Pion decay constant

2012-08-06T15:41:45Z

132.199.99.7:

{{Expert-subject|Physics|date=February 2009}}
{{Lie groups |Physics}}

The connection between [[particle physics]] and [[representation theory]] is a natural connection, first noted in the 1930s by [[Eugene Wigner]],<ref>Wigner received the [[Nobel Prize in Physics]] in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles"; see also [[Wigner's theorem]], [[Wigner's classification]].</ref> between the properties of [[elementary particle]]s and the structure of [[Lie groups]] and [[Lie algebras]]. According to this connection, the different [[quantum state]]s of an elementary particle give rise to an [[irreducible representation]] of the [[Poincaré group]]. Moreover, the properties of the various particles, including their [[energy spectrum|spectra]], can be related to representations of Lie algebras, corresponding to "approximate symmetries" of the universe.

== General picture ==

In [[quantum mechanics]], any particular particle (with a given momentum distribution, location distribution, spin state, etc.) is written as a [[vector space|vector]] (or "[[bra-ket notation|ket]]") in a [[Hilbert space]] H. To help understand what types of particles can exist, it is important to classify the possibilities for H, and their properties. The particle is more precisely characterized by the associated ''[[projective space|projective]]'' Hilbert space '''P'''H, since two vectors that differ by a scalar factor (or in physics terminology, two "kets" that differ by a "[[phase factor]]") correspond to the same physical [[quantum state]].

Let ''G'' be the ''symmetry group of the universe'' – that is, the set of symmetries under which the laws of physics are invariant. (For example, one element of ''G'' is the simultaneous translation of all particles and fields forward in time by five seconds.) Starting with a particular particle in the state ket <math>|p_0\rangle</math>, and a symmetry transformation ''g'' in ''G'', it is possible to apply the symmetry transformation to the particle to get a new state ket <math>|p_g\rangle=g|p_0\rangle</math>. For this picture to be consistent, it is necessary that '''P'''H is a [[projective representation|projective group representation]] of ''G''. (For example, this condition guarantees that applying a symmetry transformation, then applying its inverse transformation, will restore the original quantum state.)

Therefore, any given particle is associated with a unique [[projective representation|representation]] of ''G'' on a projective vector space '''P'''H. (We say the particle "lies in", or "transforms as" the representation.) In many important cases, it can be shown that the particle is also (more specifically) associated with a [[group representation]] of ''G'' on the underlying (non-projective) space H.<ref name=WeinbergCh2Appendix/> [[Wigner's Theorem]] proves that it is a [[unitary representation]], or possibly anti-unitary.<ref name=WeinbergCh2Appendix>See Weinberg (1995), Chapter 2 appendix A and B.</ref>

So we conclude that each type of particle corresponds to a representation of ''G'', and if we can classify the group representations of ''G'', we will have much more information about the possibilities and properties of H, and hence what types of particles can exist.

== Poincaré group ==
The group of translations and [[Lorentz transformations]] form the [[Poincaré group]], and this group is certainly a subgroup of ''G'' (neglecting [[general relativity]] effects, or in other words, in [[flat space]]). Hence, any representation of ''G'' will in particular be a representation of the Poincaré group. [[Representation theory of the Poincaré group|Representations of the Poincaré group]] are in many cases characterized by a nonnegative [[mass]] and a half-integer [[spin (physics)|spin]] (see [[Wigner's classification]]); this can be thought of as the reason that particles have quantized spin. (Note that there are in fact other possible representations, such as [[tachyon]]s, [[infraparticle]]s, etc., which in some cases do not have quantized spin or fixed mass.)

== Other symmetries ==
[[File:Standard Model charges.svg|300px|right|thumb|The pattern of [[weak isospin]]s, [[weak hypercharge]]s, and [[color charge|color]] charges (weights) of all known elementary particles in the [[Standard Model]], rotated by the [[weak mixing angle]] to show electric charge roughly along the vertical.]]

While the [[spacetime symmetries]] in the Poincaré group are particularly easy to visualize and believe, there are also other types of symmetries, called [[internal symmetry|internal symmetries]]. One example is [[color charge|color]] [[SU(3)]], an exact symmetry corresponding to the continuous interchange of the three [[quark]] colors.

== Approximate symmetries ==

Although the above symmetries are believed to be exact, other symmetries are only approximate.

===Hypothetical example===
As an example of what an approximate symmetry means, suppose we lived inside an infinite [[ferromagnet]], with magnetization in some particular direction. An experimentalist in this situation would find not one but two distinct types of electrons: one with spin along the direction of the magnetization, with a slightly lower energy (and consequently, a lower mass), and one with spin anti-aligned, with a higher mass. Our usual [[SO(3)]] rotational symmetry, which ordinarily connects the spin-up electron with the spin-down electron, has in this hypothetical case become only an ''approximate'' symmetry, relating ''different types of particles'' to each other.

===Lie algebras versus Lie groups===
Many (but not all) symmetries or approximate symmetries, for example the ones above, form [[Lie groups]]. Rather than study the [[Representation theory#Lie groups|representation theory]] of these Lie groups, it is often preferable to study the closely related [[Representation theory#Lie algebras|representation theory]] of the corresponding Lie algebras, which are usually simpler to compute.

===General definition===
In general, an approximate symmetry arises when there are very strong interactions that obey that symmetry, along with weaker interactions that do not. In the electron example above, the two "types" of electrons behave identically under the [[strong force|strong]] and [[weak force]]s, but differently under the [[electromagnetic force]].

===Example: isospin symmetry===
{{main|Isospin}}
An example from the real world is [[isospin|isospin symmetry]], an [[SU(2)]] group corresponding to the similarity between [[up quark]]s and [[down quark]]s. This is an approximate symmetry: While up and down quarks are identical in how they interact under the [[strong force]], they have different masses and different electroweak interactions. Mathematically, there is an abstract two-dimensional vector space
:<math>\text{up quark} \rightarrow \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad \text{down quark} \rightarrow \begin{pmatrix} 0 \\ 1 \end{pmatrix}</math>
and the laws of physics are ''approximately'' invariant under applying a determinant-1 [[unitary transformation]] to this space:<ref name=Thomson>[http://www.hep.phy.cam.ac.uk/~thomson/partIIIparticles/handouts/Handout_7_2011.pdf Lecture notes by Prof. Mark Thomson]</ref>
:<math>\begin{pmatrix} x \\ y \end{pmatrix} \mapsto A \begin{pmatrix} x \\ y \end{pmatrix}, \quad \text{where } A \text{ is in } SU(2)</math>
For example, <math>A=\begin{pmatrix} 0&1 \\ -1&0 \end{pmatrix}</math> would turn all up quarks in the universe into down quarks and vice-versa. Some examples help clarify the possible effects of these transformations:
*When these unitary transformations are applied to a [[proton]], it can be transformed into a [[neutron]], or into a superposition of a proton and neutron, but not into any other particles. Therefore, the transformations move the proton around a two-dimensional space of quantum states. The proton and neutron are called an "[[isospin multiplet|isospin doublet]]", mathematically analogous to how a [[spin-½]] particle behaves under ordinary rotation.
*When these unitary transformations are applied to any of the three [[pion]]s ({{SubatomicParticle|Pion0}}, {{SubatomicParticle|Pion+}}, and {{SubatomicParticle|Pion-}}), it can change any of the pions into any other, but not into any non-pion particle. Therefore, the transformations move the pions around a three-dimensional space of quantum states. The pions are called an "[[isospin multiplet|isospin triplet]]", mathematically analogous to how a spin-1 particle behaves under ordinary rotation.
*These transformations have no effect at all on an [[electron]], because it contains neither up nor down quarks. The electron is called an isospin singlet, mathematically analogous to how a spin-0 particle behaves under ordinary rotation.

In general, particles form [[isospin multiplet]]s, which correspond to irreducible representations of the [[Special unitary group#Lie algebra|Lie algebra SU(2)]]. Particles in an isospin multiplet have very similar but not identical masses, because the up and down quarks are very similar but not identical.

===Example: Flavour symmetry===
Isospin symmetry can be generalized to [[flavour symmetry]], an [[SU(3)]] group corresponding to the similarity between [[up quark]]s, [[down quark]]s, and [[strange quark]]s.<ref name=Thomson/> This is, again, an approximate symmetry, violated by quark mass differences and electroweak interactions—in fact, it is a poorer approximation than isospin, because of the strange quark's noticeably higher mass.

Nevertheless, particles can indeed be neatly divided into groups that form irreducible representations of the [[Special unitary group#Lie algebra|Lie algebra SU(3)]], as first noted by [[Murray Gell-Mann]] and independently by [[Yuval Ne'eman]] (see [[eightfold way (physics)|the eightfold way]]).

== See also ==
*[[Lie algebra]]
*[[Lie group]]
*[[Poincaré group]]
*[[Representation theory]]:
**[[Representation theory#Lie algebras|Of Lie algebras]]
**[[Representation theory#Lie groups|Of Lie groups]]
**[[Representation theory of the Poincaré group|Of the Poincaré group]]
*[[Special unitary group]]
*[[Symmetry]]

== References ==
*Coleman, Sidney (1985) ''Aspects of Symmetry: Selected Erice Lectures of Sidney Coleman''. Cambridge Univ. Press. ISBN 0-521-26706-4.
*Georgi, Howard (1999) ''Lie Algebras in Particle Physics''. Reading, MA: Perseus Books. ISBN 0-7382-0233-9.
* Hall, Brian C., (2006) ''Lie Groups, Lie Algebras, and Representations: An Elementary Introduction''. Springer. ISBN 0-387-40122-9.
*Sternberg, Shlomo (1994) ''Group Theory and Physics''. Cambridge Univ. Press. ISBN 0-521-24870-1. Especially pp. 148–150.
*{{cite book | author=[[Steven Weinberg]] | title=The Quantum Theory of Fields, Volume 1: Foundations | publisher=Cambridge Univ. Press | year=1995 | isbn=0-521-55001-7}} Especially appendices A and B to Chapter 2.

== Notes ==
{{reflist}}

== External links ==
*John C. Baez & John Huerta, ''The Algebra of Grand Unified Theories'', {{arXiv|0904.1556}}

{{DEFAULTSORT:Particle Physics And Representation Theory}}
[[Category:Lie algebras]]
[[Category:Particle physics]]
[[Category:Representation theory of Lie groups]]
[[Category:Theoretical physics]]
[[Category:Conservation laws]]
[[Category:Quantum field theory]]

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