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Phase space
Phase space of a dynamical system with focal stability. In mathematics and physics , a phase space is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space. For mechanical systems , the phase space usually consists of all possible values of position and momentum variables (i.e. the cotangent space of configuration space ).A phase diagram in physical chemistry , engineering , mineralogy , and materials science is a type of chart used to show conditions at which thermodynamically distinct phases can occur at equilibrium .
Phase diagram
Chaos theory is a field of study in mathematics , with applications in several disciplines including physics , engineering , economics , biology , and philosophy . Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect . Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general. [ 1 ] This happens even though these systems are deterministic , meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. [ 2 ] In other words, the deterministic nature of these systems does not make them predictable. [ 3 ] [ 4 ] This behavior is known as deterministic chaos, or simply chaos .
Chaos theory
The Structure of Scientific Revolutions by Thomas S. Kuhn is an analysis of the history of science , published in 1962 by the University of Chicago Press .
The Structure of Scientific Revolutions
Wave function
A wave function or wavefunction is a probability amplitude in quantum mechanics describing the quantum state of a particle and how it behaves. Typically, its values are complex numbers and, for a single particle, it is a function of space and time. The laws of quantum mechanics (the Schrödinger equation ) describe how the wave function evolves over time.
Wave function collapse
In quantum mechanics , wave function collapse (also called collapse of the state vector or reduction of the wave packet ) is the phenomenon in which a wave function —initially in a superposition of several different possible eigenstates —appears to reduce to a single one of those states after interaction with an observer . [ 1 ] It is the essence of measurement in quantum mechanics , and connects the wave function with classical observables like position and momentum . In classical terms, it is the reduction of all possible physical states to a single possibility which is measured by the observer.
Quantum state
In quantum physics , quantum state refers to the state of a quantum system . A quantum state is given as a vector in a vector space , called the state vector . The state vector theoretically contains statistical information about the quantum system. For example, when dealing with the energy spectrum of the electron in a hydrogen atom , the relevant state vector is given by the principal quantum number
Artist's representation of a black hole The black hole information paradox is a paradox that results from the combination of quantum mechanics and general relativity . It suggests that physical information could permanently disappear in a black hole , allowing many physical states to evolve into the same state.
Black hole information paradox
Causal dynamical triangulation (abbreviated as CDT ) invented by Renate Loll , Jan Ambjørn and Jerzy Jurkiewicz , and popularized by Fotini Markopoulou and Lee Smolin , is an approach to quantum gravity that like loop quantum gravity is background independent .
Causal dynamical triangulation
Black hole complementarity is a conjectured solution to the black hole information paradox , proposed by Leonard Susskind and Larus Thorlacius [ 1 ] and Gerard 't Hooft . [ 2 ] Ever since Stephen Hawking suggested information is lost in evaporating black hole once it passes through the event horizon and is inevitably destroyed at the singularity and that this can turn pure quantum states into mixed states , some physicists have wondered if a complete theory of quantum gravity might be able to conserve information with a unitary time evolution . But how can this be possible if information can't escape the event horizon without traveling faster than light?
Black hole complementarity
Quantum gravity ( QG ) is a field of theoretical physics that develops scientific models . Its goal is to unify quantum mechanics , which describes three of the four known fundamental interactions , with general relativity , which describes the fourth fundamental interaction, gravity . The ultimate aim is to develop a quantum gravity theory that would unify all fundamental interactions into a single mathematical framework. Such a model would describe all known observable interactions in the universe at both subatomic and cosmological scales.
Quantum gravity
Induced gravity (or emergent gravity ) is an idea in quantum gravity that space-time background emerges as a mean field approximation of underlying microscopic degrees of freedom, similar to the fluid mechanics approximation of Bose–Einstein condensates . The concept was originally proposed by Andrei Sakharov in 1967. Sakharov observed that many condensed matter systems give rise to emergent phenomena which are identical to general relativity . For example, crystal defects can look like curvature and torsion in an Einstein-Cartan spacetime .
Induced gravity
Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum.