Special Relativity
  • The principle of special relativity which states that the laws of physics take the same form in all inertial frames
  • The speed of light (c) in a vacuum is the same for all observers
  • Causality is preserved: It connects the cause with the effect
    Nothing can travel faster than the speed of light
  • Energy and mass are equivalent: E = m c2
  • Time run slower in moving objects
  • Space is shorter in moving objects

General Relativity (GR)
Fundamental theory of nature at large scales
  • Background independence
    Geometry of space and time is a dynamical entity: space and time are no fixed strutures
    NO fixed space-time backfround
  • Gravity is not a force – it is a property of space-time itself
  • Evolution of geometry (gravity) = evolution of matter
  • Matter and energy determine the geometry of spacetime
  • The curvature of space and time influence the way that matter move
  • The matter influence the curvature of space-time
  • Diffeomorphism invariance implies that no coordinate system is better than another
    NO preferred coordinate systems
    NO location according to a set coordinates
    NO depend on the lattice
  • Field equations should be generally covariant
    they should take the same general form in any coordinate system
  • The principle of general relativity that states that the laws of physics take the same form in all reference systems
  • Gravitational time dilation
  • Distortions of the geometry should propagate ( gravitational waves)
  • If enough mass is concentrated should form a black hole

Quantum Mechanics
Fundamental theory of nature at small scales and low energies
  • Background dependent (Newton's absolute space and time)
    Fixed background space and time geometry
  • Quantization of energy (Planck's constant), momentum
  • Wave-particle duality
  • Uncertainty principle: limit to the precision with which pairs of physical properties (position-mommentum and time-energy) can be known
  • In quantum field theory (QFT) (Interactions among particles are interactions among quantum fields)
    Field theory of quntum mechanics and special relativity
    QED and QCD and Electroweak interaction
  • Two formulations:Matrix mechanics and Wave mechanics

The Standard Model
    the combination of two schemes:
  • the electroweak force
    quantum electrodynamics (quantum theory of the electromagnetic field)
    Weak force
  • quantum chromodynamics (QCD)

Kaluza-Klein theory
  • a theory of general relativity in 5-spacetime dimensions
  • curl up one of the dimensions into a circle
  • 4-dimensional theory of general relativity plus electromagnetism
    electromagnetism is a U(1) gauge theory (the group of rotations around a circle)

The holographic principle
  • All information in the bulk of a space is coded at its border
  • gravity in a three-dimensional volume should emerge from the two dimensions of the surface surrounding the volume
    the universe may really have two spacial dimensions
  • Theoretical work that may confirm this principle:
    The entropy of a black hole is proportional to the surface area, not its volume
    AdS/CFT correspondence in String Theory
    Calculations that are hard in one version of the theory may actually be easy in the other one

Espectral dimension
  • The dimension that a random walker experiences in a diffusion process
    measure the probability for him to return to one point
    Each point in the population has a certain number of closest neighbors
    The smaller the average return probability the higher the number of dimensions
  • how things spread through a medium over time
Loop Quantum Gravity (LQG)
  • Background independent ( Leibniz's relationist -like GR theory)
    NO background(lattice) of absolute space and time (only relationships between elementary entities)
    Time is a change in the relationships
    It is a quantum theory of gravity(it is not a TOE)
    Space is discrete (granular, quantize)
    There is no meaning to distance at scales smaller than the Planck length
  • LQG use the Ashtekar variables of canonical GR (ADM formalism and its Hamiltonian formulation of GR)

  • In LQG, at each instant of time, geometry is concentrated on graphs
  • A graph is a network of one dimensional structures, oriented lines which are linked together at their end point
  • Spin network= nodes + lines + arrows(orientation) + labels(half-integral spin numbers)
    spin network = quantum state of space at a certain point in time
  • Each element of the graph is dual to an element of the space
  • node (zero-dimensional point or dot) + lines (one-dimensional) = three-dimensional volume (3-0)=3 dimensions + two-dimensional faces orthogonal to the line(3-1)=2 dimensions
  • labelling the graph:
    node number = volumen
    line number = surface area
  • Eigenvalue of the area operator (only a discrete set of special values of the order of the Plank area (10-33)2 cm2)
  • Wavefunction over geometries (probability of having one space-time geometry) the Wheeler-DeWitt equation
  • Schrödinger wavefunction over particles (probability of a particle is either here or there)

  • loops (they are the physical space) are quantum excitations of the lines of force of the gravitational field (Yang-Mills fiels) (low-energy approximation gravitons)
  • State of space is a network of intersection loops
  • NO location of the net
    location ON the net
    loops ON loops
  • Einstein's gravitational field is replaced by the Ashtekar connection field (loop variables)
  • loops are finite elements of space

  • Spin networks: networks of loops
  • Nodes of spin networks:represented by dots
  • Nodes are the points at which loops intersect :elementary quanta of volume
    V=(ħ G/c3)3/2
  • Links of the net: elementary surface the area elements:represented by lines joining the nodes
    A=16πV2/3 √(j(j+1)) j=integers or half-integers
  • Region of space is a quantum superposition of spin-network states
  • Dynamics in the region: Wheeler-DeWitt equation
  • Loop: a sequence of links
  • Areas and volumes are quantized in multiples of the Barbero-Immirzi parameter
  • Resolution of the Big Bang singularity by a Big Bounce

    Spin foam
  • Spin foam is a topological structure made out of two-dimensional faces
  • Spacetime is a superposition of spinfoams
  • The boundary of a spinfoam is a spin network
  • The black hole entropy ( Bekenstein-Hawking entropy -S = A / 4-) is obtained using the Immirzi parameter
  • the lines of the spin network become planes
    the nodes becomes lines
    A slice through a spin foam at a particular time yields a spin network

    Quantum spin dynamics (QSD)
  • The values of the spin labels can increase or decrease (even disappear) over time
    Matter could only exit on lines of the spin-network
    Empty region are no space at all

Causal Dynamical Triangulations (CDT)
  • it proposes a topological substrate (spin-foam) made of simplexes
    a 0-simplex is a point
    a 1-simplex is a line segment
    a 2-simplex is a triangle
    a 3-simplex is a tetrahedron
    a 4-simplex, is a pentachoron
  • Spatial curvature emerges from how those simplexes can join together
  • Near the Planck scale the structure of spacetime
    shows to be 2-d
    reveals a fractal structure on slices of constant time
    it is supposed to be constantly changing due to quantum fluctuations
  • Theory uses a triangulation process which varies dynamically and follows deterministic rules
    • it divides spacetime into tiny triangular sections
    • it allows only those configurations in which the timelines of all joined edges of simplices agree
    • a k-simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions
    • the space that emerged was one of two degenerate forms
      it either has infinite dimensions or just one
    • if edges of simplexes can only be joined when their arrows of time are pointing in the same direction
    • the space emerges correctly
    • time has an arrow of time because it has an unmixed state at one end (or point) and a mixed state at the other
      when you have an entropy gradient along a time-like curve, you have an arrow of time
      only join edges if arrows of time are the same

Causal Sets
  • Spacetime is discrete
  • Spacetime events are related by a partial order (causality relations)
  • The building blocks of space-time are simple mathematical points that are connected by links
    With each link pointing from past to future (causality)
    Causality = an earlier point can affect a later one, but not vice versa
    The volume of a spacetime region is the number of points in that region
    Order + Number = Geometry

Spontaneous Dimensional Reduction
  • At smallest scates space-time is 1+1 dimensional
  • Gravity requires 3 dimensions in order to propagate
    Gravity cannot propagate atsmall scales
  • In 1-dimensional space the force( F ∝ 1/r0) remains unchanged to distance
    No infinite forces at small distances
    In 2 and 3-dimensional space, the force (∝ 1/r1 and ∝ 1/r2) diminish over distance becoming infinite at r=0
    The effective dimensionality of the space depends on the length scale

Randall–Sundrum (RS1 and RS2) models
  • the real world is a five-dimensional anti-de Sitter space (AdS)
    the elementary particles except for the graviton are localized on a (3 + 1)-dimensional brane or branes
  • RS1, has a finite size for the extra dimension with two branes, one at each end
    a finite five-dimensional bulk
  • the Planckbrane (where gravity is a relatively strong force; also called "Gravitybrane" has positive brane energy)
    the Tevbrane (our home with the Standard Model particles; also called "Weakbrane" has negative brane energy)
    the two branes are separated in the fifth dimension by approximately 16 units (from 10-33cm to 10-17cm)
    warped spacetime along the fifth dimension
  • RS2, one brane has been placed infinitely far away
    so that there is only one brane left in the model

String theory
  • Background dependent
  • Elementary particles are one dimension strings
  • String can vibrate
  • The vibrations give rise to the fundamental properties of elementary particles (mass, charge, spin...)
  • Different types of vibrations give rise to differents types of particles
  • Strings can be open or closed
  • Strings can be oriented and unoriented
  • Closed strings have periodic boundary conditions
  • Conditions of the endpoints of an open string:
    Neumann boundary conditions: free endpoints
    Dirichlet boundary conditions: endpoint confined to (attached to) a D-brane
  • Gravitons are vibrational states of closed strings (not attached to D-brane)
  • Elementary particles are made of open strings bound to the D-brane
  • Strings has tension
  • The 6 additional dimensions are curled up into tiny complicated shapes ( a Calabi-Yau surface)
    The 6 extra dimensions are very small and invisible at low energies
  • It is a theory of everything
    matter and fundamental forces (including quantum gravity)
  • Bosonic string theory: Only bosons in 26 (25 space + 1 time) dimensions
  • With supersymmetry that relates bosons (interger-spin) and fermions (half-integer spin)
    We have supersymmetric string theory or Superstring theory in 10 (9 space +1 time) dimensions
  • The anti-de Sitter/conformal field theory (AdS/CFT) correspondence is the equivalent between a string theory or supergravity in a AdS and CFT in a boundary with d-1 dimensions
  • S-duality and T-duality unifies all five superstring theories in M-theory in 11 dimensions
  • Five strings theories:
    • Type I
    • Type IIA
    • Type IIB
    • SO(32) heterotic
    • E8xE8 heterotic
  • Heterotic string:
    Closed string
    superstring and bosonic string
  • Type I string: unoriented closed and open strings
  • Type II string: oriented closed strings

  • S-duality
    Strongly interacting particles in one theory is equivalent to a weakly interacting particles in a different theory
    Type I and SO(32) heterotic
    M-theory and E8xE8 heterotic
    M-theory and Type IIA
    Type IIB with itself
  • T-duality
    A string propagating aroung a circle of radius R is equivalent to a string propagating around a circle of radius 1/R
    SO(32) heterotic and E8xE8 heterotic
    Type IIA and Type IIB

  • The force carrier particles of the weak, the strong, and the electro-magnetic forces are strings with end points that confine them to their D-branes
  • The gravitons are strings with closed loops and completely free to move from one membrane to another membrane
  • D0-brane is a point
    D1-brane (D-string) is a line
    D2-brane is a plane
    D-9 is a D-brane with 9 spatial dimension
    D-1 all the space and time coordinates are fixed: instanton or D-instanton
    D0 all the spatial coordinates are fixed: D0-brane or D-particle
  • 11-dimensional spacetime: upper limit is D10-brane (M-theory)
  • 10-dimensional spacetime: upper limit is D9-brane (Superstrings theories)

Brane World Scenario
  • the entire Universe is an eleven dimensional bulk composed of
    an infinite number of ten dimensional branes
  • Each brane is consists of
    a macroscopic 4D spacetime and a compacted
    microscopic 6D quantum world (a Calabi-Yau space)
    1D bulk
  • Three of the four fundamental forces (strong, weak, electromagnetism and their associated particles) are represented by open strings
    Gravity (and gravitons) are represented by closed strings (loops)
  • Open strings are attached to their respective branes
    closed strings (gravity) are free to move between branes
  • An infinite number of parallel universes is literally just a millimeter away
    branes in the bulk of the multiverse
  • gravity appears very weak at macroscopic distances because its intensity is spread in the large extra dimensions known as the bulk
  • The observable universe could be a 1+3 surface (the brane) embedded
    in a 1+3+d-dimensional spacetime (the bulk)
    Standard Model particles and fields trapped on the brane
    gravity is free to access the bulk
    At least one of the extra spatial dimensions is very large relative to the Planck scale (not compacted)
    non-compacted spatials dimensions must be extremely large, possibly infinite
  • Big bang: Collision of our brane (or universe) with another brane in the bulk