energy levels are occupied by particles. These particles are always present and in the absence of electromagnetic field are distributed homogeneously all over space filling up the so called ” Dirac sea ”. Dirac sea cannot be observed experimentally and its energy and charge can be removed. Thus, since the total energy of the non-observable vacuum is given as the summation over all the negative energy states:

4 X X E0 = − ωp , (3.105) p r=3 we simply subtract it from the latter Hamiltonian:

ˆ ˆ He = H − E0 2 4 ! X X † X † = ωp αˆ (p, r)ˆα(p, r) + ωp 1 − αˆ (p, r)ˆα(p, r) p r=1 r=3 2 4 ! X X † X † = ωp αˆ (p, r)ˆα(p, r) + ωp αˆ(p, r)ˆα (p, r) p r=1 r=3 2 2 ! X X X ˆ = ωp nˆp,r + ωp nep,r (3.106) p r=1 r=1

32 (r) where in the last step we introduced the number opearator for holes for the state ψp , r = 3, 4. This operator is defined as ˆ † nep,r = 1 − nˆp,r =α ˆ(p, r)ˆα (p, r). (3.107) ˆ The hole operator, nep,r, has also eigenvalues 1,0 and represents the absence or not of particles in Dirac sea. These holes are interpreted as antiparticles and in these terms, the Hamiltonian operator, (3.106), has physical meaning and is also a well-defined positive definite quantity. Thus, the physical vacuum is now redefined as the state which contains neither particles nor antiparticles.

nˆp,r |0i = 0 ⇒ αˆ(p, r) |0i for r = 1, 2 (3.108) ˆ † nep,r |0i = 0 ⇒ αˆ (p, r) |0i for r = 3, 4 (3.109) Now we clearly see that our ﬁrst interpretation ofα ˆ†(p, r),α ˆ(p, r) as creation and annihilation operators is incorrect. In fact the operatorsα ˆ(p, r),α ˆ†(p, r) are the annihilation operators for the cases of r = 1, 2 and r = 3, 4. And the operatorsα ˆ†(p, r),α ˆ(p, r) are the creation operators for r = 1, 2 and r = 3, 4 respectively. For the sake of simplicity we introduce separate notation for the particle and hole operators. In addition we rename the unit Dirac spinors. Thus, u(p, s) will be used for the spinors with positive energy values and υ(p, s) for the negative ones. They are connected to the spinors wr(p) as follows:

w1(p) = u(p, +s) ,

w2(p) = u(p, −s) ,

w3(p) = υ(p, −s) ,

w4(p) = υ(p, +s) . (3.110) The orthogonality relations between u(p, s) and υ(p, s) are easily derived from the orthogonality properties of Dirac unit spinors, wr(p) (3.86). Therefore we obtain the relations: ω u†(p, s) u(p, s0) = υ†(p, s) υ(p, s0) = p δss0 , m u†(−p, s) υ(p, s0) = υ†(−p, s) u(p, s0) = 0 . (3.111) Accordingly we introduce new particle and hole operators: ˆb(p, +s) =α ˆ(p, 1) , ˆb(p, −s) =α ˆ(p, 2) , dˆ†(p, −s) =α ˆ(p, 3) , dˆ†(p, +s) =α ˆ(p, 4) . (3.112) The latter is a canonical transformation since it does not change the form of the anti-commutation relations. Now expressed in terms of the new operators, the anti-commutation relations read as

nˆ ˆ† 0 0 o 3 0 b(p, s), b (p , s ) = δss0 δ (p − p ) , (3.113)

n ˆ ˆ† 0 0 o 3 0 d(p, s), d (p , s ) = δss0 δ (p − p ) , (3.114) where all the other combinations of operators do vanish. We also switched again in the continuous description. Now, using the new notation the expanded ﬁeld operator reads as

1/2 X Z d3p m ψˆ(x, t) = ˆb(p, s)u(p, s) e−ip·x + dˆ†(p, s)υ(p, s) e+ip·x . (3.115) (2π)3/2 ω s p Therefore, the Hamiltonian operator takes the form Z X 3 ˆ† ˆ ˆ† ˆ Hˆ = d p ωp b (p, s)b(p, s) + (d (p, s)d(p, s) . (3.116) s Summarizing, the interpretation of the new operators reads as ˆb† : creator of particle, ˆb : annihilator of particle, dˆ† : creator of antiparticle, dˆ : annihilator of antiparticle.

33 The vacuum state |0i deﬁned as earlier according to the new notation is given as

ˆb(p, s) |0i = 0 , dˆ(p, s) |0i = 0 for s = ±1 . (3.117)

Thus, the Fock space can be constructed by the repeated action of creation operators ˆb†(p, s), dˆ†(p, s) vacuum state |0i. These states will be anti-symmetric and they automatically respect Pauli’ s exclusion principle. In Fock space the state vectors are normalized to unity and the scalar product is given by

0 0 n , n ... np1,s, np1,s ... = δ 0 δ 0 ... (3.118) p1,s ep1,s e np1,s, np1,s ,nep1,s,,nep1,s The total negative energy can also be discarded in a more elegant way. We follow the prescription of normal ordering. In the latter, one splits the ﬁeld operator into two parts. One part involves annihilation operators (positive frequency) and the other involves creation operators (negative frequency).

ψˆ = ψˆ(+) + ψˆ(−) . (3.119)

As a consequence, the expansion of ψ into plane waves (3.3) acting on vacuum, |0i leads to the following useful properties:

ψˆ(+) |0i = 0 , ψˆ(−)† |0i = 0 . (3.120)

In normal ordering one moves all positive frequency contributions to the right and the negative ones to the left. In addition, since the Dirac ﬁeld has anti-commuting character we add a minus sign for every ordering step we do. For example the normal ordering of two operatorsχ ˆ, ψˆ holds as:

: χˆ ψˆ : =χ ˆ(+)ψˆ(+) +χ ˆ(−)ψˆ(−) +χ ˆ(−)ψˆ(+) − ψˆ(−)χˆ(+) . (3.121)

In normal ordering the Hamiltonian operator immediately takes the physical form (without the negative energy contribution): Z Z 3 ˆ† ˆ X 3 ˆ† ˆ ˆ† ˆ Hˆ = d x : ψ (−iα · ∇ + βm) ψ : = d p ωp b (p, s)b(p, s) + (d (p, s)d(p, s) . (3.122) s And thus, we complete our approach to the basic concepts of Dirac ﬁeld quantization.

34 Chapter 4

Energy-Momentum Tensor

In this chapter we deal with the Energy-Momentum tensor in the cases of fluids and fields. The energy- momentum tensor is of great importance in particle physics and astrophysics. It describes the internal properties of an energy-mass distribution and in the framework of General Relativity it is considered the source of curvature of space-time. Each of its components has a specific physical interpretation. The component T 00 represents the density of energy, , the elements T 0i correspond to the energy flux across the xi-surface and T i0 give the momentum density. Since c = 1, the momentum density has to be equal to the energy flux. Finally, the spatial components T ij represent the flux of momentum across the j-surface or in other words the pressure and shear stress. Now let us study the case of a fluid with every part of its matter elements moving with velocity υ. An observer who also moves with velocity υ will observe the fluid to be momentarily static, with no spatial momentum components. Therefore the spatial components, T ij, have to be symmetric. Otherwise it would imply the elements are whirling around inside the fluid. In addition to the above we conclude that in this rest-frame of reference T µν is symmetric, and therefore T µν has to be symmetric in every frame of reference.

4.1 Relativity-EMT-Perfect Fluid

A perfect fluid is a medium with spherical symmetry (isotropic pressure) in which shear stresses and heat conduction are absent. This kind of fluid that moves with velocity υ is associated with an energy-momentum tensor which is defined in General Relativity as T µν = −P gµν + (P + ) uµuν , (4.1) where and P are the energy density and pressure respectively, in the rest frame of the fluid (υ = 0). Now, uµ is the four-velocity defined in special relativity as the derivative of xµ vector with respect of the proper time dτ: dxµ uµ = . (4.2) dτ Proper time, dτ, is connected with the the regular time component, x0 = t with the following relation dt 1 = γ = √ . (4.3) dτ 1 − υ2 Thus, the four-velocity components can be written as dxµ dxµ dt uµ = = = γ (1, υ) , (4.4) dτ dt dτ and uµ = γ (1, −υ) . (4.5) µ Therefore, it follows that u uµ = 1. The energy density, , and pressure, P , in rest frame are obtained using (4.1) in terms of the diagonal temporal component, T 00, and the trace, T ii, of the spatial elements of T µν : = T 00 , (4.6) 1 P = T ii . (4.7) 3 In next chapters these quantities will be replaced with the expectation values of the ground state, hT 00i and hT iii .

35 4.2 EMT- Klein-Gordon Field

In this section we examine the case of Klein-Gordon scalar ﬁeld. Here, the energy-momentum tensor is given by (2.32) using the Lagrange density (2.60). Since ∂L = ∂µφ the energy-momentum tensor is given by ∂(∂µφ) T µν = ∂µφ ∂ν φ − gµν L. (4.8)

In addition the symmetry of the metric tensor, gµν , guarantees the symmetry of T µν :

T νµ = ∂ν φ∂µφ − gνµL = ∂µφ∂ν φ − gµν L = T µν . (4.9)

Let us now compute explicitly each of the matrix elements of T µν , starting with T 00: 1 T 00 = (∂ φ)2 − g00 (∂ φ)2 − (∇φ)2 + m2φ2 0 2 0 1 = π2 + (∇φ)2 + m2φ2 2 = H . (4.10)

This result is expected since as Noether’ s theorem states, the volume integral of T 00 represents the total conserved energy, E, of the system, (2.34). Now, the energy ﬂux, T 0i = T i0, across the xi−surface is given by

T 0i = ∂0φ ∂iφ − g0iL |{z} =0 0i = −π∂iφ → T = −π∇φ , (4.11) therefore the momentum of the ﬁeld, P , is given by Z P = − d3x T 0i = −π∇φ . (4.12)

The pressure and shear stresses are expressed in terms of the spatial components as:

xx 2 yy 2 zz 2 T = (∂xφ) + L ,T = (∂yφ) + L ,T = (∂zφ) + L , (4.13) and

xy yx xz zx yz zy T = T = ∂xφ ∂yφ , T = T = ∂xφ ∂zφ , T = T = ∂yφ ∂zφ . (4.14)

4.3 EMT- Dirac Field

We proceed with the computation of Dirac ﬁeld’s energy-momentum tensor. We do so by deploying equations (2.32), (2.110) keeping in mind that in this case, the independent ﬁelds are ψ and ψ†. Hence, we obtain:

µν ∂L ν ∂L ν † µν T = ∂ ψ + † ∂ ψ − g L . (4.15) ∂(∂µψ) ∂(∂µψ ) However, since the second term vanishes the form simpliﬁes into:

T µν = iψγ¯ µ∂ν ψ − gµν L (4.16) which we can simplify even more by taking into account the equation of motion (2.104) of the Dirac ﬁeld, ψ, and hence discarding the second term. Thus we obtain:

T µν = iψγ¯ µ∂ν ψ . (4.17)

We clearly see that T µν is not symmetrical. Now let us compute the matrix elements of T µν . Once again, T 00 stands for the energy density of our system, hence the Hamilton density, H:

T 00 = iψ†ψ˙ = ψ† (−iα · ∇ + mβ) ψ = H (4.18)

36 and T 0i stand for the momentum density: T 0i = iψ†∂iψ † 0i † = −iψ ∂iψ → T = −iψ ∇ψ (4.19) written as a vector in space. Following Noether’ s theorem the momentum of the ﬁeld, P , is given as the expectation value of the momentum operator, Pˆ = −i∇, as: Z P = d3x ψ†(x)Pˆψ(x) . (4.20)

In general there is a way to derive a symmetric energy-momentum tensor, involving an additional term with vanishing four-divergence, so the conserved current is not altered. However we will not examine this case. Now, the rest of components are easily obtained through simple calculations: T i0 = iψγ¯ iψ˙ , (4.21)

xx † 1 yy † 2 zz † 3 T = −iψ α ∂xψ , T = −iψ α ∂yψ , T = −iψ α ∂zψ, (4.22) and

xy † 1 yx † 2 xy yx † T = −iψ α ∂yψ , T = −iψ α ∂xψ , T − T = −iψ (α × ∇ψ)z , (4.23) yz † 2 zy † 3 yz zy † T = −iψ α ∂zψ , T = −iψ α ∂yψ , T − T = −iψ (α × ∇ψ)x , (4.24) zx † 3 xz † 1 zx xz † T = −iψ α ∂xψ , T = −iψ α ∂zψ , T − T = −iψ (α × ∇ψ)y . (4.25) Now, as a consequence of equations(4.25) the stresses in the Dirac ﬁeld do not vanish, hence whirling forces are implied.

4.4 EMT- Massive Vector Boson Field

Consequently, we deal with the massive vector boson field. In this case we focus on constructing a symmetrical form of T µν rather than computing every single matrix element. Also note that the photon field is omitted thanks to its major similarities with the massive vector boson field. Hence, moving on the energy-momentum tensor , it is given through the Lagrange density (2.132), and the relation ∂(F F γδ) γδ = −F µα . (4.26) ∂(∂µAα) Remember that we assume an absence of sources, jµ = 0. Therefore, we substitute (4.26) to (2.32) which leads to:

µν µα ν µν T = −F ∂ Aα − g L 1 1 = gµν F F αβ − gµν m2A Aα − F µα∂ν A . (4.27) 4 αβ 2 α α However, this tensor clearly is not symmetric and as we stated in the earlier, we are able to replace T µν with its equivalent symmetric one. We do that, by adding a term which involves the four-divergence of a 3-rank tensor which is anti-symmetric with respect to the ﬁrst two indexes, χσµν = −χµσν . This way, the the four-divergence of the new tensor, Teµν , vanishes: µν µν σµν ∂µTe = ∂µT + ∂µ∂σχ = 0 (4.28) | {z } anti-symmetric and the conserved quantities are not altered either: Z Z Z Z ν 3 0ν 3 0ν 3 00ν 3 i0ν ν Pe = d x Te = d x T + d x ∂0χ + d x ∂iχ = P (4.29) | {z } | {z } χ00ν =0 surface terms where in the last step we applied Gauss theorem of divergence and deployed the fact that in surface our ﬁelds vanish. Therefore we follow this prescription to symmetrize the energy-momentum tensor T µν . Hence we add µσ ν µν the anti-symmetric term ∂σ(F A ) to T which leads to

µν µν µσ ν Te = T + ∂σ(F A ) µν µσ ν µσ ν = T + (∂σF )A + F ∂σA . (4.30)

37 Now, using the Proca equation we introduced in Ch1, (2.137), and the anti-symmetry of F µσ we deduce the relation µσ σµ 2 µ ∂σF = −∂σF = +m A (4.31) in addition, we exploit the useful relation

µσ ν µα ν µ σ ν µ ν α F ∂σA − F ∂ Aα = F σ∂ A − F α∂ A µ σ ν µ ν σ = F σ∂ A − F σ∂ A µ σ ν ν σ = F σ (∂ A − ∂ A ) µ σν µσ ν = F σF = F Fσ (4.32) thus, the new energy-momentum tensor takes its ﬁnal form, a form that is completely symmetrical: 1 1 T µν = gµν F F αβ − gµν m2A Aα + m2AµAν + F µσF ν . (4.33) 4 αβ 2 α σ

38 Chapter 5

Quantum Hadrodynamics - QHD

A successful model for the description of nuclear matter is one that is able to accurately describe high density matter (neutron star) and also predict correctly the properties of matter observed in normal densities. In addition, in high densed systems the particles reach energies comparable to their rest masses, and therefore relativistic phenomena arise. Thus, the study of high density systems must include the concepts of quantum mechanics, Lorentz covariance, electromagnetic gauge invariance and microscopic causality in the many-body system. For the above reasons we conclude that the most suitable candidate should be a theory compatible with relativistic quantum field theory. A well established theory among physicists is Quantum Chromodynamics (QCD). QCD describes the interaction between quarks through the exchange of gluons. It is a fundamental theory and at first glance could be a strong candidate for the description of nuclear matter. However, in nuclear experiments quark and gluon degrees of freedom are not observed but hadronic ones are. Hadrons are particles with internal structure which are subdivided to two categories: baryons and mesons. Baryons are fermions that consist of three quarks (protons, neutrons), while mesons are bosons that contain a quark- anti-quark pair. In this work we study the interaction of nucleons by deploying hadronic degrees of freedom, using the the relativistic quantum theory of hadrons (QHD) pronounced by John Walecka in 1974. Quantum Hadrodynamics is a relativistic quantum field theory best suited for the description of nuclear matter. The main idea is that nucleon-nucleon interaction is mediated by mesons. Additionally, QHD is an effective field theory, rather than a fundamental one since hadrons are composite particles. Now, in QHD, as in QCD, one has to surpass a lot of computational difficulties and thus approximation have to be made. However, since the coupling constants of QHD are large we cannot use perturbation expansion in terms of coupling constants. For this reason, we incorporate the approximation of relativistic mean-field theory. QHD is a model made to describe nuclear matter and as a model it needs additional experimental input data to constrain its validity. In particular, in QHD the coupling constants between different meson and nucleon fields are the unknown parameters. The usual method to determine these parameters is to fit the calculated properties of nuclear matter to the experimentally observed values. These parameter sets differ from each other since authors tend to alter the formalism.

39 5.1 QHD-I

In the framework of QHD the parameter sets vary among others, where the original and simplest one is QHD- I, or also known as the σ - ω model. The latter models the nucleon-nucleon interaction with the exchange of neutral (isoscalar) scalar σ-mesons and neutral (isoscalar) vector ω-mesons. The nucleons included in the QHD-I are protons and neutrons (baryons). It is well established that nucleus and nuclear matter is best studied using σ-, ω- mesons. In general the σ- mesons give rise to a strong attractive central force and a spin-orbit force in the interaction between nucleons, while in contrast the ω-mesons cause a strong repulsive central force and a spin-orbit force (same sign as the spin-orbit force associated with σ-meson). Summarizing, the nucleon-nucleon interaction in QHD-I, is accomplished with aid of three ﬁelds, the baryon (proton, neutron), scalar σ-meson and vector ω-meson ﬁelds. In addition, note that in QHD-I the electromagnetic properties of nuclear matter are omitted (no charged mesons are included) and the masses of proton and neutron are considered identical. Let us now proceed to the construction of the Lagrange density, L. Firstly, we have to introduce the notation we will use:

• ψ, ψ† represent the baryon field and conjugate baryon field respectively • V represents the ω-vector meson field • φ represents the σ-scalar meson field • mω and ms are the masses of the mesons and M denotes the nucleon mass • gυ and gs are the vector and scalar coupling constants • Vµν = ∂µVν − ∂ν Vµ . 1 The description of spin- 2 baryons is carried by Lagrange density of Dirac’ s field, and the scalar σ- field and vector ω- field are described by the additional terms of Klein-Gordon and massive vector boson fields (Proca). Since the Lagrange density has to be a Lorentz scalar quantity the scalar field, φ, has to couple to the scalar density of the baryon field, ψψ¯ , and the vector field V µ, has to couple to the conserved baryon current, ψγ¯ µψ. Another thing we have to take into account is that the scalar σ-field causes an attractive force and the vector ω-field a repulsive one. Thus the general form of the Lagrange density is:

L = T − V |{z} |{z} kinetic term potential 1 1 1 = ψ¯(x)(iγµ∂ − M) ψ(x) + ∂ φ(x)∂µφ(x) − m2φ2(x) − V V µν + m2 V (x)V µ(x) µ 2 µ s 4 µν 2 ω µ ¯ µ ¯ + gsφ(x)ψ(x)ψ(x) − gυV (x)ψ(x)γµψ(x) (5.1) | {z } | {z } attractive force repulsive force or expressed in a simpler form:

¯ h µ µ i L = ψ(x) γµ i∂ − gυV (x) − M − gsφ(x) ψ(x) 1 1 1 = + ∂ φ(x)∂µφ(x) − m2φ2(x) − V V µν + m2 V (x)V µ(x) . (5.2) 2 µ s 4 µν 2 ω µ Therefore we can now compute the Euler-Lagrange equations of motion of our system. For this purpose we ﬁrst have to calculate a few useful partial derivative expressions. Starting with : ∂L = γ0iγ ∂µψ(x) − γ0Mψ(x) + g φ(x)γ0ψ(x) − g V µ(x)γ0γ ψ(x) ∂ψ† µ s υ µ 0 µ µ = γ iγµ∂ ψ(x) − Mψ(x) + gsφ(x)ψ(x) − gυV (x)γµψ(x) , (5.3) ∂L † = 0 , (5.4) ∂(∂µψ ) and moving on the derivatives involving scalar ﬁeld φ:

∂L 2 ¯ ∂µL µ = −msφ(x) + gsψ(x)ψ(x) , = ∂ φ(x) , (5.5) ∂φ ∂(∂µφ)

40 and the vector ﬁeld V µ:

∂L 1 2 ∂ α ∂ α ¯ = mω V (x)Vα(x) − gυ V (x)ψ(x)γαψ(x) ∂Vµ 2 ∂Vµ ∂Vµ 2 µ ¯ µ = mωV (x) − gυψ(x)γ ψ(x) , (5.6) ∂L 1 ∂ αβ = − VαβV ∂(∂ν Vµ) 4 ∂(∂ν Vµ) 1 αβ ∂ ακ βλ ∂ = − V (x) Vαβ(x) + Vαβ(x)g g Vκλ(x) 4 ∂(∂ν Vµ) ∂(∂ν Vµ) 1 αβ ∂ κλ ∂ = − V (x) ∂αVβ − ∂βVα + V (x) ∂κVλ − ∂λVκ 4 ∂(∂ν Vµ) ∂(∂ν Vµ) 1 h i = − δ ν δ µ − δ ν δ µ V αβ(x) + V κλ(x) δ ν δ µ − δ ν δ µ 4 α β β α κ λ λ κ 1 = − V νµ − V µν + V νµ − V µν = V µν . (5.7) 4 Thus, employing equations (5.4), (5.5), (5.7) and substituting them into Euler-Lagrange equations we obtain the following non-linear coupled diﬀerential equations:

µ 2 ¯ ∂µ∂ φ(x) + msφ(x) = gsψ(x)ψ(x) (5.8) µν 2 ν ¯ ν ∂µV + mωV (x) = gυψ(x)γ ψ(x) (5.9) µ µ [γµ (i∂ − gυV (x)) − (M − gsφ(x))] ψ(x) = 0 . (5.10)

As we see, the first equation is nothing more but the Klein-Gordon equation of the scalar field with an additional source term, the baryon scalar density, ψ¯(x)ψ(x). The second one, the Proca equation, describes the vector field having as a source term the coupling of it to the conserved baryon current, ψ¯(x)γν ψ(x). Finally the third one is the modified Dirac equation for the baryon field, with a shifted mass (M − gsφ(x)) due to the scalar field and an additional term for the vector-baryon field interaction. These equations are too complicated to be solved and for this reason we deploy the approximation of Relativistic Mean-Field theory.

5.2 Relativistic Mean-Field Approximation

As the title indicates , the approximation is based on the replacement of the unknown fields with their mean- values. In relativistic mean-field approximation (RMF) the meson field operators are replaced with classical fields, and in particular with their ground state (|Φi) expectation values in the following a way:

ˆ ˆ φ −→ hΦ| φ |Φi = hφi = φ0 , (5.11)

Vˆµ −→ hΦ| Vˆµ |Φi = hVµi = δµ0V0 . (5.12)

To justify this, we consider a box with volume V where we fill it uniformly with B baryons. Since we know the volume, the baryon density, ρB, is also known. If we start shrinking the box, the baryon density, ρB, enlarges and as a consequence the source terms on the right-hand of the meson field equations of motion (5.10) become large too. This way the meson field operators and the source terms can be replaced with their ground state expectation values. Another consideration is that the system has to be at rest, in zero temperature (T = 0 K), meaning that the baryons occupy energy levels till a maximum energy, the Fermi energy. In the static system the baryon flux, ψ¯(x)γiψ(x), is zero (there is no baryon current) since there is no preferable spatial direction of the baryons. Under these conditions, in the static, uniform system the meson fields φ0, V0, are independent of space-time which greatly simplifies the problem. As we mentioned earlier, in RMF approximation the source terms are replaced with the expectation values of the baryon field operators. These operators are evaluated with respect to the ground state, |Φi of the system. Since the baryon field gives rise to negative energy solutions the expectation values have to normal-ordered.

ψ¯(x)ψ(x) −→ hΦ| : ψ¯ˆ(x)ψˆ(x) : |Φi = hψψ¯ i (5.13) ψ¯(x)γµψ(x) −→ hΦ| : ψ¯ˆ(x)γµψˆ(x) : |Φi = hψγ¯ 0ψi = hψ†ψi . (5.14)

The normal ordered expectation values are chosen so the contribution of negative energy states is ignored. This is the no Dirac-sea approximation for the modiﬁed Dirac equation and was introduced in previous chapter.

41 µ Substituting the x-independent φ0 and V into Euler-Lagrange equations, (5.10), they are greatly simpliﬁed: 2 ¯ msφ0 = gshψψi (5.15) 2 † mωV0 = gυhψ ψi (5.16) µ [iγµ∂ − gυγ0V0 − (M − gsφ0)] ψ = 0 (5.17) where the spatial components V i vanish as a consequence of the vanishing baryon ﬂux, ψγ¯ iψ:

2 i ¯ i mωV = gυhψγ ψi = 0 . (5.18) Thus the Lagrange density of RMF QHD-I takes the simpler form: h i 1 1 L = ψ¯(x) iγ ∂µ − g γ V − M − g φ ψ(x) − m2φ2 + m V 2 . (5.19) RMF µ υ 0 0 s 0 2 s 0 2 ω 0

The energy-momentum tensor for RMF QHD-I Lagrange density, LRMF is easily computed as 1 1 T µν = iψγ¯ µ∂ν ψ − gµν − m2φ2 + m2 V 2 . (5.20) RMF 2 s 0 2 ω 0 Therefore, we calculate the energy density, , and pressure, P , of QHD-I: D E = T 00 D E 1 1 = i ψγ¯ 0∂0ψ − g00 − m2φ2 + m2 V 2 2 s 0 2 ω 0 D E 1 1 = ψ†iψ˙ + m2φ2 − m2 V 2 , (5.21) 2 s 0 2 ω 0 1 D E P = T ii 3 1 D E 1 1 1 = i ψγ¯ i∂iψ − gii − m2φ2 + m2 V 2 3 3 2 s 0 2 ω 0 1 D E 1 1 = −i ψ†αi∂ ψ − m2φ2 + m2 V 2 . (5.22) 3 i 2 s 0 2 ω 0

5.3 Baryon Field - Expectation values

¯ † † ˙ † i Now, in order to compute the expectation values hψψi, hψ ψi, hψ iψi and hψ α ∂iψi we have to construct an explicit representation of the baryon ﬁeld operator, ψˆ. Thus we expand ψˆ in Fourier decomposition in terms of a set of classical functions. For this purpose we have to solve the Dirac equation for the QHD-I. Firstly, let us manipulate a little bit (5.17):

0 i 0 ∗ iγ ∂0 + iγ ∂i − gυγ V0 − m ψ = 0 0 0 0 i 0 ∗ γ iγ ∂0 + iγ α ∂i − gυγ V0 − m ψ = 0 ˙ ∗ iψ = − iα · ∇ + gυV0 + m β ψ ˙ iψ = HˆDψ (5.23)

∗ where we introduced the modiﬁed mass, m = M − gsφ0, shifted due to the presence of the scalar ﬁeld φ0, ∗ and the Hamilton Dirac operator: HˆD = α · pˆ + gυV0 + m β. Now as we see, HˆD commutes with momentum operator pˆ and therefore plane-waves must be eigenfunctions of HˆD. Thus we try solutions of the form

ψ(x, t) = ω(k) ei(k·x−t) (5.24) where we switched to wave-length notation, k, by employing De Broglie’ s momentum-wavevector relation, p = ~ k. ω(k) is a 4-component spinor, similar to the Dirac single particle’ s case, which can be written as φ ω(k) = (5.25) χ with φ and χ being 2-component spinors. Now, ω(k) is x-independent and thus its insertion to (5.23) leads to the algebraic equation:

∗ ω = α · k + gυV0 + m β ω (5.26)

42 or written in matrix form

∗ φ gυV0 + m σ · k φ = ∗ . (5.27) χ σ · k gυV0 − m χ Therefore, we obtain the coupled equations:

∗ φ = (gυV0 + m ) φ + σ · k χ (5.28) ∗ χ = (gυV0 − m ) χ + σ · k φ (5.29) which can be solved either in terms of χ or φ: σ · k σ · k χ = ∗ φ and φ = ∗ χ . (5.30) e + m e − m where e = − gυV0 is the modiﬁed energy. Now, again similarly to Dirac’ s single particle, we substitute one of the solutions to the other and thus we obtain the relation: k2 ( + m∗) χ = χ e ∗ e − m ∗ ∗ 2 (e + m )(e − m ) χ = k χ 2 ∗2 2 e = m + k (5.31) Therefore the Baryon Dirac equation allows positive and negative energy solutions which are symmetric around gυV0:

± p ∗2 2 (k) = gυV0 ± m + k . (5.32)

Let us now extract the exact form of ω-spinors. Due to the similarities to Dirac ’s single particle we know that the ﬁrst two spinors, ω1, ω2 represent positive energy solutions φr ωr(k) = σ·k for r = 1, 2 (5.33) ∗ φr e+m with φr taking the form: 1 0 φ = and φ = . (5.34) 1 0 2 1

On the contrary, the spinors attributed with negative energy are redeﬁned by replacing k to −k, and are given by

−σ·k −m∗ χr ωr(k) = e for r = 1, 2 (5.35) χr with 1 0 χ = and χ = . (5.36) 1 0 2 1

Now, before we proceed to the quantization of the Baryon Dirac ﬁeld, we introduce the new notation for the spinors, like we did in Dirac’ s ﬁeld quantization, :

ω1(k) = u(k, +s) ,

ω2(k) = u(k, −s) ,

ω3(k) = υ(k, −s) ,

ω4(k) = υ(k, +s) . (5.37)

The new spinors u(k, s) and υ(k, s) satisfy similar orthogonality and completeness relations with (3.87). In addition, we impose the normalization condition:

† 0 † 0 u(k, s) u(k, s ) = υ(k, s) υ(k, s ) = δss0 (5.38) (5.39)

43 Thus, their ﬁnal normalized form is given by: r r + m∗ φ − m∗ −σ·k e s e −m∗ χs u(k, s) = σ·k for s = 1, 2 υ(k, s) = e for s = 2, 1 . (5.40) 2 ∗ φs 2 χs e e+m e As a consequence we obtain the relations between the Dirac adjoint spinors:

∗ 0 0 m u¯(k, s)u(k, s ) =υ ¯(k, s)υ(k, s ) = δss0 (5.41) ωk √ 2 ∗2 ˆ where ωk = k + m . At this point, we are able to derive the exact form of the baryon ﬁeld operator ψ. We expand it in terms of the solutions regarding the QHD-I Dirac equation. Therefore:

Z 3 X d k +i(k·x−+t) † −i(k·x+−t) ψˆ(x, t) = ˆb(k, s) u(k, s)e k + dˆ (k, s) υ(k, s)e k (5.42) (2π)3/2 s where we included the particle and anti-particle creation and annihilation operators that satisfy the same anti-commutation relations as in (3.114). Moreover, the adjoint ﬁeld operator is given by:

Z 3 † X d k † † −i(k·x−+t) † +i(k·x+−t) ψˆ (x, t) = ˆb (k, s) u (k, s)e k + dˆ(k, s) υ (k, s)e k . (5.43) (2π)3/2 s Since we study a uniform static system at zero temperature, our ground state is obviously not the vacuum. We consider ground state, |Φi, the state that is ﬁlled with a particular amount of particles and no anti-particles. The maximum energy of the system is the Fermi energy that corresponds to a Fermi wave-number, kf . All the other positive energy states with wave-length below kf are also ﬁlled. Therefore the ground state, |Φi, in the no-sea approximation, has the following properties: ˆ ˆ ne(k, s) |Φi = 0 → d(k, s) |Φi = 0 ∀ |k| , (5.44) ˆ nˆ(k, s) |Φi = 0 → b(k, s) |Φi = 0 ∀ |k| > kf , (5.45) ˆ† b (k, s) |Φi = 0 ∀|k| < kf . (5.46)

Moving on computing the corresponding expectation values, we deploy the expression of baryon operator, ψˆ, and we take into account the rules of normal ordering and the properties of the ground state |Φi. In normal ordering one moves all annihilation operators to the right side and all creation operators to the left, where every ordering step adds a minus sign. Thus we start with the normal ordering of ψˆ†ψˆ

1 Z Z 0 + + † X 0 † † +i(k −k)·x +i( − 0 )t : ψˆ ψˆ : = dk dk ˆb ˆb 0 0 u u 0 0 e e k k (2π)3 k,s k ,s k,s k ,s ss0 † † † 0 + − ˆ ˆ −i(k +k)·x +i(k − 0 )t + bk,sdk0,s0 uk,sυk0,s0 e e k † 0 − + ˆ ˆ +i(k +k)·x +i(k − 0 )t + dk,sbk0,s0 υk,suk0,s e e k † † 0 − − ˆ ˆ −i(k −k)·x +i( 0 −k )t − dk0,s0 dk,sυk,sυk0,s0 e e k (5.47) where we switched to more simple notation. Therefore the expectation value takes the form:

hψ†ψi = hΦ| : ψˆ†ψˆ : |Φi

44 For this purpose we have to take into account that the different baryon states are vectors in Fock space. These vectors satisfy orthogonality relations similar to Dirac’ s filed. They are normalized to unity and their inner product is similar to (3.118). ˆ† ˆ 0 Therefore let us examine the first inner product, hΦ| bk,sbk0,s0 |Φi, where |k | < kf , otherwise it has to vanish. ˆ In the ground state, |Φi, all the energy states below f are occupied by particles. Moreover bk0,s0 |Φi represents a state where is equal to |Φi with exception that there is an absence of a particle with k0-wave vector and s0 spin projection, which we will denote as: ˆ |Φ − 1k0,s0 i = bk0,s0 |Φi . (5.50) ˆ† Now, since the operator bk,s creates a particle with k-wave vector and s spin projection two cases arise. In the 0 first one where |k| < kf , there is a creation of a particle where all the energy states except the one with k -wave 0 ˆ† ˆ vector and s spin projection are filled, and thus thanks to (5.46) every one of the states bk,sbk0,s0 |Φi has to vanish except the case where k = k0 and s = s0 are satisfied simultaneously. In the second case, |k| > kf , the energy states are empty and thus the creation operator leads to a new state where we denote as ˆ† ˆ ˆ† bk,sbk0,s0 |Φi = bk,s |Φ − 1k0,s0 i = |Φ − 1k0,s0 + 1k,si (5.51) which is a different vector in Fock space and thus the inner product vanishes due to orthogonality: ˆ† ˆ hΦ| bk,sbk0,s0 |Φi = hΦ|Φ − 1k0,s0 + 1k,si = 0 . (5.52) Therefore we can summarize these cases in a more compact form as ˆ† ˆ 3 0 0 hΦ| bk,sbk0,s0 |Φi = δ (k − k ) δss0 ∀ |k|, |k | < kf (5.53) while the rest of inner products vanish in a similar fashion. Hence: ˆ† ˆ† ˆ ˆ hΦ| bk,sdk0,s0 |Φi = 0 & hΦ| dk,sbk0,s0 |Φi = 0 . (5.54) Thus, using the above result, the expectation value hψ†ψi is given by hψ†ψi = hΦ| : ψˆ†ψˆ : |Φi

1 Z Z 0 + + X 0 3 0 † +i(k −k)·x +i( − 0 )t = dk dk δ (k − k ) δ 0 u u 0 0 e e k k (2π)3 ss k,s k ,s ss0 X 1 Z = dk u† u ·1 · 1 (2π)3 k,s k,s s | {z } =1 " # 1 Z X = dk (2π)3 s | {z } =γ

Z kf γ 2 = 3 4π dk k (2π) 0 γ = k3 6π2 f = ρ (5.55) where we introduced γ as the spin-degeneracy of the baryon. The ρ quantity represents the baryon number density when maximum |k| equals kf . Its motivation comes from the time component of the conserved current, j0 = ψγ¯ 0ψ = ψ†ψ. Now the next expectation value is computed in a similar way: hψψ¯ i = hΦ| : ψ¯ˆψˆ : |Φi

1 Z Z 0 + + X 0 † +i(k −k)·x +i( − 0 )t = dk dk hΦ| ˆb ˆb 0 0 |Φi u¯ u 0 0 e e k k (2π)3 k,s k ,s k,s k ,s ss0 † † 0 + − ˆ ˆ −i(k +k)·x +i(k − 0 )t + hΦ| bk,sdk0,s0 |Φi u¯k,sυk0,s0 e e k 0 − + ˆ ˆ +i(k +k)·x +i(k − 0 )t + hΦ| dk,sbk0,s0 |Φi υ¯k,suk0,s e e k † 0 − − ˆ ˆ −i(k −k)·x +i( 0 −k )t − hΦ| dk0,s0 dk,s |Φi υ¯k,sυk0,s0 e e k X 1 Z = dk u¯ u (2π)3 k,s k,s s | {z } ∗ m /ωk γ Z kf k2m∗ = dk √ . (5.56) 2 2 ∗2 2π 0 k + m

45 ˙ Moving on hψ†iψ˙i, we ﬁrst derive the expression of the operator iψˆ :

Z 3 ˙ X d k 2 + +i(k·x−+t) † 2 − −i(k·x+−t) iψˆ = ˆb u − i e k + dˆ υ − i e k (2π)3/2 k,s k,s k k,s k,s k s Z 3 X d k + +i(k·x−+t) † − −i(k·x+−t) = ˆb u e k + dˆ υ e k (5.57) (2π)3/2 k,s k,s k k,s k,s k s and therefore the normal ordered expectation values takes the form ˆ hψ†iψ˙i = hΦ| : ψˆ†iψ˙ : |Φi

1 Z Z 0 + + X 0 † † + +i(k −k)·x +i( − 0 )t = dk dk hΦ| ˆb ˆb 0 0 |Φi u u 0 0 e e k k + vanishing terms (2π)3 k,s k ,s k,s k ,s k0 ss0 | {z } 3 0 δ (k−k ) δss0 X 1 Z = dk + (2π)3 k s Z kf γ 2 p 2 ∗2 = 2 dk k gυV0 + k + m 2π 0 Z kf γ 2p 2 ∗2 = gυV0 ρ + 2 dk k k + m . (5.58) 2π 0

† For the ﬁnal expectation value, hψ − iα · ∇ ψi, we ﬁrst compute a few expressions related with the uk,s baryon spinors that will turn out to be very useful. Using ,(5.41) uk,s and uk,2 are given by

 1   1  r ∗ r ∗ e + m  0  e + m  0  uk,1 =  k3  & uk,1 =  k1−ik2  . (5.59) 2 ∗ 2 ∗ e  e+m  e  e+m  k1+ik2 −k3 ∗ ∗ e+m e+m Consequently we have to compute the following expression:

r r 2 + m∗ φ + m∗ k e 0 σ · k 1 e +m∗ φ1 α · k uk,1 = σ·k = e (5.60) 2 σ · k 0 ∗ φ1 2 σ · kφ e e+m e 1 and therefore

 k2  +m∗ ∗ e ∗ 2 2 2 † + m 0 + m k k k e 1 , 0 , k3 , k1−ik2   e uk,1 α · k uk,1 = +m∗ +m∗   = ∗ + ∗ = . 2e e e  k3  2e e + m e + m e k1 + ik2 (5.61)

In the same fashion we derive the expression regarding uk,2 spinor

2 † k uk,2 α · k uk,2 = . (5.62) e or expressed in a more compressed form:

2 † k uk,s α · k uk,s = δss0 (5.63) e Therefore by employing the above relations we proceed to calculation of hψ† − iα · ∇ψi. However, let us ﬁrst calculate the action of − iα · ∇ to ﬁeld operator ψˆ:

Z 3 X d k +i(k·x−+t) † −i(k·x+−t) − iα · ∇ ψˆ = ˆb α · k u e k − dˆ α · k υ e k . (5.64) (2π)3/2 k,s k,s k,s k,s s

46 Thus, D E ψ† − iα · ∇ψ = hΦ| : ψˆ† − iα · ∇ψˆ : |Φi

1 Z Z 0 + + X 0 † † +i(k −k)·x +i( − 0 )t = dk dk hΦ| ˆb ˆb 0 0 |Φi u α · k u e e k k (2π)3 k,s k ,s k,s k,s ss0 | {z } | {z } δ3(k−k0) δ 0 k2 ss δss0 e + vanishing terms X 1 Z k2 = dk √ (2π)3 2 ∗2 s k + m γ Z kf k4 = dk √ (5.65) 2 2 ∗2 2π 0 k + m where we used eq.(5.63) we derived earlier. With this, we completed the calculations for all the required quantities regarding φ0, V0, and P . However the problem now lies on extracting the exact form of the involved integrals.

5.4 Parameter Set, Equation of State, Observables

The values of the particle masses used for this study, for QHD-I formalism [8], are presented in Table 5.1.

Mproton Mneutron ms mυ 939 939 520 783

Table 5.1: Masses of particles (in MeV) in QHD-I.

In addition, the other unknown quantities are the coupling constants gs, gυ. They are given in terms of Cs and Cυ which are deﬁned as

2 2 2 2 M 2 2 M Cs = gs 2 & Cυ = gυ 2 . (5.66) ms mυ

2 2 The values of Cs, Cυ are given in Table 5.2, next to the corresponding values of gs , gυ. These values of Cs, Cυ are chosen because they give better description of the saturation properties of nuclear matter.

2 2 2 2 Cs Cυ gs gυ 357.4 273.8 109.6 190.4

Table 5.2: Coupling constants for the QHD-I.

Let us now take advantage of the previous section’ s results. Substituting (5.55), (5.56)to the Euler-Lagrange equations in RMF approximation, (5.17), the equations of motion for the ﬁelds φ0, V0 reduce to

Z 2 ∗ gs γ k m φ0 = dk √ , (5.67) 2 2 2 ∗2 ms 2π k + m gυ V0 = 2 ρ . (5.68) mω Moreover employing (5.58), (5.65), the energy density and pressure of the system take the form

Z kf 1 2 2 1 2 2 γ 2p 2 ∗2 = msφ0 − mωV0 + gυV0 ρ + 2 dk k k + m , (5.69) 2 2 2π 0 1 1 γ Z kf k4 P = − m2φ2 + m2 V 2 + dk √ . (5.70) s 0 ω 0 2 2 ∗2 2 2 6π 0 k + m

Therefore we now proceed on computing the symmetry energy coeﬃcient, α4 which was previously deﬁned as

2 ∂ ρn − ρp α4 = where t ≡ . (5.71) ∂t2 ρ t=0 ρ

47 The quantity /ρ represents the energy per nucleon and t is variable that shows the asymmetry (between protons and neutrons) of the system. Now, thanks to (5.55) we can similarly extract the number densities for protons and neutrons in terms of their respective Fermi- wavelengths, kp and kn: k3 k3 ρ = p & ρ = n γ = 2 . (5.72) p 3π2 n 3π2 Where the total density of the system is given as the summation

ρ = ρp + ρn . (5.73)

For the case of symmetric nuclear matter, ρn equals ρp and therefore t = 0. As a consequence the Fermi- wavelengths kp and kn are equal too. Thus the total density for t = 0 is given by k3 ρ = 2 × f . (5.74) 0 3π2

Now since the only surviving terms of that contribute to α4 are the ones that involve the integrals between wavelengths we deﬁne the isospin-dependent energy per nucleon as

Z kp Z kn 1 2p 2 ∗2 1 2p 2 ∗2 s/ρ = 2 dk k k + m + 2 dk k k + m . (5.75) ρπ 0 ρπ 0

To calculate the partial derivative (5.71), s/ρ has to be ρ-, t-dependent. For this purpose we have to express kp and kn in terms of ρ, t. It can easily be shown that these relations are: h3π2 i1/3 k = ρ (1 − t) = k (ρ, t) , (5.76) p 2 p h3π2 i1/3 k = ρ (1 + t) = k (ρ, t) . (5.77) n 2 n Therefore Z kp Z kn ∂ s 1 ∂kp ∂ 2p 2 ∗2 1 ∂kn ∂ 2p 2 ∗2 = 2 dk k k + m + 2 dk k k + m ∂t ρ ρπ ∂t ∂kp 0 ρπ ∂t ∂kn 0 2 q 2 q ρπ 1 1 2 2 ∗2 ρπ 1 1 2 2 ∗2 = − 2 2 kp kp + m + 2 2 kn kn + m 2 kp ρπ 2 kn ρπ 1q 1q = k2 + m∗2 − k2 + m∗2 (5.78) 2 n 2 p and thus the second derivative is given as   ∂2 1 k ∂k k ∂k s = n n − p p 2 p q  ∂t ρ 2 k2 + m∗2 ∂t 2 ∗2 ∂t n kp + m   1 k ρπ2 1 k ρπ2 1 = n + p p 2 ∗2 2 q 2  2 k + m 2 kn 2 ∗2 2 kp n kp + m   1 1 1 2 1 1 2 =  p ρπ + q ρπ  . (5.79) 2 2kn k2 + m∗2 2kp 2 ∗2 n kp + m

Finally calculating the derivative for t = 0, kp = kn = kf   ∂2 1 2ρ π2 1 s = 0 2  q  ∂t ρ t=0 2 kf 2 ∗2 kp + m 1 π2 2k3 1 = f 2 q 2 kf 3π 2 ∗2 kp + m 2 1 kf = q (5.80) 3 2 ∗2 kp + m and thus α4 is given by 2 kf α4 = . (5.81) q 2 ∗2 6 kf + m

48 5.5 Alternative Method - Expectation Values

In this section we introduce an alternative method for deriving the normal-ordered expectation values we computed in the previous sections. The method was motivated by N. K. Glendenning, and it suggests that the expectation value of an operator Γˆ with respect to the ground state in the many-body system can be written as the continuum summation the single-particle expectation value, Z ¯ 3 ¯ (ψΓψ)k,s = d x ψk,s(x)Γˆψk,s(x) . (5.82) where k denotes the wavelength-vector of the particle and s denotes its spin orientation. Thus the expectation value in the many-body system is given by X Z dk hψ¯Γψi = (ψ¯Γψ) Θ[µ − ] . (5.83) (2π)3 k,s k s where √ ∗2 2 • k = gυV0 ± m + k , is the positive single particle energy, since we consider the ground state in no-sea approximation. • µ is the chemical potential which at T = 0 k equals the Fermi energy . kf • Θ[µ − k] is a step function where ( 1 if |k| ≤ kf Θ[µ − k] = . (5.84) 0 if |k| > kf

The main idea is that all the information about Γ or all the possibilities for Γ can be found in the Baryon Dirac Hamiltonian, HˆD:

∗ HˆD = −iα · ∇ + gυV0 + m β . (5.85)

Let us now take the single-particle expectation value of HˆD.

† † † (ψ HDψ)k,s = (ψ kψ)k,s = k(ψ ψ)k,s = k (5.86)

† R 3 † where (ψ ψ)k,s = d x ψk,s(x)ψk,s(x) = 1 arises from the normalization of the Baryon spinors, (5.39). Conse- † † quently, we take the derivative of (ψ HDψ)k,s with respect to any variable ζ, of (ψ HDψ)k,s: ∂ ∂H ∂ψ† ∂ψ (ψ†H ψ) = (ψ† D ψ) + ( H ψ) + (ψ†H ) ∂ζ D k,s ∂ζ k,s ∂ζ D k,s D ∂ζ k,s ∂H ∂ψ† ∂ψ = (ψ† D ψ) + ( ψ) + (ψ† ) ∂ζ k,s k ∂ζ k,s k ∂ζ k,s ∂H ∂ = (ψ† D ψ) + (ψ†ψ) ∂ζ k,s k ∂ζ k,s ∂H = (ψ† D ψ) . (5.87) ∂ζ k,s Therefore, we can obtain any single-particle expectation value by choosing the appropriate ζ for the derivative ¯ † ˙ of HD to yield Γ. Let us now compute the expectation values of the previous section, hψψi, hψ iψi and hψ†(−iα · ∇)ψi. ¯ † 0 (ψψ)k,s = (ψ γ ψ)k,s ∂H = (ψ† D ψ) ∂M k,s ∂ = ∂M k m∗ = √ , (5.88) k2 + m∗2 Hence, employing the general expression (5.83), hψψ¯ i takes the form: Z ∗ ¯ X dk m hψψi = √ Θ[µ − k] (2π)3 2 ∗2 s k + m γ Z kf dk k2m∗ = √ . (5.89) 2 3 2 ∗2 2π 0 (2π) k + m

49 Similarly, the expectation value regarding the energy density, hψ†iψ˙i is as calculated as follows

† ˙ † (ψ iψ)k,s = (ψ HDψ)k,s † = k(ψ ψ)k,s

= k (5.90)

Hence,

X Z dk hψ†iψ˙i = Θ[µ − ] (2π)3 k k s X Z dk p = g V + k2 + m∗2 Θ[µ − ] (2π)3 υ 0 k s Z kf γ 2p 2 ∗2 = gυV0 ρ + 2 dk k k + m . (5.91) 2π 0

Finally, hψ†(−iα · ∇)ψi is given as

† ¯ (ψ (−iα · ∇)ψ)k,s = (ψ(γ · k)ψ)k,s ¯ = (ψγψ)k,s · k ∂H = (ψ† D ψ) · k ∂k k,s ∂ = k · k ∂k k · k = √ (5.92) k2 + m∗2 and thus Z 2 D † E X dk k ψ (−iα · ∇)ψ = √ Θ[µ − k] (2π)3 2 ∗2 s k + m γ Z kf k4 = dk √ . (5.93) 2 2 ∗2 2π 0 k + m

50 Chapter 6

Advanced QHD Parameter Sets

Unfortunately the QHD-I parameter set does not produce accurate values of nuclear properties such as the compressibility, K. Thus, in this chapter we take a step forward and study advanced QHD parameter sets. The starting point was 1997, where J. Boguta and A. R. Bodmer suggested the inclusion of self-coupling of the scalar σ-meson field in the QHD Lagrange density. In addition, they included the charged (isovector) vector ρ-meson triplet (ρ0, ρ±). Now, protons and neutrons with respect to strong interaction are treated equally. They are considered to be the different projections of the nucleon in isospin space. Therefore the addition of the ρ-triplet leads to better distinction between the baryons and thus more accurate values of the symmetry energy. Due to their charge, the interaction between ρ-mesons and protons is different than the one with neutrons. The ρ-mesons are included in the QHD Lagrange density with the additional coupling between the ρ-meson and the conserved isospin density. Various other parameter sets expanded the previous ideas. For instance, the NL-SH parameter set was formulated to obtain better description of nuclei radii of neutron-rich nuclei. Later, in 1994, Y. Sugahara and H. Toki introduced the self-coupling of the ω-meson field and produced the TM-1, TM-2 parameter sets regarding the light and heavy unstable nuclei respectively. G. A. Lalazissis et al. to (among st others) formulated the NL3 parameter set, in which the ρ-mesons and the self-coupling of σ-meson are included to achieve better description of nuclei with large isospin asymmetry. PK1 parameter set is an improvement of TM-1 obtained by fitting of the properties in a wider spectrum of heavy nuclei. B. Todd-Rutel and J. Piekarewicz proposed the coupling between ρ-meson and ω-meson field and formulated the FSUGold parameter set. In general, in all the above parameter sets the photon field was also included. However in this work the formalism aims the study of neutron stars. Now, neutron stars are considered to be charge neutral. This is justified due to the electromagnetic forces being of greater magnitude than gravitational forces, meaning that a charged neutron star would overcome gravitational forces.

51 6.1 Formalism and Conventions

Now, the most general Lagrange density compatible with QHD-I, NL-SH, TM-1,TM-2, NL-3, PK-1 and FSUGold is the following h g i L = ψ¯(x) γµ i∂ − g V (x) − ρ τ · b (x) − M − g φ(x) ψ(x) µ υ µ 2 µ s 1 1 κ 3 λ 4 + ∂ φ ∂µφ(x) − m2φ2(x) − g φ(x) − g φ(x) 2 µ 2 s 3! s 4! s 1 1 ζ 2 − V µν V + m2 V µ(x)V (x) + g2V µ(x)V (x) 4 µν 2 ω µ 4! υ µ 1 1 − bµν b + m2bµ(x) · b (x) 4 µν 2 ρ µ 2 µ 2 µ + Λυ gυV (x)Vµ(x) gρb (x) · bµ(x) . (6.1)

The notation is similar with the one of the previous chapter with the addition of a few new concepts:

• ψ is the baryon field isodouplet (protons and neutrons) • V represents the ω-vector meson field • φ represents the σ-scalar meson field • mω, ms and mρ are the masses of the mesons and M denotes the nucleon mass • gυ, gs and gρ are the vector, scalar and isovector coupling constants • Vµν = ∂µVν − ∂ν Vµ • bµν = ∂µbν − ∂ν bµ • bµ denotes the Lorentz vector field with three isospin components of the ρ- meson fields:

µ µ µ µ b = b1 , b2 , b3 . (6.2)

µ 0 ± The third component, b3 represents the neutral ρ meson while the charged ρ mesons are expressed in terms µ µ of b1 and b2 as follows: 1 bµ = √ bµ ± bµ . (6.3) ± 2 1 2 τ = τ1, τ2, τ3 is the isospin operator and is given in terms of the Pauli matrices:

0 1 0 −i 1 0 τ = , τ = , τ = . (6.4) 1 1 0 2 i 0 3 0 −1

The proton and the neutron can be interpreted as the diﬀerent projections of nucleon in isospin space. The projection is carried by the operator τ3 with eigenvalues +1 and −1 for the proton and neutron respectively:

τ3 |pi = +1 |pi & τ3 |ni = −1 |ni (6.5) where,

1 0 |pi = & |ni = . (6.6) 0 1

The baryon ﬁeld isodouplet ψ is constructed by taking the tensor product between the vectors of 2-dimensional isospin space and the Dirac spinors for 4-dimensional spinor space as follows

ψ(x) = ψD ⊗ |pi + ψD ⊗ |ni (6.7) and thus

ψ 0 ψ ψ(x) = p + = p (6.8) 0 ψn ψn which is a 8 × 1 spinor. Therefore the isospin operator is given by the tensor product

τ = τ ⊗ 14×4 , (6.9)

52 with the third component projection operator given by 1 0 14×4 0 14×4 0 τ3 = ⊗ = . (6.10) 0 −1 0 14×4 0 −14×4

Now, in a similar fashion the Dirac γ matrices have to be redeﬁned to be compatible with 2-dimensional isospin vectors. Thus

µ µ µ 1 0 γ 04×4 γ = γD ⊗ = µ . (6.11) 0 1 04×4 γ

Now, by employing the following relations

∂L ¯ 2 κ 3 2 λ 4 3 ∂L µ = gsψψ − msφ − gs φ − gs φ , = ∂ φ , (6.12) ∂φ 2! 3! ∂(∂µφ) ∂L ¯ ν 2 ν ζ 4 ν α ∂L µν = −gυψγ ψ + mωV + gυV V Vα , = −V ∂Vν 3! ∂(∂µVν ) 2 µ 2 ν +2Λgρ b · bµ gυV (6.13) ∂L gρ ¯ ν 2 ν 2 α 2 ν ∂L µν = − ψγ τ ψ + mρb + Λgυ V Vα gρb , = −b , (6.14) ∂bν 2 ∂(∂µbν )

∂L 0 µ gρ L † = γ γ i∂µ − gυVµ − τ · bµ − M − gsφ ψ , † = 0 , (6.15) ∂ψ 2 ∂(∂µψ ) the Euler-Lagrange equations of motion are obtained κ λ ∂ ∂µφ + m2φ + g3φ2 + g4φ3 = g ψψ¯ (6.16) µ s 2! s 3! s s ζ ∂ V µν + m2 V ν + V ν V αV + 2Λg2bµ · b g2V ν = g ψγ¯ ν ψ (6.17) µ ω 3! α ρ µ υ υ g ∂ bµν + m2bν + Λg2V αV g2bν = ρ ψγ¯ ν τ ψ (6.18) µ ρ υ α ρ 2 g γµi∂ − g V − ρ τ · b − M − g φψ = 0 . (6.19) µ υ µ 2 µ s In literature there are two dominant conventions to express the QHD Lagrange density. These conventions are important when the various parameter sets are compared with each other. The way we expressed the Lagrange density in (6.1), which is used in the original QHD formalism, is considered to be the Walecka convention, in contrast to the other convention known as the Ring convention. Now, there are a few differences between these conventions. First of all in Walecka convention the scalar field, φ, is always treated as a positive quantity while in Ring convention it is a negative one. Secondly, the baryon field ’s first four components, in Walecka convention, describe the proton field where the last four describe the neutron field, which contradicts Ring convention where the first four components refer to neutron field and the last four to the proton field. Lastly, in Walecka convention the coupling between the ρ-meson and the isospin density has an additional factor of half where in Ring convention it is implied. Table 6.1 shows the exact differences between these conventions.

53 Property Walecka Ring

∗ ∗ ∗ Reduced mass, m m = M − gsφ m = M + gsφ

Coupling between the ρ- field ¯ µ ¯ µ ψγ (gυ/2)τ · bψ ψγ gυτ · bψ and the baryon field, gρ Third-order self-coupling constant κ g in the scalar meson field 2

Fourth-order self-coupling constant λ g in the scalar meson ﬁeld 3

Coupling between the ρ- Λ Not applicable and the ω- meson ﬁelds υ

ψ ψ Baryon ﬁeld, ψ p n ψn ψp

Table 6.1: Diﬀerences between Walecka and Ring conventions

In this work we primary use Walecka convention but at the same time we present the calculated expressions in terms of the Ring convention too. Various parameter sets examine diﬀerent coupling and thus the coupling constants often appear only in one the conventions. It should be noted that for the majority of parameter sets the masses of proton and neutron are treated as equal quantities (939 MeV). However there are exceptions such as the PK-1 parameter set. This can be added in the formalism by replacing the mass terms as follows

∗ ∗ m 04×4 mp 04×4 ∗ → ∗ . (6.20) 04×4 m 04×4 mn

6.2 Relativistic Mean-Field Approximation

The RMF approximation for these advanced QHD parameter sets is motivated in a similar fashion with QHD-I. We assume a static, uniform system of baryons in which we enlarge the known number density ρB and source terms. This way we are able to replace the meson ﬁeld operators with their ground state expectation values, as ˆ ˆ φ −→ hΦ| φ |Φi = hφi = φ0 (6.21)

Vˆµ −→ hΦ| Vˆµ |Φi = hVµi = δµ0V0 ˆ ˆ bµ −→ hΦ| bµ |Φi = hbαµi = δµ0δα3b0 . (6.22)

The first two components of the isovector field, b1 and b2, vanish since they can be expressed in terms of the raising and lowering operators of the charged ρ-field (6.3) meaning that the expectation values vanish due to i i the orthogonality of the states. Now, the reason we discarded the spatial V , b3 components arises from the uniform, static system where the baryon current ψγ¯ iψ and isospin current ψγ¯ iτ ψ vanish. This is shown by i i solving the Euler-Lagrange equations for the RMF with respect to V , b3. In addition the baryon sources in the no-sea approximation are replaced with their ground state normal ordered expectation values as follows:

54 Therefore the Euler-Lagrange equations of motion in RMF approximation (Walecka convention) are given by

2 gs h ¯ κ 2 λ 3i gsφ0 = 2 hψψi − (gsφ0) + (gsφ0) (6.26) ms 2! 3! 2 gυ hD † E ζ 3 2i gυV0 = 2 ψ ψ − (gυV0) − 2Λ(gρV0)(gρb0) (6.27) mω 6 gρ h1D † E 2 i gρb0 = 2 ψ τ3ψ − Λ(gυV0) (gρb0) (6.28) mρ 2 h g i 0 = iγµ∂ − g γ0V − ρ τ γ0b − (M − g φ ) ψ (6.29) µ υ 0 2 3 0 s 0 where in Ring convention the corresponding equations hold as

2 ¯ 2 3 msφ0 = −gshψψi − g2φ0 − g3φ0 (6.30) 2 D † E 3 mωV0 = gυ ψ ψ − c3V0 (6.31)

2 D † E mρb0 = gρ ψ τ3ψ (6.32)

h µ 0 0 i 0 = iγ ∂µ − gυγ V0 − gρτ3γ b0 − (M + gsφ0) ψ . (6.33)

i i The vanishing of V and b3 is easily justiﬁed through the following equations h ζ i V i m2 + V αV + 2Λg2g2bµ · b = 0 (6.34) ω 3! α ρ υ µ | {z } 6=0 i 2 2 2 2 b3 mρ + 2ΛυgυgρV0 = 0 . (6.35) | {z } 6=0

Thus, the Lagrange density in RMF approximation is given by g L = ψ¯ iγµ∂ − g γ0V − ρ τ γ0b − m∗ ψ RMF µ υ 0 2 3 0 1 κ λ 1 − m2φ2 − (g φ )3 − (g φ )4 + m2 V 2 2 s 0 3! s 0 4! s 0 2 ω 0 ζ 1 + (g2V )2 − m2b2 + Λ (g V )2(g b )2 (6.36) 4! υ 0 2 ρ 0 υ ρ 0 ρ 0 ∗ where m = M − gsφ0 is obviously the reduced mass. Consequently we obtain the energy-momentum tensor for the RMF approximation as

µν ¯ µ ν µν TRMF = ψiγ ∂ ψ − g Le (6.37) where Le is given by

1 2 2 κ 3 λ 4 1 2 2 ζ 2 2 2 1 2 2 Le = − m φ − (gsφ0) − (gsφ0) + m V + (g V ) − m b 2 s 0 3! 4! 2 ω 0 4! υ 0 2 ρ 0 2 2 + Λυ(gρV0) (gρb0) . (6.38)

And hence the expressions for the energy density and pressure of the system read as D E = T 00 D E = ψiγ¯ 0∂0ψ − hLie D E = ψ†iψ˙ − hLie , (6.39) 1D E P = T ii 3 1D E = ψiγ¯ i∂iψ + hLie 3 1D E = ψ†(−iα · ∇)ψ + hLie . (6.40) 3

55 6.3 Expectation Values

We can compute the normal ordered expectation values by following the same prescription as in the previous chapter, by the quantization of the baryon ﬁeld. However we choose to deploy the alternative method we introduced in (5.83). For this purpose we ﬁrst have to extract the energy values of our system starting with baryon Dirac equation: g γ0 · iγ0∂ + iγi∂ − g γ0V − ρ τ γ0b − m∗ ψ = 0 0 i υ 0 2 3 0 g − iα · ∇ + g V + ρ τ b + m∗β ψ = iψ˙ (6.41) υ 0 2 3 0 ˙ HˆDψ = iψ (6.42)

ˆ ˆ gρ ∗ where in the last step we introduced the Dirac Hamiltonian, HD = α · k + gυV0 + 2 τ3b0 + m β. Similarly, since ˆ [HˆD, k] = 0, we look for plane wave solutions of the form ψ(x, t) = ω(k)ei(k·x−t) (6.43) where ω(k) is 8 × 1 spinor that contains the ωp(k), ωn(k) component Dirac spinors: ω (k) ω(k) = p , (6.44) ωn(k) the substitution of (6.43) to the Dirac equation leads to the algebraic equation g ω(k) = α · k + g V + ρ τ b + m∗β ω(k) (6.45) υ 0 2 3 0 which after some algebra takes simpliﬁes to ω H ω p = Dp p (6.46) ωn HDn ωn where HDp and HDn are 4 × 4 matrices deﬁned as g H = α · k + g V + ρ b + m∗β (6.47) Dp υ 0 2 0 g H = α · k + g V − ρ b + m∗β (6.48) Dn υ 0 2 0 or expressed in a more compact form as ( g +1, p H = α · k + g V + ρ τ b + m∗β where τ = . (6.49) D υ 0 2 3 0 3 −1, n

In addition, familiar manipulations lead to the positive and negative energy values for proton and neutron as g p ± = g V + ρ τ b ± k2 + m∗2 . (6.50) p,n υ 0 2 3 0 Thus, we now proceed on evaluating the expectation values. However in this case we ﬁrst have to split the † proton and neutron contributions in the single particle expectation value, (ψ Γψ)k,s, Z † 3 † ˆ (ψ Γψ)k,s = d x ψk,s(x)Γψk,s(x)

† † = (ψpΓψp)k,s + (ψnΓψn)k,s , (6.51) and therefore the general form for the expectation value of the many-body system is given as X Z dk X Z dk hψ†Γψi = (ψ†Γψ ) Θ[µ − ] + (ψ†Γψ ) Θ[µ − ]. (6.52) (2π)3 p p k,s kp (2π)3 p n k,s kn s s Now, by straight forward application of (6.52) we compute hψ†ψi, X Z dk X Z dk hψ†ψi = (ψ†ψ ) Θ[µ − ] + (ψ† ψ ) Θ[µ − ] (2π)3 p p k,s kp (2π)3 n n k,s kn s | {z } s | {z } =1 =1 1 1 = k3 + k3 3π2 p 3π2 n = ρp + ρn . (6.53)

56 Similarly, since ∗ ¯ † ∂HD ∂ + m (ψpψp)k,s = (ψp ψp)k,s = p = √ , (6.54) ∂M ∂M k2 + m∗2 ∗ ¯ † ∂HD ∂ + m (ψnψn)k,s = (ψn ψn)k,s = n = √ . (6.55) ∂M ∂M k2 + m∗2 Then it follows that 1 Z kp k2m∗ 1 Z kn k2m∗ hψψ¯ i = dk √ + dk √ . (6.56) 2 2 ∗2 2 2 ∗2 π 0 k + m π 0 k + m

† The evaluation of hψ τ3ψi is a little tricky. The action of τ3 on the 8 × 1 baryon spinor ψ is to split the contributions of protons and neutrons to positive and negative as 14×4 0 ψp ψp τ3 · ψ = = (6.57) 0 −14×4 ψn −ψn and thus the single particle expectation value is given as

† † † (ψ τ3ψ)k,s = (ψp ψp)k,s − (ψn ψn)k,s (6.58) One can summarize this result into

( † † (ψpτ3ψp)k,s = +1 (ψ τ3ψ)k,s = † . (6.59) (ψnτ3ψn)k,s = −1

Hence, the expectation value of the many-body system is given by X Z dk X Z dk hψ†τ ψi = (ψ†τ ψ ) Θ[µ − ] + (ψ† τ ψ ) Θ[µ − ] 3 (2π)3 p 3 p k,s kp (2π)3 n 3 n k,s kn s | {z } s | {z } =+1 =−1 1 1 = k3 − k3 3π2 p 3π2 n = ρp − ρn . (6.60)

Moving the expectation value, hψ†iψ˙i, the method is identical with the case of QHD-I. The single particle expectation value is given by † ˙ † † (ψ iψ)k,s = (ψpHDψp)k,s + (ψnHDψn)k,s

= p + n (6.61) and therefore in the many-body system it takes the form Z X dk gρ p hψ†iψ˙i = g V + b + k2 + m∗2 Θ[µ − ] (2π)3 υ 0 2 0 kp s Z X dk gρ p + g V − b + k2 + m∗2 Θ[µ − ] (2π)3 υ 0 2 0 kn s Z kp gρ 1 2p 2 ∗2 = gυV0ρ + b0(ρp − ρn) + 2 dk k k + m 2 π 0 Z kn 1 2p 2 ∗2 + 2 dk k k + m (6.62) π 0 and ﬁnally, due the following expressions

† † (ψ (−iα · ∇ )ψ)k,s = (ψ α · k ψ)k,s ∂H ∂H = (ψ† D ψ ) · k + (ψ† D ψ ) · k p ∂k p k,s n ∂k n k,s ∂ ∂ = p · k + n · k (6.63) ∂k ∂k the corresponding expectation value in many-body system reads as

D E 1 Z kp k4 1 Z kn k4 ψ†(−iα · ∇)ψ = dk √ + dk √ . (6.64) 2 2 ∗2 2 2 ∗2 π 0 k + m π 0 k + m

57 6.4 Parameter Sets, Equation of State, Observables

In Tables.6.2, 6.3 we present the coupling constants and masses used for models in Walecka convention while in Tables.6.4, 6.5 are given the ones in Ring convention. The Ring NL3 parameter set is taken from [9] whereas the parameter sets in Walecka convention are from [10]. The input data for PK1 models are taken from [11] while the NL-SH and TM1, TM2 were taken from [12] and [13] respectively. The parameter sets for FUSGold model and NL3 in Walecka convention are given in [10].

2 2 2 Model gs gυ gρ κ(MeV) λ ζ Λυ NL3 104.3871 165.5854 79.6000 3.8599 -0.01591 0.00 0.00 S271 81.1071 116.7655 85.4357 6.6834 -0.01580 0.00 0.00 Z271 49.4401 70.6689 90.2110 6.1696 +0.15634 0.06 0.00 FSUGold 112.1996 204.5469 138.4701 1.4203 +0.0238 0.0600 0.0300

Table 6.2: Coupling constants for all parameter sets with respect to Walecka convention. These coupling constants are dimensionless except for the case of κ which has dimension of MeV

Model Mproton Mneutron ms mω mρ NL3 939 939 508.1940 782.5 763 S271 939 939 505.000 783 763 Z271 939 939 465.000 783 763 FSUGold 939 939 491.5000 783 763

Table 6.3: Particle masses (MeV) of paramaeter sets with respect to Walecka convention.

−1 Model gs gυ gρ g2 (fm ) g3 g4 NL3 10.217 12.868 4.474 -10.431 -28.885 0.00 PK1 10.3222 13.0131 4.5297 -8.1688 -9.9976 55.636 TM1 10.0289 12.6139 4.6322 -7.2325 0.6183 71.3075 TM2 11.4694 14.6377 4.6783 -4.4440 4.6076 84.5318 NL-SH 10.444 12.945 4.383 -6.9099 -15.8337 0.00

Table 6.4: Coupling constants for all parameter sets with respect to Ring convention. These coupling constants −1 are dimensionless except for the case of g2 which has dimension of fm

Now, by taking into account the evaluated results of the previous section regarding the expectation values, † ¯ † hψ ψi, hψψi and hψ τ3ψi the equations of motion of the ﬁelds (6.33) in Walecka convention reduce to

2 Z kp 2 ∗ Z kn 2 ∗ gs 1 k m 1 k m gsφ0 = dk √ + dk √ 2 2 2 ∗2 2 2 ∗2 ms π 0 k + m π 0 k + m 2 gs κ 2 λ 3 + 2 − (gsφ0) + (gsφ0) (6.65) ms 2! 3! 2 gυ ζ 3 2 gυV0 = 2 ρp + ρn − (gυV0) − 2Λ(gρV0)(gρb0) (6.66) mω 6 2 gρ 1 2 gρb0 = 2 (ρp − ρn) − Λ(gυV0) (gρb0) . (6.67) mρ 2

58 Model Mproton Mneutron ms mω mρ NL3 939 939 508.194 782.501 763.0000 PK1 939.5731 938.2796 514.0891 784.254 763 TM1 938 938 511.198 783.0 770.0 TM2 938 938 526.443 783.0 770.0 NL-SH 939 939 526.059 783.00 763.00

Table 6.5: Particle masses (MeV) of paramaeter sets with respect to Ring convention.

In a similar fashion the ﬁeld equations of motion in Ring convention take the following form.

Z kp 2 ∗ Z kn 2 ∗ 2 1 k m 1 k m m φ0 = −gs dk √ + dk √ s 2 2 ∗2 2 2 ∗2 π 0 k + m π 0 k + m 2 3 − g2φ0 − g3φ0 (6.68) 2 3 mωV0 = gυ(ρp + ρn) − c3V0 (6.69) 2 mρb0 = gρ(ρp − ρn) . (6.70)

Consequently we derive the energy density and pressure through (6.40), in Walecka convention

1 κ λ 1 = m2φ2 + (g φ )3 + (g φ )4 − m2 V 2 2 s 0 3! s 0 4! s 0 2 ω 0 ζ 1 − (g2V 2)2 − m2b2 − Λ (g V )2(g b )2 + g V (ρ + ρ ) 4! υ 0 2 ρ 0 υ ρ 0 ρ 0 υ 0 p n Z kp gρ 1 2p 2 ∗2 + b0(ρp − ρn) + 2 dk k k + m 2 π 0 Z kn 1 2p 2 ∗2 + 2 dk k k + m , (6.71) π 0 1 κ λ 1 P = − m2φ2 − (g φ )3 − (g φ )4 + m2 V 2 2 s 0 3! s 0 4! s 0 2 ω 0 ζ 1 + (g2V 2)2 + m2b2 + Λ (g V )2(g b )2 4! υ 0 2 ρ 0 υ ρ 0 ρ 0 1 Z kp k4 1 Z kn k4 + dk √ + dk √ , (6.72) 2 2 ∗2 2 2 ∗2 3π 0 k + m 3π 0 k + m while the these quantities in Ring convention are given by 1 g g 1 c = m2φ2 + 2 φ3 + 3 φ4 − m2 V 2 − 3 V 4 2 s 0 3 0 4 0 2 ω 0 4 0 1 − m2b2 + g V (ρ + ρ ) 2 ρ 0 υ 0 n p Z kp 1 2p 2 ∗2 − gρb0(ρp − ρn) + 2 dk k k + m , (6.73) π 0 1 g g 1 c 1 P = − m2φ2 − 2 φ3 − 3 φ4 + m2 V 2 + 3 V 4 + m2b2 2 s 0 3 0 4 0 2 ω 0 4 0 2 ρ 0 1 Z kp k4 1 Z kn k4 + dk √ + dk √ . (6.74) 2 2 ∗2 2 2 ∗2 3π 0 k + m 3π 0 k + m At this point we have to compute the symmetry energy coefficient of the system. For this purpose we have to extract the isospin dependent energy per nucleon, s/ρ, which we do by examining the equations of motion, (6.33). Clearly, the scalar field, φ0, is isospin independent, while the ω-meson field has an implied dependence through the coupling with the ρ-meson field which is isospin dependent. Therefore the isospin dependent energy

59 per nucleon is given by

1 b2 Λ 1 b s = − m2 0 − υ (g V )2(g b )2 + 0 (ρ − ρ ) ρ 2 ρ ρ ρ υ 0 ρ 0 2 ρ p n Z kp Z kn 1 2p 2 ∗2 1 2p 2 ∗2 + 2 dk k k + m + 2 dk k k + m ρπ 0 ρπ 0 b2 1 = 0 m2 + 2Λ (g V )2g2 − g b t ρ ρ υ υ 0 ρ 2 ρ 0 Z kp Z kn 1 2p 2 ∗2 1 2p 2 ∗2 + 2 dk k k + m + 2 dk k k + m . (6.75) ρπ 0 ρπ 0

Now, using (6.67) we express b0 in terms of t as

1 gρ(ρp − ρn) b0 = 2 2 2 2 mρ + 2Λυgρ(gυV0)

1 gρ(−t)ρ = 2 2 2 (6.76) 2 mρ + 2Λυgρ(gυV0) and thus the isospin energy per nucleon is

2 2 2 2 s 1 gρ(−t)ρ mρ + 2Λυ(gυV0) gρ 1 gρ(−t)ρ = − 2 2 2 − tgρ 2 2 2 ρ 2 mρ + 2Λυgρ(gυV0) 2ρ 2 mρ + 2Λυgρ(gυV0) Z kp Z kn 1 2p 2 ∗2 1 2p 2 ∗2 + 2 dk k k + m + 2 dk k k + m ρπ 0 ρπ 0 2 2 t ρ gρ = 2 2 2 8 mρ + 2Λυgρ(gυV0) Z kp Z kn 1 2p 2 ∗2 1 2p 2 ∗2 + 2 dk k k + m + 2 dk k k + m . (6.77) ρπ 0 ρπ 0 Therefore the symmetry energy coeﬃcient is given by

2 1 ∂ s α4 = 2 ∂t2 ρ t=0 k3 g2 k2 = f ρ + f (6.78) 2 2 2 2 q 12π mρ + 2Λυgρ(gυV0) 2 ∗2 6 kf + m where in the last step we used the baryon density at saturation, ρ0, (5.74). We can now obtain the symmetry energy coeﬃcient in Ring convention from (6.78) by discarding the coupling constant Λυ (note that we recovered the omitted 1/2 factor in the term of isospin density in Ring convention). Thus:

k3 g 2 k2 α = f ρ + f . (6.79) 4 2 q 12π mρ 2 ∗2 6 kf + m

60 Chapter 7

Results

The results for this chapter were taken from the bibliography, [1]. In the latter, the author generated them with a computer program written in FORTRAN90 in order to solve the equations of motion for QHD formalism, using a variety of parameter sets. Firstly we present the relation between the binding energy and diﬀerent Fermi wave-lengths kf around the −1 saturated density, Fig. 7.1, where it corresponds to kf = 1.3 fm . The parameter sets included in the diagram are QHD-I, NL3, PK1 and FSUGold. It is clear that the parameter sets behave diﬀerent at high densities ( above saturated density).

Figure 7.1: Relation between binding energy per nucleon and Fermi wave-length for a variety of parameter sets

Consequently we present results calculated in NL3, PK1 and FSUGold parameter sets regarding neutron star properties. Initially, a neutron star can be approximated as it is composed entirely of neutrons. With the inclusion of the beta-equilibrium effects the maximum neutron star mass is reduced and thus the equation of state is softened. The beta-equilibrium effects lead to different other particles. These particles, such as muons, are energetically favorable to occupy first the lower and then the high energy muon states and thus the energy increases more smoothly as the pressure increases. In Fig7.2 we show the softening of the equation of state due to the inclusion of additional particles in description of neutron star matter.

61 Figure 7.2: Softening of the EoS due to the inclusion of additional particles in description of neutron star matter.

The softening of the equation of state causes a shifting in the Mass-Radius relationship as calculated in FSUGold parameter set, Fig7.3.

Figure 7.3: Mass-radius relationship due to the softening of EoS.

Crustal effects in neutron star description do not affect the total neutron star mass in comparison to the description where crustal effects are absent. However, the radii of neutron stars, in general, are larger when crustal effects are added. The crustal effects do not affect the central pressure of the neutron star which corresponds to the total overlying mass. The inclusion of crustal effects lowers the pressure gradient near the boundary of the star and as a consequence the radius of the star enlarges. In Fig7.4 we show the neutron star mass-radius relationship for an equation of state based on the FSUGold parameter set. In the latter different descriptions for the outer crust and a polytropic equation of state for the inner crust have been included. Fig7.5 shows the neutron star mass against the central density of the same calculation.

62 Figure 7.4: Mass-radius relationship due to the inclusion of crustal eﬀects.

Figure 7.5: Plot of Mass against central density.

63 Bibliography

[1] Jacobus Petrus Willem Diener, Relativistic Mean Field Theory Applied to the Study of Neutron Star Prop- erties

[2] Walter Greiner, Joachim Reinhardt, D.A. Bromley, Field Quantization, Springer, (1996). [3] John Dirk Walecka, Theoretical Nuclear And Subnuclear Physics, World Scientiﬁc Publishing Company, (2004). [4] N. K. Glendenning,Compact Stars, 2nd edition, Springer, (2000).

[5] Walter Greiner, B. Muller, Quantum Mechanics, Symmetries, Springer, (1990). [6] Solution to Dirac equation for a free particle, retrieved from: http : //www.nyu.edu/classes/tuckerman/quant.mech/lectures/lecture 7/node1.html [7] Stress tensor, retrieved from: https : //en.wikipedia.org/wiki/Stress tensor

[8] B. D. Serot and J. D. Walecka, Int. J. Mod. Phys. E 6, 515 (1997). [9] G. A. Lalazissis, J. Konig, and P. Ring, Phys. Rev. C 55, 540 (1997). [10] B. G. Todd-Rutel and J. Piekarewicz, Phys. Rev. Lett. 95, 122501 (2005).

[11] W. Long, J. Meng, N. Van Giai, and S. Zhou, Phys. Rev. C 69, 034319 (2004). [12] M. M. Sharma, M. A. Nagarajan, and P. Ring, Phys. Lett. B312, 337 (1993). [13] Y. Sugahara and H. Toki, Nuc. Phys. A579, 557 (1994). [14] Sources and Types of Gravitational Waves, retrieved from:https : //www.ligo.caltech.edu/page/gw − sources [15] Neutron star, retrieved from:https : //en.wikipedia.org/wiki/Neutron star [16] F. Weber, Prog. Pat. Nucl. Phys. 54, 193 (2005).