I found the following books very interesting:
 

'QED' by Richard P. Feynman An almost maths-free demonstration of the basic principle governing quantum physics. This principle explains some strange and also familiar phenomena in a surprising way. The reasons for these astonishing workings of quantum mechanics are still a puzzle, even though some people will try to make you believe that everything is in order. It is not...

'The Feynman lectures on physics', Vols 2 and 3 This is a fantastic read on an intermediate level with emphasis on common sense and intuitive understanding. Feynman was one of the most insightful physicists ever and had a passion for explaining things.

'Course of Theoretical Physics', Vols 1-3, by Landau & Lifshitz Reference for classical mechanics, fields and QM. Everything is worked out independently of what everyone else says, and you often find the best (and briefest...) explanations in these books.

'The Principles of Quantum Mechanics' by P.A.M. Dirac Still a very good introduction to the standard approach to quantum mechanics. It is pure and honest genius, covering both non-relativistic QM and the first equations of Quantum Field Theory.

'Mathematical Foundations of Quantum Mechanics' by G. Mackey For clarity it is often important to know what is maths and what physics. In chapter 2-2 Mackey develops the mathematical setup used tacitly in QM from a set of axioms. Each new concept is carefully introduced to see how it works. Checking the calculations is fairly hard work, but the resulting clarification is worth it.

'Group theory and physics', by S. Sternberg Group representations are an essential part of modern physics. This book shows reasons behind physical results in atomic physics, elementary particles and field theory with rigorous mathematical arguments. However, watch out for typographical errors and the somewhat nonchalant notation.

'The Quantum Theory of Fields', Vol. 1, by S. Weinberg This follows on conveniently from quantum mechanics and 'Group theory and physics'. Emphasis is on understanding foundations rather than problem solving. The amount to learn is massive; in places it is advisable to consult other books as well.

'Introducing Einstein's Relativity' by Ray D'Inverno This must be the most accessible and quick introduction to General Relativity available. The style is geared towards self-study through reading in several stages and exercises which are (with a few exceptions) very useful. For additional background I found the following literature helpful.

'Gauge theory and variational principles' by D. Bleecker Differential geometry has revolutionized classical field theory and the interpretation of time and space. This book presents the modern approach in terms of fibre bundles and its application to gauge field theories, cleary developing the necessary mathematical apparatus along the way. See also Naber: Topology, Geometry, and Gauge Fields - Interactions for interesting topological results.

'Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics' by R.M. Wald This firsthand introduction carefully reformulates quantum fields to obtain results which remain valid on a non-flat background. As a byproduct many mathematically doubtful methods disappear. In terms of physics this leads for instance to a derivation of the mysterious Unruh radiation.

Mathematics:

'Introduction to functional analysis' by A.E. Taylor My supervisor in England recommended studying the first few chapters of this book "to get started in mathematics". It is very good for learning the proper meaning of vectors, dual spaces etc.

'Lehrbuch der Topologie' by H. Seifert and W. Threlfall Homology is a concept needed for multidimensional integration (results of de Rham). This book gives the best first introduction. It turns out that algebraic topology is an important ingredient of modern research in many different areas.

'The topological constraints on analysis' by R. Bott, AMS video Although I only understand some of this lecture it makes me think that this is what mathematics is really about... Absolutely fascinating and also full of anecdotes from this masterful gentleman.

'Differential geometry' by R.W. Sharpe (Prerequisites) Most modern texts on this subject are frustrating. In this book successive stages of generalization are covered from Euclidean and non-Euclidean geometry via homogenous spaces up to connections in principal fibre bundles. For the beginner this slow approach is better, but one should switch to a standard textbook as soon as possible. See S. Sternberg: Lectures on Differential Geometry.

'Spin geometry' by H. Lawson, M.-L. Michelsohn The use of Clifford algebras in differential geometry is a technical improvement similar to the transition from vector calculus to differential forms, but also gives new insights. Reading selectively from this book you can get the essentials about Dirac operators, Hodge theory, index theorems and applications.

'Characteristic Classes' by J.W. Milnor and J.D. Stasheff This book is about practical application of algebraic topology via vector bundles. It is like attending a very interesting lecture and made me study the details of various techniques (see topology books). Additionally, chapter XII in Kobayashi/Nomizu sheds light on the subject.

'Topology and Analysis' by B. Booss and D. Bleecker Three chapters lead to the Atiyah-Singer index theorem. Chapter I and II on elliptic operators are very good for self-study, however the third part moves quickly through K-theory and cohomology and does not really provide enough explanation. Yet the summaries of the three existing approaches for the proof (one is sketched a bit better) are helpful for deciding whether to delve deeper into the subject.

'General Topology' by J.L. Kelley Reference for point set topology and analysis from first principles. The only thing missing is a chapter on sigma-algebras and measures (see e.g. Rudin:"Principles of mathematical analysis" as a supplement). For the more 'geometric' aspects see Dugundji: "Topology".

'Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren' by J. von Neumann, 1929 (in 'Collected Works', Vol. 2, or online) I read this in conjunction with von Neumann's book on QM. The good thing is that all arguments are elementary and given in detail which makes it a very tangible introduction to linear operators in Hilbert space.

'Functional Analysis' by W. Rudin This book presents linear operator theory on graduate level (i.e. about 3 years university maths). Very systematic. It's interesting to see how the original ideas of von Neumann and Stone were developed. For a very detailed treatment, see Dunford & Schwartz. See also books by J.B. Conway, Reed & Simon, and Akhiezer & Glazman.

'A first course in abstract algebra' by J.B. Fraleigh Essential algebra easy to understand. This is what you need as a beginner. The standard advanced reference is S. Lang: 'Algebra'.

'Introduction to stochastic integration' by K.L. Chung and R.J. Williams This is a very quick way to the fundamental Itô formula. As reference for basic definitions and various other subjects see Kallenberg: Foundations of modern probability.

'A course in mathematics for students of physics', Vol. 2, by Bamberg and Sternberg A lot of classical physics uses old-fashioned and outdated mathematics and is still taught that way. This is the opposite. One of the highlights is an elucidation of electrostatics using cohomology.

'Generalized functions', Vol. 1, by Gelfand and Shilov Hard to find derivations of results involving distributions used in physics. See also Hörmander: The analysis of partial differential operators, Vol.I

'Lehrbuch der Funktionentheorie' by W.F. Osgood, 1912/1924 (archived here) Very old book about complex analysis written by Osgood in German (back then the standard language in mathematics). For a more recent introduction to complex analysis see Brown & Churchill: Complex variables and applications.

'Formes différentielles' by H. Cartan This book gives a nice introduction to multilinear and differential forms.

'Leçons sur l'intégration' by H. Lebesgue, Gauthier-Villars, 1904 (archived here) What is a Lebesgue integral? In this historic book the inventor explains the 'new' method of integration. For a quick modern introduction to Lebesgue integration, see "Principles of mathematical analysis" by W. Rudin.

'Die Idee der Riemannschen Fläche' by H. Weyl, 1913 Old but lucid introduction to complex manifolds.

'The Penguin Dictionary of Mathematics' This nicely written pocket reference gives a quick definition of a large variety of mathematical concepts and is great fun to browse in. I found it very helpful as an overview of maths for a beginner.

'Encyclopaedia of Mathematics', 10 Volumes or one CD ROM This is the best for a quick guide to advanced mathematical subjects up to the 1990s. For more details, the cited references are always a very good choice.

'Advanced Engineering Mathematics' by E. Kreyszig Good reference for engineering maths. I also liked Kreyszig's book on classical differential geometry.

'Vector Analysis' by M. Spiegel  Reference for results in R3-vector integral and differential calculus used in old style physics. "Second year calculus" by D. Bressoud gives modern counterparts.

Other sciences:

'Nature' Every week this journal is full of exciting new developments in all branches of science and technology. Many groundbreaking results are first published in Nature.

'Neuroscience' by Bear, Connors, Paradiso  Modern introduction to brain research and neurology. Of particular interest to electrical engineers and physicists are the structure of neurons and the chemistry of nerve conduction, the processing of images and sound, control of reflexes, motion and mood, as well as the development and plasticity of brain functions.

  (last revised 16.5.9)

  German version