Is there any $SU(\infty)$ gauge theory in quantum field theory? - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-10-18T09:38:00Z https://physics.stackexchange.com/feeds/question/203123 https://creativecommons.org/licenses/by-sa/4.0/rdf https://physics.stackexchange.com/q/203123 3 Is there any $SU(\infty)$ gauge theory in quantum field theory? kryomaxim https://physics.stackexchange.com/users/72314 2015-08-28T20:22:00Z 2015-08-30T21:26:18Z <p>The groups $U(N)$ and $SU(N)$ are the most important Lie groups in quantum field theory. The most popular are the $U(1),SU(2),SU(3)$ groups (these gauge groups form the Standard model). But is there mentioned a $SU(\infty)$ gauge theory in physics literature?</p> <p>An example of such a theory could be the following: May be $g \in SU(\infty)$ smooth and for a function $f(x,y)$ with spacetime coordinate $x$ and the new $SU(\infty)$ degree of freedom $y$ it holds $gf(x,y) = \int d^4y (g(x,y,y')f(x,y'))$. Now it is straighforward to define a gauge connection and the gauge field strength. </p> <p>In more non-theoretical words: Some quantum states have degeneracies and these degeneracies are based on a special symmetry (operator) that exists in a quantum system. If now the degeneracy symmetry operator is unitary and <strong>local</strong> symmetry one can define a gauge theory. Was this concept used in quantum mechanics or does such a concept makes sense?</p> <p>Another interesting case is this: One can perform the following switch of coordinates $g(x,y,y') = g(x,x-y,x-y')$ and hence the generators $T_a(y,y')$ defined by $g(x,y,y') = \sum_a g_a(x)T_a(y,y')$ become dependent on the spacetime coordinate. Another question: Is it possible to define spacetime dependent generators of a Lie algebra?</p> https://physics.stackexchange.com/questions/203123/-/203159#203159 3 Answer by Mitchell Porter for Is there any $SU(\infty)$ gauge theory in quantum field theory? Mitchell Porter https://physics.stackexchange.com/users/1486 2015-08-29T01:26:25Z 2015-08-29T01:26:25Z <p>There are apparently several thousand references to "SU(\infty)" on arxiv.org, and some of them are definitely talking about gauge fields or Yang-Mills. </p> <p>I suspect that some of the time, this will just be a way of talking about the large N limit of SU(N), i.e., not referring to a literal SU(∞) field theory, but rather the N→∞ limit of some quantity in SU(N) field theory. </p> https://physics.stackexchange.com/questions/203123/-/203424#203424 4 Answer by Qmechanic for Is there any $SU(\infty)$ gauge theory in quantum field theory? Qmechanic https://physics.stackexchange.com/users/2451 2015-08-30T21:26:18Z 2015-08-30T21:26:18Z <p>Comments to the question (v2):</p> <ol> <li><p>The idea to consider the <a href="http://en.wikipedia.org/wiki/1/N_expansion" rel="nofollow">planar large $N_c\to \infty$ limit</a> in $SU(N_c)$ QCD goes back to Ref. 1. </p></li> <li><p>In light-cone <a href="http://ncatlab.org/nlab/show/M2-brane" rel="nofollow">membrane theory</a>, pioneered in Ref. 2, the group $SU(\infty)$ is naturally identified with area-preserving diffeomorphisms ${\rm SDiff}_0(T^2)$ on the torus $T^2$ connected to the identity.</p></li> <li><p>Concretely, OP's proposal resembles a Fourier series expansion of an extra (compact) spacetime dimension. Such exercises are customary in string theory.</p></li> </ol> <p>References:</p> <ol> <li><p>G. 't Hooft, <em>A planar diagram theory for strong interactions,</em> <a href="http://dx.doi.org/10.1016/0550-3213(74)90154-0" rel="nofollow">Nucl. Phys. B72 (1974) 461</a>. </p></li> <li><p>J. Goldstone, unpublished; J. Hoppe, MIT Ph.D. Thesis, 1982. </p></li> </ol>