A soft introduction to physics for mathematicians who don't know the first thing about physics

Question

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 good introduction to the mathematics of those subjects, what I require is different: what provides a soft and readable introduction to the (many) concepts and theories out there, such that the mathematics involved in it is in comfortable generality. What makes this a "for mathematicians" question is that a standard soft introduction will also assume that the reader is uncomfortable with the word "manifold" or certainly "sheaf" and "Lie algebra". So I'm looking for the benefit of scope and narrative, together with a presumption of mathematical maturity.

N.B. If your roadmap is several books, that is also very welcome.

Answer 1

I haven't looked at it yet, but Michael Spivak's Physics for Mathematicians: Mechanics I was published a month or so ago. It seems very interesting. Here's an excerpt from the preface:

" I want to explore the working of elementary physics ... which I have always found so hard to fathom.[...] I have written this work in order to learn the subject myself, in a form that I find comprehensible.[...] By physics I mean ... well, physics, what physicists mean by physics, i.e., the actual study of physical objects ... (rather than the study of symplectic structures on cotangent bundles, for example)."

Some lecture notes covering what presumably turned into the first few chapters was put online some time ago here:

http://alpha.math.uga.edu/~shifrin/Spivak_physics.pdf (elementary mechanics from a mathematican's viewpoint)

Answer 2

If you really know nothing about physics I suggest you begin with any text book on physics for undergrad. Easy to read, it will introduce the main usual suspects. After, you'll ask again :)

I am not sure that jumping from nothing to quantum mechanics, or even worse quantum fields theory, would be wise, like jumping from nothing in math to algebraic geometry or K-Theory.

After that, it depends of course at what level of mathematical physics you want to stop. I will illustrate this with some examples:

Question: What is the "mass" of an isolated dynamical system?

Math Answer: It is the class of cohomology of the action of the group of Galilee, measuring the lack of equivariance of the moment map, on a symplectic manifold representing the isolated dynamical system.

Another question: Why in general relativity $E = mc^2$?

Math Answer: Because the group of Poincaré has no cohomology

Another, other question: What is the theorem of decomposition of motions around the center of gravity?

Math Answer: Let $(M,\omega)$ be a symplectic manifold with an hamiltonian action of the group of Galilee, if the "mass" of the system is not zero (in the sense above) then $M$ is the symplectic product or $({\bf R}^6, {\rm can})$, representing the motions of the center of gravity, by another symplectic manifold $(M_0,\omega_0)$, representing the motions around the center of gravity. The group of Galillee acting naturally on $\bf R^6$ and $SO(3) \times {\bf R}$ on $M_0$.

Another, other, other question: What are the constants of motions?

Math Answer: Let $(M,\omega)$ be a pre-symplectic manifold with an hamiltonian action of a Lie group $G$, then the moment map is constant on the characteristics of $\omega$, that is the integral manifolds of the vector distribution $x \mapsto \ker(\omega_x)$.

These answers are the mathematical versions of physics classical constructions, but it would be very difficult to appreciate them if you have no pedestrian introduction of physics. You may enjoy also Aristotles' book "Physics", as a first dish, just for tasting the flavor of physics :)

After that, you will be able to appreciate also quantum mechanics, but this is another question.

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Addendum

Just before entering in the modern world of physics I would suggest few basic lectures for the winter evenings, near the fireplace (I'm sorry I write them down in french because I read them in french).

• Platon, Timée, trad. Émile Chambry.

• Aristote, La Physique, Éd. J. Vrin.

• Maïmonide, Le Guide des Égarés, Éd. Maisonneuve & Larose. (the part about time as an accident of motion, accident of the thing. Very deep and modern thoughts).

• Giordano Bruno, Le Banquet des Cendres, Éd. L’éclat.

• Galileo Galilei, Dialogue sur les Deux Grands Systèmes du Monde, Éd. Points.

• Albert Einstein, La Relativité, Éd. Payot.

• Joseph-Louis Lagrange, Mécanique Analytique, Éd. Blanchard.

• Felix Klein, Le Programme d’Erlangen, Éd. Gauthier-Villars.

• Jean-Marie Souriau, Structure des Systèmes Dynamiques, Éd. Dunod.

• Victor Guillement & Shlomo Sternberg, Geometric Asymptotics, AMS Math Books

• François DeGandt Force and Geometry in Newton Principia.

Answer 3

Dolgachev has some lecture notes for an introduction to physics course he taught to math graduate students. Certainly it presumes mathematical maturity.

Answer 4

Let me suggest a reading plan :

• The Feynman's books are a pleasure to read and provide great insights into basic physics.
• For classical mechanics, "Mathematical methods of Classical mechanics" by V.I. Arnold and "Mechanics" by Landau-Lifshitz.
• for quantum mechanics, "Mathematical foundations of quantum mechanics" - Von Neumann. another nice book on quantum mechanics is by R.Shanker.
• for statistical physics, the two volumes by Landau are my favorites.
• For General Relativity, R.M. Wald and more mathematically inclined is "Large Scale structures of Space time" by Hawking-Ellis.
• For QFT and String Theory, read AMS book "Quantum Fields and Strings for mathematicians". It contains beautiful lectures by experts in the field addressed to mathematicians.
• Another good book is Clay monograph "Mirror Symmetry" by Hori et al.It starts with classical mechanics, moves through quantum mechanics to QFT, String Theory.
• Ivan Mirkovic has nice lecture notes here http://www.math.umass.edu/~mirkovic/ the notes on string theory. It also has discussion on classical and quantum mechanics.

Answer 5

I recommend the book The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penrose.

It tries to touch almost all areas in physics, including the hot ones. Penrose emphasizes the mathematical part (especially the geometric interpretations), and avoids to be superficial (many scientific writers, when trying to make the things easier, use misleading metaphors). One warning is to be careful that sometimes he expresses his personal viewpoint, which is not always mainstream. But it is clear when he does this, and he is very careful to make justice to the mainstream viewpoint, by presenting it very well.

This book is indeed a road map to modern physics, and I recommend to anyone interested in this area to read it at least once, and then to consult the chapters of interest as they need.

Answer 6

First of all "physics" is rather general. You're more likely to find good books on more specific topics like special relativity, quantum mechanics, etc.

Second, if you have the time I would encourage you to read physics books that are written for physicists, not for mathematicians. There are numerous differences in terminology and worldview between the physics and mathematics community, even when the underlying subject matter is in some sense the same. It's very valuable for a mathematician to be able to read and understand recent physics arxiv postings, and the only way to do this is to go through some (perhaps accelerated) version of physics grad school.

Here are some physics books which I have enjoyed. The list is of course constrained by my own limited experience.

• Electricity and Magnetism, Berkeley Physics Course Vol. II by Edward M. Purcell. This book presupposes knowledge of special relativity, but I thought is was really great when I read it as an undergraduate.

• Feynman lectures on physics. Not mathematically sophisticated, but very readable and also covers many different topics.

• The Quantum Theory of Fields, volume 1 by Steven Weinberg. I found this book to be much less impenetrable (from the point of view of a mathematician who foolishly stopped taking physics courses when he was an undergraduate) than the typical QFT textbook.

• Quantum Field Theory in a Nutshell by Anthony Zee. This book omits a lot of details and emphasizes the big picture. It's a great companion to a more detailed book on QFT.

Answer 7

As someone who knows a fair amount of math and has never had a physics course, I felt that Sudbery's book on Quantum Mechanics and the Particles of Nature was written specifically for me.

Answer 8

I really liked Feynman's popular QED book:

http://en.wikipedia.org/wiki/QED:_The_Strange_Theory_of_Light_and_Matter

and really hated anything by Schwinger.

For Russian readers, there is a really nice little book by Faddeev (I asked him whether he wanted it translated once, and he demurred that it would need more work, but I respectfully disagree).

Answer 9

The Feynman lectures are good, but one of the main things which separates physics from mathematics is the role of experiment and observation. Physics is not just a matter of getting the formulae and models right, but also of testing mathematical models against observations to see whether they stand up or break down in "the real world". Part of the role of mathematical models is to give physicists some guidance on potentially fruitful places to look.

So it rather depends whether you are looking at mathematical/theoretical physics as a mathematician/theoretician would understand it, or whether you are looking to understand the role of mathematics in physics as a discipline.

Answer 10

Edit: The list below fits not that good to the requirements you describe, but the texts there are what I found helpfull. If you can read German books, I would recommend W. Greiner's "Theoretische Physik", which explains basically all the needed mathematics. Usefull too may be J. Baez' "Gauge Fields, Knots and Gravity", which contains a "rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell's equations on arbitrary spacetimes. The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations".

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I found Novikov, Shifman, Vainshtein, Zarkhavov's "ABC of Instantons" very good and helpfull to enter the 'physicist's mindspace'.

And then, I found F. C.'s recommendation to Nahm's very fascinating "Conformal Field Theory and Torsion Elements of the Bloch Group" very good, e.g. Nahm writes "readable for mathematicians", "much of this article is aimed at mathematicians who want to see quantum field theory in an understandable language .... all computations should be easily reproducible by the reader". Nahm's issue is a strange connection between some quantum field theories and algebraic K-theory and he hopes, his article could stimulate mathematicians to become interested in these exciting topic. A forthcoming article by Zagier on "quantum modular forms" may relate to that too.

Very interesting too is Nahm's article on the very strange and puzzling history of quantum field theory and string theory, which makes mathematicians so much headaches.

Connes/Marcolli's book"Noncommutative Geometry, Quantum Fields and Motives" contains a very readable introduction in quantum field theory, renormalization etc., Marcolli's "Feynman motives" a chapter "Perturbative Quantum Field Theory and Feynman Diagrams".

Rabin hold a very readable "Introduction to Quantum Field Theory for Mathematicians" in this conference.

Answer 11

Hermann Weyl's The Theory of Groups and Quantum Mechanics is an great read to learn about quantum mechanics (and math!) if you are a mathematician.

Answer 12

Here is a list of books I find useful that present some physical topics from a mathematical viewpoint. Sadly I don't know a good reference for electromagnetism, quantum field theory or statistical physics.

Arnold, Mathematical methods of classical mechanics.

Woodhouse, Special relativity.

Woodhouse, General relativity.

Woodhouse, Geometric Quantization.

Answer 13

Jeffrey Rabin has written a lightning-fast introduction to physics designed for exactly the audience you describe: people with "the mathematical background of a first-year graduate student," but "[no] prior knowledge of physics beyond F = ma."

It's a bit single-minded, because Rabin's ultimate goal is quantum field theory, but it hits most of the important subjects in modern physics, including:

• Classical mechanics (Newtonian, Lagrangian, and Hamiltonian formalisms)
• Classical field theory
• The Lorentz group (presumably that means some special relativity?)
• Quantum mechanics (not sure how in-depth this section is, but it's better than nothing)

The only glaring omissions I can see are classical electromagnetism, statistical mechanics, and general relativity.

You can find Rabin's introduction in the book Geometry and Quantum Field Theory.

Answer 14

Nobel Prize winner Gerard 't Hooft has a page called "How to become a GOOD Theoretical Physicist", with lots of useful links and book references.

Answer 15

There are two outstanding books which I found very readable (I belong to the class of mathematicians who have great difficulties reading physics books and papers):

1. Landau and Lifshitz, Mechanics, and

2. Faddeev and Yakubovskii, Lectures on quantum mechanics for mathematics students.

Answer 16

Hmmm, the first thing that occurs to me is that mathematicians need to learn about "time", because something as fundamental as "conservation of energy" is not directly to be found in mathematics in its physics form. The two are connected by what is usually known as "Noether's theorem" and so this provides a more manageable question: where can mathematicians genuinely learn about the role of symmetry principles in physics? This starts getting us somewhere, but observe what goes on: the traditional route goes through calculus of variations in some form, and that is a theory not in Bourbaki.

So the deal looks like this to me: do we want to "bridge the gap" between contemporary mathematics and contemporary physics on the way that hits the Zeno paradox? Or do we want to invoke a bisection method and claim that it works? In the first, the question "are we doing real physics yet?" has the status of the kids in the back of the car asking "are we nearly there yet?": you only get anywhere close to the destination long after you stop asking. And probably if you have to be told what is "real" physics you aren't even close. The second idea seems more promising. If I just said "find a readable introduction to moment maps and find out how they work, and you will have grasped a Bourbaki-type intermediate between mathematics for its own sake and mainstream Newtonian dynamics, avoiding calculus of variations, with use of symmetry", it seems to me that I have communicated something. I don't know the second thing about physics (which might be how you would know that you had quantised a system) but what I have said might be a first thing,

Answer 17

I add another one I would recommend, separately to allow the votes to sort them: Mikio Nakahara, Geometry, topology, and physics. I quote from Google Books:

Differential geometry and topology have become essential tools for many theoretical physicists. In particular, they are indispensable in theoretical studies of condensed matter physics, gravity, and particle physics. Geometry, Topology and Physics, Second Edition introduces the ideas and techniques of differential geometry and topology at a level suitable for postgraduate students and researchers in these fields. The second edition of this popular and established text incorporates a number of changes designed to meet the needs of the reader and reflect the development of the subject. The book features a considerably expanded first chapter, reviewing aspects of path integral quantization and gauge theories. Chapter 2 introduces the mathematical concepts of maps, vector spaces, and topology. The following chapters focus on more elaborate concepts in geometry and topology and discuss the application of these concepts to liquid crystals, superfluid helium, general relativity, and bosonic string theory. Later chapters unify geometry and topology, exploring fiber bundles, characteristic classes, and index theorems. New to this second edition is the proof of the index theorem in terms of supersymmetric quantum mechanics. The final two chapters are devoted to the most fascinating applications of geometry and topology in contemporary physics, namely the study of anomalies in gauge field theories and the analysis of Polakov's bosonic string theory from the geometrical point of view. Geometry, Topology and Physics, Second Edition is an ideal introduction to differential geometry and topology for postgraduate students and researchers in theoretical and mathematical physics.

Answer 18

(some) standard treatments (for physicists) are:

At the most elementary level (taken before or in parallel with a standard calculus sequence) there is for instance the The M.I.T. Introductory Physics Series by Anthony Philip French (volumes on: mechanics, electromagnetism, vibrations and waves, special relativity,...)

undergraduate (upper-level; start here if you know the math)

Goldstein Classical Mechanics, Griffiths Introduction to electrodynamics, Griffiths Quantum Mechanics, Griffiths Introduction to elementary particles, Reif Fundamentals of Statistical and Thermal Physics

graduate

Arnol'd Mathematical Methods of Classical Mechanics, Jackson Electrodynamics, Sakurai Modern Quantum Mechanics, Kardar Statistical Physics of [...] (two volumes), J.Negele H.Orland Quantum Many-particle Systems, Wald General Relativity, M.Peskin D.Schroeder, An Introduction to Quantum Field Theory ...

As well as the older series by Sommerfeld or Landau and the text on electrodynamics by Smythe.

Higher-level courses in mechanics is (often) geared toward teaching the math needed to study modern physics. That's one of the primary motivations. Goldstein (quantum mechanics), Arnol'd (gauge field theories) etc.

Answer 19

As a more advanced introduction I like and recommend A unified grand tour of theoretical physics by Ian D. Lawrie

I quote from Google Books:

A unified account of the principles of theoretical physics, A Unified Grand Tour of Theoretical Physics, Second Edition stresses the inter-relationships between areas that are usually treated as independent. The profound unifying influence of geometrical ideas, the powerful formal similarities between statistical mechanics and quantum field theory, and the ubiquitous role of symmetries in determining the essential structure of physical theories are emphasized throughout. This second edition conducts a grand tour of the fundamental theories that shape our modern understanding of the physical world. The book covers the central themes of space-time geometry and the general relativistic account of gravity, quantum mechanics and quantum field theory, gauge theories and the fundamental forces of nature, statistical mechanics, and the theory of phase transitions. The basic structure of each theory is explained in explicit mathematical detail with emphasis on conceptual understanding rather than on the technical details of specialized applications. The book gives straightforward accounts of the standard models of particle physics and cosmology.

Answer 20

I like Folland's Quantum Field Theory: a tourist guide for mathematicians. (although it might not be as mathematically soft as the OP had in mind) In the first chapters he quickly deals with classical mechanics, special relativity and quantum mechanics, so that he can focus on QFT straight away.

Answer 21

I have seen that most answers (based, perhaps, on the structure of the original question), have verged around quantum mechanics and relativity. I may suggest, however, the following book, which may be used fairly well together with Feynman's lectures in physics: "Physics and Partial Differential Equations, Volume I", by Tatsien Li and Teihu Qin. Translated by Yachun Li. SIAM, ISBN, 978-1-611972-26-9

I am pretty sure this book has many of the aspects upon the spirit of the original question (including assuming some level of mathematical maturity, without going to deep into the details of subjects like group theory), since it is about how some mathematical models are derived from known physics, what is the mathematics important around them (including transformation into other useful forms, theorem proving, and so on), and other aspects which show nicely how mathematics can tell a lot about physics, once it is settled into a mathematical model.

Granted, this book mentions a lot of concepts unfamiliar to a pure mathematician, hence my recommendation to take it along Feynman's lectures.

I am listing the title of the chapters and one or two important issues covered within them (the list is not comprehensive, but it does gives a hint of what could you expect of the book in terms of subjects; also, I have chosen on purpose among the issues with tendency to mathematical reasoning):

1. Electrodynamics Proof of Gauss's Law Proof of Ampére theorem Mathematical structure of Maxwell's equations

2. Fluid dynamics some lemmas concerning the convexity of some functions, which lead to the conclusion that other functions are also convex, with respect to their arguments. Theorem 2.1 [certain] first-order system [...] can be turned into a first-order symmetric hyperbolic system [...] through a transformation of unknown variables (I mention this theorem, because the proof is decomposed into the part of necessity and the part of sufficiency) Theorem 2.2 concerning saying that certain gases may have compressive shock waves, given some entropy conditions. This theorem is also proven in segments, as in "per cases".

3. Magnetohydrodynamics systems of magnetohydrodynamic equations, assuming infinite conductivity. Here one can begin to see the consequences of assuming limit cases.
   Mathematical structure of magnetohydrodynamics system

4. Reacting fluid dynamics Mathematical structure of the system of reacting fluid dynamics

5. Elastic mechanics Theorem 5.1 demands the existence of a certain matrix with some algebraic properties, and this is proven using some geometrical concepts. Mathematical relationship between stress and deformation.

Appendix A. (some elementary definitions and useful formulas concerning tensors; several theorems are stated and proved in there)

Appendix B. Overview of thermodynamics. This includes the Legendre transform, which roughly speaking, it is a geometrical transformation, which preserves some other mathematical-physical properties.

Answer 22

As I mentioned in a comment, if you're interested in learning physics you should really consider giving a good amount of time to classical mechanics, for instance with Spivak's book which is great.

After that you could take a look at eletromagnetism, i recommend A. O. Barut's "Electrodynamics and Classical Theory of Fields and Particles", though it should probably be supplemented by a more standard reference, such as Griffiths' "Introduction to Electromagnetism".

There have been good indications for quantum mechanics, and I don't want to be repetitive, but for General Relativity a good place is "Semi-Riemmanian Manifolds - with applications to Relativity" by Barret O'Neill, which does not presupposes differential geometry but gets to the most famous theorems in Relativity by the end, as well as giving a good description of the theory in general

Answer 23

Open a book on partial differential equations that doesn't cover theory (Only the methods of solutions) . About 90& of classical physics and much of quantum physics is about solving differential equations . After you finish differential equations , read Jackson electrodynamics book to understand classical field theory .

Answer 24

As an introduction to quantum mechanics, try Mathematical Structure of Quantum Mechanics, by F. Strocchi. I also like this because it starts out describing (classical) physics in terms of observables and experiments, which is a crucial prerequisite to understanding how physicists think.

Another good book is Hilbert Space and Quantum Logic by David W. Cohen. But this is not really an "introduction to physics" book, rather an "introduction to making sense of experiments with bizarre results".

For classical mechanics, I quite enjoyed John R. Taylor's Classical Mechanics, which is a very clear, self-contained text. It covers Newtonian mechanics, the calculus of variations, electromagnetism and various models for things like air resistance. It does not go into the structural details that mathematicians like, but on the other hand, it does not spend too much time introducing calculus (or worse, avoiding calculus and saying crazy "intuitive" things!).

Answer 25

Let me make a few comments on learning physics for someone with a good background in mathematics.

Physics relies on two pillars: a formal logical system like Euclidean geometry (of Euclid, not modern) and using experiments to find causal relationships.

However, the process of going from assumptions (axioms) to conclusions is not as formal in physics as it is in modern mathematics. This makes it harder to be rigorous in physics, the same way it was hard to be rigorous about the fifth postulate in Euclidean geometry. However, a physicist is extremely rigorous in a different way.

The most important feature of a formal system is that it should be consistent; that is, there shouldn't be any contradictions within. For a physicist, the outside world/universe is this consistent system. Mathematics and the related symbolism are a language for communication/narration of the underlying physical phenomena.

Most importantly, unlike mathematicians who have set-theoretic objects, physicists' objects exist in experiments. Any attempt to avoid the thousands of experiments that give physicists a refined conception of objects like light, heat, atoms, mu-mesons, ... can lead to folly.

On a concrete level, I suggest two excellent books at two opposite levels: A high-school textbook and Folland's QFT.

Answer 26

You have to learn the following :

1 - Learn path integrals . You can do whatever you want using path integrals . The Ising model , potts model , Quantum field theory , String theory , stochastic processes etc etc. It's very important . This book http://www.amazon.com/Integrals-Quantum-Mechanics-Statistics-Financial/dp/9814273554 contains a plethora of applications of path integrals .

2- Dynamics. You have to know the basics of things like calculus of variations , Hamiltonians , Lagrangians , legendre transformations , The principle of least action etc . The best resource of this in my opinion is http://www.amazon.com/Mathematical-Classical-Mechanics-Graduate-Mathematics/dp/0387968903/ref=sr_1_1?s=books&ie=UTF8&qid=1426831652&sr=1-1&keywords=mathematical+methods+of+classical+mechanics after you learn this , You can exercise using these tools by solving problems in any classical mechanics textbook .

You can't understand path integrals with understanding lagrangians and hamiltonians .

3 - Representation theory : To understand particle physics

4 - For classical electrodynamics , Landau's Classical field theory is most important. If you know the lagrangian and the hamiltonian ,you should read it. The idea is that you start from the principle that physics is invariant under lorentz transformations. This means , Spacetime is a four-dimensional semi-riemannian flat manifold with a metric $g_{\mu \nu}$. Expressions like $d^4x$ and $A_{\mu}A^{\mu}$ are invariant under lorentz transformation. So you can put these quantities in the action. At the end you can derive maxwell's equations. Most of classical electrdynamics is how to solve these maxwell's equations in various situations.

5- Learn GTR which is easy if you know riemannian geometry. Basically it says that Curvature of the spacetime manifold=Stress-Energy tensor . Read Wald's general relativity textbook.

4 - Also , read popular books on physics. Books by stephen hawking , leonard susskind , Brian greene etc.

5 - Watch lectures on physics on youtube (e.g by leonard susskind). There are very interesting lectures at http://www.perimeterinstitute.ca/video-library/

When reading physics textbooks , I found that I spend a lot of time doing stupid calculations like multiplying matrices , solving systems of linear equations etc so software like mathematica and matlab can be big time-savers .

Answer 27

Feynman lectures are what i used. Still the best way to brush up on physics for any mathematician.