Mathematician Dave Morrison is giving a colloquium talk tomorrow at the KITP with the provocative title How Much Mathematics Does A Theoretical Physicist Need To Know? It should soon be available for viewing on the KITP web-site, and I’m looking forward to seeing what he has to say.

I’m not at all sure myself how much mathematics a theoretical physicist needs to know, it certainly depends on what they’re trying to do. But there does seem to me to be a well-defined list of what mathematics goes into our current most fundamental physical theories, and anyone who hopes to work on extending these should start by learning these subjects, which include (besides the classical mathematical physics of PDE’s, Fourier analysis, complex analysis):

Riemannian geometry

More general geometry of principal and vector bundles: connection, curvature, etc.

Spinor geometry

Lie groups and representation theory

deRham cohomology

I’m sure others have different ideas about this….

**Update:** Dave Morrison’s talk is now on-line here. He began his talk my noting that it had been advertised here on “Not Even Wrong”, and he put up a slide of my posting and people’s comments as an example of people’s lists of what mathematics theoretical physicists should know. He did say that that his talk wasn’t intended to provide such a list, but rather various comments about how physicists can fruitfully interact with mathematicians.

He began by giving several examples of people who had to construct new mathematics to do physics: Newton, Fourier, Heisenberg, and Gell-Mann. David Gross correctly objected that SU(3) representation theory was already known before Gell-Mann started using it, even though at first Gell-Mann wasn’t aware of this. As for more recent interactions, he mainly mentioned the connection between the index theorem and anomalies, as well as various math related to the quantum hall effect. For some reason he decided not to go into the relation of string theory and mathematics, which has been quite fruitful. He did say that he still believes there is some unknown more fundamental way of thinking about string theory that will involve now unknown mathematics. His general advice to physicists was that they should be willing to acquire mathematical tools as needed, but should be aware that if they ask a mathematician questions, they are likely to get answers of too great generality. He ended his talk early, opening the floor to a long discussion.

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I would say you have the majority of what is necessary (you probably want some functional analysis as well). Then again there have been several successful theorists who’s knowledge in certain of those subjects is sorely lacking (read nonexistant).

In general people adapt pretty quickly in physics to what they need to know for the problem at hand. For instance, (a subject close to your heart), I have a few friends working on the landscape and they are more or less pouring over extremely sophisticated books on probability and analysis (lots of Bayesian statistics etc).

So what does a theoretical physicist need above all else? Well, pretty much he/she needs to be a quick learner.

I agree that physicists are quick learners. I come from the mathematical side and I’m often amazed how quick physicists assimilate concepts that took me ages to understand. The problem is that they often turn these concepts into something I don’t recognize ðŸ™‚

But of course it is most useful to have both insights. By the way, I am very grateful to the physicists who occasionally write formulae in clean mathematical notations, it saves me to do the job myself and helps me a lot in understanding what they say.

I see a problem if Bayesian statistics is considered extremely sophisticated ðŸ˜Ž

As you said Peter, depends very much on what we want to do, but I’d like to add ‘dynamical systems’, a glaring omission IMHO. There’s a nice little book “A First Course in Dynamics” by Hasselblatt & Katok for example.

Times have moved on from when J.M. Ziman claimed to have ‘contrived to give the appearance of doing research in theory .. with little more serious analytic equipment than could be learnt from … Dirac’s Quantum Mechanics’, for more than twenty years. Is this an altogether good thing?

Frankly, this choice of topics rather surprises me. I am a theoretical particle physicist, but I was trained as both a physicist and a mathematician. Although I understand all the topics listed, I have almost never found the geometrical ones useful in my research. (I do agree with Lie groups and algebras are ubiquitous though.) If one is interested in incorporating gravity into a theory, then there is surely reason for working with differential geometry. However, there is plenty of interesting work that can be done with no (or almost no) reference to gravitational interactions. What I work on is undoubtedly “physics beyond the standard model,” yet it is done mostly within the context of quantum field theory. In fact, the only time I have ever had occasion to use any tools of differential geometry, it was of a much deeper and more complex variety than anything listed here. (And that line of investigation turned out to be a dead end anyway.)

This brings me to my second point. I wonder what exactly it should mean in this context to “know” a subject. My experience has certainly not been that physicists have an easy time mastering sophisticated concepts in mathematics. Most theoretical physicists possess a set of tools drawn from higher mathematics; however, their understanding of the underlying structure of what they’re doing is usually minimal. For example, take Lie theory, which is something every theoretical physicist really does need to be familiar with. In graduate school, I took three classes in this subject; however, for the practicalities of physics research, I could have gotten away with just one, the most elementary class in Lie algebra structure. That is about the level of understanding that I see in most physicists.

There is one area of topology that I think theoretical physicists do need to learn, and that is homotopy theory. This has applications in standard model physics and is a much more general subject area than DeRham cohomology. The mathematician in me says that ideally, physicists working on topological issues should master the full machinery of homology theory as well, but this is just not going to happen. Homotopy theory is very intuitive though, and it can be used to analyze just about all the issused that are likely to occur in physical theories living on manifolds.

This is an interesting idea, because lots of things that condensed matter theorists might consider important aren’t here. For example, there is no discussion of numerical techniques, optimization, graph theory, etc.

I’ve used all of that a lot more as a condensed matter theorist than I have Riemannian geometry.

Does theoretical physicist in this context really mean theoretical particle physicist – that is, the only theoretical physicists are those working to extend the standard model?

robert Says:

Only if you believe that anybody’s really going anywhere… or aren’t they still just trying to rationalize-away the flaw that prevented Dirac from unifying QM and GR…?

http://www.lns.cornell.edu/spr/2005-06/msg0069755.html

Why did you omit dynamical systems and topology?

I would encourage knowing (real) analysis (spectral theory in particular and maybe advanced probability theory) and modern PDE’s but I understand that they have so much overlap with everything else that you might not need to take a course to learn them.

I assume there is a mantra for theoretical physicists:

Must … learn … more … math!

A little symplectic topology; how to calculate characteristic classes, what is an almost complex structure and when, and so on. Goes with the homotopy theory. Any finitely presented group is the fundamental group of some symplectic manifold; stick that in your landscape and smoke it.

Just to clarify. I think different kinds of theorists studying different aspects of theoretical physics need different kinds of mathematics. The list I gave was one for the specific case of particle theorists interested in how to better understand the standard model and find some way of improving upon it.

Some people mentioned more topology, especially Brett who mentioned homotopy theory. I’m very fond of topology myself, and had to resist putting K-theory on the list. I still think though that deRham cohomology is the basic thing most theorists should know about. It gives specific integral formulas for topological invariants, and most of the topological invariants that come about in standard model related QFTs can be computed in terms of cohomology, not needing homotopy (especially important is Chern-Weil theory, which computes topological invariants of bundles in terms of connections and curvature). There are some places where the fact that a fundamental group is Z2 is important, but otherwise cohomology covers most of the topology physicists need.

I’m surprised that nobody has mentioned this story yet:

When Weisskopf was asked “how much mathematics does a theoretical physicist need to know?”, his answer was:

More!What “new mathematics” was “constructed” by Heisenberg in order “to do physics” ?

Bear in mind that according to http://mooni.fccj.org/~ethall/quantum/quant.htm

“… Hilbert suggested to Heisenberg that he find the differential equation that would correspond to his matrix equations. Had he taken Hilbert’s advice, Heisenberg may have discovered the SchrÃ¶dinger equation before SchrÃ¶dinger. …”.

Tony Smith

http://www.valdostamuseum.org/hamsmith/

On Differential geometry(DG) in particle physics, I think tranditional particle physicist’s language is not easy to understand. DG can make many topics more clear. The examples include

Neother theorem’s presentation

angular momentum for spin nonzero particle

which both involve the Lie derivative concepts and Lie group

“… several examples of people who had to construct new mathematics to do physics: Newton, …”

Peter, Newton uses standard geometry of 300 BC (Euclid, “Elements of Geometry” stuff) in his Principia, not calculus. There is no evidence he invented calculus for the physics in Principia. Newton also delved into Biblical numerology and alchemy. Archimedes’ “Method” shows that Archimedes used a type of calculus to work out the volume of a sphere and cylinder, but merely used that as scaffolding to help him work out a geometrical proof. Although both Archimedes and Newton worked on calculus, both were careful to express all results for physics classically. They didn’t need to provide critics with ammunition to sneer at by taking a sum of an infinite number of infidesimal slices to get a result. It was better to use geometry which people understood and respected, not a newfangled approach.

The same happened with Einstein. Professor Morris Kline describes the situation after 1911, when Einstein began to search for more sophisticated mathematics to build gravitation into space-time geometry:

â€˜Up to this time Einstein had used only the simplest mathematical tools and had even been suspicious of the need for “higher mathematics”, which he thought was often introduced to dumbfound the reader. However, to make progress on his problem he discussed it in Prague with a colleague, the mathematician Georg Pick, who called his attention to the mathematical theory of Ricci and Levi-Civita. In Zurich Einstein found a friend, Marcel Grossmann (1878-1936), who helped him learn the theory; and with this as a basis, he succeeded in formulating the general theory of relativity.â€™

(M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1990, vol. 3, p. 1131.)

I would add to the list of interesting mathematics some new items (I confess immediately that they relate to my personal favorite theories;_)).

No one has mentioned number theory yet. The zeros of Riemann’s Zeta seem to be closely related to quantum chaotic systems and conformal symmetry and Riemann hypothesis could relate in a very deep manner. The generalization of quantum physics to p-adic number fields and fusion of quantum physics in different number fields to single super structure is highly attractive idea with which I have worked for a more than decade now. The notion of primeness is incredibly general and the direct analogy with the notion of elementary particle might serve as an inspiration for an imaginative theoretician. For instance, the construction of infinite primes can be interpreted as an iterated second quantization of super-symmetric arithmetic quantum theory.

There has been no mention of infinite-dimensional geometries, which might be of interest for string model builders. Loop space Kaehler geometries are essentially unique from the mere requirement that they exist mathematically as shown by Freed. Kac Moody symmetries as isometries guarantee the existence of Riemann connection. Curvature scalar is however infinite which suggests that strings are quite not enough.

This inspires the idea that physics might be unique from the mere existence of Kaehler geometry and corresponding spinor structure for infinite-dimensional configuration space, “the world of classical worlds”. My own bet is that the space of 3-surfaces in certain uniquely determined 8-D imbedding space is the correct guess and leads to a physics unique from its mere mathematical existence. This approach is a diametric opposite for the M-theory approach where imbedding space can be almost anything and leads to landscape problem.

A third fascinating branch of mathematics not yet mentioned relates to von Neumann algebras. The so called hyper-finite factors of type II_1 are obtained by requiring that the infinite-dimensional unit matrix has unit trace. The Clifford algebra of a separable Hilbert space realizes this algebra. The Clifford algebra of spinors of an infinite-dimensional “world of classical worlds” can be regarded as direct integral of these factors. Hyper-finite type II_1 factors have fascinating connections with conformal field theories, knot and braid theory, 3-manifold invariants, etc..

In wave mechanics factors I_n, n=1,..,infty, appear. In algebraic quantum field theory factors of type III_1 appear and possess very counter-intuitive properties. Could it be that hyper-finite factors of type II_1 provide the solution to the problems of QFT via the geometrization of the fermionic Fock algebra in terms of gamma matrix algebra for the world of classical worlds?

Matti Pitkanen

Another question.

“How much experimental physics does a theoretical physicist need to know?”

This brings to mind the tale of the 3 physicists in the hot air balloon who realize they are lost. They come over a hill and see some campers. They shout down: “where are weeeeee”.

No answer, while they continue to drift, until finally they are just about out of sight, comes the reply “in a balloooon”.

One physist says to the others, “they must be mathematicians. It took them forever to answer, the answer was manifestly true, but it was totally useless.”

When asked why he became a mathematician, Polya is said to have replied:

It seems appropriate to mention this well-known quote, which I have seen attributed to Dirac, Wigner and Einstein, and perhaps others:

“God is a mathematician.”

Of course, this statement has a dual formulation, which I came up with myself:

“Physics is divine mathematics.”

“Like the crest of a peacock, like the gem on the head of a snake,

so is mathematics at the head of all knowledge.”

-Vedanga Jyotisa (c. 500 BC, India)

Not really on topic, but did any of the UK readers see the Horizon program about Stephen Hawking on BBC2 last night? It featured, amongst many others, Leonard Susskind as Hawking’s bete noir in disagreeing about information loss in black holes. The producers of the program seemed to be expecting Hawking to keep churning out brilliant ideas despite (a) being over 60 and (b) having a nasty wasting disease. This seemed a bit unfair to me.

Maybe we just need to learn a different type of trigonometry …

Advice from a bygone age: J.E. Littlewood responded to a stduent’s request for a background reading list with a curt

‘Nothing is neccessary – or sufficient’