It might be a bit overwhelming for someone who really only wants to focus on the physical applications, but many of the technical details of the background material that are often omitted in other texts are presented here in full. The second part is focused on physical applications, mostly to classical gauge theories. The text assumes some basic familiarity with manifolds, but not much else. A "standard introductory book" on differential geometry, translated to the language of physicists. Isham is careful to point out where mathematical notions that he introduces are used in physics, which is nice for those who prefer not to lose track of the physical relevance of it all.

Covers all the basics up to fiber bundles in about pages. Quickly gets to more advanced topics including moduli spaces, spinors and supermanifolds all within the first pages in the first part, dedicated to mathematics. The second part is dedicated to physics and includes e.

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Though this is pretty much a "general introduction" book of the type I said I wouldn't include, I've decided to violate that rule. This book is Russian, and the style of Russian textbooks is very physical and interesting for physics students, in my opinion. Furthermore, the book does not focus on either differential geometry or topology, but covers both briefly , which is also good for physics students. The second volume covers more advanced topics such as Chern classes. The go-to book for mathematical prerequisites for e. I personally think it's terrible because it doesn't explain anything properly, but I guess it's good to learn buzzwords.

This book is not very physical, but seems very nice if you're really trying to get a good grip on the math. It takes its time to properly, I hope develop all the theory in order: fundamental group, homology, cohomology and higher homotopy groups are all introduced, before fiber bundles and then Morse theory and topological defects are treated!!

The final chapter is on Yang-Mills theories, discussing instantons and monopoles. After about pages of preparatory mathematics including, besides the standard topics, Frobenius theory and foliations, which is nice! Couldn't resist putting this in: The original classic on spinors, by the discoverer himself. Kind of outdated in e. Deligne et al. The two volumes cover about pages, with contributions from famous mathematicians and physicists alike Deligne, Witten Covers lots of advanced topics in physics from a mathematical perspective, and includes exercises.

Because the OP mentions solitons I thought this might be interesting to mention: Basic topology and geometry is assumed to be known, and a lot of physically interesting topics are covered such as monopoles, kinks, spinors on manifolds etc. This book seems fascinating for those who are really trying to get into the more difficult parts of gauge theory.

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Topics covered include topological field theories knots invariants, Floer homology etc , anomalies and conformal field theory. This book is completely dedicated to the theory of twistors: The last part is about applications to gauge theory.

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This book presents, in a concise and direct manner, the appropriate mathematical formalism and fundamentals of differential topology and differential geometry together with essential applications in many branches of physics. Don't hurry ramanujan, learn basic mathematical methods first from Sadri-Hassani's "Mathematical Physics" for instance. Then the standard reference for you to learn grad-level mathematics would be Nakahara's "Geometry, Topology and Physics". If you think it's too much, you're right; this is a very serious advanced topic.

But if you want to quickly pick some basic ideas, check out the 10th chapter of Ryder's "Quantum Field Theory". An advanced and physically oriented discussion would be found in Coleman's "Aspects of Symmetry". Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Book covering differential geometry and topology for physics Ask Question. Asked 7 years, 3 months ago. Active 7 months ago.

## Differential Geometrical Methods in Theoretical Physics - Google книги

Viewed 12k times. It's easy and fun, and it gets you going on nontrivial stuff without a lot of prerequisites or worrying about what a physicist would consider pathological cases. However, in terms of what might be useful for physics I would recommend either: Nakahara's "Geometry, Topology and Physics" Naber's "Topology, Geometry and Gauge Fields: Foundations" Personally, I haven't read much of Nakahara, but I've heard good things about it, although it may presuppose too many concepts.

Nilay Kumar. However, judging by the absolutely ridiculous price on Amazon Nakahara is nice. Schwarz seemed good at first glance, but I havent read it. Apart from the basic definitions and so on, one of the most applied concepts is Homotopy. It is beautiful in itself, and it formalizes the concept of winding numbers to higher dimension. In physics it is commonly used to enumerate the topological solitons present in your theory. There are others, but I found Homotopy to be very important and useful. Nash and Sen is a good book, too. Schreiber: ncatlab. Bleecker - Gauge Theory and Variational Principles Starts with a very brief treatment of tensor calculus, fiber bundles etc, quickly moving on to physical topics such as Dirac fields, unification of gauge fields and spontaneous symmetry breaking.

Bredon - Topology and Geometry After seeing the added message by QuanticMan, requesting answers not to shy away from going beyond the intentions of the OP and to recommend books with geometrical intuition, this book immediately sprang to mind. In the preface, Bredon states: While the major portion of this book is devoted to algebraic topology, I attempt to give the reader some glimpses into the beautiful and important realm of smooth manifolds along the way, and to instill the tenet that the algebraic tools are primarily intended for the understanding of the geometric world.

Burke - Applied Differential Geometry Starts with about pages of mathematical tools from tensors to forms and then delves into applications: From the heat equation to gauge fields and gravity. Cahill - Physical Mathematics This is a really basic book, that does much more than just topology and geometry: It starts off with linear algebra, spends a lot of time on differential equations and eventually gets to e. According to Aristotelian physics , the circle was the perfect form of motion, and was the intrinsic motion of Aristotle's fifth element —the quintessence or universal essence known in Greek as aether for the English pure air —that was the pure substance beyond the sublunary sphere , and thus was celestial entities' pure composition.

The German Johannes Kepler [—], Tycho Brahe 's assistant, modified Copernican orbits to ellipses , formalized in the equations of Kepler's laws of planetary motion. An enthusiastic atomist, Galileo Galilei in his book The Assayer asserted that the "book of nature" is written in mathematics.

Galilei's book Discourse on Two New Sciences established the law of equal free fall as well as the principles of inertial motion, founding the central concepts of what would become today's classical mechanics. Descartes sought to formalize mathematical reasoning in science, and developed Cartesian coordinates for geometrically plotting locations in 3D space and marking their progressions along the flow of time. Christiaan Huygens was the first to use mathematical formulas to describe the laws of physics, and for that reason Huygens is regarded as the first theoretical physicist and the founder of mathematical physics.

Isaac Newton — developed new mathematics, including calculus and several numerical methods such as Newton's method to solve problems in physics. Newton's theory of motion, published in , modeled three Galilean laws of motion along with Newton's law of universal gravitation on a framework of absolute space —hypothesized by Newton as a physically real entity of Euclidean geometric structure extending infinitely in all directions—while presuming absolute time , supposedly justifying knowledge of absolute motion, the object's motion with respect to absolute space.

Having ostensibly reduced the Keplerian celestial laws of motion as well as Galilean terrestrial laws of motion to a unifying force, Newton achieved great mathematical rigor, but with theoretical laxity. In the 18th century, the Swiss Daniel Bernoulli — made contributions to fluid dynamics , and vibrating strings. The Swiss Leonhard Euler — did special work in variational calculus , dynamics, fluid dynamics, and other areas. Also notable was the Italian-born Frenchman, Joseph-Louis Lagrange — for work in analytical mechanics : he formulated Lagrangian mechanics and variational methods.

A major contribution to the formulation of Analytical Dynamics called Hamiltonian dynamics was also made by the Irish physicist, astronomer and mathematician, William Rowan Hamilton Hamiltonian dynamics had played an important role in the formulation of modern theories in physics, including field theory and quantum mechanics.

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The French mathematical physicist Joseph Fourier — introduced the notion of Fourier series to solve the heat equation , giving rise to a new approach to solving partial differential equations by means of integral transforms. Into the early 19th century, the French Pierre-Simon Laplace — made paramount contributions to mathematical astronomy , potential theory , and probability theory. In Germany, Carl Friedrich Gauss — made key contributions to the theoretical foundations of electricity , magnetism , mechanics , and fluid dynamics.

In England, George Green published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism in , which in addition to its significant contributions to mathematics made early progress towards laying down the mathematical foundations of electricity and magnetism. A couple of decades ahead of Newton's publication of a particle theory of light, the Dutch Christiaan Huygens — developed the wave theory of light, published in By , Thomas Young 's double-slit experiment revealed an interference pattern, as though light were a wave, and thus Huygens's wave theory of light, as well as Huygens's inference that light waves were vibrations of the luminiferous aether , was accepted.

Jean-Augustin Fresnel modeled hypothetical behavior of the aether.

Michael Faraday introduced the theoretical concept of a field—not action at a distance. Midth century, the Scottish James Clerk Maxwell — reduced electricity and magnetism to Maxwell's electromagnetic field theory, whittled down by others to the four Maxwell's equations. Initially, optics was found consequent of [ clarification needed ] Maxwell's field. Later, radiation and then today's known electromagnetic spectrum were found also consequent of [ clarification needed ] this electromagnetic field.

The English physicist Lord Rayleigh [—] worked on sound. The Irishmen William Rowan Hamilton — , George Gabriel Stokes — and Lord Kelvin — produced several major works: Stokes was a leader in optics and fluid dynamics; Kelvin made substantial discoveries in thermodynamics ; Hamilton did notable work on analytical mechanics , discovering a new and powerful approach nowadays known as Hamiltonian mechanics. Very relevant contributions to this approach are due to his German colleague Carl Gustav Jacobi — in particular referring to canonical transformations. The German Hermann von Helmholtz — made substantial contributions in the fields of electromagnetism , waves, fluids , and sound.

In the United States, the pioneering work of Josiah Willard Gibbs — became the basis for statistical mechanics. Fundamental theoretical results in this area were achieved by the German Ludwig Boltzmann Together, these individuals laid the foundations of electromagnetic theory, fluid dynamics, and statistical mechanics. By the s, there was a prominent paradox that an observer within Maxwell's electromagnetic field measured it at approximately constant speed, regardless of the observer's speed relative to other objects within the electromagnetic field.

Thus, although the observer's speed was continually lost [ clarification needed ] relative to the electromagnetic field, it was preserved relative to other objects in the electromagnetic field. And yet no violation of Galilean invariance within physical interactions among objects was detected.

As Maxwell's electromagnetic field was modeled as oscillations of the aether , physicists inferred that motion within the aether resulted in aether drift , shifting the electromagnetic field, explaining the observer's missing speed relative to it. The Galilean transformation had been the mathematical process used to translate the positions in one reference frame to predictions of positions in another reference frame, all plotted on Cartesian coordinates , but this process was replaced by Lorentz transformation , modeled by the Dutch Hendrik Lorentz [—].

In , experimentalists Michelson and Morley failed to detect aether drift, however. It was hypothesized that motion into the aether prompted aether's shortening, too, as modeled in the Lorentz contraction. It was hypothesized that the aether thus kept Maxwell's electromagnetic field aligned with the principle of Galilean invariance across all inertial frames of reference , while Newton's theory of motion was spared.

In the 19th century, Gauss 's contributions to non-Euclidean geometry , or geometry on curved surfaces, laid the groundwork for the subsequent development of Riemannian geometry by Bernhard Riemann — Austrian theoretical physicist and philosopher Ernst Mach criticized Newton's postulated absolute space.

## Differential Geometric Methods in Theoretical Physics

In , Pierre Duhem published a devastating criticism of the foundation of Newton's theory of motion. Refuting the framework of Newton's theory— absolute space and absolute time —special relativity refers to relative space and relative time , whereby length contracts and time dilates along the travel pathway of an object. In , Einstein's former professor Hermann Minkowski modeled 3D space together with the 1D axis of time by treating the temporal axis like a fourth spatial dimension—altogether 4D spacetime—and declared the imminent demise of the separation of space and time.

Einstein initially called this "superfluous learnedness", but later used Minkowski spacetime with great elegance in his general theory of relativity , [11] extending invariance to all reference frames—whether perceived as inertial or as accelerated—and credited this to Minkowski, by then deceased.

General relativity replaces Cartesian coordinates with Gaussian coordinates , and replaces Newton's claimed empty yet Euclidean space traversed instantly by Newton's vector of hypothetical gravitational force—an instant action at a distance —with a gravitational field. The gravitational field is Minkowski spacetime itself, the 4D topology of Einstein aether modeled on a Lorentzian manifold that "curves" geometrically, according to the Riemann curvature tensor , in the vicinity of either mass or energy.

Under special relativity—a special case of general relativity—even massless energy exerts gravitational effect by its mass equivalence locally "curving" the geometry of the four, unified dimensions of space and time. Another revolutionary development of the 20th century was quantum theory , which emerged from the seminal contributions of Max Planck — on black-body radiation and Einstein's work on the photoelectric effect.

This revolutionary theoretical framework is based on a probabilistic interpretation of states, and evolution and measurements in terms of self-adjoint operators on an infinite dimensional vector space. That is called Hilbert space , introduced in its elementary form by David Hilbert — and Frigyes Riesz , and rigorously defined within the axiomatic modern version by John von Neumann in his celebrated book Mathematical Foundations of Quantum Mechanics , where he built up a relevant part of modern functional analysis on Hilbert spaces, the spectral theory in particular. Paul Dirac used algebraic constructions to produce a relativistic model for the electron , predicting its magnetic moment and the existence of its antiparticle, the positron.

From Wikipedia, the free encyclopedia. Application of mathematical methods to problems in physics. Main articles: Lagrangian mechanics and Hamiltonian mechanics. Main article: Partial differential equations. Main article: Quantum mechanics. Main articles: Theory of relativity and Quantum field theory. Main article: Statistical mechanics.

Archived from the original on Retrieved Depending on the ratio of these two components, the theorist may be nearer either to the experimentalist or to the mathematician. In the latter case, he is usually considered as a specialist in mathematical physics. Frenkel, as related in A. Filippov, The Versatile Soliton , pg Birkhauser, Good theory is like a good suit. Thus the theorist is like a tailor. Frenkel, as related in Filippov , pg J De mechanisering van het wereldbeeld.

Meulenhoff, Amsterdam.