In mathematics, the gamma function is one of the extensions of the factorial function with its argument shifted down by 1, to real and complex numbers. Derived by Daniel Bernoulli, if n is a positive integer, Γ =! Although other extensions do exist, this particular definition is the most useful; the gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral: Γ = ∫ 0 ∞ x z − 1 e − x d x This integral function is extended by analytic continuation to all complex numbers except the non-positive integers, yielding the meromorphic function we call the gamma function, it has no zeroes, so the reciprocal gamma function 1/Γ is a holomorphic function. In fact the gamma function corresponds to the Mellin transform of the negative exponential function: Γ = The gamma function is a component in various probability-distribution functions, as such it is applicable in the fields of probability and statistics, as well as combinatorics.
The gamma function can be seen as a solution to the following interpolation problem: "Find a smooth curve that connects the points given by y =! at the positive integer values for x."A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that describes the curve, in which the number of operations does not depend on the size of x. The simple formula for the factorial, x! = 1 × 2 × … × x, cannot be used directly for fractional values of x since it is only valid when x is a natural number. There are speaking, no such simple solutions for factorials. A good solution to this is the gamma function. There are infinitely many continuous extensions of the factorial to non-integers: infinitely many curves can be drawn through any set of isolated points; the gamma function is the most useful solution in practice, being analytic, it can be characterized in several ways. However, it is not the only analytic function which extends the factorial, as adding to it any analytic function, zero on the positive integers, such as k sin mπx, will give another function with that property.
A more restrictive property than satisfying the above interpolation is to satisfy the recurrence relation defining a translated version of the factorial function, f = 1, f = x f, for x equal to any positive real number. But this would allow for multiplication by any periodic analytic function which evaluates to one on the positive integers, such as ek sin mπx. There's a final way to solve all this ambiguity: Bohr–Mollerup theorem states that when the condition that f be logarithmically convex is added, it uniquely determines f for positive, real inputs. From there, the gamma function can be extended to all real and complex values by using the unique analytic continuation of f. See Euler's infinite product definition below where the properties f = 1 and f = x f together with the asymptotic requirement that limn→+∞! nx / f = 1 uniquely define the same function. The notation Γ is due to Legendre. If the real part of the complex number z is positive the integral Γ = ∫ 0 ∞ x z − 1 e − x d x converges and is known as the Euler integral of the second kind.
Using integration by parts, one sees that: Γ = ∫ 0 ∞ x z e − x d x = 0 ∞ + ∫ 0 ∞ z x z − 1 e − x d x = lim x →
Zero-point energy is the difference between the lowest possible energy that a quantum mechanical system may have, the classical minimum energy of the system. Unlike in classical mechanics, quantum systems fluctuate in their lowest energy state due to the Heisenberg uncertainty principle; as well as atoms and molecules, the empty space of the vacuum has these properties. According to quantum field theory, the universe can be thought of not as isolated particles but continuous fluctuating fields: matter fields, whose quanta are fermions, force fields, whose quanta are bosons. All these fields have zero-point energy; these fluctuating zero-point fields lead to a kind of reintroduction of an aether in physics, since some systems can detect the existence of this energy. However this aether cannot be thought of as a physical medium if it is to be Lorentz invariant such that there is no contradiction with Einstein's theory of special relativity. Physics lacks a full theoretical model for understanding zero-point energy.
Physicists Richard Feynman and John Wheeler calculated the zero-point radiation of the vacuum to be an order of magnitude greater than nuclear energy, with a single light bulb containing enough energy to boil all the world's oceans. Yet according to Einstein's theory of general relativity any such energy would gravitate and the experimental evidence from both the expansion of the universe, dark energy and the Casimir effect show any such energy to be exceptionally weak. A popular proposal that attempts to address this issue is to say that the fermion field has a negative zero-point energy while the boson field has positive zero-point energy and thus these energies somehow cancel each other out; this idea would be true. However, the LHC at CERN has so far found no evidence to support supersymmetry. Moreover, it is known that if supersymmetry is valid at all, it is at most a broken symmetry, only true at high energies, no one has been able to show a theory where zero-point cancellations occur in the low energy universe we observe today.
This discrepancy is known as the cosmological constant problem and it is one of the greatest unsolved mysteries in physics. Many physicists believe that "the vacuum holds the key to a full understanding of nature"; the term zero-point energy is a translation from the German Nullpunktsenergie. The terms zero-point radiation or ground state energy are sometimes used interchangeably; the term zero-point field can be used when referring to a specific vacuum field, for instance the QED vacuum which deals with quantum electrodynamics or the QCD vacuum which deals with quantum chromodynamics. A vacuum can be viewed not as the combination of all zero-point fields. In quantum field theory this combination of fields is called the vacuum state, its associated zero-point energy is called the vacuum energy and the average energy value is called the vacuum expectation value called its condensate. In classical mechanics all particles can be thought of as having some energy made up of their potential energy and kinetic energy.
Temperature, for example, arises from the intensity of random particle motion caused by kinetic energy. As temperature is reduced to absolute zero, it might be thought that all motion ceases and particles come to rest. In fact, kinetic energy is retained by particles at the lowest possible temperature; the random motion corresponding to this zero-point energy never vanishes as a consequence of the uncertainty principle of quantum mechanics. The uncertainty principle states that no object can have precise values of position and velocity simultaneously; the total energy of a quantum mechanical object is described by its Hamiltonian which describes the system as a harmonic oscillator, or wave function, that fluctuates between various energy states. All quantum mechanical systems undergo fluctuations in their ground state, a consequence of their wave-like nature; the uncertainty principle requires every quantum mechanical system to have a fluctuating zero-point energy greater than the minimum of its classical potential well.
This results in motion at absolute zero. For example, liquid helium does not freeze under atmospheric pressure regardless of temperature due to its zero-point energy. Given the equivalence of mass and energy expressed by Einstein's E = mc2, any point in space that contains energy can be thought of as having mass to create particles. Virtual particles spontaneously flash into existence at every point in space due to the energy of quantum fluctuations caused by the uncertainty principle. Modern physics has developed quantum field theory to understand the fundamental interactions between matter and forces, it treats every single point of space as a quantum harmonic oscillator. According to QFT the universe is made up of matter fields, whose quanta are fermions, force fields, whose quanta are bosons. All these fields have zero-point energy. Recent experiments advocate the idea that particles themselves can be thought of as excited states of the underlying quantum vacuum, that all properties of matter are vacuum fluctuations arising from interactions of the zero-point field.
The idea that "empty" space can have an intrinsic energy associated to it, that there is no such thing as
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of quantities to compensate for effects of their self-interactions. However if it were the case that no infinities arise in loop diagrams in quantum field theory, it can be shown that renormalization of mass and fields appearing in the original Lagrangian is necessary. For example, an electron theory may begin by postulating an electron with an initial mass and charge. In quantum field theory a cloud of virtual particles, such as photons and others surrounds and interacts with the initial electron. Accounting for the interactions of the surrounding particles shows that the electron-system behaves as if it had a different mass and charge than postulated. Renormalization, in this example, mathematically replaces the postulated mass and charge of an electron with the experimentally observed mass and charge.
Mathematics and experiments prove that positrons and more massive particles like protons, exhibit the same observed charge as the electron - in the presence of much stronger interactions and more intense clouds of virtual particles. Renormalization specifies relationships between parameters in the theory when parameters describing large distance scales differ from parameters describing small distance scales. In high-energy particle accelerators like the CERN Large Hadron Collider the concept named pileup occurs when undesirable proton-proton collisions interact with data collection for simultaneous, nearby desirable measurements. Physically, the pileup of contributions from an infinity of scales involved in a problem may result in further infinities; when describing space-time as a continuum, certain statistical and quantum mechanical constructions are not well-defined. To define them, or make them unambiguous, a continuum limit must remove "construction scaffolding" of lattices at various scales.
Renormalization procedures are based on the requirement that certain physical quantities equal observed values. That is, the experimental value of the physical quantity yields practical applications, but due to their empirical nature the observed measurement represents areas of quantum field theory that require deeper derivation from theoretical bases. Renormalization was first developed in quantum electrodynamics to make sense of infinite integrals in perturbation theory. Viewed as a suspect provisional procedure by some of its originators, renormalization was embraced as an important and self-consistent actual mechanism of scale physics in several fields of physics and mathematics. Today, the point of view has shifted: on the basis of the breakthrough renormalization group insights of Nikolay Bogolyubov and Kenneth Wilson, the focus is on variation of physical quantities across contiguous scales, while distant scales are related to each other through "effective" descriptions. All scales are linked in a broadly systematic way, the actual physics pertinent to each is extracted with the suitable specific computational techniques appropriate for each.
Wilson clarified which are redundant. Renormalization is distinct from regularization, another technique to control infinities by assuming the existence of new unknown physics at new scales; the problem of infinities first arose in the classical electrodynamics of point particles in the 19th and early 20th century. The mass of a charged particle should include the mass-energy in its electrostatic field. Assume that the particle is a charged spherical shell of radius re; the mass–energy in the field is m em = ∫ 1 2 E 2 d V = ∫ r e ∞ 1 2 2 4 π r 2 d r = q 2 8 π r e, which becomes infinite as re → 0. This implies that the point particle would have infinite inertia, making it unable to be accelerated. Incidentally, the value of re that makes m em equal to the electron mass is called the classical electron radius, which turns out to be r e = e 2 4 π ε 0 m e c 2 = α ℏ m e c ≈ 2.8 × 10 − 15 m, where α ≈ 1 / 137 is the fine-structure constant, ℏ / ( m
Physics is the natural science that studies matter, its motion, behavior through space and time, that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, its main goal is to understand how the universe behaves. Physics is one of the oldest academic disciplines and, through its inclusion of astronomy the oldest. Over much of the past two millennia, chemistry and certain branches of mathematics, were a part of natural philosophy, but during the scientific revolution in the 17th century these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, the boundaries of physics which are not rigidly defined. New ideas in physics explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics enable advances in new technologies.
For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have transformed modern-day society, such as television, domestic appliances, nuclear weapons. Astronomy is one of the oldest natural sciences. Early civilizations dating back to beyond 3000 BCE, such as the Sumerians, ancient Egyptians, the Indus Valley Civilization, had a predictive knowledge and a basic understanding of the motions of the Sun and stars; the stars and planets were worshipped, believed to represent gods. While the explanations for the observed positions of the stars were unscientific and lacking in evidence, these early observations laid the foundation for astronomy, as the stars were found to traverse great circles across the sky, which however did not explain the positions of the planets. According to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, all Western efforts in the exact sciences are descended from late Babylonian astronomy.
Egyptian astronomers left monuments showing knowledge of the constellations and the motions of the celestial bodies, while Greek poet Homer wrote of various celestial objects in his Iliad and Odyssey. Natural philosophy has its origins in Greece during the Archaic period, when pre-Socratic philosophers like Thales rejected non-naturalistic explanations for natural phenomena and proclaimed that every event had a natural cause, they proposed ideas verified by reason and observation, many of their hypotheses proved successful in experiment. The Western Roman Empire fell in the fifth century, this resulted in a decline in intellectual pursuits in the western part of Europe. By contrast, the Eastern Roman Empire resisted the attacks from the barbarians, continued to advance various fields of learning, including physics. In the sixth century Isidore of Miletus created an important compilation of Archimedes' works that are copied in the Archimedes Palimpsest. In sixth century Europe John Philoponus, a Byzantine scholar, questioned Aristotle's teaching of physics and noting its flaws.
He introduced the theory of impetus. Aristotle's physics was not scrutinized until John Philoponus appeared, unlike Aristotle who based his physics on verbal argument, Philoponus relied on observation. On Aristotle's physics John Philoponus wrote: “But this is erroneous, our view may be corroborated by actual observation more than by any sort of verbal argument. For if you let fall from the same height two weights of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a small one, and so, if the difference in the weights is not considerable, that is, of one is, let us say, double the other, there will be no difference, or else an imperceptible difference, in time, though the difference in weight is by no means negligible, with one body weighing twice as much as the other”John Philoponus' criticism of Aristotelian principles of physics served as an inspiration for Galileo Galilei ten centuries during the Scientific Revolution.
Galileo cited Philoponus in his works when arguing that Aristotelian physics was flawed. In the 1300s Jean Buridan, a teacher in the faculty of arts at the University of Paris, developed the concept of impetus, it was a step toward the modern ideas of momentum. Islamic scholarship inherited Aristotelian physics from the Greeks and during the Islamic Golden Age developed it further placing emphasis on observation and a priori reasoning, developing early forms of the scientific method; the most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn al-Haytham, in which he conclusively disproved the ancient Greek idea about vision, but came up with a new theory. In the book, he presented a study of the phenomenon of the camera obscura (his thousand-year-old
Quantum field theory
In theoretical physics, quantum field theory is a theoretical framework that combines classical field theory, special relativity, quantum mechanics and is used to construct physical models of subatomic particles and quasiparticles. QFT treats particles as excited states of their underlying fields, which are—in a sense—more fundamental than the basic particles. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding fields; each interaction can be visually represented by Feynman diagrams, which are formal computational tools, in the process of relativistic perturbation theory. As a successful theoretical framework today, quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century, its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory — quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure.
A second major barrier came with QFT's apparent inability to describe the weak and strong interactions, to the point where some theorists called for the abandonment of the field theoretic approach. The development of gauge theory and the completion of the Standard Model in the 1970s led to a renaissance of quantum field theory. Quantum field theory is the result of the combination of classical field theory, quantum mechanics, special relativity. A brief overview of these theoretical precursors is in order; the earliest successful classical field theory is one that emerged from Newton's law of universal gravitation, despite the complete absence of the concept of fields from his 1687 treatise Philosophiæ Naturalis Principia Mathematica. The force of gravity as described by Newton is an "action at a distance" — its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with Richard Bentley, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else, not material, operate upon and affect other matter without mutual contact."
It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields — a numerical quantity assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered a mathematical trick. Fields began to take on an existence of their own with the development of electromagnetism in the 19th century. Michael Faraday coined the English term "field" in 1845, he introduced fields as properties of space having physical effects. He argued against "action at a distance", proposed that interactions between objects occur via space-filling "lines of force"; this description of fields remains to this day. The theory of classical electromagnetism was completed in 1862 with Maxwell's equations, which described the relationship between the electric field, the magnetic field, electric current, electric charge. Maxwell's equations implied the existence of electromagnetic waves, a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the speed of light.
Action-at-a-distance was thus conclusively refuted. Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in atomic spectra, nor for the distribution of blackbody radiation in different wavelengths. Max Planck's study of blackbody radiation marked the beginning of quantum mechanics, he treated atoms, which absorb and emit electromagnetic radiation, as tiny oscillators with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as quantum harmonic oscillators; this process of restricting energies to discrete values is called quantization. Building on this idea, Albert Einstein proposed in 1905 an explanation for the photoelectric effect, that light is composed of individual packets of energy called photons; this implied that the electromagnetic radiation, while being waves in the classical electromagnetic field exists in the form of particles. In 1913, Niels Bohr introduced the Bohr model of atomic structure, wherein electrons within atoms can only take on a series of discrete, rather than continuous, energies.
This is another example of quantization. The Bohr model explained the discrete nature of atomic spectral lines. In 1924, Louis de Broglie proposed the hypothesis of wave-particle duality, that microscopic particles exhibit both wave-like and particle-like properties under different circumstances. Uniting these scattered ideas, a coherent discipline, quantum mechanics, was formulated between 1925 and 1926, with important contributions from de Broglie, Werner Heisenberg, Max Born, Erwin Schrödinger, Paul Dirac, Wolfgang Pauli.:22-23In the same year as his paper on the photoelectric effect, Einstein published his theory of special relativity, built on Maxwell's electromagnetism. New rules, called Lorentz transformation, were given for the way time and space coordinates of an event change under changes in the observer's velocity, the distinction between time and space was blurred.:19 It was proposed that all physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations.
Two difficulties remained. Observationally, the Schrödinger equation underlying q
Number theory is a branch of pure mathematics devoted to the study of the integers. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of objects made out of integers or defined as generalizations of the integers. Integers can be considered either as solutions to equations. Questions in number theory are best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may study real numbers in relation to rational numbers, for example, as approximated by the latter; the older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory"; the use of the term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.
The first historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 contains a list of "Pythagorean triples", that is, integers such that a 2 + b 2 = c 2. The triples are too large to have been obtained by brute force; the heading over the first column reads: "The takiltum of the diagonal, subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity 2 + 1 = 2, implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and reordered by c / a for actual use as a "table", for example, with a view to applications, it is not known whether there could have been any. It has been suggested instead. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, Babylonian algebra was exceptionally well developed. Late Neoplatonic sources state.
Much earlier sources state that Pythagoras traveled and studied in Egypt. Euclid IX 21–34 is probably Pythagorean. Pythagorean mystics gave great importance to the even; the discovery that 2 is irrational is credited to the early Pythagoreans. By revealing that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; this forced a distinction between numbers, on the one hand, lengths and proportions, on the other hand. The Pythagorean tradition spoke of so-called polygonal or figurate numbers. While square numbers, cubic numbers, etc. are seen now as more natural than triangular numbers, pentagonal numbers, etc. the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period. We know of no arithmetical material in ancient Egyptian or Vedic sources, though there is some algebra in both; the Chinese remainder theorem appears as an exercise in Sunzi Suanjing There is some numerical mysticism in Chinese mathematics, unlike that of the Pythagoreans, it seems to have led nowhere.
Like the Pythagoreans' perfect numbers, magic squares have passed from superstition into recreation. Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-m
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia