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Quantum mechanics and dirac equation

Schrodinger equation   | Matrix mechanics   | Dirac Notation   |   Quantum electrodynamics    

Theory of relativity

Special theory of relativity   |   General theory of relativity
the dirac equation

Abstract

Mathematics and physics both contain abstract elements. Physics is dependable upon mathematics for its logical and consistent foundation. Physics is approximately true and its truth is deduced from empirical evidence. We know physics is the study of matter and motion but in deep level all that is concerned are the abstract world of human sense data. Physics has made matter less material like psychology has made mind less mental. These are the things to be discussed in the abstractness of physics. Dirac equation is a kind of law that is very crucial to understanding subatomic world. Philosophy can be quite lengthy but the most important aspect of it was the part related to the prediction of anti-matter. Dirac was a hero and heroes live forever. He was the professor of mathematics in oxford in the position which Newton once held. We know him as much as we know Newton.
The Dirac equation supports precise solutions for a freely dynamic electron and for the case of the Coulomb potential, but other exact solutions also exist. For instance, one can obtain energy levels for an electron in a continous magnetic field. According to the non-relativistic Schrödinger theory, we have a similar structure of energy states, which are known as Landau levels. Fundamental physics before quantum was all about 2nd order differential equations. The Schrodinger equation was first order in time and second order in space which was clearly not going to work with special relativity. To make it compatible, Dirac made it a first order equation with four components where each component evolved causally. Two of these components were for the spin components of Pauli-Schroedinger. Two were mysterious that he thought might be the proton. Later they became the prediction for antimatter (but that took a bit of fanegaling to get that to work). Technically these other components have negative mass not positive mass with reversed charge. Through the formal and ugly machinations of QFT, it becomes the basic fields of electrons and positrons. I once wrote an article using the Dirac matrices as a substitute for the gravitational metric so that they were the keeper of all geometric information. In this sense, one could view the Dirac equation as the most bare example of gravity’s influence on particles (and vice versa).

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Brief excursion into quantum mechanics

Maxwell equations showed that electromagnetic radiation is composed of waves. This wave carries energy. When applied to microwave woven , the radiation emitted by the hot walls must have whole number of wave peaks and thoughs. The wavelength of a wave is the distance between two peaks or two thoughs. A wave with specific wavelength has specific number of wave peaks between the walls of the woven. But as there is an infinite possibilities of such combination, the energy of the radiation is supposed to be infinite. Clearly this is not the case.
This is also known as ultraviolate catastrophy. To overcome this problem, Max Plank came with his theory of quanta, known as Plank's law.
He hypothesized that energy of any radiation comes in discrete lumps. Such lump of energy is proportional to frequencies. So energy can stay only in an amount of hf(frequency) or an integer multiple of it. There can not be any fractional part of this quantity of hf. It can be explained with an ATM booth transaction.
Suppose you went to withdraw some money from ATM booth : say 1530 USD. And ATM booth has only denominations of 1000 $ and 500 $. So you have two option: one is that you type in 1530 or you type the nearest value 1500. If you type 1530 , you will be notified by the computer that this transaction can not be made. This is because machine can only gives out money that is integer multiple of the denominations. If the amount to be recieved is less than the denomination, the machine can not transfer any more balance. Transactions can only be made with integer multiple of denominations. No fraction of those is allowed. This is what Max Plank's law tells about radiation and energy.

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Relativistic quantum mechanics

Quantum field theory is the union of Einstein’s special relativity and quantum mechanics. It forms the foundation of what scientists call the standard model, which is a theoretical framework that describes all known particles and interactions with the exception of gravity. There is no time like the present to learn it—the Large Hadron Collider (LHC) being constructed in Europe will test the final pieces of the standard model (the Higgs mechanism) and look for physics beyond the standard model. In addition quantum field theory forms the theoretical underpinnings of string theory, currently the best candidate for unifying all known particles and forces into a single theoretical framework. Quantum field theory is also one of the most difficult subjects in science. Unfortunately, learning quantum field theory entails some background in physics and math. The bottom line is, I assume you have it. The background I am expecting includes quantum mechanics, some basic special relativity, some exposure to electromagnetics and Maxwell’s equations, calculus, linear algebra, and differential equations. If you lack this background do some studying in these subjects and then give this website a try. Now let’s forge ahead and start learning quantum field theory.


The Terms

i - Imaginary number i=\sqrt{-1}

\hbar - Planck's constant divided by 2\pi  [Value: 1.051 × 10-34 m2 kg s-1]

 c - The velocity of light [Value: 3 × 108 m s-1]

\gamma^\mu - Dirac matrices

\partial_\mu - 4-gradient, \partial_\mu=\left(\frac{1}{c}\frac{\partial}{{\partial}t},\boldsymbol{\nabla}\right)

m - Rest mass of the electron [Value: 0.511 MeV / c2]

\psi -  Wavefunction of the system - the probability amplitude for different configurations of the system at different times. Also known as the quantum state, this is the most complete description that can be given to a physical system.


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Dirac equation is one of formulations of relativistic quantum mechanics. The Dirac equation is perhaps the most powerful equation of modern physics. It has wide range of applications and uses. The most important prediction was that of anti-particle.
There are several obvious dominance of the dirac equation in comparison to its nonrelativistic counterpart, Schrödinger equation (or the Pauli equation for particles with spin 1/2). First of all, the Dirac equation is compatible with the special theory of relativity, because the proper orthochronous Poincaré group has a representation by symmetry transformations in the state space associated with the Dirac equation.
On the other hand, the Dirac equation shows some strange effects. It modifies relativistic kinematics in a quite unexpected way through the appearance of negative energies (or negative masses). The energy according to the Dirac equation is not bounded from below (>=). This causes the usual free variational methods for computing energy eigenvalues to fail. Therefore one might think of considering replacements of the Dirac equation by equations with energy bounded from below. The principle The energy- momentum equation describes how mass is altered as the velocity of a body increases. There are two kinds of masses: one is proper mass and other is relativistic mass. Proper mass is the mass which is measured in the rest frame of the material body. As the velocity of the body increases, the kinetic energy is added as extra mass. The change of kinetic energy is the same as change of mass so that the relativistic energy relation reduces to classical energy conservation law in low velocity limit.
One could convert the classical relativistic energy-momentum relation E=c^2p^2+m^2c^4
into a quantum-mechanical equation just in the usual way, that is, by replacing the momentum p with a differential operator -iħ▿ acting on suitable wave functions. This leads to the Hamiltonian H=−c2ℏ2Δ+m2c4. The Schrödinger equation with this Hamiltonian is called the square-root Klein-Gordon equation because of its formal resemblence to a square-root of the Klein-Gordon equation (Oscar Klein and Walter Gordon in fact had little to do with the square-root equation). Unfortunately, the meaning of the square-root Klein-Gordon equation is shadowed by the following points.
a) The Hamiltonian incorporates the square-root of a differential operator. It is no problem to define this operator with the help of the Fourier transformation and to investigate its properties, but the resulting operator H is non-local. This means that in order to compute its action on a wave function at some point x, one needs to know the values of the wave function at all other places.
b) Wave function describes certain wavepacket that behaves almost like wave. A wave packet is a superposition of multiple waves. The more localized the wave packet is , the wider the spread of its wavelength is. So it has large number of component waves. More localized wave has larger range of frequencies. Each individual wave corresponding to individual frequency has a momentum , described by De Broglie equation p=h/λ . More localization decrease the value of position but it spreads the frequency spectrum. More and more frequencies contribute to the spectrum.
the dirac equation

The average value of all the wavelength will correspond to average value of the momentum of particle. We have a loss of information about the momentum as we no longer have a definite momentum but a collection of momenta. We can only calculate the average of all momenta. So descrease in position increases the width of momentum distribution. By distribution I mean many momentum values or data. This is the same thing as saying that the more definite the position is, the more uncertain the momentum is. Only sine wave has a definite momentum. Every other waves are a combination of multiple sine waves. There is always a trade-off between the position and momentum. Th is fact corresponds exactly to properties of Fourier transformation.
dirac equation

This uncertainty relation is also applicable for energy and time pair also. The energy can never be zero as zero seems to be a very precise number. Th is is why particle and antiparticle can be created inside the vacuum. The energy of vacuum is not zero so that there is always a fluctuation in it. This is called quantum fluctuation. Near event horizon of black holes particle and antiparticle always appear out of nothing. These particles can either enter black hole or can escape from event horizon. Stephen Hawking famously retorted "black hole ain't so black".

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Frequency is what is actually observed in any quantum system. It corresponds to the electron jumping from one orbit to another. So the position of electron is dependeble on the radiation frequency. This is the idea behind the development of Heisenberg's matrix mechanics which ultimately led him to discover uncertainty principle.
The spin of elementary particles is not described by the square-root Klein-Gordon equation. The solutions of the square-root Klein-Gordon equation are scalar wave functions. Real electrons have spin and should be described by a matrix-wave equation.

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Paul Dirac

Probably the greatest English physicist since Sir Isaac Newton, Paul Adrien Maurice Dirac was born on 8th August 1902 in Bristol. His father Charles came to England from Geneva and married Florence Holten. After studying electrical engineering at Bristol University Paul eventually secured a place at St John's College, Cambridge, later becoming Lucasian Professor of Mathematics and a Fellow of the Royal Society. He maintained his own version of quantum theory and was accredited the Nobel Prize for Physics and the Order of Merit. He got married to Margit Balazs and had two daughters. For many years he was a research professor in Florida USA; he died on 20th October 1984 at Tallahassee, where he is buried. Paul dirac was a legend. He was the genius of tweentieth century. He is the superhero of theoretical physics

Matrix

This matrix is not the sci-fi movie matrix. It is a mathematical object. Matrix is a collection of numbers arranged in rows and columns. So every element of a matrix can be indexed by two numbers which is different from numbers inside the matrix. Suppose you want to solve simultaneous system of linear equations of three or more variables using computer programm. You need to track the coeeficient s appearing in the equations in order : from the first variable to second and to third and so on. So it it better to specify a grid with row one for first equation and row two for second equation and so on. When you specify multiple row s, then multiple columns will be needed. This grid is the matrix of numbers. concept of tensor depen d's on this properties of matrix. If you multiply two vectors you get a matrix or second rank tensor. Tensor have the additional properties that it's components transform in specific ways.

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Pauli equation

In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into consideration the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be utilized where particles are hurling at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927. Pauli equation can be stated to be related to an extra term added to the right hand side of Schrodinger equation.


the dirac equation

The term on the right side is stern-gerlach term which describes the spin orientation of atoms with one valence electron. Valence electron is the electron in the outermost shell around nucleus , which take s part in chemical bonding. Here atom with one valence bond is considered. The term is calculated by the dot product of Pauli matrices with magnetic field B.

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Pauli was a mega boss and genius of geniuses. Few scienetists have as much reputation as him. We know him for his famous "exclusion principle" which make it possible to arrange all the elements in a periodic table.

Explanation

Dirac first derived relativistic quantum theory which is the reconciliation of quantum mechanics with special theory of relativity. Schrodinger equation breaks down for particle moving with high velocity and it also did not distinguish same particle with different spin.
Spin is a physical quantity which describes certain properties of sub-atomic particle. The story began when Pauli gave his "Exclusion Principle". It states that no two elctrons in an orbital can have same set of four quatum numbers in a quantum system. The spin of electron is the fourth quantum number which uniquely determines state of electron. Spin is very analogous to angular momentum, which takes discrete values. When a charge spins it has a certain way of aligning itself with external magnetic field. Spin of a particle like electron makes it behave certain way in the presence of magnetic field. Dirac equation describes perfectly particles with different spin. Dirac spinor is associated with the spin of electron.
Dirac successfully made Schrodinger equation consistent with relativity by altering the Hamiltonian operator of Schrodinger equation. The wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Original equation proposed by Dirac is :


the dirac equation

New elements in this equation are the 4 × 4 matrices αk and β, and the four-component wave function ψ. There are four components in ψ because the representation of it at any given point in configuration space is a bispinor. It is construed as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron (see below for further discussion).
The 4 × 4 matrices αk and β are all Hermitian and have squares equal to the identity matrix:
The single symbolic equation thus untangle into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. These matrices and the form of the wave function have a deep mathematical bearing. The algebraic structure exhibited by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th-century work of the German mathematician Hermann Grassmann in his Lineale Ausdehnungslehre (Theory of Linear Extensions).
The equation described above is the modification of Schrodinger equation. Shrodinger equation satisfies a wave function. The wave function gives the probability distribution for the value of each observables like position or momentum. This probability gives the likelihood of finding a partcle at certain place and time.
Spinor was first defined by Pauli as two component wave function , each of which indicates particular spin of electron. Dirac introduced , as a necessity to accommodate negative energy , another pair of components to the wave function. This four component wave function is called spinors. Wave function is usually a complex quantity. Dirac spinors are two set of vectors , each of which corresponds to unique spin quantum number. These Dirac spinors came naturally from the solution of the Dirac equation. spinors are elements of four component complex vector space. They are not like usual vector and tensors which transform in certain way.


dirac equation

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Prediction of anti particle

Dirac equation explains the spin 1/2 particles and gives the foundation of Quantum electrodynamics. Moreover it predicted the existence of positron which can have negative mass. When treating particle to have negative energy in this way , each particle field seems to have negative counterpart. Proton has its.counterpart anti-proton, neutrino has its counterpart anti-neutrino, atom as a whole have anti-atom and universe seems to have anti-universe counterpart too. When positron collide with its negative counterpart , annihilation occurs. if this annihilation occurs at low energy , gamma rays are produced. So if you meet your counterpart from distant part of the universe someday you should be serious about shaking your hand with him because both of you can be annihilated. Positron is the negative energy solution of dirac equation.


the dirac equation


the dirac equation


the dirac equation

The electron and positron field ψ has corresponding expression , which solve the dirac equation. The field here has the similar meaning that at each point in space and time it has a value. It is better called relativistic scalar function which does not have usual role in determining probability density like wave function φ The sigma (σ) in the γ matrices(4X4) are pauli 2X2 matrices. u and v are spinors each of which is grouped with two distinct values: one for spin up (+1) and one for spin down(0). That is why they are named spinors.



dirac equation

Klein -Gordon equation

The manifested field in Klein -Gordon equation is φ which represent energetic particles. The Lagrangian formulation is the kinetic energy of the field subtracted the potential energy as the quantity m multiplied by the complex conjugate of the field(φ +) and φ itself;. Applying euler - lagrange equation for classical path we get the Klein _ Gordon equation. It is quantized version of relativistic energy-momentum relation. Each of the four components ψ1, ψ2, ψ3, ψ4 of the original four-dimensional vector of wave functions (spinors) individually satisfies the Klein-Gordon equation.


the dirac equation

Dirac equation implies Klein Gordon equation. They are each other's complementary. If dirac equation holds , then the other also holds. Any solution of the free Dirac equation is, component-wise, a solution of the free Klein–Gordon equation. They can have same kind of solution for the field φ . The key idea behind the quantum field theory is that of quantum harmonic oscillator. It has a Hamiltonian as described by annihilation(a+) and creation operator (a).


Dirac equation

Derivation of Dirac equation

A simple derivation of dirac equation can now be illustrated. Theory of Relativity threw up some road blocks when quantum mechanics was first developed, especially for the particles physicists wanted to look at most, electrons or fermions. For zero spin particles, including relativity appears to be simple. The classical kinetic energy Hamiltonian for a particle in free space, is


Dirac equation

The above can be replace with Einstein's relativistic expression

Dirac equation

where m is the rest mass and and mc^2 correspond to energy of it. Now we can squaree operator in both sides of Schrodinger equation

Schrodinger equation

to get rid of square root. Finally we get the standard form of dirac equation making it consistent with relativity.
Dirac equation

Hole theory

The negative E solutions to the equation are troublesome, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to turn down the negative-energy solutions. Since they exist, we cannot simply disregard them, for once we include the interaction between the electron and the electromagnetic field, any electron kept in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy. Real electrons obviously do not behave in this way, or they would disappear by radiating energy in the form of photons.
To adapt with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are filled up. This description of the vacuum as a "sea" of electrons is called the Dirac sea. Since the Pauli exclusion principle prohibit electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be restricted from decaying into negative-energy eigenstates.
If an electron is restricted from simultaneously occupying positive-energy and negative-energy eigenstates, then the charcterstics known as Zitterbewegung, which arises from the interference of positive-energy and negative-energy states, would have to be regarded to be an unphysical prediction of time-dependent Dirac theory. This conclusion may be inferred from the explanation of hole theory given in the previous paragraph. Recent results have been published in Nature [R. Gerritsma, G. Kirchmair, F. Zaehringer, E. Solano, R. Blatt, and C. Roos, Nature 463, 68-71 (2010)] in which the Zitterbewegung feature was simulated in a trapped-ion experiment. This experiment affects the hole interpretation if one infers that the physics-laboratory experiment is not merely a audit on the mathematical correctness of a Dirac-equation solution but the measurement of a real effect whose detectability in electron physics is still beyond reach.
Dirac further argued that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle. The hole possesses a positive energy since energy is needed to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but Hermann Weyl claimed that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually interpreted as the positron, experimentally discovered by Carl Anderson in 1932.
It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons have to be annealed by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "jellium" background so that the total electric charge density of the vacuum is zero. In quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (turning an occupied negative-energy electron state into an unfilled positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it.
In certain applications of condensed matter physics, however, the underlying ideas of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, posseses electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively charged ionic lattice of the material.
Soulution to clasical electromagnetism again
A good question is then why a particle at rest does not fall down to the negative levels radiating the excess energy as electromagnetic radiation. Dirac suggested the following solution. In what we call vacuum, all the negative energy states are occupied by electrons.
If a singular electron is placed in this vacuum it cannot fall into a negative energy level because of the Pauli exclusion principle. We also redefine the energy such that it is zero for our vacuum, we can do this as we only can measure energy differences. Assume that we have this vacuum and add energy, for instance in the form of electromagnetic radiation. We then can lift an electron from the negative “sea” to a positive energy level. This requires at least 2mc2 . The electron will leave a hole in the negative sea. This hole will behave as a positive real particle with positive mass and velocity and momentum in the same direction. Conversely, an electron and a hole can disintegrate each other radiating electromagnetic radiation. One problem with Dirac’s solution is that the theory stops being a one-particle theory. For instance, we should take into account the interactions between the negative sea electrons. Secondly the theory becomes asymmetric with respect to electrons and positrons. The problem is solved by introducing quantized fields. We should also note that for the Klein-Gordon equation, that describes particles with spin zero (i.e. bosons) we don’t have the Pauli exclusion principle and we cannot use Dirac’s solution.

Notes and additional comments

Atom:
The basic building block of all normal matter, consisting of a nucleus (which is itself composed of positively-charged protons and zero-charged neutrons) orbited by a cloud of negatively-charged electrons, so that the positive charge is exactly balanced by the negative charge and the atom as a whole is electrically neutral. Atoms range from about 32 to about 225 picometers in size (a picometer is a trillionth of a meter). A typical human hair is about 1 million carbon atoms in width.
Atom is an mini universe- a solar system to be precise. The interaction between the atom and the outside world is fuzzy. What goes on inside the atom is very mysterious. The nucleus consists of proton and neutron. The strong nuclear force that binds these particle inside the nucleus comes from the reduced mass of the proton and neutrons. The energy can be found by Einstein mass energy equivalence formula. Strong nuclear force is described by quantum field theory. Nucleus is the heart of atom. Electron revolves around the nucleus like planets revolves around the sun. The physical picture of atom with fewer electrons is easy to visualize. As the number of electron increases , the complexity of atom also increases. It is hard to imagine that no two electrons collide with themselves although their orbits may intersect. Their orbits may be circular or elliptic.
Albert Einstein first proved the existence of atoms observing their Brownian movements. Atoms are hard to see with bare eye. Almost five million atoms can fit together on the tip of a ball pen. Democritus first gave the concept of atoms as an indivisible element of matter. The laws of chemistry were developed using the theory of atom put forward by Dalton. Later modern quantum theory were developed in order to understand what is going inside the atom. There are four quantum numbers that truly specify the state of electron inside the atom. First one is principle Quantum number, second is azimuthal quantum number, thrid is magnetic quantum number and the fourth is spin quantum number. These four quantum numbers need some preliminary explanation :
1. Principle quantum number: the principle Quantum number describes the orbits where electron stays with certain energy. It is represented by n=1, 2, 3 and so on
2. Secondary or azimuthal quantum number describes subshell that electron might occupy in certain condition. it is also called orbital angular momentum. For example in hydrogen atom there is only one subshell of principle Quantum number n =1. In heliumn there are two subshells 1s1 and 1s2.
3. Magnetic quantum number describes the specific orbital (cloud) of the subshells. The value of each magnetic quantum number can be found by projection of angular momentum into specific coordinate axes. This magnetic quantum number has certain mathematical expression.
4. The spin quantum number describes the intrinsic spin of electrons inside atom. It is much like angular momentum but it is intrinsic like mass and charge. It is due to spin quantum number that the electron gets aligned in certain direction when an external magnetic field is applied. Spin is , thus, a vector quantity which has magnitude and direction.

Quantum mechanical spin

Quantum mechanical spin is a subtle concept . Especially in quantum field theory , where particles are viewed as dots,, it is hard to imagine what "spinning" would actually mean. What really happens is that experiments shows that particles can possess an intrinsic property which is exactly analogous to angular momentum. Moreover , uantum theory shows, and experiments confirm, that particle will generally have angular momentum that is integer multiple of planck constant h. Since classical spinning particles possess an intrinsic angular momentum ( one that is not immutable - it changes as the rotational speed changes ), theoreticians have borrowed the the name "spinning " and applied it to quantum situation. Hence the name "spinning angular momentum". While "spinning like a top" provides a reasonable mental image , it is more accurate to think that particles are defined not only by their mass or charge , nuclear charges but also by instrinsic and immutable spin angular momentum.
All of the matter particles (their antimatter partner as well) have spin equal to that of electron. In the language of trade, physicists say that matter particles all have "spin-1/2" where the value 1/2 is , roughly speaking the quantum-mechanical measure of how quickly the particle rotates. Photons, weak gauge bosons, gluons all have spin that is twice that of matter particles . They have spin 1.

Electron and geometry

Einstein's law of general relativity in empty space (G(uv) = 0) does not seem to have any reference to the constitution of an empty continuum. It is a law of material structure showing what dimensions a specified coolection of molecules must take up in order to adjust itself to equilibrium with the surrounding conditions of the world.
In particular , electrons must make these adjustments, and it is suggested elsewhere that the symmetry of an electron and its equality with other electrons are not substantial facts, but consequences of the method of measurements. One can not explain of an author for not doing everything, but at this point most readers will feel a desire for some discussion of the theory of measurement. The elementary meaning of measurement of lengths is derived from superposition of supposedly rigid body. A rigid body, as Dr Whitehead has pointed out, is primarily one which seems rigid, such as steel bar in contradistinction to a piece of putty. When I say that a body seems "rigid" , I mean it does not alter its shape and size. This, so far as it can be relied upon, implies some constant relation to the human body: if the eye and the hand grew at the same rate as the "rigid" body, it would look and feel as if it were changingng. But if other objects in our immediate environment did not grow meanwhile , we should infer that we and our measure had grown. There would, however, be no meaning in the supposition that all bodies are bigger in certain places than ther are in certain others; at least, if we suppose the alteration to be in a fixed ratio. If we do not add this proviso, there is a good meaning in the supposition ; in fact, we do actually believe that all bodies are bigger at equator than at the North pole, except such are too small to be visible or palpable. When we say that length of an object at the equator is one metre, we do not mean that its length is that which the standard metre would have if we have moved it from Paris to the equator. But the expansion of bodies with temperature would have difficult if it had not been possible to bring bodies of different temperatures into the same neighborhood and measure them before their temperature becomes equal; it would also have been difficult if all bodies had expanded equally when their temperature rose. These elementary considerations, along with many others, make rigidity an ideal, which actual bodies approach without attaining . Mere superposition thus ceases to give measure of length: it still gives the comparison between the two bodies concerned, but not of either with the standard length unit of length. To obtain the latter, we have to adjust the immediate results of the operation of measuring, by means of a mass of physical theory. If the measures which we obtain are mutually consistent , that is all we can ask; but it is possible that a change in physical theory might have given other measures which would have been also consistent.


Pure Mathematics


The formalist interpretation of mathematics is by no means new, but for our purposes we may ignore its older forms. As presented by Hilbert, for example in the sphere of number, it consists in leaving the integers undefined, but asserting concerning them such axioms as shall make possible the deduction of the usual arithmetical propositions. That is to say, we do not assign any meaning to our symbols 0, 1, 2, . . . except that they are to have certain properties enumerated in the axioms. These symbols are, therefore, to be regarded as variables. The later integers may be defined when 0 is given, but 0 is to be merely something having the assigned characteristics. Accordingly the symbols 0, 1, 2, . . . do not represent one definite series, but any progression whatever. The formalists have forgotten that numbers are needed, not only for doing sums, but for counting. Such propositions as “There were 12 Apostles” or “London has 6,000,000 inhabitants” cannot be interpreted in their system. For the symbol “0” may be taken to mean any finite integer, without thereby making any of Hilbert’s axioms false; and thus every number-symbol becomes infinitely ambiguous. The formalists are like a watchmaker who is so absorbed in making his watches look pretty that he has forgotten their purpose of telling the time, and has therefore omitted to insert any works. There is another difficulty in the formalist position, and that is as regards existence. Hilbert assumes that if a set of axioms does not lead to a contradiction, there must be some set of objects which satisfies the axioms; accordingly, in place of seeking to establish existence theorems by producing an instance, he devotes himself to methods of proving the self-consistency of his axioms. For him, “existence”, as usually understood, is an unnecessarily metaphysical concept, which should be replaced by the precise concept of non-contradiction. Here, again, he has forgotten that arithmetic has practical uses. There is no limit to the systems of non-contradictory axioms that might be invented. Our reasons for being specially interested in the axioms that lead to ordinary arithmetic lie outside arithmetic, and have to do with the application of number to empirical material. This application itself forms no part of either logic or arithmetic; but a theory which makes it a priori impossible cannot be right. The logical definition of numbers makes their connection with the actual world of countable objects intelligible; the formalist theory does not. The intuitionist theory, represented first by Brouwer and later by Weyl, is a more serious matter. There is a philosophy associated with the theory, which, for our purposes, we may ignore; it is only its bearing on logic and mathematics.
Reverting back to logic, it is seen that logic is vital to mathematics. Logic have a bearing upon the thuth of mathematical statements , which pure mathematics uses as a notion apart from logical constants. Logic has many usage in constructing physical theories. General theory of relativity is a patchwork of logical notions and mathematics. Physics , from the time of Einstein got new style from the adoption of the axiomatic approach like as mathematics. The propotions of mathematics are analytics whereas those of physics are synthetic. Analytic propositions are the propositions of pure mathematics. Their truth depends on the logical grounds and they serves only to elucidate the meaning already implicit in the subjects. . On the other hand synthetic propositions refer to the emirical facts. For example , the sum of three angles of a triangle is two right angles. This is an example of analytic proposition. The fact that the sum is two right triangle is true is established on logical grounds , i.e those of euclid's axioms and propostions. On the other hand all bachelors are unmarried is a synthetic proposition. It is true by virtue of empirical facts.


Geometry

Geometry is classified as two kinds : first the metrical geometry and second the projective geometry. The former involves metrical ideas whereas the later involves projective transformation, which is somewhat difficult concept.
To begin with metrical geometry it is seen that conditions of free mobility isessential to all measurement of space. It finds the analytical expression of these conditions in the existence of a space-constant or constant measure of curvature, which is equivalent to homogeneity of space. This is its firt axiom.
The second axiom is that space has a finite number of dimensions. The point in space can be repsented by a finite number of dimensions , which can be called the coordinates of the point.
The third axioms assumes the concept of distance. There is a unique distance relationship between two points in space. This relationship can be expressed with an equation.
These are the axioms of metrical geometry which uses the notion of metric or distance measurement explicitly. And the distance uses the notion of quantity. In projective geometry we do not have any quanitative comparisons. Only qualitative relations and identity is presupposed in it. Although metrical geometry is prior to projective geometry. In it , two point uniquely detrmine a line and conversely two lines can uniquely represent a point. Lines and points are equivalent in qualitative perspective. This duality is the basis of projective geometry. Full account of projective geometry deserves another chapter, which at the moment best left.
All the geometries , surprisingly, have a number of notions and axioms. From these axioms , all the propositions can be deduced logically. We know this very well , from Euclidean system. All geometries , alike, deal with spatial relations and magintudes. Points , which are infinite division of space , have zero dimenion. Relation between two points give the quantitative desription of distance. Now in three dimensions, three points not in a stright line determine a plane, in four dimensions four points not in a plane determine a unique three dimensional figure and in five , five points and so on. This process continues in metrical geometry as the number of dimensions increases. But in projective geometry this does not happen. At some time the new point included must be one the points mentioned before so that the in (n+1) th step the number of dimensions reduces to n. This is a speciality of projective geometry.

Local gauge invariance

The Lagrangian density for a free Dirac particle is given by L = i∂µ Ψγ µ Ψ + mΨΨ
Using Lagrange’s equations ∂µ ∂L ∂( µ Ψ ) − ∂L ∂Ψ = 0
we get the Dirac equation:
∂µ iγ µ ( Ψ ) − mΨ = iγ µ ∂µ ( − m)Ψ = 0
A local gauge transformation is a transformation where we do the replacement Ψ →Ψ e iqΛ(x) ,Ψ →Ψ e −iqΛ(x)
It is easily seen the the Lagrangian above is not invariant under such a transformation because of the coordinate dependence on the gauge function Λ
Exercise. Show this.
However, if we replace the derivative according to
∂µ→ Dµ = ∂µ− iqAµ
the covariant derivative, the new Lagrangian is invariant under a local gauge transformation provided that the vector field Aµ transforms according to
Aµ → Aµ + ∂µΛ
Proof:
L = iDµ
Ψγ µ Ψ + mΨΨ = i ∂µ ( − iqAµ )Ψγ µ
Ψ + mΨΨ L → iΨ e −iΛq γ µ e +iΛq iqΨ ∂µΛ + ∂µ Ψ − iqAµ Ψ − iqΨ ∂ ( µΛ) + mΨΨ
We see that the offending derivative of Λ disappears and thus the new Lagrangian is invariant.
The Dirac equation derived from the modified Lagrangian is then
iγ µDµ ( − m)Ψ = iγ µ ∂µ ( ( − iqAµ ) − m)Ψ = 0 The extra term in the Lagrangian is an interaction term between the electron and the vector field −qΨγ µ Ψ Aµ . We have earlier seen that Ψγ µ Ψ is the (four-dimensional) probability current so with the identification q = e we identify the quantity Jµ = −eΨγ µ Ψ with the electrical current and Aµ with the electromagnetic field. We can add a term to generate the differential equation for the electromagnetic field
1/4 F µν Fµν , F µν = ∂µ Aν − ∂ν Aµ
The equation for the electromagnetic field will be ∂µ ∂µAν =!Aν = Jν
that is precisely the equation you get from Maxwell’s equations. Holaa..

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