Quantum mechanics

Uncertainty principle

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In this Feynman diagram, an electron and positron annihilate producing a virtual photon that becomes a quark-antiquark pair. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Quantum mechanics (QM or quantum theory) is a physical science dealing with the behavior of Matter and Energy on the scale of Atoms The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The positrons or antielectron is the Antiparticle or the Antimatter counterpart of the Electron. Annihilation is defined as "total destruction" or "complete obliteration" of an object having its root in the Latin nihil (nothing In Physics, a virtual particle is a particle that exists for a limited time and space introducing uncertainty in their energy and momentum due to the Heisenberg Uncertainty In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. Then one radiates a gluon. Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the   In the field of solid-state physics similar diagrams are also used, where typically the photon is replaced by a phonon.

A Feynman diagram is a tool invented by American physicist Richard Feynman for performing scattering calculations in quantum field theory. A physicist is a Scientist who studies or practices Physics. Physicists study a wide range of physical phenomena in many branches of physics spanning Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum Scattering is a general physical process whereby some forms of Radiation, such as Light, Sound or moving particles for example are forced to deviate from In quantum field theory (QFT the forces between particles are mediated by other particles Particles are represented by lines, which can be drawn in various ways depending on the type of particle being depicted. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made A point where lines connect to other lines is called an interaction vertex, or vertex for short. Lines fall into three categories: internal lines (which connect two vertices), incoming lines (which extend from "the past" to a vertex and represent the initial noninteracting state) and outgoing lines (which extend from a vertex to "the future" and represent the final noninteracting state). Most commonly the bottom of the diagram represents the past and the top of the diagram represents the future.

Feynman diagrams are a pictorial representation of a term in a perturbative expansion of the scattering amplitude for the experiment defined by the incoming and outgoing lines. The word term is from the Latin terminus "boundary line limit" from the Indo-European root ter- "peg post boundary" In Quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system The scattering amplitude describes the amplitude of an outgoing elementary spherical wave relative to a plane incoming wave scattered on a point size particle In some quantum field theories (notably quantum electrodynamics), one can obtain an excellent approximation of the scattering amplitude from a few terms of the perturbative expansion, corresponding to a few simple Feynman diagrams with the same incoming and outgoing lines connected by different vertices and internal lines. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics.

The method, although originally invented for particle physics, has somewhat informally been adopted in solid-state physics, where the behavior of phonons may be expressed in analogy to that of photons, for example in the theory of superconductivity.

Feynman diagrams are frequently confused with spacetime diagrams and bubble chamber images because of their visual similarity, but the connection is weak. The Minkowski diagram was developed in 1908 by Herman Minkowski and provides an illustration of the properties of space and time in the Special theory of relativity A bubble chamber is a vessel filled with a superheated transparent Liquid (most often Liquid hydrogen) used to detect electrically charged Feynman diagrams are merely graphs; there is no concept of position or space in a Feynman diagram, and there is no concept of time aside from the distinction between incoming and outgoing lines. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. Additionally, only a collection of Feynman diagrams can be said to represent any given particle interaction; particles do not choose a particular diagram each time they interact.

Motivation and history

In this diagram, a kaon, made of an up and anti-strange quark, decays weakly into three pions, with intermediate steps involving a W boson and a gluon. In Particle physics, a kaon (/ˈkeɪɒn/ also called K-meson and denoted) is any one of a group of four Mesons distinguished by the fact that they The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature In Particle physics, pion (short for pi meson) is the collective name for three Subatomic particles, and. The W and Z bosons are the Elementary particles that mediate the Weak force. Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the

The problem of calculating scattering cross sections in particle physics reduces to summing over the amplitudes of all possible intermediate states (each corresponding to one term in the perturbation expansion which is known as the Dyson series). Scattering is a general physical process whereby some forms of Radiation, such as Light, Sound or moving particles for example are forced to deviate from In nuclear and Particle physics, the concept of a cross section is used to express the likelihood of interaction between particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In Quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system In Scattering theory, the Dyson series is a Perturbative series and each term is represented by Feynman diagrams This series diverges asymptotically These states can be represented by Feynman diagrams, which are much easier to keep track of than frequently tortuous calculations. Feynman showed how to calculate diagram amplitudes using so-called Feynman rules, which can be derived from the system's underlying Lagrangian. The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system Each internal line corresponds to a factor of the corresponding virtual particle's propagator; each vertex where lines meet gives a factor derived from an interaction term in the Lagrangian, and incoming and outgoing lines provide constraints on energy, momentum, and spin. In Physics, a virtual particle is a particle that exists for a limited time and space introducing uncertainty in their energy and momentum due to the Heisenberg Uncertainty In Quantum mechanics and Quantum field theory, the propagator gives the Probability amplitude for a particle to travel from one place to another in a given In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin A Feynman diagram is therefore a symbolic notation for the factors appearing in each term of the Dyson series. In Scattering theory, the Dyson series is a Perturbative series and each term is represented by Feynman diagrams This series diverges asymptotically

However, being a perturbative expansion, nonperturbative effects do not show up in Feynman diagrams. In Quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system

In addition to their value as a mathematical tool, Feynman diagrams provide deep physical insight to the nature of particle interactions. Particles interact in every way available; in fact, intermediate virtual particles are allowed to propagate faster than light. The probability of each final state is then obtained by summing over all such possibilities. This is closely tied to the functional integral formulation of quantum mechanics, also invented by Feynman–see path integral formulation. You may also be looking for Functional integration (neurobiology or Functional integration (sociology. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral.

The naïve application of such calculations often produces diagrams whose amplitudes are infinite, which is undesirable in a physical theory. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness The problem is that particle self-interactions are erroneously ignored. The technique of renormalization, pioneered by Feynman, Schwinger, and Tomonaga compensates for this effect and eliminates the troublesome infinite terms. In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection Julian Seymour Schwinger ( February 12, 1918 &ndash July 16, 1994) was an American Theoretical physicist. Sin-Itiro Tomonaga or Shinichirō Tomonaga (朝永 振一郎 Tomonaga Shin'ichirō, March 31, 1906 After such renormalization, calculations using Feynman diagrams often match experimental results with very good accuracy.

Feynman diagram and path integral methods are also used in statistical mechanics. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics

Penguin diagrams

Example of a penguin diagram

John Ellis was the first to refer to a certain class of Feynman diagrams as penguin diagrams, due in part to their shape, and in part to a legendary bar-room bet with Melissa Franklin. John Ellis is a British theoretical physicist born in 1946 in London Melissa Franklin is an experimental particle physicist and professor at Harvard University. According to John Ellis:^[1]

Thorsten Ohl's paper on generating Feynman diagrams with LaTeX (see the external links) illustrates their penguin-like shape. Thorsten Ohl is a theoretical particle physicist at University of Würzburg. LaTeX (ˈleɪtɛ

In 1991 and 1994, the CLEO collaboration provided the first experimental evidence for these processes.

Alternative names

Murray Gell-Mann always referred to Feynman diagrams as Stückelberg diagrams, after a Swiss physicist, Ernst Stückelberg, who devised a similar notation. Murray Gell-Mann (born September 15, 1929) is an American Physicist who received the 1969 Nobel Prize in physics for his work This article is about the physicist for his grandfather the Swiss artist see Ernst Alfred Stueckelberg Ernst Carl Gerlach Stueckelberg ( ^[2]

Historically they were also called Feynman-Dyson diagrams or Dyson graphs^[3].

Interpretation

Feynman diagrams are really a graphical way of keeping track of deWitt indices, much like Penrose's graphical notation for indices in multilinear algebra. Physics often deals with classical models where the dynamical variables are a collection of functions {φα}α over a d-dimensional space/spacetime Manifold M In Mathematics and Physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten visual depiction of Multilinear functions In Mathematics, multilinear algebra extends the methods of Linear algebra. There are several different types for the indices, one for each field (this depends on how the fields are grouped; for instance, if the up quark field and down quark field are treated as different fields, then there would be different type assigned to both of them but if they are treated as a single multicomponent field with "flavors", then there would only be one type). The edges, (i. e. , propagators) are tensors of rank (2,0) in deWitt's notation (i. In Quantum mechanics and Quantum field theory, the propagator gives the Probability amplitude for a particle to travel from one place to another in a given History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually e. , with two contravariant indices and no covariant indices), while the vertices of degree n are rank n covariant tensors which are totally symmetric among all bosonic indices of the same type and totally antisymmetric among all fermionic indices of the same type and the contraction of a propagator with a rank n covariant tensor is indicated by an edge incident to a vertex (there is no ambiguity in which "slot" to contract with because the vertices correspond to totally symmetric tensors). In Multilinear algebra, a tensor contraction is an operation on one or more Tensors that arises from the natural pairing of a finite- Dimensional The external vertices correspond to the uncontracted contravariant indices.

A derivation of the Feynman rules using Gaussian functional integrals is given in the functional integral article. You may also be looking for Functional integration (neurobiology or Functional integration (sociology. You may also be looking for Functional integration (neurobiology or Functional integration (sociology.

Each Feynman diagram on its own does not have a physical significance. It's only the infinite sum over all possible (bubble-free) Feynman diagrams which gives physical results. Unfortunately, this infinite sum is only asymptotically convergent. In Mathematics an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which

Mathematical details

Main article: Feynman graph

A Feynman diagram for beta decay. A Feynman graph is a graph suitable to be a Feynman diagram in a particular application of Quantum field theory. In Nuclear physics, beta decay is a type of Radioactive decay in which a Beta particle (an Electron or a Positron) is emitted The straight lines in the diagrams represent fermions, while the wavy line represents virtual bosons. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein

A Feynman diagram can be considered as a graph. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. When considering a field composed of particles, the edges will represent (sections of) particle world lines; the vertices represent virtual interactions. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In physics the world line of an object is the unique path of that object as it travels through 4- Dimensional Spacetime. For other uses see Vertex. In Graph theory, a vertex (plural vertices) or node is the fundamental unit out Interaction is a kind of action that occurs as two or more objects have an Effect upon one another Since only certain interactions are permitted, the graph is constrained to have only certain types of vertices. The type of field of an edge is its field label; the permitted types of interaction are interaction labels.

The value of a given diagram can be derived from the graph; the value of the interaction as a whole is obtained by summing over all diagrams.

See also

• Stückelberg-Feynman interpretation
• Invariance mechanics

References

1. ^ [hep-ph/9510397] ITEP Lectures in Particle Physics
2. ^ The Jaguar and the Fox - 00.07
3. '^ Gribbin, John and Mary. to most kinds of particles, there is an associated antiparticle with the same Mass and opposite Electric charge. In physics invariance mechanics, in its simplest form is the rewriting of the laws of Quantum field theory in terms of invariant quantities only Richard Feynman: A Life in Science, Penguin-Putnam, 1997 Ch 5.

• Gerardus 't Hooft, Martinus Veltman, Diagrammar, CERN Yellow Report 1973, online
• David Kaiser, Drawing Theories Apart: The Dispersion of Feynman Diagrams in Postwar Physics, Chicago: University of Chicago Press, 2005. ISBN 0-226-42266-6
• Martinus Veltman, Diagrammatica: The Path to Feynman Diagrams, Cambridge Lecture Notes in Physics, ISBN 0-521-45692-4 (expanded, updated version of above)

External links

• Feynman diagram page at SLAC
• AMS article: "What's New in Mathematics: Finite-dimensional Feynman Diagrams"
• WikiTeX supports editing Feynman diagrams directly in Wiki articles. The Stanford Linear Accelerator Center ( SLAC) is a United States Department of Energy National Laboratory operated by Stanford University under
• Drawing Feynman diagrams with FeynDiagram C++ library that produces PostScript output.
• Feynman Diagram Examples using Thorsten Ohl's Feynmf LaTeX package.
• JaxoDraw A Java program for drawing Feynman diagrams.