Mathematical physics refers to the development ofmathematical methods for application to problems inphysics. TheJournal of Mathematical Physics defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories".[1] An alternative definition would also include those mathematics that are inspired by physics, known asphysical mathematics.[2]
The usage of the term "mathematical physics" is sometimesidiosyncratic. Certain parts of mathematics that initially arose from the development ofphysics are not, in fact, considered parts of mathematical physics, while other closely related fields are. For example,ordinary differential equations andsymplectic geometry are generally viewed as purely mathematical disciplines, whereasdynamical systems andHamiltonian mechanics belong to mathematical physics.John Herapath used the term for the title of his 1847 text on "mathematical principles of natural philosophy", the scope at that time being "the causes of heat, gaseous elasticity, gravitation, and other great phenomena of nature".[4]
The term "mathematical physics" is sometimes used to denote research aimed at studying and solving problems in physics orthought experiments within a mathematicallyrigorous framework. In this sense, mathematical physics covers a very broad academic realm distinguished only by the blending of some mathematical aspect and theoretical physics aspect. Although related totheoretical physics,[5] mathematical physics in this sense emphasizes the mathematical rigour of the similar type as found in mathematics.
On the other hand, theoretical physics emphasizes the links to observations andexperimental physics, which often requires theoretical physicists (and mathematical physicists in the more general sense) to useheuristic,intuitive, or approximate arguments.[6] Such arguments are not considered rigorous by mathematicians.
Such mathematical physicists primarily expand and elucidate physicaltheories. Because of the required level of mathematical rigour, these researchers often deal with questions that theoretical physicists have considered to be already solved. However, they can sometimes show that the previous solution was incomplete, incorrect, or simply too naïve. Issues about attempts to infer the second law ofthermodynamics fromstatistical mechanics are examples.[citation needed] Other examples concern the subtleties involved with synchronisation procedures in special and general relativity (Sagnac effect andEinstein synchronisation).
The effort to put physical theories on a mathematically rigorous footing not only developed physics but also has influenced developments of some mathematical areas. For example, the development of quantum mechanics and some aspects offunctional analysis parallel each other in many ways. The mathematical study ofquantum mechanics,quantum field theory, andquantum statistical mechanics has motivated results inoperator algebras. The attempt to construct a rigorous mathematical formulation ofquantum field theory has also brought about some progress in fields such asrepresentation theory.
There is a tradition of mathematical analysis of nature that goes back to the ancient Greeks; examples includeEuclid (Optics),Archimedes (On the Equilibrium of Planes,On Floating Bodies), andPtolemy (Optics,Harmonics).[7][8] Later,Islamic andByzantine scholars built on these works, and these ultimately were reintroduced or became available to the West in the12th century and during theRenaissance.
In the first decade of the 16th century, amateur astronomerNicolaus Copernicus proposedheliocentrism, and published a treatise on it in 1543. He retained thePtolemaic idea ofepicycles, and merely sought to simplify astronomy by constructing simpler sets of epicyclic orbits. Epicycles consist of circles upon circles. According toAristotelian physics, the circle was the perfect form of motion, and was the intrinsic motion of Aristotle'sfifth element—the quintessence or universal essence known in Greek asaether for the Englishpure air—that was the pure substance beyond thesublunary sphere, and thus was celestial entities' pure composition. The GermanJohannes Kepler [1571–1630],Tycho Brahe's assistant, modified Copernican orbits toellipses, formalized in the equations of Kepler'slaws of planetary motion.
An enthusiastic atomist,Galileo Galilei in his 1623 bookThe Assayer asserted that the "book of nature is written in mathematics".[9] His 1632 book, about his telescopic observations, supported heliocentrism.[10] Having made use of experimentation, Galileo then refuted geocentriccosmology by refuting Aristotelian physics itself. Galileo's 1638 bookDiscourse on Two New Sciences established the law of equal free fall as well as the principles of inertial motion, two central concepts of what today is known asclassical mechanics.[10] By the Galileanlaw of inertia as well as the principle ofGalilean invariance, also called Galilean relativity, for any object experiencing inertia, there is empirical justification for knowing only that it is atrelative rest orrelative motion—rest or motion with respect to another object.
René Descartes developed a complete system of heliocentric cosmology anchored on the principle of vortex motion,Cartesian physics, whose widespread acceptance helped bring the demise of Aristotelian physics. Descartes used mathematical reasoning as a model for science, and developedanalytic geometry, which in time allowed the plotting of locations in 3D space (Cartesian coordinates) and marking their progressions along the flow of time.[11]
Christiaan Huygens, a talented mathematician and physicist and older contemporary of Newton, was the first to successfully idealize a physical problem by a set of mathematical parameters inHorologium Oscillatorum (1673), and the first to fully mathematize a mechanistic explanation of an unobservable physical phenomenon inTraité de la Lumière (1690). He is thus considered a forerunner oftheoretical physics and one of the founders of modern mathematical physics.[12][13]
The prevailing framework for science in the 16th and early 17th centuries was one borrowed fromAncient Greek mathematics, where geometrical shapes formed the building blocks to describe and think about space, and time was often thought as a separate entity. With the introduction of algebra into geometry, and with it the idea of a coordinate system, time and space could now be thought as axes belonging to the same plane. This essential mathematical framework is at the base of all modern physics and used in all further mathematical frameworks developed in next centuries.
By the middle of the 17th century, important concepts such as thefundamental theorem of calculus (proved in 1668 by Scottish mathematicianJames Gregory) and finding extrema and minima of functions via differentiation using Fermat's theorem (by French mathematicianPierre de Fermat) were already known before Leibniz and Newton.[14]Isaac Newton (1642–1727) developedcalculus (althoughGottfried Wilhelm Leibniz developed similar concepts outside the context of physics) andNewton's method to solve problems in mathematics and physics. He was extremely successful in his application ofcalculus and other methods to the study of motion. Newton's theory of motion, culminating in hisPhilosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) in 1687, modeled three Galilean laws of motion along with Newton'slaw of universal gravitation on a framework ofabsolute space—hypothesized by Newton as a physically real entity of Euclidean geometric structure extending infinitely in all directions—while presumingabsolute time, supposedly justifying knowledge of absolute motion, the object's motion with respect to absolute space.[15] The principle of Galilean invariance/relativity was merely implicit in Newton's theory of motion. 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.[16]
In the 18th century, the SwissDaniel Bernoulli (1700–1782) made contributions tofluid dynamics, andvibrating strings. The SwissLeonhard Euler (1707–1783) did special work invariational calculus, dynamics, fluid dynamics, and other areas. Also notable was the Italian-born Frenchman,Joseph-Louis Lagrange (1736–1813) for work inanalytical mechanics: he formulatedLagrangian mechanics) and variational methods. A major contribution to the formulation of Analytical Dynamics calledHamiltonian dynamics was also made by the Irish physicist, astronomer and mathematician,William Rowan Hamilton (1805–1865). Hamiltonian dynamics had played an important role in the formulation of modern theories in physics, including field theory and quantum mechanics. The French mathematical physicistJoseph Fourier (1768 – 1830) introduced the notion ofFourier series to solve theheat equation, giving rise to a new approach to solving partial differential equations by means ofintegral transforms.
A couple of decades ahead of Newton's publication of a particle theory of light, the DutchChristiaan Huygens (1629–1695) developed the wave theory of light, published in 1690. By 1804,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 theluminiferous aether, was accepted.Jean-Augustin Fresnel modeled hypothetical behavior of the aether. The English physicistMichael Faraday introduced the theoretical concept of a field—not action at a distance. Mid-19th century, the ScottishJames Clerk Maxwell (1831–1879) reduced electricity and magnetism to Maxwell's electromagnetic field theory, whittled down by others to the fourMaxwell's equations. Initially, optics was found consequent of[clarification needed] Maxwell's field. Later, radiation and then today's knownelectromagnetic spectrum were found also consequent of[clarification needed] this electromagnetic field.
By the 1880s, 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 objectsin the electromagnetic field. And yet no violation ofGalilean invariance within physical interactions among objects was detected. As Maxwell's electromagnetic field was modeled as oscillations of theaether, physicists inferred that motion within the aether resulted inaether drift, shifting the electromagnetic field, explaining the observer's missing speed relative to it. TheGalilean 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 onCartesian coordinates, but this process was replaced byLorentz transformation, modeled by the DutchHendrik Lorentz [1853–1928].
In 1887, experimentalists Michelson and Morley failed to detect aether drift, however. It was hypothesized that motioninto the aether prompted aether's shortening, too, as modeled in theLorentz contraction. It was hypothesized that the aether thus kept Maxwell's electromagnetic field aligned with the principle of Galilean invariance across allinertial frames of reference, while Newton's theory of motion was spared.
Austrian theoretical physicist and philosopherErnst Mach criticized Newton's postulated absolute space. MathematicianJules-Henri Poincaré (1854–1912) questioned even absolute time. In 1905,Pierre Duhem published a devastating criticism of the foundation of Newton's theory of motion.[16] Also in 1905,Albert Einstein (1879–1955) published hisspecial theory of relativity, newly explaining both the electromagnetic field's invariance and Galilean invariance by discarding all hypotheses concerning aether, including the existence of aether itself. Refuting the framework of Newton's theory—absolute space and absolute time—special relativity refers torelative space andrelative time, wherebylength contracts andtime dilates along the travel pathway of an object.
Cartesian coordinates arbitrarily used rectilinear coordinates. Gauss, inspired by Descartes' work, introduced the curved geometry, replacing rectilinear axis by curved ones. Gauss also introduced another key tool of modern physics, the curvature. Gauss's work was limited to two dimensions. Extending it to three or more dimensions introduced a lot of complexity, with the need of the (not yet invented) tensors. It was Riemman the one in charge to extend curved geometry to N dimensions. In 1908, Einstein's former mathematics professorHermann Minkowski, applied the curved geometry construction to model 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.[17] Einstein initially called this "superfluous learnedness", but later usedMinkowski spacetime with great elegance in hisgeneral theory of relativity,[18] 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 withGaussian coordinates, and replaces Newton's claimed empty yet Euclidean space traversed instantly by Newton'svector of hypothetical gravitational force—an instantaction at a distance—with a gravitationalfield. The gravitational field isMinkowski spacetime itself, the 4Dtopology of Einstein aether modeled on aLorentzian manifold that "curves" geometrically, according to theRiemann curvature tensor. The concept of Newton's gravity: "two masses attract each other" replaced by the geometrical argument: "mass transform curvatures ofspacetime and free falling particles with mass move along a geodesic curve in the spacetime" (Riemannian geometry already existed before the 1850s, by mathematiciansCarl Friedrich Gauss andBernhard Riemann in search for intrinsic geometry and non-Euclidean geometry.), in the vicinity of either mass or energy. (Under special relativity—a special case of general relativity—even massless energy exerts gravitational effect by itsmass equivalence locally "curving" the geometry of the four, unified dimensions of space and time.)
Another revolutionary development of the 20th century wasquantum theory, which emerged from the seminal contributions ofMax Planck (1856–1947) (onblack-body radiation) and Einstein's work on thephotoelectric effect. In 1912, a mathematicianHenri Poincare publishedSur la théorie des quanta.[19][20] He introduced the first non-naïve definition of quantization in this paper. The development of early quantum physics followed by a heuristic framework devised byArnold Sommerfeld (1868–1951) andNiels Bohr (1885–1962), but this was soon replaced by thequantum mechanics developed byMax Born (1882–1970),Louis de Broglie (1892–1987),Werner Heisenberg (1901–1976),Paul Dirac (1902–1984),Erwin Schrödinger (1887–1961),Satyendra Nath Bose (1894–1974), andWolfgang Pauli (1900–1958). This revolutionary theoretical framework is based on a probabilistic interpretation of states, and evolution and measurements in terms ofself-adjoint operators on an infinite-dimensional vector space. That is calledHilbert space (introduced by mathematiciansDavid Hilbert (1862–1943),Erhard Schmidt (1876–1959) andFrigyes Riesz (1880–1956) in search of generalization of Euclidean space and study of integral equations), and rigorously defined within the axiomatic modern version byJohn von Neumann in his celebrated bookMathematical Foundations of Quantum Mechanics, where he built up a relevant part of modern functional analysis on Hilbert spaces, thespectral theory (introduced byDavid Hilbert who investigatedquadratic forms with infinitely many variables. Many years later, it had been revealed that his spectral theory is associated with the spectrum of the hydrogen atom. He was surprised by this application.) in particular. Paul Dirac used algebraic constructions to produce a relativistic model for theelectron, predicting itsmagnetic moment and the existence of its antiparticle, thepositron.
List of prominent contributors to mathematical physics in the 20th century
^Definition from theJournal of Mathematical Physics."Archived copy". Archived fromthe original on 2006-10-03. Retrieved2006-10-03.{{cite web}}: CS1 maint: archived copy as title (link)
^Quote: " ... a negative definition of the theorist refers to his inability to make physical experiments, while a positive one... implies his encyclopaedic knowledge of physics combined with possessing enough mathematical armament. 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.", Ya. Frenkel, as related in A.T. Filippov,The Versatile Soliton, pg 131. Birkhauser, 2000.
^Quote: "Physical theory is something like a suit sewed for Nature. Good theory is like a good suit. ... Thus the theorist is like a tailor." Ya. Frenkel, as related in Filippov (2000), pg 131.
^Pellegrin, P. (2000). Brunschwig, J.; Lloyd, G. E. R. (eds.). "Physics".Greek Thought: A Guide to Classical Knowledge:433–451.
^abImre Lakatos, auth, Worrall J & Currie G, eds,The Methodology of Scientific Research Programmes: Volume 1: Philosophical Papers (Cambridge: Cambridge University Press, 1980), pp213–214,220
^Minkowski, Hermann (1908–1909), "Raum und Zeit" [Space and Time], Physikalische Zeitschrift, 10: 75–88. Actually the union of space and time was implicit in Descartes's work first, with space and time being represented as axis of coordinates, and in Lorentz transformation later, but its physical interpretation was still hidden to common sense.
^Salmon WC & Wolters G, eds,Logic, Language, and the Structure of Scientific Theories (Pittsburgh: University of Pittsburgh Press, 1994), p125
^McCormmach, Russell (Spring 1967). "Henri Poincaré and the Quantum Theory".Isis.58 (1):37–55.doi:10.1086/350182.S2CID120934561.
^Irons, F. E. (August 2001). "Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms".American Journal of Physics.69 (8):879–84.Bibcode:2001AmJPh..69..879I.doi:10.1119/1.1356056.
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Adam, John A. (2017),Rays, Waves, and Scattering: Topics in Classical Mathematical Physics, Princeton University Press.,ISBN978-0-691-14837-3
Bloom, Frederick (1993),Mathematical Problems of Classical Nonlinear Electromagnetic Theory, CRC Press,ISBN0-582-21021-6
Boyer, Franck; Fabrie, Pierre (2013),Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Springer,ISBN978-1-4614-5974-3
Colton, David; Kress, Rainer (2013),Integral Equation Methods in Scattering Theory, Society for Industrial and Applied Mathematics,ISBN978-1-611973-15-0
Galdi, Giovanni P. (2011),An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems (2nd ed.), Springer,ISBN978-0-387-09619-3
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Kirsch, Andreas; Hettlich, Frank (2015),The Mathematical Theory of Time-Harmonic Maxwell's Equations: Expansion-, Integral-, and Variational Methods, Springer,Bibcode:2015mttm.book.....K,ISBN978-3-319-11085-1
Knauf, Andreas (2018),Mathematical Physics: Classical Mechanics, Springer,ISBN978-3-662-55772-3
Lechner, Kurt (2018),Classical Electrodynamics: A Modern Perspective, Springer,ISBN978-3-319-91808-2
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