Niemeier lattice: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>Headbomb
m Various citation cleanup (identifiers mostly), replaced: | id={{MR|1938666}} → | mr=1938666 (5) using AWB
 
en>Pythagoras0
 
Line 1: Line 1:
The '''fermion doubling problem''' is a problem that is encountered when naively trying to put [[fermionic field]]s on a [[lattice field theory|lattice]]. It consists in the appearance of spurious states, such that one ends up having 2<sup>''d''</sup> fermionic particles (with ''d'' the number of discretized dimensions) for each original fermion. In order to solve this problem, several strategies are in use, such as [[Wilson fermion]]s and [[staggered fermion]]s.
This has been said that an addiction to cigarette smoking is among one of, if not the most challenging dependence on strike. We’ve all read the “quit ice cold turkey” idiom well before. Happily for several cigarette smokers finding it difficult throughout the world, tobacco users desperate to stop smoking cigarettes have some different alternatives now. It is no easy feat escaping the icy proper grip of craving, but clearette electronic cigarettes can easily assist a smoker inside their experience.<br><br>Not every person is all set to say their ultimate farewell to pure nicotine. But even if they are not, clearette electronic cigarettes could play a role in the curing and process of recovery. Without absolutely everyone believes from what scope, you will find basically no skepticism that e-cigarette smoking can be a better alternative option to conventional combustible smoking. But if you are in a position to kick the habit for better, here are some good ideas (or more to the stage, causes) to why clearette electronic cigarettes are more and top-quality beneficial than the nicotine patch and quitting smoking cool turkey.<br><br>1. Vaping seems like Tobacco smoking.<br><br>Whether you are seeking a hypnotist, a diet plan, or perhaps a miraculous spell to curb your desire to cigarette smoke, you’re still gonna obtain that need. That deep-seated required to pull the best brand. And E-cigarettes have fun with on that need. If you need to smoke cigarettes, if you are like you will need to cigarette smoke, then practice it. But rather than grabbing that carcinogen riddled keep of loss of life, take an excellent, easy to use electronic cigarette. If you are among those smokers that just doesn’t know what to do what together with his/her hands and fingers, in comparison to the e-cig is good for you. It is possible to draw in the delicious vapors with no taking in the numerous contaminants located in regular combustible cigarette smoking. E-cigs are definitely the upcoming plus the tip for beating your smoking habit.<br><br>2. No Odour, No Problem<br><br>Any cigarette smoker, whether or not they can be a sequence cigarette smoker or otherwise, is aware all to properly the challenge of smelling like smoking. They are really odorless. That's among the key reselling points on most e-tobacco cigarettes. No more does a smoke enthusiast should settle for the continual surroundings of stagnant smoke adhering to them. Because clearette electronic cigarettes tend not to use up, their is no burning off or ashy odour. They heat vapour. Tasty, odorless vapor. You now never must package up and venture out within the cool winter’s nights when you like to light up. Rather you could be seated directly on the sofa with your family with no stressing about 2nd-fretting hand smoke or  [http://clearettereview.com/how-do-electronic-cigarettes-work/ vapor smoke] smelling like an ashtray. Also, your no-smoke enthusiast spouse will not cringe at any time time you lean in for a kiss.<br><br>3. No Tar residue<br><br>Have you any idea what tar does for your requirements? Tar residue isn’t meant to be in the body. It yellows your the teeth and your fingernails. None of us really wants to go walking about with yellowish features. This also damages your epidermis. Who requirements that?<br><br>4. They’re just less difficult<br><br>You no longer must “bum a light” or maybe “bum a cig.” You by no means need to light up an e-cigarette simply because they provide the warming up supply internal. And just as before, there is no need to shift each and every time you would like to cigarette smoke, particularly a cigarette smoke-cost-free setting. No eliminating, no light up, no problem.
 
==Mathematical overview==
The action of a [[free field|free]] [[Dirac fermion]] in ''d'' dimensions,<ref group="note">As the lattice discretization is always defined in Euclidean spacetime, we will suppose the suitable [[Wick rotation]] has been performed. Herefore, no difference will be made between covariant and cotravariant indices.</ref> of [[mass]] ''m'', and in the continuum (i.e. without discretization) is commonly given as
:<math> S = \int d^dx \; \bar\psi (\partial\!\!\!/ + m) \psi \;.</math>
Here, the [[Feynman slash notation]] was used to write
:<math> \gamma_\mu a^{\mu} = a\!\!\!/ </math>
where γ<sub>μ</sub> are the [[gamma matrices]]. When this action is discretized on a cubic lattice, the fermion field ψ(''x'') is replaced with a discretized version ψ<sub>''x''</sub>, where ''x'' now denotes the lattice site. The derivative is replaced by the [[finite difference]]. The resulting action is now:<ref name=wiese>{{Cite journal | last1 = Chandrasekharan | last2 = Wiese | title = An Introduction to chiral symmetry on the lattice | journal = Prog. Part. Nucl. Phys. | volume = 53 | pages = 373 | date = 2004 | url = http://www.sciencedirect.com/science/article/pii/S0146641004000948 | doi = 10.1016/j.ppnp.2004.05.003|arxiv = hep-lat/0405024 |bibcode = 2004PrPNP..53..373C }}</ref>
:<math> S = a^d \sum_{x,\mu} \frac1{2a} (\bar\psi_x\gamma_\mu\psi_{x+\hat\mu} - \bar\psi_{x+\hat\mu}\gamma_\mu\psi_x) + a^d \sum_x m\bar\psi_x\psi_x \;,</math>
where ''a'' is the lattice spacing and <math>\hat\mu</math> is the vector of length ''a'' in the μ direction. When one computes the inverse fermion propagator in [[momentum space]], one readily finds:<ref name="wiese" />
:<math> S^{-1}(p) = m + \frac ia \sum_\mu \gamma_\mu \sin(p_\mu a) \;. </math>
Due to the finite lattice spacing the momenta ''p''<sub>μ</sub> have to be inside the [[Brillouin zone]], which is typically taken to be the interval [−{{Pi}}/''a'',+{{Pi}}/''a''].
 
When simply taking the limit ''a'' → 0 in the above inverse propagator, one recovers the correct continuum result. However, when instead expanding this expression around a value of ''p''<sub>μ</sub> where one or more of the components are at the corners of the Brillouin zone (i.e. equal to {{Pi}}/''a''), one finds the same continuum form again, although the sign in front of the gamma matrix can change.<ref>{{cite arxiv | last = Gupta | eprint = hep-lat/9807028 | title = Introduction to lattice QCD | year = 1998}}</ref><ref group="note">Due to these changes in sign, the [[chiral anomaly]] cancels exactly, which is not in agreement with phenomenology.</ref> This means that, when one of the components of the momentum is near {{Pi}}/''a'', the discretized fermion field will again behave like a continuum fermion. This can happen with all ''d'' components of the momentum, leading to —with the original fermion with momentum near the origin included— 2<sup>''d''</sup> different "tastes" (in analogy to [[flavor (particle physics)|flavor]]).<ref group="note">As the action of scalars contains second derivatives, a similar procedure in this case would lead to a quadratic inverse propagator, which does not have these doublers.</ref>
 
==The Nielsen–Ninomiya theorem==
[[Holger Bech Nielsen|Nielsen]] and Ninomiya proved a theorem<ref>{{Cite journal | last1 = Nielsen | last2 = Ninomiya | authorlink1 = Holger Bech Nielsen | title = Absence of Neutrinos on a Lattice | journal = Nucl. Phys. B | volume = 185 | year = 1981 | pages = 20 | doi = 10.1016/0550-3213(81)90361-8|bibcode = 1981NuPhB.185...20N }}<br/>{{Cite journal | last1 = Nielsen | last2 = Ninomiya | authorlink1 = Holger Bech Nielsen | title = No go theorem for regularizing chiral fermions | journal = Phys. Lett. B | volume = 105 | year = 1981 | pages = 219 | doi = 10.1016/0370-2693(81)91026-1|bibcode = 1981PhLB..105..219N }}</ref> stating that a local, real, free fermion lattice action, having [[chiral symmetry|chiral]] and translational invariance, necessarily has fermion doubling. The only way to get rid of the doublers is by violating one of the presuppositions of the theorem —for example:
* [[Wilson fermion]]s explicitly violate chiral symmetry, giving an infinitely high mass to the doublers which then decouple.
* So-called "[[perfect fermions|perfect lattice fermions]]" have a nonlocal action.
* [[Staggered fermion]]s
* [[Twisted mass fermions]]
* [[Ginsparg–Wilson fermions]]
* [[Domain wall fermions]]
* [[Overlap fermions]]
 
==See also==
* [[Staggered fermion]]s: a way to reduce the number of doublers
* [[Optical phonon#Acoustic and optical phonons|Acoustic and optical phonons]]: a similar phenomenon in solid state crystals
 
==Notes and references==
;Notes
<references group="note" />
 
;References
{{Reflist|close=1}}
 
[[Category:Lattice models]]
[[Category:Fermions]]

Latest revision as of 16:26, 28 June 2014

This has been said that an addiction to cigarette smoking is among one of, if not the most challenging dependence on strike. We’ve all read the “quit ice cold turkey” idiom well before. Happily for several cigarette smokers finding it difficult throughout the world, tobacco users desperate to stop smoking cigarettes have some different alternatives now. It is no easy feat escaping the icy proper grip of craving, but clearette electronic cigarettes can easily assist a smoker inside their experience.

Not every person is all set to say their ultimate farewell to pure nicotine. But even if they are not, clearette electronic cigarettes could play a role in the curing and process of recovery. Without absolutely everyone believes from what scope, you will find basically no skepticism that e-cigarette smoking can be a better alternative option to conventional combustible smoking. But if you are in a position to kick the habit for better, here are some good ideas (or more to the stage, causes) to why clearette electronic cigarettes are more and top-quality beneficial than the nicotine patch and quitting smoking cool turkey.

1. Vaping seems like Tobacco smoking.

Whether you are seeking a hypnotist, a diet plan, or perhaps a miraculous spell to curb your desire to cigarette smoke, you’re still gonna obtain that need. That deep-seated required to pull the best brand. And E-cigarettes have fun with on that need. If you need to smoke cigarettes, if you are like you will need to cigarette smoke, then practice it. But rather than grabbing that carcinogen riddled keep of loss of life, take an excellent, easy to use electronic cigarette. If you are among those smokers that just doesn’t know what to do what together with his/her hands and fingers, in comparison to the e-cig is good for you. It is possible to draw in the delicious vapors with no taking in the numerous contaminants located in regular combustible cigarette smoking. E-cigs are definitely the upcoming plus the tip for beating your smoking habit.

2. No Odour, No Problem

Any cigarette smoker, whether or not they can be a sequence cigarette smoker or otherwise, is aware all to properly the challenge of smelling like smoking. They are really odorless. That's among the key reselling points on most e-tobacco cigarettes. No more does a smoke enthusiast should settle for the continual surroundings of stagnant smoke adhering to them. Because clearette electronic cigarettes tend not to use up, their is no burning off or ashy odour. They heat vapour. Tasty, odorless vapor. You now never must package up and venture out within the cool winter’s nights when you like to light up. Rather you could be seated directly on the sofa with your family with no stressing about 2nd-fretting hand smoke or vapor smoke smelling like an ashtray. Also, your no-smoke enthusiast spouse will not cringe at any time time you lean in for a kiss.

3. No Tar residue

Have you any idea what tar does for your requirements? Tar residue isn’t meant to be in the body. It yellows your the teeth and your fingernails. None of us really wants to go walking about with yellowish features. This also damages your epidermis. Who requirements that?

4. They’re just less difficult

You no longer must “bum a light” or maybe “bum a cig.” You by no means need to light up an e-cigarette simply because they provide the warming up supply internal. And just as before, there is no need to shift each and every time you would like to cigarette smoke, particularly a cigarette smoke-cost-free setting. No eliminating, no light up, no problem.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Niemeier_lattice&oldid=239345"