This essay is a natural continuation of my article explaining Why Matrix theory contains membranes.
The fuzzy sphere and the fuzzy torus – two of the allowed shapes that membranes in Matrix theory may possess – were the simplest examples of "noncommutative geometries". What does it mean? What is noncommutative about them?
If you have an ordinary shape, space, or a manifold, the functions \(f(x^\mu)\) and \(g(x^\mu)\) defined on these shapes commute with each other; it means that their ordering in the product doesn't matter,\[
f(x^\mu) g(x^\mu) = g(x^\mu) f(x^\mu).
\] After all, they are just numbers at each point. For noncommutative geometries, \(=\) has to be replaced by \(\neq\); the identity above no longer holds. We have already seen that in the case of the fuzzy torus and the fuzzy sphere. Because functions are at least locally written as functions of coordinates, the noncommutative property of the noncommutative geometry effectively means that the coordinates don't commute with each other. We're going to understand this statement in the simplest possible context, the case of 2 coordinates on a plane.
In the case of the fuzzy torus, we emulated the functions \(\exp(i\sigma_1)\) and \(\exp(i\sigma_2)\) by large enough matrices \(U,V\), the clock and shift operators. They satisfied\[
UV = VU \cdot \exp(2\pi i / N)
\] where \(N\) is large and it is the size of the matrices \(U,V\). If \(N\) is truly large, we may write the large matrices \(X^i\), nine of them, as some functions of \(U,V\) and these matrices rather faithfully encode functions \(x^i(\sigma_1,\sigma_2)\) on a large torus that is "almost commutative".
What is the "local geometry" of this torus? Can't we make it infinitely large? Yes, we can.
As I have already mentioned, the property above – that \(U,V\) nearly commute with each other – may be rewritten as\[
UV - VU = (\exp(2\pi i / N) - 1) VU \sim \frac{2\pi i}{N} VU
\] which is small: it scales like \(1/N\). Now, try to understand how these things behave in the fuzzy vicinity of the point \((\sigma_1,\sigma_2)=(0,0)\) of the torus, a randomly chosen point. Near that point, our identification reduces to\[
U\sim 1+i\sigma_1,\quad V\sim 1+i\sigma_2.
\] I have simply Taylor-expanded the exponentials defining \(U,V\) in our dictionary. So the commutator relationship for \(U,V\) says (because \(1\) commutes with everything)\[
[i\sigma_1,i\sigma_2] \sim \frac{2\pi i}{N} VU \sim \frac{2\pi i}{N}.
\] In other words,\[
[\sigma_1\sqrt{N},\sigma_2\sqrt{N}] = -2\pi i.
\] The sign came from the two \(i\) factors; I have rescaled the \(\sigma_{1,2}\) coordinates by \(\sqrt{N}\) because it's natural to expect that the size \(N\) matrices will be able to describe a membrane whose "number of cells" (a few more comments about it will appear at the end of this article) i.e. area scales like \(N\) which means that the radius of the sphere naturally scales with \(\sqrt{N}\).
At any rate, we see that near the chosen point of the large torus, the commutator of the two coordinates equals an imaginary \(c\)-number. There's an obvious analogy with the defining commutator behind the quantum mechanical uncertainty principle,\[
[x,p]=i\hbar.
\] It's not just an analogy; it's an exact mathematical isomorphism! (However, you must realize that the physical interpretation is different: field theories on noncommutative geometries may be classical, lacking any fancy quantum features.) Needless to say, I could derive the very same thing from the fuzzy sphere, too. Consider the neighborhood of the North pole of the unit sphere, i.e. points with \(z\sim 1\). The two-dimensional neighborhood is described by the coordinates \(x,y\) which we represented, up to some coefficients, by the matrices for \(J_x\) and \(J_y\) in a large, \(N\)-dimensional irreducible representation of \(SU(2)\). But because\[
[J_x,J_y] = i\hbar J_z
\] and because \(J_z\) is approximately equal to \(J\), a constant, near the Northern pole, we again learn that \([x,y]\) is equal to \(i\) times a constant. Various coordinates may be scaled in such a way that the constants agree in all the situations. We're led to consider two-dimensional geometry (the large, flat limit of all the special cases) whose coordinates obey\[
[x,y] = i\theta.
\] The commutator is nonzero; that's why we deal with a noncommutative geometry. It's the same commutator as the commutator \([x,p]=i\hbar\) defining the phase space in quantum mechanics; I have used a more general constant \(\theta\) instead of \(\hbar\) but the mathematics is clearly isomorphic. You should note that in noncommutative geometry, the constant \(\theta\) in the commutator has units of \({\rm Length}^2\) so it defines a length scale. At distances much longer than \(\theta\) i.e. \(L\gg \sqrt{\theta}\), the commutator proportional to \(\theta\) may be neglected relatively to the (squared) distances and the geometry reduces to a commutative one. However, at distances comparable to and shorter than \(\sqrt{\theta}\), the noncommutativity matters a great deal.
Can we define classical field theories and quantum field theories whose fields don't live on the ordinary geometries but that live on this noncommutative plane?
Theories on the noncommutative plane
Of course, we can. We will ultimately see that such theories may be defined by the same Lagrangians of field theory that we know; the only novelty is that all the pointwise products of fields in the Lagrangian, \(f(x,y)g(x,y)\), have to be replaced with a new kind of a product which is no longer pointwise and no longer commutative, the star-product \(f(x,y)*g(x,y)\) – well, the particular explicit type of the star-product we will play with is known as the Moyal product.
This star-product is nothing else than an operation between two ordinary functions that perfectly emulates the non-commutative multiplication of operators in quantum mechanics.
But before we get there, it may be a good idea to learn something about the space of linear operators that act on the Hilbert space of a particle moving on a line, the Hilbert space on which we know the operators \(\hat x\) and \(\hat p\), the most widespread undergraduate model of one-dimensional quantum mechanics. How can we classify all linear operators on this Hilbert space?
Well, we may choose a basis of this Hilbert space and define an operator \(\hat M\) by its matrix elements\[
M_{ij} = \bra{\psi_i} \hat M \ket{\psi_j}.
\] We may either choose a discrete (infinite but countable) basis, e.g. the eigenstates of the harmonic oscillator Hamiltonian; or we may choose a continuous basis in which case \(M_{ij}\) should really be written as \(M(i,j)\) because the arguments are continuous. But we don't really want to deal with the harmonic oscillator or any other ad hoc discrete or continuous basis; we would prefer to describe all allowed operators as some kinds of functions of \(\hat x,\hat p\). How can it be done? How many such linearly independent operators are there?
If \(\hat x,\hat p\) commuted with one another, it would look like all possible operators we may construct are simply functions of \(\hat x,\hat p\). But they don't commute so the concept of a function is more subtle. For example, the product is a simple function of two arguments but if the two arguments don't commute with one another, the definition of the function based on the word "product" is ambiguous because it doesn't tell us how to order the factors.
However, this problem is very small, from an appropriate point of view. Consider all polynomial operators you may have. They're linear combinations of the following monomials:\[
1,\, x,\, p,\, xp,\, px,\, xxx,\, xxp,\, \dots
\] I have suppressed hats and let's forget about them. Are all these monomials independent? Well, the first four operators are surely independent of each other. However, the fifth one, \(px\), is almost the same thing as \(xp\); they only differ by \(i\hbar\) which is a multiple of the operator \(1\) that has already appeared in the list. So if you organize the basis of the operators in this way, only one ordering should be included as the quantum representative of any classical product. All the other orderings are linearly dependent; they differ from your chosen ordering by terms that have already been included before.
(The previous terms have been processed in the same way so we may also achieve all orderings of those simpler guys.)
If you think about the previous paragraph, it actually explains why you may represent generic and "smooth enough" operators on the 1D quantum mechanical Hilbert space by functions of \(x\) and \(p\). All we need to make this idea specific is to define a particular dictionary between operators and functions. Let's start with the map assigning an operator to a \(c\)-number-valued function; the inverse function will be fully determined immediately as inverse functions often are.
The right map is given by the symmetric ordering,\[
\widehat{Op}(x^m p^{n-m}) = \frac{1}{n!} \sum_{\pi\in S_n}^{n!\text{ permutations}} \pi [ \hat x\hat x\cdots \hat p\hat p\hat p ].
\] We're averaging all possible permutations \(\pi\) of the linear factors \(\hat x\) and \(\hat p\). For a second, I restored the hats to emphasize that there are no hats in the argument of the left hand side but there are hats on the right hand side. Because of my previous arguments, this dictionary is enough to express any operator as \(Op\) of a "classical" function. Note that if the function is real, the operator will be Hermitian.
Now, we would like to define a star-product of ordinary functions so that this product mimics the noncommutative product of the corresponding operators; but we don't have to translate from functions to operators and back. In order words, we want to define the operation \(*\) such that\[
\widehat{Op} (f(x,p)*g(x,p)) = \widehat{Op}(f(x,p))\cdot \widehat{Op}(g(x,p)).
\] If we multiply two functions by the star-product and translate the result into an operator, it's the same thing as if we first translate the two functions into operators and then we multiply these two operators by the regular noncommutative "dot" product (i.e. composition or matrix product, if we choose a basis) of these operators.
A funny thing is that given this general definition of the star-product and with our convenient "symmetric ordering" dictionary between functions and operators, we may write down the formula for the star-product explicitly. The result turns out to be\[
f(\vec x)*g(\vec x) =\qquad\\ \qquad = \exp(\frac{i}{2}\theta_{mn}\pfrac{}{a_m}\pfrac{}{b_n}) f(\vec x+\vec a) g(\vec x + \vec b)|_{\vec a=\vec b=0}
\] If you don't know how the exponential of the product of partial derivatives is defined, just use the Taylor expansion for the exponential. However, as you gain some familiarity with mathematics, you should be able to deal with the exponential "more directly", using many of its mathematical properties, and avoid the Taylor expansion at all times.
Because \(\partial/\partial a_m\) differentiates with respect to the coordinates that the function on the left side from the star-product depends on, \(\partial / \partial a_m\) is often written as \(\overleftarrow{\partial}_m\). For a similar reason, \(\overrightarrow{\partial}_n\equiv \partial / \partial b_n\). So you often find partial derivatives with arrows above them (in both directions); this notation is rather intuitive and self-explanatory.
I have introduced an antisymmetric object \(i\theta_{mn}=[x_m,x_n]\) derived from the commutators of coordinates we want to emulate. This new notation made the formalism general enough so that it may be used for higher-dimensional manifolds and not just two-dimensional ones. However, note that the number of coordinates has to be even if all the coordinates are supposed to be actively involved in the nonzero commutators; they're really paired into pairs.
With this explicit definition of the star-product, you should be able to calculate the star-product of the monomials \(x^a p^b\) and \(x^c p^d\) and verify that the star-product is linked to the operator product via the function-operator dictionary. However, it may be easier to work with "plane wave" functions \(\exp(i\alpha x + i \beta p)\) instead of the monomials. Everything about the star-product is then included in its bilinearity and its action on the two plane waves\[
\exp(i\alpha x) * \exp(i\beta p) = e^{-i\alpha\beta\theta_{12}/2} \exp(i\alpha x+i\beta p),\\
\exp(i\beta p) * \exp(i\alpha x) = e^{+i\alpha\beta\theta_{12}/2} \exp(i\alpha x+i\beta p).
\] The functions of only \(x\) star-commute with each other; the same comment holds for functions of \(p\) and nothing else (I am sometimes using \(x,y\), sometimes \(x,p\), sometimes \(x_1,x_2\) for the coordinates, but it's always meant to be the same thing; also, \(\theta\equiv\theta_{12}\)). However, the displayed equations above show that the star-product isn't commutative due to the reversed sign in the \(c\)-number-valued exponent on the right hand side.
Feynman rules and softer UV behavior
The star-product was defined using an exponential of \(i\) times a product of partial derivatives. If the functions are plane waves – i.e. energy-momentum eigenstates, if we think about plane waves in quantum field theory – the partial derivatives simply become the energy-momentum. It means that in the momentum representation, the star-product differs from the ordinary product by a simple extra factor of the form\[
\exp( \frac{i \theta_{mn}}{2} p_m q_n )
\] where \(p_m\), \(q_n\) are some momentum vectors of propagators that end in a given vertex. Note that the interaction vertex is exactly the place where the star-product shows its muscles. The Feynman rules mean that the chaotically oscillating function above is added to the integrand. This fact actually makes the integrals corresponding to the Feynman diagrams more convergent than before: the leading divergence is cancelled by the oscillating phase.
(Here, we construct the Feynman diagrams in such a way that a single interaction term with the star-product yields a single vertex. We could also expand the exponential of the derivatives and produce infinitely many, increasingly irrelevant vertices out of the Taylor expansion of the exponential. But it would be a cumbersome thing to do: we would treat the noncommutative field theories as generic theories with infinitely many higher-derivative corrections i.e. generic non-local theories and we would fail to use the special properties of noncommutative geometries, properties that make it a rather conservative generalization of ordinary commutative geometries.)
OK, let me finally say what we want to do. We want to study theories such as\[
\LL = \frac 12 \partial_\mu \phi * \partial^\mu \phi - \frac A2 \phi * \phi + \frac B3 \phi*\phi*\phi.
\] I've shifted \(\phi\) to get rid of the first-order term; the absolute, zeroth-order term is a vacuum energy term that is inconsequential in all non-gravitational theories, of course. So the action above may be viewed as the most general action for a scalar field with at most cubic interactions. It may be shown that the bilinear terms – those with a single star only, such as the "noncommutative" Klein-Gordon kinetic term above – actually may be replaced by the ordinary products. This part of the action isn't affected by the star-deformation which means that the propagators are unaffected, too.
The Feynman rules for this scalar theory with a cubic interaction – and indeed, there's nothing that would prevent us from including Dirac or gauge fields – are almost the same ones as the Feynman rules for the same theory with the commuting star-product. The only novelty is the extra phase of the type indicated above that is associated with each vertex. I don't want to discuss the precise formulae here; you need to study and calculate those things yourself if you want to become a real practitioner if not a complex one.
Cool new solitons
In quantum field theories on commutative geometries, it would be rather stupid to drop the kinetic terms in the search for solutions to the equations of motion. By doing so, we would only find the minima of the potential with no interesting dependence on the spacetime.
However, the star-product does contain some derivatives so we may actually play with a very interesting theory in which the potential terms dominate and the kinetic terms are negligible. In effect, this hierarchy appears if we study the "infinite noncommutativity limit", i.e. exactly the opposite limit than the limit that gives us the commutative geometry. The equation of motion resulting from the Lagrangian above, in the limit in which the kinetic terms may be dropped, is something like \[
A\phi = B \phi*\phi.
\] Up to the star-product appearing in every term, the structure of the equation reproduces the equation \(V'(\phi)=0\), i.e. the equation that is classically solved by the stationary points of the potential. But up to a normalization or, equivalently, for \(A=B=1\), this is nothing else than an equation for a projection operator (if you realize that the star-product is just a secret way to write the product of operators in quantum mechanics). And such an equation has beautiful solutions that project you onto \(1\)-dimensional or \(k\)-dimensional Hilbert spaces corresponding to \(1\) or \(k\) "cells" in the phase space.
The projection operator arises as the special case of an operator \(\widehat{Op}(\phi)\) whose eigenvalues are \(0\) and \(1\). More generally, i.e. for a more general potential that isn't necessarily cubic, there may be several allowed eigenvalues that are nothing else than the solutions of \(V'(\phi)=0\), i.e. the list of stationary points of the potential.
You may effectively imagine that the two-dimensional spatial plane in this Klein-Gordon theory is divided to "cells" just like the usual phase space in quantum mechanics. (Recall that a single cell in the simple phase space has the area equal to \(h=2\pi\hbar\).) And each "cell" may be told to live in a particular stationary point of the classical potential – without any "smooth transitions" between these points! This cool basic insight was made in the 2000 pioneering paper by Gopakumar, Minwalla, and Strominger.
They have also showed that if you do include the kinetic terms but with a small coefficient, so that they may be viewed as a small perturbation, the projection operators on a harmonic oscillator ground state (or several states) are preferred. Without the kinetic terms, the projection operators on any states and subspaces would be equally good but this ambiguity is lifted if you include the kinetic terms by the methods generalizing "degenerate perturbation theory" in quantum mechanics.
Field theories on noncommutative geometries have lots of other interesting properties, including the UV-IR mixing (a surprising relationship between dynamics at short distances and dynamics at long distances) but I can't tell you everything in a blog entry.
Getting noncommutative field theories from open strings in string theory
In 1999, Seiberg and Witten wrote one of the most famous papers by this couple (aside from their analyses of the \(\NNN=2\) gauge theories in \(d=4\) published in the early and mid 1990s). They showed that noncommutative geometry isn't just nice and natural; it is actually included in some of the simplest vacua of string theory.
In most cases, when we want to deduce the quantum field theories that approximate string theory, we take the limit of distances much longer than the string scale. The massive string excitations may be neglected and we get back to ordinary commutative point-like-particle field theories.
However, if you study the low-energy dynamics of open string modes (note that open strings have to be attached to a D-brane), you may also include a \(B\)-field, a two-form potential field that arises as the "antisymmetric part of the metric tensor" out of closed strings, if you allow me an extravagant description, and which produces a three-form field strength. The type II fundamental strings themselves are electrically charged under this new generalization of an electric field with an extra Lorentz index; the NS5-branes are the dual magnetic objects (that are magnetically charged under the same field).
A surprising thing is that this \(B\)-field which superficially looks like just another point-like-particle local field (one that arises from the quantization of closed strings, much like the graviton) actually has far-reaching consequences for the open string dynamics: it makes the quantum field theories for the open strings noncommutative!
This embedding of noncommutative geometry into string theory isn't just a religious procedure that gives us another reason to worship noncommutative geometry. Instead, it has very tangible consequences, something that string theory is famous for. For example, Seiberg and Witten have found an interesting equivalence between some noncommutative gauge theories and some commutative but non-Abelian ones, and so on. This equivalence depends on the fact that within the D-brane, the \(B\)-field (coming from closed strings) may be replaced by a magnetic field (coming from open strings); only their sum matters for the open string dynamics (and only the sum is gauge-invariant).
Lower-dimensional D-branes from higher-dimensional ones
Because of this equivalence, the noncommutativity may be produced from an ordinary magnetic field. There's a funny thing about D-branes. If you consider a D2-brane with several units of a magnetic flux, these units will behave just like several D0-branes! Similarly, you may produce D\((p-2)\)-branes dissolved in D\(p\)-branes by the magnetic flux. If you also consider things like \(F\wedge F\wedge \cdots \wedge F\) instead of just the magnetic flux \(F\), you may reduce the dimensionality of the branes by any even number. For example, a single instanton inside a D\(p\)-brane carries the same (Ramond-Ramond) charges as a D\((p-4)\)-brane. The instanton configuration of the D-brane's gauge field literally is a lower-dimensional D-brane, one that has been dissolved within the higher-dimensional D-brane so that its size (i.e. the characteristic size of the instanton) may be changed.
With an appropriate magnetic flux, you may produce lower-dimensional branes out of higher-dimensional ones. And vice versa: you may interpret higher-dimensional branes as "composites" of many lower-dimensional ones. However, it's no "discrete LEGO building"; the noncommutative geometry and all of its functional, inherently continuous defining relations are totally vital for such relationships.
In fact, we may finally understand where the counting of \(N\) units of area of the "fuzzy torus" and "fuzzy sphere" membranes in Matrix theory came from. The same objects appear as D2-branes in type IIA string theory and they may be decomposed to many D0-branes. Each "cell of the phase space" that is identified with the D2-brane carries a single unit D0-brane charge. Let me also say that noncommutative geometry has played an important role in string field theory – unstable D-branes of lower dimensions may be obtained as noncommutative solitons living within higher-dimensional unstable D-branes.
One of the general morals of the story is that noncommutative geometry – and string theory in which the noncommutative geometry is contained – allows us a more fuzzy and more flexible perspective on the number of dimensions in the spacetime. If you have a lower-dimensional theory with fields that transform as large enough matrices etc., you may end up with physically indistinguishable dynamics from the situation in which you have just a few fields (and no large matrices) in a higher-dimensional space.
Deconstruction is yet another inequivalent example of the "emergent character" of dimensions in string theory (and in clever quantum field theories it merges into a cool family).
Noncommutative geometry for open-string fields etc. is rather well understood. It has certain features – and it violates certain intuitive expectations – in similar ways as ways we expect in quantum gravity. Just to be sure, there was no gravity in the Klein-Gordon or gauge theories with the star-product above. But the spacetime underlying quantum gravity has at least a comparable degree of fuzziness as the spacetime in noncommutative theories. However, it's not an isomorphism so we must be cautious. In many respects, noncommutative geometries defined by the star-products and the "fuzzy spacetime" in quantum gravity differ. In particular, higher-than-two-dimensional noncommutative geometries inevitably break the rotational symmetry. On the other hand, quantum gravity may (and usually does) perfectly preserve the rotational (and Lorentz) symmetry. Also, noncommutative geometry doesn't respect any holography; it's still "volume-extensive", unlike quantum gravity.
Another broader lesson from this story is that it is often useful to consider "the opposite limits" from those we know very well. We are often consumed by some prejudices, expectations, and inequalities (forcing us to spend our lives in various "limits") but it is often useful to "think different" and investigate the consequences of the opposite inequalities from those that have been studied in hundreds of papers. Once we acquire the experience from various mutually exclusive limits, we often learn a lot about the general theory and the situations "in the middle", too.
Finally, if a theory (e.g. a quantum field theory with a star-product) looks so natural that we may be surprised why it's not included in a broader "confederation" of interesting theories such as string theory, the explanation may be and often is that the "confederate theory" actually does contain these natural theories. We must just look more carefully.
And that's the memo.
The fuzzy sphere and the fuzzy torus – two of the allowed shapes that membranes in Matrix theory may possess – were the simplest examples of "noncommutative geometries". What does it mean? What is noncommutative about them?
If you have an ordinary shape, space, or a manifold, the functions \(f(x^\mu)\) and \(g(x^\mu)\) defined on these shapes commute with each other; it means that their ordering in the product doesn't matter,\[
f(x^\mu) g(x^\mu) = g(x^\mu) f(x^\mu).
\] After all, they are just numbers at each point. For noncommutative geometries, \(=\) has to be replaced by \(\neq\); the identity above no longer holds. We have already seen that in the case of the fuzzy torus and the fuzzy sphere. Because functions are at least locally written as functions of coordinates, the noncommutative property of the noncommutative geometry effectively means that the coordinates don't commute with each other. We're going to understand this statement in the simplest possible context, the case of 2 coordinates on a plane.
In the case of the fuzzy torus, we emulated the functions \(\exp(i\sigma_1)\) and \(\exp(i\sigma_2)\) by large enough matrices \(U,V\), the clock and shift operators. They satisfied\[
UV = VU \cdot \exp(2\pi i / N)
\] where \(N\) is large and it is the size of the matrices \(U,V\). If \(N\) is truly large, we may write the large matrices \(X^i\), nine of them, as some functions of \(U,V\) and these matrices rather faithfully encode functions \(x^i(\sigma_1,\sigma_2)\) on a large torus that is "almost commutative".
What is the "local geometry" of this torus? Can't we make it infinitely large? Yes, we can.
As I have already mentioned, the property above – that \(U,V\) nearly commute with each other – may be rewritten as\[
UV - VU = (\exp(2\pi i / N) - 1) VU \sim \frac{2\pi i}{N} VU
\] which is small: it scales like \(1/N\). Now, try to understand how these things behave in the fuzzy vicinity of the point \((\sigma_1,\sigma_2)=(0,0)\) of the torus, a randomly chosen point. Near that point, our identification reduces to\[
U\sim 1+i\sigma_1,\quad V\sim 1+i\sigma_2.
\] I have simply Taylor-expanded the exponentials defining \(U,V\) in our dictionary. So the commutator relationship for \(U,V\) says (because \(1\) commutes with everything)\[
[i\sigma_1,i\sigma_2] \sim \frac{2\pi i}{N} VU \sim \frac{2\pi i}{N}.
\] In other words,\[
[\sigma_1\sqrt{N},\sigma_2\sqrt{N}] = -2\pi i.
\] The sign came from the two \(i\) factors; I have rescaled the \(\sigma_{1,2}\) coordinates by \(\sqrt{N}\) because it's natural to expect that the size \(N\) matrices will be able to describe a membrane whose "number of cells" (a few more comments about it will appear at the end of this article) i.e. area scales like \(N\) which means that the radius of the sphere naturally scales with \(\sqrt{N}\).
At any rate, we see that near the chosen point of the large torus, the commutator of the two coordinates equals an imaginary \(c\)-number. There's an obvious analogy with the defining commutator behind the quantum mechanical uncertainty principle,\[
[x,p]=i\hbar.
\] It's not just an analogy; it's an exact mathematical isomorphism! (However, you must realize that the physical interpretation is different: field theories on noncommutative geometries may be classical, lacking any fancy quantum features.) Needless to say, I could derive the very same thing from the fuzzy sphere, too. Consider the neighborhood of the North pole of the unit sphere, i.e. points with \(z\sim 1\). The two-dimensional neighborhood is described by the coordinates \(x,y\) which we represented, up to some coefficients, by the matrices for \(J_x\) and \(J_y\) in a large, \(N\)-dimensional irreducible representation of \(SU(2)\). But because\[
[J_x,J_y] = i\hbar J_z
\] and because \(J_z\) is approximately equal to \(J\), a constant, near the Northern pole, we again learn that \([x,y]\) is equal to \(i\) times a constant. Various coordinates may be scaled in such a way that the constants agree in all the situations. We're led to consider two-dimensional geometry (the large, flat limit of all the special cases) whose coordinates obey\[
[x,y] = i\theta.
\] The commutator is nonzero; that's why we deal with a noncommutative geometry. It's the same commutator as the commutator \([x,p]=i\hbar\) defining the phase space in quantum mechanics; I have used a more general constant \(\theta\) instead of \(\hbar\) but the mathematics is clearly isomorphic. You should note that in noncommutative geometry, the constant \(\theta\) in the commutator has units of \({\rm Length}^2\) so it defines a length scale. At distances much longer than \(\theta\) i.e. \(L\gg \sqrt{\theta}\), the commutator proportional to \(\theta\) may be neglected relatively to the (squared) distances and the geometry reduces to a commutative one. However, at distances comparable to and shorter than \(\sqrt{\theta}\), the noncommutativity matters a great deal.
Can we define classical field theories and quantum field theories whose fields don't live on the ordinary geometries but that live on this noncommutative plane?
Theories on the noncommutative plane
Of course, we can. We will ultimately see that such theories may be defined by the same Lagrangians of field theory that we know; the only novelty is that all the pointwise products of fields in the Lagrangian, \(f(x,y)g(x,y)\), have to be replaced with a new kind of a product which is no longer pointwise and no longer commutative, the star-product \(f(x,y)*g(x,y)\) – well, the particular explicit type of the star-product we will play with is known as the Moyal product.
This star-product is nothing else than an operation between two ordinary functions that perfectly emulates the non-commutative multiplication of operators in quantum mechanics.
But before we get there, it may be a good idea to learn something about the space of linear operators that act on the Hilbert space of a particle moving on a line, the Hilbert space on which we know the operators \(\hat x\) and \(\hat p\), the most widespread undergraduate model of one-dimensional quantum mechanics. How can we classify all linear operators on this Hilbert space?
Well, we may choose a basis of this Hilbert space and define an operator \(\hat M\) by its matrix elements\[
M_{ij} = \bra{\psi_i} \hat M \ket{\psi_j}.
\] We may either choose a discrete (infinite but countable) basis, e.g. the eigenstates of the harmonic oscillator Hamiltonian; or we may choose a continuous basis in which case \(M_{ij}\) should really be written as \(M(i,j)\) because the arguments are continuous. But we don't really want to deal with the harmonic oscillator or any other ad hoc discrete or continuous basis; we would prefer to describe all allowed operators as some kinds of functions of \(\hat x,\hat p\). How can it be done? How many such linearly independent operators are there?
If \(\hat x,\hat p\) commuted with one another, it would look like all possible operators we may construct are simply functions of \(\hat x,\hat p\). But they don't commute so the concept of a function is more subtle. For example, the product is a simple function of two arguments but if the two arguments don't commute with one another, the definition of the function based on the word "product" is ambiguous because it doesn't tell us how to order the factors.
However, this problem is very small, from an appropriate point of view. Consider all polynomial operators you may have. They're linear combinations of the following monomials:\[
1,\, x,\, p,\, xp,\, px,\, xxx,\, xxp,\, \dots
\] I have suppressed hats and let's forget about them. Are all these monomials independent? Well, the first four operators are surely independent of each other. However, the fifth one, \(px\), is almost the same thing as \(xp\); they only differ by \(i\hbar\) which is a multiple of the operator \(1\) that has already appeared in the list. So if you organize the basis of the operators in this way, only one ordering should be included as the quantum representative of any classical product. All the other orderings are linearly dependent; they differ from your chosen ordering by terms that have already been included before.
(The previous terms have been processed in the same way so we may also achieve all orderings of those simpler guys.)
If you think about the previous paragraph, it actually explains why you may represent generic and "smooth enough" operators on the 1D quantum mechanical Hilbert space by functions of \(x\) and \(p\). All we need to make this idea specific is to define a particular dictionary between operators and functions. Let's start with the map assigning an operator to a \(c\)-number-valued function; the inverse function will be fully determined immediately as inverse functions often are.
The right map is given by the symmetric ordering,\[
\widehat{Op}(x^m p^{n-m}) = \frac{1}{n!} \sum_{\pi\in S_n}^{n!\text{ permutations}} \pi [ \hat x\hat x\cdots \hat p\hat p\hat p ].
\] We're averaging all possible permutations \(\pi\) of the linear factors \(\hat x\) and \(\hat p\). For a second, I restored the hats to emphasize that there are no hats in the argument of the left hand side but there are hats on the right hand side. Because of my previous arguments, this dictionary is enough to express any operator as \(Op\) of a "classical" function. Note that if the function is real, the operator will be Hermitian.
Now, we would like to define a star-product of ordinary functions so that this product mimics the noncommutative product of the corresponding operators; but we don't have to translate from functions to operators and back. In order words, we want to define the operation \(*\) such that\[
\widehat{Op} (f(x,p)*g(x,p)) = \widehat{Op}(f(x,p))\cdot \widehat{Op}(g(x,p)).
\] If we multiply two functions by the star-product and translate the result into an operator, it's the same thing as if we first translate the two functions into operators and then we multiply these two operators by the regular noncommutative "dot" product (i.e. composition or matrix product, if we choose a basis) of these operators.
A funny thing is that given this general definition of the star-product and with our convenient "symmetric ordering" dictionary between functions and operators, we may write down the formula for the star-product explicitly. The result turns out to be\[
f(\vec x)*g(\vec x) =\qquad\\ \qquad = \exp(\frac{i}{2}\theta_{mn}\pfrac{}{a_m}\pfrac{}{b_n}) f(\vec x+\vec a) g(\vec x + \vec b)|_{\vec a=\vec b=0}
\] If you don't know how the exponential of the product of partial derivatives is defined, just use the Taylor expansion for the exponential. However, as you gain some familiarity with mathematics, you should be able to deal with the exponential "more directly", using many of its mathematical properties, and avoid the Taylor expansion at all times.
Because \(\partial/\partial a_m\) differentiates with respect to the coordinates that the function on the left side from the star-product depends on, \(\partial / \partial a_m\) is often written as \(\overleftarrow{\partial}_m\). For a similar reason, \(\overrightarrow{\partial}_n\equiv \partial / \partial b_n\). So you often find partial derivatives with arrows above them (in both directions); this notation is rather intuitive and self-explanatory.
I have introduced an antisymmetric object \(i\theta_{mn}=[x_m,x_n]\) derived from the commutators of coordinates we want to emulate. This new notation made the formalism general enough so that it may be used for higher-dimensional manifolds and not just two-dimensional ones. However, note that the number of coordinates has to be even if all the coordinates are supposed to be actively involved in the nonzero commutators; they're really paired into pairs.
With this explicit definition of the star-product, you should be able to calculate the star-product of the monomials \(x^a p^b\) and \(x^c p^d\) and verify that the star-product is linked to the operator product via the function-operator dictionary. However, it may be easier to work with "plane wave" functions \(\exp(i\alpha x + i \beta p)\) instead of the monomials. Everything about the star-product is then included in its bilinearity and its action on the two plane waves\[
\exp(i\alpha x) * \exp(i\beta p) = e^{-i\alpha\beta\theta_{12}/2} \exp(i\alpha x+i\beta p),\\
\exp(i\beta p) * \exp(i\alpha x) = e^{+i\alpha\beta\theta_{12}/2} \exp(i\alpha x+i\beta p).
\] The functions of only \(x\) star-commute with each other; the same comment holds for functions of \(p\) and nothing else (I am sometimes using \(x,y\), sometimes \(x,p\), sometimes \(x_1,x_2\) for the coordinates, but it's always meant to be the same thing; also, \(\theta\equiv\theta_{12}\)). However, the displayed equations above show that the star-product isn't commutative due to the reversed sign in the \(c\)-number-valued exponent on the right hand side.
Feynman rules and softer UV behavior
The star-product was defined using an exponential of \(i\) times a product of partial derivatives. If the functions are plane waves – i.e. energy-momentum eigenstates, if we think about plane waves in quantum field theory – the partial derivatives simply become the energy-momentum. It means that in the momentum representation, the star-product differs from the ordinary product by a simple extra factor of the form\[
\exp( \frac{i \theta_{mn}}{2} p_m q_n )
\] where \(p_m\), \(q_n\) are some momentum vectors of propagators that end in a given vertex. Note that the interaction vertex is exactly the place where the star-product shows its muscles. The Feynman rules mean that the chaotically oscillating function above is added to the integrand. This fact actually makes the integrals corresponding to the Feynman diagrams more convergent than before: the leading divergence is cancelled by the oscillating phase.
(Here, we construct the Feynman diagrams in such a way that a single interaction term with the star-product yields a single vertex. We could also expand the exponential of the derivatives and produce infinitely many, increasingly irrelevant vertices out of the Taylor expansion of the exponential. But it would be a cumbersome thing to do: we would treat the noncommutative field theories as generic theories with infinitely many higher-derivative corrections i.e. generic non-local theories and we would fail to use the special properties of noncommutative geometries, properties that make it a rather conservative generalization of ordinary commutative geometries.)
OK, let me finally say what we want to do. We want to study theories such as\[
\LL = \frac 12 \partial_\mu \phi * \partial^\mu \phi - \frac A2 \phi * \phi + \frac B3 \phi*\phi*\phi.
\] I've shifted \(\phi\) to get rid of the first-order term; the absolute, zeroth-order term is a vacuum energy term that is inconsequential in all non-gravitational theories, of course. So the action above may be viewed as the most general action for a scalar field with at most cubic interactions. It may be shown that the bilinear terms – those with a single star only, such as the "noncommutative" Klein-Gordon kinetic term above – actually may be replaced by the ordinary products. This part of the action isn't affected by the star-deformation which means that the propagators are unaffected, too.
The Feynman rules for this scalar theory with a cubic interaction – and indeed, there's nothing that would prevent us from including Dirac or gauge fields – are almost the same ones as the Feynman rules for the same theory with the commuting star-product. The only novelty is the extra phase of the type indicated above that is associated with each vertex. I don't want to discuss the precise formulae here; you need to study and calculate those things yourself if you want to become a real practitioner if not a complex one.
Cool new solitons
In quantum field theories on commutative geometries, it would be rather stupid to drop the kinetic terms in the search for solutions to the equations of motion. By doing so, we would only find the minima of the potential with no interesting dependence on the spacetime.
However, the star-product does contain some derivatives so we may actually play with a very interesting theory in which the potential terms dominate and the kinetic terms are negligible. In effect, this hierarchy appears if we study the "infinite noncommutativity limit", i.e. exactly the opposite limit than the limit that gives us the commutative geometry. The equation of motion resulting from the Lagrangian above, in the limit in which the kinetic terms may be dropped, is something like \[
A\phi = B \phi*\phi.
\] Up to the star-product appearing in every term, the structure of the equation reproduces the equation \(V'(\phi)=0\), i.e. the equation that is classically solved by the stationary points of the potential. But up to a normalization or, equivalently, for \(A=B=1\), this is nothing else than an equation for a projection operator (if you realize that the star-product is just a secret way to write the product of operators in quantum mechanics). And such an equation has beautiful solutions that project you onto \(1\)-dimensional or \(k\)-dimensional Hilbert spaces corresponding to \(1\) or \(k\) "cells" in the phase space.
The projection operator arises as the special case of an operator \(\widehat{Op}(\phi)\) whose eigenvalues are \(0\) and \(1\). More generally, i.e. for a more general potential that isn't necessarily cubic, there may be several allowed eigenvalues that are nothing else than the solutions of \(V'(\phi)=0\), i.e. the list of stationary points of the potential.
You may effectively imagine that the two-dimensional spatial plane in this Klein-Gordon theory is divided to "cells" just like the usual phase space in quantum mechanics. (Recall that a single cell in the simple phase space has the area equal to \(h=2\pi\hbar\).) And each "cell" may be told to live in a particular stationary point of the classical potential – without any "smooth transitions" between these points! This cool basic insight was made in the 2000 pioneering paper by Gopakumar, Minwalla, and Strominger.
They have also showed that if you do include the kinetic terms but with a small coefficient, so that they may be viewed as a small perturbation, the projection operators on a harmonic oscillator ground state (or several states) are preferred. Without the kinetic terms, the projection operators on any states and subspaces would be equally good but this ambiguity is lifted if you include the kinetic terms by the methods generalizing "degenerate perturbation theory" in quantum mechanics.
Field theories on noncommutative geometries have lots of other interesting properties, including the UV-IR mixing (a surprising relationship between dynamics at short distances and dynamics at long distances) but I can't tell you everything in a blog entry.
Getting noncommutative field theories from open strings in string theory
In 1999, Seiberg and Witten wrote one of the most famous papers by this couple (aside from their analyses of the \(\NNN=2\) gauge theories in \(d=4\) published in the early and mid 1990s). They showed that noncommutative geometry isn't just nice and natural; it is actually included in some of the simplest vacua of string theory.
In most cases, when we want to deduce the quantum field theories that approximate string theory, we take the limit of distances much longer than the string scale. The massive string excitations may be neglected and we get back to ordinary commutative point-like-particle field theories.
However, if you study the low-energy dynamics of open string modes (note that open strings have to be attached to a D-brane), you may also include a \(B\)-field, a two-form potential field that arises as the "antisymmetric part of the metric tensor" out of closed strings, if you allow me an extravagant description, and which produces a three-form field strength. The type II fundamental strings themselves are electrically charged under this new generalization of an electric field with an extra Lorentz index; the NS5-branes are the dual magnetic objects (that are magnetically charged under the same field).
A surprising thing is that this \(B\)-field which superficially looks like just another point-like-particle local field (one that arises from the quantization of closed strings, much like the graviton) actually has far-reaching consequences for the open string dynamics: it makes the quantum field theories for the open strings noncommutative!
This embedding of noncommutative geometry into string theory isn't just a religious procedure that gives us another reason to worship noncommutative geometry. Instead, it has very tangible consequences, something that string theory is famous for. For example, Seiberg and Witten have found an interesting equivalence between some noncommutative gauge theories and some commutative but non-Abelian ones, and so on. This equivalence depends on the fact that within the D-brane, the \(B\)-field (coming from closed strings) may be replaced by a magnetic field (coming from open strings); only their sum matters for the open string dynamics (and only the sum is gauge-invariant).
Lower-dimensional D-branes from higher-dimensional ones
Because of this equivalence, the noncommutativity may be produced from an ordinary magnetic field. There's a funny thing about D-branes. If you consider a D2-brane with several units of a magnetic flux, these units will behave just like several D0-branes! Similarly, you may produce D\((p-2)\)-branes dissolved in D\(p\)-branes by the magnetic flux. If you also consider things like \(F\wedge F\wedge \cdots \wedge F\) instead of just the magnetic flux \(F\), you may reduce the dimensionality of the branes by any even number. For example, a single instanton inside a D\(p\)-brane carries the same (Ramond-Ramond) charges as a D\((p-4)\)-brane. The instanton configuration of the D-brane's gauge field literally is a lower-dimensional D-brane, one that has been dissolved within the higher-dimensional D-brane so that its size (i.e. the characteristic size of the instanton) may be changed.
With an appropriate magnetic flux, you may produce lower-dimensional branes out of higher-dimensional ones. And vice versa: you may interpret higher-dimensional branes as "composites" of many lower-dimensional ones. However, it's no "discrete LEGO building"; the noncommutative geometry and all of its functional, inherently continuous defining relations are totally vital for such relationships.
In fact, we may finally understand where the counting of \(N\) units of area of the "fuzzy torus" and "fuzzy sphere" membranes in Matrix theory came from. The same objects appear as D2-branes in type IIA string theory and they may be decomposed to many D0-branes. Each "cell of the phase space" that is identified with the D2-brane carries a single unit D0-brane charge. Let me also say that noncommutative geometry has played an important role in string field theory – unstable D-branes of lower dimensions may be obtained as noncommutative solitons living within higher-dimensional unstable D-branes.
One of the general morals of the story is that noncommutative geometry – and string theory in which the noncommutative geometry is contained – allows us a more fuzzy and more flexible perspective on the number of dimensions in the spacetime. If you have a lower-dimensional theory with fields that transform as large enough matrices etc., you may end up with physically indistinguishable dynamics from the situation in which you have just a few fields (and no large matrices) in a higher-dimensional space.
Deconstruction is yet another inequivalent example of the "emergent character" of dimensions in string theory (and in clever quantum field theories it merges into a cool family).
Noncommutative geometry for open-string fields etc. is rather well understood. It has certain features – and it violates certain intuitive expectations – in similar ways as ways we expect in quantum gravity. Just to be sure, there was no gravity in the Klein-Gordon or gauge theories with the star-product above. But the spacetime underlying quantum gravity has at least a comparable degree of fuzziness as the spacetime in noncommutative theories. However, it's not an isomorphism so we must be cautious. In many respects, noncommutative geometries defined by the star-products and the "fuzzy spacetime" in quantum gravity differ. In particular, higher-than-two-dimensional noncommutative geometries inevitably break the rotational symmetry. On the other hand, quantum gravity may (and usually does) perfectly preserve the rotational (and Lorentz) symmetry. Also, noncommutative geometry doesn't respect any holography; it's still "volume-extensive", unlike quantum gravity.
Another broader lesson from this story is that it is often useful to consider "the opposite limits" from those we know very well. We are often consumed by some prejudices, expectations, and inequalities (forcing us to spend our lives in various "limits") but it is often useful to "think different" and investigate the consequences of the opposite inequalities from those that have been studied in hundreds of papers. Once we acquire the experience from various mutually exclusive limits, we often learn a lot about the general theory and the situations "in the middle", too.
Finally, if a theory (e.g. a quantum field theory with a star-product) looks so natural that we may be surprised why it's not included in a broader "confederation" of interesting theories such as string theory, the explanation may be and often is that the "confederate theory" actually does contain these natural theories. We must just look more carefully.
And that's the memo.
Noncommutative magic of the star-product
![Noncommutative magic of the star-product]()
Reviewed by DAL
on
June 11, 2012
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