course assignment topic quantum electrodynamics

3II.BIOGRAPHIES OF PHYSICISTS...3III.BACKGROUND OF QED...5IV.DEVELOPMENT OF THE THEORY...51.Nuclear force electrodynamics theory Feynman, Schwinger...62.Theory of quantum renormalization

UNIVERSITY OF ECONOMICS AND BUSINESS

COURSE ASSIGNMENT

Student: Nguyen Huong Giang Student’s ID: 22055812 Instructor: Dr Nguyen Quoc Hung

HANOI, 2023

TABLE OF CONTENTS

ABSTRACT 3

I INTRODUCTION 3

II BIOGRAPHIES OF PHYSICISTS 3

III BACKGROUND OF QED 5

IV DEVELOPMENT OF THE THEORY 5

V ANALYSIS 6

1 Nuclear force electrodynamics theory (Feynman, Schwinger) 6

2 Theory of quantum renormalization electrodynamics (Tomonaga) 10

3 Mathematical formulation 11

VI IMPLICATIONS OF THE NOBEL PRIZE IN PHYSICS IN 1965 17

VII CONCLUSION 18

REFERENCE 20

TABLE OF FIGURES

Figure 1: The elementary components of Feynman Diagrams for Quantum Electrodynamics 7

Figure 2: Compton scattering 8

Figure 3: Addition of probability amplitudes as complex numbers 9

Figure 4: Multiplication of probability amplitudes as complex numbers 9

Figure 5: One-loop contribution to the vacuum polarization function 17

Figure 6: One-loop contribution to the electron self-energy function 17

Figure 7: One-loop contribution to the vertex function 17

This scientific report provides an overview of the groundbreaking research that led to theNobel Prize in Physics in 1965, awarded to Sin-Itiro Tomonaga, Julian Schwinger, andRichard Feynman for their contributions to the development of quantum electrodynamics(QED) The report discusses the limitations of the old model of QED, which only includedthe exchange of individual photons, and the groundbreaking discoveries made by Tomonaga,Schwinger, and Feynman regarding the complex nature of electron-electron scattering and theexchange of multiple photons The report also highlights the importance of the new QEDmodel in accurately describing high-energy physics phenomena The report concludes byemphasizing the significance of the Nobel Prize-winning research in advancing the field ofphysics and contributing to our understanding of the fundamental principles of the universe

I INTRODUCTION

The Nobel Prize in Physics is one of the five Nobel Prizes established by Swedishinventor Alfred Nobel Nobel was known for his contributions in fields such as chemistry,physics, medicine, literature, and peace After his death in 1896, the idea of creating awards

to honor significant contributions in these fields was proposed and implemented by the NobelFoundation The Nobel Prize in Physics is awarded annually to honor significantachievements in physics Initially, the award was given to those who made significantcontributions to inventions in the field of physics, but it has since expanded to include thosewho have made important contributions to discoveries and theories in the field In 1965, theNobel Prize in Physics was awarded to three physicists - Richard P Feynman, JulianSchwinger, and Tomonaga Shin'ichirō - for their pioneering work in the field of quantumelectrodynamics (QED) QED is the study of the interaction between electromagneticradiation and matter and is one of the most successful and accurate theories in physics Thecontributions of Feynman, Schwinger, and Tomonaga to QED were groundbreaking, and theirwork laid the foundation for many advances in the field, including the development of theStandard Model of particle physics In this report, I will explore the contributions of thesethree Nobel laureates to the field of QED, their impact on modern physics and technology,and the significance of their work in the broader context of scientific discovery We willexamine the historical context of their discoveries, the methodology and techniques used intheir research, and the implications and applications of their findings Overall, the work ofFeynman, Schwinger, and Tomonaga represents a major milestone in the history of physics,and their contributions have had a profound impact on our understanding of the universe andthe fundamental laws that govern it By exploring their discoveries and their enduring legacy,

we can gain a deeper appreciation for the power and potential of scientific exploration and

II BIOGRAPHIES OF PHYSICISTS

Richard Feynman (1918-1988) was an American physicist, and one of the three

scientists awarded the Nobel Prize in Physics in 1965 He was born in New York andgraduated from the Massachusetts Institute of Technology (MIT) in 1939 He then continuedhis studies and research at Princeton University, where he received his Ph.D in 1942 DuringWorld War II, Feynman worked at the Manhattan Project and participated in the development

of the atomic bomb After the war, Feynman returned to Princeton and later worked atCornell University He made significant contributions to the theory of nuclear physics,quantum electrodynamics, and quantum optics Feynman also developed a unique method forcalculating quantum electrodynamics effects, called the Feynman diagram In addition,Feynman was a talented and renowned educator He wrote many books on physics and othersubjects but is best known for "The Feynman Lectures on Physics," a collection of hislectures at the California Institute of Technology, where he taught from 1950 to 1981 Thisbook has become an important teaching resource in the field of physics and is considered one

of the greatest works on physics in the 20th century

Julian Schwinger (1918-1994) was an American physicist, and one of the three

scientists awarded the Nobel Prize in Physics in 1965 He was born in New York andgraduated from Columbia University in 1936 He then continued his studies and research atHarvard University, where he received his Ph.D in 1939 Schwinger made significantcontributions to quantum electrodynamics and was one of the first to develop therenormalization theory in nuclear physics Schwinger taught at Harvard University from 1945

to 1972 and then moved to the University of California, Los Angeles He made significantcontributions to the theory of quantum electrodynamics and nuclear physics and helped createimportant computational tools to study the interactions between particles In addition,Schwinger also developed some applications of quantum electrodynamics theory for otherproblems, such as statistical physics and quantum statistical physics He also developed somenew research methods to solve problems in nuclear physics and quantum optics

Tomonaga Shin'ichirō (1906–1979) was a Japanese physicist and the last of the three

scientists to win the Nobel Prize in Physics in 1965 He was born in Tokyo and graduatedfrom Kyoto University in 1929 He then continued his studies and research at the University

of Leipzig in Germany and received his Ph.D in 1938 Tomonaga returned to Japan andbecame a professor at the University of Tokyo, where he continued his research on the theory

of electromagnetic interactions with matter and electrons and made significant contributions

to quantum electrodynamics He developed the renormalization method of calculation tosolve problems of infinity in quantum electrodynamics theory Tomonaga also helpeddetermine how the quantum electrodynamics theory and Albert Einstein's theory of relativity

could be combined to form a relativistic quantum electrodynamics theory He also developedsome applications of this theory in other problems, including the theory of nuclear structure.

III BACKGROUND OF QED

Scientists needed a quantum-mechanical explanation of light when they observedphenomena that could not be explained with the classical theory of electromagnetic radiationdeveloped by British physicist James Clerk Maxwell in the 1860s This classical theorypredicted the behavior of light as waves of vibrating electric and magnetic fields However, in

1887, German physicist Heinrich Hertz's discovery of the photoelectric effect showed that theenergy of the electrons produced by light was only dependent on the wavelength of the light,not its intensity German physicist Max Planck's further work on light suggested that it maycome in tiny packets, or quanta, of energy, while Einstein proposed in 1905 that light could

be composed of particles called photons

Despite initial skepticism from most physicists, American physicist Arthur Compton'sdiscovery of the Compton effect in 1923 provided evidence that light has momentum, aproperty usually associated with particles, not electromagnetic waves This led physicists todevelop a new description of light and its interaction with particles, while also working ondeveloping the important new description of matter known as quantum mechanics Many ofthe same physicists, including Planck, Einstein, Danish physicist Niels Bohr, and Germanphysicist Arnold Sommerfeld, worked on both problems from 1900 to 1922 The resultingquantum-mechanical explanation of light and matter describes both in terms of waves andparticles

IV DEVELOPMENT OF THE THEORY

In 1926, German physicists Max Born, Werner Heisenberg, and Ernst Pascual Jordanpublished the first Quantum Electrodynamics (QED) theory, which explained that the energyand momentum of the electric and magnetic fields in a light ray come in bundles calledphotons British physicist Paul A M Dirac applied the rules of the quantum theory toelectromagnetic radiation in 1927, resulting in a theory that explained how atoms emit andabsorb photons Dirac also constructed a description of electrons in 1928 that was consistentwith both quantum mechanics and the special theory of relativity, which was important inreconciling quantum mechanics and relativity since descriptions involving photons andvelocities near the speed of light must involve special relativity Dirac's equation predictedthe existence of antimatter

In the 1930s, physicists such as Heisenberg, Wolfgang Pauli, and J Robert Oppenheimeradded more corrections to QED to make the theory more accurate, but their correctionsintroduced some troubling infinite terms in the equations of QED Physicists struggled foralmost two decades to remove the infinite terms from QED equations and keep the theory

In the late 1940s, Willis Lamb and Robert Retherford discovered the Lamb shift, whichshowed that the interaction between light and electrons was more complicated thanpreviously believed This led to physicists redefining QED's description of the electron in aprocess called renormalization, which removed the infinite terms plaguing the theory andmade QED equations match the new experimental results.

Physicists made changes to QED to account for the Lamb shift, resulting in a set ofequations that required the addition of a series of terms, each of which violated the specialtheory of relativity In the early 1950s, Richard Feynman, Julian Schwinger, and TomonagaShin'ichirō developed versions of QED that were consistent with the special theory ofrelativity Feynman's method allowed physicists to represent particle interactions with simplediagrams, called Feynman diagrams Later, Freeman Dyson showed that the two approachesproduced the same results, and that Feynman's approach could be derived from the equations

of Schwinger and Tomonaga Feynman, Schwinger, and Tomonaga won the 1965 Nobel Prize

in physics for their work with QED, which has been one of the most successful theories ofmodern physics, showing remarkable agreement with experimental results

V ANALYSIS

1 Nuclear force electrodynamics theory (Feynman, Schwinger)

Towards the end of his life, Richard Feynman delivered a series of lectures on QuantumElectrodynamics (QED) for the general public These lectures were later transcribed andpublished as the book "QED: The Strange Theory of Light and Matter" in 1985 This book isconsidered a classic non-mathematical exposition of QED, and Feynman presented threebasic actions that form the foundation of QED

The first action is the movement of a photon from one place and time to another Thesecond action is the movement of an electron from one place and time to another The thirdaction is the emission or absorption of a photon by an electron at a specific place and time.These three actions are visually represented by the three basic elements of Feynmandiagrams: a wavy line for the photon, a straight line for the electron, and a junction of twostraight lines and a wavy one for a vertex representing the emission or absorption of a photon

by an electron

Feynman diagrams are a form of visual shorthand that allow physicists to representcomplex interactions between particles and provide a way to calculate the probabilities ofdifferent outcomes Feynman's presentation of QED in "QED: The Strange Theory of Lightand Matter" has become a significant contribution to the popular understanding of quantummechanics and the nature of light and matter

Figure 1: The elementary components of Feynman Diagrams for Quantum

Electrodynamics

Richard Feynman introduced a unique shorthand for numerical quantities known asprobability amplitudes in addition to the visual shorthand for the actions in QuantumElectrodynamics (QED) These amplitudes are complex numbers that describe the probability

of a particular event occurring in a quantum system The probability of a specific outcome iscalculated by squaring the absolute value of the total probability amplitude

Feynman developed a shorthand notation to represent the probability amplitudesassociated with the movement of particles For a photon moving from one place and time A toanother place and time B, the associated probability amplitude is written as P (A to B) inFeynman's notation This quantity depends solely on the momentum and polarization of thephoton

Similarly, for an electron moving from place C to time D, the associated probabilityamplitude is written as E (C to D) However, this quantity not only depends on themomentum and polarization of the electron but also on a constant known as n, whichFeynman referred to as the "bare" mass of the electron This constant is related to theelectron's measured mass but is not identical to it

Finally, Feynman introduced a quantity known as j to describe the probability amplitudefor an electron to emit or absorb a photon This quantity is sometimes referred to as the

"bare" charge of the electron and is a constant related to the electron's measured charge e butnot identical to it

Suppose we have an electron at a specific place and time, labeled A, and a photon at adifferent place and time, labeled B A common question in physics is to determine theprobability of finding the electron at a later time and a different place, labeled C, and the

photon at yet another place and time, labeled D The most straightforward way to achieve this

is for the electron to move from A to C, and the photon to move from B to D These aresimple actions known as elementary processes To calculate the probability of both processeshappening together, we need to know the probability amplitudes of each sub-process, denoted

as E (A to C) and P (B to D) We can then estimate the overall probability amplitude bymultiplying these two values, using rule b) Once we have the estimated overall probabilityamplitude, we can calculate the estimated probability by squaring it

There are alternative ways in which the end result of finding the electron at C and thephoton at D could occur For example, the electron could move to a different place and time,labeled E, where it absorbs the photon before moving on and emitting a new photon at F Theelectron then moves to C while the new photon moves to D To calculate the probability ofthis complex process, we need to know the probability amplitudes of each individual actioninvolved, including three electron actions, two photon actions, and two vertexes (oneemission and one absorption) We can estimate the total probability amplitude by multiplyingthe probability amplitudes of each action for any chosen positions of E and F To find theactual probability, we need to add up all the probability amplitudes for all possible positions

of E and F, which in practice requires integration

Another possibility is that the electron first moves to a different place and time, labeled

G, where it emits a photon that moves to D, while the electron moves on to H, where itabsorbs the first photon, before finally moving on to C This process is known as Comptonscattering

Figure 2: Compton scattering

a Probability amplitudes

Quantum mechanics presents a significant departure from traditional probabilitycalculations While probabilities are still represented by real numbers, as they are in our daily

lives, the way in which probabilities are calculated is different In quantum mechanics,probabilities are computed as the square of the absolute value of probability amplitudes,which are complex numbers.

For a given process, if two probability amplitudes, v, and , are involved, the probabilityw

of the process will be given either by

or

The theory of complex numbers employs addition and multiplication as commonoperations, which are illustrated in the figures To find the sum of two complex numbers, thesecond arrow's starting point is placed at the end of the first arrow The resulting sum isrepresented by a third arrow that goes directly from the beginning of the first arrow to the end

of the second arrow When multiplying two complex numbers represented by arrows, thelength of the product arrow is equal to the product of the two lengths The direction of theproduct arrow is determined by adding the angles that each of the two arrows have beenturned through relative to a reference direction This yields the angle by which the productarrow is turned relative to the reference direction

Figure 3: Addition of probability amplitudes as complex numbers

Figure 4: Multiplication of probability amplitudes as complex numbers

An important detail associated with the polarization of electrons is that electrons arefermions and follow Fermi-Dirac statistics As a result, if we have the probability amplitudefor a complex process involving multiple electrons, we must also include the complementaryFeynman diagram in which two electron events are exchanged In this case, the resultingamplitude is the reverse, or negative, of the first For example, consider the case of twoelectrons starting at A and B and ending at C and D The amplitude would be calculated as the

"difference" between E (A to D) × E (B to C) and E (A to C) × E (B to D) This is in contrast

to our everyday idea of probabilities, where we would expect the amplitude to be a sum

b, Propagators

To complete the calculation, it is necessary to determine the probability amplitudes forthe photon and the electron, denoted as P (A to B) and E (C to D), respectively Theseprobability amplitudes are obtained by solving the Dirac equation, which describes thebehavior of the electron's probability amplitude, and the Maxwell's equations, which describethe behavior of the photon's probability amplitude These solutions are known as Feynmanpropagators To facilitate understanding and communication, the Feynman notation is oftentranslated into a notation commonly used in standard literature:

where a shorthand symbol such as x{A} stands for the four real numbers that give thetime and position in three dimensions of the point labeled A

c, Mass renormalization

In the early days of Feynman's approach, a significant problem arose that impededprogress for two decades Although the approach was based on three fundamental "simple"actions, the rules required that all possible Feynman diagrams with the given endpoints must

be taken into account when calculating the probability amplitude for an electron to movefrom point A to point B This meant that there could be multiple ways for the electron totravel, such as emitting and absorbing photons at various points along the way This resulted

in a fractal-like situation where a line could break up into a collection of "simple" lines, andeach of these lines could be further composed of simpler lines, and so on infinitely Thiscomplexity presented a significant challenge to handle The situation became even morechallenging when it was found that the simple correction mentioned above led to infiniteprobability amplitudes, which was a disaster To address this issue, the technique ofrenormalization was developed over time However, Feynman himself remained dissatisfiedwith the solution, describing it as a "dippy process." Despite the difficulties, Feynman'sapproach has revolutionized the field of quantum mechanics and continues to be a valuabletool for understanding the behavior of subatomic particles