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Coordinate Basis

Coordinate basis

In mathematics, the term holonomic may occur with several different meanings.

Holonomic basis

A holonomic basis for a manifold is a set of basis vectors ek for which all Lie derivatives vanish: :[e_j,e_k]=0 Some authors call a holonomic basis a coordinate basis, and an anholonomic basis a non-coordinate basis. See also Jet bundle.

Holonomic system

In Classical Mechanics a system may be defined as holonomic if all the constraints of the system are holonomic. For a constraint to be holonomic it must be expressible as a function: f(x_1, x_2, x_3, ... x_n, t) = 0 . Examples of holonomic systems are: the simple pendulum [√(x² + y²) – L =0]; rigid bodies. See also nonholonomic system.

Holonomic system (D-modules)

In the Mikio Sato school of D-module theory, holonomic system has a further, technical meaning. Roughly speaking, with a D-module considered as a system of partial differential equations on a manifold, a holonomic system is a highly over-determined system, such that the solutions locally form a vector space of finite dimension (instead of the expected dependence on some arbitrary function). Such systems have been applied, for example, to the Riemann-Hilbert problem in higher dimension, and to quantum field theory. Category:Classical mechanics Category:Algebraic topology Category:Differential topology Category:Mathematical disambiguation

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Quantity

This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. : :Number – Natural number – Integers – Rational numbers – Real numbers – Complex numbers – Hypercomplex numbers – Quaternions – Octonions – Sedenions – Hyperreal numbers – Surreal numbers – Ordinal numbers – Cardinal numbers – p-adic numbers – Integer sequences – Mathematical constants – Number names – Infinity – Base

Structure

:Pinning down ideas of size, symmetry, and mathematical structure. : :Abstract algebra – Number theory – Algebraic geometry – Group theory – Monoids – Analysis – Topology – Linear algebra – Graph theory – Universal algebra – Category theory – Order theory – Measure theory

Space

:A more visual approach to mathematics. : :Topology – Geometry – Trigonometry – Algebraic geometry – Differential geometry – Differential topology – Algebraic topology – Linear algebra – Fractal geometry

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

See also list of mathematics history topics :History of mathematics – Timeline of mathematics – Mathematicians – Fields medal – Abel Prize – Millennium Prize Problems (Clay Math Prize) – International Mathematical Union – Mathematics competitions – Lateral thinking – Mathematical abilities and gender issues

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

See also


- Mathematical game
- Mathematical problem
- Mathematical puzzle
- Puzzle

Bibliography


- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links


- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์



Basis vectors

In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. More precisely, a basis of a vector space is a set of linearly independent vectors that span the whole space.

Definition

Let B be a subset of a vector space V. A linear combination is a finite sum of the form : a_1 v_1 + \cdots + a_n v_n, \, where the vk are different vectors from B and the ak are scalars. The vectors in B are linearly independent if the only linear combinations adding up to the zero vector have a_1 = \cdots = a_n = 0\,. The set B is a generating set if every vector in V is a linear combination of vectors in B. Finally, B is a basis if it is a generating set of linearly independent vectors.

Properties

Again, B denotes a subset of a vector space V. Then,B is a basis if and only if any of the following equivalent conditions is met:
- B is a minimal generating set of V, i.e., it is a generating set but no proper subset of B is.
- B is a maximal set of linearly independent vectors, i.e., it is a linearly independent set but no other linearly independent set contains it as a proper subset.
- Every vector in V can be expressed as a linear combination of vectors in B in a unique way. One can prove that every vector space has a basis. For spaces that cannot be finitely generated, Zorn's lemma is needed for the proof. All bases of a vector space have the same cardinality (number of elements), called the dimension of the vector space. The latter result is known as the dimension theorem.

Examples


- The vectors e1, e2, ..., en are linearly independent and generate Rn. Therefore, they form a basis for Rn and the dimension of Rn is n. This basis is called the standard basis.
- Let V be the real vector space generated by the functions et and e2t. These two functions are linearly independent, so they form a basis for V.
- Let R[x] denote the vector space of real polynomials; then (1, x, x2, ...) is a basis of R[x]. The dimension of R[x] is therefore equal to aleph-0.

Basis extension

Between any linearly independent set and any generating set there is a basis. More formally: if L is a linearly independent set in the vector space V and G is a generating set of V containing L, then there exists a basis of V that contains L and is contained in G. In particular (taking G = V), any linearly independent set L can be "extended" to form a basis of V. These extensions are not unique.

Proving that a set is a basis

As an easy example, let us show that the vectors (1,1) and (-1,2) form a basis for R2. The following proof methods require increasing amounts of sophistication and decreasing amounts of effort.

By brute force

We have to prove that these two vectors are linearly independent and that they generate R2. Part I: To prove that they are linearly independent, suppose that there are numbers a,b such that: : a(1,1)+b(-1,2)=(0,0). \, Then:
:
(a-b,a+2b)=(0,0) \,
  and  
a-b=0 \;
  and  
a+2b=0. \,
Subtracting the first equation from the second, we obtain:
:
3b=0 \;
  so  
b=0. \,
And from the first equation then:
: a=0. \, Part II: To prove that these two vectors generate R2, we have to let (a,b) be an arbitrary element of R2, and show that there exist numbers x,y such that:
: x(1,1)+y(-1,2)=(a,b). \, Then we have to solve the equations:
: x-y=a \, : x+2y=b. \, Subtracting the first equation from the second, we get:
:
3y=b-a, \,
          and then :
y=(b-a)/3, \,
        and finally : x=y+a=((b-a)/3)+a. \,

By the dimension theorem

Since (-1,2) is clearly not a multiple of (1,1) and since (1,1) is not the zero vector, these two vectors are linearly independent. Since the dimension of R2 is 2, the two vectors already form a basis of R2 without needing any extension.

By the invertible matrix theorem

Simply compute the determinant :\det\begin1&-1\\1&2\end=3\neq0. Since the above matrix has a nonzero determinant, its columns form a basis of R2. See: invertible matrix.

Ordered bases

A basis is just a set of vectors with no given ordering. For many purposes it is convenient to work with an ordered basis. For example, when working with a coordinate representation of a vector it is customary to speak of the "first" or "second" coordinate, which makes sense only if a ordering is specified for the basis. For finite-dimensional vector spaces one typically indexes a basis by the first n integers. Suppose V is an n-dimensional vector space over a field F. A choice of an ordered basis for V is equivalent to a choice of a linear isomorphism from the coordinate space Fn, with its standard basis, to V. To see this, let :A : Fn → V be a linear isomorphism. Define an ordered basis for V by : vi = A(ei) for 1 ≤ i ≤ n where is the standard basis for Fn. Conversely, given any ordered basis for V define a linear map A : Fn → V by :A(x) = \sum_^n x_i v_i It is not hard to check that A is an isomorphism. Thus ordered bases for V are in 1-1 correspondence with linear isomorphisms Fn → V.

Related notions

The phrase Hamel basis is sometimes used to refer to a basis as defined above, in which the fact that all linear combinations are finite is crucial. A set B is a Hamel basis of a vector space V if every member of V is a linear combination of just finitely many members of B. In Hilbert spaces and other Banach spaces, there is a need to work with linear combinations of infinitely many vectors. In an infinite-dimensional Hilbert space, a set of vectors orthogonal to each other can never span the whole space via their finite linear combinations. What is called an orthonormal basis is a set of mutually orthogonal unit vectors that "span" the space via sometimes-infinite linear combinations. Except in the finite-dimensional case, this concept is not purely algebraic, and is distinct from a Hamel basis; it is also more generally useful. An orthonormal basis of an infinite-dimensional Hilbert space is therefore not a Hamel basis. In topological vector spaces, quite generally, one may define infinite sums (infinite series) and express elements of the space as certain infinite linear combinations of other elements. To keep clear the distinction of bases using finite and infinite combination, vector space bases are called Hamel bases if the context requires it, and the vector space dimension is also known as Hamel dimension.

Example

In the study of Fourier series, one learns that the functions ∪ are an "orthonormal basis" of the set of all complex-valued functions that are quadratically integrable on the interval [0, 2π], i.e., functions f satisfying :\int_0^ \left|f(x)\right|^2\,dx<\infty. These functions are linearly independent, and every function that is quadratically integrable on that interval is an "infinite linear combination" of them. That means that :\lim_\int_0^\left|\left(a_0+\sum_^n a_k\cos(kx)+b_k\sin(kx)\right)-f(x)\right|^2\,dx=0 for suitable coefficients ak, bk. But most quadratically integrable functions cannot be represented as finite linear combinations of these basis functions, which therefore do not comprise a Hamel basis. Every Hamel basis of this space is much bigger than this merely countably infinite set of functions. Hamel bases of spaces of this kind are of little if any interest; orthonormal bases of these spaces are important to Fourier analysis.

See also


- Linear algebra
- Linear combination Category:Linear algebra

Jet bundle

In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Historically, jet bundles are attributed to Ehresmann, and were an advance on the method (prolongation) of Elie Cartan, of dealing geometrically with higher derivatives, by imposing differential form conditions on newly-introduced formal variables. Jet bundles are sometimes called sprays, although sprays usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifolds.) More recently, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations. Consequently, the jet bundle is now recognized as the correct domain for a geometrical covariant field theory and much work is done in general relativistic formulations of fields using this approach.

Jets

Let (\mathcal, \pi, \mathcal) be a fiber bundle and let p \in \mathcal. Let I=(I(1),I(2),\cdots,I(m)) be an ordered m-tuple. Let \Gamma(\pi)\, denote the set of all local sections whose domain contains p\,. We define : Define the local sections \sigma, \eta \in \Gamma(\pi) to have the same r-jet at p\, if :\left.\frac\right|_ = \left.\frac\right|_ The relation that two maps have the same r\,-jet is an equivalence relation and the equivalence class with representative \sigma\, is denoted j^_\sigma. p\, is the source of j^_\sigma. \sigma(p)\, is the target of j^_\sigma.

Jet Manifolds

The r^\, jet manifold of \pi\, is the set :\ and is denoted J^\pi\,. We may define projections \pi_\, and \pi_\, called the source and target projections respectively, by : If 1 \leq k \leq r, then the k-jet projection is the function \pi_\, defined by : From this definition, it is clear that \pi_ = \pi \circ \pi_ and that if 0 \leq m \leq k, then \pi_ = \pi_ \circ \pi_. It is conventional to regard \pi_=id_\,, the identity map on J^\pi \, and to identify J^\pi\, with \mathcal. The functions \pi_, \pi_\, and \pi_\, are smooth surjective submersions. submersion A co-ordinate system on \mathcal will generate a co-ordinate system on J^\pi\,. Let (U,u)\, be an adapted co-ordinate chart on \mathcal, where u = (x^, u^)\,. The induced co-ordinate chart (U^, u^)\, on J^\pi\, is defined by : where : and the n \left( ^C_ -1\right)\, functions :u^_:U^ \longrightarrow \mathbb\, are specified by :u^_(j^_\sigma) = \left.\frac\right|_ and are known as the derivative co-ordinates. Given an atlas of adapted charts (U,u)\, on \mathcal, the corresponding collection of charts (U^,u^)\, is a finite-dimensional C^\, atlas on J^\pi\,.

Jet Bundles

Since the atlas on each J^\pi\, defines a manifold, the triples (J^\pi, \pi_, J^\pi), (J^\pi, \pi_, \mathcal)\, and (J^\pi, \pi_, \mathcal)\, all define fibered manifolds. In particular, if (\mathcal, \pi, \mathcal)\, is a fiber bundle, the triple (J^\pi, \pi_, \mathcal)\, defines the r^\, jet bundle of \pi\,. If W \subset \mathcal\, is an open submanifold, then : J^\left(\pi|_\right) \cong \pi^_(W)\, If p \in \mathcal\,, then the fiber \pi^_(p)\, is denoted J^_\pi\,. Let \sigma\, be a local section of \pi\, with domain W \subset \mathcal\,. The r^\, jet prolongation of \sigma\, is the map j^\sigma:W \longrightarrow J^\pi\, defined by : (j^\sigma)(p) = j^_\sigma \, Note that \pi_ \circ j^\sigma = id_ \,, so j^\sigma\, really is a section. In local co-ordinates, \sigma\, is given by : \left(\sigma^, \frac\right) \qquad 1 \leq |I| \leq r \, We identify j^\sigma\, with \sigma\,.

Example

If \pi\, is the trivial bundle (\mathcal \times \mathbb, pr_, \mathcal), then there is a canonical diffeomorphism between the first jet bundle J^\pi\, and T^\mathcal \times \mathbb . To construct this diffeomorphism, for each \sigma \in \Gamma_(\pi)\, write \bar = pr_ \circ \sigma \in C^(W)\,. Then, whenever p \in W \, : j^_\sigma = \ \, Consequently, the mapping : is well-defined and is clearly injective. Writing it out in co-ordinates shows that it is a diffeomorphism, because if (x^,u)\, are co-ordinates on \mathcal \times \mathbb, where u=id_\, is the identity co-ordinate, then the derivative co-ordinates u_\, on J^\pi\, correspond to the co-ordinates \partial_\, on T^\mathcal\,. Likewise, if \pi\, is the trivial bundle (\mathbb \times \mathcal, pr_, \mathbb), then there exists a canonical diffeomorphism between J^\pi\, and \mathbb \times T\mathcal\,

Contact Forms

A differential 1-form \theta\, on the space J^\pi\, is called a contact form (ie. \theta \in \Lambda_^\pi\,) if it is pulled back to the zero form on \mathcal\, by all prolongations. In other words, if \theta \in \Lambda^J^\pi\,, then \theta \in \Lambda_^\pi_\, iff, for every open submanifold W \subset \mathcal\, and every \sigma \in \Gamma_(\pi)\,, :(j^\sigma)^\theta = 0\,

Example

Let us consider the case (\mathcal,\pi,\mathcal), where \mathcal \simeq \mathbb^ and \mathcal \simeq \mathbb. Then, (J^\pi, \pi, \mathcal) defines the first jet bundle, and may be co-ordinated by (x,u,u_)\,, where : for all p \in \mathcal and \sigma \in \Gamma_(\pi)\,. A general 1-form on J^\pi\, takes the form :\theta = a(x, u, u_)dx + b(x, u, u_)du + c(x, u,u_)du_\, A section \sigma \in \Gamma_(\pi)\, has first prolongation j^\sigma = (u,u_) = \left(\sigma(p), \left.\frac\right|_\right)\,. Hence, (j^\sigma)^ \theta\, can be calculated as : This will vanish for all sections \sigma\, iff c=0\, and a = -b\sigma'(x)\,. Hence, \theta=b(x, \sigma(x), \sigma'(x))\theta_\, must necessarily be a multiple of the basic contact form \theta_=du-u_dx\,. Proceeding to the second jet space J^\pi\, with additional co-ordinate u_\,, such that :u_(j^_\sigma)=\left.\frac\right|_ = \sigma(x)\, a general 1-form has the construction : \theta = a(x, u, u_,u_)dx + b(x, u, u_,u_)du + c(x, u, u_,u_)du_ + e(x, u, u_,u_)du_\, This is a contact form iff : which implies that e=0\, and a=-b\sigma'(x)-c\sigma(x)\,. Therefore, \theta\, is a contact form iff :\theta = b(x, \sigma(x), \sigma'(x))\theta_ + c(x, \sigma(x), \sigma'(x))\theta_\, where \theta_ = du_ - u_dx\, is the next basic contact form (Note that here we are identifying the form \theta_\, with its pull-back (\pi_)^\theta_\, to J^\pi\,). In general, providing x,u, \in \mathbb\,, a contact form on J^\pi\, can be written as a linear combination of the basic contact forms :\theta_ = du_ - u_dx \qquad k=0, \ldots, r-1\, where u_(j^\sigma)= \left.\frac\right|_\,. Similar arguments lead to a complete characterization of all contact forms. In local coordinates, every contact one-form on J^\pi\, can be written as a linear combination :\theta = \sum_^ P_^\theta_^\, with smooth coefficients P^_(x^,u^)\, of the basic contact forms :\theta_^ = du^_ - u^_dx^\, |I|\, is known as the order of the contact form \theta_^. Note that contact forms on J^\pi\, have orders at most r\,. Contact forms provide a characterization of those local sections of \pi_\, which are prolongations of sections of \pi\,. Let \psi \in \Gamma_(\pi_)\,, then \psi = j^\sigma\, where \sigma \in \Gamma_(\pi)\, iff \psi^(\sigma|_)=0, \forall \sigma \in \Lambda_^\pi_\,

Vector Fields

A general vector field on the total space \mathcal, co-ordinated by (x,u) \equiv (x^,u^)\,, is :V \equiv \rho^(x,u)\frac + \phi^(x,u)\frac\, A vector field is called horizontal, meaning all the vertical coefficients vanish, if \phi^=0\,. A vector field is called vertical, meaning all the horizontal coefficients vanish, if \rho^=0\,. For fixed (x,u)\,, we identify : V_ \equiv \rho^(x,u) \frac + \phi^(x,u) \frac\, having co-ordinates (x,u,\rho^,\phi^)\,, with an element in the fiber T_\mathcal of T\mathcal over (x,u) \in \mathcal, called a tangent vector in T\mathcal. A section : is called a vector field on \mathcal with V = \rho^(x,u) \frac + \phi^(x,u) \frac\, and \psi \in \Gamma(T\mathcal)\,. The jet bundle J^\pi\, is co-ordinated by (x,u,w) \equiv (x^,u^,u_^)\,. For fixed (x,u,w)\,, identify : having co-ordinates (x,u,w,v^_, v^_,\ldots,v^_)\,, with an element in the fiber T_(J^\pi)\, of T(J^\pi)\, over (x,u,w) \in J^\pi\,, called a tangent vector in T(J^\pi)\,. Here, v^_, v^_,\ldots,v^_\, are real-valued functions on J^\pi\,. A section : is a vector field on J^\pi\,, and we say \Psi \in \Gamma(T(J^\pi))\,.

Partial Differential Equations

Let (\mathcal,\pi,\mathcal) be a fiber bundle. An r^\, order partial differential equation on \pi\, is a closed embedded submanifold \mathcal of the jet manifold J^\pi\,. A solution is a local section \sigma \in \Gamma_(\pi)\, satisfying j^_\sigma \in \mathcal, \forall p \in \mathcal. Let us consider an example of a first order partial differential equation.

Example

Let \pi\, be the trivial bundle (\mathbb^ \times \mathbb, pr_, \mathbb^)\, with global co-ordinates (x^, x^, u^)\,. Then the map F:J^\pi \longrightarrow \mathbb\, defined by :F = u^_u^_ - 2x^u^\, gives rise to the differential equation :S = \

Function (mathematics)

In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). The concept of a function is fundamental to virtually every branch of mathematics and every quantitative science. The terms function, mapping, map and transformation are usually used synonymously. The term operation is frequently used for binary functions; functions whose domain is a set of functions, or a vector space, are often called operators (see also operator (programming)).

Intuitive introduction

Essentially, a function is a "rule" or procedure that assigns an "output" value to each given "input" value. The following are examples of functions:
- In a group of people, each person has a favorite colour—from the set of red, orange, yellow, green, cyan, blue, indigo, or violet. Here, the input is the person, and the output is one of the 8 colours. The favorite colour is a function of the person. For example, John has favorite colour red, while Kim has favorite colour violet. Note that more than one person may be associated with a given colour (e.g., John and Kim may both like red), but one person cannot have more or less than one favorite color.
- A stone is dropped from different stories of a tall building. The dropped stone may take 2 seconds to fall from the second story, and 4 seconds to fall from the 8th story. Here, the input is the story, and the output is the number of seconds. The relevant function describes the relationship between the time it takes the stone to reach the ground and the story. (See acceleration) The "rule" defining a function can be specified by a formula, a relationship, or simply a table listing the outputs against inputs. The most important feature of a function is that it is consistent, or deterministic, always producing the same output from a given input. In this way, a function may be thought of as a mechanism or "machine" (a "black box") consistently converting a given valid input into its unique associated output. In certain technical contexts, the input is often called the argument of the function, and the output the value of the function. A very common type of function occurs when the argument (input) and the value (output) are both numbers, the functional relationship is expressed by a formula, and the value (output) of the function is obtained by direct substitution of the argument into the formula. Consider for example :f(x)=x^ which for any number x, assigns to x the associated value the square of x. A straightforward generalization is to allow functions depending on several arguments. For instance, :g(x,y) = xy is a function which takes the input, two expressions x and y, and assigns to it its product (output), xy. It might seem that this is not really a function as we described above, because this "rule" depends on two inputs. However, if we think of the two inputs together as a single pair (x, y), then we can interpret g as a function -- the argument (unified single input) is the ordered pair (x, y), and the function value (output) is xy. Such functions whose input consists of ordered pairs are called "binary" or "2-ary". In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on earth over time: this is a function which takes location and time as arguments and gives as output value the temperature at the indicated location at the indicated moment in time. We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that every element of the domain is associated to precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, the function concept is fundamental to virtually every branch of mathematics and the quantitative sciences.

History

As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, and these measurements are the basis of calculus. The word function was later used by Euler during the mid-18th century to describe an expression or formula involving various arguments, e.g. f(x) = sin(x) + x3. During the 19th century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis). By broadening the definition of functions, mathematicians were then able to study "strange" mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only theoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from functional analysis have shown that these functions are in some sense "more common" than differentiable functions. Such functions have since been applied to the modeling of physical phenomena such as Brownian motion. Towards the end of the 19th century, mathematicians started trying to formalize all of mathematics using set theory, and they sought to define every mathematical object as a set. Dirichlet and Lobachevsky independently and almost simultaneously gave the modern "formal" definition of function (see formal definition below). In this definition, a function is a special case of a relation. In most cases of practical interest, however, the differences between the modern definition and Euler's definition are negligible. The notion of function as a rule for computing, rather than a special kind of relation, has been formalized in mathematical logic and theoretical computer science by means of several systems, including the lambda calculus, the theory of recursive functions and the Turing machine.

Formal definition

Formally a function f from a set X to a set Y, written f : X → Y, is an ordered triple (X, Y, G(f)), where G(f) is a subset of the cartesian product X × Y, such that for each x in X, there is a unique y in Y such that the ordered pair (x, y) is in G(f). X is called the domain of f, Y is called the codomain of F, and G(f) is called the graph of f. For each "input value" x in the domain, the corresponding unique "output value" y in the codomain is denoted by f(x). Equivalently a function f can be defined as a relation between X and Y which satisfies: # f is total, or entire: for all x in X, there exists a y in Y such that x f y (x is f-related to y), i.e. for each input value, there is at least one output value in Y. # f is many-to-one, or functional: if x f y and x f z, then y = z. i.e., many input values can be related to one output value, but one input value cannot be related to many output values. A relation between X and Y that satisfies condition (1) is a multivalued function. Every function is a multivalued function, but not every multivalued function is a function. A relation between X and Y that satisfies condition (2) is a partial function. Every function is a partial function, but not every partial function is a function. In this encyclopedia, the term "function" will mean a relation satisfying both conditions (1) and (2), unless otherwise stated. Consider the following three examples:
image:notMap1.png This relation is total but not many-to-one; the element 3 in X is related to two elements b and c in Y. Therefore, this is a multivalued function, but not a function.
image:notMap2.png This relation is many-to-one but not total; the element 1 in X is not related to any element of Y. Therefore, this is a partial function, but not a function.
image:mathmap2.png This relation is both total and many-to-one, and so it is a function from X to Y. Note that the emphasis is on "-to-one" as "many" may actually mean "one". The function can be given explicitly by specifying its graph G(f) = or as :f(x)=\left\

Nonholonomic system

In physics and mathematics, a nonholonomic system is a system in which a return to the original internal configuration does not guarantee return to the original system position. In other words, unlike with a holonomic system, the outcome of a nonholonomic system is path-dependent. For example, when riding a two-wheeled cart, a return to the original internal (wheel) configuration does not guarantee return to the original system (cart) position. Cars, bicycles and unicycles are all examples of nonholonomic systems. The branch of mathematics dealing with nonholonomic systems is known as sub-Riemannian geometry. Category:Algebraic topology Category:Differential topology Category:Classical mechanics





Quantum field theory

Quantum field theory (QFT) is the application of quantum mechanics to fields. It provides a theoretical framework, widely used in particle physics and condensed matter physics, in which to formulate consistent quantum theories of many-particle systems, especially in situations where particles may be created and destroyed. Non-relativistic quantum field theories are needed in condensed matter physics— for example in the BCS theory of superconductivity. Relativistic quantum field theories are indispensable in particle physics (see the standard model), although they are known to arise as effective field theories in condensed matter physics.

Why quantum field theory

Quantum field theory originated in the problem of computing the power radiated by an atom when it dropped from one quantum state to another of lower energy. This problem was first examined by Max Born and Pascual Jordan in 1925. In 1926, Max Born, Werner Heisenberg and Pascual Jordan wrote down the quantum theory of the electromagnetic field neglecting polarization and sources to obtain what would today be called a free field theory. In order to quantize this theory, they used the canonical quantization procedure. In 1927, Paul Dirac gave the first consistent treatment of this problem. Quantum field theory followed unavoidably from a quantum treatment of the only known classical field, ie, electromagnetism. The theory was required by the need to treat a situation where the number of particles changes. Here, one atom in the initial state becomes an atom and a photon in the final state. It was obvious from the beginning that the quantum treatment of the electromagnetic field required a proper treatment of relativity. Jordan and Wolfgang Pauli showed in 1928 that commutators of the field were actually Lorentz invariant. By 1933, Niels Bohr and Leon Rosenfeld had related these commutation relations to a limitation on the ability to measure fields at space-like separation. The development of the Dirac equation and the hole theory drove quantum field theory to explain these using the ideas of causality in relativity, work that was completed by Wendell Furry and Robert Oppenheimer using methods developed for this purpose by Vladimir Fock. This need to put together relativity and quantum mechanics was a second motivation which drove the development of quantum field theory. This thread was crucial to the eventual development of particle physics and the modern (partially) unified theory of forces called the standard model. In 1927 Jordan tried to extend the canonical quantization of fields to the wave function which appeared in the quantum mechanics of particles, giving rise to the equivalent name second quantization for this procedure. In 1928 Jordan and Eugene Wigner found that the Pauli exclusion principle demanded that the electron field be expanded using anti-commuting creation and annihilation operators. This was the third thread in the development of quantum field theory— the need to handle the statistics of multi-particle systems consistently and with ease. This thread of development was incorportated into many-body theory, and strongly influenced condensed matter physics and nuclear physics.

What QFT is

Just as quantum mechanics deals with operators acting upon a (separable) Hilbert space, QFT also deals with operators acting upon a Hilbert space. However, in the case of QFT, the operators are generated by what is known as operator-valued fields, that is, operators which are parametrized by a spacetime point. Intuitively, this means that operators can be localized. This definition applies even to the cases of theories which aren't quantizations, and as such, is pretty general. This is sometimes stated as "position is an operator in QM but is a parameter in QFT" but this statement, while accurate, can be very misleading. QM deals with particles and one of the properties of a particle is its position as a function of time and in QM, this becomes the position operator as a function of time (it's constant in the Schrodinger picture and varying in the Heisenberg picture). QFT, on the other hand, deals with fields on a fundamental level and particles only emerge as localized excitations (aka quanta aka quasiparticles) of the ground state (aka the vacuum) and it's precisely these quantum fields which correspond to the operator valued functions. Put more simply, instead of looking at the operators generated by :\hat(t) and \hat(t), we now look at operators generated by :\hat(\mathbf,t) And just as in QM, we may work in the Schrödinger picture, the Heisenberg picture or the interaction picture (in the context of perturbation theory). Only the Heisenberg picture is manifestly Lorentz covariant. The energy is given by the Hamiltonian operator, which can be generated from the quantum fields, and corresponds to the generator of infinitesimal time translations. (the condition that the generator of infinitesimal time translations can be generated by the quantum fields rules out many unphysical theories, which is a good thing) We further assume that this Hamiltonian is bounded from below and has a lowest energy eigenstate (this rules out theories which are unstable and have no stable solutions, which is also a good thing), which may or may not be degenerate. (although there are physical QFTs which have a lower bound to the Hamiltonian but don't have a lowest energy eigenstate, like N=1 super QCD theories with too few quarks...) This lowest energy eigenstate is called the vacuum in particle physics and the ground state in condensed matter physics. (QFT appears in the continuum limit of condensed matter systems) This simple explanation of what QFT really is, is often obscured in treatments which jump straight to the path integral approach, which is a good computational technique but often obscures the underlying ideas.

Technical statement

Quantum field theory corrects several limitations of ordinary quantum mechanics, which we will briefly discuss. The Schrödinger equation, in its most commonly encountered form, is : \left[ \frac + V(\mathbf) \right] |\psi(t)\rang = i \hbar \frac |\psi(t)\rang where |\psi\rang denotes the quantum state (notation) of a particle with mass m, in the presence of a potential V. The first problem occurs when we seek to extend the equation to large numbers of particles. As described in the article on identical particles, quantum mechanical particles of the same species are indistinguishable, in the sense that the state of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are extremely complicated to write. For example, the general quantum state of a system of N bosons is written as : |\phi_1 \cdots \phi_N \rang = \sqrt \sum_ |\phi_\rang \cdots |\phi_ \rang where |\phi_i\rang are the single-particle states, N_j is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases. Large numbers of particles are needed in condensed matter physics where typically the number of particles is on the order of Avogadro's number, approximately 1023. The second problem arises when trying to reconcile the Schrödinger equation with special relativity. It is possible to modify the Schrödinger equation to include the rest energy of a particle, resulting in the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy eigenvalues which extend to –∞, so that there seems to be no easy definition of a ground state. Such inconsistencies occur because these equations neglect the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory. This problem brings to the fore the notion that a consistent relativistic quantum theory, even of a single particle, must be a many particle theory.

Quantizing a classical field theory

Canonical quantization

Quantum field theory solves these problems by consistently quantizing a field. By interpreting the physical observables of the field appropriately, one can create a (rather successful) theory of many particles. Here is how it is: 1. Each normal mode oscillation of the field is interpreted as a particle with frequency f. 2. The quantum number n of each normal mode (which can be thought of as a harmonic oscillator) is interpreted as the number of particles. The energy associated with the particle is therefore = (n+1/2)hf. With some thought, one may similarly associate momenta and position of particles with observables of the field. Having cleared up the correspondence between fields and particles (which is different from non-relativistic QM), we can proceed to define how a quantum field behaves. Two caveats should be made before proceeding further: #Each of these "particles" obeys the usual uncertainty principle of quantum mechanics. The "field" is an operator defined at each point of spacetime. #Quantum field theory is not a wildly new theory. Classical field theory is the same as classical mechanics of an infinite number of dynamical quantities (say, tiny elements of rubber on a rubber sheet). Quantum field theory is the quantum mechanics of this infinite system. The first method used to quantize field theory was the method now called canonical quantization (earlier known as second quantization). This method uses a Hamiltonian formulation of the classical problem. The later technique of Feynman path integrals uses a Lagrangian formulation. Many more methods are now in use; for an overview see the article on quantization.

Canonical quantization for bosons

Suppose we have a system of N bosons which can occupy mutually orthogonal single-particle states |\phi_1\rang, |\phi_2\rang, |\phi_3\rang, and so on. The usual method of writing a multi-particle state is to assign a state to each particle and then impose exchange symmetry. As we have seen, the resulting wavefunction is an unwieldy sum of N! terms. In contrast, in the second quantized approach we will simply list the number of particles in each of the single-particle states, with the understanding that the multi-particle wavefunction is symmetric. To be specific, suppose that N=3, with one particle in state |\phi_1\rang and two in state|\phi_2\rang. The normal way of writing the wavefunction is : \frac \left[ |\phi_1\rang |\phi_2\rang |\phi_2\rang + |\phi_2\rang |\phi_1\rang |\phi_2\rang + |\phi_2\rang |\phi_2\rang |\phi_1\rang \right] In second quantized form, we write this as : |1, 2, 0, 0, 0, \cdots \rangle which means "one particle in state 1, two particles in state 2, and zero particles in all the other states." Though the difference is entirely notational, the latter form makes it easy for us to define creation and annihilation operators, which add and subtract particles from multi-particle states. These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta. However, these operators literally create and annihilate particles with a given quantum state. The bosonic annihilation operator a_2 and creation operator a_2^\dagger have the following effects: : a_2 | N_1, N_2, N_3, \cdots \rangle = \sqrt \mid N_1, (N_2 - 1), N_3, \cdots \rangle : a_2^\dagger | N_1, N_2, N_3, \cdots \rangle = \sqrt \mid N_1, (N_2 + 1), N_3, \cdots \rangle We may well ask whether these are operators in the usual quantum mechanical sense, i.e. linear operators acting on an abstract Hilbert space. In fact, the answer is yes: they are operators acting on a kind of expanded Hilbert space, known as a Fock space, composed of the space of a system with no particles (the so-called vacuum state), plus the space of a 1-particle system, plus the space of a 2-particle system, and so forth. Furthermore, the creation and annihilation operators are indeed Hermitian conjugates, which justifies the way we have written them. The bosonic creation and annihilation operators obey the commutation relation : \left[a_i , a_j \right] = 0 \quad,\quad \left[a_i^\dagger , a_j^\dagger \right] = 0 \quad,\quad \left[a_i , a_j^\dagger \right] = \delta_ where \delta stands for the Kronecker delta. These are precisely the relations obeyed by the "ladder operators" for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator. The final step toward obtaining a quantum field theory is to re-write our original N-particle Hamiltonian in terms of creation and annihilation operators acting on a Fock space. For instance, the Hamiltonian of a field of free (non-interacting) bosons is :H = \sum_k E_k \, a^\dagger_k \,a_k where E_k is the energy of the k-th single-particle energy eigenstate. Note that :a_k^\dagger\,a_k|\cdots\rangle=N_k| \cdots, N_k, \cdots \rangle.

Second quantization for fermions

It turns out that the creation and annihilation operators for fermions must be defined differently, in order to satisfy the Pauli exclusion principle. For fermions, the occupation numbers N_i can only take on the value 0 or 1, since particles cannot share quantum states. We then define the fermionic annihilation operators c and creation operators c^\dagger by : c_j | N_1, N_2, \cdots, N_j = 0, \cdots \rangle = 0 : c_j | N_1, N_2, \cdots, N_j = 1, \cdots \rangle = (-1)^ | N_1, N_2, \cdots, N_j = 0, \cdots \rangle : c_j^\dagger | N_1, N_2, \cdots, N_j = 0, \cdots \rangle = (-1)^ | N_1, N_2, \cdots, N_j = 1, \cdots \rangle : c_j^\dagger | N_1, N_2, \cdots, N_j = 1, \cdots \rangle = 0 The fermionic creation and annihilation operators obey an anticommutation relation, : \left\ = 0 \quad,\quad \left\ = 0 \quad,\quad \left\ = \delta_ One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.

Significance of creation and annihilation operators

When we re-write a Hamiltonian using a Fock space and creation and annihilation operators, as in the previous example, the symbol N, which stands for the total number of particles, drops out. This means that the Hamiltonian is applicable to systems with any number of particles. Of course, in many common situations N is a physically important and perfectly well-defined quantity. For instance, if we are describing a gas of atoms sealed in a box, the number of atoms had better remain a constant at all times. This is certainly true for the above Hamiltonian. Viewing the Hamiltonian as the generator of time evolution, we see that whenever an annihilation operator a_k destroys a particle during an infinitesimal time step, the creation operator a_k^\dagger to the left of it instantly puts it back. Therefore, if we start with a state of N non-interacting particles then we will always have N particles at a later time. On the other hand, it is often useful to consider quantum states where the particle number is ill-defined, i.e. linear superpositions of vectors from the Fock space that possess different values of N. For instance, it may happen that our bosonic particles can be created or destroyed by interactions with a field of fermions. Denoting the fermionic creation and annihilation operators by c_k^\dagger and c_k, we could add a "potential energy" term to our Hamiltonian such as: :V = \sum_ V_q (a_q + a_^\dagger) c_^\dagger c_k This describes processes in which a fermion in state k either absorbs or emits a boson, thereby being kicked into a different eigenstate k+q. In fact, this is the expression for the interaction between phonons and conduction electrons in a solid. The interaction between photons and electrons is treated in a similar way; it is a little more complicated, because the role of spin must be taken into account. One thing to notice here is that even if we start out with a fixed number of bosons, we will generally end up with a superposition of states with different numbers of bosons at later times. On the other hand, the number of fermions is conserved in this case. In condensed matter physics, states with ill-defined particle numbers are also very important for describing the various superfluids. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers.

Field operators

We can now define field operators that create or destroy a particle at a particular point in space. In particle physics, these are often more convenient to work with than the creation and annihilation operators, because they make it easier to formulate theories that satisfy the demands of relativity. Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator \phi(\mathbf) is :\phi(\mathbf) \equiv \sum_ e^ a_ The bosonic field operators obey the commutation relation : \left[\phi(\mathbf) , \phi(\mathbf) \right] = 0 \quad,\quad \left[\phi^\dagger(\mathbf) , \phi^\dagger(\mathbf) \right] = 0 \quad,\quad \left[\phi(\mathbf) , \phi^\dagger(\mathbf) \right] = \delta^3(\mathbf - \mathbf) where \delta(x) stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators. It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say :H = - \frac \sum_i \nabla_i^2 + \sum_ U(|\mathbf_i - \mathbf_j|) where the indices i and j run over all particles, then the field theory Hamiltonian is :H = - \frac \int d^3\!r \; \phi(\mathbf)^\dagger \nabla^2 \phi(\mathbf) + \int\!d^3\!r \int\!d^3\!r' \; \phi(\mathbf)^\dagger \phi(\mathbf')^\dagger U(|\mathbf - \mathbf'|) \phi(\mathbf) \phi(\mathbf) This looks remarkably like an expression for the expectation value of the energy, with \phi playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.

Quantization of classical fields

So far, we have shown how one goes from an ordinary quantum theory to a quantum field theory. There are certain systems for which no ordinary quantum theory exists. These are the "classical" fields, such as the electromagnetic field. There is no such thing as a wavefunction for a single photon in classical electromagnetisim, so a quantum field theory must be formulated right from the start. The essential difference between an ordinary system of particles and the electromagnetic field is the number of dynamical degrees of freedom. For a system of N particles, there are 3N coordinate variables corresponding to the position of each particle, and 3N conjugate momentum variables. One formulates a classical Hamiltonian using these variables, and obtains a quantum theory by turning the coordinate and position variables into quantum operators, and postulating commutation relations between them such as :\left[ q_i , p_j \right] = \delta_ For an electromagnetic field, the analogue of the coordinate variables are the values of the electrical potential \phi(\mathbf) and the vector potential \mathbf(\mathbf) at every point \mathbf. This is an uncountable set of variables, because \mathbf is continuous. This prevents us from postulating the same commutation relation as before. The way out is to replace the Kronecker delta with a Dirac delta function. This ends up giving us a commutation relation exactly like the one for field operators! We therefore end up treating "fields" and "particles" in the same way, using the apparatus of quantum field theory.

Path integral methods

The axiomatic approach

There have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. These attempts fall into two broad classes. The first class of axioms (most notably the Wightman, Osterwalder-Schrader, and Haag-Kastler systems) tried to formalize the physicists' notion of an "operator-valued field" within the context of functional analysis. These axioms enjoyed limited success. It was possible to prove that any QFT satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the PCT theorems. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory (e.g. quantum chromodynamics) satisfied these axioms. Most of the theories which could be treated with these analytic axioms were physically trivial: restricted to low-dimensions and lacking in interesting dynamics. In the 1980s, a second wave of axioms were proposed. These axioms (associated most closely with Atiyah and Segal, and notably expanded upon by Witten, Borcherds, and Kontsevich) are more geometric in nature, and more closely resemble the path integrals of physics. They have not been exceptionally useful to physicists, as it is still extraordinarily difficult to show that any realistic QFTs satisfy these axioms, but have found many applications in mathematics, particularly in representation theory, algebraic topology, and geometry. Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. In fact, one of the Clay Millenium Prizes offers $1,000,000 to anyone who proves the existence of a mass gap in Yang-Mills theory. It seems likely that we have not yet understood the underlying structures which permit the Feynman path integrals to exist.

Renormalization

The essence of quantum field theory is renormalization. A single particle state in quantum field theory incorporates within it multiparticle states. This is most simply demonstrated by examining the evolution of a single particle state in the interaction picture— ::|\psi(t)\rangle = e^ |\psi(0)\rangle = \left[1+iH_It-\frac12 H_I^2t^2 -\frac iH_I^3t^3 + \frac1H_I^4t^4 + \cdots\right] |\psi(0)\rangle. Taking the overlap with the initial state, one retains the even powers of HI. These terms are responsible for changing the number of particles during propagation, and are therefore quintessentially a product of quantum field theory. Corrections such as these are incorporated into wave-function renormalization and mass renormalization. Similar corrections to the interaction Hamiltonian, HI, include vertex renormalization, or, in modern language, effective field theory.

Gauge theories

gauge theory

Supersymmetry

supersymmetry

Beyond local field theory

History

More details can be found in the article on the history of quantum field theory. Quantum field theory was created by Dirac when he attempted to quantize the electromagnetic field in the late 1920s. The early development of the field involved Fock, Jordan, Pauli, Heisenberg, Bethe, Tomonaga, Schwinger, Feynman, and Dyson. This phase of development culminated with the construction of the theory of quantum electrodynamics in the 1950s. Gauge theory was formulated and quantized, leading to the unification of forces embodied in the standard model of particle physics. This effort started in the 1950s with the work of Yang and Mills, was carried on by Martinus Veltman and a host of others during the 1960s and completed during the 1970s by the work of Gerard 't Hooft, Frank Wilczek, David Gross and David Politzer. Parallel developments in the understanding of phase transitions in condensed matter physics led to the study of the renormalization group. This in turn led to the grand synthesis of theoretical physics which unified theories of particle and condensed matter physics through quantum field theory. This involved the work of Michael Fisher and Leo Kadanoff in the 1970s which led to the seminal reformulation of quantum field theory by Kenneth Wilson. The study of quantum field theory is alive and flourishing, as are applications of this method to many physical problems. It remains one of the most vital areas of theoretical physics today, providing a common language to many branches of physics.

See also


- List of quantum field theories
- Feynman path integral
- Quantum chromodynamics
- Quantum electrodynamics
- Schwinger-Dyson equation

Suggested reading


- Wilczek, Frank ; Quantum Field Theory, Review of Modern Physics 71 (1999) S85-S95. Review article written by a master of Q.C.D., [http://nobelprize.org/physics/laureates/2004/wilczek-autobio.html Nobel laureate 2003]. Full text available at : [http://fr.arxiv.org/abs/hep-th/9803075 hep-th/9803075]
- Ryder, Lewis H. ; Quantum Field Theory (Cambridge University Press, 1985), [ISBN 0-521-33859-X] Highly readable textbook, certainly the best introduction to relativistic Q.F.T. for particle physics.
- Zee, Anthony ; Quantum Field Theory in a Nutshell, Princeton University Press (2003) [ISBN 0-691-01019-6].
- Peskin, M and Schroeder, D. ;An Introduction to Quantum Field Theory (Westview Press, 1995) [ISBN 0201503972]
- Weinberg, Steven ; The Quantum Theory of Fields (3 volumes) Cambridge University Press (1995). A monumental treatise on Q.F.T. written by a leading expert, [http://nobelprize.org/physics/laureates/1979/weinberg-lecture.html Nobel laureate 1979].
- Loudon, Rodney ; The Quantum Theory of Light (Oxford University Press, 1983), [ISBN 0198511558]
- Siegel, Warren ; [http://insti.physics.sunysb.edu/%7Esiegel/errata.html Fields] (also available from arXiv:hep-th/9912205)
- 't Hooft, Gerard ; The Conceptual Basis of Quantum Field Theory, Handbook of the Philosophy of Science, Elsevier (to be published). Review article written by a master of gauge theories, [http://nobelprize.org/physics/laureates/1999/thooft-autobio.htmlNobel laureate 1999]. Full text available in [http://www.phys.uu.nl/~thooft/lectures/basisqft.pdfpdf].
- Srednicki, Mark ; [http://gabriel.physics.ucsb.edu/~mark/qft.html Quantum Field Theory] Category:Quantum mechanics ko:양자 마당 이론 ja:場の量子論