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"Yet nature is made better by no mean
But nature makes that mean : so, over that art
Which you say adds to nature, is an art,
That nature makes."
William Shakespear, A winters tale.
"Energy makes it go and entropy tells where to go"

# Quantum field theory

Path integral   |   S Matrix   |   Quantum electrodynamics

# Theory of relativity

Special theory of relativity   |   General theory of relativity

### s matrix and scattering matrix

In physics, the s matrix or scattering matrix correlates the initial state and the final state of a physical system evolving a scattering process. It is utilized in quantum mechanics, scattering theory and quantum field theory (QFT). More formally, in the language of QFT, the s matrix is defined as the unitary matrix connecting sets of asymptotically free particle states (the in-states and the out-states) in the Hilbert space of quantum physical states. A multi-particle state is assumed to be free (non-interacting) if it transforms under Lorentz transformations as a tensorial product, or direct product in physics parlance, of one-particle states as prescribed by equation (1) below. Asymptotically free then implies that the state has this appearance in either the distant past or the distant future.
While the s-matrix may be defined for any background (spacetime) that is asymptotically solvable and has no event horizons, it has a simplistic structure in the case of the Minkowski spacetime. In this special case, the Hilbert space is a space of irreducible unitary representations of the inhomogeneous Lorentz group (the Poincaré group); the s matrix is the evolution operator between time equal to minus infinity (the distant past), and time equal to plus infinity (the distant future). It is defined only in the limit of zero energy density (or infinite particle separation distance).
It can be proven that if a quantum field theory in Minkowski spacetime has a mass gap, the state in the asymptotic past and in the asymptotic future are both described by Fock spaces.

#### Motivation

In high-energy particle physics we are very interested in computing the probability for distinct outcomes in scattering experiments. These experiments can be broken down into three separate stages:
a)collide together a collection of incoming particles (usually two particles with high energies).
b)Allowing the incoming particles to interact with themselves. These interactions may change the types of particles present (e.g. if an electron and a positron (anto-electron) annihilate they may produce two photons).
c)Measuring the mass of resulting outgoing particles.
The process by which the incoming particles are converted (through their interaction) into the outgoing particles is called scattering process. For particle physics, a physical theory of these processes must be able to systematically compute the probability for different outgoing particles when different incoming particles collide with different energies.
The S-matrix in quantum field theory achieves exactly this goal. It is assumed that the small-energy-density approximation is valid in these cases.

"failure is the pillar of success.."

### s matrix

An operator (a matrix) describing the process of transference of a quantum-mechanical system from one state into another under their interactions (scattering).
Under process of scattering, the system moves from one quantum state, the initial one (one may relate it to the time t=−∞), into another, the final one (related to t=+∞). If one denotes the set of quantum numbers describing the initial (final) state by i (j), then the scattering amplitude (the square of whose modulus defines the probability of a given scattering) can be written as S(ij). The collection or set of all scattering amplitudes forms a table with two inputs, and is called the scattering matrix S. Each of these scattering amplitudes or transition amplitudes represents a Feynman diagram which contributes to its corresponding scattering amplitude perturbatively. Feynman diagram tells how particles interact with each other. Thus it is the same thing as the initial assumption of s-matrix.
Determining scattering matrices is a core problem in quantum mechanics and quantum field theory. The scattering matrix contains complete information about the behaviour of a system, provided that one knows not only the numerical values, but also the analytical properties of its elements. In particular, its poles specify the bound states of the system (and thus the discrete energy levels). The most important property of a scattering matrix follows from the basic principles of quantum theory: it must be unitary.

##### Explanation

The difference between Heisenberg picture and Schrodinger picture of atom is the time dependency of observables and operators. Fundamentally two pictures developed by Heisenberg and Schrodinger is equivalent. They both describe same phenomena in two different ways. To switch into the interaction picture, we divide the Schrödinger picture Hamiltonian into two parts:

A little comparison can be made with a chart.

The interaction part , as seen , is the representation of two states that interact through Hamiltonian H(0,s) which is time dependent. S-matrix is developed based on this interaction Hamiltonian H(0,s). mIf the Hamiltonian has explicit time-dependence (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly time-dependent terms with H(1,S) leaving H(0,S) time-independent. S-matrix or scattering matrix is the transformation matrix between incoming and outgoing states of particles waves. This is the intermediate representation between Schrodinger picture and Heisenberg picture. In Heisenberg picture the state vector is constant or time-independent but in Schrodinger's picture this is not the case. In interaction picture both the state vector and operator carry time dependence of the observables. The interaction part of the Hamiltonian is dependent of time. Total Hamiltonian is the sum of free part H(0) and an interaction part V. So the interaction picture is :

--- (1)

Let us give a short description of each step taken to reach the final form of S - matrix. The first step denotes the projection of the initial state Φ(i) on the final state Φ(f). This gives the elements of S matrix and defines an operator S. In the next step the evolution operator U(,) is used and comparing the two results we see S-operator is the evolution operator U(a,a). The final S(fi) matrix is the projection of initial state onto final state Ψ(i) by the operator S which is defined above. The s - matrix is similar to transition probability amplitude in quantum mechanics. The elements of S-matrix represents scattering amplitudes. Scattering amplitudes are various ways in which wave interact with themselves or with particles.

"you can watch me, mock me and try to block me but you can not stop me.."

We are dealing with atoms, quarks, electrons and other almost non-existent particles here.."

### Perturbation theory

Any Hamiltonian contains kinetic and potential parts. Interaction means the potential part of the Hamiltonian. Sometimes a part of interaction can be treated exactly together with the kinetic part. They form an approximative or "non perturbed" Hamiltonian. The rest of interaction is then treated perturbatively and is called "a perturbation". Interaction of the atomic electron with an external magnetic field is such part of the total Hamiltonian.
Interaction Hamiltonian is a part of total Hamiltonian that contains time-dependent term containing function referring to some other physical system that is not part of the system being described but interacts with it. For example, in
H = p^2/2m + U - μE(t)
the term U is potential energy, and -μE(t) is the interaction hamiltonian because E(t) is the electric field which is not described by the Hamiltonian but interacts with the system.
Time-independent perturbation theory is one of two categories of perturbation theory, the other being time-dependent perturbation. In time-independent perturbation theory the perturbation Hamiltonian is static (i.e., possesses no time dependence). Time-independent perturbation theory was presented by Erwin Schrödinger in a 1926 paper, shortly after he developed his theories in wave mechanics. In this paper Schrödinger referred to earlier work of Lord Rayleigh, who investigated harmonic vibrations of a string perturbed by small inhomogeneities. This is why this perturbation theory is often regarded as Rayleigh–Schrödinger perturbation theory.

Dyson equation is the equation relating electron's bare energy and its self-energy. The energy that a particle has as a result of changes that it itself causes in its environment defines self-energy. This is defined as a loop in the diagram below. The energy that electron creates come beck to it partly due to interaction.

A bare particle is an excitation of an elementary quantum field. Such a particle is not identical to the particles observed in the experiments: the real particles are dressed particles that also include additional particles surrounding the bare one. The energy of bare particle is called bare energy. So the equation above is expressed in terns of self-energy , bare energy and dressed energy.
The Schwinger–Dyson equations (SDEs), or Dyson–Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between Green functions in quantum field theories (QFTs). They are also known as the Euler–Lagrange equations of quantum field theories, since they are the equations of motion corresponding to the Green's function. They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite tower of SDEs(ordinary differential equations).

Formally, Dyson series is the solution to the Schrodinger equation for the matrix elements of time-evolution operator in interaction picture. The integral equation given above is the Schrodinger equation. We can easily get to the next equation by integrating and recombining terms. The solution comes with iterative process. The final solution seems to be a colllection of terms U(n) {n=0,1,2, 3 ..) which are summed. Each term contributes to the final solution perturbatively.
However its great utility comes in perturbation theory where each terms in the series can be put in one to one correspondence with Feynman diagrams and in principle computed to any desired accuracy (that's an oxymoron because perturbation series are asymptotically divergent). So in my opinion, Dyson series is an incredibly useful tool to handle perturbation series covariantly.

"What one calls God, another calls it the laws of physics"

## Non-perterbative theory

In nor-perturbative theory the solution is one which can not be described as a power series expansion of some function. In mathematics the non-pertubative functions also refers to these functions which can not be expanded as a power series. As example of such fucntion is the function describing Swinger-effect whereby a strong field may decay into electron-positron pairs . The exact function is

This function can not be expanded as a Tylor series in quantity e ( charge). This is a non-pertuabative process.

### CROSS SECTION IN CLASSICAL MECHANICS

To understand the concept of a cross section in classical mechanics first, consider the following situation: A beam of particles approaches the target under a certain impact parameter b. According to this impact parameter, dN of the particles are scattered into an angle interval [θ, θ+dθ] per unit time. To normalise, the number n of particles passing a unit area orthogonal to the direction of the beam per unit time is introduced. Then,

dσ = dN/n
is the corresponding classical cross section. Obviously it has dimensions of an area. It is completely determined by the scattering center and the incoming particles.
CROSS SECTION IN QUANTUM MECHANICS -

Consider now the scattering of incoming particles on some fixed target, the classical configuration. The cross section, usually denoted by σ, is defined as the transition rate per scatter seed in the target, per normalised incident flux of scatterers. The transition rate is the transition probability per unit time.
A cross section is therefore a measure of the effective surface area seen by the impinging particles, and as such is expressed in units of area. The cross section of two particles (i.e. observed when the two particles are colliding with each other) is a measure of the interaction event between the two particles. The cross section is proportional to the probability that an interaction will occur; for example in a simple scattering experiment the number of particles scattered per unit of time (current of scattered particles Ir) depends only on the number of incident particles per unit of time (current of incident particles Ii), the characteristics of target (for example the number of particles per unit of surface N), and the type of interaction.

## Lamb shift

The Lamb shift is the difference of energy between two energy levels of Hydrogen 2s1 and 2P1 , which was not predicted by Dirac equation. Dirac equation states these two energy level should contain same energy.

The J is here the total angular momentum which is the sum of spin quantum number and magnetic quantum number. The calculation of actual value of Lamb shift is done with the help of QED (quantum electrodynamics) equations.

HERTZ’S DYNAMICS

We have seen that Newton’s Laws are wholly lacking in selfevidence— so much so, indeed, that they contradict the law of causation in a form which has usually been held to be indubitable. We have seen also that these laws are specially suggestive of the law of gravitation. In order to eliminate what, in elementary Dynamics, is specially Newtonian, from what is really essential to the subject, we shall do well to examine some attempts to re-state the fundamental principles in a form more applicable to such sciences as Electricity. For this purpose the most suitable work seems to be that of Hertz.
The fundamental principles of Hertz’s theory are so simple and so admirable that it seems worth while to expound them briefly. His object, like that of most recent writers, is to construct a system in which there are only three fundamental concepts, space, time and mass. The elimination of a fourth concept, such as force or energy, though evidently demanded by theory, is difficult to carry out mathematically. Hertz seems, however, to have overcome the difficulty in a satisfactory manner. There are, in his system, three stages in the specification of a motion. In the first stage, only the relations of space and time are considered: this stage is purely kinematical. Matter appears here merely as a means of establishing, through the motion of a particle, a oneone correlation between a series of points and a series of instants. At this stage a collection of n particles has 3n coordinates, all so far independent: the motions which result when all are regarded as independent are all the thinkable motions of the system. But before coming to kinetics, Hertz introduces an intermediate stage. Without introducing time, there are in any free material system direct relations between space and mass, which form the geometrical connections of the system. (These may introduce time in the sense of involving velocities, but they are independent of time in the sense that they are expressed at all times by the same equations, and that these do not contain the time explicitly.) Those among thinkable motions which satisfy the equations of connection are called possible motions. The connections among the parts of a system are assumed further to be continuous in a certain well-defined sense (p. 89). It then follows that they can be expressed by homogeneous linear differential equations of the first order among the coordinates. But now a further principle is needed to discriminate among possible motions, and here Hertz introduces his only law of motion, which is as follows:
“Every free system persists in its state of rest or of uniform motion in a straightest path.”
This law requires some explanation. In the first place, when there are in a system unequal particles, each is split into a number of particles proportional to its mass. By this means all particles become equal. If now there are n particles, their 3n coordinates are regarded as the coordinates of a point in space of 3n dimensions. The above law then asserts that, in a free system, the velocity of this representative point is constant, and its path from a given point to another neighbouring point in a given direction is that one, among the possible paths through these two points, which has the smallest curvature. Such a path is called a natural path, and motion in it is called a natural

## Perturbation in calculating fine structure energy deviation

Fine structure describes the splitting of energy of atomic orbital due to the spin-orbital interaction and relativistic energy correction to the non-relativistic Shrodinger equation. The fine structure energy can be obtained by using perturbation theory. To perform this calculation one must add three terms to the hamiltonian. One is the leading order correction relativistic correction to kinetic energy, the second is the spin-orbital correction and the third is the Darwin term coming from quantum fluctuating motion.
The gross structure assumes that kinetic energy term of the energy takes the same as classical one.

But the relativistic energy will be

Using tylor's expansion we get

The first order correction then can be taken as

Using this as perturbation we can find first order energy correction due to the relativistic effect

a(0) is the Bohr's radius and n is the principle quantum number, l is the azimuthal quantum number, r is the distance of electron from the nucleus. Therefore the first order correction of energy of elctron of Hydrogen atom is

Spin-orbital coupling of elctron is given by

There is one last term in the non-relativistic expansion of Dirac equation. This term is called Darwin term as it was first derived by scientist Darwin.

Spin and orbital angular momentum of electron are very sophisticated concepts. Effects due to these are very minute yet very important for atomic physics. An electron is like the earth which is rotating and spinning at the same time. In case of electron the effect can only be observed applying an external magnetic field. Electron responds to the magnetic field by aligning itself properly. The avearage spin orbital coupling is measured by integrating over the entire atom .

The Bohr's radius changes as the square of the orbit quantum number n.

Atomic physics are interesting and hard at the same time. Don't be scared. You can understand the mathematics.

Schrodinger's equation created a revolution in quantum physics. Now radial Schrodinger's equation will be derived.
Lagrangian is

It is subject to normalization contraint

Variating it (subject to the normalization condition) we get:

Which gives the Schrödinger equation assuming the surface integral disappears.
Note: to apply the variation Δ correctly, one uses the definition:

The weak formulation is obtained from the above by substituting Δψ to v (the test function) so we get:

There are two ways to derive Radial Schrodinger's equation : one is from lagrangian and other is from equation itself. We will adopt the second.

The way to solve it is to decouple the equation into radial and angular parts by writing the Laplace operator in spherical coordinates as:

Substituting ψ(x)=R(ρ)Y(θ,φ) into the Schrödinger equation yields:

Using the fact that L^2Y=l(l+1)Y we can cancel Y and we get the radial Schrödinger equation:

Now we derive it from lagrangian.
We need to convert the Lagrangian to spherical coordinates. In order to easily make sure we do things covariantly, we start from the action (which is a scalar):

where we used the following properties of spherical harmonics:

We now minimize the action (subject to the normalization ∫ ρ^2 R^2dρ = 1) to obtain the radial equation:

So the radial Schrodinger equation becomes

## Weak formulation

The weak formulation is obtained from the action above by substituting Δ R to v (the test function) so we get:

We can also start from the equation itself, multiply by a test function v:

We integrate it. Normally we need to be using ρ^2dρ in order to integrate covariantly, but the above equation was already multiplied by ρ^2 (i.e. strictly speaking, it is not coordinate independent anymore), so we only integrate by dρ:

After integration by parts:

Where a is the end of the domain (the origin is at 0). The boundary term is zero at the origin, so we get:

We normally want to have the boundary term 1/2ρ^2R'(a)v(a) equal to zero. This is equivalent to either letting R'(a) = 0 (we prescribe the zero derivative of the radial wave function at a) or we set v(a)=0 (which corresponds to zero Dirichlet condition for R, i.e. setting R(a)=0). We can also write all the formulas using the Dirac notation:

Then normalization is:

The operator &hat; H can be written as:

and we obtain the FE formulation by expanding |R> = ∑_j R_j |j> (note that the basis |ketj> is not orthogonal, so in particular ∑_j |j> j != 1):

This is a generalized eigenvalue problem. In the special case of an orthonormal basis, = Δ_ij (which FE is not), we reach:

## Variational formulation of dirac equation

The QED lagrangian density is

Where

We will treat the fields as classical fields, so we get the classical wave Dirac equation, after plugging this Lagrangian into the Euler-Lagrange equation of motion:

Notice that the Lagrangian appears to be zero for the solution of Dirac equation (e.g. the extremum of the action). This has nothing to do with the variational principle itself, it's just a coincindence.
In this section we are only interested in the Dirac equation, so we write the Lagrangian as:

where we introduced the potential function by V = ceA_0. We also could have done the same manipulation to the dirac equation itself and we would get the same expression:

The corresponding eigenvalue problem is

Now we will develop radial dirac equation from dirac equation itself.
The manipulations are well known, one starts by writing the Dirac spinors using the spin angular functions and radial components P and Q:

and putting this into the Dirac equation one obtains:

So one obtains the following radial equations:

## You are beginning to believe....

The total hamiltonian is then given by

First term is the Coulomb interaction.
The total effect is then calculated by adding up the individual components.

### Reference materials:

Feynman's lectures on physics (elementary)
Quantum electrodynamics and s matrix (springer)
s matrix and quantum eqlectrodynamics by Greiner W & Reinhardt J.
Quantum mechanics
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf
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