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
In geometry and physics, spinors are elements of a vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight rotation. However, when a sequence of such small rotations is composed to form an overall final rotation, the resulting spinor transformation depends on which sequence of small rotations was used: unlike vectors and tensors, a spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360°; this property characterizes spinors. It is possible to associate a similar notion of spinor to Minkowski space in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or "spin", of the electron and other subatomic particles. Spinors are characterized by the specific way.
They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved. There are two topologically distinguishable classes of paths through rotations that result in the same overall rotation, as famously illustrated by the belt trick puzzle; these two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class, it doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class. In mathematical terms, spinors are described by a double-valued projective representation of the rotation group SO. Although spinors can be defined purely as elements of a representation space of the spin group, they are defined as elements of a vector space that carries a linear representation of the Clifford algebra.
The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, in applications the Clifford algebra is the easiest to work with. After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices, matrices that satisfy a set of canonical anti-commutation relations; the spinors are the column vectors. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what constitutes a "column vector", involves the choice of basis and gamma matrices in an essential way; as a representation of the spin group, this realization of spinors as column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even.
What characterizes spinors and distinguishes them from geometric vectors and other tensors is subtle. Consider applying a rotation to the coordinates of a system. No object in the system itself has moved, only the coordinates have, so there will always be a compensating change in those coordinate values when applied to any object of the system. Geometrical vectors, for example, have components that will undergo the same rotation as the coordinates. More broadly, any tensor associated with the system has coordinate descriptions that adjust to compensate for changes to the coordinate system itself. Spinors do not appear at this level of the description of a physical system, when one is concerned only with the properties of a single isolated rotation of the coordinates. Rather, spinors appear when we imagine that instead of a single rotation, the coordinate system is rotated between some initial and final configuration. For any of the familiar and intuitive quantities associated with the system, the transformation law does not depend on the precise details of how the coordinates arrived at their final configuration.
Spinors, on the other hand, are constructed in such a way that makes them sensitive to how the gradual rotation of the coordinates arrived there: they exhibit path-dependence. It turns out that, for any final configuration of the coordinates, there are two inequivalent gradual rotations of the coordinate system that result in this same configuration; this ambiguity is called the homotopy class of the gradual rotation. The belt trick puzzle famously demonstrates two different rotations, one through an angle of 2π and the other through an angle of 4π, having the same final configurations but different classes. Spinors exhibit a sign-reversal that genuinely depends on this homotopy class; this distinguishes them from other tensors, none of which can feel the class. Spinors can be exhibited as concrete objects using a choice of Cartesian coordinates. In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to the thre
The Road to Reality
The Road to Reality: A Complete Guide to the Laws of the Universe is a book on modern physics by the British mathematical physicist Roger Penrose, published in 2004. It covers the basics of the Standard Model of particle physics, discussing general relativity and quantum mechanics, discusses the possible unification of these two theories; the book discusses the physical world. Many fields that 19th century scientists believed were separate, such as electricity and magnetism, are aspects of more fundamental properties; some texts, both popular and university level, introduce these topics as separate concepts, reveal their combination much later. The Road to Reality reverses this process, first expounding the underlying mathematics of space–time showing how electromagnetism and other phenomena fall out formed; the book is just over 1100 pages, of which the first 383 are dedicated to mathematics—Penrose's goal is to acquaint inquisitive readers with the mathematical tools needed to understand the remainder of the book in depth.
Physics enters the discussion on page 383 with the topic of space–time. From there it moves on to fields in spacetime, deriving the classical electrical and magnetic forces from first principles. Energy and conservation laws appear in the discussion of Lagrangians and Hamiltonians, before moving on to a full discussion of quantum physics, particle theory and quantum field theory. A discussion of the measurement problem in quantum mechanics is given a full chapter; the book ends with an exploration of possible ways forward. The final chapters reflect Penrose's personal perspective, which differs in some respects from what he regards as the current fashion among theoretical physicists, he is skeptical about string theory. He is optimistic about twistor theory, he holds some controversial views about the role of consciousness in physics, as laid out in his earlier books. Alfred A. Knopf, February 2005, hardcover, ISBN 0-679-45443-8 Vintage Books, 2005, softcover, ISBN 0-09-944068-7 Vintage Books, 2006, softcover, ISBN 0-09-944068-7 Vintage Books, 2007, softcover, ISBN 0-679-77631-1 Site with errata and solutions to some exercises from the first few chapters.
Not sponsored by Penrose. Archive of the Road to Reality internet forum, now defunct. Solutions for many Road to Reality exercises
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups; the compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type; the term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups. The classical groups form most useful part of the subject of linear Lie groups. Most types of classical groups find application in modern physics. A few examples are the following; the rotation group SO is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group O is a symmetry group of spacetime of special relativity.
The special unitary group SU is the symmetry group of quantum chromodynamics and the symplectic group Sp finds application in hamiltonian mechanics and quantum mechanical versions of it. The classical groups are the general linear groups over R, C and H together with the automorphism groups of non-degenerate forms discussed below; these groups are additionally restricted to the subgroups whose elements have determinant 1, so that their centers are discrete. The classical groups, with the determinant 1 condition, are listed in the table below. In the sequel, the determinant 1 condition is not used in the interest of greater generality; the complex classical groups are SO and Sp.. A group is complex according to; the real classical groups refers to all of the classical groups since any Lie algebra is a real algebra. The compact classical groups are the compact real forms of the complex classical groups; these are, in turn, SO and Sp.. One characterization of the compact real form is in terms of the Lie algebra g.
If g = u + iu, the complexification of u if the connected group K generated by exp: X ∈ u is a compact, K is a compact real form. The classical groups can uniformly be characterized in a different way using real forms; the classical groups are the following: The complex linear algebraic groups SL, SO, Sp together with their real forms. For instance, SO∗ is a real form of SO, SU is a real form of SL, SL is a real form of SL. Without the determinant 1 condition, replace the special linear groups with the corresponding general linear groups in the characterization; the algebraic groups in question are Lie groups, but the "algebraic" qualifier is needed to get the right notion of "real form". The classical groups are defined in terms of forms defined on Rn, Cn, Hn, where R and C are the fields of the real and complex numbers; the quaternions, H, do not constitute a field. However, it is still possible to define matrix quaternionic groups. For this reason, a vector space V is allowed to be defined over C, as well as H below.
In the case of H, V is a right vector space to make possible the representation of the group action as matrix multiplication from the left, just as for R and C. A form φ: V × V → F on some finite-dimensional right vector space over F = R, C, or H is bilinear if φ = α φ β, ∀ x, y ∈ V, ∀ α, β ∈ F. and if φ = φ + φ + φ + φ, ∀ x 1, x 2, y 1, y 2 ∈ V. It is called sesquilinear if φ = α ¯ φ β, ∀ x, y ∈ V, ∀ α, β ∈ F. and if: φ = φ (
Sir Roger Penrose is an English mathematical physicist and philosopher of science. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford and Emeritus Fellow of Wadham College, Oxford. Penrose has made contributions to the mathematical physics of general cosmology, he has received several prizes and awards, including the 1988 Wolf Prize for physics, which he shared with Stephen Hawking for the Penrose–Hawking singularity theorems. Born in Colchester, Roger Penrose is a son of psychiatrist and geneticist Lionel Penrose and Margaret Leathes, the grandson of the physiologist John Beresford Leathes and his Russian wife, Sonia Marie Natanson, who had left St. Petersburg in the late 1880s, his uncle was artist Roland Penrose. Penrose is the brother of physicist Oliver Penrose and of chess Grandmaster Jonathan Penrose. Penrose attended University College School and University College, where he graduated with a first class degree in mathematics. In 1955, while still a student, Penrose reintroduced the E. H. Moore generalised matrix inverse known as the Moore–Penrose inverse, after it had been reinvented by Arne Bjerhammar in 1951.
Having started research under the professor of geometry and astronomy, Sir W. V. D. Hodge, Penrose finished his PhD at Cambridge in 1958, with a thesis on "tensor methods in algebraic geometry" under algebraist and geometer John A. Todd, he devised and popularised the Penrose triangle in the 1950s, describing it as "impossibility in its purest form", exchanged material with the artist M. C. Escher, whose earlier depictions of impossible objects inspired it. Escher's Waterfall, Ascending and Descending were in turn inspired by Penrose; as reviewer Manjit Kumar puts it: As a student in 1954, Penrose was attending a conference in Amsterdam when by chance he came across an exhibition of Escher's work. Soon he was trying to conjure up impossible figures of his own and discovered the tribar – a triangle that looks like a real, solid three-dimensional object, but isn't. Together with his father, a physicist and mathematician, Penrose went on to design a staircase that loops up and down. An article followed and a copy was sent to Escher.
Completing a cyclical flow of creativity, the Dutch master of geometrical illusions was inspired to produce his two masterpieces. Having become a reader at Birkbeck College, London it was in 1964 that, in the words of Kip Thorne of Caltech, "Roger Penrose revolutionised the mathematical tools that we use to analyse the properties of spacetime"; until work on the curved geometry of general relativity had been confined to configurations with sufficiently high symmetry for Einstein's equations to be soluble explicitly, there was doubt about whether such cases were typical. One approach to this issue was by the use of perturbation theory, as developed under the leadership of John Archibald Wheeler at Princeton; the other, more radically innovative, approach initiated by Penrose was to overlook the detailed geometrical structure of spacetime and instead concentrate attention just on the topology of the space, or at most its conformal structure, since it is the latter — as determined by the lay of the lightcones — that determines the trajectories of lightlike geodesics, hence their causal relationships.
The importance of Penrose's epoch-making paper "Gravitational collapse and space-time singularities" was not only its result. It showed a way to obtain general conclusions in other contexts, notably that of the cosmological Big Bang, which he dealt with in collaboration with Dennis Sciama's most famous student, Stephen Hawking, it was in the local context of gravitational collapse that the contribution of Penrose was most decisive, starting with his 1969 cosmic censorship conjecture, to the effect that any ensuing singularities would be confined within a well-behaved event horizon surrounding a hidden space-time region for which Wheeler coined the term black hole, leaving a visible exterior region with strong but finite curvature, from which some of the gravitational energy may be extractable by what is known as the Penrose process, while accretion of surrounding matter may release further energy that can account for astrophysical phenomena such as quasars. Following up his "weak cosmic censorship hypothesis", Penrose went on, in 1979, to formulate a stronger version called the "strong censorship hypothesis".
Together with the BKL conjecture and issues of nonlinear stability, settling the censorship conjectures is one of the most important outstanding problems in general relativity. From 1979 dates Penrose's influential Weyl curvature hypothesis on the initial conditions of the observable part of the universe and the origin of the second law of thermodynamics. Penrose and James Terrell independently realised that objects travelling near the speed of light will appear to undergo a peculiar skewing or rotation; this effect has come to be called Penrose -- Terrell rotation. In 1967, Penrose invented the twistor theory which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature. Penrose is well known for his 1974 discovery of Penrose tilings, which are formed from two tiles that can only tile the plane nonperiodically, are the first tilings to exhibit fivefold rotational symmetry. Penrose developed these ideas based on