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“I believe that ideas such as absolute certitude, absolute exactness, final truth, etc.
are figments of the imagination which should not be admissible in any field of
science... This loosening of thinking seems to me to be the greatest blessing which
modern science has given to us.”

--Max Born

Quantum mechanics

Matrix mechanics   |   Dirac Equation   |   Renormalization    

Theory of relativity

Special theory of relativity   |   General theory of relativity
feynman qoute

Quantum electrodynamics feynman

Quantum electrodynamics has its lair in quantum mechanics. So understanding quantum mechanics is a prerequisite to understand quantum electrodynamics. Let's look at the Bohr - Sommerfield theory :
feynman qoute

The laws of quantum electrodynamics

1. the amplitude that an atomic system will absorb a photon during the process of transition from one state to another is exactly the same as the amplitude that the transition is made under the influence of external potention equal to the electromagnetic wabe representing that photon, provided (a) the classical wave is normalized to represent an energy density equal to hw times the probability desnsity per cubic cm to find the photon. (b) the real classical wave is split into two complex waves : one e^(iwt) and other is e^(-iwt) and only the -iwt part is kept. and (c) potential acts only once in perturbation : that is only terms to first order in electromagnetic field strength should be retained.
Replacing the word absorbed by emission in rule 1 requires only to change -wt into wt.
2. the number of states per cubic centimeter of a given polarization is d^3 K/(2π)^3
3. Photons obey Bose-Einstein statistics. That is , the states of a given collection of photons must be symmetric(exchange of photons and amplitudes). Also statistical weight of a state of n identical photons must is 1 instead of n!.
Thus in general a photon can be represented by a solution of Maxwell equations if properly normalized.
Schrodinger showed that particle behaves like waves. He developed his wave equation describing wave function as a probability of finding a particle in a certain place at a certain time.

Main ingredients

quantum elctrodynamics feynman

"On a microscopic level, all four forces are not forces in the usual sense of the word. The way that physics today explains the forces of nature is by exchange of gauge particles. Gauge particles are particles which are exchanged between other particles that form the genuine constituents of matter (quarks and leptons). So when an electron repels or attracts another electron or positron, what happens is that there is a "force-carrying field" between them. In that particular case, it is actually a field composed of photons! Photons are the mediators of the electromagnetic interaction, and particles interacting electromagnetically constantly exchange photons between them (those photons can not be "seen" in the usual sense, but that is another story). Now when an electron emits or absorbs a photon, it more or less stays the same, only its momentum and spin might change."

Because electromagnetism is a field theory, the result of QED was a quantum field theory — a quantum theory that possess a value at every point in space. You can think that the mathematics of such a theory was intimidating, to say the least, even to those trained in physics and mathematics.


Quantum electrodynamics is the first true theory of quantum field theory. This is the theory of electromagnetic interaction with matter. In other words it explains the electromagnetic force in quantum mechanical way. Dirac developed his relativistic quantum field equation which laid the foundation of this impressive branch of theoretical physics. It is often called the "Jewel of physics". The electromagnetic force can be thought of exchange of particles called photons. Quantum electrodynamics is concerned with this aspect of interaction between electrons and photons which is the quantized form of electromagnetism.
The basic ingredient of quantum electrodynamics is field. As stated, photons are excitation states of electromagnetic field. How does the excitation of field is interpreted as particles in quantum field theory? Idea is quite simple. Suppose you have a string of length L. Now excite the strings to vibrate transversely. The field φ (x,t) will represent the excitation of the spring. According to quantum field theory there are infinite numbers of springs attached to every point in space. As the spring vibrate the energy of the string creates particle according to Einstein's mass energy equivalence principle. This is the fundamental concept of all quantum field theories.
In quantum field theory the field ψ is itself an operator. It is operator valued distribution. To every space time point x it associates an operator ψ(x) acting on the Hilbert space of quantum field theory. It has a similar role like the position and momentum operators in quantum mechanics. Two specific field operators do not commute.


quantum electrodynamics operators
This relation holds true because quantum field φ(x,t) is contructed from annihilation and creating operator. So fields are themselves operators. This is how it is formed :

quantum electrodynamics feynman

The classical view of elecromanetic force relies on the concept of electric and magnetic field lines extending outwards from charged particles. The force fields permeates everywhere and the force that these force field creates diminishes as the inverse square of distance from the charged particles. What could be the quantum mechanical description of the electromagetic force? The quantum mechanical view is that there are countless number of particle and antiparticles created from the energy of the field around the charged particles. These particles surround the charged particle like fog or mist surrounding an object. This phenomena of particle and antiparticle creation is called quantum fluctuation. As we move closer and closer to electron we become less subject to these virtual particles. That results in an increase of visibility of charged particle. So the upshot is that as the distance decreases the force increases. In case of strong force and weak force, the magnitude of the force diminishes as the distance increases.
Quantum field theory is formulated using two basic approach. Classical mechanics is quantized and then combined with special theory of relativity.



Quantum field theory diagram


quantum electrodynamics

The equation above seems to be somewhat complicated. Let us try another one :

quantum electrodynamics equation

The lagrangian of electrodynamics are invariant under local gauge transformation. The Noether current density J(u) can be written in terms of dirac field &psi of electron. The potentoial A(u) transforms in certain way with the derivative of some function of space x. This corresponds to the phase transformation of dirac field φ. That is the interpretation of local gauge symmetry. In field theory, the gauge transformation is the change of system field configurations. Under this kind of change you still get the same results. Here are the examples of gauge transformations :

gauge transformation
A gauge symmetry is one in which the group transformations can have spacetime dependence ; in this case the potential A and field ψ are transformed into different location in space and different moment in time.
α is the coupling constant. Its value captures the likelihood thatone particle will fire a force-carrying bullet and that the second partcle will receive it. For particles such as electrons, governed by electromagnetic force, experimental measurements have determined that the coupling constant , associated with photon-bullet , is about .0073.

The arbitrary number of gauge functionS f(x) corresponds to U(1) gauge freedom of this thoery. Gauge fixing can be done in various ways.

In the picture below the basic form of feynman diagram for interaction between electrons and photons are depicted. Feynman diagrams are space-time diagrams which tell how many way particles can interact with themselves or others. Here two electrons interact with each other by emitting a photon which is represented by wavy line. And after the interaction they scatter in two different directions.


feynman diagram quantum electrodynamics

The horizontal and vertical lines are space and time axis. This is the basic feynman diagram of interaction between photon and electron.
Also (and perhaps more important from a physics standpoint), a quantum field theory (at least those that seem to match our real world) quickly reaches infinity if distances become too small. To see how these infinities can pop out, consider both the fact that electromagnetic forces get larger at small distances (infinitely larger at infinitely small distances) and also the distance and momentum relationship from the uncertainty principle of quantum mechanics.
Even talking about the instances where two electrons are incredibly near to each other (such as within a Planck length) becomes effectively impossible in a world governed by quantum physics.
By quantizing electromagnetics, as QED does, Feynman, Schwinger, and Tomonaga were able to use the theory despite these infinities. The infinities were still present, but because the virtual photon meant that the electrons didn’t need to get so close to each other, there weren’t as many infinities, and the ones that were left didn’t enter physical predictions.

scattering theory quantum electrodynamics

"Science may set limit to knowledge but it should not set limit to imagination.."

A propagator in quantum field theory is a function that assigns a probability amplitude to a particle for a change of space and time coordinate. By assigning probability amplitudes for different kinds of lines in Feynman diagram can help us solve specific Quantum electrodynamics prolem i.e the interaction of electron with photon. For Klein Gordon equation the propagator is the representative of the field (φ).


propagator


propagator

The propagator that helps solve the equation is Δ(x-y) which value can be calculated using the integral of the form as given above. The propagator Δ(x-y) assigns the probability amplitude of a particle for the path from point x to y. The green function which solves the equation plays the role of propagator in quantum electrodynamics. Here is an explicit example for solving green function to find propagator

propagator

OR
feynmanpropagator

At point x the field φ contain's the electron's properties like energy , momentum and gives other quantum mechanical behavior. We generally call these properties the electron's state function. Similar condition applies to point y. So what we can infer is that quantum behaviour of electron changes from point x to point y. It may be others particles like fermions, photons etc.
Two electron might come from different direction and they annihilate by producing a photon. The photon can again create a electron pair. In this way fundamental interaction of photon and electron can be found by Feynman diagram. There are four lines representing four different paths of two electrons. Each line can be assigned a probability amplitude by propagator function K. Node 5 and 6 are called vertex. The wavy line is the propagator for virtual particle. δ(+) represents the probability amplitude of photon's propagation. Here s(56) represented the distance in space and time that the photon travels ; S^2 = x^2 -t^2 ;


Schrodinger equation

Three different probability amplitudes for three Feynman diagram are found out to be :
Schrodinger equation

In Feynman’s hands, then, the diagram for electron - electron scattering stood in for the mathematical expression (itself written in terms of the abbreviations K+ and δ+):

mathematical expression

In the picture above the parameter p is the momentum , m is the mass , η is the Minkowski metric, γ is another matrix of specific values and four dimensional dirac delta function is used at vertex point.

"I can live with doubt and uncertainty and not knowing. I think it is more interesting to live not knowing that to have answers which may be wrong..Richard Feynman"

Feynman diagrams are a powerful tool for making calculations in quantum theory. As in any quantum-mechanical calculation, the currency of interest is a complex number, or “amplitude,” whose absolute square yields a probability. For example, A(t, x) might represent the amplitude that a particle will be found at point x at time t; then the probability of finding the particle there at that time will be |A(t, x)|^2.
In QED, the amplitudes are composed of a few basic ingredients, each of which has an associated mathematical expression. To illustrate, I might write: —amplitude for a virtual electron to travel undisturbed from x to y: B(x,y); —amplitude for a virtual photon to travel undisturbed from x to y: C(x,y); and —amplitude for electron and photon to scatter: eD.
Here e is the charge of the electron, which governs how strongly electrons and photons will interact.
Feynman introduced his diagrams to keep track of all of these possibilities. The rules for using the diagrams are fairly straightforward: At every “vertex,” draw two electron lines meeting one photon line. Draw all of the topologically distinct ways that electrons and photons can scatter. Then build an equation: Substitute factors of B(x,y) for every virtual electron line, C(x,y) for every virtual photon line, eD for every vertex and integrate over all of the points involving virtual particles. Because e is so small (e^2~ 1/137, in appropriate units), diagrams that involve fewer vertices tend to contribute more to the overall amplitude than complicated diagrams, which contain many factors of this small number. Physicists can thus approximate an amplitude, A, by writing it as a series of progressively complicated terms.
For example, consider how an electron is scattered by an electromagnetic field. Quantum-mechanically, the field can be described as a collection of photons. In the simplest case, the electron (green line) will scatter just once from a single photon (red line) at just one vertex (the blue circle at point x0):


physics with feynman diagram

Only real particles appear in this diagram, not virtual ones, so the only contribution to the amplitude comes from the vertex. But many more things can happen to the hapless electron. At the next level of complexity, the incoming electron might shoot out a virtual photon before scattering from the electromagnetic field, reabsorbing the virtual photon at a later point:


physics with feynman diagram

In this more complicated diagram, electron lines and photon lines meet in three places, and hence the amplitude for this contribution is proportional to e^3. Still more complicated things can happen. At the next level of complexity, seven distinct Feynman diagrams enter:


physics with feynman diagram

As an example, we may translate the diagram at upper left into its associated amplitude:


physics with feynman diagram

The total amplitude for an electron to scatter from the electromagnetic field may then be written :


physics with feynman diagram

and the probability for this interaction is |A|^2. Robert Karplus and Norman Kroll first attempted this type of calculation using Feynman’s diagrams in 1949; eight years later several other physicists found a series of algebraic errors in the calculation, whose correction only affected the fifth decimal place of their original answer. Since the 1980s, Tom Kinoshita (at Cornell) has gone all the way to diagrams containing eight vertices—a calculation involving 891 distinct Feynman diagrams, accurate to thirteen decimal places!—D.K.
There are other interpretations of Feynman diagram. One can be done through s-matrix:
The probability amplitude for a transition of a quantum system from the initial state |i⟩ to the final state | f ⟩ is given by the matrix element
S(fi) = < f|S|i >
In the canonical (first quantization ) quantum field theory the S-matrix is represented within the interaction picture by the perturbation series in the powers of the interaction Lagrangian,


perturbation series feynman diagram
where L(v) is the interaction Lagrangian and T signifies the time-ordered product of operators.
perturbation series lagrangian
A Feynman diagram is a graphical representation of a term in the Wick's expansion of the time-ordered product in the nth order term S(n) of the S-matrix. The second order perturbation term in S-matrix is

second order term
This term also represents a feynman diagram.
where N signifies the normal-product of the operators and (±) takes care of the possible sign change when commuting the fermionic operators to bring them together for a contraction (a propagator).
Some notations useful in QED are :

notations in QED

For explanation of γ see Dirac equation
Finally we can formulate the Lagrangian of the electromagnetic interaction as algebraic sum of three individual components. Ψ is electron field. F is electromagnetic field and vertex factor is the last component composed of γ, and potential A.

Schrodinger equation
We have at last the necessary recipe which expains quantum electrodynamics!!! Oh one thing is important to remember. It is Dirac spinor
Dirac spcinor equation

Some equations of special relativity explained

Mass in Newtonian mechanics was a concept which was asummed constant feature of matter. But in relativity mass is a function of velocity. It increases as the velocity of the object increase. The mathematical relationship is as follows:
mass dialation equation
Photon has no rest mass which is measured in the rest frame of photon. But it has relativistic energy and momentum. So it is considered a particle in quantum electrodynamics. When it hits another particle like electron it imparts some of its momentum to the electron. Quantum electrodynamics is a relativistic quantum mehanical theory which has been the most successful theory.
So the metric used in Quantum electrodynamics is the flat Minkowsky metric tensor g(uv) which has the following components:

metric tensor equation
When spacetime is curved we use riemannian metric which has somewhat complicated expression.
So far quantum theory of gravity have not been successful. All that are concerned here are quantum mechanics and special relativtiy. All theories of physics are deduced from very few principles or rules. For example quantum electrodynamics calculations are made using three rules described earlier : photon propagator, electron propagator, electron and photon vertex function. These three rules theoretically completes the theory. Although QED has been tested and verified within an accuracy of thirteen decimal places.

Quantum mechanics of Schrodinger and Bohr

No equation in the history had created as much revolution as Schrodinger's equation. It describes the subatomic particles with an mathematical entity named wave function ψ(x,t).
quantum mechanics for dummies

"Like the silicon chips of more recent years, the Feynman diagram was bringing computation to the masses."

More Feynman Diagrams

Feynman diagrams which are representations of spacetime digrams of particles contribute to overall aplitude of a process like scattering of electrons. Each diagram has a probability amplitude. The more complex a digram become the less it contribute to overall amplitude. The probability of interaction in such diagramm becomes less as the order of the diagram increases. Feynman diagrams of many body perturbation process would be like this :
many body perturbation

Different Feynman diagrams can also be drawn by using this equation of perturbation theory.
perturbation method

With this matrix element T(if) which describes the transition between initial and final states we can draw rules for Feynman diagrams as follows :

Feynman Rules

Electron vertex correction

Electron vertex when forming a loop contributes infinite quantity. This is corrected in the following way :

Electron vertex

The Yukawa Potential

As we have seen feynman diagram represents scattering amplitude between various particles. But if we thing this phenomena in terms of Newtonian mechanics we have to say that some force or potential energy must kick off particles to scatter in various directions. The scattering amplitude of feynman diagram when translated into Newtonian potential is known as The Yukawa Potential . The full mathematical expression is :

Yukawa Potential quantum electrodynamics
There is still a better way to derive the Yukawa Potential.

Yukawa Potential quantum electrodynamics

Standard Model

Standard model is a framework to describe particle physics. It is still based on the principles of quantum field theory. Quantum electrodynamics as we have seen obeys gauge symmetry. This guage local symmetry is represented as U(1). U(1) is unitary group. It has all complex numbers and is also called the circle group. It is the group representation of electromagnetic force carrier photon. On the other hand SU(3) and SU(2) represent special unitary groups of 3X3 and 2X2 matrices. The standard model is the representation SU(3)XSU(2)XU(1). It can be put in this form below:

standard model
U(1) group contains all the complex numbers in a circle. It is a trivial group. Every group contains an identity element e so that if b and c are two elements of a group and (.) is group operation then
e.b = b.e
and b.e = c
that is b is inverse of c. Each element of a group has exactly one inverse element. An example can make it clear :

gd3 group
Group homomorphisms are functions that preserve group structure. A function a -> G -> H between two groups G(,.) and H(. *) is called homomorphisms if a(gk) = a(g)(a(k) for all g, k belonging to G and H respectively.
Various groups and their properties can be listed with a table given below :

various group
Group theory can be irritating topics to study. This is how

various group
Galious proved that there is no general closed form solution in terms of radicals for abgebraic equation of order five or more.

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"Everything is a copy of a copy of a copy..."

Geometric series

The perturbative method used in quantum electrodynamics relies upon concept of geometric series :
A series is geometric if its terms have a common ratio with each other. Here is an example :
geometric progression

Notes and additional comments

If you wanted to answer the question of what's truly fundamental in this Universe, you'd need to investigate matter and energy on the smallest possible scales. If you attempted to split particles apart into smaller and smaller constituents, you'd start to notice some extremely funny things once you went smaller than distances of a few nanometers, where the classical rules of physics still apply.

On even smaller scales, reality starts behaving in strange, counterintuitive ways. We can no longer describe reality as being made of individual particles with well-defined properties like position and momentum. Instead, we enter the realm of the quantum: where fundamental indeterminism rules, and we need an entirely new description of how nature works. But even quantum mechanics itself has its failures here. They doomed Einstein's greatest dream — of a complete, deterministic description of reality — right from the start. Here's why.


quantum description quantum electrodynamics

If we lived in an entirely classical, non-quantum Universe, making sense of things would be easy. As we divided matter into smaller and smaller chunks, we would never reach a limit. There would be no fundamental, indivisible building blocks of the Universe. Instead, our cosmos would be made of continuous material, where if we build a proverbial sharper knife, we'd always be able to cut something into smaller and smaller chunks.
That dream went the way of the dinosaurs in the early 20th century. Experiments by Planck, Einstein, Rutherford and others showed that matter and energy could not be made of a continuous substance, but rather was divisible into discrete packets, known as quanta today. The original idea of quantum theory had too much experimental support: the Universe was not fundamentally classical after all.


quantum description

For perhaps the first three decades of the 20th century, physicists struggled to develop and understand the nature of the Universe on these small, puzzling scales. New rules were needed, and to describe them, new and counterintuitive equations and descriptions. The idea of an objective reality went out the window, replaced with notions like:
a)probability distributions rather than predictable outcomes,
b)wavefunctions rather than positions and momenta,
c)Heisenberg uncertainty relations rather than individual properties.
The particles describing reality could no longer be described solely as particle-like. Instead, they had elements of both waves and particles, and behaved according to a novel set of rules.

quantum description

quantum description

But the way you permitted your system to evolve depended on time, and the notion of time is different for different observers. This was the first existential crisis to face quantum physics. We say that a theory is relativistically invariant if its laws don't change for different observers: for two people moving at different speeds or in different directions. Formulating a relativistically invariant version of quantum mechanics was a challenge that took the greatest minds in physics many years to overcome, and was finally achieved by Paul Dirac in the late 1920s.


special relativity

which describes realistic particles like the electron, and also accounts for: anti-matter, intrinsic angular momentum (a.k.a., spin), magnetic moments, the fine structure properties of matter, and the behavior of charged particles in the presence of electric and magnetic fields. This was a great leap forward, and the Dirac equation did an excellent job of describing many of the earliest known fundamental particles, including the electron, positron, muon, and even (to some extent) the proton, neutron, and neutrino.


comptron scattering

But it couldn't account for everything. Photons, for instance, couldn't be fully described by the Dirac equation, as they had the wrong particle properties. Electron-electron interactions were well-described, but photon-photon interactions were not. Explaining phenomena like radioactive decay were entirely impossible within even Dirac's framework of relativistic quantum mechanics. Even with this enormous advance, a major component of the story was missing.
The big problem was that quantum mechanics, even relativistic quantum mechanics, wasn't quantum enough to describe everything in our Universe.


electric field

Think about what happens if you put two electrons very close to one another. If you're thinking classically, you'll think of these electrons as each generating an electric field, and also a magnetic field if they're in motion. Then the other electron, seeing the field(s) generated by the first one, will experience a force as it interacts with the external field. This works both ways, and in this way, a force is exchanged.
This would work just as well for an electric field as it would for any other type of field: like a gravitational field. Electrons possess mass as well as charge, so if you place them in a gravitational field, they'd respond based on their mass the same way their electric charge would force them to respond to an electric field. Even in General Relativity, where mass and energy curve space, that curved space is continuous, just like any other field.

"Nothing is mightier than an idea whose time has come"


electric field

The problem with this type of formulation is that the fields are on the same level as position and momentum are under a classical treatment. Fields push on particles located at certain positions and change their momenta. But in a Universe where positions and momenta are indeterminate, and need to be treated like operators rather than a physical quantity with a value, we're short-changing ourselves by allowing our treatment of fields to remain classical.


fabric of spacetime

That was the big advance of the idea of quantum field theory, or its related theoretical advance: second quantization. If we treat the field itself as being quantum, it also becomes a quantum mechanical operator. All of a sudden, processes that weren't predicted (but are observed) in the Universe, like: a)matter creation and annihilation,
b)radioactive decays,
c)quantum tunneling to create electron-positron pairs,
d)and quantum corrections to the electron magnetic moment,
all made sense.

feynman diagram

Although physicists typically think about quantum field theory in terms of particle exchange and Feynman diagrams, this is just a calculational and visual tool we use to attempt to add some intuitive sense to this notion. Feynman diagrams are incredibly helpful, but they're a perturbative (i.e., approximate) approach to calculating, and quantum field theory often yields fascinating, unique results when you take a non-perturbative approach.
But the motivation for quantizing the field is more fundamental than that the argument between those favoring perturbative or non-perturbative approaches. You need a quantum field theory to successfully explain the interactions between not merely particles and particle or particles and fields, but between fields and fields as well. With quantum field theory and further progresses in their applications, everything from photon-photon scattering to the strong nuclear force was now explicable.

neutrino interaction

At the same time, it became immediately clear why Einstein's approach to unification would never work. Motivated by Theodr Kaluza's work, Einstein became enamored with the idea of unifying General Relativity and electromagnetism into a single scheme. But General Relativity has a fundamental limitation: it's a classical theory at its core, with its notion of continuous, non-quantized space and time.
If you refuse to quantize your fields, you doom yourself to missing out on important, intrinsic properties of the Universe. This was Einstein's fatal error in his unification attempts, and the reason why his stategy towards a more fundamental theory has been entirely (and justifiably) abandoned. Einstein struggled through his entire life to find a unified field theory. He published several papers but all of those were found to be misleading.

Electron and its interaction

Professor Eddington who proved Einstein's general theory of relativity is concerned with measurement by direct comparison.
"The statement that radius of curvature is a constant length requires more consideration before its full significance is appreciated. Length is not absolute and the result can mean only constant relaTIVE to some material standard of length used in all our measuement and in particular measurements which verify G(uv) = &lamda;g(uv) [vacuum equation].
Electrons behave like photon. So if you through electrons through double slit which was used in Young's experiment you will see interference pattern. But wen you place a detector in one of the slits the inteference pattern disappears. Very strange , isn't it?
Let me first explain how events are explained in quantum electrodynamics. Phebimena like existence of electrons and electron traveling from place to place have probabilities assigned to those. Even electron scattering event is interpreted in terms of probability. The probablity is the square of modulus of a complex quantity. Suppose we have two source X and Y and two detectors A and B. We want to calculate the probability that two electrons from X and Y end up at A and B. The laws of probability is that combined probabilty of two events which happen independently is the multiplication of probabilities of the two events. That is P(AB) = P(A)P(B). This law will help us interpret the event of the two electrons goint from X and Y to A and B.
The probability of this particular event can be found in two ways: one possibility is that an electron can go from X to A and other can go from Y to B. The second possibility is that one can go from X to B and the other can go from Y to A. These two possibilities are mutually independent.
Now the probability that one electron can go from X to A is .5 ( as it can either reach A or can not reach A). Let the clock hand move 5 unit at the time the electron reach detector A. Similarly the other probability that other electron can reach B is .5. The clock hand will move 5 unit in this case also. So the probability that the two electrons reach point A and B is .5X(.5) = .25. The total clock hand movement is now 5+5 = 10 .


light path calculation

Here the clock hand's displacement is interpreted as the phase of arrow which represents the probability. If we multiply two phasor the arguments or angles add up. This we know from law of complex number.

complex multiplication

Similarly we can compute probability for other possibility. The result will be the same. Probability amplitude is fundamental in electron interaction in quantum electrodynamics. There are ways of calculating the probability that a particular event will happen. First we have to draw an arrow on a piece of paper according to the rules as follows :
1. The probability of an event is the square of the arrow length. It is called "probility anplitude". For example if the arrow length is .4 its probability amplitude is .16.
2. For drawing an arrow for an event that can happen in multiple ways : draw an arrow for each ways , and then combine them bu hooking the head of one arrow with the tail of another. The final arrow is drawn by hooking the tail of the first arrow with the head of the last arrow. The final arrow is the one whose square gives the probability of the entire event.
Now another example of light travelling from one place to another is depicted according to the above rules :


light path calculation

Each path the light could go (in the simplified diagram above) is shown at the top, with a point on the graph below it showing the time it take the photon to go from the source to that point on the mirror, and to the photomultiplier. Below the graph is the direction of each arrow, and at the bottom is the result of adding all the arrows. It is evident that the major contribution to the final arrow's length is made by the arrows E to I, whose directions are nearly the same because the timing of their paths are nearly the same. This also happens to be where the total time is least. It is therefore approximately right to say that light goes where the time taken is least

Two photon pair annihilation

Another phenomena which is simillar to compton scattering is the photon pair annihilation. Two photons are necessary (in the outgoing radiation) to maintain conservation of energy and momentum when the annihilation takes place in the ansence of external potential. The interaction can be diagrammed as shown in the figure below.

light path calculation
Photons carry energy and momentum according to relativistic energy-momentum formula even though photon does not have any rest mass. Its rest mass is zero.
To explain the phenomena of photon pair annihilation we need to define differential cross section in scattering events. Cross section is defined as the transition probability between incoming and outgoing particle. An electron can capproach a nucleus abnd bounce off it to go in different direction. So some kind of solid angle is concerned through which this scattering happens. So the cross section of scattering is defined as the transition probability per particle per unit time divided per unit solid angle. It is usually called differential cross section. The full mathematical expression will be :

differential cross section
Where T(if) is linearly dependent on S-matrix S(if). And solid angle can be defined like this.
differential cross section

Classical versus Yukawa potential


classical versus Yukawa interaction

Few concepts and terms of Quantum field theory

On Shell
In physics, particularly in quantum field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell.
In quantum field theory, virtual particles are termed off shell because they do not fullfil the condition required by the energy–momentum relation; real exchange particles do satisfy this relation and are termed on shell (mass shell). In classical mechanics for instance, in the action formulation, extremal solutions to the variational principle fall under "on shell representation" and the Euler–Lagrange equations give the on-shell equations. Noether's theorem regarding differentiable symmetries of physical action and conservation laws is another on-shell theorem.
Vacuum polarization

In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field creates virtual electron–positron pairs that change the distribution of charges and currents that generated the original electromagnetic field. It is also sometimes termed as the self-energy of the gauge boson (photon).

vacuum polarization diagram
vacuum polarization tensor
The vacuum polarization is quantified by the vacuum polarization tensor Πμν(p) which describes the dielectric effect as a function of the four-momentum p carried by the photon. Thus the vacuum polarization depends on the momentum transfer, or in other words, the dielectric constant is scale dependent. In particular, for electromagnetism we can write the fine structure constant as an effective momentum-transfer-dependent quantity; to first order in the corrections, we have fine struction constant

A bit of quantum history

In its original form, Planck's postulate was not so far reaching as it is in the form we have given. Planck's initial work was done by treating, in detail, the behavior of the electrons in the walls of the blackbody and their coupling to the electromagnetic radiation within the cavity. This coupling leads to the same factor v 2 we obtained in (1-12) from the more general arguments due to Rayleigh and Jeans. Through this coupling, Planck related the energy in a particular frequency component of the blackbody radiation to the energy of an electron in the wall oscillating sinusoidally at the same frequency, and he postulated only that the energy of the oscillating particle is quantized. It was not until later that Planck accepted the idea that the oscillating electromagnetic waves were themselves quantized, and the postulate was broadened to include any entity whose single coordinate oscillates sinusoidally. At first Planck was unsure whether his introduction of the constant h was only a mathematical device or a matter of deep physical significance. In a letter to R. W. Wood, Planck called his limited postulate "an act of desperation." "I knew," he wrote, "that the problem (of the equilibrium of matter and radiation) is of fundamental significance for physics; I knew the formula that reproduces the energy distribution in the normal spectrum; a theoretical interpretation had to be found at any cost, no matter how high." For more than a decade Planck tried to fit the quantum idea into classical theory. With each attempt he appeared to retreat
from his original boldness, but always he generated new ideas and techniques that quantum theory later adopted. What appears to have finally convinced him of the correctness and deep significance of his quantum hypothesis was its support of the definiteness of the statistical concept of entropy and the third law of thermodynamics. It was during this period of doubt that Planck was editor of the German research journal Annalen der Physik. In 1905 he received Einstein's first relativity paper and stoutly defended Einstein's work. Thereafter he became one of young Einstein's patrons in scientific circles, but he resisted for some time the very ideas on the quantum theory of radiation advanced by Einstein that subsequently confirmed and extended Planck's own work. Einstein, whose deep insight into electromagnetism and statistical mechanics was perhaps unequalled by anyone at the time, saw as a result of Planck's work the need for a sweeping change in classical statistics and electromagnetism. He advanced predictions and interpretations of many physical phenomena which were later strikingly confirmed by experiment.

Pair production

photons lose their energy in interactions with matter, namely the process of pair production. Pair production is also an excellent example of the conversion of radiant energy into rest mass energy as well as into kinetic energy. In this process, illustrated schematically in Figure 2-12, a high energy photon loses all of its energy hv in an encounter with a nucleus, creating an electron and a positron (the pair) and endowing them with kinetic energies. A positron is a particle which is identical in all of its properties with an electron, except that the sign of its charge (and of its magnetic moment) is opposite to that of an electron; a positron is a positively charged electron. In pair production the energy taken by the recoil of the nucleus is negligible because it is so massive, and thus the balance of total relativistic energy in the process is simply hv = E_ + E+ = (moc^2 + K _ ) + (m oc^2 + K+) = K_ + K+ + 2m0c^2 --- (2-15) In this expression E _ and E + are the total relativistic energies, and K _ and K + are the kinetic energies of the electron and positron, respectively. Both particles have the same rest mass energy m oc 2. The positron is produced with a slightly larger kinetic energy than the electron because the Coulomb interaction of the pair with the positively charged nucleus leads to an acceleration of the positron and a deceleration of the electron. In analyzing this process here we ignore the details of the interaction itself, considering only the situation before and after the interaction. Our guiding principles are the conservation of total relativistic energy, conservation of momentum, and conservation of charge. From these conservation laws, it is not difficult to show that a photon cannot simply disappear in empty space, creating a pair as it vanishes.
pair production from photon
presence of the massive nucleus (which can absorb momentum without appreciably affecting the energy balance) is necessary to allow both energy and momentum to be conserved in the process. Charge is automatically conserved, the photon having no charge and the created pair of particles having no net charge. From (2-15) we see that the minimum, or threshold, energy needed by a photon to create a pair is 2m 0c2 or 1.02 MeV (1 MeV = 106 eV), which is a wavelength of 0.012 A. If the wavelength is shorter than this, corresponding to an energy greater than the threshold value, the photon endows the pair with kinetic energy as well as rest mass energy. The pair production phenomenon is a high-energy one, the photons being in the very short x-ray or y-ray regions of the electromagnetic spectrum (see Figure 2-4), where their energies by are equal to or greater than 2m oc2. As we shall see in the next section, experimental results demonstrate that the absorption of photons in interaction with matter occurs principally by the photoelectric process at low energies, by the Compton effect at medium energies, and by pair production at high energies. Electron-positron pairs are produced in nature by cosmic-ray photons and in the laboratory by bremsstrahlung photons from particle accelerators. Other particle pairs, such as proton and antiproton, can be produced as well if the initiating photon has sufficient energy. Because the electron and positron have the smallest rest mass of known particles, the threshold energy of their production is the smallest. Experiment verifies the quantum picture of the pair production process. There is no satisfactory explanation whatever of this phenomenon in classical theory.

Solution of the dirac equation of free particle

Dirac equation of a free particle come in the following form :
free dirac equation
here u is the component of four dimensional Dirac spinor.
free dirac spinor
u will be used to define propagators.

Zero-point energy and Casimir effect

Zero point energy is the lowest energy value that a quantum-mechanical system can have. It has this following mathematical expression.
zero-point-energy
According to Heisenberg's uncertainty principle energy of vacuum can not be zero. Zero is a very precise quantity when it comes to quantum mechanics. Heisenberg's uncertainty tells that nothing can be too precise . So in vacuum particles and anti-particles are always created out of nothing. This is called vacuum fluctuation.
Casimir effect is related to this phenomena of quantum fluctuation. It is a phenomena that is commonly presented as an evidence for existence of zero point energy. Vacuum zero-point energy contains contributions from all wavelengths of light. As the distance between two plate decreases the energy also decreases as less and less wavelengths contributes.

casimir effect

Russell's account of physics


Reference materials:

Quantum electrodynamics simply explained
A briefer history of time by S. Hawking
quantum electrodynamics Hawking
Quantum mechanics
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf
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