Quantum Field Theory Applied in Compton Scattering

Quantum Field Theory Applied In Compton Scattering

Aristotle University of Thessaloniki Faculty of Physics Institute of Theoretical Physics

Gkiatas Dimitrios

Supervisor : Nikolaos D. Vlachos

March 2016 Contents

Abstract in English 4

Abstract in Greek 5

1 Classical Field Theory 7 1.1 Relativistic Notation ...... 8 1.2 Mathematical Background ...... 10 1.3 Lagrangian Formalism ...... 12 1.4 The Hamilton Formalism ...... 14 1.5 Symmetries and Conservation Laws ...... 16

2 The Klein-Gordon Field 21 2.1 Canonical Quantization ...... 21 2.2 The Real Klein-Gordon eld ...... 22 2.3 The Bosonic Fock Space ...... 29 2.4 The Complex Klein-Gordon Field ...... 31 2.5 Microcausality ...... 35 2.6 The Scalar Feynman Propagator ...... 37

3 The Dirac Field 40 3.1 The Dirac Equation ...... 40 3.2 Covariance of the Dirac Equation ...... 41 3.3 Classical Dirac Field Theory ...... 43 3.4 Fermionic Fock Space ...... 44 3.5 Canonical Quantization of the Dirac Field ...... 45 3.6 Plane Wave Expansion of the Field Operator ...... 46 3.7 The Dirac Field Feynman Propagator ...... 50

4 The Electromagnetic Field 53 4.1 The Maxwell Equations ...... 53 4.2 Plane Wave Expansion of the Electromagnetic Field ...... 56 4.3 Lagrangian Density and Conserved Quantities ...... 57 4.4 Quantization of the Electromagnetic Field ...... 58 4.5 Canonical Quantization of the Lorenz Gauge ...... 59 4.6 Feynman Propagator for Photons ...... 63

5 Interacting Fields 65 5.1 The Dirac Picture ...... 65 5.2 The Time-Evolution Operator ...... 67

2 CONTENTS 3

5.3 The S-Matrix ...... 69 5.4 Wick's Theorem ...... 70 5.5 Coupling Photons to Fermions (QED) ...... 73 5.6 Feynman Rules in QED ...... 73 5.7 Cross Section ...... 76 5.8 Spin Sums ...... 78 5.9 Trace Technique ...... 79

6 Compton Scattering 80 6.1 Feynman Diagrams ...... 80 6.2 Compton Scattering Dierential Cross Section ...... 82 6.3 Energy Shift Relation ...... 84 6.4 Klein-Nishina Formula ...... 85

Bibliography 90 Abstract in English

This thesis, provides an extensive introduction to Quantum Field Theory with an application to Compton Scattering, studying for 3 chapters the canonical quantization of almost every free eld, i.e Klein-Gordon-Real and Complex-eld, Dirac eld and Elec- tromagnetic eld. Prior to the eld quantization throughout the 1st Chapter a short introduction of Classical eld theory from the aspect of Lagrangian formalism takes place. During the 5th Chapter we discuss interacting elds and more specically the coupling of fermions to photons (QED). In addition, we deduce the expressions of quantities such as the probability rate and scattering cross section which we will use in our application. Concluding this chapter, we state the Feynman translation rules and Feynman rules in momentum space and correlate them with Feynman diagrams and the probability amplitude respectively. In the 6th Chapter, we use the tools proven before to draw the Feynman diagrams and nd the Klein-Nishina formula-unpolarized scattering cross section- for the process of Compton Scattering

0 γ(k) + e(pi) → γ(k ) + e(pf )

4 Abstract in Greek

Η παρούσα πτυχιακή εργασία ασχολείται με τίτλο ¨Κβαντική Θεωρια Πεδίου με Εφαρμογή στην Σκέδαση Compton’ αποτελεί μία εισαγωγή στην Κβαντική Θεωρία Πεδίου μελετώντας στα πρώτα 4 κεφάλαια τα βασικά ελεύθερα πεδία, ενώ στο 5ο μελετάμε την αλληλεπίδραση αυτών καταλήγοντας στο 6ο κεφάλαιο στον υπολογισμό της ενεργού διατομής για την σκέ- δαση Compton. Αρχικά στο 1ο κεφάλαιο γίνεται μία εισαγωγή στην κλασσική θεωρία πεδίου από την πλευρά του Λαγρανζιανού Φορμαλισμού, μέσω της οποίας εξάγονται οι πεδιακές εξισώσεις οι οποίες και αποτελούν γενίκευση των κλασσικών εξισώσεων Euler-Lagrange συναρτή- σει της Λαγκρανζιανής πυκνότητας. Στην συνέχεια μελετάμε τον Χαμιλτονιανό φορμαλ- ισμό υπολογίζοντας τις αγκύλες Poisson μεταξύ συζυγών μεγεθών και τελικά εξάγουμε την γενικότερη μορφή της εξίσωσης συνέχειας από την οποία και προκύπτουν οι διατηρήσιμες ποσότητες (θεώρημα Noether), αναφέροντας κάποιες βασικές συμμετρίες που πρέπει να υπακούει κάθε σχετικιστική θεωρία (αναλοιωσιμότητα κάτω από χωρικές μεταθέσεις και Lorentz covariance). Το 2ο κεφάλαιο ασχολείται τόσο με το πραγματικό όσο και με το μιγαδικό πεδίο Klein- Gordon. Πιο συγκεκριμένα, βρίσκουμε όλες τις πεδιακές ποσότητες που αναφέραμε και υπολογίσαμε στο 1ο κεφάλαιο (συζυγές πεδίο, Χαμιλτονιανή πυκνότητα κ.λ.π). Κατόπιν, κβαντώνουμε τα πεδία αντικαθιστώντας τις συζυγείς πεδιακές ποσότητες με τελεστές και τις αγκύλες Poisson με μεταθετικές σχέσεις και υπολογίζουμε τους αντίστοιχους τελεστές. Προχωρώντας, αναπτύσσουμε τους πεδιακούς τελεστές γύρω από κλασσικές βάσεις επιπέδων κυμάτων εισάγοντας τους τελεστές δημιουργίας και καταστροφής αφού πρώτα πούμε δύο λόγια για τον χώρο Fock τον οποίο ορίζουν (αναπαράσταση αριθμού σωματιδίων). Τέλος βρίσκουμε κάποιες από τις διατηρήσιμες ποσότητες (Διανυσμα ορμής, Χαμιλτονιανή και Φορτίο) και υπολογίζουμε τον διαδότη Feynman για το μιγαδικό βαθμωτό πεδίο αφού πρώτα επιβεβαιώσουμε πως η θεωρία μας υπακούει στην μικροαιτιότητα. Στο 3ο κεφάλαιο μελετάμε το πεδίο Dirac. Πρώτα από όλα κάνουμε μία εισαγωγή στην εξίσωση Dirac και τα Dirac spinors που αποτελούν τις κυματοσυναρτήσεις της θεω- ρίας από την σκοπιά της σχετικιστικής κβαντομηχανικής. Επιπλέον, αποδεικνύουμε την αναλοιωσιμότητα της θεωρίας κάτω από μετασχηματισμούς Lorentz (Lorentz covarience). Μετά συζητάμε την κλασσική θεωρία πεδίου καθώς και τις διατηρήσιμες ποσότητες που προκύπτουν από την θεωρία μας και ορίζουμε τον φερμιονικό πλέον χώρο Fock στον οποίο και εργαζόμαστε κατά την κβάντωση του κλασσικού πεδίου Dirac η οποία επιτυγχάνεται μέσω της επιβολής κατάλληλων αντιμεταθετικών σχέσεων μεταξύ των συζυγών πεδιακών τελεστών. Ωστόσο, σε αυτή την περίπτωση επιλέγουμε να κάνουμε ανάπτυγμα των πε- διακών τελεστών σε επίπεδα κύματα που είναι λύσεις της ελεύθερης εξίσωσης Dirac και παρατηρούμε την εμφάνιση δύο ειδών τελεστών δημιουργίας και καταστροφής που ερμη- νεύουμε πως ανταποκρίνονται σε φερμιονικά σωμάτια και αντισωμάτια. Κλείνοντας, βρίσκ- ουμε τον φερμιονικό διαδότη Feynman.

5 6 CONTENTS

Στο 4ο κεφάλαιο, αναπτύσσουμε την κβάντωση του ηλεκτρομαγνητικού πεδίου. Ξεκ- ινώντας από τις εξισώσεις Maxwell απουσία ηλεκτρικών και μαγνητικών πηγών τις οποίες και γράφουμε σε ανταλλοίωτη τανυστική μορφή συναρτήσει του διανυσματικού και βαθμωτού δυναμικού, επικεντρωνόμαστε στην μελέτη της συμμετρίας βαθμίδας, που παίζει καθοριστικό ρόλο στην κβάντωση, αναφέροντας τόσο την βαθμίδα Coulomb όσο και την βαθμίδα Lorenz. Παρακάτω κάνουμε ανάπτυγμα του 4-διάστατου πλέον διανυσματικού δυναμικού σε επίπεδα κύματα ορίζοντας τις τέσσερις αρχικά καταστάσεις πόλωσης που αποτελούν τους εκάσ- τοτε βαθμούς ελευθερίας του πεδίου. Τέλος κβαντώνουμε το ηλεκτρομαγνητικό πεδίο εισά- γοντας τις ανταλλοίωτες μποζονικές μεταθετικές σχέσεις μέσω των οποίων υπολογίζουμε τον τελεστή Χαμιλτον, όπου εργαζόμενοι στην βαθμίδα Lorenz μέσω του Gupta-Bleuler πε- ριορισμού εξασφαλίζουμε την απουσία αρνητικού μέτρου στον χώρο Hilbert επιβεβαιώνοντας τους 2 εγκάρσιους βαθμούς ελευθερίας του φωτονίου. Η συζήτηση για τα ελεύθερα πεδία κλείνει με την εξαγωγή και ερμηνεία των διαφόρων όρων του φωτονικού διαδότη (εγκάρσιος, Coulomb και υπολειπόμενος). Το πέμπτο κεφάλαιο αναφέρεται στα αλληλεπιδρώντα κβαντικά πεδία, με στόχο την εύρεση ενός συστηματικού τρόπου εφαρμογής της θεωρίας διαταραχών, με σκοπό την εξ- αγωγή παρατηρήσιμων ποσοτήτων και πιο συγκεκριμένα του πίνακα σκέδασης (scattering matrix). Από την εικόνα Heisenberg που χρησιμοποιούσαμε μέχρι τώρα, μεταβαίνουμε στην εικόνα της αλληλεπίδρασης (Dirac). Στη συνέχεια, αποδεικνύουμε την σχέση του Dyson και συζητάμε το θεώρημα του Wick. ΄vΕπειτα ασχολούμαστε με την Κβαντική Ηλεκτρο- δυναμική εξάγοντας την Λαγκρανζιανή πυκνότητα και μελετάμε την διαδικασία της σκέδασης. Κατόπιν, βρίσκουμε την έκφραση της ενεργούς διαφορικής διατομής, και παραθέτουμε τους κανόνες μετατόπισης και Feynman στον χώρο ορμής, συσχετίζοντάς τους με τα διαγράμματα Feynman και πιθανότητα μετάπτωσης αντίστοιχα. Στο τελευταίο κεφάλαιο, εφαρμόζουμε τις τεχνικές που αναπτύξαμε στον σχεδιασμό των διαγραμμάτων Feynmam και στον υπολογισμό της διαφορικής ενεργούς διατομής για τη σκέδαση Compton, η οποία περιλαμβάνει τη σκέδαση φωτονίου από ηλεκτρόνιο

0 γ(k) + e(pi) → γ(k ) + e(pf ) Chapter 1

Classical Field Theory

In this chapter we will take the elds at each point as the dynamical variables and quantize them directly. This approach generalizes the classical mechanics of a system of particles and its quantization, to a continuous system, i.e the elds. A eld is a quantity dened at every point in space and time (~x,t). While classical mechanics deals with a nite number of generalized coordinates

qr(t), r = 1, 2, ...., n (1.1) in eld theory we are interested in the dynamics of elds denoted by the generic name

1 φr(~x,t) , r = 1, 2, ..., N (1.2) where both r and ~x are considered as labels. Therefore, each point of a (nite or innite) region in space will be associated with some continuous eld variable as I have stated above. This obviously constitutes a system with an innite number of degrees of freedom. Consequently, our dynamic variables are the values of the eld φ(x) at every point in space, instead of the nite coordinates qr(t). Next, we will introduce the Lagrangian (more specically, the Lagrangian density (L)) for which the eld equations follow by means of Hamilton's Principle. In addition, we will also introduce the momenta conjugate to the elds and impose canonical commutation relations directly on the elds and the conjugate momenta. This process, provides a systematic quantization procedure for any classical eld theory derivable from a Lagrangian. Finally, we will derive the general form of Noether's Theorem, thus, calculating the current density and nding conserved quantities that follows from some fundamental transformations. Moreover, given that the eld theory will be developed in a relativistic form, the need of discussing some relativistic notations is essential.

1One can immediately notice that the concept of position has been relegated from a dynamical variable in classical mechanics to a mere label in eld theory

7 8 CHAPTER 1. CLASSICAL FIELD THEORY 1.1 Relativistic Notation

First of all, throughout this thesis, we will use the natural units system

~ = c = 1 Following the standard convention we shall write

xµ = (ct, ~x), (µ = 0, 1, 2, 3) (1.3) in order to indicate the space-time four-vector with the time component x0 = ct and the space coordinates xj(j = 1, 2, 3). The components of four-vectors will be labeled by Greek indices, while the spatial three-vectors by latin indices. Furthermore, denoting the metric tensor that describes the Minkowski space-time with components

 1 0 0 0   0 −1 0 0  gµν = diag{1, −1, −1, −1} =   (1.4)  0 0 −1 0  0 0 0 −1

µ one can dene the covariant vector xµ from the contravariant x as follows

ν xµ = gµνx (1.5) where the summation convention - repeated Greek indices one contravariant and one covariant are summed - has been used. Thus, one can easily conclude that the components of the covariant four-vector equal to xµ = (ct, −~x). Let's consider that we have two inertial reference frames S(xµ) and S0(x0µ) in the 4-dimensional Minkowski space-time where S0 is moving with constant speed v in space relative to S. Using a Lorentz Transformation:

µ 0µ µ ν (1.6) x → x = Λ νx one can express the results from one inertial frame in terms of the other. To be more precise, in (1.6) we have expressed the space-time coordinates of S0(x0µ) in terms of S(xµ) reference's frame. This transformation helps the comparison of results made in one inertial frame with those gained from another. Another very useful property is that a Lorentz Transformation

µ 0µ µ ν x → x = Λ νx leaves the scalar product, which is a scalar quantity

µ 0 2 2 x xµ = (x ) − ~x (1.7) invariant. One can immediately comprehend this property considering the fact that every scalar quantity is coordinate independent. The four-dimensional generalization of the gradient operator ∇~ transforms like a four- vector. To be more precise, if φ(x) is a scalar function, so is ∂φ δφ = δxµ ∂xµ 1.1. RELATIVISTIC NOTATION 9

hence the quantity ∂φ ≡ ∂ φ ∂xµ µ is a covariant four-vector. Similarly,

∂φ ≡ ∂µφ ∂xµ is a contravariant four-vector. Therefore, writing in an elegant way the components of 4-divergence

∂ Covariant 4-Divergence : ∂ = , ∇~ (1.8) µ ∂x0

∂ Contravariant 4-Divergence : ∂µ = , −∇~ (1.9) ∂x0 Finally, through hackneyed calculations one can show that the scalar product of 4- divergence - which is invariant under Lorentz Transformations - is equal to the d'Alembertian operator ∂2 ∂ ∂µ = − ∇~ 2 = (1.10) µ ∂t2 10 CHAPTER 1. CLASSICAL FIELD THEORY 1.2 Mathematical Background

Before we derive the eld equations of motions, it is necessary to state some mathematical notions in order to smoother the transition from classical mechanics to eld theory. To do so, we are going to present a brief and non rigorous discussion of the denition and some properties of functionals. In mathematics, a functional can be considered as a mapping from a linear space of functions M = {y(x): x ∈ R} to the eld of real or complex numbers, and it is customary to denote a functional dependence by square brackets as follows

J[y(x)] : M → R or C As one can easily notice from the denition, a functional depends solely on the function y(x) that determines the space of functions. One should also note that the functional is independent of the coordinate x itself. The type of functionals that we will deal with have the general form of: Z 3 J[u(x, y, z)] = d xf(x, y, z, u, ux, uy, uz) (1.11) (V ) Functionals are of great interest in physics- both in classical mechanics and eld theory- given that one can derive the Euler-Lagrange equations of motions in classical mechanics and eld equations in eld theory, by nding the dierential equation the independent variables must satisfy so that the functional has a stationary value. A very important concept one should comprehend is the variation of functional (δJ) or function (δf(x)). In order to visualize and simplify the notion of variation, rst of all let us consider the functional

Z x2 J[y(x)] = f(x, y(x), y0(x))dx x1 that along the curve y(x) acquires a stationary value. Let us also suppose, the mono- parametric family of curves, that represents the variation of curves in respect of y(x), having the form Y (x, ) = y(x) + δ(x − k) (1.12) The variation of y(x) is dened as the dierence between the mono-parametric family of curves, and y(x), thus expressing the deviation from y(x)

δy = Y (x, ) − y(x) = δ(x − k) (1.13)

Proportionally, one can dene the variation of J[y(x)], as

δJ[y(x)] =J[Y (x, )] − J[y(x)] =J[y(x) + δ(x − k)] − J[y(x)] =J[y(x) + δy] − J[y(x)] (1.14)

Furthermore, by performing Taylor expansion around δy → 0 and considering only 1st order corrections one can nd

1 δJ δJ[y(x)] = J[y(x) + (δy → 0)] − J[y(x)] + δy 1! δy δy→0 1.2. MATHEMATICAL BACKGROUND 11

δJ δJ[y(x)] = (1.15) δy(k) where one can immediately conclude that the functional variation equals to zero when the functional has a stationary value, hence

Z x2 Z x2 δJ = 0 ⇒ δ f(x, y(x), y0(x))dx = 0 ⇒ δf(x, y(x), y0(x))dx = 0 x1 x1 It is very common to confuse the notion of variation (δ) with that of symbol of dierentiation (d), but, they are quite dierent. On one hand, the dierential refers to innitesimal variations that occur on the same curve for a given variation of the independent variable. On the other hand, the variation (δ) refers to innitesimal variations of the function between to adjacent curves (Y (x, ), y(x)), with the same value of the independent variable (x). Moreover, by taking the variation of the derivative of a function, one can nd that those two quantities commute. More specically

dy dY (x, ) dy d d δ = − = (Y (x, ) − y(x)) = (δy) dx dx dx dx dx thus, d δ, = 0 (1.16) dx which is a very useful relationship that I am about to use extensively throughout this chapter. Grasping the concept of functional variation, one can introduce the functional derivative with respect to its function dependence, having in mind the equation (1.15), as follows: δJ[y] J[y(x) + δ(x − k)] − J[y(x)] = lim (1.17) δy(k) →0 which expresses how the value of the functional J[y(x)] is changed when the value of the function y(x) is varied at the point x. The functional derivative may obeys many of the rules of ordinary dierential, but, it is very important that one should not view those two quantities as equal. In addition, as the dening relation of the functional derivative , we will use the Z δJ[y] δJ[y] = dx δy(x) (1.18) δy(x) where one can verify its validity by substituting the variation δy = δ(x − k), and nding the same result for the functional variation as in (1.15). 12 CHAPTER 1. CLASSICAL FIELD THEORY 1.3 Lagrangian Formalism

In this section, by introducing the concept of Lagrangian density we will address the new form of eld equations that we will use throughout this thesis. Turning our interest to the deduction of the eld equations of motion, from classical mechanics and given that the Lagrange function expresses a mapping from the space of functions (i.e elds) to the real numbers, one can conclude that the Lagrange function is a functional of the elds. Therefore, we are going to denote it as

˙ L(t) = L[φr(~x,t), φr(~x,t)], for r = 1, 2, ...., N (1.19)

where we have assumed that the Lagrange function depends on the values of φr and ˙ φr at all points in space, which is logical, considering the analogous case in classical mechanics with those quantities describing the position and velocity. In order to make this dependence more clear, one can compare the quantities that measure the displacement and the location of a point, by introducing the following dictionary Classical Mechanics Classical Field Theory

qi(t) → φ(~x,t) ˙ q˙i(t) → φ(~x,t) i → x ˙ L(qi(t), q˙i(t)) → L[φ(~x,t), φ(~x,t)] where one can immediately spot the functional character of the Lagrangian in classical eld theory. However, in order to depart from the abstract notion of functional derivatives and insert the well understood concept of partial dierential it is necessary to write down the Lagrangian as a volume integral over a density function called Lagrangian density L, considering local eld theories. Subsequently, our Lagrangian can be written down as Z 3 ˙ L(t) = d x L(φr(~x,t), ∇φr(~x,t), φr(~x,t) (1.20)

It is vital to note some assumptions made from us in order to dene the Lagrangian

density. First of all, as one can see the Lagrangian density depends on the eld φr, its ˙ temporal (φr) and spatial derivatives (∇φr). There is no restriction for the Lagrangian density to depend on higher derivatives of φr also, but this occurrence would have had negative consequences on the deduced Euler-Lagrange eld equations in respect to the Lagrangian density, because higher order terms would have been gained, which is un- pleasant. Secondly, the addition of spatial derivatives dependence in the Lagrangian density has a deeper meaning. When constructing a new theory, the Lagrangian density must be invariant under specic transformations in order to obey their symmetries. In any relativistic eld theory, the Lagrangian density must be invariant under Lorentz transformation, hence, the need of inserting spatial derivatives dependence is crucial in order to keep this quantity invariant when going from one reference frame to another. Finally, by studying local theories, we exclude dependence of other variables (y 6= x) in the Lagrangian density. Using the Lagrangian density, the action can be written as

Z t2 Z t2 Z 3 ˙ S = dtL(t) = dt d xL(φr, ∇φr, φr(x)) t1 t1 1.3. LAGRANGIAN FORMALISM 13

Therefore, the variation of action considering Hamilton's principle gives rise to the relation

Z t2 Z 3 ˙ δS =δ dt d xL(φr, ∇φr, φr(x)) t1 Z t2 Z 3 ˙ = dt d x δL(φr, ∇φr, φr(x)) t1 Z t2 Z ∂L ∂L ∂L = dt d3x δφ (x) + δ(∇φ (x)) + δ(φ˙ (x)) r r ˙ r t1 ∂φr(x) ∂(∇φr(x)) ∂(φr(x)) =0 where it is very important to point out that we have used the convention

φr(~x,t) = φr(x) Next, considering that both the temporal and spatial derivatives commute with the variation as proven in (1.16) we nd

[δ, ∂t]φr = 0 ⇒ δ(∂tφr) = ∂t(δφr)

[δ, ∇]φr = 0 ⇒ δ(∇φr) = ∇(δφr) Consequently, considering the relations stated above one can nd

Z t2 Z ∂L ∂L ∂L 0 = dt d3x δφ (x) + ∇(δφ (x)) + ∂ (δφ (x)) r r ˙ t r t1 ∂φr(x) ∂(∇φr(x)) ∂(φr(x)) However, in order to perform integration by parts in the second and third term we should rewrite them as follows ∂L ∂L ∂L ∇(δφr(x)) =∇ δφr(x) − ∇ δφr(x) ∂(∇φr(x)) ∂(∇φr(x)) ∂(∇φr(x)) ∂L ∂L ∂L ∂ (δφ (x)) = ∂ δφ (x) − ∂ δφ (x) ˙ t r t ˙ r t ˙ r ∂(φr(x)) ∂(φr(x)) ∂(φr(x)) Substituting the relations gained above in the integral we nd

Z ∂L ∂L ∂L 0 = d4x − ∇ − ∂ δφ (x) t ˙ r ∂φr(x) ∂(∇φr(x)) ∂(φr(x)) Z t2 ~x2 Z t2 ∂L 3 ∂L + dt δφr(x) + d x δφr(x) t ∂(∇φr(x)) ∂(φ˙ (x) 1 ~x1 r t1 However, given that we have assumed known endpoints of a line in the 4-dimensional space-time, thus knowledge about those points with space-time coordinates

µ and µ x1 = (~x1, t1) x2 = (~x2, t2) means that the variation of the eld function at those coordinates is zero. More specically

δφr(~x1, t) = δφr(~x2, t) = 0

δφr(~x,t1) = δφr(~x,t2) = 0 14 CHAPTER 1. CLASSICAL FIELD THEORY

Subsequently, the value of the last two integrals is zero, hence we have

Z ∂L ∂L ∂L d4x − ∇ − ∂ δφ (x) = 0 t ˙ r ∂φr(x) ∂(∇φr(x)) ∂(φr(x) Finally, expressed in terms of the Lagrangian density, the Euler-Lagrange eld equation reads ∂L ∂L ∂L − ∇ − ∂ = 0 (1.21) t ˙ ∂φr(x) ∂(∇φr(x)) ∂(φr(x) and using the relativistic covariant notation

∂L ∂L − ∂µ = 0 (1.22) ∂φr(x) ∂(∂µφr(x)) where as I have stated in (1.8) we have used the covariant 4-divergence.

1.4 The Hamilton Formalism

In order to quantize this classical theory stated above by the canonical formalism of non relativistic quantum mechanics one must introduce conjugate variable by dealing with Hamilton's formalism.To apply Hamilton's formalism to a eld theory, one rst has to dene a momentum that is canonically conjugate to the eld function. In correspondence to classical mechanics, we dene the canonically conjugate eld by the functional derivative δL(t) π (x) = (1.23) r ˙ δφr(x) whence, in respect to the Lagrangian density

∂L π (x) = (1.24) r ˙ ∂φr(x) Next, we can introduce the Hamiltonian functional through the Legendre transformation that reads Z 3 h ˙ i H(t) = d x πr(x)φr(x) − L(t) (1.25) hence, having in mind the functional character of Hamiltonian one can also introduce the concept of Hamiltonian density as follows Z 3 H(t) = d x H(πr, φr, ∇πr, ∇φr) (1.26)

where ˙ ˙ H(πr, φr, ∇πr, ∇φr) = πr(x)φr(x) − L(φr(x), ∇φr(x), φr(x)) (1.27) Finally, it is very useful to study the role of Poisson brackets in eld theory. More

specically, given two functionals F [φr, πr] and G[φr, πr] one can dene their Poisson bracket as Z δF δG δF δG {F,G} = d3x − (1.28) δφr(x) δπr(x) δπr(x) δφr(x) which is the generalization of classical mechanic's Poisson bracket denition. In order to nd the Poisson brackets between the eld and conjugate eld functions, we 1.4. THE HAMILTON FORMALISM 15

need to calculate the functional derivatives of those quantities in respect to themselves and their conjugate. A clever way to do so is by writing down those functions as functionals depending on themselves. To be more precise

Z 3 0 0 0 φr(~x,t) = d x φr(~x , t)δ(~x − ~x ) (1.29a) Z 3 0 0 0 πr(~x,t) = d x πr(~x , t)δ(~x − ~x ) (1.29b)

where ~x,t are considered as the parameters of the functional. Consequently, dealing with the eld functional, and taking its variation we nd

Z δφ (~x,t) 3 0 r 0 (1.30) δφr(~x,t) = d x 0 δφr(~x , t) δφr(~x , t) Thus, juxtaposing (1.29a) and (1.30) relations we can nd the functional derivative of the eld. Following the same steps we nd the functional derivative of the conjugate eld also, hence having

δφ (~x,t) r 0 (1.31a) 0 =δ(~x − ~x ) δφr(~x , t) δπ (~x,t) r 0 (1.31b) 0 =δ(~x − ~x ) δπr(~x , t) We already know, given that the eld functional is independent of its conjugate, that the former's (or latter's) functional derivative in respect to the latter (former) is zero

δφr(~x,t) δπr(~x,t) (1.32) 0 = 0 = 0 δπr(~x , t) δφr(~x , t) Afterwards, it is very easy to calculate their Poisson brackets, a step essential in order to depart from classical to quantum eld theory. According to equation (1.28) we have

Z 0 0 0 3 00 δφr(~x,t) δπr(~x , t) δφr(~x,t) δπr(~x , t) {φr(~x,t), πr(~x , t)} = d x 00 00 − 00 00 δφr(~x , t) δπr(~x , t) δπr(~x , t) δφr(~x , t) Z = d3x δ(~x − ~x00) δ(~x0 − ~x00)

=δ(~x − ~x0)

Therefore, we obtain the Poisson brackets between the eld and its conjugate eld. In summary we nd

0 0 {φr(~x,t), πr(~x , t)} =δ(~x − ~x ) (1.33a) 0 0 {φr(~x,t), φr(~x , t)} ={πr(~x,t), πr(~x , t)} = 0 (1.33b)

It is very important to note that the Poisson brackets stated above hold true only for equal time t as can be seen from the way we derived them. In addition, the index in the eld and conjugate eld functions remained constant, thus we have been considering a specic eld and not a summation of independent elds. 16 CHAPTER 1. CLASSICAL FIELD THEORY 1.5 Symmetries and Conservation Laws

In this section we will explore the connection between symmetry transformation and conservation laws in eld theory. Conservation Laws, i.e the existence of quantities which do not change in time, regardless of the dynamical evolution of a system, play a crucial role in the construction of our theory. To be more precise, conservation of energy, momentum and angular momentum are fundamental laws that our theory must guarantee. Conservation laws are the results from the existence of various symmetries. For each continuous transformation of the coordinates and the elds, under which physics does not change, the existence of a conserved quantity can be deduced. The mathematical foundation for this connection is known as Noether's Theorem. We are about to prove this theorem in the most general form, therefore, for any continuous symmetry its conserved quantity will be derived as a special case.

Noether's Theorem. In order to deduce the most general form of the continuity equation, from which we will obtain the conserved quantities, we have to consider the following innitesimal transformations, corresponding to the

i) Change in the coordinates xµ 0 (1.34) xµ = xµ + δxµ ii) Change in the eld φr(x)

0 0 (1.35) φr(x ) = φr(x) + δφr(x) iii) As a result change in the Lagrangian density L(x)

L0(x0) = L(x) + δL(x) (1.36)

µ where for our convenience, we have denoted the Lagrangian density as L(x) = L(φr(x), ∂ φr(x)) and inserting the primed quantities into the original Lagrangian density we get L0(x0) = 0 0 µ 0 0 . However, it is very important to note that the variation above con- L(φr(x ), ∂ φr(x )) sists of two dierent transformations: a) the change in coordinates and b) the change of the eld. In order to simplify our calculations it is very useful to introduce a modied variation, which deals with the eld transformation only, called total variation given by the relation

˜ 0 (1.37) δφr(x) = φr(x) − φr(x)

The reason for introducing this notion is because the total variation (δ˜) commutes with the covariant 4-divergence (∂µ) as we will show, whereas the local variation (δ) does not. More specically

∂ h˜ i ∂ 0 δφr(x) = [φr(x) − φr(x)] ∂xµ ∂xµ ∂φ0 (x) ∂φ (x) = r − r ∂xµ ∂xµ ∂φ (x) =δ˜ r ∂xµ 1.5. SYMMETRIES AND CONSERVATION LAWS 17

Therefore [δ,˜ ∂µ] = [∂µ, δ˜] = 0 (1.38) However, it is crucial to mention the relation connecting those two variations, which is ˜ ∂φr(x) δφr(x) = δφr(x) − δxµ (1.39) ∂xµ and holds true only for rst order approximation, that is sucient for us given that we are considering innitesimal transformations only. Now, we will study the consequences that follows from the fact that our innitesimal transformations, leave the action integral invariant, which is the most general case. Furthermore, we have Z Z δS = d4x0 L0(x0) − d4x L(x) (Ω0) (Ω) =0 where Ω0 denotes the same integration region as Ω but expressed in the new coordinates x0. Next, introducing the variation of Lagrangian density from (1.36) we nd Z Z Z d4x0 δL(x) + d4x0 L(x) − d4x L(x) = 0 (1.40) Ω0 Ω0 Ω But, we need to express the integration dierential element d4x0 = dx00dx01dx02dx03 in terms of d4x = dx0dx1dx2dx3. To do so we use the relation ∂(δxµ) d4x0 = 1 + d4x (1.41) ∂xµ where, given that we have considered innitesimal transformations in our calculations we have omitted greater than 1st order mixed or pure dierentials. Substituting (1.41) in (1.40) we get Z Z µ Z 4 4 ∂(δx ) 4 0 = d x δL(x) + d x 1 + µ L(x) − d x L(x) Ω Ω ∂x Ω Z Z ∂(δxµ) 4 4 (1.42) = d x δL(x) + d x µ L(x) Ω Ω ∂x Introducing now the total variation of the Lagrangian density δ˜L = L0(x) − L(x) (1.43) we know that it is related with the local variation given by (1.36) through ˜ ∂L(x) δL(x) = δL(x) − δxµ (1.44) ∂xµ Consequently, inserting (1.44) in (1.42) then Z Z ∂(δx ) 0 = d4x δL(x) + d4x L(x) µ ∂xµ Z Z Z 4 ˜ 4 ∂L(x) 4 ∂(δxµ) = d x δL(x) + d x δxµ + d x L(x) ∂xµ ∂xµ Z Z 4 ˜ 4 ∂ = d x δL(x) + d x (L(x)δxµ) ∂xµ Z 4 ˜ ∂ = d x δL(x) + (L(x)δxµ) (1.45) ∂xµ 18 CHAPTER 1. CLASSICAL FIELD THEORY

where given that the total variation of the Lagrangian density can be expanded in terms µ of its independent variables L(x) ≡ L(φr(x)∂ φr(x)), hence ˜ ∂L ˜ ∂L ˜ µ δL = δφr + µ δ(∂ φr) ∂φr ∂(∂ φr) Moreover, having in mind (1.38) and performing product dierentiation we nd for the total variation of the Lagrangian density ˜ ∂L ˜ ∂L µ ˜ δL = δφr + µ ∂ (δφr) ∂φr ∂(∂ φr) ∂L ˜ µ ∂L ˜ µ ∂L ˜ = δφr + ∂ µ δφr − ∂ µ δφr ∂φr ∂(∂ φr) ∂(∂ φr) ∂L ∂L ∂L µ ˜ µ ˜ (1.46) = − ∂ µ δφr + ∂ µ δφr ∂φr ∂(∂ φr) ∂(∂ φr) where one can immediately notice that the term inside the rst square brackets is the eld equations of motions that our eld satises, therefore it is equal to zero. Thus, substituting (1.46) to (1.45) and recalling the equation that connects the total to the local variation of the eld we get Z 4 µ ∂L ˜ 0 = d x ∂ µ δφr + L(x)δxµ ∂(∂ φr) Z 4 µ ∂L ∂φr = d x ∂ µ δφr − δxν + L(x)δxµ ∂(∂ φr) ∂xν Now, since the range of integration can be chosen arbitrarily the integrand itself can vanish, thus acquiring the equation of continuity

µ ∂ jµ(x) = 0 (1.47) where the Noether current is given by

∂L ∂L ∂φ r ν (1.48) jµ(x) = µ δφr − µ ν + gµνL(x)δxµ δx ∂(∂ φr) ∂(∂ φr) ∂x In order to nd the conserved quantity, we need to integrate the continuity equation over a 3-dimensional space and use Gauss' Theorem, hence Z 3 µ 0 = d x∂ fµ(x) (V ) Z I d 3 ~ = d xf0(x) + d~s · f(x) dx0 (V ) (S) Z d 3 dG = d x f0(x) = dt (V ) dt where Z 3 G = d xf0(x) (1.49) (V ) is the conserved quantity which has a constant value in time. 1.5. SYMMETRIES AND CONSERVATION LAWS 19

Next, we will study several important applications of Noether's Theorem considering foundamental transformations that every theory must obey, and we will also nd the conserved quantities for each case.

Invariance under Translation. We will now study the invariance under translations

x0µ = xµ + µ (1.50) where the local variation of the eld is zero, thus,

0 0 (1.51) φr(x ) = φr(x) Recalling equation (1.48) the 4-current density is given by

∂L ∂φ r ν (1.52) (jµ(x))ν = − µ − gµνL(x) , ν = 0, 1, 2, 3 ∂(∂ φr) ∂xν Eliminating the constant factor ν from the dierential conservation law (1.47) we have ∂ Tµν = 0 (1.53) ∂xµ where the quantity ∂L ∂φr (1.54) Tµν = µ ν − gµνL(x) ∂(∂ φr) ∂x is called the canonical energy momentum tensor. Thus, we can immediately conclude - considering the fact that ν = 0, 1, 2, 3 - that there are 4 conserved quantities, generated from (1.49), which are the total energy E ~ and total momentum vector P = (P1,P2,P3) of the eld conguration, which can be summarized in the 4-momentum Pν as follows Z ~ 3 Pν = (E, P ) = d x T0ν = const. , ν = 0, 1, 2, 3 (1.55)

Lorentz Invariance. Four-dimensional space-time is considered to be isotropic in respect to rotations also. Therefore, as in any relativistic theory, the action is required to be invariant under Lorentz transformations, which include: a) ordinary rotations in space and b) velocity transformations as well. A Lorentz transformation can be represented as follows µ µ µ (1.56) Λ ν = δ ν + δω ν where µ = innitesimally small. δω ν As I have already stated, through a Lorentz transformation one can go from a frame S with basis {xµ} to another frame S0 with basis {x0µ}

µ µ 0µ Λ ν : {x } → {x } and from (1.56) we can nd

0µ µ ν x =Λ νx µ µ ν =(δν + δω ν )x µ µν =x + δω xν (1.57) 20 CHAPTER 1. CLASSICAL FIELD THEORY

which denotes an innitesimal rotation in the 4-D space-time. However, we should note that the matrix δωµν depends on the rotation angles and is anti-symmetric δωµν = −δωµν The proof of its anti-symmetric property comes from the fact that under a Lorentz transformation the scalar product is invariant

0µ 0 µ µν µν x xµ = x xµ ⇒ δω = −δω

The transformed eld function will depend linearly with the 4-vector xµ as one can see from

0 0 µ φr(x ) =φr(Λ νx) µ µ ν =φr(x + δω ν x ) µ µ ν =φ(x ) + δω ν x ∂µφr(x) where this dependence can be written down as

1 φ0 (x0) = φ (x) + δω (Iµν) φ (x) (1.58) r r 2 µν rs s where the quantities Iµν are the innitesimal generators of the Lorentz Transformation. They can be chosen to be anti-symmetric Iµν = −Iµν, thus existing 6 innitesimal generators that 3 of them corresponds to 3-D rotations, while the other 3 to Lorentz boosts. Next, inserting the transformation relations given by (1.58) and (1.57) in the density current relation (1.48) we get

1 ∂L νλ νλ (1.59) jµ(x) = µ δωνλ(I )rsφs(x) − Tµνδω xλ 2 ∂(∂ φr) where using the relations

1 δωνλ = δω[νλ] = (δωνλ − δωλν) 2 σκ δωνλ = gσνgλκδω and after some trivial calculations we can obtain for the current density under Lorentz transformations, the relation 1 j (x) = δνλM (x) (1.60) µ 2 µνλ where ∂L (1.61) Mµνλ(s) = µ (Iνλ)rsφs(x) − Tµνxλ + Tµλxν ∂(∂ φr) While the conserved quantity can be readily deduced for µ = 0, that equals to

Z ∂L M (x) = d3x T x − T x + (I ) φ (x) (1.62) νλ 0λ ν 0ν λ ˙ νλ rs s ∂φr where this quantity plays the role of the tensor of angular momentum. Chapter 2

The Klein-Gordon Field

2.1 Canonical Quantization

Before diving ourselves into the study of the Klein-Gordon eld, it is essential to introduce the concept of canonical quantization, which is an important step towards the path of eld quantization. In quantum mechanics, canonical quantization is a recipe that transforms the Hamil- tonian formalism of classical dynamics to quantum theory. Through this path, we can immediately replace the generalized coordinated qr and their conjugate momenta pr as operators. The result of the process stated above is that the Poisson brackets of those quantities will be replaced by their commutation relations as follows

[qr, qs] = [pr, ps] (2.1a)

[qr, ps] = iδrs (2.1b) In eld theory we do the same thing, but now our generalized coordinates are the

eld function φr(~x,t) ≡ φr(x) and the conjugate eld πr(~x,t) ≡ πr(x). Therefore, one can consider the quantum eld as an operator valued function (function that provides operators) in space-equaltime, which satises the equal time commutation relations

0 0 [φr(~x,t), πr(~x , t)] =iδ(~x − ~x ) (2.2a) 0 0 [φr(~x,t), φr(~x , t)] =[πr(~x,t), πr(~x , t)] = 0 (2.2b) obtained just by replacing the Poisson brackes found in classical eld theory, given by equations (1.33a) and (1.33b) with commutators, inserting the imaginary number i and recalling that we are working on the natural unit system, hence ~ = 1. An important point is that the commutation relations are equal time, thus hold true for time t, due to the fact that we are working on the Heisenberg picture, where all the ˆ ˆ operators φr(~x,t) = φr(x) and πˆr(~x,t) =π ˆr(x) vary in time according to Heisenberg's equation

dO(t) i = [O(t),H] (2.3) dt but they have no explicit time dependence. On the other hand, the eigenfunctions |Φi being time independent form a basis on the Hermite space.

21 22 CHAPTER 2. THE KLEIN-GORDON FIELD 2.2 The Real Klein-Gordon eld

The simplest example of a relativistic eld theory deals with spin-0 particles described by the Klein-Gordon equation. In this section, we will study the simplest form of the Klein-Gordon eld which we will quantize. The Lagrangian density of a real spin-0 eld φ(x) = φ(~x,t) with mass m, recalling that we are working on the natural units system, reads

1 1 L(x) = ∂µφ∂ φ − m2φ2 (2.4) 2 µ 2 1 = gµν(∂ φ)(∂ φ) − m2φ2 2 ν µ where inserting this Lagrangian density in the eld equations given by (1.22)

∂L ∂L − ∂µ = 0 ∂φr(x) ∂(∂µφr(x)) we obtain the Klein-Gordon equation

2 ( + m )φ = 0 (2.5)

where: µ ∂2 ~ 2 is the four dimensional Laplace operator. = ∂µ∂ = ∂t2 − ∇ It is vital here to point out that given that we are dealing with a scalar eld φ(x), then δφ(x) = 0. This means that the angular momentum tensor given by the equation (1.62) has a zero spin part, justifying the fact that the Lagrangian density stated above describes spin-0 particles. In addition, it is well known that spin-0 particles are bosons, therefore our theory must obey the Bose-Einstein statistics. As a result, we will use the equal time commutators (instead of anti-commutators used for fermions) to quantize our eld operators. Next, in order to calculate the conjugate eld and the Hamiltonian that rises from our Lagrangian density, it is useful to expand the four-divergence in the Lagrangian as follows 1 L(x) = (∂ φ)(∂ φ) − (∇φ)(∇φ) − φ2 2 t t Hence through trivial calculations we nd

∂L π(x) = = φ˙(x) (2.6) ∂φ˙ for the conjugate eld, and

H(x) =π(x)φ˙(x) − L(x) 1 = π2(x) + (∇φ(x))2 + m2φ(x)2 2 for the Hamiltonian density. Therefore, the Hamilton function will be given by

1 Z H(t) = d3x π2(x) + (∇φ(x))2 + m2φ(x)2 (2.7) 2 2.2. THE REAL KLEIN-GORDON FIELD 23

Subsequently, to quantize our theory, we need to follow the standard procedure, where one replaces the eld and conjugate eld functions by operators as follows

φ(~x,t) →φˆ(~x,t) ≡ φˆ(x) π(~x,t) →πˆ(~x,t) ≡ πˆ(x) and applies the equal-time commutation relations (ETCR) - recalling the fact that the theory describes bosons -

[φˆ(~x,t), πˆ(~x0, t)] =iδ(~x − ~x0) (2.9a) [φˆ(~x,t), φˆ(~x0, t)] =[ˆπ(~x,t), πˆ(~x0, t)] = 0 (2.9b) This procedure is often called as second quantization. Imposing those operators in the Hamilton function (2.7), we nd 1 Z h i Hˆ (t) = d3x πˆ2(x) + (∇φˆ(x))2 + m2φˆ(x)2 (2.10) 2 Furthermore, given that we are working on the Heisenberg representation, where the operators evolve in time according to the Heisenberg equation given by (2.3), therefore for the time evolution of eld and conjugate eld operators we have ˙ h i φˆ = − i [φˆ(~x,t), Hˆ (~x0, t)] =π ˆ(~x,t) (2.11a) h i πˆ˙ = − i [ˆπ(~x,t), Hˆ (~x0, t)] = (∇2 − m2)φˆ(~x,t) (2.11b) where in order to deduce (2.11b) we have used the fact that

[φˆ(~x,t), ∇0φˆ(~x0, t)] = ∇0[φˆ(~x,t), φˆ(~x0, t)] = 0 [ˆπ(~x,t), ∇0φˆ(~x0, t)] = ∇0[ˆπ(~x,t), φˆ(~x0, t)] = ∇δ(~x − ~x0) Finally according to the equations found above, we conclude that the eld operator satises the Klein-Gordon equation also. More specically ¨ φˆ(~x,t) = (∇2 − m2)φˆ(~x,t) (2.12) During the procedure of eld quantization we have introduced the notions of eld and its conjugate eld operator, where they are dened over an abstract space of state vectors |Ψi that do not change in time. Therefore in order to nd a representation, we begin by choosing a complete set of classical wave functions u(~x). The eld operator can be represented in terms of a generalized Fourier decomposition with respect to this set of functions as follows Z ˆ 3 φ(~x,t) = d p u~p(~x)ˆa~p(t) (2.13)

where the time evolution of the operator is denoted by the operators aˆ~p(t). Next, the eld operator φ(~x,t) will be expanded into a plane wave basis as follows

i~p·~x u~p(~x) = Npe (2.14) where, inserting the set of basis in (2.13) one can nd Z ˆ 3 i~p·~x φ(~x,t) = d x Npe aˆ~p(t) (2.15) 24 CHAPTER 2. THE KLEIN-GORDON FIELD

Substituting (2.15) into (2.12) we nd the dierential equation

¨ 2 2 aˆ~p(t) = −(~p + m )ˆa~p(t) which one can immediately notice that the time evolution of an operator in the Klein- Gordon eld obeys the dierential equation of the Harmonic oscillator. Consequently, the solution of the dierential equation stated above equals to

(1) −iωpt (2) iωpt (2.16) aˆ~p(t) =a ˆ~p e +a ˆ~p e where the operators (1) and (2) are constant in time and the frequency is dened as aˆ~p aˆ~p

p 2 2 ωp = + ~p + m However, recalling the fact that we are dealing with real scalar elds φ(x), one can conclude that the relation φ(x) = φ∗(x) must hold true for the eld, thereby the eld operator must be Hermitian, i.e

φˆ†(~x,t) = φˆ(~x,t) (2.17) Thus, the eld and its hermitian adjoint operator are given by Z h i ˆ 3 i~p·~x (1) −iωpt (2) iωpt (2.18a) φ(~x,t) = d p Npe aˆ~p e +a ˆ~p e Z † † ˆ† 3 −i~p·~x (1) iωpt (2) −iωpt (2.18b) φ (~x,t) = d p Npe aˆ~p e + aˆ~p e

and inserting (2.18a) and (2.18b) in (2.17) we can nd for the time independent coecients † (1) (2) (2.19) aˆ~p = a−~p One can conrm the validity of the relation found above by substituting it in (2.18b), thus getting Z h i ˆ† 3 −i~p·~x (2) iωpt (1) −iωpt φ (~x,t) = − d p Npe aˆ−~pe +a ˆ−~pe Z h i 3 i~p·~x (1) −iωpt (2) iωpt = d p Npe aˆ~p e +a ˆ~p e =φˆ(~x,t)

∗ where we have used that N−p = Np , ωp = ω−p , (Np) = Np. Moreover, the nal expression of the eld and conjugate eld operators reads Z h i ˆ 3 i(~p·~x−ωpt) † −i(~p·~x−ωpt) (2.20a) φ(~x,t) = d p Np aˆ~pe +a ˆ~pe Z h i 3 i(~p·~x−ωpt) † −i(~p·~x−ωpt) (2.20b) πˆ(~x,t) = d p Np(−iωp) aˆ~pe − aˆ~pe

† where we have set: (1) and † (1) (2). In addition, (2.20b) came from aˆ~p =a ˆ~p aˆ~p = aˆ~p =a ˆ−~p ˙ the relation πˆ(~x,t) = φˆ(~x,t). One can immediately highlight the fact that the relations 2.2. THE REAL KLEIN-GORDON FIELD 25

found above depend on both positive and negative frequency values, which is a dening characteristic for relativistic theories. Furthermore, we will nd the commutation relations that and † satisfy. We expect aˆ~p aˆ~p them to fulll the algebra typical for creation and annihilation operators (we will try to clarify this concepts in the next section), i.e h i † 0 (2.21a) aˆ~p, aˆ~p0 = δ(~p − ~p ) h i † 0† (2.21b) [ˆa~p, aˆ~p0 ] = aˆ~p, aˆ~p = 0 To check whether those commutation relations hold true, we will conrm that they satisfy (2.9a), which our theory obeys. To do so, and in order to simplify our calculation we will use the relativistic product quantity

µ p · x = pµx = ωpt − ~p · ~x As a result the eld and its conjugate operators take the form Z h i ˆ 3 −ip·x † ip·x φ(~x,t) = d p Np aˆ~pe +a ˆ~pe Z 0 3 0 0 0 h −ip0·x0 † ip0·x0 i πˆ(~x , t) = d p Np(−iωp) aˆ~p0 e − aˆ~p0 e

For their commutation relation we nd Z Z ˆ 0 3 3 0 h −ip·x † ip·x −ip0·x0 † ip0·x0 i [φ(~x,t), πˆ(~x , t)] = d p d p NpNp0 (−iωp0 ) aˆ~pe +a ˆ~pe , aˆ~p0 e − aˆ~p0 e Z Z 3 3 0 h 0 −i(p·x−p0·x0) 0 i(p·x−p0·x0)i = d p d p NpNp0 (iωp0 ) δ(~p − ~p )e + δ(~p − ~p)e Z Z 3 3 0 0 h −i(p·x−p0·x0) i(p·x−p0·x0)i = d p d p NpNp0 (iωp0 )δ(~p − ~p ) e + e Z 3 2 h i~p·(~x−~x0) −i~p·(~x−~x0)i = d pNp (iωp) e + e Z Z 3 2 i~p·(~x−~x0) 3 2 −i~p·(~x−~x0) = i d pNp ωpe + i d pNp ωpe Z 3 2 i~p·(~x−~x0) = 2i d pNp ωpe

However, we know that the delta function can be dened through the relation

3 Z d p 0 δ(~x − ~x0) = ei~p·(~x−~x ) (2π)3 therefore, comparing those two equations we can easily nd that when the normalization

factor Np equals to 1 Np = (2.22) p 3 2ωp(2π) Then, we derive the desired commutation relations for the eld and its conjugate

3 Z d p 0 [φˆ(~x,t), πˆ(~x0, t)] = i ei~p·(~x−~x ) = iδ(~x − ~x0) (2π)3 26 CHAPTER 2. THE KLEIN-GORDON FIELD

Thus, the commutation relations for the operators stated in equations (2.21a) and (2.21b) hold true, hence he have assumed the correct algebra. In our study we have employed the continuum normalization, when constructing the plane waves given by the equation

1 −i(ωpt−~p·~x) u~p(~x,t) = e (2.23) p 3 2ωp(2π) However, we can switch to a discrete formulation by using box normalization, hence, by imposing periodic boundary conditions at the surface of a cube having volume V = L3 we take discrete values of the momentum 2π p = ~l with ~l = (l , l , l ) ~l L x y z while the corresponding plane waves must form an orthonormal basis, i.e

u 0 , u = δ 0 (2.24) ~pl ~pl ~pl,~pl and in order to describe the discreteness of space included in the box, one should replace the integral with a summation over the discrete values of momentum. As a result, the commutation relation given by (2.21a) takes the form

aˆ , aˆ 0 = δ 0 (2.25) ~pl ~pl ll One should point out here that the operators † and have the function to create and a~p aˆ~p annihilate particles, described by the wave function u~p(~x,t). Moreover, given that plane waves are momentum eigenstates, then the particles described by the plane wave basis expansion have well dened momentum. Consequently, by using other wave functions as basis in our expansion (spherical waves), we can describe particles having another well dened quantity (angular momentum). Now, we shall express the Hamiltonian in terms of the creation and annihilation operators. Inserting (2.23) in the eld and conjugate eld operators, gives us

Z h i ˆ 3 † ∗ (2.26) φ(~x,t) = d p aˆ~pu~p(~x,t) +a ˆ~pu~p(~x,t) Z h i 3 † ∗ (2.27) πˆ(~x,t) = − i d p ωp aˆ~pu~p(~x,t) − aˆ~pu~p(~x,t) therefore, substituting them in the Hamiltonian given by (2.10) and neglecting the symbol of operator over a we nd 1 Z h i Hˆ (t) = d3x πˆ2(x) + (∇φˆ(x))2 + m2φˆ(x)2 2 Z Z Z 1 3 3 0 h † ∗ i 3 h † ∗i = d x − d p ω 0 a 0 u 0 − a u 0 d pω a u − a u 2 p ~p ~p ~p0 ~p p ~p ~p ~p ~p Z Z 3 0 0 h † ∗ i 3 ∗ − d p ~p a~p0 u~p0 − a~p0 u~p0 d p~p a~pu~p − a~pu~p Z Z 2 3 0 h † ∗ i 3 h † ∗i + m d p a~p0 u~p0 + a~p0 u~p0 d p a~pu~p + a~pu~p 2.2. THE REAL KLEIN-GORDON FIELD 27

But, we need to consider the equations Z 1 3 ∗ 0 (2.28a) d xu~p0 u~p = δ(~p − ~p ) 2ωp Z 3 1 −2iωpt 0 d xu~p0 u~p = e δ(~p + ~p ) (2.28b) 2ωp Therefore for the three terms above in the integral we have Z ω2 3 p −2iωpt † † † † 2iωpt (i) = d p a−~pa~pe − a~pa~p − a~pa~p + a−~pa~pe 2ωp Z 2 3 |~p| −2iωpt † † † † 2iωpt (ii) = d p a−~pa~pe − a~pa~p − a~pa~p − a−~pa~pe 2ωp Z 2 3 m −2iωp † † † † 2iωpt (iii) = d p a−~pa~pe + a~pa~p + a~pa~p + a−~pa~pe 2ωp while combining those equations and inserting them in the Hamiltonian we obtain the nal result after some calculations 1 Z Hˆ (t) = d3p ω aˆ aˆ† +a ˆ†aˆ (2.29) 2 p ~p ~p ~p ~p where from the commutation relation (2.21a) setting ~p = ~p0 we can easily nd that † ~ † † ~ [ˆa~p, a~p] = δ(0) ⇒ aˆ~paˆ~p =a ˆ~paˆ~p + δ(0) and substituting in the Hamiltonian (2.29) we have 1 Z h i Hˆ (t) = d3p ω 2ˆa aˆ† + δ(~0) (2.30) 2 p ~p ~p As we can see two dierent kinds on innities exist. The rst one is due to the delta function that exists, and the second one appears when we consider high frequencies, thus ωp → ∞. The reason for the existence of the rst innity is the innitely large space that we have assumed through our continuum formulation. To prevent this innity from appearing we switch back to the discrete formulation stated before, conning our system into a large cube volume V = L3. Then, rewriting our Hamiltonian using the commutation relations in (2.25) we nd ˆ X † 1 H = ω~p aˆ aˆ~p + (2.31) l ~pl l 2 ~l where, we have eliminated the delta divergence from any state (even the vacuum one). On the other hand, the divergence at high frequencies still exists, but, it arises through our assumption that our theory is valid to arbitrarily short distances, hence arbitrarily high energies. This is not right, our integral should be cut-o at high energy, high momentum in order to reect the range that our theory must have. We can also justify the harmless of this divergence by studying the vacuum energy X 1 E = ω 0 2 ~pl ~l Recalling that physical observables are measured in respect to some base, then one can modify the divergent base, thus redening the Hamiltonian in order to eliminate this strong divergence. We can do that by dening a new Hamiltonian as ˜ˆ ˆ H = H − E0 28 CHAPTER 2. THE KLEIN-GORDON FIELD

A more elegant way to do so is by splitting our eld and conjugate eld operators in two parts containing positive (exp (−iωpt)) and negative (exp (iωpt)) frequencies. Then, we can introduce the concept of normal ordering of those operators. The normal product of two φˆ and χˆ operators, is dened as a product in which the parts with negative frequency generally stand to the left of the parts with positive frequency, i.e

h i N φˆχˆ = φˆ(−)χˆ(−) + φˆ(−)χˆ(+) +χ ˆ(−)φˆ(+) + φˆ(+)χˆ(+)

Splitting the eld operators in a positive and negative frequency part, we have Z Z ˆ 3 3 † ∗ φp = d p aˆ~pu~p + d p aˆ~pu~p ˆ(+) ˆ(−) =φp + φp Z Z 3 3 † ∗ πˆp = d p (−iωp)ˆa~pu~p + d p (−iωp)ˆa~pu~p

(+) (−) =ˆπp +π ˆp Therefore, dening the new Hamiltonian as the normal ordered product of the eld operators and using the property stated above we nd

1 Z 2 H˜ˆ = d3x N πˆ2 + ∇φˆ + m2φˆ2 2 Z 3 † = d p ωpaˆ~paˆ~p

In order to study the momentum of our quantum eld it is necessary to remember that the momentum 4-vector as we have dened it in the 1st chapter is given by the relation Z Z ~ 3 3 Pν = (E, P ) = d x T0ν(x) = d x (π(x)∂µφ(x) − g0µL(x))

then the components of the momentum vector are given by the relation

Z ∂φ(x) P i = − d3x π(x) , i = 1, 2, 3 ∂xi Consequently, quantizing the notion of total momentum found through Noether's Theo- rem and inserting in our eld theory, we will dene the momentum operator as

ˆ Z P~ = − d3x πˆ(~x,t)∇φˆ(~x,t) (2.32)

However, given that the eld and conjugate eld operators do not commute, during quantization there is no suciently good reason to explain why one should multiply those terms in this order, therefore the need to symmetrize this operator arises. In addition, symmetrization ensures that the operator is Hermitian. Hence, we have

ˆ 1 Z h i P~ = − d3x πˆ(~x,t)∇φˆ(~x,t) + ∇φˆ(~x,t)ˆπ(~x) (2.33) 2 2.3. THE BOSONIC FOCK SPACE 29

Finally, by expanding the eld operators into plane waves like before and having in mind (2.28a) and (2.28b) equations, and substituting in (2.33) while taking the normal-ordered product (to avoid divergences) one can easily deduce

Z ˆ 1 † † P~ = d3p ~p a a + a a (2.34) 2 ~p ~p ~p ~p Z 3 † (2.35) = d p ~p aˆ~paˆ~p

As one can see from the equations

h ˆ i H, aˆ~p = − ωpaˆ~p h ˆ †i † H, aˆ~p =ωpaˆ~p one can construct energy eigenstates by acting the creation operator aˆ~p on the vacuum state |0i. Therefore, we dene the momentum eigenstate as

† (2.36) |pi =a ˆ~p |0i

where we have shown that the state |pi has momentum ~p when the normal-ordered total momentum operator is acted upon it.

2.3 The Bosonic Fock Space

It is necessary to clarify the concept of Fock space on which we have already dened our commutation relations (algebra), and interpreted the operators and † as annihilation aˆ~p aˆ~p and creation operators. One can prove the validity of those statements, working on Fock space. In order to simplify and make easier to visualize our results, we will work according to the discrete formulation considering the Hamiltonian gained in (2.31) after normal-ordering and in the most general form

ˆ X † H = ωiaˆi aˆi ~i This Hamiltonian describes the total energy of particles distributed over various excited

states ui having energies ωi. In general, we dene the total particle number operator as

ˆ X X † (2.37) N = nî = aî aî i i 30 CHAPTER 2. THE KLEIN-GORDON FIELD

The particle number operators ni for dierent i-states commute among each other, since the commutation relations in general can be written as

h † i aˆi(t), aˆj(t) =δij

h † † i [ˆai(t), aˆj(t)] = aˆi (t), aˆj(t) = 0

then,

h † † i [ˆni, nˆj] = aˆi aˆi, aˆjaˆj

h † † i † h † i = aî , aˆjaˆj aî +a î aî, aˆjaˆj † † = − aˆjaîδij +a î δijaˆj † † =âi aˆj − aî aˆj =0

Taking the time evolution of the total particle number operator, one can nd

˙ h i Nˆ = − i N,ˆ Hˆ X = − i [ˆni, nˆj] ωj = 0 i,j thus, given that we are working on the Heisenberg picture, the total particle number operator is a conserved quantity and expresses the conservation of the number of particles. Furthermore, given that the total particle number commutes with the Hamiltonian, we can nd a common set of eigenstates characterizing the number of particles for all the i-states

nˆi |n1, n2, . . . , ni,...i = ni |n1, n2, . . . , ni,...i (2.38)

The state vectors |n1, n2, . . . , ni,...i form a basis of the Hilbert space of second quantization in the particle-number representation. This Hilbert space is also called the Fock space. In other words, a Fock state can be considered as one with a well dened number of particles (quanta). It is obvious that not every element of Fock space can be considered as a Fock state (superposition of Fock states). Normalizing the basis in Fock space

0 0 hn , n ,... |n , n ,...i = δ 0 δ 0 , ··· (2.39) 1 2 1 2 n1,n1 n2,n2 we nd for the total particle number operator

ˆ X N |n1, n2,...i = ni |n1, n2,...i i

=n |n1, n2,...i

Finally, we will calculate the inuence that the operators and † have when they act aˆ~p aˆ~p upon a state vector |n1, n2,...i, by calculating the number of particles that exist in the new state † . aˆi |n1, n2,...i 2.4. THE COMPLEX KLEIN-GORDON FIELD 31

Nai |n1, n2,...i =(n − 1)ˆai |n1, n2,...i (2.40b) One can write down those equations in a more elegant way as follows

h ˆ †i † N, aî =âi h ˆ i N, aî = − aî Moreover, we should highlight that the whole set of state vectors can be constructed based on a single state which does not contain any particles, acting the creation operator. In quantum eld theory this state is called vacuum state, denoted by |0i and dened as

aˆi |0i = 0 ∀i = 1, 2, ...

2.4 The Complex Klein-Gordon Field

The real Klein-Gordon eld was found to describe a collection of spin-0 particles of identical type. It is relatively easy to generalize this to particles having an internal degree of freedom. The simplest generalization introduces a doublet of particles and anti-particles that can be described by going over to complex elds, thus, φˆ 6= φˆ†. The Lagrangian density for the complex eld is given by

µ ∗ 2 ∗ L(x) = (∂ φ )(∂µφ) − m φ φ (2.41) where the eld φ and its complex φ∗ are treated as independent elds. One can nd the conjugate and its complex elds equal to ∂L π(~x,t) = = φ˙∗(~x,t) (2.42a) ∂φ˙(~x,t) ∂L π∗(~x,t) = = φ˙(~x,t) (2.42b) ∂φ˙∗(~x,t) 32 CHAPTER 2. THE KLEIN-GORDON FIELD

Consequently, we nd for the Hamiltonian Z h i H(t) = d3x π(~x,t)φ˙(~x,t) + π∗(~x,t)φ˙∗(~x,t) − L Z = d3x [π(~x,t)π∗(~x,t) + π∗(~x,t)π(~x,t) − L(x)] Z = d3x π(~x,t)π∗(~x,t) + ∇φ∗(~x,t)∇φ(~x,t) + m2φ∗(~x,t)φ(~x,t) (2.43)

Quantization of the theory can be achieved by replacing those four quantities with operators as follows

φ(x) →φˆ(~x,t) φ∗(x) →φˆ†(~x,t) π(x) →πˆ(~x,t) π∗(x) →πˆ†(~x,t) that are required to obey the commutation relations h i h i φˆ(~x,t), πˆ(~x0, t) = φˆ†(~x,t), πˆ†(~x0, t) = iδ(~x − ~x0) (2.44a) h i h i h i φˆ(~x,t), φˆ†(~x0, t) = πˆ(~x,t), πˆ†(~x0, t) = φˆ(~x,t), πˆ†(~x0, t) = φˆ†(~x,t), πˆ(~x0, t) = 0 (2.44b) Next, performing Fourier analysis around a plane wave basis as we did on the real Klein- Gordon eld we can nd for the eld and its complex conjugate operators Z h i ˆ 3 ˆ† ∗ (2.45a) φ(~x,t) = d p aˆ~pu~p(~x,t) + b~pu~p(~x,t) Z h i ˆ† 3 † ∗ ˆ (2.45b) φ (~x,t) = d p aˆ~pu~p(~x,t) + b~pu~p(~x,t)

Recalling that the eld operator is no longer Hermitian in our theory, therefore φ 6= φ†, we ˆ cannot nd a relation connecting the coecients aˆ~p and b~p, hence they are to be treated as independent, dening their own algebra each, and constituting independent creation and annihilation operators. Since they are independent and obey the same algebra, their dening commutation relations must be given by h i h i † ˆ ˆ† 0 (2.46a) aˆ~p, aˆ~p = b~p, b~p0 = δ(~p − ~p ) h i h i h i h i ˆ ˆ† † ˆ † ˆ† (2.46b) aˆ~p, b~p0 = aˆ~p, b~p0 = aˆ~p, b~p0 = aˆ~p, b~p0 = 0 h i h i h i ˆ ˆ ˆ† ˆ† † † (2.46c) b~p, b~p0 = [ˆa~p, aˆ~p0 ] = b~p, b~p0 = aˆ~p, aˆ~p0 = 0 Moreover, expressing the conjugate eld operator and its Hermitian conjugate in terms of creation and annihilation operators, inserting in (2.42a) and (2.42b), (2.45a) and (2.45b) we nd Z h i 3 † ∗ ˆ (2.47a) πˆ(~x,t) = i d p ωp aˆ~pu~p(~x,t) − b~pu~p(~x,t) Z h i † 3 ˆ† ∗ (2.47b) πˆ (~x,t) = i d p ωp b~pu~p(~x,t) − aˆ~pu~p(~x,t) 2.4. THE COMPLEX KLEIN-GORDON FIELD 33

Substituting in the Hamiltonian (2.43) and recalling (2.28a) and (2.28b) equations we nd that its given by Z ˆ 3 † ˆ†ˆ (2.48) H(t) = d p ωp aˆ~paˆ~p + b~pb~p which expresses the complex Klein-Gordon eld Hamiltonian. However, in order to visualize and interpret our Hamiltonian in terms of the particle number operators, we need to nd the normal-ordered Hamiltonian, Z h i ˆ 3 † ˆ†ˆ H(t) = d p ωp N aˆ~paˆ~p + b~pb~p Z 3 h † ˆ†ˆ i = d p ωp aˆ~paˆ~p + b~pb~p Z h i 3 (a) (b) (2.49) = d p ωp nˆ~p +n ˆ~p where as I have already stated the two independent creation operators correspond to dierent sets of particles that we may call particles a represented by aˆ~p operator with ~p ˆ momentum, and particles b represented by b~p operator and ~p momentum. The correlation between them will be elucidated afterwards. Furthermore, (a) is the a-particles number nˆ~p operator and (b) is the b-particles number operator, that describes the number of a or b nˆ~p particles that exist in the state with momentum ~p. Similarly, the momentum operator becomes Z h i Z h i ~ˆ 3 † ˆ†ˆ 3 (a) (b) (2.50) P = d p ~p aˆ~paˆ~p + b~pb~p = d p ~p nˆ~p +n ˆ~p Again, the Fock space can be constructed starting from the vacuum state where neither a nor b particles exist, however this time given that there are two sets of annihilation and creation operators, we will also have two types of states corresponding to each particle

φ0 =φeia ' (1 + ia)φ (2.51) φ∗0 =φ∗e−ia ' (1 − ia)φ∗ (2.52) Inserting them in (2.41), we obtain the new Lagrangian density after the transformations, that reads

0 µ ∗0 0 2 ∗0 0 L (x) = (∂ φ )(∂µφ ) − m φ φ After some trivial calculations and keeping only terms of rst order regarding the constant phase a, one observes that L0(x) = L(x) 34 CHAPTER 2. THE KLEIN-GORDON FIELD

Consequently, the invariance of the Lagrangian density under phase transformations re- veals according to the Noether's Theorem the existence of a new conserved quantity Z Z 3 3 ∗ ∗ G = d x j0(x) = d x [π(x)δφ(x) + π (x)δφ (x)] Z =i d3x [π(x)φ(x) − π∗(x)φ∗(x)]

given that δφˆ(x) = ia and δφˆ∗(x) = −ia. Nevertheless, we will dene our conserved quantity as Z Q = −i d3x [π(x)φ(x) − π∗(x)φ∗(x)] (2.53) which we will call the charge. In order to quantize this quantity, thus replacing it with an operator, given that the eld and conjugate eld commutation relations hold, we cannot order the operators from classical theory. As we did in the momentum operator (real Klein-Gordon eld), the solution is given by symmetrizing the charge operator. Consequently, the charge operator will be given by

i Z Qˆ = − d3x πˆφˆ + φˆπˆ − πˆ†φˆ† − φˆ†πˆ† (2.54) 2 Considering the relations expressing the eld, conjugate eld, their Hermitian quantities, (2.28a) and (2.28b) we get

1 Z Qˆ = d3p aˆ†aˆ +a ˆ aˆ† − ˆb ˆb† − ˆb†ˆb (2.55) 2 ~p ~p ~p ~p ~p ~p ~p ~p By adopting normal-ordering, we nd

1 Z h i Qˆ = d3pN aˆ†aˆ +a ˆ aˆ† − ˆb ˆb† − ˆb†ˆb 2 ~p ~p ~p ~p ~p ~p ~p ~p Z 3 † ˆ†ˆ = d p aˆ~paˆ~p − b~pb~p Z h i 3 (a) (b) (2.56) = d p nˆ~p − n~p

It can be seen combining equations (2.49) and (2.56) that the time evolution of the charge operator equals to

˙ h i h i Qˆ = −i Q,ˆ Hˆ = 0 ⇒ Q,ˆ Hˆ = 0 (2.57) indicating that the charge operator is a conserved quantity in our quantized theory. From equation (2.56) we conclude the existence of two types of particles with the same mass and opposite charge. Consequently, the operator † creates a particle having momentum aˆ~p and charge , whereas the operator ˆ†, creates a particle having momentum and ~p +1 b~p ~p charge −1. The particles b are called antiparticles of a. 2.5. MICROCAUSALITY 35 2.5 Microcausality

Up to now, our discussion of canonical eld quantization has been based on the equal time commutation relations between the eld and its conjugate operators, having arbitrary spatial separation. However, in order to acquire a general picture of the commutation relations, the need of nding them in a relativistic form that does not single out the time coordinate arises. Consequently, we will study the complex Klein-Gordon ˆ ˆ eld starting from the commutation relations between the eld operators φ(~x,x0) ≡ φ(x) ˆ† ˆ† and φ (~y, y0) ≡ φ (y) for unequal times x0, y0 h i i∆(x − y) = φˆ(x), φˆ†(y) (2.58)

This ∆−function is often called as Pauli-Jordan function, where we have taken into account the homogeneous character of space-time by implying that the commutation relation depends solely on the dierence x−y. Recalling the eld and its conjugate expansion in respect to the plane waves basis found before, we can nd an explicit expression for the ∆(x − y) as follows h i i∆(x − y) = φˆ(x), φˆ†(y) Z Z 3 3 0 h ˆ† ∗ † ∗ ˆ i = d p d p aˆ~pu~p(x) + b~pu~p(x) , aˆ~p0 u~p0 (y) + b~p0 u~p0 (y) Z 3 ∗ ∗ = d p u~p(x)u~p(y) − u~p(x)u~p(y) Z 3 2 −ip·(x−y) ip·(x−y) = d p Np e − e Z 3 d p 1 −ip·(x−y) ip·(x−y) = 3 e − e (2π) 2ωp In summary, we have found two equivalent equations for the Dirac-Jordan function

Z d3p 1 −ip·(x−y) ip·(x−y) (2.59) ∆(x − y) = − i 3 e − e (2π) 2ωp Z 3 d p sin [p · (x − y)] (2.60) = − 3 (2π) ωp

One of the most important properties of the ∆−function is that it is Lorentz invariant. One can verify this statement by writing it in a four-dimensional form, thus obtaining

Z d4p i∆(x − y) = sgn(p )δ(p2 − m2)e−ip·(x−y) (2.61) (2π)3 0

p 2 2 where ωp = + p + m d4p = d3p dp = the 4-momentum element ( 0 +1 for p0 > 0 sgn(p0) = the sign function −1 for p0 < 0 One can immediately notice that this expression is Lorentz Invariant given that it is composed entirely out of scalar quantities. In addition, one can derive the equal time 36 CHAPTER 2. THE KLEIN-GORDON FIELD commutation relations, if one nds rst the partial derivative of the Dirac-Jordan function at equal times

∂ ∆(x − y) = −δ(~x − ~y) ∂x 0 x0→y0 Then h i h ˙ i φˆ(~x,t), πˆ(~y, t) = φˆ(x), φˆ†(y) = iδ(~x − ~y)

Next, nding the Dirac-Jordan function that corresponds to equal times, we see from (2.59) that it takes the form

Z d3p sin [~p · (~x − ~y)] ∆(0, ~x − ~y) = 3 = 0 (2π) ωp given that the integrand is an odd function in respect to ~p. Consequently we nd h i φˆ(~x,t), φˆ†(~y, t) = 0 (2.62)

However, one can expand this commutation relation due to the Lorentz invatiance of the Dirac-Jordan function as follows

∆(x − y) = 0 for (x − y)2 < 0 (2.63)

corresponding to space-like separation of two events, thus ensuring that our theory is causal. This is the most important property and assures that in our theory all space-like separated observable operators commute

2 [O1(x), O2(y)] = 0 for (x − y) < 0 (2.64) i.e, measurements at two points that have space-like separation-cannot get into contact through transmission of light signals (outside of the light cone)- do not inuence each other. This is also known as microcausality. 2.6. THE SCALAR FEYNMAN PROPAGATOR 37 2.6 The Scalar Feynman Propagator

As we will see later, one of the most important quantities in Quantum Field Theory is the Feynman Propagator, which for the charged Klein-Gordon eld that we are studying is dened as the vacuum expectation value of the time-ordered product of the eld operators h i ˆ ˆ† i∆F (x − y) = h0|T φ(x)φ (y) |0i (2.65)

where the symbol T denotes that the product is arranged in chronological order, thus dening the time-ordered product, given that we are dealing with bosons, as h i ˆ ˆ† ˆ ˆ† ˆ† ˆ T φ(x)φ (y) = φ(x)φ (y)Θ(x0 − y0) + φ (y)φ(x)Θ(y0 − x0)

( +1 for t1 > t2 where Θ(t1 − t2) = is the step function. 0 for t1 < t2 In order to evaluate the Feynman Propagator (2.65) we have to study the cases

i) For x0 > y0, then

ˆ ˆ† i∆F (x − y) = h0|φ(x)φ (y)|0i = h0|φˆ(+)(x)φˆ†(−)(y)|0i (2.66) where in order to obtain the last result we have separated the eld and its conjugate operators in respect to positive and negative frequencies given by the relations Z Z ˆ(+) 3 , ˆ(−) 3 ˆ† ∗ φ (x) = d p aˆ~pu~p(x) φ (x) = d p b~pu~p(x) Z Z ˆ†(+) 3 ˆ , ˆ†(−) 3 † ∗ φ (x) = d p b~pu~p(x) φ (x) = d p aˆ~pu~p(x)

through the use of φˆ(+)(x) |0i = φˆ†(+)(y) |0i = 0 and h0| φˆ(−)(x) = h0| φˆ†(−)(y) = 0 we have reached equation (2.66). Next, rewriting the terms in (2.66) so that we will create a commutation relation in the expectation value we have

Z d3p eip·(x−y) i∆F (x − y) = 3 (2π) 2ωp (−) =i∆ (x − y) for x0 < y0 38 CHAPTER 2. THE KLEIN-GORDON FIELD

To sum up, for the Feynman Propagator we have found

(R d3p e−ip·(x−y) for 3 x > y (2π) 2ωp 0 0 i∆ (x − y) = 3 (2.67) F R d p eip·(x−y) for 3 x < y (2π) 2ωp 0 0 Combining those results, in order to obtain a more general relation for the Propagator, by taking into account both cases, then

(+) (−) i∆F (x − y) =Θ(x0 − y0)i∆ (x − y) + Θ(y0 − x0)i∆ (x − y) Z d3p 1 −ip·(x−y) ip·(x−y) (2.68) = 3 Θ(x0 − y0)e + Θ(y0 − x0)e (2π) 2ωp Next, to nd a more elegant and useful in our calculations expression for the Feynman

Propagator, the need of converting the last term in an integral in respect to p0, arises. To do so, one needs to recall the Cauchy's theorem of residues, that reads as follows

I X f(z)dz = 2πi Res [f(zj)] (C) j where: (C) is the boundary of a closed surface gives the residue for simple poles. Res [f(z0)] = limz→zj [(z − zj)f(zj)] Simplifying our factor by setting ~p → −~p in the second term, we have

1 −ip·(x−y) ip·(x−y) = Θ(x0 − y0)e + Θ(y0 − x0)e 2ωp 1 −iωp(x0−y0) iωp(x0−y0) −i~p·(~x−~y) = Θ(x0 − y0)e + Θ(y0 − x0)e e 2ωp Consequently, it is relatively easy to show, having in mind the Cauchy residue theorem that our relation turns out to be equal to

I −ip0(x0−y0) dp0 e (2.69) (c) 2πi (p0 − ωp)(p0 + ωp)

One can immediately notice that on the real axis at points p0 = ∓ωp there are two simple poles, thus, in order to circumvent those anomalies, it is essential to break down the integral, and using Jordan's Lemma we obtain I Z Z f(p0)dp0 = f(p0)dp0 + f(p0)dp0 (c) (CR) Γ() Z = f(p0)dp0 Γ() where we have eliminated the rst integral taking its limit (R → ∞). Therefore, our integral depends on the path followed on the real axis in order to avoid the anomalies. As a result, dierent paths (contours) will give dierent values of our integral. Here, choosing the contour depicted in the picture below, we can write the integral in the compact form

Z dp e−ip0(x0−y0) − 0 (2.70) Γ() 2πi (p0 − ωp)(p0 + ωp) 2.6. THE SCALAR FEYNMAN PROPAGATOR 39

Figure 2.1: Path chosen for Scalar Feynman Propagator

To obtain the result (2.70), we had to nd the semicircle that we will choose to close iθ our integral. To answer this question, one should write p0 = p0e , hence substituting in the exponential

−ip0(x0−y0) −ip0z0 −ip0z0 cos θ ip0z0 sin θ e =e = e e → 0 for p0 → ∞ ( sin (θ) < 0 for x0 > y0 lower half eip0 sin θ → 0 ⇒ sin(θ) > 0 for x0 < y0 upper half Finally, the Feynamn Propagator can be written in the 4-momentum form

Z 4 −ip·(x−y) d p e (2.71) ∆F (x − y) = 4 2 2 Γ() (2π) p − m In addition, one can nd the same result, by shifting the poles as shown in the picture, thus getting

Z d4p e−ip·(x−y) ∆ (x − y) = (2.72) F (2π)4 p2 − m2 + i From (2.72) one can show (avoiding the singular- ities), that the Feynman Propagator is the Green function of the Klein-Gordon equation. To be more precise

Z d4p (∂2 − ∇2 + m2)∆ (x − y) = − e−ip·(x−y) t F (2π)4 = − δ4(x − y) Moreover, one can see the meaning of the Feynman propagator, that even from its denition expresses Figure 2.2: Shifting the poles for the probability amplitude describing a process by Scalar Feynman Propagator which a particle starts at point 1 with space-time co-

ordinates x1 and propagates to point 2 with space- time coordinates x2. One can address this procedure through the relation

ˆ ˆ† i∆F (x2 − x1)Θ(t2 − t1) = h0|φ(~x2, t2)φ (~x1, t1)|0i Θ(t2 − t1) ˆ However, one can also claim that an antiparticle at 2 (described by b~p) was created and moved to 1 where it was annihilated, as presented by

ˆ† ˆ i∆F (x2 − x1)Θ(t1 − t2) = h0|φ (~x1, t1)φ(~x2, t2)|0i Θ(t1 − t2) Both processes are included in the Feynman Propagator. Chapter 3

The Dirac Field

3.1 The Dirac Equation

Having treated the quantization of the Klein-Gordon eld, we will now study particles with spin- 1 , which are described by the Dirac Equation. The Dirac equation for massive 2 spin- 1 particles reads 2 ∂φ i = ~a · ∇ + m2β ψ (3.1) ∂t 1 2 3 where ~a = {a , a , a } = −{a1, a2, a3}. However, in order for (3.1) to give the correct energy-momentum relation for the relativistic free particle E2 = ~p2 + m2, then it must satisfy the Klein-Gordon equation (2.5). Substituting (3.1) in (2.5) we nd that the matrices a, β must obey the relations

aiaj + ajai =2δij (3.2a)

aiβ + βai =0 (3.2b) 2 2 (3.2c) ai = β = I thus establishing an algebra for the ψ matrices. In order to ensure the Hermiticity of the Dirac Hamiltonian in (3.1), then † , † ai = ai β = β In addition, due to (3.2c) one can notice that the only possible eigenvalues of a, β are ±1. Furthermore, from the anticommutation relations follow that the trace of each matrix

has to be zero (Tr[ai] = Tr[β] = 0). Combining those two facts one can deduce that each matrix ai, β must possess as many positive as negative eigenvalues, thus has to be of even dimension. The smallest dimension for which the requirements hold true is N = 4. In the standard matrix representation those two matrices have the form 0 σi I2 0 ai = , β = , i = 1, 2, 3 (3.3) σi 0 0 −I2 where σi are the Pauli matrices. Therefore, our wavefunction will be described by a four-component vector called Dirac Spinor and symbolized as   ψ1  ψ2  ψa(x) =   , a = 1, 2, 3, 4 (3.4)  ψ3  ψ4

40 3.2. COVARIANCE OF THE DIRAC EQUATION 41 3.2 Covariance of the Dirac Equation

A proper relativistic theory has to be Lorentz covariant, thus its form has to be invariant under a transition from one inertial system to another, denoted as

0 ν ν µ (x ) = Λ µx where the orthogonality relations for Lorentz transformations given by

µ σ σ Λ νΛµ = δν hold true. From the relation stated above, one can nd that

µ det(Λ ν) = ±1 Thus, one can dene proper Lorentz transformations as those that the determinant of the transformation coecients equals to

µ det(Λ ν) = +1 These proper Lorentz transformations can be obtained by an innite number of successive innitesimal Lorentz transformations. On the other hand, improper Lorentz transformations contain a reection either in space or time and cannot be obtained through successive application of innitesimal transformations. Their determinant equals to

µ det(Λ ν) = −1 In the following considerations and throughout this thesis, it is much more convinient to denote the Dirac equation in its four-dimensional notation in order to highlight the symmetry between time and space coordinates. Therefore, after some trivial calculations (3.1) takes the elegant form µ (iγ ∂µ − m) ψ(x) = 0 (3.5) with the denitions

γ0 = β , γi = βai , i = 1, 2, 3

Using this notation, one can nd a unied formulation regarding the anticommutation relations that the gamma-matrices satisfy

µ ν µ ν ν µ µν (3.6) {γ , γ } = γ γ + γ γ = 2g I4

that consist the Dirac algebra. In addition, one can easily prove that γi matrices are anti-Hermitian and Unitary, i.e

γi−1 = γi† , γi† = −γi , i = 1, 2, 3 while γ0 is Hermitian and Unitary

γ0−1 = γ0† , γ0† = γ0 42 CHAPTER 3. THE DIRAC FIELD

In the standard matrix representation , the γµ matrices can be written down explicitly as follows i i 0 σ , 0 I2 0 , γ = i γ = i = 1, 2, 3 −σ 0 0 −I2 Finally introducing the Feynman dagger notation one can set that

µ ∂/ = γ ∂µ so the Dirac equation reads (i∂/ − m)ψ(x) = 0 (3.7) Next, we will construct the Lorentz transformation that the Dirac spinors ψ(x) and ψ0(x0) obey. It is required to be linear since both the Dirac equation as well as the Lorentz transformation are linear in space-time coordinates. Hence, we will assume the form

ψ0(x0) =S(Λ)ψ(x) =S(Λ)ψ(Λ−1x0) where µ denotes the matrix of the Lorentz transformation. is a matrix Λ = Λ ν S(Λ) 4 × 4 which is a factor of the parameters of the Lorentz transformation and operates upon the four components of the Dirac spinor. Given the invariance of phyical laws for all inertial systems, then an inverse operator S−1(Λ) must exist that enables the observer A (ψ(x)) to construct his Dirac spinor from observer's B (ψ0(x0)) Dirac spinor. Consequently, ψ(x) =S−1(Λ)ψ0(x0) =S−1(Λ)ψ0(Λx) Our goal is to construct the Lorentz transformation matrix (S) for the Dirac spinors. Starting from the Dirac equation of observer A

µ (iγ ∂µ − m) ψ(x) = 0 and expressing the Dirac spinor by means of observer's B viewpoint we nd

µ −1 −1 0 0 iγ S (Λ)∂µ − mS (Λ) ψ (x ) = 0 where using the unitary property of the Lorentz transformation matrix −1 S(Λ)S (Λ) = I4 we obtain

µ −1 0 0 iS(Λ)γ S (Λ)∂µ − m ψ (x ) = 0 Next, recalling the vector transformation of the 4-divergence under proper Lorentz transformations ν 0 , the previously stated equation turns our to be equal to ∂µ = Λ µ∂ν µ −1 ν 0 0 0 i S(Λ)γ S (Λ)Λ µ ∂ν − m ψ (x ) = 0 This equation has to be identical with the Dirac equation, since the equations of motion must be invariant under Lorentz transformations, thus ν −1 ν µ (3.8) S(Λ)γ S (Λ) = Λµ γ which is the equation that determines the S-matrix. Therefore, we have proven that (3.8) holds true for proper and orthocronous ( 0 ) and improper (discrete) Lorentz Λ 0 > 0 transformations given that there is no dependence on the determinant. Hence, once we have shown the existence of a solution S(Λ) for (3.8), we will have proven the covariance of the Dirac equation. 3.3. CLASSICAL DIRAC FIELD THEORY 43 3.3 Classical Dirac Field Theory

As I have already stated, the covariant relativistic notation for the Dirac equation of spin- 1 massive particles, reads 2

µ (iγ ∂µ − m)ψ(x) = 0

where ψ(x) = ψ(~x,t) is the classical eld of the Dirac wave function and the γ-matrices satisfy the Dirac algebra. The Lagrangian density leading to the Dirac equation has the bilinear form

¯ µ L(x) =ψ(iγ ∂µ − m)ψ (3.9a) =iψ†ψ˙ + iψ†~a · ∇ψ − mψ†βψ (3.9b)

where 0 , 2 , 0 and ¯ † 0. We will treat the Dirac spinors β = γ β = I4 ~a = γ ~γ ψ = ψ γ ψ and ψ† as independent elds, each having four components. Consequently, inserting the Lagrangian density in the equation of motion and dierentiating in respect to each independend eld, we get

µ (iγ ∂µ − m)ψ =0 (3.10a) ¯ µ←− ψ(iγ ∂ µ + m) =0 (3.10b) where the arrow indicates that the partial derivative acts on the function to the left. One can notice that variation of the Lagrangian density in respect to ψ¯ led to the known Dirac equation (3.10a) and variation of the Lagrangian density in respect to ψ led to (3.10b) which is the Hermitian conjugate of (3.10a). The canonically conjugate elds are

∂L † πψ = = iψ (3.11a) ∂ψ˙ ∂L πψ† = = 0 (3.11b) ∂ψ˙ † Furthermore, the Hamiltonian density equals to

˙ ˙ † H(x) =πψ(x)ψ(x) + πψ† (x)ψ (x) − L(x) =ψ†(−i~a · ∇ + βm)ψ (3.12)

The Hamiltonian is given by Z H(t) = d3xH(x) Z = d3x ψ†(x)(−i~a · ∇ + βm) ψ(x) (3.13)

Finally, it is easy to prove that under global phase transformations

ψ0 =ψeia ' (1 + ia)ψ ψ†0 =ψ†e−ia ' (1 − ia)ψ† 44 CHAPTER 3. THE DIRAC FIELD

the Dirac Lagrangian density is invariant, i.e

L0(x) = L(x) (3.14)

Therefore, the current density vector jµ(x) equals to ¯ jµ(x) = eψγµψ (3.15) and the resulting conserved quantity (as shown in the complex Klein-Gordon eld) corresponding to the total charge equals to Z Q = e d3x ψ†ψ (3.16)

where we have inserted the electric charge unit in order to coincide (3.15) with the electrical current density of the Dirac theory.

3.4 Fermionic Fock Space

It is well known that fermions are particles found in nature that obey the Pauli exclusion principle and have antisymmetric wave functions. It is possible to change the formulation in order to apply on fermions too. One can achieve that by replacing the commutation relations among the eld operators by anticommutation relations. Thus

n o ψˆ(~x,t), ψˆ†(~x0, t) =δ(~x − ~x0) (3.17a) n o n o ψˆ(~x,t), ψˆ(~x0, t) = ψˆ†(~x,t), ψˆ†(~x0, t) = 0 (3.17b)

In order to study the consequences that the anticommutation relations have, and construct the Fock space (as we did for bosons), we expand the eld operators in respect to a set of known basis. Then, the time dependent coecients will satisfy the anticommutation relations n o † (3.18a) aî, aˆj =δij n o † † (3.18b) {aî, aˆj} = aî , aˆj = 0

However, this time we can nd that the square of the operators stated above equals to 2 zero 2 † . As a result, the particle number operator is restricted as follows aˆi = aˆi = 0

2 † † (ˆni) =âi aîaî aî † =âi aî

=ˆni

One can nd acting the operators 2 on Fock states for their eigenvalues ( ) nˆi, nˆi ni

2 ni = ni ⇒ ni = 0, 1 3.5. CANONICAL QUANTIZATION OF THE DIRAC FIELD 45

i.e we are dealing with Fermi-Dirac statistics. In addition, given that the relations

[ˆni, aî] = − aî (3.19a) h i † † (3.19b) nî, aî =âi -where we have used that [AˆB,ˆ Cˆ] = Aˆ{B,ˆ Cˆ}−{A,ˆ Cˆ}Bˆ-hold true, then one can interpret the operators a,ˆ aˆ† as annihilation and creation operators. Furthermore, we can easily construct the Fock space for fermions as follows

n1 n2 † † (3.20) |n1, n2, n3,...i = aˆ1 aˆ2 · · · |0i where n1, n2,... are either 0 or 1. The state vector in which one particle is in the state i is † |1ii =a î |0i For the two-particle states when i 6= j we have † † † † |1i, 1ji =a î aˆj |0i = −aˆjaî |0i = − |1j, 1ri thus, ensuring the antisymmetric character of the state under particle interchange, whereas for i = j 2 † |2ii = aî |0i = 0 where we have proved that those particles must obey the Pauli exclusion principle.

3.5 Canonical Quantization of the Dirac Field

As we already know, quantization of the Dirac eld can be achieved by replacing the Dirac spinors ψ(~x,t), ψ†(~x,t) with the operators ψˆ(~x,t), ψˆ(~x,t). Given that we are dealing with the Dirac equation that describes electrons, therefore we will choose to use the anticommutation relations for the eld operators n o ˆ ˆ† 0 0 (3.21a) ψa(~x,t), ψβ(~x , t) =δaβδ(~x − ~x ) n o n o ˆ ˆ 0 ˆ† ˆ† 0 (3.21b) ψa(~x,t), ψβ(~x , t) = ψa(~x,t), ψβ(~x , t) = 0 Since we are in the Heisenberg picture, then the equation of motion for the eld operator reads ˙ h i ψˆ(~x,t) = − i ψˆ(~x,t),H(x0) Z h i = − d3x0 ψˆ(~x,t), ψˆ†(~x0, t)~a · ∇0ψˆ(~x0, t) + imψˆ†(~x0, t)βψˆ(~x0, t) Z h i = − d3x0 δ(~x − ~x0)~a · ∇0ψˆ(~x0, t) + imδ(~x − ~x0)βψˆ(~x0, t)

= − (~a · ∇ + imβ) ψˆ(~x,t) where we have used the identity [A,ˆ BˆCˆ] = {A,ˆ Bˆ}Cˆ − Bˆ{A,ˆ Cˆ}. Multiplying by γ0 we obtain (i∂/ − m)ψˆ(~x,t) = 0 which is the free Dirac equation. Therefore, the quantized eld operator satises the free Dirac equation. 46 CHAPTER 3. THE DIRAC FIELD 3.6 Plane Wave Expansion of the Field Operator

We will expand the eld operator ψˆ(~x,t) in a complete set of classical wave functions. Consequently, we will use the free plane wave solution of the Dirac equation

r m (r) −3/2 −ir(ωpt−~p·~x) (3.22) Ψ~p = (2π) wr(~p)e ωp

where wr(~p) is the Dirac unit spinors. The index r goes from 1 to 4 expressing four independent solutions. For r = 1, 2 we have chosen the solutions with positive energy p 2 2 r = 1 → E+ = ωp = ~p + m , whereas for r = 3, 4 we have chosen the negative energy p 2 2 solutions r = −1 → E− = −ωp = − ~p + m . In the standard matrix representation the plane waves satisfy the Dirac equation

(r) (i∂/ − m)Ψ~p = 0 Inserting the plane wave expansion in the Dirac equation we nd that the Dirac unit spinors satisfy the equation

(p/ − rm)wr(~p) = 0 (3.23) In addition, the Dirac unit spinors possess the orthogonality and completeness properties, necessary in order to ensure the proper normalization of the plane waves. More specically from the orthogonality relation

† 0 ωp w 0 ( ~p)w ( ~p) = δ 0 r r r r m rr one can prove the orthogonality of the plane waves s Z 2 Z †(r0) (r) 1 m 0 3 i(r0 ω~p0 −rωp)t † 0 3 −i(r0 ~p −r ~p)·~x d x Ψ~p0 (x)Ψ~p (x) = 3 e wr0 (~p )wr(~p) d x e (2π) ωpωp0 (2π)3 i(r0 ωp0 −rωp)t † 0 0 =√ e wr0 (~p )wr(~p)δ(r0 ~p − r~p) ωpωp0 0 =δ(~p − ~p )δrr0 (3.24) Therefore, using the plane wave Dirac solutions as a basis for the Dirac eld operator ψˆ(~x,t) and its hermitian conjugate ψ†(~x,t), we nd

4 X Z d3p r m ψˆ(~x,t) = aˆ (~p)ω (~p)e−irp·x (3.25a) (2π)3/2 ω r r r=1 p 4 X Z d3p r m ψˆ†(~x,t) = aˆ†(~p)w†(~p)eirp·x (3.25b) (2π)3/2 ω r r r=1 p

Since, (3.24) holds, then the operators † satisfy the anticommutator relations aˆr(~p), aˆr(~p) n o † 0 0 (3.26a) aˆr(~p), aˆr0 (~p ) = δ(~p − ~p )δrr0 n o 0 † † (3.26b) {aˆr(~p), aˆr0 (~p )} = aˆr(~p), aˆr0 (~p) = 0 3.6. PLANE WAVE EXPANSION OF THE FIELD OPERATOR 47

Next, from (3.13) we can nd the Hamilton operator as follows

Z Hˆ = d3x ψˆ†(~x,t)(−i~a · ∇ + βm) ψˆ(~x,t) Z Z Z X 3 3 0 † 0 3 †(r0) (r) = d p d p aˆr0 (~p )ˆar(~p) d x Ψ~p0 (−i~a · ∇ + βm)Ψ~p r,r0 Z Z Z X 3 3 0 † 0 3 †(r0) (r) = d p d p aˆr0 (~p )rωpaˆr(~p) d x Ψ~p0 Ψ~p r,r0 4 Z X 3 † = d p rωpaˆr(~p)ˆar(~p) r=1 where in order to obtain the previous equation, we have used the relation

(r) (r) (r) (−i~a · ∇ + βm)Ψ~p (x) = i∂0Ψ~p (x) = rωpΨ~p (x) that follows from the Dirac equation. Separating the positive and negative energies in the Hamilton operator we have

Z " 2 4 # ˆ 3 X † X † (3.27) H = d p ωpaˆr(~p)ˆar(~p) − ωpaˆr(~p)ˆar(~p) r=1 r=3 One can notice that the Hamiltonian (3.27) in not bounded below, meaning that the total energy can drop beyond any bound, hence obtaining lower and lower states without any limitation which is unacceptable. Dirac introducing the Dirac's hole picture avoided that problem, postulating that all the negative energy states are lled, thus only the positive ones are accessible. The lled negative energy states are referred to as the Dirac Sea. According to this concept, in the vacuum state all the levels of negative energy are occupied. In addition, given that we measure only energy (and charge) dierences, one can subtract their energy in the Hamiltonian as follows. First of all, the non observable and divergent vacuum state is given by

Z 4 3 X E0 = − d p ωp r=3 Next, we can avoid this inconsistency by redening our Hamiltonian

ˆ 0 ˆ H =H − E0 (3.28) Z " 2 4 # 3 X † X † = d p ωp aˆr(~p)âr(~p) + aˆr(~p)âr(~p) r=1 r=3 Z " 2 4 # 3 X X = d p ωp nˆr(~p) + n¯ˆr(~p) (3.29) r=1 r=3 where † is the particle ( ) number operator related to the state nˆr(~p) =a ˆr(~p)âr(~p) r = 1, 2 (r) and ˆ † is the number operator of the holes ( ) related Ψ~p (x) n¯r(~p) =a ˆr(~p)âr(~p) r = 3, 4 to the state (r) . Now, (3.29) expresses a well dened positive valued Hamiltonian. Ψ~p (x) 48 CHAPTER 3. THE DIRAC FIELD

Interpreting the holes as antiparticles one can dene the vacuum state as one where neither particles nor antiparticles exist

nˆr(~p) |0i = 0 → aˆr(~p) |0i = 0 for r = 1, 2 † for n¯ˆr(~p) |0i = 0 → aˆr(~p) |0i = 0 r = 3, 4

Consequently, the operator aˆr(~p) will be called annihilation operator for particles (r = ) and creation operator for antiparticles ( ), while the operator † will be 1, 2 r = 3, 4 aˆr(~p) called annihilation operator for antiparticles (r = 3, 4) and creation operator for particles (r = 1, 2). In order to simplify our results it is convenient to introduce separate operators for the particles and antiparticles. Hence, we will seek plane wave solutions of the free Dirac equation of the form

ψ(+)(x) = Nu(~p)e−ip·x positive energy ψ(−)(x) = Nv(~p)eip·x negative energy

Inserting them in the Dirac equation, we nd that they must satisfy the relations

(p/ − m)u(~p) = 0 (p/ + m)v(~p) = 0

In the rest frame pµ = (m,~0) we nd that there are two linearly independent u and v solutions that read

 1   0   0   0  ~  0  ~  1  ~  0  ~  0  u1(m, 0) =   , u2(m, 0) =   , u3(m, 0) =   , u4(m, 0) =    0   0   1   0  0 0 0 1

However, for each energy (positive or negative) solution there are two possible spin eigenvalues, thus we can classify even more each state by introducing another operator. One can prove this by introducing the Helicity operator

 1 0 0 0  ˆ ˆ ~ ~ ~p ˆ ~ ~  0 −1 0 0  Λs = Ω · = Sz = Ωz =   2 |~p| 2 2  0 0 1 0  0 0 0 −1 which is the projection of the spin onto the direction of momentum. It is relatively easy (in the rest frame of the particle) to notice that it has eigenvalues ±~/2. We will use the names of us(~p) and vs(~p) for the unit Dirac spinors corresponding to the positive and negative energy solutions respectively. They are related with wr(~p) as

w1(~p) =us(~p)

w2(~p) =u−s(~p)

w3(~p) =u−s(~p)

w4(~p) =us(~p) 3.6. PLANE WAVE EXPANSION OF THE FIELD OPERATOR 49

Next, we will introduce the following new operators

ˆ bs(~p) =a ˆ1(~p) ˆ b−s(~p) =a ˆ2(~p) ˆ† d−s(~p) =a ˆ3(~p) ˆ† ds(~p) =a ˆ4(~p) that they satisfy the same anticommutation relations that read

nˆ ˆ† 0 o 0 bs(~p), bs0 (~p ) = δ(~p − ~p )δss0 n ˆ ˆ† 0 o 0 ds(~p), ds0 (~p ) = δ(~p − ~p )δss0 while the other anticommutators vanish. Inserting that notation in the eld and hermitian conjugate, they take the form

X Z d3p r m h i ψˆ(~x,t) = ˆb (p)u (p)e−ip·x + d†v (p)eip·x (2π)3/2 ω s s s s s p X Z d3p r m h i ψˆ†(~x,t) = ˆb†(p)u†(p)eip·x + d v†(p)e−ip·x (2π)3/2 ω s s s s s p After some trivial calculations one can nd that the xed Hamiltonian in the new notation equals to

Z h i ˆ 3 ˆ† ˆ ˆ† ˆ H = d p ωp bs(p)bs(p) + ds(p)ds(p) Z 3 = d p ωp nˆs(p) + n¯ˆs(p) (3.32)

where in order to compute that result we have used the orthogonality relations that u and v must obey, that read

† † ωp u 0 (p)u (p) = v 0 (p)v (p) = δ 0 s s s s m ss † † us0 (−p)vs(p) = vs0 (−p)us(p) = 0 One can notice that

b† : creation operator of a particle b : annihilation operator of a particle d† : creation operator of an antiparticle d : annihilation operator of an antiparticle Using these operators, the Fock space can be constructed starting from the vacuum state |0i dened as ˆ ˆ bs(p) |0i = 0 , ds(p) |0i = 0 through repeated application of the creation operators. 50 CHAPTER 3. THE DIRAC FIELD

Furthermore, the charge operator equals to Z Qˆ =e d3x ψˆ†(x)ψˆ(x) e Z h i = d3x ψˆ†(x)ψˆ(x) − ψˆ(x)ψˆ†(x) 2 Z X 3 hˆ† ˆ ˆ† ˆ i =e d p bs(p)bs(p) − ds(p)ds(p) s Z X 3 † (3.33) =e d p nˆs(p) − n¯ˆs(p) s Therefore, particles carry +e charge, whereas antiparticles carry −e charge.

3.7 The Dirac Field Feynman Propagator

Like in the Klein-Gordon eld, the Feynman propagator for the Dirac Field can be dened as the vacuum expectation value of the time-ordered product of the eld operators

h ˆ ¯ˆ i iSF aβ(x − y) = h0|T ψa(x)ψβ(y) |0i (3.34)

where a, β = 1, 2, 3, 4 are Dirac indices. However, given that we are dealing with fermions the time ordered product of the eld operators is given by

h ˆ ¯ˆ i ˆ ¯ˆ ¯ˆ ˆ T ψa(x)ψβ(y) = Θ(x0 − y0)ψa(x)ψβ(y) − Θ(y0 − x0)ψβ(y)ψa(x) ( +1 for t1 > t2 where Θ(t1 − t2) = is the step function. 0 for t1 < t2 Therefore, inserting the expanded time ordered product in the Feynman propagator (3.34) we have ˆ ¯ˆ ¯ˆ ˆ iSF aβ(x − y) = Θ(x0 − y0) h0|ψa(x)ψβ(y)|0i − Θ(y0 − x0) h0|ψβ(y)ψa(x)|0i (3.35) One can notice that in the Dirac eld case, the time ordered product has a minus sign, which is a result of the anticommutation relations that fermions should obey (a + sign would violate microcausality). Next, separating the eld operators in positive and negative frequency parts we have

X Z d3p r m X Z d3p r m ψˆ(+)(~x,t) = ˆb (p)u (p)e−ip·x , ψˆ(−)(~x,t) = dˆ†(p)v (p)eip·x (2π)3/2 ω s s (2π)3/2 ω s s s p s p X Z d3p r m X Z d3p r m ψ¯ˆ(+)(~x,t) = ˆb†(p)¯u (p)eip·x , ψ¯ˆ(−)(~x,t) = dˆ†(p)¯v (p)eip·x (2π)3/2 ω s s (2π)3/2 ω s s s p s p and

ψˆ(+)(~x,t) |0i = ψ¯ˆ(+)(~x,t) |0i = 0 h0| ψˆ(−)(~x,t) = h0| ψ¯ˆ(−)(~x,t) = 0 3.7. THE DIRAC FIELD FEYNMAN PROPAGATOR 51

Inserting the plane wave expansion in the eld operators and taking into account the relations stated above, we have

i) For x0 > y0 we nd

(−) ¯ˆ ˆ iSF aβ(x − y) = − h0|ψβ(y)ψa(x)|0i ¯ˆ(+) ˆ(−) = − h0|ψβ (y)ψa (x)|0i n ¯ˆ(+) ˆ(−) o = − h0| ψβ (y), ψa (x) |0i Z d3p m X = − eip·(x−y) v¯β(p)va(p) (2π)3/2 ω s s p s Z 3 ip·(x−y) d p e (3.37) = − 3 (p/ − m)aβ (2π) 2ωp In order to obtain the last results we have used the completeness relations that hold for the unit Dirac spinors and read

X p/ + m ua(p)¯uβ(p) = s s 2m s aβ X p/ − m va(p)¯vβ(p) = s s 2m s aβ Substituting (3.36) and (3.37) in (3.34) we nd

(+) (−) iSF aβ(x − y) =Θ(x0 − y0)iSF aβ(x − y) − Θ(y0 − x0)iSF aβ(x − y) Z 3 d p 1 −ip·(c−y) ip·(x−y) = 3 Θ(x0 − y0)e (p/ + m)aβ − Θ(y0 − x0)e (p/ − m)aβ (2π) 2ωp Z d3p e−ip·(x−y) Z d3p eip·(x−y) / =(i∂x + m)aβ Θ(x0 − y0) 3 + Θ(y0 − x0) 3 (2π) 2ωp (2π) 2ωp (+) (−) =(i∂/x + m)aβ Θ(x0 − y0)i∆ (x − y) + Θ(y0 − x0)i∆ (x − y)

=(i∂/x + m)aβ∆F (x − y) (3.38) where we have found the Feynman propagator for the Dirac eld in respect to the Feyn- man propagator of the scalar complex Klein-Gordon eld given by (2.68). However, in 52 CHAPTER 3. THE DIRAC FIELD

order to acquire the result found above we have used the relations

−ip·(x−y) −ip·(x−y) (i∂/x + m)aβe =(p/ + m)aβe ip·(x−y) ip·(x−y) −(i∂/x + m)aβe =(p/ − m)aβe which is a pretty straightforward calculation. In addition, during our calculations we have assumed that the slashed divergence does not aect the step function, hence we have arbitrarily used the term (i∂/x + m)aβ as a common factor. This is the case because the term that appears when the slashed divergence acts on the step function Θ(x0 − y0) is eliminated BY the term that appears from the other step function Θ(y0 − x0) when the slashed divergence acts upon it. We can easily prove it as follows

(+) 0 (+) 0 (+) i∂/xΘ(x0 − y0)∆ (x − y) = iγ ∂x0 Θ(x0 − y0)∆ (x − y) = iγ δ(x0 − y0)∆ (x − y) (−) 0 (−) 0 (−) i∂/xΘ(y0 − x0)∆ (x − y) = iγ ∂x0 Θ(y0 − x0)∆ (x − y) = −iγ δ(x0 − y0)∆ (x − y) Combining those two equations we have

(+) (−) i∂/xΘ(x0 − y0)∆ (x − y) + i∂/xΘ(y0 − x0)∆ (x − y) = 0 (+) (−) =iγ δ(x0 − y0) ∆ (x − y) − ∆ (x − y) 0 =iγ δ(x0 − y0)∆(x − y) =0

where ∆(x − y) is the Dirac-Jordan function, which as we have proven satises the microcausality, thus the quantity vanishes for equal times. Finally, substituting the scalar Feynman Propagator given by (2.72) we nd that the Feynman Propagator for the Dirac Field equals to

4 Z d p (p/ + m)aβ S (x − y) = e−ip·(x−y) (3.39) F aβ (2π)4 p2 − m2 + i Consequently, given that the Feynman Propagator in momentum space consists Fourier pairs with the space-time Feynman Propagator , then it is given by

1 SF aβ(p) = (3.40) p/ − m + i Chapter 4

The Electromagnetic Field

In eld theory, particles of spin-1 are described as the quanta of vector elds. Such vector bosons play a crucial role as the mediators in particle physics. In the following section we will discuss the wave equations of the massless spin-1 eld, i.e the Maxwell Equations.

4.1 The Maxwell Equations

The Electromagnetic phenomena in the absence of any source can be described through the celebrated Maxwell Equations

∇ · E~ = 0 (4.1a) ∇ · B~ = 0 (4.1b) ∂E~ ∇ × B~ = (4.1c) ∂t ∂B~ ∇ × E~ = − (4.1d) ∂t

where E~ (~x,t) = (E1,E2,E3), B~ (~x,t) = (B1,B2,B3) are the three dimensional vector elds called the electric and magnetic eld strength. It is well known that the set of Maxwell's Equations can be conned into a covariant form by introducing the antisymmetric tensor of rank 2, called the Faraday's eld strength tensor F µν,

 0 −E1 −E2 −E3  1 3 2 µν  E 0 −B B  F =   (4.2)  E2 B3 0 −B1  E3 −B2 −B1 −E3

Using Faraday's tensor, we can write the Maxwell's equations in the compact form of

µν ν ∂µF = j (4.3a) ∂λF µν + ∂νF λµ + ∂µF νλ = 0 (4.3b) where the 4-current is given by jν = (ρ,~j) = (0,~0) in the absence of any sources. Ex- panding the 4-dimensional notation,(4.3a) leads to (4.1a) and (4.1b), while expanding the Bianchi identity given by (4.3b) leads to (4.1c) and (4.1d), thus obtaining the full set

53 54 CHAPTER 4. THE ELECTROMAGNETIC FIELD

of Maxwell's Equations. In addition taking the four-divergence of (4.3a) one can reach the continuity equation that reads

ν ∂νj = 0 (4.4) In the Framework of quantum theory, for the fundamental dynamical variable, the vector potential Aµ(x) = (A0(x), A~(x)) is employed. The relation connecting it with the vector elds strength is

B~ =∇ × A~ (4.5a) ∂A~ E~ = − − ∇A0 (4.5b) ∂t and the Faraday tensor in respect to the vector potential is given by

F µν = ∂µAν − ∂νAµ (4.6) We can conrm that the Faraday tensor dened in (4.6) satises the Bianchi identity (4.3b). However, introducing the vector potential we have decreased the number of the conned Maxwell Equations to 1, as one can notice by inserting the Faraday tensor dened above in (4.3a), hence obtaining the second order wave equation

ν ν µ (4.7) A − ∂µ(∂ A ) = 0 Separating the space and time coordinates we have the following set of equations

~ 0 ~ ~ (4.8a) A − ∇(∂0A + ∇ · A) =0 2 0 ~ −∇ A − ∂0(∇ · A) =0 (4.8b) It is known that the vector potential Aµ(x) is not an observable quantity and not uniquely determined. To be more precise, a local gauge transformation that reads

A0µ(x) = Aµ(x) + ∂µΛ(x) (4.9) i.e adding a four-divergence of an arbitrary scalar function Λ(~x,t) in our vector potential, does not modify the Faraday eld tensor

0 Fµν = Fµν As a result, we get the same Maxwell equations. In order to calculate the quantity of interest one restricts the vector potential to obey a particular condition, i.e xes the gauge. However, regardless the gauge we are working on, the observable quantities have to be gauge invariant. Several gauges are being used, having dierent names. We are going to deal with

µ Lorenz Gauge : ∂µA = 0 Coulomb Gauge : ∇ · A~ = 0 One can naively assume given its 4-dimensional denition, that the vector potential has four independent degrees of freedom. However, this is not the case. Recalling (4.8b)

0 =∇ · E~ 2 0 ~ = − ∇ A − ∂0(∇ · A) 4.1. THE MAXWELL EQUATIONS 55

we know that this dierential equation has the solution

Z d3x0 ∂ (∇ · A~(x)) A0(x) = 0 (4.10) 4π |~x − ~x0| where we have used that 2 1 . In conclusion, we have expressed the time ∇ |~x| = −4πδ(~x) (zeroth) component of the vector potential in terms of its spatial components. Therefore, A0(x) is not independent and the vector potential has only 3 degrees of freedom and it is not a dynamical variable. However, given that our theory describes photons that have 2 degrees of freedom (their polarization states), a further constraint must be imposed (gauge xing).

Lorenz Gauge. The condition for the Lorenz gauge is the only Lorentz covariant condition. Any given vector potential can be made to satisfy the Lorenz condition if (4.9) is chosen in such a way that Λ(x) satises the equation µ Λ(x) = −∂µA (x) 0µ One can easily prove that by inserting this equation in (4.9), hence nding ∂µA = 0 which is the Lorenz condition. When Aµ(x) satises the Lorenz condition, we say that the potential is in Lorenz gauge. However, Aµ(x) is not completely determined. In fact, we might perform another gauge transformation

0 0 Aµ(x) → Aµ(x) = Aµ(x) + ∂µΛ (x) with

0 Λ (x) = 0 then the new eld 0 (dierent from the rst one) also satises the Lorenz gauge Aµ(x) 0µ condition ∂µA (x) = 0. In this case, the equations of motion reduce to the wave equation ν A (x) = 0 which is the Klein-Gordon equation for massless particles. But, since the Lorenz gauge is relativistically invariant, the theory formulated in this gauge is explicitly covariant, however it does not uniquely determine the vector potential.

Coulomb Gauge. This gauge requires that the 3-dimensional divergence vanishes, i.e the vector potential is spatially transverse, with polarization vectors orthogonal to the direction of propagation. Recalling (4.10) in the Coulomb gauge our rst constraint takes the form of A0(x) = 0. Consequently, in this gauge there are two explicit constraints A0(x) =0 ∇ · A~(x) =0 making the Coulomb gauge ideal to describe the photon eld with its 2 degrees of freedom. However, there are several disadvantages that follow from its denition. First of all, the Coulomb gauge is not Lorentz covariant. Secondly, just like the Lorenz gauge, the Coulomb gauge does not uniquely determine the vector potential. One can prove this by noticing that the Coulomb gauge holds true if we replace A~(x) → A~0(x) = ∇ × A~(x) ⇒ ∇ · A~0(x) = 0 56 CHAPTER 4. THE ELECTROMAGNETIC FIELD 4.2 Plane Wave Expansion of the Electromagnetic Field

In this section we are going to expand the vector eld Aµ(x) in terms of plane waves and dene the 4-dimensional polarization states. To begin with, expanding the vector eld one acquires

~ µ µ ~ −(ωkt−k·~x) (4.11) A (x) = Nkλ(k)e p where ~ 2 2. Here µ ~ with are the 4-dimensional polarization ωk = k + m λ(k) λ = 0, 1, 2, 3 vectors. One observes that there are 4 linearly independent µ ~ , 3 space-like and one λ(k) time-like. In addition, we pick the normalization ~ µ ~ (4.12) µλ(k)λ(k) = gλλ0

0 where for the time-like polarization states λ = λ = 0 then gλλ0 = +1, whereas for the 0 space-like polarization states λ = λ = 1, 2, 3, gλλ0 = −1. In the massless case, in order to construct the polarization vectors, we need to choose a convenient Lorentz frame of reference. Next, out of the three space-like polarization states, we pick two of them λ = 1, 2 to lie transverse to the momentum k, i.e µ ~ ~ (4.13a) 1 (k) = 0,~1(k) µ ~ ~ (4.13b) 2 (k) = 0,~2(k) with µ ~ µ ~ (4.14) kµ1 (k) = kµ2 (k) = 0 thus, ensuring their transverse property. However, in order to dene the µ ~ that should 3 (k) be perpendicular to the other two states hence expressing the longitudinal component in the 4-D space-time, one needs to choose a specic frame of reference. For simplicity in our calculations we will choose the reference dened from the time-like unit vector

µ ~ µ (4.15) 0 (k) = n = (1, 0, 0, 0) In this frame of reference, the longitudinal polarization has the components

~ ! µ ~ k (4.16) 3 (k) = 0, |~k| where µ ~ µ ~ (4.17) kµ0 (k) = −kµ3 (k) = k · n Therefore, for the transverse and longitudinal polarizations, we can conrm that they are orthogonal, given that they satisfy the relation ~ ~ ~r(k) · ~s(k) = 0 for r, s = 1, 2, 3 (4.18) In addition, we can write the longitudinal polarization in the covariant form

kµ − (k · n)nµ µ(~k) = (4.19) 3 p(k · n)2 − k2 4.3. LAGRANGIAN DENSITY AND CONSERVED QUANTITIES 57

unifying the 4 components in one equation and ensuring the proper space-like norm 2 µ ~ . 3 (k) = −1 Finally, except from the orthogonality, the completeness relations between the polarization states for the massless case in the special Lorentz frame, now read

3 X µ ~ ν ~ µν (4.20) gλλλ(k)λ(k) = g λ=0 where gµν = diag{+1, −1, −1, −1} the covariant Minkowski metric tensor.

4.3 Lagrangian Density and Conserved Quantities

The Lagrangian density of the electromagnetic eld in the absence of any sources reads as follows 1 L(x) = − F F µν (4.21) 4 µν We can calculate that the canonical energy-momentum eld operator T µν µν ∂L ν µν T = ∂ Aσ(x) − g L(x) (4.22) ∂(∂µAσ(x)) using the identity

∂(F F aβ) aβ = 4F µσ ∂(∂µAσ(x)) equals to

1 T µν = gµνF F aβ + F σµ(∂νA (x)) (4.23) 4 aβ σ In addition, using (4.3a) one can conrm the conservation of energy and momentum as follows 1 ∂ T µν = gµν∂ (F F aβ) + ∂ [F σµ(∂νA (x))] µ 4 µ aβ µ σ 1 = ∂ν(F F aβ) + (∂ F σµ)(∂νA (x)) + ∂ ∂νA (x)F σµ 4 aβ µ σ µ σ 1 = (∂νF )F aβ + ∂ν∂ A (x)F σµ 2 aβ µ σ 1 = − ∂ν (∂ A (x) + ∂ A (x)) F aβ 2 a β β a =0

On the last step we have used the antisymmetric property of the Faraday tensor. It is useful to say that the existence of sources would have resulted in the violation of the conservation of energy . 58 CHAPTER 4. THE ELECTROMAGNETIC FIELD 4.4 Quantization of the Electromagnetic Field

As I have already stated, the vector eld Aµ(x) has two superuous degrees of freedom, which have to be eliminated. In the following, we will discuss the two important procedures of Covariant Quantization in order to ensure the Lorentz covariance and gauge invariance of our theory. Thus, we will quantize the electromagnetic eld in the Lorenz gauge. First of all, computing the conjugate eld, one nds

∂L 0 (4.24a) π (x) = 0 = 0 ∂(∂0A (x)) ∂L i 0i i (4.24b) π (x) = 0 = −F = E ∂(∂0A (x)) Next, computing the Hamiltonian one obtains Z 3 0 H(t) = d x πi(x)∂0A (x) − L(x) Z 1 = d3x −F ∂ Aµ(x) + F F µν 0µ 0 4 µν Z 1 = d3x E~ · (E~ + ∇A ) − E2 − B2 0 2 Z 1 = d3x E2 + B2 + ∇ · (EA~ ) − A (∇ · E~ ) 2 0 0 Z 1 = d3x E2 + B2 − A (∇ · E~ ) (4.25) 2 0 where the second spatial term (divergence) dropped out given its surface contribution. However, in order to ensure the demand of Lorentz covariance, the need of nding a non zero value for the conjugate eld π0(x) arises. To do so, we will redene our Lagrangian density adding an extra term, hence introducing the modied Lagrangian density L0(x)

1 1 L0(x) = − F F µν − ζ(∂ Aσ(x))2 (4.26) 4 µν 2 σ where ζ is a freely chosen parameter called gauge xing term. One can show that the ν modied Lagrangian satises the same equations of motion A (x) = 0 as the previously non-modied one, when implementing the Lorenz gauge in both cases (in classical eld theory). For the new modied Lagrangian density, the conjugate eld equals to

0 σ π (x) = − ζ(∂σA (x)) (4.27a) i 0i σ π (x) = − F (x) − ζ(∂σA (x)) (4.27b) where as one can see from (4.27a) the zeroth component of the conjugate eld is not zero, but dependent on the gauge xing term. One can show that for the new modied Lagrangian density, the Euler-Lagrange equations of motion takes the general form

ν ν µ (4.28) A (x) − (1 − ζ)∂ (∂µA (x)) = 0 4.5. CANONICAL QUANTIZATION OF THE LORENZ GAUGE 59

We can notice that in the special case where ζ = 1, the equations of motion take the same form as those that follow from the Lorenz condition, which is the massless version of the Klein-Gordon equation. Consequently, throughout this chapter we will choose for the gauge xing parameter the value ζ = 1. It is customary to denote this value as the Feynman gauge. Hence, in our case, the Lagrangian density takes the form

1 1 L (x) = − F µνF − (∂ Aσ(x))2 (4.29) F 4 µν 2 σ Expanding the Feynman gauge Lagrangian in its terms one can simplify it even more

1 1 L (x) = − (∂ A (x))(∂µAν(x)) + ∂ [A (x)(∂νAµ(x)) − (∂ Aν(x))Aµ(x)] F 2 µ ν 2 µ ν ν where the last term (four-divergence) does not contribute in the calculation of the equations of motions, so we can ignore it and dene a new simple Lagrangian density in the Feynman gauge that reads

1 L0 (x) = − (∂ A (x)) (∂µAν(x)) (4.30) F 2 µ ν The conjugate eld now takes the form

∂L0 (x) µ F 0 (4.31) π (x) = µ = −∂ Aµ(x) ∂(∂0A (x)) In addition, the new Hamiltonian density equals to

µ 0 HF (x) =π (x)∂0Aµ(x) − LF (x) 1 1 = − πµ(x)π (x) − (∇A (x)) · (∇Aν(x)) (4.32) 2 µ 2 ν 4.5 Canonical Quantization of the Lorenz Gauge

Now we can proceed to the canonical quantization of the electromagnetic eld. We will implement for the eld and conjugate eld operators the bosonic covariant commutation relations that read h i Aˆµ(~x,t), πˆν(~x0, t) =igµνδ(~x − ~x0) (4.33a) h i Aˆµ(~x,t), Aˆν(~x0, t) = [ˆπµ(~x,t), πˆν(~x0, t)] = 0 (4.33b)

We know that the quanta of the electromagnetic eld are the photons. In order to describe them, the need to expand the eld operator Aˆµ(x) into plane waves

Z 3 3 µ d k X h µ −ik·x † µ ik·xi Aˆ (x) = aˆ~ e +a ˆ (~k)e (4.34) p 3 kλ λ ~kλ λ 2ωk(2π) λ=0 The conjugate eld is given by

Z r 3 µ 3 ωk X h µ −ik·x † µ ik·xi π (x) = i d k aˆ~ (~k)e − aˆ (~k)e (4.35) 2(2π)3 kλ λ ~kλ λ λ=0 60 CHAPTER 4. THE ELECTROMAGNETIC FIELD

We can nd that the Fourier coecients satisfy the following commutation relations

h † i ~ ~ 0 aˆ~ 0 0 , aˆ = − g 0 δ(k − k ) (4.36a) k λ ~kλ λλ h † † i aˆ~ 0 0 , aˆ~ = aˆ , aˆ = 0 (4.36b) k λ kλ ~k0λ0 ~kλ

In order to prove their validity we will show they lead to the initial commutation relations (4.33a).

Z 3 Z r h µ ν 0 i X d k 3 0 ωk0 µ ν Aˆ (~x,t), πˆ (~x , t) = i d k (~k) 0 p 3 2(2π)3 λ λ λλ0 2ωk(2π) h −ik·x ik·x −ik·x0 † ik0·x0 i aˆ~ e +a ˆ~ e , aˆ~ 0 0 e − aˆ e kλ kλ k λ ~k0λ0 Z 3 X d k µ ν h i~k·(~x0−~x) i~k·(~x−~x0)i = i (~k) 0 (~k) g 0 e + g 0 e 2(2π)3 λ λ λλ λλ λ,λ0 Z 3 " # d k X µ ν i~k·(~x−~x0) X µ ν i~k·(~x0−~x) =i g 0 (~k) 0 (~k)e + g 0 (~k) 0 (~k)e 2(2π)3 λλ λ λ λλ λ λ λλ0 λλ0 µν 3 3 ig Z d k 0 Z d k 0 = ei~k·(~x −~x) + ei~k·(~x−~x 2 (2π)3 (2π)3 =igµνδ(~x − ~x0)

where we have used the completeness relation (4.20). Similarly we can prove the rest of the initial commutation relations that involve eld operators, hence we have assumed the correct commutator relations for their coecients. It is useful to notice that the commutation relations for the space-like polarization states (λ = 1, 2, 3) are identical with the bosonic commutation relations, while in the commutation relations that correspond to the time-like polarization state, a (-) sign appears. Later on, we will see the results caused by this anomaly. However, we can nd the Hamilton operator as the normal-ordered Hamilton density as follows

Z Hˆ (t) = d3x Hˆ(x) 1 Z = − d3x N [ˆπµ(x)ˆπ (x) + (∇Aν(x)) · (∇A (x))] 2 µ ν

At this stage of calculation it is easier to compute each term separately and combine the results. Before, we dive in the calculations, it is convenient to state some basic relations that will become a very helpful tool towards our nal goal

3 Z d k 0 e−i(k+k )·x =δ(~k + ~k0)e−i(ωk+ωk0 t (2π)3 3 Z d k 0 e−i(k−k )·x =δ(~k − ~k0)e−i(ωk−ωk0 t (2π)3 4.5. CANONICAL QUANTIZATION OF THE LORENZ GAUGE 61

For the rst term we nd Z Z r Z r X ωk ωk0 d3xπˆµ(x)ˆπ (x) = − d3k d3k0 · µ 2(2π)3 2(2π)3 λλ0 Z 3 h µ −ik·x † µ ik·xi d x aˆ~ (~k)e − aˆ (~k)e · kλ λ ~kλ λ h µ ~ 0 −ik0·x † µ ~ 0 ik0·xi aˆ~ 0 0 0 (k )e − aˆ 0 (k )e k λ λ ~k0λ0 λ Z ω h 3 k X µ −2iωkt = − d k aˆ aˆ (~k) 0 (−~k)e − 2 ~kλ −~kλ0 λ µλ λλ0 † † − g 0 aˆ~ aˆ − g 0 aˆ aˆ~ 0 + λλ kλ ~kλ0 λλ ~kλ kλ i † µ ~ ~ 2iωkt +a ˆ aˆ ~ 0 (k) 0 (−k)e ~kλ −kλ λ µλ while for the second term we nd Z Z 3 Z 3 0 3 ˆµ ˆ X d k ~ d k ~ 0 d x(∇A (x)) · (∇Aµ(x)) = − p k p k · 3 0 3 λλ0 2ωk(2π) 2ωk (2π) Z 3 h µ −ik·x † µ ik·xi d x aˆ~ (~k)e − aˆ (~k)e · kλ λ ~kλ λ

h 0 0 µ ~ 0 −ik ·x † µ ~ 0 ik ·xi aˆ~ 0 0 0 (k )e − aˆ 0 (k )e k λ λ ~k0λ0 λ Z ω h 3 k X µ −2iωkt = − d k −aˆ aˆ (~k) 0 (−~k)e − 2 ~kλ −~kλ0 λ µλ λλ0 † † − g 0 aˆ~ aˆ − g 0 aˆ aˆ~ 0 − λλ kλ ~kλ0 λλ ~kλ kλ i † µ ~ ~ 2iωkt − aˆ aˆ ~ 0 (k) 0 (−k)e ~kλ −kλ λ µλ where adding them and taking the normal product where the creation operator has been moved to the left one can nd that the Hamilton operator is given by the equation

Z " 3 # 3 X † † Hˆ (t) = d k ω aˆ aˆ~ − aˆ aˆ~ (4.37) k ~kλ kλ ~k0 k0 λ=1

† Consequently the interpretation of the operators aˆ , aˆ~ as creation and annihilation ~kλ kλ operators hold true for the transverse (λ = 1, 2) and longitudinal (λ = 3) polarization states, but not for the scalar photons. To be more specic, one will encounter this problem while trying to construct the Fock space for photons. As always, we dene the vacuum state |0i as

aˆ~kλ |0i = 0 where no photons exist, normalized as h0|0i = 1. We can create Fock states acting the creation operator upon the vacuum as follows

n † ~kλ |n~ i = N aˆ |0i kλ ~kλ

† Furthermore, the norm of one-photon Fock state |1~ i =a ˆ |0i equals to kλ ~kλ ~ (4.38) h1~kλ|1~kλi = −gλλ0 δ(0) 62 CHAPTER 4. THE ELECTROMAGNETIC FIELD

and for scalar photos has a negative value. Consequently we have found that the norm of the state containing a scalar photon is negative? In order to circumvent this diculty, we shall impose the Lorenz condition, so our theory becomes equavalent to Maxwell's equations. Unfortunately

ˆµ ∂µA (x) 6= 0

ˆµ ˆν 0 given that its commutator with the eld operator does not vanish [∂µA (~x), A (~x )] 6= 0. As a result, the Lorenz condition cannot be taken as an operator identity. Instead, in order to dene our Hilbert space, we will use this constraint as a condition to dene our admissible state vectors |Φi. To do so, we will separate the eld operator as

ˆ ˆ(+) ˆ(−) (4.39) Aµ(x) = Aµ (x) + Aµ (x) and dene our physical states through the equation

µ ˆ(+) (4.40) ∂ Aµ (x) |Φi = 0 known as the Gupta-Bleuler condition. This condition ensures that

µ ˆ µ ˆ µ ˆ(−) hΦ|∂ Aµ(x)|Φi = hΦ|∂ Aµ(x)|Φi + hΦ|∂ Aµ (x)|Φi =0

Next, using the plane wave expansion, we nd that the positive frequency eld operator equals to

Z 3 3 (+) d k X h µ −ik·xi Aˆ (x) = aˆ~ (~k)e (4.41) µ p 3 kλ λ 2ωk(2π) λ=0 Thus, the Gupta-Bleuler constraint becomes

µ ˆ(+) 0 =∂ Aµ |Φi Z 3 3 d k X h µ −ik·xi = aˆ~ (~k)e |Φi p 3 kλ λ 2ωk(2π) λ=0 3 X µ µ ~ = k λaˆ~kλ(k) |Φi λ=0 The equation found above using (4.14) and (4.17) takes the form

(4.42) aˆ~k0 − aˆ~k3 |Φi = 0 and computing the expectation values of their number operators

† † hΦ|aˆ aˆ~ |Φi = hΦ|aˆ aˆ~ |Φi ~k0 k0 ~k3 k3 one can nd that they are equal. 4.6. FEYNMAN PROPAGATOR FOR PHOTONS 63

The expectation value of the Hamiltonian operator now reads

Z " 2 # 3 X † † † hΦ|Hˆ |Φi = d k ω hΦ| aˆ aˆ~ |Φi + hΦ|aˆ aˆ~ |Φi − hΦ|aˆ aˆ~ |Φi (4.43) k ~kλ kλ ~k3 k3 ~k0 k0 λ=1 Z 2 3 X † = d k ω hΦ|aˆ aˆ~ |Φi (4.44) k ~kλ kλ λ=1 Z 2 3 X (4.45) = d k ωk n~kλ λ=1 where only the transverse photons make a contribution to the energy, hence we have gotten rid of the negative energies, ensuring only positive possible energy expectation valued outcomes. From this result one can deduce that in free space, observable quantities will involve transverse photons only. Consequently, through the imposition of the Gupta- Bleuler condition we have created a Hilbert space where in the physical states, pseudo photons (longitudinal and scalar) does not contribute, hence we have eliminated the negative valued norms. In summary, we have separated the full Hilbert space to a section that respects the Lorenz condition (Gupta-Bleuler condition) and contains admissible physical states, and the part where the Gupta-Bleuler constraint does not hold.

4.6 Feynman Propagator for Photons

As always, the Feynman Propagator for the massless photon eld can be computed as the expectation value of the vacuum state for the time ordered eld operator product h i µν ˆµ ˆν iDF (x − y) = h0|T A (x)A (y) |0i Z 3 " 3 # d k 1 X µ ν −ik·(x−y) ik·(x−y) 0 ~ ~ = 3 (−gλλ )λ(k)λ(k) Θ(x0 − y0)e + Θ(y0 − x0)e (2π) 2ωp λ=1 Z d3k = − gµν Θ(x − y )e−ik·(x−y) + Θ(y − x )eik·(x−y) 2π)3 0 0 0 0 where we have used the completeness relation (4.20). But recalling the scalar Feynman Propagator given by (2.67) leading to (2.72) we can express the photon propagaror in respect to the scalar Feynman propagator, hence obtaining

Z d4k e−ik·(x−y) Dµν(x − y) = −gµν (4.46) F (2π)4 k2 + i while in momentum space the photon propagator in the Feynman gauge equals to gµν Dµν(k) = − (4.47) F k2 + i In order to nd the contribution of each term, the need of separating its polarization states is essential

" 2 # 1 X Dµν = µ(~k)ν (~k) + µ(~k)ν(~k) − µ(~k)ν(~k) F k2 + i λ λ 3 3 0 0 λ=1 64 CHAPTER 4. THE ELECTROMAGNETIC FIELD

Considering (4.19) which is the covariant form of the longitudinal polarization state in the special Lorentz frame, after some straightforward calculations we nd for the photon Feynman propagator in momentum space

" 2 # 1 X k2nµnν kµkν − (kµnν + knunµ)(k · n) Dµν(k) = µ(~k)ν (~k) + + F k2 + i λ λ (k · n)2 − k2 (k · n)2 − k2 λ=1 µν µν µν (4.48) =DF (trans)(k) + DF (Coul)(k) + DF (res)(k) where

2 µν X µ ~ ν ~ (4.49) DF (trans)(k) = λ(k)λ(k) λ=1 is the transverse part.

nµnν Dµν (k) = (4.50) F (Coul) (k · n)2 − k2 is the Coulomb part. Finally

1 kµkν − (kµnν + knunµ)(k · n) Dµν (k) = (4.51) F (res) k2 + i (k · n)2 − k2 is the remainder part that does not contribute in the calculation of observable quantities. It is very interesting to perform a more detailed study in the Coulomb part of the photon propagator. In the special Lorentz frame nµ = (1, 0, 0, 0), the Coulomb part (in momentum space) reads

δ δ Dµν (k) = µ0 ν0 F (Coul) |~k|2 where in the coordinate space it takes the form

Z d4p Dµν (x − y) = e−ik·(x−y)Dµν (k) F (Coul) (2π)4 F (Coul) Z dk Z d3k ei~k·(~x−~y) 0 −ik0·(x0−y0) =δµ0δν0 e (2π) (2π)3 |~k|2 δ(x − y ) =δ δ 0 0 µ0 ν0 4π|~x − ~y| The equation found above describes an instantaneous interaction having the Coulomb potential form. Consequently, the Coulomb interaction arises from the combined propagation of the longitudinal and scalar photons. Leaving out these degrees of freedom, one would also remove the Coulomb interaction. However, the covarianr photon propagator we have found in (4.47), automatically takes into consideration all four types of photons. Chapter 5

Interacting Fields

The previous chapters were dedicated to the study of free elds. In this chapter we will start to examine more complicated theories that include interaction terms. They will take the form of extra interacting parts (higher order terms) in the Lagrangian density, thus deviating from its bilinear form.

5.1 The Dirac Picture

As we know from Quantum Mechanics, a theory can be formulated in dierent but equivalent representations or pictures that dier depending on how the time-dependence is treated. The two basic pictures are, the Schrodinger picture - operators are constant while state vectors have a time-dependence - and the Heisenberg picture - state vectors are constant and operators are time-dependent - in which we were currently working. First of all, in the Heisenberg picture, the operators must obey the Heisenberg equation of motion ˆ h ˆ ˆ i i∂tOH (t) = OH (t), H (5.1) A formal solution for the time dependence of an operator in the Heisenberg picture can be easily written down as

ˆ iHtˆ ˆ −iHtˆ OH (t) = e OH (0)e (5.2) The state vectors in the Heisenberg representation are constant having the form (5.3) |ψ(t)iH = |ψ(0)iH = |ψiH On the other hand, in the Schrodinger picture, the state vectors satisfy the Schrodinger equation of motion that reads ∂ |ψ(t)i i S = Hˆ |ψ(t)i (5.4) ∂t S having the solution

65 66 CHAPTER 5. INTERACTING FIELDS

while in the Schrodinger picture the operators are constant, hence ˆ ˆ ˆ OS(t) = OS(0) = OS (5.7) Consequently, the operators in the Schrodinger representation are connected to the operators in the Heisenberg picture through the transformation

ˆ −iHtˆ ˆ iHtˆ OS = e OH (t)e (5.8) which is unitary, given that ˆ† ˆ . OSOS = I However, in order to ensure the invariance of commutation or anticommutation relations when one moves from one representation to another, they must be connected through a canonical transformation. Therefore, if two operators in the Heisenberg picture obey the (anti)-commutation relations

h ˆ ˆ i ˆ AH , BH = CH ± then in the Schrodinger picture they take the form h i h i ˆ ˆ −iHtˆ ˆ iHtˆ −iHtˆ ˆ iHtˆ AS, BS = e AH e , e BH e ± ± −iHtˆ ˆ iHtˆ −iHtˆ ˆ iHtˆ −iHtˆ ˆ iHtˆ −iHtˆ ˆ iHtˆ =e AH e e BH e ± e BH e e AH e −iHtˆ ˆ ˆ ˆ ˆ iHtˆ =e AH BH ± BH AH e h i −iHtˆ ˆ ˆ iHtˆ =e AH , BH e ± iHtˆ ˆ iHtˆ =e CH e ˆ =CS therefore, the (anti)-commutation relations remain invariant under the transformation claimed above. The interaction or Dirac picture is a mixture of the two representations, used extensively in perturbation theory. To begin with, we separate the Hamiltonian into two parts ˆ ˆ ˆ H = H0 + HI (t) (5.9) ˆ ˆ where H0 is the Hamiltonian of free elds and HI (t) is the time dependent perturbation we have inserted. Therefore, performing the transformation

ˆ −iH0t (5.10) |ψ(t)iS = e |ψ(t)iD in the Schrodinger equation ∂ |ψ(t)i h i i S = Hˆ + Hˆ (t) |ψ(t)i (5.11) ∂t 0 I S we obtain the Schrodinger equation in the Dirac picture as follows

∂ h ˆ i h i ˆ i e−iH0t |ψ(t)i = Hˆ + Hˆ (t) e−iH0t |ψ(t)i ⇒ ∂t D 0 I D ∂ |ψ(t)i ˆ ˆ i D =eiH0tHˆ (t)e−iH0t |ψ(t)i ∂t 1 D 5.2. THE TIME-EVOLUTION OPERATOR 67

whence

∂ |ψ(t)i i D = Hˆ (t) |ψ(t)i (5.12) ∂t D D The Hamilton operator in the Dirac picture is related with the Hamilton operator in the Schrodinger representation through the unitary transformation

ˆ ˆ ˆ iH0t ˆ −iH0t HD(t) = e H1(t)e

In summary, the transformations connecting Schrodinger to Dirac representation for state vectors and operators read

ˆ ˆ ˆ iH0t ˆ −iH0t OD(t) = e OS(t)e (5.13a) ˆ iH0t (5.13b) |ψ(t)iD = e |ψ(t)iS

5.2 The Time-Evolution Operator

In our previous discussion, we have found that the Schrodinger equation in the Dirac picture is given by (5.12)

∂ |ψ(t)i i D = Hˆ (t) |ψ(t)i ∂t D D However, in order to solve this dierential equation, we will dene the time-evolution ˆ operator or Dyson operator U(t, t0) as follows

ˆ (5.14) |ψ(t)iD = U(t, t0) |ψ(t0)iD that describes the time evolution of state vectors in the Dirac representation. Next, substituting this equation in (5.12) we get

ˆ ˆ ˆ i∂tU(t, t0) = HDU(t, t0) (5.15)

which describes the time evolution of the time evolution operator. The dierential equation can be easily solved, thus obtaining the solution

Z t ˆ ˆ 0 0 U(t, t0) = exp −i HD(t )dt (5.16) t0 It is important to state that the time evolution operator must satisfy the condition

ˆ U(t0, t0) = I However, in order to nd an analytic solution for the Dyson operator we will write down its integral equation

Z t ˆ 0 ˆ 0 ˆ 0 (5.17) U(t, t0) = I + (−i) dt HD(t )U(t , t0) t0 68 CHAPTER 5. INTERACTING FIELDS

which was obtained by integrating (5.15) from t to t0. For small perturbations this equation can be solved by iteration as follows

Z t ˆ 0 ˆ 0 ˆ 0 Un+1(t, t0) = I + (−i) dt HD(t )Un(t , t0) t0 hence, acquiring the innite Neuman series

Z t1 ˆ ˆ U(t, t0) = I+(−i) dt1HD(t1) t0 Z t Z t1 2 ˆ ˆ +(−i) dt1 dt2HD(t1)HD(t2) t0 t0 +higher order terms

One can simplify even further the series using a procedure introduced by Dyson, in which our goal is to nd a way so all of our integrals have the same upper and lower boundaries. To do so, we will deal with the second order term in the series

Z t Z t1 ˆ ˆ dt1 dt2HD(t1)HD(t2) t0 t0 We already know that our integration extends over the upper and lower orthogonal tri-

angle depending on whether t1 > t2 or t2 > t1 on the t2 − t1 plane depicted below. Given that we have assumed t1 > t2, then our integral

Z t Z t1 ˆ ˆ (a) = dt1 dt2HD(t1)HD(t2) t0 t0 cuts the upper triangle as shown. We want to modify our integral, so it extends throughout the whole square. To do so, we change (a) so it cuts intervals

parallel to the t1 axis as follows

Z t Z t ˆ ˆ (a) = dt2 dt1HD(t1)HD(t2) = (b) t0 t2

Next, setting t ↔ t in (b) we get 1 2 Figure 5.1: Integration Space Z t Z t ˆ ˆ (b) = dt1 dt2HD(t2)HD(t1) = (c) t0 t1

It is important to note that changing t1 ↔ t2 and given that we have assumed (t1 > t2), we consider (t2 > t1), hence our last integral is over the lower half. However, we can note that adding those two integrals (b) and (c) our integration extends through the whole square

Z t Z t Z t Z t 1 ˆ ˆ ˆ ˆ (a) = dt2 dt1HD(t1)HD(t2) + dt1 dt2HD(t2)HD(t1) 2 t0 t2 t0 t1 5.3. THE S-MATRIX 69

that represents half of the square. Therefore, we can write our initial integral in the compact form

Z t Z t1 Z t Z t ˆ ˆ 1 h ˆ ˆ i dt1 dt2HD(t1)HD(t2) = dt1 dt2T HD(t1)HD(t2) t0 t0 2 t0 t0 ( h i Hˆ (t )Hˆ (t ) for t > t where T Hˆ (t )Hˆ (t ) = D 1 D 2 1 2 D 1 D 2 ˆ ˆ HD(t2)HD(t1) for t1 < t2 which is the Dyson's formula. The result found above means that we have ordered the second term of the Dyson operator in a chronological way. In particular, when t1 > t2 like we have assumed, the integral is calculated through the upper orthogonal triangle, whereas when t2 > t1 in the lower one. Finally, the Dyson operator using the Dyson's formula reads

∞ X (−i)n Z t Z t h i Uˆ(t, t ) = + dt ··· dt T Hˆ (t ) ··· Hˆ (t ) (5.18) 0 I n! 1 n D 1 D n n=1 t0 t0 5.3 The S-Matrix

The scattering matrix (S-matrix) is a fundamental notion in Quantum Field Theory, that denotes the probability amplitude for a process in which the system makes a transition from an initial to a nal state under the inuence of an interaction. In the Dirac representation, the proper tool to use in order to compute the S-matrix is the Dyson operator. Let |ψ(t)i denotes the time dependent state vector in the interaction picture. In the limit t → −∞ our system is at the initial state |ii, i.e

lim |ψ(t)i = |ii (5.19) t→−∞ The S-matrix is dened as the projection of the time-dependent state vector onto the free nal state hf| ˆ Sfi = hf|S|ii It is rather important to observe that we have assumed our system's initial and nal states are eigenstates of the free Hamiltonian. Expressing in respect to the evolution operator we have

ˆ Sfi = lim lim hf|U(t2, t1)|ii t2→+∞ t1→−∞ hence the S-matrix is connected to the time-evolution operator through the relation

Sˆ = Uˆ(+∞, −∞) and using the Dyson formula we nd

∞ X (−i)n Z ∞ Z ∞ h i Sˆ = + dt ··· dt T Hˆ (t ) ··· Hˆ (t ) (5.20) I n! 1 n D 1 D n n=1 −∞ −∞

which is a unitary operator Sˆ†Sˆ = I. 70 CHAPTER 5. INTERACTING FIELDS 5.4 Wick's Theorem

h ˆ ˆ i Given that the time-ordered product T HD(x1) ··· HD(xn) represents an innite series of terms classied in chronological order, their evaluation can become quite a di- cult task. In this section we will develop tools which will simplify our computing process. Wick's theorem tells us how to go from time ordered products to normal-ordered products in our calculations. Let us rst recall the denition of the normal-ordered product. A eld operator is split into "positive-frequency" part (ψˆ(+)) and a "negative-frequency" part (φˆ(−)). For a scalar eld the decomposition reads

φˆ(x) = φˆ(+)(x) + φˆ(−)(x)

The normal product of two bosonic elds is dened as

h i N φˆ(x)φˆ(y) =φˆ(−)(x)φˆ(−)(y) + φˆ(+)(x)φˆ(−)(y) +φˆ(+)(x)φˆ(+)(y) + φˆ(−)(y)φˆ(+)(x) (5.21)

while decomposing the fermionic eld operator (ψˆ(x))

ψˆ(x) = ψˆ(+)(x) + ψˆ(−)(x)

the normal product of two fermionic elds now reads

h i N ψˆ(x)ψˆ(y) =ψˆ(−)(x)ψˆ(−)(y) + ψˆ(+)(x)ψˆ(−)(y) +ψˆ(+)(x)ψˆ(+)(y) − ψˆ(−)(y)ψˆ(+)(x) (5.22)

ˆ In the following the generic name ΦA(x) will be employed for both bosonic and fermionic operators. Next, we will derive a systematic formalism in order to evaluate the time- ordered product of an arbitrary number of operators. Consequently, our nal goal is to nd an expression for the time-ordered product in respect to normal-product. However, in order to grasp the procedure essential to prove the relation we will nd the form for the time-ordered product of two terms and subsequently generalize our result for arbitrary time-ordered product terms.

Firstly, evaluating the time-ordered product of two eld operator, considering t1 > t2, then

hˆ ˆ i ˆ ˆ T ΦA(x1)ΦB(x2) =ΦA(x1)ΦB(x2) ˆ (−) ˆ (−) ˆ (+) ˆ (−) =ΦA (x1)ΦB (x2) + ΦA (x1)ΦB (x2) ˆ (+) ˆ (+) ˆ (−) ˆ (+) +ΦA (x1)ΦB (x2) + ΦA (x1)ΦB (x2)

Now, the second term (which is not normal-ordered) can be written as

ˆ (+) ˆ (−) hˆ (+) ˆ (−) i ˆ (−) ˆ (+) ΦA (x1)ΦB (x2) = ΦA (x1), ΦB (x2) + ABΦB (x2)ΦA (x1) ± 5.4. WICK'S THEOREM 71

( +1 for bosons where AB = −1 for fermions But, the fact that the (anti)commutators of free elds are c-numbers1, implies that the (anti)commutators can be expressed as the vacuum expectation value, hence

Inserting the result in our initial equation, we nd

ˆ (+) ˆ (−) hˆ ˆ i ˆ (−) ˆ (+) ΦA (x1)ΦB (x2) = h0|T ΦA(x1)ΦB(x2) |0i + ABΦB (x2)ΦA (x1) hence obtaining the relation between normal-ordered and time-ordered product after some trivial calculations that read

hˆ ˆ i h ˆ ˆ i hˆ ˆ i T ΦA(x1)ΦB(x2) = N ψ(x)ψ(y) + h0|T ΦA(x1)ΦB(x2) |0i

which holds true for . t1 ≷ t2 It is obvious that both time-ordered and normal-ordered products dier by a commutator which is a c-number. Since the vacuum expectation value of the time-ordered product of operators occurs frequently in our calculations, it is given a special name. To be more precise, one denes the notion of contraction of two operators as

ˆ ˆ hˆ ˆ i ΦA(x1)ΦB(x2) = h0|T ΦA(x1)ΦB(x2) |0i

Furthermore it is relatively easy to expand the concept of contraction in more complicated expressions, which we will meet extensively, i.e within the argument of the normal product. More specically h i h i N AˆBˆCˆDˆEˆF...ˆ Kˆ LˆM...ˆ = N AˆBˆF...ˆ Kˆ M...ˆ CˆEˆDˆLˆ (5.23) where = ±1 depending on the permutation of the fermionic operators. Using the denition of contraction, we obtain the nal result for the time-ordered product of two eld operators

hˆ ˆ i h ˆ ˆ i hˆ ˆ i T ΦA(x1)ΦB(x2) = N ψ(x)ψ(y) + N ΦA(x1)ΦB(x2) (5.24)

This result can be generalized to products of arbitrary complexity. This is the central viewpoint of Wick's Theorem that states

1The term c-number (classical or commuting number) refers to real or complex numbers that obey all the laws of arithmetic including commutativity. In particular, a c-number is "a random variable represented by a scalar multiple of the identity operator". Through this denition one can understand the reason of computing the expectation value of a c-number quantity. A c-number is used to distinguish from operators called q-numbers (quantum or queer numbers) that do not commute in general. 72 CHAPTER 5. INTERACTING FIELDS

Wick's Theorem. The time-ordered product of a set of operators can be decomposed into the sum of the corresponding contracted normal products. All contractions of operator pairs that possibly can arise enter this sum.

For any collection of operators, A,ˆ B,...ˆ Zˆ according to Wick's Theorem we can nd that their time-ordered product equals to h i h i T AˆBˆC...ˆ Yˆ Zˆ =N AˆBˆC...ˆ Yˆ Zˆ h i h i h i +N AˆBˆC...ˆ Yˆ Zˆ + N AˆBˆC...ˆ Yˆ Zˆ + ... + N AˆBˆC...ˆ Yˆ Zˆ h i h i h i +N AˆBˆCˆD...ˆ Yˆ Zˆ + N AˆBˆCˆD...ˆ Yˆ Zˆ + ... + N AˆBˆC...ˆ Wˆ XˆYˆ Zˆ

+ higher contractions (5.25)

However, it is useful to recall some time-ordered products of several free elds found before and relate them with the newly introduced concept of contraction. In particular, for a Scalar Field we have found h i ˆ ˆ† ˆ ˆ† φ(x)φ (y) = h0|T φ(x)φ (y) |0i = i∆F (x − y) (5.26)

where the Feynman scalar propagator is given by

Z d4p e−i·(x−y) ∆ (x − y) = F (2π)4 p2 − m2 + i For the Dirac Field we nd

ˆ ¯ˆ h ˆ ¯ˆ i ψa(x)ψβ(y) = h0|T ψa(x)ψβ(y) |0i = iSF (x − y) (5.27)

where the spinor Feynman propagator is given by

Z −ip·(x−y) 4 4 e SF (x − y) = d ](2π) p/ − m + i Finally, foe the electromagnetic eld

ˆ ˆ h ˆ ˆ i Aµ(x)Aν(y) = h0|T Aµ(x)Aν(y) |0i = iDµν(x − y) (5.28)

while, the photon propagator in the Feynman gauge equals to

Z d4k g D (x − y) = − µν e−ik·(x−y) µν (2π)4 k2 + i Consequently, one can readily deduce the fact that the contraction of the eld operators is identical to the Feynman propagator. 5.5. COUPLING PHOTONS TO FERMIONS (QED) 73 5.5 Coupling Photons to Fermions (QED)

In this section we shall exclude the Lagrangian density that rises in QED, hence when coupling photons to electrons. In order to insert the electromagnetic interaction in the Dirac equation and preserve the gauge invariance simultaneously, we are going to use the prescription of minimal coupling. To do so, we replace the covariant 4-divergence with

∂µ → Dµ = ∂µ + iqAµ(x) (5.29) which is the gauge covariant derivative, hence obtaining

(iD/ − m)ψ(x) = 0 Substituting the covariant derivative, we nd

(i∂/ − m)ψ(x) = qA/(x)ψ(x) (5.30) One can observe that the rst term corresponds to the free Dirac equation, whereas the second expresses the electromagnetic interaction mixing the vector potential with the Dirac spinor. Therefore, the new Lagrangian now reads 0 (5.31) LD(x) = LD + Lint where ¯ LD(x) =ψ(x) i∂/ − m ψ(x) (5.32a) ¯ Lint = − qψ(x)A/(x)ψ(x) (5.32b) Finally, the Lagrangian density of QED is written as

LQED(x) =LD(x) + Lem(x) + Lint(x) 1 =ψ¯(x) i∂/ − m ψ(x) − F F µν − qψ¯(x)A/(x)ψ(x) (5.33) 4 µν which is invariant under local gauge transformations 0 (5.34a) Aµ(x) →Aµ(x) = Aµ(x) + ∂µΛ(x) ψ(x) →ψ0(x) = e−iqΛ(x)ψ(x) (5.34b) ψ¯(x) →ψ¯0(x) = eiqΛ(x)ψ¯ (5.34c)

5.6 Feynman Rules in QED

Now, we will study the theory of Quantum Electrodynamics (QED). Our goal is to show that the Feynman rules of QED can be recovered in a systematic way in quantum eld theory. Firstly we have found that the QED Lagrangian is given by

LQED(x) = LD(x) + Lem(x) + Lint(x) where ¯ LD(x) =ψ(x) i∂/ − m ψ(x) 1 L (x) = − F F µν − (∂ Aµ(x))2 em 4 µν µ ¯ Lint(x) = − qψ(x)A/(x)ψ(x) 74 CHAPTER 5. INTERACTING FIELDS

In order to quantize our theory, we will employ the normal-ordering prescription, i.e for the interaction term

h ¯ˆ ˆ i Lint(x) = − qN ψ(x)A/(x)ψ(x)

The Hamiltonian density describing the interaction is given by

ˆ Hint = − Lint h i = − qN ψ¯ˆ(x)A/(x)ψ(ˆx) (5.35)

As a result, the Dyson formula in the S-matrix now reads

∞ X (−i)n Z h i Sˆ = + d4x ··· d4x T Hˆ (x ) ··· Hˆ (x ) I n! 1 n int 1 int n n=1 ∞ X (−iq)n Z h h i h ii = + d4x ··· d4x T N ψ¯ˆ(x )A/(x )ψˆ(x ) ···N ψ¯ˆ(x )A/(x )ψˆ(x ) I n! 1 n ! 1 1 n n n n=1 (5.36)

were we can nd its analytic expression using Wick's Theorem. Next, we will write down the plane wave expansion found in the previous chapters for each eld operator

Z d3p r m X h i ψˆ(x) = ˆb (p)u (p)e−ip·x + dˆ†(p)v (p)eip·x (2π)3/2 E s s s s p s Z d3p r m X h i ψ¯ˆ(x) = ˆb†(p)¯u (p)e−ip·x + dˆ (p)¯v (p)eip·x (2π)3/2 E s s s s p s Z 3 h i ˆ d k X ~ −ik·k † ~ ik·x Aµ(x) = aˆ~ µλ(k)e +a ˆ µλ(k)e p 3 kλ ~kλ 2ωk(2π) λ

In the evaluation of the S-matrix element, the terms that will contribute are those that does not eliminate the expectation value. As we know, each order of the S-matrix expansion will result to a set of dierent processes regarding fermions and photons. Processes in QED are contained within the second order of the S-matrix operator Sˆ(2). Consequently, the need of expanding the Sˆ(2) matrix element in terms using Wick's Theorem is essential. The Sˆ(2) operator reads

(−iq)2 Z h h i h ii Sˆ(2) = d4x d4x T N ψ¯ˆ(x )A/(x )ψˆ(x ) N ψ¯ˆ(x )A/(x )ψˆ(x ) (5.37) 2! 1 2 1 1 1 2 2 2

In order to simplify this expression even further using Wick's theorem we will state another rule regarding contraction: contractions between operators with the same argument 5.6. FEYNMAN RULES IN QED 75

(4-vector) do not contribute. Therefore, our QED S-matrix takes the form

(−iq)2 Z h i Sˆ(2) = d4x d4x N ψ¯ˆ(x )γµψˆ(x )ψ¯ˆ(x )γνψˆ(x )Aˆ (x )Aˆ (x ) (a) 2! 1 2 1 1 2 2 µ 1 ν 2 (−iq)2 Z h i + d4x d4x N ψ¯ˆ(x )γµψˆ(x )ψ¯ˆ(x )γνψˆ(x )Aˆ (x )Aˆ (x ) (b) 2! 1 2 1 1 2 2 µ 1 ν 2 (−iq)2 Z h i + d4x d4x N ψ¯ˆ(x )γµψˆ(x )ψ¯ˆ(x )γνψˆ(x )Aˆ (x )Aˆ (x ) (c) 2! 1 2 1 1 2 2 µ 1 ν 2 (−iq)2 Z h i + d4x d4x N ψ¯ˆ(x )γµψˆ(x )ψ¯ˆ(x )γνψˆ(x )Aˆ (x )Aˆ (x ) (d) 2! 1 2 1 1 2 2 µ 1 ν 2 (−iq)2 Z h i + d4x d4x N ψ¯ˆ(x )γµψˆ(x )ψ¯ˆ(x )γνψˆ(x )Aˆ (x )Aˆ (x ) (e) 2! 1 2 1 1 2 2 µ 1 ν 2 (−iq)2 Z h i + d4x d4x N ψ¯ˆ(x )γµψˆ(x )ψ¯ˆ(x )γνψˆ(x )Aˆ (x )Aˆ (x ) (f) 2! 1 2 1 1 2 2 µ 1 ν 2 (−iq)2 Z h i + d4x d4x N ψ¯ˆ(x )γµψˆ(x )ψ¯ˆ(x )γνψˆ(x )Aˆ (x )Aˆ (x ) (g) 2! 1 2 1 1 2 2 µ 1 ν 2 (−iq)2 Z h i + d4x d4x N ψ¯ˆ(x )γµψˆ(x )ψ¯ˆ(x )γνψˆ(x )Aˆ (x )Aˆ (x ) (h) 2! 1 2 1 1 2 2 µ 1 ν 2 Each of those 8 terms expresses a process, but not everyone represent a physical one. In order to correlate those terms with the specic process that they represent and visualize them, an analytic expansion can become very tedious, we will introduce a graphic notation through a series of Feynman translation rules depicted below.

Figure 5.2: Feynman Translation Rules

Using those rules, it is relatively easy to nd the physical QED process each term corresponds to. They will also become a powerful tool in order to draw the Feynman diagrams that correspond to the Compton Scattering in the next chapter. Therefore, the Feynman translation rules are essential in order to draw the Feynman diagrams. Each such diagram is in 1-1 correspondence with the terms in the matrix- element expansion. In particular, after drawing the Feynman diagrams for a specic physical process, one can evaluate the matrix-element Sfi that corresponds to this particular process. To do so, we need to state some rules so we can map each part of the Feynman diagram to a mathematical concept. These sets of rules are called Feynman Rules. To be more precise, we will mention the Feynman Rules in Momentum Space. 76 CHAPTER 5. INTERACTING FIELDS

Feynman Rules in Momentum Space.

• All the relevant topologically distinct Feynman diagrams with n vertices are drawn. Each internal line is assigned a momentum variable ki.

• To each vertex, we write down a factor of −iqγµ

Internal Photon line µν • iDF (ki)

• Internal Fermion line iSF aβ(pi)

• External Fermion line Npus(p) (incoming electron) Npv¯s(p) (incoming positron) Npu¯s(p) (outgoing electron) Npvs(p) (outgoing positron)

External Photon line µ ~ (incoming photon) • Nkλ(k) 0µ ~ (outgoing photon) Nkλ (k)

All momenta of the internal lines are integrated over R d4p • (2π)4

4 (4) P P • Each vertex is associated with a factor (2π) δ ( ki − ko) P i o P where ki denotes all momenta owing into the vertex, whereas ko denotes all momenta owing out of the vertex.

• Each closed fermion loop leads to a factor of -1.

The normalization factors for the Dirac and photon wave functions are

r m Np = 3 (2π) Ep s 1 Nk = 3 (2π) 2ωk

5.7 Cross Section

We consider a scattering process in which particles with four momenta pi = (Ei, ~pi) collide and produce nal particles with momenta 0 0 0 . In order to unify our pf = (Ef , ~pf ) result we will use for the transition S-matrix element the generic relation, through which we can dene the Feynman amplitude (M). However, the transition probability is given by the square of this quantum amplitude. Therefore, we can write

! ! s ! r N N 0 4 X 0 X Y i Y f (5.38) Sfi = i(2π) δ pf − pi 0 Mfi 2VEi 2VE f i i f f where Ni = 1 for spin-0 bosons and photons, Ni = 2m for spin-1/2 particles and V = (2π)3 is the volume of the nite box we conne our system. Trying to square this quantity 5.7. CROSS SECTION 77

we encounter the problem of the square delta function. One can easily circumvent that problem conning the system, hence having nite t, V , as follows ! ! 4 X 0 X X 0 X (2π) δ pf − pi = lim lim δtV pf − pi t→∞ V →∞ f i f i Z t Z 0 3 h X 0 X i = lim lim dt d x exp ix · pf − pi t→∞ V →∞ 0 (V ) P 0 P exp ix · p − pi − 1 = lim lim f t→∞ V →∞ P 0 P pf − pi

Squaring the quantity found above we get

P " !#2 sin2 k·x 4 X 0 X 2 (2π) δ pf − pi = lim lim 2 t→∞ V →∞ P k f i 2 ! 4 X 0 X =(2π) V tδ pf − pi f i with errors that tend to zero. Hence the transition probability per unit time which is our desired quantity now reads

|S |2 w = fi t ! ! 1 1 4 X 0 X Y Y 2 (5.39) =V (2π) δ pf − pi 0 |Mfi| 2VEi 2VE f i i f f having the time independent form known from the time dependent perturbation theory. This quantity expresses the transition probability rate. To obtain the transition probability to a group of nal states within the intervals 0 0 0 we must multiply by (~pf , ~pf + d~pf ) V d3p0 the number of these states Q f . Furthermore, we know that the dierential cross f (2π)3 section is given by transition rate (5.40) dσ = incident ux ! ! (2π)4 1 d3p0 X 0 X Y Y f 2 (5.41) = δ pf − pi 3 0 |Mfi| |urel| 2VEi (2π) 2E f i i f f where the incident ux |urel| and is the relative velocity of the colliding particles. j = V urel In addition, it is helpful to mention that (5.41) holds true for any Lorentz frame in which the colliding particles move collinearly. Two very important examples of such frames are the center of mass (CoM) and the laboratory frame-where ~pi or ~pf = ~0-which we will use in our application. Finally, it is crucial to note that the relativistic invariance of the cross-section formula follows from the Lorentz covariance of d3p also. 2E 78 CHAPTER 5. INTERACTING FIELDS 5.8 Spin Sums

In many experiments the colliding particles are unpolarized (electrons) and the polarization of the nal-state particles is not detected. As a result, during our calculations 2 we must average |Mfi| over the initial polarization states (spin for electrons) and sum it over all nal polarization states, hence obtaining the unpolarized cross section. There is an elegant way to perform these computations leading to the unpolarized cross section. Firstly, we consider the Feynman amplitude of the form

M =u ¯s(pf )Γur(pi) (5.42) where the Γ-matrix consists out of slashed quantities and subsequently Dirac γ-matrices. We will have to evaluate the quantity

2 2 1 X X X = |M|2 2 r=1 s=1 where we have averaged over initial spins and summed over nal spins. Next, dening

Γ˜ = γ0Γ†γ0

we can write

1 X X h i X = [¯u (p )Γu (p )] u¯ (p )Γ˜u (p ) 2 s f r i r i s f r s Writing down the Dirac indices explicitly for each term,we have

1 X X h i X = [¯u (p )Γ u (p )] u¯ (p )Γ˜ u (p ) 2 sa f aβ rβ i rγ i γδ sδ f r s " # " # 1 X X = u (p )¯u (p ) Γ u (p )¯u (p ) Γ˜ 2 sδ f sa f aβ rβ i rγ i γδ s r p/ + m p + m 1 f /i ˜ = Γaβ Γγδ 2 2m δa 2m βγ 1 p/ + m p/ + m = Tr f Γ i Γ˜ 2 2m 2m where we have used the completeness relation that holds true for the unit Dirac spinors. Consequently, we have concluded that the unpolarized cross section will be proportional to the trace of some matrices product. Therefore, in the next section we will mention some basic properties of the traces, involving γ-matrices and subsequently slashed quantities. 5.9. TRACE TECHNIQUE 79 5.9 Trace Technique

In this section, we will briey state some basic rules and extremely useful relations in evaluating the trace of the product of γ-matrices or slashed quantities. First of all, for any n×n matrices U and V

Tr(UV ) = Tr(VU)

If the product (A/1A/2 ··· A/ν) with ν = 2k + 1 then for their trace we have

Tr[A/1A/2 ··· A/ν] = 0 (5.43) but in the case where ν = 2k then for their trace we nd

Tr[A/1A/2 ··· A/ν] = Tr[A/ν ··· A/2A/1] (5.44) In addition, the trace of the product of two and four slashed quantities using the anti- commutativity of the γ-matrices can be found equal to

Tr[A/B/ ] =2(A · B) (5.45) Tr[A/B/ C/ D/] =4 [(A · B)(C · D) − (A · C)(B · D) + (A · D)(B · C)] (5.46)

Finally, using the Dirac algebra of γ-matrices -{γµγν} = 2gµν- we can move slashed quantities, i.e

A/B/ = 2A · B − B/ A/ (5.47) and in the specic case where A · B = 0, then

A/A/ =A2 (5.48) A/B/ = − B/ A/ (5.49) Chapter 6

Compton Scattering

The name Compton Scattering refers to the scattering of photons by free electrons.

In the QED language an incoming photon with momentum k and polarization vector λ is absorbed by an electron with initial momentum pi, and a second photon with four- 0 momentum k and polarization vector λ0 is emitted. This reaction can be represented schematically as

0 γ(k) + e(pi) → γ(k ) + e(pf )

6.1 Feynman Diagrams

The terms of the (2)-matrix that represents Compton scattering are (2) and (2) S Sb Sc which are equivalent if one renames x1 ↔ x2 and µ ↔ ν. Consequently, our term of interest is

(−iq)2 Z h i S(2) = d4x d4x N ψ¯ˆ(x )γµψˆ(x )ψ¯ˆ(x )γνψˆ(x )Aˆ (x )Aˆ (x ) b 2! 1 2 1 1 2 2 µ 1 ν 2 (−iq)2 Z h i = d4x d4x N ψ¯ˆ(x )A/(x )S (x − x )A/(x )ψˆ(x ) 2! 1 2 1 1 F 1 2 2 2 However, the initial and nal states are constructed as

|ii =ˆb† (p )ˆa† (~k) |0i si i λ ˆ† † ~ 0 |fi =b (p )ˆa 0 (k ) |0i sf f λ given that in our initial and nal state we have an electron and a photon. Therefore, we need to nd the correct dependence of the normal-ordered eld operators in order to obtain a non-vanishing vacuum expectation value. In particular our central term is

h ¯ˆ ˆ i N ψ(x1)A/(x1)SF (x1 − x2)A/(x2)ψ(x2)

Given that our eld operators are normal-ordered, then there are two possible combinations for non-vanishing result, where each of them represents a Feynman diagram for the Compton scattering process, constructed using the Feynman "translation rules". The combinations, with the corresponding Feynman diagrams are presented below.

80 6.1. FEYNMAN DIAGRAMS 81

i) Ψ(¯ˆ x ) →Ψ¯ˆ (−)(x ) ˆb† (p )eipf ·x1 1 1 ∼ si f ˆ ˆ (+) ˆ −ipi·x2 Ψ(x2) →Ψ (x2) ∼ bsi (pi)e (−) 0 † ~ 0 ik ·x1 A/(x1) →A/ (x1) ∼ aˆλ0 (k )e (+) ~ −ik·x2 A/(x2) →A/ (x2) ∼ aˆλ(k)e

ii) Ψ(¯ˆ x ) →Ψ¯ˆ (−)(x ) ˆb† (p )eipf ·x1 1 1 ∼ si f ˆ ˆ (+) ˆ −ipi·x2 Ψ(x2) →Ψ (x2) ∼ bsi (pi)e (+) † ~ −ik·x1 A/(x1) →A/ (x1) ∼ aˆλ(k)e (−) 0 ~ 0 ik ·x2 A/(x2) →A/ (x2) ∼ aˆλ0 (k )e However, we should note that we cannot perform the exchange Ψ¯ˆ (−) ↔ Ψ¯ˆ (+) or Ψˆ (+) ↔ Ψˆ (−) because then we will be dealing with positrons, hence having Compton scattering of a positron. Figure 6.1: Feynman Diagrams for In order to evaluate the S-matrix element for Compton Scattering each Feynman diagram, we will use the Feynman Rules in momentum space and add the result because there are not fermionic permutations. Thus, for each diagram we nd s 2 2 s 2 (2) iq m 1 (4π) 4 0 µ ~ ν ~ 0 i) S1 = − (2π) δ(pi + k − pf − k )λ(k)λ0 (k )¯usf (pf )γµSF (pi + k)γνusi (pi) 2 0 2V Epi Epf ωkωk s 2 2 s 2 iq m 1 (4π) 4 0 µ ~ ν ~ 0 (1) = − (2π) δ(pi + k − pf − k )λ(k)λ0 (k )Mµν 2 0 2V Epi Epf ωkωk and for the second diagram we have s 2 2 s 2 (2) iq m 1 (4π) 4 0 ν ~ µ ~ 0 i) S2 = − (2π) δ(pi + k − pf − k )λ(k)λ0 (k )¯usf (pf )γνSF (pi + k)γµusi (pi) 2 0 2V Epi Epf ωkωk s 2 2 s 2 iq m 1 (4π) 4 0 ν ~ µ ~ 0 (2) = − (2π) δ(pi + k − pf − k )λ(k)λ0 (k )Mµν 2 0 2V Epi Epf ωkωk Finally adding the result we nd the the Matrix-element for Compton scattering equals to s s iq2m2 1 (4π)2 (2) 4 0 ν ~ µ ~ 0 (6.1) SComp = − (2π) δ(pi + k − pf − k )λ(k)λ0 (k )Mµν 2 0 2V Epi Epf ωkωk where

0 Mµν =¯usf (pf )[γµSF (pi + k)γν + γνSF (pi − k )γµ] usi (pi) " # 1 1 =¯us (pf ) γµ γν + γν γµ us (pi) (6.2) f p/ + k/ − m p − k/0 − m i i /i which is the Compton Tensor. 82 CHAPTER 6. COMPTON SCATTERING 6.2 Compton Scattering Dierential Cross Section

Now, suppose that in the initial state we have an electron with momentum pi = in the spin state and a photon with momentum ~ and polariza- (Epi , ~pi) usi (pi) k = (ωk, k) ~ tion vector λ(k), and the corresponding quantities for the nal state are an electron with 0 0 0 momentum , and a photon with momentum 0 ~ , 0 ~ . pf = (Epf , ~pf ) usf (pf ) k = (ωk , k ) λ (k ) Furthermore, given that we are working in the laboratory frame of reference, where the initial electron stands still, then it is important to mention the relations

~ 2 2 (6.3a) pi = (m, 0) →pi = m k2 = k02 =0 (6.3b) To sum up our previous results, we have written down in (6.1) the S-matrix element for Compton scattering and in (6.2) the Compton tensor as s 2 2 s 2 (2) iq m 1 (4π) 4 0 ν ~ µ ~ 0 SComp = − (2π) δ(pi + k − pf − k )λ(k)λ0 (k )Mµν 2 0 2V Epi Epf ωkωk " # 1 1 Mµν =¯us (pf ) γµ γν + γν γµ us (pi) f p/ + k/ − m p − k/0 − m i i /i In addition, the 4-momentum conservation law for this process as can be seen from the second vertex in the Feynman diagrams reads

0 k + pi = k + pf (6.4) Consequently, using (6.3a) (6.3b) and inserting them in the Compton tensor we can modify the quantity to take the form

" 0 # p/ + k/ + m p/ − k/ + m M =u ¯ (p ) γ i γ − γ i γ u (p ) (6.5) µν sf f µ ν ν 0 µ si i 2pi · k 2pi · k We know that the dierential cross-section is given by (5.41), bu given that we want to nd the transition rate to a group of nal photon and electron states, then we need to integrate over nal photon and electron states, hence

Z 2 3 3 0 |Sfi| V V d pf V d k (6.6) dσ = 3 3 T |urel| (2π) (2π)

where ~urel = ~c − ~ue is the relative velocity of the photons in respect to the electron. Inserting the Compton tensor in the dierential cross section we nd

q4 m2(4π)2 Z d3p d3k0 0 µ ν 2 f (6.7) dσ = δ(pf + k − pi − k) |λ0 Mµνλ| 2 0 (2π) Epi Epf |urel| Epf 2ωk In the laboratory frame of reference, the relative velocity takes the form

|urel| = |c − ue| = |c| = 1 Moreover, we will use the covariant expression for the Lorentz covariant measure as follows

3 Z +∞ d pf 4 2 2 = d pf δ(pf − m )Θ(pf0 ) 2Ep f pf0 =−∞ 6.2. COMPTON SCATTERING DIFFERENTIAL CROSS SECTION 83

where is a step function. Next, substituting that relation in (6.7), one obtains Θ(pf0 )

2q4m2 Z d3k0 0 µ ν 2 4 2 2 (6.8) dσ = δ(pf + k − pi − k) |λ0 Mµνλ| d pf δ(pf − m )Θ(p0) 0 Epi ωk ωk while, rewriting the nal momentum dierential and the photon dispersion relation as

3 0 ~ 0 2 d k =|k | dωk0 dΩk0 02 ~ 0 2 2 k = 0 →|k | = ωk0 the dierential cross section takes the form

4 Z Z 2q m µ ν 2 4 0 2 2 0 0 0 dσ = dΩk |λ0 Mµνλ| mωk dωk d pf δ(pf + k − pi − k)δ(pf − m )Θ(pf0 ) Epi ωk (6.9) where, in order to simplify te dierential cross-section, we need to evaluate the last integral.

Evaluating the Second Integral. The integral, after dropping out the delta function related with the dierential f the nal momentum (dpf ) is given by

Z +∞ 0 2 2 (i) = m ωk0 dωk0 Θ(m − ωk0 − ωk)δ (pi − k − k ) − m (6.10) ωk0 =0 The argument in the delta function can be simplied as

0 2 0 0 (pi + k + k ) =(pi + k − k )(pi + k − k ) 2 0 2 0 0 0 02 =pi + pi · k − pi · k + k · pi + k − k · k − k · pi − k · k + k 2 2 02 0 0 =pi + k + k + 2k · pi − 2k · pi − 2k · k 2 ~ ~ 0 =m + 2mωk − 2mωk0 − 2ωkωk0 + 2k · k 2 =m + 2m(ωk − ωk0 ) − 2ωkωk0 (1 − cos θ) where θ is the scattering angle. Inserting in (6.10), we have

Z ∞ (i) =m ωk0 dωk0 Θ(m − ωk0 + ωk)δ[2m(ωk − ωk0 ) − 2ωkωk0 (1 − cos θ)] 0 Z ωk+m =m ωk0 dωk0 δ[2m(ωk − ωk0 ) − 2ωkωk0 (1 − cos θ)] (6.11) 0 ( 1 for ωk0 < ωk + m where the upper limit has changed, due to Θ(m − ωk0 + ωk) = 0 for ωk0 > ωk + m In order to annalytically compute the integral we will use the property

Z f(x) f(x)δ[g(x)]dx = dg(x) dx g→0 84 CHAPTER 6. COMPTON SCATTERING

For our integral

f(x) = ωk0

g(x) = 2m(ωk − ωk0 ) − 2ωkωk0 (1 − cos θ) dg(x) = − 2m − 2ω (1 − cos θ) dx k ω − ω 0 g(x) = 0 → 1 − cos θ = m k k ωkωk0 hence, nding

mω 0 (i) = k 2|m + ωk(1 − cos θ)| 2 ω 0 = k (6.12) 2ωk Finally, substituting (6.12) in (6.9) we have for the scattering cross section

2 dσ 2 ωk0 µ ν 2 =r 2 |λ0 Mµνλ| dΩk0 ωk 2 2 ! ω 0 / 0 (p/ + k/ + m)/ / 0 (p/ − k/ + m)/ =r2 k u¯ (p ) λ i λ − λ i λ u (p ) (6.13) 2 sf f 0 si i ωk 2pi · k 2pi · k where r2 = q4.

6.3 Energy Shift Relation

In order to compute the relation that determines the energy shift of the recoiled photon,rstly, we need to recall that we are working in the laboratory frame of reference (6.3a) and constrained by the 4-momentum conservation law (6.4). Combining those equations, we nd that in the laboratory frame, the nal electron 3-momentum and the energy of the photon in the nal state are equal to ~ ~ 0 ~pf =k − k (6.14a) E2 =m2 + ~p2 = m2 + (~k − ~k0)2 (6.14b) pf f Taking the square of the nal state electron 4-momentum we get p2 =E2 − |~p |2 f pf f 2 2 2 =m + |~pf | − |~pf | =m2 Thus, the 4-momentum conservation law becomes

2 0 2 m =(pi + k − k ) 2 =m + 2m(ωk − ωk0 ) − 2ωkωk0 (1 + cos θ) nding that the energy shift of the scattered photon due to recoil reads as

mωk ωk0 = (6.15) m + ωk(1 − cos θ) 6.4. KLEIN-NISHINA FORMULA 85 6.4 Klein-Nishina Formula

In our following discussion, we will nd the unpolarized scattering cross section in Compton scattering, i.e the scattering cross section for unpolarized electrons. To do so, we need to average over the initial spins and sum over the nal electron spin stated. In particular

dσ¯ 1 X dσ = (6.16) dΩk0 2 dΩk0 si,sf In order to simplify our calculations, we will use the familiar trace technique to eliminate the electron spinors through spin summation.

Electron Spin Sums. As we have proven, we will introduce the well known Γ-matrix as follows

2 2 2 ! dσ¯ r ω 0 X X /λ0 (p/ + k/ + m)/λ /λ0 (p/ − k/ + m)/λ = k u¯ (p ) i − i u (p ) 2 sf f 0 si i dΩk0 2ωk 2pi · k 2pi · k sf si 2 2 r ωk0 X X ¯ = 2 u¯sf (pf )Γusi (pi) u¯si (pi)Γusf (pf ) 2ωk sf si 2 2 r ωk0 X X ¯ = 2 usf (pf )¯usf (pf ) Γ [usi (pi)¯usi (pi)]Γ 2ωk sf si 2 2 r ωk0 X X ¯ = 2 usf δ(pf )¯usf a(pf ) Γaβ [usiβ(pi)¯usiγ(pi)]Γγδ 2ωk sf si " # 2 2 p/ + m p + m r ωk0 f /i ¯ = 2 Γaβ Γγδ 2ωk 2m δa 2m βγ 2 2 p + m r ω 0 / p/ + m k f i ¯ (6.17) = 2 Tr Γ Γ 2ωk 2m 2m where the Γ elements equal to

/ 0 (p/ + k/ + m)/ / (p/ − k/ + m)/ 0 λ i λ λ i λ (6.18a) Γ = − 0 2pi · k 2pi · k

/ (p/ + k/ + m)/ 0 / 0 (p/ − k/ + m)/ ¯ λ i λ λ i λ (6.18b) Γ = − 0 2pi · k 2pi · k However, we can simplify the Γ-matrices form using the Dirac algebra that the γ-matrices satisfy {γµ, γν} = 2gµν and applying it in the Γ-matrix moving the p to the right, i.e /i p =p γµγν /i/λ iµ λν µν ν µ =piµ(2g − γ γ )λν =2p · − p i λ /λ/i and performing the same technique in the Γ¯-matrix, but moving p to the left , i.e /i p = 2p · − p /λ/i i λ /i/λ 86 CHAPTER 6. COMPTON SCATTERING

(6.18b) and (6.18b) become

0 2p · / + / k// 2p · 0 / − / k/ / i λ λ0 λ0 λ i λ λ λ λ0 (6.19a) Γ = − 0 2pi · k 2pi · k 2p · / + / k// 2p · / − / k/0/ ¯ i λ λ0 λ λ0 i λ λ0 λ0 λ (6.19b) Γ = − 0 2pi · k 2pi · k In order to simplify even further our calculations, we will choose a convenient gauge in which the polarization vectors are orthogonal to the initial momentum pi. Consequently, the polarization vectors must have the form

λ · pi = 0 (6.20a)

λ0 · pi = 0 (6.20b) Therefore, in the laboratory frame, external photons are of the form µ , µ λ = (0,~λ) λ0 = (0,~λ0 ) Using (6.20a) and (6.20b), the Γ-matrices given by (6.19a) and (6.19b) become

/ k// / k/0/ λ0 λ λ λ0 (6.21a) Γ = + 0 2pi · k 2pi · k / k// / k/0/ ¯ λ λ0 λ0 λ (6.21b) Γ = + 0 2pi · k 2pi · k Inserting the in (6.17) we have to evaluate the trace of

2 2 dσ¯ r ω 0 A A A + A k 1 2 3 4 (6.22) = 2 2 2 + 0 2 + 0 dΩk0 8ωkm (2pi · k) (2k · pi) (2pi · k)(2pi · k ) where h i A = Tr (p + m) k/ (p + m) k/ (6.23a) 1 /f /λ0 /λ /i /λ /λ0 h i A = Tr (p + m) k/0 (p + m) k/0 (6.23b) 2 /f /λ /λ0 /i /λ0 /λ h i A = Tr (p + m) k/ (p + m) k/0 (6.23c) 3 /f /λ0 /λ /i /λ0 /λ h i A = Tr (p + m) k/0 (p + m) k/ (6.23d) 4 /f /λ /λ0 /i /λ /λ0

However, we can go from A1 → A2 and from A3 → A4 through the transformation

λ ↔ λ0 (6.24a) k ↔ − k0 (6.24b)

This transformation holds true for both cases given that the total Feynman amplitude remains unchanged under this transformation (crossing symmetry). Next, we will compute each of the four quantities. However, we should mention some relations that we will use extensively throughout the calculations

0 0 0 k · k = k · k = k · λ =k · λ0 = pi · λ = pi · λ0 = 0 2 λ · λ = λ0 · λ0 = − 1 , pi · pi = m 6.4. KLEIN-NISHINA FORMULA 87

Evaluating A1. For the rst trace we nd h i A = Tr (p + m) k/ (p + m) k/ 1 /f /λ0 /λ /i /λ /λ0 h i = Tr (p + m) k/ p k/ + m(p + m) k/ k/ /f /λ0 /λ/i/λ /λ0 /f /λ0 /λ/λ /λ0 h i = Tr p k/ p k/ + m2 Tr k/ k/ /f /λ0 /λ/i/λ /λ0 /λ0 /λ/λ /λ0 h i = Tr p k/ p k/ /f /λ0 /λ/i/λ /λ0 h i = Tr p k/p k/ /f /λ0 /λ /i /λ/λ0 h i =2(p · k) Tr p k/ i /f /λ0 /λ /λ/λ0 h i = − 2(p · k) Tr p k/ i /f /λ0 /λ/λ /λ0 h i =2(p · k) Tr p k/ i /f /λ0 /λ0 h i =2(p · k) Tr p (2 0 · k − k/ ) i /f λ /λ0 /λ0 h i h i =4(p · k)( 0 · k) Tr p + 2(p · k) Tr p k/ i λ /f /λ0 i /f 0 2 =8(pi · k) (k · pi) + 2(k · λ0 ) where, in order to obtain the nal result we have used the relations

0 k · pf =k · pi

λ0 · pf =λ0 · k deduced from 4-momentum energy conservation as follows

0 0 λ0 · pi + λ0 · k = λ0 · pf +λ0 · k ⇒ k · pf = k · pi 2 0 2 (pi + k) = (pf + k ) ⇒λ0 · pf = λ0 · k Therefore, we have found

0 2 A1 = 8(pi · k) (k · pi) + 2(k · λ0 ) (6.25)

Evaluating A2. Performing the transformation (6.24a), (6.24b) in (6.25) we obtain

0 0 2 A2 = 8(pi · k ) (k · pi) + 2(k · λ) (6.26)

Evaluating A3. h i A = Tr (p + m) k/ (p + m) k/0 3 /f /λ0 /λ /i /λ0 /λ

0 But from 4-momentum conservation law we have pf = pi + k − k , thereby nding h i A = Tr (p + k/ − k/0 + m) k/ (p + m) k/0 3 /i /λ0 /λ /i /λ0 /λ h i h i = Tr (p + m) k/ (p + m) k/0 + Tr (k/ − k/0) k/ (p + m) k/0 /i /λ0 /λ /i /λ0 /λ /λ0 /λ /i /λ0 /λ (1) (2) =A3 + A3 88 CHAPTER 6. COMPTON SCATTERING

Consequently, we need to compute each trace term.

Evaluating (1). Firstly, we will calculate the trace A3 h i A(1) = Tr (p + m) k/ (p + m) k/0 3 /i /λ0 /λ /i /λ0 /λ hh i i = Tr 2p · k + k/(m − p ) (p + m)k/0 i /λ0 /λ /λ0 /λ /i /i /λ0 /λ h i h i = Tr 2p · k (p + m)k/0 + Tr k/(m − p )(m + p )k/0 i /λ0 /λ /i /λ0 /λ /λ0 /λ /i /i /λ0 /λ h i =2(p · k) Tr (p + m)k/0 i /λ0 /λ /i /λ0 /λ h i =2(p · k) Tr p k/0 i /λ0 /λ/i /λ0 /λ h 0 i =2(p · k) Tr 2 0 · − p k/ i λ λ /λ/λ0 /i /λ0 /λ h h 0 i h 0ii =2(p · k) 2( 0 · ) Tr p k/ − Tr p k/ i λ λ /i /λ0 /λ /i

Evaluating (2). For the second trace we nd A3 h i A(2) = Tr (k/ − k/0) k/ (p + m) k/0 3 /λ0 /λ /i /λ0 /λ h i = Tr k/ k/ (p + m) k/0 − Tr k/0 k/ (p + m) k/0 /λ0 /λ /i /λ0 /λ /λ0 /λ i /λ0 /λ h 0 i 0 h 0i =2( 0 · k) Tr k/ (p + m) k/ − 2( · k ) Tr k/ (p + m) k/ λ /λ /i /λ0 /λ λ /λ0 /λ /i /λ0 h 0 i 0 h 0i =2( 0 · k) Tr k/ p k/ − 2( · k ) Tr k/ p k/ λ /λ/i/λ0 /λ λ /λ0 /λ/i/λ0 h 0i 0 h 0 i =2( 0 · k) Tr k/ p k/ − 2( · k ) Tr k/ p k/ λ /λ /λ/i/λ0 λ /λ/i/λ0 /λ0 h 0i 0 h 0i =2( 0 · k) Tr k/p k/ − 2( · k ) Tr k/ p k/ λ /i/λ0 λ /λ/i Therefore, in summary we have found

(1) h h 0 i h 0ii A =2(p · k) 2( 0 · ) Tr p k/ − Tr p k/ (6.27a) 3 i λ λ /i /λ0 /λ /i (2) h 0i 0 h 0i A =2( 0 · k) Tr k/p k/ − 2( · k ) Tr k/ p k/ (6.27b) 3 λ /i/λ0 λ /λ/i

Final Evaluation of A3. In order to nd the nal expression for the third term we will use the (5.46) mentioned in the previous chapter, hence acquiring

(1) 0 2 (6.28a) A3 =8(pi · k)(pi · k ) 2(λ0 · λ) − 1 (2) 0 2 2 0 (6.28b) A3 =8(pi · k ) (k · pi) − 8(λ0 · k) (k · pi) while adding them we have

0 2 0 2 2 0 A3 = 8(pi · k)(pi · k ) 2(λ0 · λ) − 1 + 8(pi · k ) (k · pi) − 8(λ0 · k) (k · pi) (6.29)

Evaluating A4. Performing the transformation (6.24a), (6.24b) in (6.25) we obtain 0 2 0 2 2 0 A4 = 8(pi · k)(pi · k ) 2(λ0 · λ) − 1 + 8(pi · k ) (k · pi) − 8(λ0 · k) (k · pi) (6.30) hence, A3 = A4. 6.4. KLEIN-NISHINA FORMULA 89

Combining the Results. To sum up for the four traces, we have found

0 2 A1 =8(pi · k) (k · pi) + 2(k · λ0 ) 0 0 2 A2 =8(pi · k ) (k · pi) + 2(k · λ) 0 2 0 2 2 0 A3 = A4 =8(pi · k)(pi · k ) 2(λ0 · λ) − 1 + 8(pi · k ) (k · pi) − 8(λ0 · k) (k · pi) Fially, inserting them in our cross section we can construct the unpolarized dierential photon scattering cross section, arriving at

2 2 0 2 0 0 2 dσ¯ r ωk0 8(pi · k) [(k · pi) + 2(k · λ0 ) ] 8(pi · k ) [(k · pi) + 2(k · λ) ] = 2 2 2 + 0 2 + dΩk0 8ωkm 4(k · pi) 4(k · pi) 0 2 0 2 2 0 16(pi · k)(pi · k ) [2(λ0 · λ) − 1] + 16(pi · k ) (k · pi) − 8(λ0 · k) (k · pi) + 0 4(k · pi)(k · pi) 2 2 0 r ωk0 k · pi k · pi 2 0 = 2 2 + 0 + 4(λ · λ ) − 2 8ωkm k · pi k · pi Finally, inserting the laboratory frame kinematics that read

k·pi = mωk 0 k ·pi = mωk0 mωk ωk0 = m + ωk(1 − cos θ) we nd

2 2 dσ¯ r ωk0 ωk0 ωk 2 0 (6.31) = 2 + + 4(λ · λ ) − 2 dΩk0 4m ωk ωk ωk0 which is the Klein-Nishina formula rst derived in 1929, that described the Compton scattering for polarized photons and unpolarized electrons. In order to ensure the validity of our result, we can see that in the limit of low photon energies (ωk → 0), then ωk0 ' ωk and the photo scattering cross section takes the form

2 dσ¯ r 2 0 = 2 (λ · λ ) dΩ 0 4m k ωk→0 which is the scattering cross section of the Thompson's scattering formula. Bibliography

[1] F. Mandl, G. Shaw, Quantum Field Theory, 2nd Edition, John Wiley and Sons In- terscience Publication, 2010

[2] Dr. David Tong, Quantum Field Theory, lecture notes, University of Cambridge, 2006

[3] Dr. Walter Greiner, Dr. Joachim Reinhardt, Field Quantization, Springer, 1996

[4] Michael E. Peskin, Daniel V. Schroeder, An Introduction to Quantum Field Theory, Perseus Books, 1999

[5] Michio Kaku, Quantum Field Theory: A Modern Introduction, Oxford University Press, 1993

[6] Claude Itzykson, Jean-Bernard Zuber, Quanrum Field Theory, McGraw-Hill Interna- tional Book Company, 1980

[7] Dr. Walter Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd Edition, Springer, 2000

[8] Dr. Walter Greiner, Dr. Joachim Reinhardt, Quantum Electrodynamics, 3rd Edition, Springer, 2002

[9] Riccardo D' Auria, Mario Trigiante, From Special Relativity to Feynman Diagrams: A Course of Theoretical Particle Physics for Beginners, Springer, 2012

[10] Eugene Merzbacher, Quantum Mechanics, 2nd Edition, Wiley International Edition, 1970

[11] Wikipedia (Wikipedia the Free Encyclopedia)

• C-number URL: http://en.wikipedia.org/wiki/C-number 2016, February 17 • Classical Field Theory URL: http://en.wikipedia.org/wiki/Classical_field_theory 2016, February 17 • Functional (Mathematics) URL: http://en.wikipedia.org/wiki/Functional_(mathematics) 2016, February 17 • Functional Derivative URL: http://en.wikipedia.org/wiki/Functional_derivative 2016, February 17

90 BIBLIOGRAPHY 91

• Gamma matrices URL: https://en.wikipedia.org/wiki/Gamma_matrices 2016, February 17 • Heisenberg Picture URL: http://en.wikipedia.org/wiki/Heisenberg_picture 2016, February 17 • Interaction Picture URL: http://en.wikipedia.org/wiki/Interaction_picture 2016, February 17 • Schrödinger picture URL: https://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture 2016, February 17 • Trace (linear algebra) URL: https://en.wikipedia.org/wiki/Trace_(linear_algebra) 2016, February 17

Figures 2.1 and 2.2 (page 39) were adopted and modied from Dr. David Tong, Quantum Field Theory, [2]. Figure 5.2 (page 75) was adopted from Dr. Walter Greiner, Dr. Joachim Reinhardt, Field Quantization, [3]. Figure 6.1 (page 81) was adopted and modied from Dr. Walter Greiner, Dr. Joachim Reinhardt, Quantum Electrodynamics [8].