## June 28, 2007

### This Week’s Finds in Mathematical Physics (Week 253)

#### Posted by John Baez

In week253, read about mysterious relations between the Standard Model, the SU(5) and SO(10) grand unified theories, the exceptional group E6, the complexified octonionic projective plane… and maybe even E8!

Here’s a lightning review of the Standard Model:

Alas, the fact that this chart looks like a square matrix seems to have no relation to any interesting physics whatsoever, except insofar as it shows that leptons and quarks come in 3 generations and the gauge bosons are something else. I know of no sense in which the $\gamma$, $g$, $W$ and $Z$ are like a “fourth generation”.

Also, this chart omits the Higgs.

But, it’s the best chart I could find when it came to simplicity, visual impact and actual information. It’s available at various places on the Fermi National Accelerator Laboratory website.

It’s possible to make a very nice chart of fermions in the SO(10) grand unified theory, and I believe such a chart can be found in Zee’s book Quantum Field Theory in a Nutshell. But, I couldn’t find a nice chart like this online. Maybe I’ll have to make one someday.

Posted at June 28, 2007 7:55 AM UTC

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### Re: This Week’s Finds in Mathematical Physics (Week 253)

Very nice summary of standard model structure and beyond!

[..] hints at a weird unification of bosons and fermions, something different from supersymmetry.

For those reading this who don’t know, one should maybe add, to avoid confusion here, that supersymmetry was never meant to make those bosons and fermions which have already been detected to be superpartners of each other. Rather, all their superpartners are hypothesized – in supersymmetric extensions of the standard model – to exist on top of that, but to have evaded detection so far due to their relatively high mass, which they are thought of as having acquired due to a “supersymmetry breaking mechanism” of one sort or another.

I wonder how all this $E_6$ magic which you mention is related to Alain Connes’ observation: that the fermions of the standard model naturally arise as

the direct sum of all inequivalent irreducible bimodules of the algebra $\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C}) \,.$

That’s another rather cute way to summarize all that information in one sentence!

Posted by: urs on June 28, 2007 2:32 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

I have a kind of hobby of trying to understand the pattern of elementary particles. I don’t count on making much progress, but at least it’s a fun way to learn more math. I’ve been meaning to think harder about Connes’ program, but haven’t gotten around to it.

So, are we saying that:

The exterior algebra $\Lambda\mathbb{C}^5$, regarded as a representation of $\mathrm{u}(1) \oplus su(2) \oplus su(3) \cong \mathrm{s}(\mathrm{u}(2) \oplus \mathrm{u}(3)) \subseteq su(5)$, is isomorphic to the direct sum of all inequivalent irreducible bimodules of $\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$

where that direct sum also becomes a representation of $\mathrm{u}(1) \oplus su(2) \oplus su(3)$ in some presumably obvious way… probably coming from the fact that it’s a left module of $\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$?

Posted by: John Baez on June 28, 2007 5:18 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

So, are we saying that: […]

My understanding is that: Yes. That’s the claim.

But I cannot really guarantee for a couple of details, which I keep forgetting since I am not working on this stuff.

For instance, I just realized that actually the algebra whose bimodules we are looking at is not exactly the one from Connes’ spectral triple $A_F := \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ but the auxiliary algebra $A_{LR} := \mathbb{C} \oplus \mathbb{H} \oplus \mathbb{H} \oplus M_3(\mathbb{C}) \,.$

See page 4 of Connes’ paper.

Then we are looking at the direct sum of all irreducible $A_{LR}$ bimodules and regard that as a $A_F$ module (and then we take three copies of that, one for each generation of fermions).

Then, on p. 5, the relation to the gauge group is described: it’s the group of unitaries in $A_F$ which have unit determinant with respect to the above action on that module.

But that’s not quite the $S(U(2)\times U(3))$ which you talked about but (proposition 2.5 on p. 5) is $U(1) \times \mathrm{SU}(2) \times \mathrm{SU}(3)$ “up to an abelian finite group”.

It is then the adjoint action of that group which gives the action of the standard model group on fermions.

Here the adjoint action is the obvious one where we remember that we have realized everything on bimodules!

Posted by: urs on June 28, 2007 7:12 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

Here are some possible hints about how to relate Connes’ work to fact that Standard Model fermions act like 10-dimensional Dirac spinors:

• Ali H. Chamseddine and Alain Connes, A dress for SM the beggar.

Abstract: The purpose of this letter is to remove the arbitrariness of the ad hoc choice of the algebra and its representation in the noncommutative approach to the Standard Model, which was begging for a conceptual explanation. We assume as before that space-time is the product of a four-dimensional manifold by a finite noncommmutative space F. The spectral action is the pure gravitational action for the product space. To remove the above arbitrariness, we classify the irreducibe geometries F consistent with imposing reality and chiral conditions on spinors, to avoid the fermion doubling problem, which amounts to have total dimension 10 (in the K-theoretic sense). It gives, almost uniquely, the Standard Model with all its details, predicting the number of fermions per generation to be 16, their representations and the Higgs breaking mechanism, with very little input. The geometrical model is valid at the unification scale, and has relations connecting the gauge couplings to each other and to the Higgs coupling. This gives a prediction of the Higgs mass of around 170 GeV and a mass relation connecting the sum of the square of the masses of the fermions to the W mass square, which enables us to predict the top quark mass compatible with the measured experimental value. We thus manage to have the advantages of both SO(10) and Kaluza-Klein unification, without paying the price of plethora of Higgs fields or the infinite tower of states.

• Ali H. Chamseddine and Alain Connes, Why the Standard Model.

Abstract: The Standard Model is based on the gauge invariance principle with gauge group U(1)× SU(2)× SU(3) and suitable representations for fermions and bosons, which are begging for a conceptual understanding. We propose a purely gravitational explanation: space-time has a fine structure given as a product of a four dimensional continuum by a finite noncommutative geometry F. The raison d’etre for F is to correct the K-theoretic dimension from four to ten (modulo eight). We classify the irreducible finite noncommutative geometries of K-theoretic dimension six and show that the dimension (per generation) is a square of an integer k. Under an additional hypothesis of quaternion linearity, the geometry which reproduces the Standard Model is singled out (and one gets k=4) with the correct quantum numbers for all fields. The spectral action applied to the product M× F delivers the full Standard Model, with neutrino mixing, coupled to gravity, and makes predictions (the number of generations is still an input).

Together with Witten’s new paper linking 3d quantum gravity to the Monster group, I get the feeling that theoretical physics may not be quite so stagnant after all!

Posted by: John Baez on June 29, 2007 12:01 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

Ali H. Chamseddine and Alain Connes, A dress for SM the beggar.

[…]

Ali H. Chamseddine and Alain Connes, Why the Standard Model

Thanks for these links. I wasn’t aware of these papers.

Now I did look at them. Unfortunately, I feel less enlightened by them then I was hoping to. But that’s my fault, not theirs.

In need to think about all this a little more.

One thing that I imagine would help me is if I could see clearer on how Connes’ setup would “decompactify” to an ordinary geometry.

Let’s see, just some simple observations.

Assuming the algebra to be the tensor product $A = C^\infty(X) \times A_F$ is of course simply assuming (in the case of ordinary geometry) a cartesian product space $X \times Y \,.$

Next, assuming the representation to be $H = L^2(X,S) \otimes H_f$ is assuming (correct me if I am wrong), in terms of ordinary geometry, that we are looking at a vector bundle over $X \times Y$ $V \to X \times Y$ which is the tensor product of the spinor bundle $S \to X$ over $X$ ($X$ is assumed to be spin and Riemannian) with some other vector bundle $E \to Y$ over $Y$: $V = \pi_X^* S \otimes \pi_Y^* E \,.$

Sorry for wasting bandwidth with these trivialities.

But now let’s see. How should we think of $E$, conceptually?

I have the following general suspicion, which I don’t know of if it’s any good or not:

maybe a good way to think of the noncommutative algebras appearing in spectral triples is as category algebras. Because, if so, we’d reobtain a nice “geometrical” picture again from these algebras, which would nicely connect with our general idea, expounded on in a couple of discussions here, that it should be helpful to think of space(time) as a category.

So, for instance, the matrix algebra $M_3(\mathbb{C})$ is just the category algebra of the groupoid $\mathrm{Codisc}(\{a,b,c\})$ which has three objects and precisely one morphism from every object to every other.

So, I feel like thinking of the algebra $M_3(\mathbb{C})$ as the algebra of functions on the triangle $\array{ && b \\ & \swarrow\nearrow && \searrow \nwarrow \\ a && \stackrel{\leftarrow}{ \rightarrow}&& c } \,.$

Maybe that’s not justified here in this context. But maybe it is.

From this point of view, we may think of a module of $M_3(\mathbb{C})$ as a vector bundle with connection. In fact, it looks like a flat vector bundle with connection over a latticized disk $D^2$, namely on the standard 2-simplex.

(In general, I think, reps of $M_n(\mathbb{C})$ would, in this logic, correspond to flat vector bundles with connection on the standard $(n-1)$-simplex.)

Of course $\mathbb{C}$ itself is the category algebra of the trivial one object one morphism category.

Is there a category such that its category algebra is the algebra of quaternions, $\mathbb{H}$? Hm, there cannot be, right?

Oh, I have to run.

Posted by: urs on June 29, 2007 5:28 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

Urs wrote:

In general, I think, reps of $M_n(\mathbb{C})$ would, in this logic, correspond to flat vector bundles with connection on the standard $(n-1)$-simplex.

Yes, that sounds right. These three algebras are isomorphic:

• The algebra of $n \times n$ complex matrices, $M_n(\mathbb{C})$.
• The category algebra of the codiscrete category on $n$ objects.
• The category algebra of the fundamental groupoid of the $(n-1)$-simplex.

So, these three sentences mean the same thing:

• $M_n(\mathbb{C})$ is Morita equivalent to $\mathbb{C}$.
• The codiscrete category on $n$ objects is equivalent to the terminal category.
• The $n$-simplex is contractible.

To get weirder algebras, like the quaternions, into the game I think we need to do something sneaky… like maybe think about Auslander–Reiten quivers. The category algebra trick you’re discussing gives a way to get algebras from quivers. The ‘Auslander–Reiten quiver’ idea is a way to get quivers from algebras. It’s all part of some big story…

Posted by: John Baez on June 30, 2007 8:16 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

we need to do something sneaky

Okay. Or something “more conceptual”. What looks sneaky now might be a consequence of general nonsense once we better understand what’s going on. here is a remark on that.

I consider myself quite happy to have found in Yan Soibelman a big shot who is actively pursuing the kind of philosophy which I have grown fond of for quite a while now. Namely (it’s really a vacuous tautology in a way, but some vacuous tautologies need to be emphasized nevertheless):

A spectral triple encodes the background geometry for a (super)particle and arises as a certain limit of 2-spectral triples which encode the backgrounds of (super)-2-particles.

(That’s of course to some extent my paraphrase.)

Yan Soibelman also stresses another aspect of this:

In as far as the algebra appearing in the spectral triple is noncommutative, it is so due to its nature as the limit of the algebra of states of a 2-particle. Here is why I am emphasizing this:

the algebra of the $n$-particle arises from pull-push along correspondence categories which encode the trivalent interaction vertex.

In its simplest form, this vertex for the 2-particle is the following category $a \to b \to c \,.$ This is naturally equipped with 3 maps from the 2-particle $x \to y$ into it: one onto the left morphism, one onto the right, and one onto the composite of both.

To get the interaction of the 2-particle, we pull states back along the first two of these maps, tensor them and pull the result back along the third.

The point is that this procedure is precisely what yields the category algebra of whatever category models the target space of the 2-particle. (This I once discussed here.)

So, I think, this is in fact one way to understand conceptually why the category algebra of the category which models target space indeed should appear as the noncommutative algebra of a spectral triple describing that target space.

But now consider this: the above interaction vertex is very delicate, in that it models the interaction of two 2-particles to take place at exactly their endpoints.

There are other way to model the interaction of two 2-particles. For instance, the idea which was originally found by Witten is given by adding to the above interaction the walking equivalence

$\array{ &&& b \\ &&& \uparrow \downarrow \\ &&& v \\ && \nearrow & & \searrow \\ & a &&&& c } \,.$ There are three obvious ways to map the 2-particle $x \to y \to z$ consisting of two halfs into this interaction vertex.

Pulling back along two such ways and then pushing forward along the remaining one yields the interaction where two 2-particles merge by summing the product of their states over all configurations in which the two 2-particles overlap with one half of their parameter space.

Depending on some of the details of how this is setup, this pull-push interaction produces – for the same category modelling target space as before – somewhat different algebras.

So this might be the right conceptual was to be “sneaky”: the algebra a certain category modelling target space gives rise to depends on how we model the interaction of our $n$-particles propagating in that space.

I’ll think about if I can find a target space category such that it reproduces Connes’s standard model algebra using the second interaction vertex mentioned above.

One issue here which one still might need to better understand is the role played by the complex numbers: do we want algebras over the field of complex numbers, or do we want the complex numbers itself come out of such a process by working just over the reals?

Posted by: urs on July 1, 2007 9:57 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

It is unclear to me whether the full simplex is relevant here or only the 1-skeleton. Similarly, does the path groupoid mean the edge path groupoid?

Btw, what is the symmetry group of the 1-skeleton with bi-directed edges?

Posted by: jim stasheff on July 3, 2007 2:12 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

I spoke of the “path groupoid” of the simplex, meaning the edge path groupoid, but I really should have said “fundamental groupoid”, meaning the quotient of the edge path groupoid where we treat homotopic paths as equal.

So, thanks for your question! I’ve used my magic superpowers to go back and rewrite history — now my comment says “fundamental groupoid” where it had said “path groupoid”. Of course this is still ambiguous until we know that we’re working with the simplex as a simplicial set, not as a topological space. So, the only paths that count are the edge paths, and we use the 2-simplices to tell when such paths are homotopic.

It is unclear to me whether the full simplex is relevant here or only the 1-skeleton.

Only the 2-skeleton is relevant: we need the 2-simplices to define the concept of homotopy between edge paths.

Btw, what is the symmetry group of the 1-skeleton with bi-directed edges?

For the $n$-simplex it’s $S_{n+1}$: all permutations of the vertices.

Posted by: John Baez on July 3, 2007 4:53 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

The splitting

e6 = so(10) ⊕ S10+ ⊕ u(1)

also hints at a weird unification of bosons and fermions, something different from supersymmetry. We’re seeing e6 as a Z/2-graded Lie algebra with so(10) ⊕ u(1) as its “bosonic” part and S10+ as its “fermionic” part. But, this is not a Lie superalgebra, just an ordinary Lie algebra with a Z/2 grading!

Wouldn’t the Coleman-Mandula theorem impede any such “unification”?

Posted by: Squark on June 28, 2007 8:58 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

Check the hypotheses of Coleman-Mandula
carefully. Some people conclude that e.g.
spin >2 particles can’t exist, but they can
mathematically if field dependent symmetries are allowed.

Posted by: jim stasheff on June 28, 2007 9:23 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

Squark wrote:

Wouldn’t the Coleman-Mandula theorem impede any such “unification”?

If you tell me what the theorem actually says — including the exact hypotheses! — I’ll answer that question.

Posted by: John Baez on June 28, 2007 9:58 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

Let me ask a different question. Is there any known example of a QFT (or a physically reasonable generalization e.g. string theory in asymptotically Minkowski spacetime) in which a bosonic symmetry doesn’t commute with the Poincare group?

Posted by: Squark on June 29, 2007 11:54 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

A slight correction: in order to exclude the Poincare group itself and the its conformal extension, require the symmetry to relate states of different spin i.e. not to commute with the square of the Pauli-Lubanski vector.

Posted by: Squark on June 29, 2007 1:17 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

Oh darn. I was going to give you a smart-aleck answer, by pointing out that the Poincaré group doesn’t commute with itself… but you just ruled that out.

Anyway, I don’t know an example of the weird thing you’re noting my $E_6$ idea would seem to require. “Seem to”, because it’s really not clear yet what physical ideas the mathematics is suggesting that we explore.

But, note that this $E_6$ idea also seems to require something even weirder: a symmetry that makes us change our viewpoint about what counts as ‘multiplication by $i$’!

$E_6$ acts as isometries of the symmetric space $(\mathbb{C} \otimes \mathbb{O})\mathrm{P}^2$. Each point of this space gives a different way of splitting $e_6$ as $so(10) \oplus S_{10}^+ \oplus u(1)$. The generator of $u(1)$ acts as multiplication by $i$ on the spinors in $S_{10}^+ \cong \mathbb{C}^{16}$. These a

So, conjugating a guy in $e_6$ by elements of the group $E_6$, we get different ways of deciding what part of this guy lives in the Lie algebra of the gauge group $so(10)$, what part lives in the spinor representation… and what part counts as ‘$i$’.

That’s pretty weird.

But, note that the spacetime symmetries are not part of this story at all, yet! The fact that $so(10)$ guys are eventually supposed to be spin-1 particles while the $S_{10}^+$ guys are supposed to be spin-1/2 particles is completely unaccounted for, so far.

So, the whole idea is only half-baked, so far — I’m just taking a strange mathematical fact and noticing that it has some relation to particle physics, and hoping maybe it could lead somewhere… but I really don’t know where, yet.

Posted by: John Baez on June 30, 2007 9:20 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

I’m just taking a strange mathematical fact and noticing that it has some relation to particle physics, and hoping maybe it could lead somewhere

Careful.. Someone did that with the beta function and now look where it’s got us.

Posted by: John Armstrong on June 30, 2007 3:16 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

John Baez wrote in part:

This gives me an excuse to say a word or two about symmetric spaces… a topic that deserves a whole week of its own!

One should know that there are a few concepts called “symmetric space”:

* this one;

* a topological space satisfying the $R_0$ regularity/separation axiom;

* another concept that I forget.

So it is an overloaded term.

Euclidean spaces and spheres are the most famous examples of symmetric spaces. If an ant decides to set up residence on a sphere, any point is just as good any other. And, if sits anywhere and looks in any direction, the view is the same as the view in the opposite direction.

In fact, in these examples, the view is the same in ~every~ direction! Is there a term for this? (More on that below.)

Later John wrote in part:

This gives a specially nice sort of homogeneous space G/H, called a “symmetric space”. This is better than your average homogeneous space.

You already used “symmetric space” and explained it a bit, so it’s odd to see it in quotes here as if it’s a new term. Did you misedit?

for each point p in [a symmetric space] there’s a map from [the space] to itself called “reflection through p”, which fixes the point p and acts as -1 on the tangent space of p.

To go back to the examples of Euclidean spaces and spheres, here there is an $O(n)$-indexed family of maps from the space to itself called “rotations about p”, each of which fixes $p$ and acts on $\mathbb{R}^n$, the tangent space of $p$, as multiplication by its index in $O(n)$. (Here $n$ is the dimension of the symmetric space as a manifold; the tangent space $T_p$ of $p$ is isomorphic to $R^n$, but this is not canonical, so it’s more fair to say that the family is indexed by $O(T_p)$. And in fact, this must be a Riemannian manifold for this to make sense.) This doesn’t seem to be the same as the “Riemannian symmetric spaces”; in fact, this extension to all of $O(n)$ seems to be independent of the condition that the reflection (and rotations?) preserve the metric.

In any case, my point is that the examples of Euclidean spaces and spheres are even more symmetric than (Riemannian) symmetric spaces in general.

–Toby

Posted by: John Baez on June 29, 2007 12:36 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

Toby Bartels wrote:

You already used “symmetric space” and explained it a bit, so it’s odd to see it in quotes here as if it’s a new term. Did you misedit?

Maybe; I’ll ponder that! Thanks.

In fact, in these examples, the view is the same in ~every~ direction! Is there a term for this? (More on that below.)

“Isotropic” might be a good name, as Miguel Carrión-Álvarez mentioned. Let’s say a Riemannian manifold is “isotropic” if for every point, the group of isometries fixing this point acts transitively on the unit tangent vectors at this point. So, there’s enough symmetry to make the view in all directions the same.

Spheres and planes are misleading examples of Riemannian symmetric spaces, since they satisfy this much stronger condition. To get an example that doesn’t, take a product of two spheres — of different radii, if you want to make it really obvious. Any product of Riemannian symmetric spaces is clearly a Riemannian symmetric space. But, a product of isotropic Riemannian symmetric spaces needn’t be isotropic… as the example I just gave shows.

The octonionic projective plane is an isotropic Riemannian symmetric space, since the group of isometries fixing a point manages to act transitively on the unit tangent vectors, even though this group is only the double cover of $SO(9)$, and it’s acting on a tangent space that looks like $\mathbb{R}^16$. It’s just a marvelous fact that this $\mathbb{R}^{16}$ is the spinor representation of $Spin(9)$, and — the really marvelous part — this group acts transitively on the unit spinors.

I’m pretty sure the complexified octonionic projective plane is not isotropic. In other words, I think $Spin(10) \times \mathrm{U}(1)$ doesn’t act transitively on the unit spinors (that is, unit vectors in $\mathbb{C}^16$).

Hmm, but I’m not actually sure.

Posted by: John Baez on June 29, 2007 1:10 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

I wrote:

I’m pretty sure the complexified octonionic projective plane is not isotropic.

According to Tony Smith it’s not. He sent me an email saying:

In Spaces of Constant Curvature, Fifth Edition (Publish or Perish 1984), Joseph A. Wolf says (pages 293-294):

$M$ is called isotropic at $x$ if $I(M)_x$ is transitive on the unit sphere in $M_x$; it is isotropic if it is isotropic at every point. …

$M$ is isotropic if and only if it is two point homogeneous. …

Let $M$ be a riemannian symmetric space. Then the following conditions are equivalent.

(i) $M$ is two point homogeneous.

(ii) Either $M$ is a euclidean space or $M$ is irreducible and of rank 1.

Since $(\mathbb{C} \otimes \mathbb{O})\mathrm{P}^2 = E_6/(SO(10) \times SO(2))$ is rank 2, it is NOT isotropic.

In the quote by Wolf, I can only guess that $I(M)_x$ is the group of isometries of $M$ while $M_x$ is the tangent space of $M$ at $x$. Similarly, I guess that ‘two point homogeneous’ means that for any $D \ge 0$, the isometry group of $M$ acts transitively on the set of pairs of points in $M$ whose distance from each other is $D$.

Posted by: John Baez on July 3, 2007 6:27 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

1. isotropic is used in e.g. material science to mean the same in all directions

2. for all D ≥ 0 seems to imply it’s irrelevant

jim

Posted by: jim stasheff on July 4, 2007 7:02 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

Jim wrote:

isotropic is used in e.g. material science to mean the same in all directions.

That’s precisely what it means here. A Riemannian symmetric space $M$ is isotropic if when you sit at any point $p$, the view in every direction is the same. If I understand the above quote by Joseph Wolf correctly, he formalizes this in two ways and shows they’re equivalent:

• $M$ is isotropic if for point $p$, the group of isometries of $M$ fixing $p$ acts transitively on the unit sphere in the tangent space at $p$. (So, $M$ has enough symmetry to make every direction look just like every other.)
• $M$ is isotropic if for all $D \ge 0$, the group of isometries of $M$ acts transitively on the set of all pairs of points that have distance $D$ from each other.

Jim writes:

for all $D \geq 0$ seems to imply it’s irrelevant

No, the distance isn’t irrelevant. Item 2. says that for any fixed distance $D \ge 0$, there’s an isometry mapping any pair of points with distance $D$ to any other pair of points with distance $D$. But, there’s never an isometry mapping a pair of points that have distance 1 to a pair of points that have distance 2. Isometries preserve distance!

Posted by: John Baez on July 5, 2007 1:31 PM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

Toby wrote:

To go back to the examples of Euclidean spaces and spheres, here there is an $\mathrm{O}(n)$-indexed family of maps from the space to itself called “rotations about $p$”, each of which fixes $p$ and acts on $\mathbb{R}^n$, the tangent space of p, as multiplication by its index in $\mathrm{O}(n)$.

In reply I gave a much weaker definition of an “isotropic” Riemannian manifold, namely one where for each point p the isometries fixing this point act transitively on the unit tangent vectors.

One can imagine a stronger definition, more like yours here, where for each point $p$ the isometries fixing this point act as all possible rotations and reflections of the tangent space at $p$. In other words, the obvious homomorphism to $\mathrm{O}(n)$ is onto.

But, this definition is so strong that I bet the only examples are spheres, Euclidean spaces and hyperbolic spaces.

Posted by: John Baez on June 29, 2007 1:33 AM | Permalink | Reply to this

### Re: This Week’s Finds in Mathematical Physics (Week 253)

While in Paris, you [and M. Mellies] may want to contact Stephane Gaubert of INRIA for information on Max-Plus Algebra and graphing with Petrie Nets as variants of game theory.