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lorentz transformations

Theory of everything summary


INTRODUCTION

Theory of everything is a physical theory that is supposed to combine general relativity with quantum mechanics. Such theory falls under the category of quantum gravity. A lot of quantum gravity theories have been proposed but none has been fruitful. The removal of singularity like big bang, which Einstein's general relativity can not explain, compels physicists to unify quantum mechanics with general relativity. All of the attemps had gone in vain so far. String theory which compares elementary particles with tiny vibrating strings has not been experimentally verified yet. The theory is mathematically consistent though. Explanation of string theory will take a lot of time. I will briefly explain all the major theories and concepts of physics that have been discovered so far, with mathematical equations. So basic understanding of some advanced mathematics is necessary.


Classical physics of Newton and Einstein

Newton's physics Newton's physics describes motion and mechanics of massive bodies likes planets , stars and comets. The equations of motion are :
equations of motionequations of motion
These equations of motion plays a vital roles in dynamics both in classical and quantum physics (in a slight distorted way). The discovery of the universal law of gravitation of Newton was an interesting event.

equations of motion
The equation of the great law is very simple indeed

equations of gravitation
With the laws of mechanics Newton's physics can be wrapped up :

equations of mechanics

Einstein

But wait , Einstein who was a clever man did not like Newton's concepts of space , time and gravity . So he modified them in a radical way. Theory of relativity was dscovered , which says we can travel into our future , even in past too.

equations of relativity
So what does the picture above mean? it means a lot like these :

equations of relativity
t= time , x, y, z = space coordinates, c = light's speed , v = observer's velocity
The most revolutionary consequences as the mass and energy equivalence relation - using which atom bombs were developed..
equations of relativityequations of relativity
Einstein did not stop even after these remarkable discoveries. He shocked the world with the bold idea that our space and time are indeed curved. Spacetime is what we need to talk about to describe the universe. In mathematical language his ideas can be put in this beautiful form :
equations of general  relativity
The field equation simply tells "space tells matter how to move and matter tells space how to curve". This is the choreography of cosmic dance between space, time , matter and energy inside it. geodesics of particles

I forgot to speak of maxwell equations without which we can not speak with our cell phones, our eye could not see. Maxwell equations describes the propagation of light or electromagnetic waves. The mathematical elegance of the theory can be easily noticed while looking at the equations of Maxwell's :

Maxwell equations
Einstein made his field equation (described avove) consistent with Maxwell equations whereas Newton's law of gravitation was not.

Quantum physics

If you enter the world of quantum mechanics you will be shocked at least if you understand it. The laws of quantum mechanics governing electrons , neutrons, quarks are different than those of classical physics. Everything needs to be expressed in terms of probabilities. This is how it is done :

Schrodinger equations

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Phase space

Phase space can be defined as “a space in which all possible states of a system are represented, with each possible state corresponding to one unique point.” A point in phase space is fixed by the positions and (conjugate) momenta of all the particles of the system.
phase space

The first principle is that you must not fool youself , you are the easiest person to fool..

Canonical transformation

To guarantee a valid transformation between (q, p, H) and (Q, P, K), we may resort to an indirect generating function approach. Both sets of variables must obey Hamilton's principle.
That is the Action Integral over the Lagrangian , obtained by the Hamiltonian via ("inverse") Legendre transformation, both must be stationary .
canonical transformation
One way for both variational integral equalities to be satisfied is to have
canonical transformation
Lagrangians are not unique: one can always multiply by a constant λ and add a total time derivative dG/dt and yield the same equations of motion. G is the generating function. In physics a generating function is a function whose derivative yields differential equations that describe dynamics of the system.
A choice of the generating function G = G_1(q, Q, t) can be found by expanding the above equation.
generating function
Since old and new coorndiates are independent the following 2N + 1 conditions must hold.
generating function
These equations define the transformation (q, p) → (Q, P) as follows. The first set of N equations
p = dG_1/dq
define relations between the new generalized coordinates Q and the old canonical coordinates (q, p). Ideally, one can invert these relations to obtain formulae for each Q_k as a function of the old canonical coordinates. Substitution of these formulae for the Q coordinates into the second set of N equations
P = -dG_1/dQ
yields analogous formulae for the new generalized momenta P in terms of the old canonical coordinates (q, p). We then invert both sets of formulae to obtain the old canonical coordinates (q, p) as functions of the new canonical coordinates (Q, P). Substitution of the inverted formulae into the final equation
K = H + dG_1/dt --- (1)
yields a formula for K as a function of the new canonical coordinates (Q, P).
In the particular case where there is a special generating function, usually denoted by S, such that the new Hamiltonian K = 0 ;
equation (1) becomes
hamiltonian jacobi equation
Where S = G has been used.
Since the new Hamiltonian K=0, the corresponding Hamilton’s equations of motion imply that the Qs and Ps are constants:
hamiltonians jacobi equation
For concreteness, consider the HJE corresponding to a particle with mass m moving in a potential V which is time-independent. It is given by:
hamiltonians jacobi equation

Quantum theory


hamiltonians jacobi equation
The fundamental entity in the quantum-mechanical framework is the concept of a quantum physical state (see Fradkin). Mathematically, it is represented by a ray in a vector space called the Hilbert space of quantum states. Two normalized vectors or pure states ψ₁ and ψ₂ belong to the same ray if the following condition is obeyed:
hilbert space
A set of states in a Hilbert space with dimension N is
{ φ_i} where i=1, 2,3, ...N
A ray ψ is the linear combination of those basis states.
ray in hilbert space

Reference materials:


Law of thermodynamics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf
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