Mathematical methods in quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.

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Where does a math person go to learn quantum mechanics?

My undergraduate advisor said something very interesting to me the other day; it was something like "not knowing quantum mechanics is like never having heard a symphony." I've been meaning to learn ...
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3answers
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Has the Lie group E8 really been detected experimentally?

A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced, "Quantum ...
70
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16answers
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Does Physics need non-analytic smooth functions?

Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is ...
70
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A soft introduction to physics for mathematicians who don't know the first thing about physics

There have been similar questions on mathoverflow, but the answers always gave some advanced introduction to the mathematics of quantum field theory, or string theory and so forth. While those may be ...
64
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0answers
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Dropping three bodies

Consider the usual three-body problem with Newtonian $1/r^2$ force between masses. Let the three masses start off at rest, and not collinear. Then they will become collinear a finite time ...
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8answers
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Theoretical physics: Why not just R^4?

You and I are having a conversation: "Okay," I say,"I think I get it. The gauge groups we know and love arise naturally as symmetries of state spaces of particles." "Something like that" "...And ...
59
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11answers
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What is an integrable system

What is an integrable system, and what is the significance of such systems? (Maybe it is easier to explain what is a non-integrable system.) In particular, is there a dichotomy between "integrable" ...
57
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28answers
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Where does a math person go to learn statistical mechanics?

The more math I read, the more I see concepts from statistical mechanics popping up -- all over the place in combinatorics and dynamical systems, but also in geometric situations. So naturally I've ...
51
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8answers
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Examples where physical heuristics led to incorrect answers?

I have always been impressed by the number of results conjectured by physicist, based on mathematically non-rigorous reasoning, then (much) later proved correct by mathematicians. A recent example is ...
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1answer
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What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...
47
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1answer
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The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly. Arguably, they are the most important such tool. Briefly, the question I'd like to ...
45
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8answers
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What is the motivation for a vertex algebra?

The mathematical definition of a vertex algebra can be found here: http://en.wikipedia.org/wiki/Vertex_operator_algebra Historically, this object arose as an axiomatization of "vertex operators" in ...
41
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11answers
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What is Quantization ?

I would like to know what quantization is, I mean I would like to have some elementary examples, some soft nontechnical definition, some explanation about what do mathematicians quantize?, can we ...
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13answers
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A reading list for topological quantum field theory?

Can you suggest a reading list, or at least a few papers that you think would be useful, for a beginner in topological quantum field theory? I know what the curvature of a connection is, know basic ...
35
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9answers
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Motivating the Laplace transform definition

In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem. The Laplace transform of a function $f(t)$, ...
34
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6answers
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Mathematical explanation of the failure to quantize gravity naively

One often hears in popular explanations of the failure to find a "Grand Unified Theory" that "Gravity goes off to infinity, but cutting off the edges gives us wrong answers", and other similar ...
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7answers
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The Unreasonable Effectiveness of Physics in Mathematics. Why ? What/how to catch?

Starting from 80-ies the ideas either coming from physics, or by physicists themselves (e.g. Witten) are shaping many directions in mathematics. It is tempting to paraphrase E. Wigner, saying about ...
32
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6answers
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The 'real' use of Quantum Algebra, Non-commutative Geometry, Representation Theory, and Algebraic Geometry to Physics

In this question, Orbicular made the following comment to Feb7 and my own answers; Please keep in mind that - even though it is stated very often - noncommutative geometry does not give "real" ...
32
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2answers
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Recent fundamental new directions in PDEs

My main interests are in modern geometry/topology, algebra and mathematical physics. I observe that there is a raising communication, language and social barrier between this community and the ...
30
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3answers
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Quantum mechanics formalism and C*-algebras

Many authors (e.g Landsman, Gleason) have stated that in quantum mechanics, the observables of a system can be taken to be the self-adjoint elements of an appropriate C*-algebra. However, many ...
29
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10answers
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Examples of non-rigorous but efficient mathematical methods in physics

There are situations of applications of mathematics in physics which seem to work well enough for physicists (for example they agree with the experimental data) but are considered unacceptable or at ...
29
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9answers
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Dimensional Analysis in Mathematics

Is there a sensible and useful definition of units in mathematics? In other words, is there a theory of dimensional analysis for mathematics? In physics, an extremely useful tool is the Buckingham Pi ...
29
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10answers
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Standard model of particle physics for mathematicians

If a mathematician who doesn't know much about the physicist's jargon and conventions had the curiosity to learn how the so called Standard Model (of particle physics, including SUSY) works, where ...
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4answers
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Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one? I've found some examples: 1) In MO-Q111339 ...
28
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5answers
818 views

are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?

I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...
27
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6answers
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A remark of Connes

In an interview (at http://www.alainconnes.org/docs/Inteng.pdf) Connes remarks that I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point ...
27
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10answers
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Is there a mathematical axiomatization of time (other than, perhaps, entropy)?

Since Euclid's axiomatization of space, we have developed a sophisticated mathematical model of space. Given a category of structures (measures), local space is modeled the spectrum of measurements ...
27
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8answers
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What is the symbol of a differential operator?

I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic ...
27
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3answers
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How much linear algebra can be done with graphs?

Let G be a finite directed acyclic graph, with sources $A=\{a_1,\ldots,a_n\}$ and sinks $B=\{b_1,\ldots,b_n\}$, with edge weights $w_{ij}$. The weight of a directed path P is the product of weights of ...
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6answers
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Book on mathematical “rigorous” String Theory?

I've been looking high and low for a mathematical Book on String Theory. The only Book I could find was "A Mathematical Introduction to String Theory" by Albeverio, Jost, Paycha and Scarlatti. I only ...
25
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3answers
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(How) is category theory actually useful in actual physics?

An answer to a recent question motivated the following question: (how) is category theory actually useful in actual physics? By "actual physics" I mean to refer to areas where the underlying ...
25
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4answers
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What is Chern-Simons theory?

What is Chern-Simons theory? I have read the wikipedia entry, but it's pretty physics-y and I wasn't really able to get any sense for what Chern-Simons theory really is in terms of mathematics. ...
24
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5answers
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Poincaré Conjecture and the Shape of the Universe

Has the solution of the Poincaré Conjecture helped science to figure out the shape of the universe?
24
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3answers
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What are D-branes, really?

In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as ...
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6answers
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Explanations for mathematicians, about the falsifiability (or not) of string theory [closed]

Like many other mathematicians, I think string theory very attractive. This theory has wonderfully influenced many new topics in mathematics (I myself have worked on one of them), but it's not the ...
23
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7answers
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Does every ODE comes from something in physics?

Not sure if this is appropriate to Math Overflow, but I think there's some way to make this precise, even if I'm not sure how to do it myself. Say I have a nasty ODE, nonlinear, maybe extremely ...
23
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3answers
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How can simple physical “proofs” of mathematical facts be made rigorous?

Mark Levi's The Mathematical Mechanic is a book of examples of how physical reasoning can be used to solve mathematical problems; another couple of examples is in this blog post at Concrete Nonsense. ...
23
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2answers
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What is the right notion of self-dual (two-dimensional) percolation in R^4?

For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. bond percolation), it is well known that there is a critical probability $0 < p_c < 1$ depending ...
22
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7answers
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Why does bosonic string theory require 26 spacetime dimensions?

I do not think it is possible really believe or experimentally check (now), but all modern physical doctrines suggest that out world is NOT 4-dimensional, but higher. The least sophisticated ...
22
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1answer
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Math and Wormholes

Hopefully, MathOverflow is the correct place for this. I had a student approach me and ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any ...
22
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3answers
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Why is a 2d TQFT formulated as a functor?

Usual mathematical formulation of a 2d (closed) TQFT is as a functor from the category of 2-dim cobordisms between 1-dim manifolds to the category of vector spaces (satisfying various properties.) ...
22
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11answers
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What kind of Lagrangians can we have?

In any physics book I've read the Lagrangian is introuced as as a functional whose critical points govern the dynamics of the system. It is then usually shown that a finite collection of ...
22
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3answers
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Topologically distinct Calabi-Yau threefolds

In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of ...
22
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6answers
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Examples of applications of the Borel-Weil-Bott theorem?

In "Quantum field theory and the Jones polynomial" (Comm. Math. Phys. 1989 vol. 121 (3) pp. 351-399), Witten writes: A representation Ri of a group G should be seen as a quantum object. This ...
22
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5answers
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Is there a mathematically precise definition of turbulence for solutions of Navier-Stokes?

Given a solution $S$ of the Navier-Stokes equations, is there a way to make mathematically precise a statement like: "$S$ is turbulent in the spacetime region $U$"? And if such a definition exists, ...
22
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3answers
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Matrix factorizations and physics

I have heard during some seminar talks that there are applications of the theory of matrix factorizations in string theory. A quick search shows mostly papers written by physicists. Are there any ...
21
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8answers
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In what ways is physical intuition about mathematical objects non-rigorous?

I'm asking this question as a mathematician who is very far removed from the Physics world, and has little to no knowledge of what math goes into it, and what math comes out of it. What I do hear is ...
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4answers
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When is a symplectic manifold equivalent to a cotangent bundle?

Let $X$ be a differentiable manifold. Its cotangent bundle $T^*X$ carries a canonical 1-form $ \alpha$ whose exterior differential $\omega = d\alpha$ endows $T^*X$ with the structure of a symplectic ...
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3answers
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Shape of snowflakes

Is there a mathematical theory that explains the shape of a snowflake? Why is it not round? Update Tree-like metric spaces appear often as limits of sequences of metric spaces (say, asymptotic cones ...
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0answers
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Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...