August 30, 2008

Imperfections, Ambiguities and Physics

Posted by David Corfield

I mentioned Lautman’s association of, on the one hand, Descartes’ argument to the existence of a perfect being (God) from an awareness of his own imperfections with, on the other, a mathematical argument to the existence of an algebraically closed field from the inability to factor polynomials in a given field, or to the existence of a simply connected space from the inability to contract all loops in a given space.

This perfection/imperfection ‘dialectic’ involves our realising from an imperfect state that there is a perfect state, and also from the nature of the imperfections what are the attributes of the perfect state.

Now does this thought have any resonance in physics? Are there specifically physical manifestations of this phenomenon? Or to the extent that we find these, such as Doron Gepner’s Galois Groups in Rational Conformal Field Theory, should we say that everything Galoisian ‘factors’ through the mathematics?

On the other hand, if we wanted a natural language description of the mathematical phenomena, is the perfection/imperfection pairing the best way, or is Galois’ own ambiguity theory not more accurate? In which case, although ambiguity might be thought an imperfection, the commonality with the cartesian situation is lessened.

Finally, are physical manifestations better described as ambiguities rather than imperfections? Don’t I remember something on ‘defects’ in nematic crystals in one of those mathematics of gauge theory textbooks? Ah yes, Topology and Geometry for Physicists. So, does the existence of a crystal defect tell us about a perfected crystal, even if physically unrealisable? Similarly, how could we describe the Aharonov-Bohm effect?

Posted at August 30, 2008 9:40 AM UTC

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Re: Imperfections, Ambiguities and Physics

While I’m asking, how about the same set of questions for duality/reciprocity. Denis was extremely helpful in showing me what was at stake mathematically.

But where do find duality/reciprocity richly exemplified outside of mathematics? In physics, e.g., electric-magnetic duality? Again is this ‘factorable’ through mathematical duality?

Posted by: David Corfield on August 30, 2008 11:49 AM | Permalink | Reply to this

Kepler and Einstein; Re: Imperfections, Ambiguities and Physics

Citing Descartes’; argument to the existence of a perfect being is on-target, because of the strangle hold such arguments had before Kepler (an elite Astrologer whose mother was a witch) determined (from his editing of Brahe’s Rudolphine Tables) that Mars had an elliptical orbit, and so it was not true that all planmets had to have “perfect” circular orbits.

As to duality, isn’t the Equivalence of Gravitation and Intertia in GR one of the most important dualities, or am I abusing the language?

Holy Roman Emperor Rudolph was an extremely interesting man, who forced Kepler and Brahe together, and was (in his day) both the major patron of Science and the major patron of the Occult. His own family became convinced that he was insane. The inventors and scientists and Mathematicians that he funded are the Who’s Who of Europe…

Posted by: Jonathan Vos Post on August 30, 2008 5:31 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

David wrote:

But where do [we] find duality/reciprocity richly exemplified outside of mathematics? In physics, e.g., electric-magnetic duality? Again is this ‘factorable’ through mathematical duality?

The duality between electric and magnetic fields is now seen as a special case of Hodge duality. In $n$-dimensional space, the Hodge star operator maps $p$-forms to $(n-p)$-forms: it’s the differential form version of Poincaré duality. The electromagnetic field strength is a 2-form $F$ built from the electric and magnetic fields. In 4d spacetime, taking the Hodge dual $\star F$ gives a new 2-form, which can also be obtained by doing these replacements:

$E \mapsto B$ $B \mapsto -E$

This explains why the vacuum Maxwell equations remain unchanged when we perform this switcheroo.

So yeah: I’d say the duality between electric and magnetic fields factors through mathematics.

But shouldn’t all sufficiently beautiful patterns in physics ultimately factor through mathematics? I thought that was the whole point of the game.

If you want a rich line of thought about duality that hasn’t fully been absorbed into modern mathematics, I suggest the duality between yin and yang in Taoist thought.

Yin and yang underly the hexagrams of the I Ching, which helped Leibniz invent binary digits, but the real idea behind yin and yang is quite different than the Boolean ‘everything is fundamentally zeros and ones’ ontology that we often hear espoused by computer scientists.

I actually think the yin-yang symbol has more to do with the idea that everything in nature is just cycling around, following the pattern of these four trig functions:

$\frac{d}{d t} sin t = cos t$

$\frac{d}{d t} cos t = -sin t$

$\frac{d}{d t} -sin t = -cos t$

$\frac{d}{d t} -cos t = sin t$

In particular, in sinusoidal motion, when you’re at your peak, that’s when you’re accelerating downhill the fastest — and when you’re at the bottom, that’s when you’re accelerating upwards fastest! I think that’s why the big white blob in the yin-yang symbol has a little black blob in it, and vice versa. Everything contains the germ of its opposite.

Understanding this sort of thing is a good way to remain calm. I think Taoism is about maintaining a kind of equilibrium in the context of change through getting a visceral understanding of very simple, general aspects of change.

But these are just some dreamy ideas of mine: I don’t think the Chinese or anyone else ever seriously tried to ‘factor yin-yang philosophy through mathematics’.

(By the way: note that differentiating $sin t$ four times gets you back where you started, as does applying the Hodge star operator four times to a 2-form in 4d spacetime. Any good mathematician should know what underlies this ‘coincidence’.)

Posted by: John Baez on September 1, 2008 11:04 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

And electric-magnetic duality brings us (or Witten at least) back to the Langlands Program.

But shouldn’t all sufficiently beautiful patterns in physics ultimately factor through mathematics? I thought that was the whole point of the game.

So it seems. But I came to this post wondering about what might be a problem for Lautman. What we have, and what he didn’t, is the notion that his perfection/imperfection ‘idea’ is capturable by a piece of mathematics, at least to the extent that the idea manifests itself in mathematics. Since his death mathematics has learned to address the idea at a level of abstraction above its manifestations. While there was no language to capture the commonality of Galoisian field extensions and Poincaréan deck transformations, it might seem plausible to take this commonality to be something beyond mathematics. And to see another of its manifestations in Descartes’s argument for God.

We could look to another field for signs of his ‘dialectical ideas’. But it looks like we agree that rather than their appearance in physics giving us a third independent sighting, the physics and maths manifestations are sitting on top of one another. And while they are extraordinarily rich, the cartesian theological manifestation was very thin.

On the other hand, perhaps the appearance of thinness in the cartesian example is illusory. Perhaps it could reveal itself to be more Galoisian then we thought. Had the full Galoisian idea appeared in Descartes’ thought, he would have had to put into association all the hierarchy of different substructures of the complex of man’s imperfections with the hierarchical of different kinds of angelic being interposed between Man and God. Apparently, in his Yad ha-Hazakah, Yesode ha-Torah, Maimonides constructs a hierarchy of angels with ten ranks. It would not be at all surprising to me if this or some other elaborate angelogical theory of the Middle Ages could be drafted into something a little more Galoisian.

It is intriguing how Lautman is drawn irrestibly to pieces of mathematics on our side of the two cultures, where category theory happily plays. What if he had opted for the Gowers-Tao side where, according to its exponents, one expects vaguer commonality, not capturable by general theory?

Posted by: David Corfield on September 2, 2008 9:22 AM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

Pseudo-Dionysius, a Christian Neoplatonist who wrote in the late fifth or early sixth century CE, described the 9-fold hierarchy of angels, 3 hierarchies of 3 levels in De coelesti hierarchia.

Aquinas explains this organisation in the Summa Theologica, First Part, Qustion 108, Article 6.

Let us then first examine the reason for the ordering of Dionysius, in which we see, that, as said above (Article [1]), the highest hierarchy contemplates the ideas of things in God Himself; the second in the universal causes; and third in their application to particular effects. And because God is the end not only of the angelic ministrations, but also of the whole creation, it belongs to the first hierarchy to consider the end; to the middle one belongs the universal disposition of what is to be done; and to the last belongs the application of this disposition to the effect, which is the carrying out of the work; for it is clear that these three things exist in every kind of operation. So Dionysius, considering the properties of the orders as derived from their names, places in the first hierarchy those orders the names of which are taken from their relation to God, the “Seraphim,” “Cherubim,” and “Thrones”; and he places in the middle hierarchy those orders whose names denote a certain kind of common government or disposition—the “Dominations,” “Virtues,” and “Powers”; and he places in the third hierarchy the orders whose names denote the execution of the work, the “Principalities,” “Angels,” and “Archangels.”

As regards the end, three things may be considered. For firstly we consider the end; then we acquire perfect knowledge of the end; thirdly, we fix our intention on the end; of which the second is an addition to the first, and the third an addition to both. And because God is the end of creatures, as the leader is the end of an army, as the Philosopher says (Metaph. xii, Did. xi, 10); so a somewhat similar order may be seen in human affairs. For there are some who enjoy the dignity of being able with familiarity to approach the king or leader; others in addition are privileged to know his secrets; and others above these ever abide with him, in a close union. According to this similitude, we can understand the disposition in the orders of the first hierarchy; for the “Thrones” are raised up so as to be the familiar recipients of God in themselves, in the sense of knowing immediately the types of things in Himself; and this is proper to the whole of the first hierarchy. The “Cherubim” know the Divine secrets supereminently; and the “Seraphim” excel in what is the supreme excellence of all, in being united to God Himself; and all this in such a manner that the whole of this hierarchy can be called the “Thrones”; as, from what is common to all the heavenly spirits together, they are all called “Angels.”

As regards government, three things are comprised therein, the first of which is to appoint those things which are to be done, and this belongs to the “Dominations”; the second is to give the power of carrying out what is to be done, which belongs to the “Virtues”; the third is to order how what has been commanded or decided to be done can be carried out by others, which belongs to the “Powers.”

The execution of the angelic ministrations consists in announcing Divine things. Now in the execution of any action there are beginners and leaders; as in singing, the precentors; and in war, generals and officers; this belongs to the “Principalities.” There are others who simply execute what is to be done; and these are the “Angels.” Others hold a middle place; and these are the “Archangels,” as above explained.

This explanation of the orders is quite a reasonable one. For the highest in an inferior order always has affinity to the lowest in the higher order; as the lowest animals are near to the plants. Now the first order is that of the Divine Persons, which terminates in the Holy Ghost, Who is Love proceeding, with Whom the highest order of the first hierarchy has affinity, denominated as it is from the fire of love. The lowest order of the first hierarchy is that of the “Thrones,” who in their own order are akin to the “Dominations”; for the “Thrones,” according to Gregory (Hom. xxiv in Ev.), are so called “because through them God accomplishes His judgments,” since they are enlightened by Him in a manner adapted to the immediate enlightening of the second hierarchy, to which belongs the disposition of the Divine ministrations. The order of the “Powers” is akin to the order of the “Principalities”; for as it belongs to the “Powers” to impose order on those subject to them, this ordering is plainly shown at once in the name of “Principalities,” who, as presiding over the government of peoples and kingdoms (which occupies the first and principal place in the Divine ministrations), are the first in the execution thereof; “for the good of a nation is more divine than the good of one man” (Ethic. i, 2); and hence it is written, “The prince of the kingdom of the Persians resisted me” (Dan. 10:13).

Aquinas then proceeds to discuss Gregory’s slightly different hierarchy. The material for a Galoisian angelology?

Posted by: David Corfield on October 18, 2008 12:02 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

Locke’s Essay on Human Understanding, Book XVI, section 12 for reasoning about more perfect beings:

…finding in all parts of the creation, that fall under human observation, that there is a gradual connexion of one with another, without any great or discernible gaps between, in all that great variety of things we see in the world, which are so closely linked together, that, in the several ranks of beings, it is not easy to discover the bounds betwixt them; we have reason to be persuaded that, by such gentle steps, things ascend upwards in degrees of perfection. It is a hard matter to say where sensible and rational begin, and where insensible and irrational end: and who is there quick-sighted enough to determine precisely which is the lowest species of living things, and which the first of those which have no life? Things, as far as we can observe, lessen and augment, as the quantity does in a regular cone; where, though there be a manifest odds betwixt the bigness of the diameter at a remote distance, yet the difference between the upper and under, where they touch one another, is hardly discernible. The difference is exceeding great between some men and some animals: but if we will compare the understanding and abilities of some men and some brutes, we shall find so little difference, that it will be hard to say, that that of the man is either clearer or larger. Observing, I say, such gradual and gentle descents downwards in those parts of the creation that are beneath man, the rule of analogy may make it probable, that it is so also in things above us and our observation; and that there are several ranks of intelligent beings, excelling us in several degrees of perfection, ascending upwards towards the infinite perfection of the Creator, by gentle steps and differences, that are every one at no great distance from the next to it.

Thanks to Brandon Watson for this reference.

Posted by: David Corfield on April 28, 2009 4:49 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

note that differentiating $sin t$ four times gets you back where you started, as does applying the Hodge star operator four times to a 2-form in 4d spacetime. Any good mathematician should know what underlies this ‘coincidence’.

How many such cases of an operation whose fourth power is the identity have a common ‘cause’?

I’ve just dug up me asking about a possible link between Wick rotation and the Fourier transform.

Posted by: David Corfield on September 3, 2008 9:28 AM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

A nifty additional example of an operation whose fourth power is the identity, closely related to Hodge duality, is the operation of taking ‘duals’ of morphisms in a fusion category (semisimple linear monoidal category where each object has a dual). If $C$ is our fusion category, then the operation of taking duals can be seen as a contravariant op-monoidal functor

(1)$* : C \rightarrow C.$

‘Taking the dual twice’ should be something like the identity, and this is precisely what a pivotal structure is — a monoidal natural isomorphism

(2)$\gamma : id \Rightarrow **.$

For instance, and roughly speaking, in the Hodge duality situation the isomorphism $\gamma$ is precisely that minus sign.

A pivotal structure is what enables you to make “closed circles” from the cups and caps and hence form “dimensions” of objects. Interestingly, whether a fusion category always admits a pivotal structure is an intricate question (see these lecture notes) and is one of the things I’m currently working on. (I am actually simultaneously trying to promote an equivalent but more streamlined and simplified paradigm of the concept of pivotal structure, which I call an even-handed structure, see here and here.)

Remarkably though, although it’s tough in general to find an isomorphism between the identity and the square of the duality functor, there is always a more-or-less canonical isomorphism between the identity and the fourth power of the duality functor:

(3)$id \Rightarrow ****.$

Even more remarkably, if you draw this out in string diagrams (for instance, see page 16 of this paper by Hagge and Hong) it turns out to be a Dirac rope trick kind of situation: you have to do two full rotations (fourth power of the duality functor $*$) to get back to where you started!

Posted by: Bruce Bartlett on September 3, 2008 11:01 AM | Permalink | Reply to this

square root of not; Re: Imperfections, Ambiguities and Physics

“How many such cases of an operation whose fourth power is the identity have a common ‘cause’?”

In Quantum Computing, but not in classical computing, we have the operation: “square root of not.”

That, to the 4th power, is logically an identity operation, preserving truth value.

I have spent some effort generalizing that, but this is not the place for the tangent.

Posted by: Jonathan Vos Post on September 3, 2008 4:37 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

I love this fourth root of unity stuff. It’s odd that nobody has yet mentioned this example:

$i^4 = 1$

I didn’t mention it because it was the answer to my puzzle! I hope the rest of you aren’t mentioning it because it’s so obvious.

The fact that the Fourier transform satisfies

$F^4 = 1$

is a direct consequence of the fact that the Fourier transform is the second quantization of the action of $i$ on $\mathbb{C}$. Irving Segal was one of the first to really understand this.

I like the idea of “square root of not”. If this operation were available in everyday language, politicians would use it all the time as they sought to halfway reverse their positions.

Of course the very meaning of “$i$” is “turning 90 degrees”, which is halfway reversing yourself. We need the complex plane because we can’t turn halfway around if we live on a mere line.

I don’t see yet how Bruce Bartlett’s remarks are related to $i^4 = 1$, but I bet they are — perhaps with marvelous consequences!

Posted by: John Baez on September 3, 2008 6:11 PM | Permalink | Reply to this

Please rotate; Re: Imperfections, Ambiguities and Physics

“… a single QCF gate is said to calculate ‘the square root of NOT.’”

American Scientist
July-August 1995
The Square Root of NOT
Brian Hayes

Weisstein, Eric W. “i.” From MathWorld–A Wolfram Web Resource

The following mathematical joke exhibits the strange way in which mathematicians think. “Rrrrrring. Operator: I’m sorry, the number you have dialed is imaginary. Please multiply by i and dial again.” A variant of this joke, actually left on one mathematician’s phone by his son, states “I’m sorry, the number you have dialed is an imaginary number. Please rotate by 90 degrees and try again.”

That is my son, referred to.

Posted by: Jonathan Vos Post on September 3, 2008 10:47 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

Hi John,

You wrote:

“the Fourier transform is the second quantization of the action of i on ℂ”

Could I trouble you for a reference for this remark? I couldn’t quite see how to work it out…

How does this fact relate to the Fourier transform as an intertwining operator?

Posted by: Minhyong Kim on September 3, 2008 11:54 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

Minhyong wrote:

Could I trouble you for a reference for this remark? I couldn’t quite see how to work it out…

Hi! Just when I thought you weren’t paying attention…

I’ll assume that you want a user-friendly explanation rather than an citable reference — my book with Segal has a chapter on the ‘Fourier–Wiener transform’ which covers this idea in vast generality, but it’s not terribly easy to read.

What leap to mind are my Fall 2003 seminar notes. On page 54 of these notes I show that

$F a^* = -i a^* F$

where

$F : L^2(\mathbb{R}) \to L^2(\mathbb{R})$

is the Fourier transform and

$a^* : L^2(\mathbb{R}) \to L^2(\mathbb{R})$

is the creation operator for the harmonic oscillator (a densely defined operator). On page 55 I show that the ground state of the harmonic oscillator is its own Fourier transform. Since we get all the other eigenstates of the harmonic oscillator by repeatedly applying the creation operator, it follows that

$F |n \rangle = (-i)^n |n \rangle$

where $|n\rangle$ is the $n$th eigenstate.

Now let’s switch to the ‘Bargmann–Segal–Fock’ representation, where $|n \rangle$ is identified with the function $z^n$ on the complex plane. This sets up an isomorphism between $L^2(\mathbb{R})$ and a certain famous Hilbert space of holomorphic functions on $\mathbb{C}$. We can use this to reinterpret $F$ as an operator on those holomorphic functions. Just to confuse you, I’ll also call that operator $F$.

Switching viewpoint this way, the equation

$F |n \rangle = (-i)^n |n \rangle$

implies that

$F z^n = (-i z)^n$

But this implies that

$(F \psi)(z) = \psi(-i z)$

for any holomorphic function $\psi : \mathbb{C} \to \mathbb{C}$ in our Hilbert space. So: in the Bargmann–Segal–Fock representation, the Fourier transform is just a quarter turn!

More generally, all sorts of symplectic transformations on the classical phase space $\mathbb{R}^{2n} \cong \mathbb{C}^n$ get represented as unitary operators on the Bargmann–Segal–Fock space of holomorphic functions on $\mathbb{C}^n$, which is isomorphic to $L^2(\mathbb{R}^n)$. This is the Segal–Shale ‘metaplectic representation’ — actually a representation of the double cover of the symplectic group. In particular, the symplectic transformation ‘multiplication by $-i$’ gets represented by an operator that corresponds to the Fourier transform on $L^2(\mathbb{R}^n)$.

The appearance of $-i$ instead of $i$ in some of these formulas is just a matter of convention.

Posted by: John Baez on September 4, 2008 5:28 AM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

I have been occasionally looking in, except the mental pain of concentrating enough to contribute anything coherent has been keeping me silent. Thanks for the explanation.

I think part of my confusion here was with how you said ‘second quantization.’ If I think of the original ‘i’ as a symplectic transformation of the symplectic manifold C, then the Fourier transform as you’ve described is the *first quantization* isn’t it? On the other hand, if I view the C as a single particle Hilbert space, I guess I should be thinking of i as some unitary operator rather than as a symplectic transformation? Perhaps I’m completely confused about the terminology?

Posted by: Minhyong Kim on September 4, 2008 1:55 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

Minhyong wrote:

I think part of my confusion here was with how you said ‘second quantization.’ If I think of the original ‘i’ as a symplectic transformation of the symplectic manifold $\mathbb{C}$, then the Fourier transform as you’ve described is the first quantization isn’t it?

Yes, you’re right. I was just slipping into standard physics jargon.

On the other hand, if I view the $\mathbb{C}$ as a single particle Hilbert space, I guess I should be thinking of $i$ as some unitary operator rather than as a symplectic transformation?

That’s true. We can view $\mathbb{C}$ either as a symplectic manifold — the classical phase space of a particle on the line — or as a Hilbert space — the Hilbert space of a quantum system with no degrees of freedom. I guess quantizing $\mathbb{C}$ deserves the name ‘first quantization’ if we take the first viewpoint, and ‘second quantization’ if we take the second viewpoint. But it’s just a matter of attitude!

Perhaps I’m completely confused about the terminology?

Well, I’m quite relaxed by now about the fact that every Hilbert space is both a classical phase space and a space of quantum states, but I achieved this bliss only after decades of agonized confusion.

A bit of detail for the people lurking:

‘Second quantization’ is standard terminology for a certain functor

$K : Hilb \to Hilb$

where just for now I’m using $Hilb$ to mean the groupoid of Hilbert spaces and unitary operators.

This functor sends any Hilbert space $H$ to its ‘Fock space’ $K H$ — the Hilbert space completion of the symmetric tensor algebra on $H$.

If apply this functor to $\mathbb{C}$, we get the Hilbert space of the quantum harmonic oscillator, which is isomorphic to $L^2(\mathbb{R})$ in a standard way (the Schrödinger representation). And if we apply this functor to

$i : \mathbb{C} \to \mathbb{C} ,$

we get the Fourier transform.

Posted by: John Baez on September 6, 2008 5:37 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

Bruce was talking about spin groups, which are double covers of rotation groups — and the need to turn some things around twice to get them back where they started. Now Minhyong got me talking about metaplectic groups, which are double covers of symplectic groups. Interesting.

Posted by: John Baez on September 4, 2008 7:29 AM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

Talking about metaplectic groups, I dare say Lautman would have liked the material discussed in this post at The Secret Blogging Seminar, which involves “a conceptual explanation of why there is no analogue of the metaplectic group over a finite field”, and a new proof of quadratic reciprocity.

Posted by: David Corfield on September 4, 2008 4:40 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

John wrote:

After all, Bruce’s observations apply even to categories that seemingly don’t involve the complex numbers or quaternions. I would like to see these number systems ‘emerge’ from string diagram considerations.

I apologize for neglecting to mention the following: for the string diagram manipulations I talked about to go through, one needs the fusion category to be defined over an algebraically closed field, because one needs to take some square roots right in the beginning. Thus one might as well assume one is working over $\mathbb{C}$. So we have:

(1)$string diagrams + \mathbb{C} = something \; whose\; 4th\; power\; is\; unity$

so the whole thing basically still rests on the magic properties of $i \in \mathbb{C}$.

Posted by: Bruce Bartlett on September 4, 2008 12:15 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

I wrote:

I don’t see yet how Bruce Bartlett’s remarks are related to $i^4 = 1$, but I bet they are — perhaps with marvelous consequences!

I think I’m seeing how it goes.

For starters, $i$ is an element of $SO(2) = U(1)$ whose fourth power is 1. Here $SO(2)$ is the group of rotations of the plane, and $U(1)$ is the unit complex numbers.

Bruce, on the other hand, is noting that dualization in string diagrams corresponds to a ‘half twist’: the element $-1$ in $SO(3)$, whose square is 1. Then he’s lifting this to an element of $SU(2)$ whose fourth power is 1. Here $SO(3)$ is the group of rotations of 3d space, and its double cover $SU(2)$ is the unit quaternions.

So, Bruce’s observation seems to involve a ‘higher-dimensional’ version of the concept of $i$, which lives in $SU(2)$ instead of $U(1)$.

But in fact there’s a standard name for a unit quaternion whose fourth power is 1: namely, $i$! But this is $i$ viewed as a unit quaternion, not just a unit complex number.

If we think about this hard enough we may learn some surprising things. After all, Bruce’s observations apply even to categories that seemingly don’t involve the complex numbers or quaternions. I would like to see these number systems ‘emerge’ from string diagram considerations.

Posted by: John Baez on September 3, 2008 11:34 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

An attempt to understand quadratic reciprocity as S-duality.

Posted by: David Corfield on October 13, 2009 4:04 PM | Permalink | Reply to this

Re: Imperfections, Ambiguities and Physics

Well, there’s one thing that is kind-of going in the opposite direction to what you’re talking about, namely computer floating point arithmetic. (You probably already know this, but maybe it forms a jumping off point for interesting further discussion).

This has many deviations from real numbers, both the “top-level” ideas like associativity of addition as well as “deeper” understanding like Dedkind cuts. Since floating point is done in terms of a finite number of discrete bits with specified transitions (albeit exact chip specific in some cases), one could understand floating point arithmetic in terms of a theory built in those discrete terms. But no-one does that: even fully trained numerical analysts (which I’m not) first figure out things in terms of real number calculations and then figure out the “imperfections” and how to compensate for them. And people like me can most of the time get away without drawing a distinction between real and floating point numbers. It’s also interesting to note that there have been some attempts to fix some of the issues with, eg, a lazy approach of keeping track of ambiguities that are “there but not known to be relevant to the particular calculation yet” and redoing calculations at greater precision when it becomes known that they will affect the result of the computation (eg, Mathematica does this in some cases, etc), but they generally aren’t used, partly from complexity and partly from computational expense.

I think part of the reason the ambiguity isn’t taken as an overwhelmingly important problem is that there are issues arising from the “perfect” mathematics, eg, trying to track the subdominant solution to a differential equation, and issues from measurements (eg, poor sensors, etc).

Posted by: bane on August 30, 2008 12:20 PM | Permalink | Reply to this

Fourier transforms

The (especially quantum) Fourier transform (and ribbon twists) are like an n=2 (ie. dimension 2) analogy of the B3 braid group at a sixth root of unity, which gives the modular group relation for braid generators. Then electric-magnetic duality for certain Fourier operators can be viewed the original Maxwellian way in terms of loops about straight conductors (point/string duality). This is closer in spirit to Langlands than to Hodge duals, which only live in a very concrete category of differential structures.

Posted by: Kea on September 4, 2008 3:25 AM | Permalink | Reply to this

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