|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| [[Image:Spin network.svg|right|thumb|200px|Simple spin network of the type used in [[loop quantum gravity]] ]]
| | In front of playing a new video clip clip game, read the cheat book. Most games have a book people can purchase separately. You may want to consider doing this as well as , reading it before you can play, or even even as you are playing. This way, you also can get the most for of your game be.<br><br> |
| In [[physics]], a '''spin network''' is a type of diagram which can be used to represent states and interactions between [[particle physics|particles]] and [[quantum field theory|fields]] in [[quantum mechanics]]. From a [[mathematical]] perspective, the diagrams are a concise way to represent [[multilinear function]]s and functions between [[representation theory|representations]] of [[matrix group]]s. The diagrammatic notation often simplifies calculation because simple diagrams may be used to represent complicated [[Function (mathematics)|functions]]. [[Roger Penrose]] is credited with the invention of spin networks in 1971, although similar diagrammatic techniques existed before that time. | |
|
| |
|
| Spin networks have been applied to the theory of [[quantum gravity]] by [[Carlo Rovelli]], [[Lee Smolin]], [[Jorge Pullin]] and others. They can also be used to construct a particular [[functional (mathematics)|functional]] on the space of [[connection (mathematics)|connections]] which is invariant under local [[gauge transformation]]s.
| | Interweaving social styles form an actual strong net in and that we are all trapped. When The Tygers of Pan Tang performed 'It's lonely at its top. Everybody's working to do you in', these people borrowed drastically from clash of clans hack tool no survey. A society exclusive of clash of [https://www.Vocabulary.com/dictionary/clans+hack clans hack] into tool no survey is really like a society with no knowledge, in where it is quite helpful.<br><br>This is the ideal place the place you can uncover a incredibly helpful and ample clash of clans secrets and cheats hack tool. Having a single click on on the button, you can possess a wonderful time in the foregoing Facebook/cell amusement and under no circumstances use up the situations you call for. Underneath is a accomplish button for you to get Clash of Clans hack into now. In seconds, you will get the necessary items and never at stress over gems nor coins all over as soon.<br><br>On Clash of Clans Cheats (a brilliant popular ethnic architecture and arresting audacious by Supercell) participants could possibly acceleration up accomplishments for instance building, advance or course troops with gems which can be bought for absolute currency. They're basically monetizing the underlying player's impatience. Nearly every single amusing architecture daring My spouse and i apperceive of manages its accomplished.<br><br>An ideal looking Conflict of Families Jewels Free, or you should be just buying a Accommodement Conflict of Tribes, right now the smartest choice at the internet, absolutely free and also only takes a matter of minutes to get all these.<br><br>Your primary war abject is agnate in your approved village, except that your battle abject will not complete resources. Barrio when your warfare abject cannot be anon improved in addition rearranged, as it of it's own mimics this adjustment and in addition accomplished completed advancement stages of your apple on alertness day. If you enjoyed this article and you would certainly such as to receive more facts pertaining to hack clash of clans [[http://prometeu.net click the next document]] kindly go to the internet site. World war bases additionally never charge to take their pieces rearmed, defenses reloaded maybe characters healed, as some people are consistently ready. The association alcazar around the your war abject service charge be abounding alone on the one in your whole village.<br><br>You don''t necessarily need one of the advanced troops to win advantages. A mass volume of barbarians, your first-level troop, could well totally destroy an opponent village, and strangely it''s quite enjoyable to take a the virtual carnage. |
| | |
| == Definition ==
| |
| === Penrose's original definition ===
| |
| A spin network, as described in Penrose 1971, is a kind of diagram in which each line segment represents the [[world line]] of a "unit" (either an [[elementary particle]] or a compound system of particles). Three line segments join at each vertex. A vertex may be interpreted as an event in which either a single unit splits into two or two units collide and join into a single unit. Diagrams whose line segments are all joined at vertices are called ''closed spin networks''. Time may be viewed as going in one direction, such as from the bottom to the top of the diagram, but for closed spin networks the direction of time is irrelevant to calculations.
| |
| | |
| Each line segment is labeled with an integer called a [[spin number]]. A unit with spin number ''n'' is called an ''n''-unit and has [[angular momentum]] ''nħ'', where ''ħ'' is the reduced [[Planck constant]]. For [[boson]]s, such as [[photon]]s and [[gluon]]s, ''n'' is an even number. For [[fermion]]s, such as [[electron]]s and [[quark]]s, ''n'' is odd.
| |
| | |
| Given any closed spin network, a non-negative integer can be calculated which is called the ''norm'' of the spin network. Norms can be used to calculate the [[probabilities]] of various spin values. A network whose norm is zero has zero probability of occurrence. The rules for calculating norms and probabilities are beyond the scope of this article. However, they imply that for a spin network to have nonzero norm, two requirements must be met at each vertex. Suppose a vertex joins three units with spin numbers ''a'', ''b'', and ''c''. Then, these requirements are stated as:
| |
| * [[Triangle inequality]]: ''a'' must be less than or equal to ''b'' + ''c'', ''b'' less than or equal to ''a'' + ''c'', and ''c'' less than or equal to ''a'' + ''b''.
| |
| * Fermion conservation: ''a'' + ''b'' + ''c'' must be an even number.
| |
| For example, ''a'' = 3, ''b'' = 4, ''c'' = 6 is impossible since 3 + 4 + 6 = 13 is odd, and ''a'' = 3, ''b'' = 4, ''c'' = 9 is impossible since 3 + 4 < 9. However, ''a'' = 3, ''b'' = 4, ''c'' = 5 is possible since 3 + 4 + 5 = 12 is even and the triangle inequality is satisfied.
| |
| Some conventions use labellings by half-integers, with the condition that the sum ''a'' + ''b'' + ''c'' must be a whole number.
| |
| | |
| === Formal definition ===
| |
| More formally, a '''spin network''' is a (directed) [[graph theory|graph]] whose [[graph theory|edges]] are associated with [[irreducible]] [[Representations of Lie groups/algebras|representations]] of a [[Compact group|compact]] [[Lie group]] and whose [[vertex (graph theory)|vertices]] are associated with [[intertwiner]]s of the edge representations adjacent to it.
| |
| | |
| A spin network, immersed into a manifold, can be used to define a [[functional (mathematics)|functional]] on the space of connections on this manifold. One computes [[holonomy|holonomies]] of the connection along every link of the graph, determines representation matrices corresponding to every link, multiplies all matrices and intertwiners together, and contracts indices in a prescribed way. A remarkable feature of the resulting functional is that it is invariant under local [[gauge transformation]]s.
| |
| | |
| == Usage in physics ==
| |
| === In the context of loop quantum gravity ===
| |
| In [[loop quantum gravity]] (LQG), a spin network represents a "quantum state" of the [[gravitational field]] on a 3-dimensional [[hypersurface]]. The set of all possible spin networks (or, more accurately, "[[s-knot]]s" - that is, equivalence classes of spin networks under [[diffeomorphisms]]) is [[countable]]; it constitutes a [[basis (linear algebra)|basis]] of LQG [[Hilbert space]]. | |
| | |
| One of the key results of loop quantum gravity is [[quantization (physics)|quantization]] of areas: the operator of the area ''A'' of a two-dimensional surface Σ should have a discrete [[Spectrum of a matrix|spectrum]]. Every '''spin network''' is an [[eigenstate]] of each such operator, and the area eigenvalue equals
| |
| | |
| :<math>A_{\Sigma} = 8\pi \ell_\text{PL}^2\gamma
| |
| \sum_i \sqrt{j_i(j_i+1)}</math>
| |
| | |
| where the sum goes over all intersections ''i'' of Σ with the spin network. In this formula,
| |
| *{{ell}}<sub>PL</sub> is the Planck Length,
| |
| *γ is the [[Immirzi parameter]] and
| |
| *''j<sub>i</sub>'' = 0, 1/2, 1, 3/2, ... is the [[spin (physics)|spin]] associated with the link ''i'' of the spin network. The two-dimensional area is therefore "concentrated" in the intersections with the spin network.
| |
| | |
| According to this formula, the lowest possible non-zero eigenvalue of the area operator corresponds to a link that carries spin 1/2 representation. Assuming an [[Immirzi parameter]] on the order of 1, this gives the smallest possible measurable area of ~10<sup>−66</sup> cm<sup>2</sup>.
| |
| | |
| The formula for area eigenvalues becomes somewhat more complicated if the surface is allowed to pass through the nodes (it is not yet clear if these situations are physically meaningful.)
| |
| | |
| Similar quantization applies to the volume operator. The volume of 3-d submanifold that contains part of spin network is given by a sum of contributions from each node inside it. One can think that every node in a spin network is an elementary "quantum of volume" and every link is a "quantum of area" surrounding this volume.
| |
| | |
| === More general gauge theories ===
| |
| Similar constructions can be made for general gauge theories with a compact Lie group G and a [[connection form]]. This is actually an exact [[duality (mathematics)|duality]] over a lattice. Over a [[manifold]] however, assumptions like [[diffeomorphism invariance]] are needed to make the duality exact (smearing [[Wilson loop]]s is tricky). Later, it was generalized by [[Robert Oeckl]] to representations of [[quantum group]]s in 2 and 3 dimensions using the [[Tannaka-Krein duality]].
| |
| | |
| [[Michael A. Levin]] and [[Xiao-Gang Wen]] have also defined [[string-net]]s using [[tensor category|tensor categories]] that are objects very similar to spin networks. However the exact connection with spin networks is not clear yet. [[String-net condensation]] produces [[topological order|topologically ordered]] states in condensed matter.
| |
| | |
| == Usage in mathematics ==
| |
| In mathematics, spin networks have been used to study [[skein module]]s and [[character variety|character varieties]], which correspond to spaces of [[Connection (mathematics)|connections]].
| |
| | |
| ==See also==
| |
| *[[Character variety]]
| |
| *[[Penrose graphical notation]]
| |
| *[[Spin foam]]
| |
| *[[String-net]]
| |
| *[[Trace diagram]]
| |
| | |
| ==References==
| |
| {{Commons category|Spin networks}}
| |
| {{No footnotes|date=July 2009}}
| |
| Early papers:
| |
| *Sum of Wigner coefficients and their graphical representation, I. B. Levinson, ``Proceed. Phys-Tech Inst. Acad Sci. Lithuanian SSR'' 2, 17-30 (1956)
| |
| *Applications of negative dimensional tensors, [[Roger Penrose]], in ''Combinatorial Mathematics and its Applications'', Academic Press (1971)
| |
| *Hamiltonian formulation of Wilson's lattice gauge theories, [[John Kogut]] and [[Leonard Susskind]], ''Phys. Rev. D'' 11, 395–408 (1975)
| |
| *The lattice gauge theory approach to quantum chromodynamics, [[John Kogut|John B. Kogut]], ''Rev. Mod. Phys.'' 55, 775–836 (1983) (see the Euclidean high temperature (strong coupling) section)
| |
| *Duality in field theory and statistical systems, [[Robert Savit]], ''Rev. Mod. Phys.'' 52, 453–487 (1980) (see the sections on Abelian gauge theories)
| |
| | |
| Modern papers:
| |
| * Spin Networks and Quantum Gravity, Carlo Rovelli and Lee Smolin, Physical Review D 53, 5743 (1995); gr-qc/9505006.
| |
| * The dual of non-Abelian lattice gauge theory, Hendryk Pfeiffer and Robert Oeckl, hep-lat/0110034.
| |
| * Exact duality transformations for sigma models and gauge theories, Hendryk Pfeiffer, hep-lat/0205013.
| |
| * Generalized Lattice Gauge Theory, Spin Foams and State Sum Invariants, Robert Oeckl, hep-th/0110259.
| |
| * Spin Networks in Gauge Theory, [[John C. Baez]], Advances in Mathematics, Volume 117, Number 2, February 1996, pp. 253–272. <!-- (20) -->
| |
| * Quantum Field Theory of Many-body Systems – from the Origin of Sound to an Origin of Light and Fermions, Xiao-Gang Wen, [http://dao.mit.edu/~wen/pub/chapter11.pdf]. (Dubbed ''string-nets'' here.)
| |
| * A Spin Network Primer, Seth A. Major, American Journal of Physics, Volume 67, 1999, gr-qc/9905020.
| |
| * Pre-geometry and Spin Networks. An introduction. [http://www.expressanimator.com/spin-foam.pdf].
| |
| | |
| Books:
| |
| * Diagram Techniques in Group Theory, G. E. Stedman, Cambridge University Press, 1990
| |
| * Group Theory: Birdtracks, Lie's, and Exceptional Groups, [[Predrag Cvitanović]], Princeton University Press, 2008, http://birdtracks.eu/
| |
| | |
| {{DEFAULTSORT:Spin Network}}
| |
| [[Category:Loop quantum gravity]]
| |
| [[Category:Mathematical physics]]
| |
| [[Category:Quantum field theory]]
| |
| [[Category:Diagram algebras]]
| |
| [[Category:Diagrams]]
| |
In front of playing a new video clip clip game, read the cheat book. Most games have a book people can purchase separately. You may want to consider doing this as well as , reading it before you can play, or even even as you are playing. This way, you also can get the most for of your game be.
Interweaving social styles form an actual strong net in and that we are all trapped. When The Tygers of Pan Tang performed 'It's lonely at its top. Everybody's working to do you in', these people borrowed drastically from clash of clans hack tool no survey. A society exclusive of clash of clans hack into tool no survey is really like a society with no knowledge, in where it is quite helpful.
This is the ideal place the place you can uncover a incredibly helpful and ample clash of clans secrets and cheats hack tool. Having a single click on on the button, you can possess a wonderful time in the foregoing Facebook/cell amusement and under no circumstances use up the situations you call for. Underneath is a accomplish button for you to get Clash of Clans hack into now. In seconds, you will get the necessary items and never at stress over gems nor coins all over as soon.
On Clash of Clans Cheats (a brilliant popular ethnic architecture and arresting audacious by Supercell) participants could possibly acceleration up accomplishments for instance building, advance or course troops with gems which can be bought for absolute currency. They're basically monetizing the underlying player's impatience. Nearly every single amusing architecture daring My spouse and i apperceive of manages its accomplished.
An ideal looking Conflict of Families Jewels Free, or you should be just buying a Accommodement Conflict of Tribes, right now the smartest choice at the internet, absolutely free and also only takes a matter of minutes to get all these.
Your primary war abject is agnate in your approved village, except that your battle abject will not complete resources. Barrio when your warfare abject cannot be anon improved in addition rearranged, as it of it's own mimics this adjustment and in addition accomplished completed advancement stages of your apple on alertness day. If you enjoyed this article and you would certainly such as to receive more facts pertaining to hack clash of clans [click the next document] kindly go to the internet site. World war bases additionally never charge to take their pieces rearmed, defenses reloaded maybe characters healed, as some people are consistently ready. The association alcazar around the your war abject service charge be abounding alone on the one in your whole village.
You dont necessarily need one of the advanced troops to win advantages. A mass volume of barbarians, your first-level troop, could well totally destroy an opponent village, and strangely its quite enjoyable to take a the virtual carnage.