Functional near-infrared spectroscopy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Functional_near-infrared_spectroscopy 

fNIRS with a Gowerlabs NTS system

Functional near-infrared spectroscopy (fNIRS), sometimes referred to as NIRS or Optical Topography (OT), is an optical brain monitoring technique which uses near-infrared spectroscopy for the purpose of functional neuroimaging. Using fNIRS, brain activity is measured by using near-infrared light to estimate cortical hemodynamic activity that occurs in response to neural activity. The use of fNIRS has led to advances in different fields such as cognitive neuroscience, clinical applications, developmental science and sport and exercise science. The signal is often compared with the BOLD signal measured by fMRI and is capable of measuring changes both in oxy- and deoxyhemoglobin concentration, but can only measure from regions near the cortical surface.

How it Works

Basic functional near infrared spectroscopy (fNIRS) abbreviations

fNIRS estimates the concentration of hemoglobin from changes in absorption of near infrared light. As light moves or propagates through the head, it is alternately scattered or absorbed by the tissue through which it travels. Because hemoglobin is a significant absorber of near-infrared light, changes in absorbed light can be used to reliably measure changes in hemoglobin concentration. Different fNIRS techniques can also use the way in which light propagates to estimate blood volume and oxygenation. The technique is safe, non-invasive, and can be used with other imaging modalities.

Oxygenated and deoxygenated hemoglobin

fNIRS is a non-invasive imaging method involving the quantification of chromophore concentration resolved from the measurement of near infrared (NIR) light attenuation or temporal or phasic changes. The technique takes advantage of the optical window in which (a) skin, tissue, and bone are mostly transparent to NIR light (700–900 nm spectral interval) and (b) hemoglobin (Hb) and deoxygenated-hemoglobin (deoxy-Hb) are strong absorbers of light .

Absorption spectra for oxy-Hb and deoxy-Hb for near-infrared wavelengths

There are different ways for infrared light to interact with the brain tissue. fNIRS focuses primarily on absorption: differences in the absorption spectra of deoxy-Hb and oxy-Hb allow the measurement of relative changes in hemoglobin concentration through the use of light attenuation at multiple wavelengths. Two or more wavelengths are selected, with one wavelength above and one below the isosbestic point of 810 nm—at which deoxy-Hb and oxy-Hb have identical absorption coefficients. Using the modified Beer-Lambert law (mBLL), relative changes in concentration can be calculated as a function of total photon path length.

Typically, the light emitter and detector are placed ipsilaterally (each emitter/detector pair on the same side) on the subject's skull so recorded measurements are due to back-scattered (reflected) light following elliptical pathways. fNIRS is most sensitive to hemodynamic changes which occur nearest to the scalp and these superficial artifacts are often addressed using additional light detectors located closer to the light source (short-separation detectors).

Modified Beer–Lambert law

Changes in light intensity can be related to changes in relative concentrations of hemoglobin through the modified Beer–Lambert law (mBLL). The Beer Lambert-law has to deal with concentration of hemoglobin. This technique also measures relative changes in light attenuation as well as using mBLL to quantify hemoglobin concentration changes.

Equipment and Software

fNIRS cap

10-20 system

fNIRS electrode locations can be defined using a variety of layouts, including names and locations that are specified by the International 10–20 system as well as other layouts that are specifically optimized to maintain a consistent 30mm distance between each location. In addition to the standard positions of electrodes, short separation channels can be added. Short separation channels allow the measurement of scalp signals. Since the short separation channels measure the signal coming from the scalp, they allow the removal of the signal of superficial layers. This leaves behind the actual brain response. Short separation channel detectors are usually placed 8mm away from a source. They do not need to be in a specific direction or in the same direction as a detector.

Software

HOMER3

HOMER3 allows users to obtain estimates and maps of brain activation. It is a set of matlab scripts used for analyzing fNIRS data. This set of scripts has evolved since the early 1990s first as the Photon Migration Imaging toolbox, then HOMER1 and HOMER2, and now HOMER3.

NIRS toolbox

This toolbox is a set of Matlab-based tools for the analysis of functional near-infrared spectroscopy (fNIRS). This toolbox defines the +nirs namespace and includes a series of tools for signal processing, display, and statistics of fNIRS data. This toolbox is built around an object-oriented framework of Matlab classes and namespaces.

AtlasViewer

AtlasViewer allows fNIRS data to be visualized on a model of the brain. In addition, it also allows the user to design probes which can eventually be placed onto a subject.

History

The conceptual origins of functional near-infrared spectroscopy date to 1977, when Frans Jöbsis demonstrated that near-infrared light could penetrate biological tissue and report on oxygenation changes in vivo. Throughout the 1980s, this observation informed early cerebral oximetry instruments used in neonatal medicine and physiology research. In the early 1990s, several research groups independently reported changes in hemoglobin concentrations during cognitive tasks, establishing fNIRS as a viable functional neuroimaging modality. Progress occurred internationally: researchers in Japan, Europe, and North America contributed to theory, hardware, and methodological refinement. The term “optical topography” was introduced in 1995 when Hitachi released one of the first practical multi-channel commercial systems. With rising sensor density, signal analysis advances, and portable configurations, fNIRS expanded rapidly into cognitive neuroscience, clinical medicine, developmental psychology, and applied research.

Diffuse Optical Spectroscopy/Imaging (DOI/DOS)

Spectroscopic techniques

Continuous wave

Continuous wave (CW) system uses light sources with constant frequency and amplitude. In fact, to measure absolute changes in HbO concentration with the mBLL, we need to know photon path-length. However, CW-fNIRS does not provide any knowledge of photon path-length, so changes in HbO concentration are relative to an unknown path-length. Many CW-fNIRS commercial systems use estimations of photon path-length derived from computerized Monte-Carlo simulations and physical models, to approximate absolute quantification of hemoglobin concentrations.

${\displaystyle \text{OD}=\ln (I_0/I)=ϵ\cdot [X]\cdot l\cdot \text{DPF}+G}$

Where ${\displaystyle \text{OD}}$ is the optical density or attenuation, ${\displaystyle I_0}$ is emitted light intensity, ${\displaystyle I}$ is measured light intensity, ${\displaystyle ϵ}$ is the attenuation coefficient, ${\displaystyle [X]}$ is the chromophore concentration, ${\displaystyle l}$ is the distance between source and detector and ${\displaystyle \text{DPF}}$ is the differential path length factor, and ${\displaystyle G}$ is a geometric factor associated with scattering.

When the attenuation coefficients ${\displaystyle ϵ}$ are known, constant scattering loss is assumed, and the measurements are treated differentially in time, the equation reduces to:

${\displaystyle \mathrm{\Delta }[X]=\mathrm{\Delta }\frac{\text{OD}}{ϵd}}$

Where ${\displaystyle d}$ is the total corrected photon path-length.

Using a dual wavelength system, measurements for HbO₂ and Hb can be solved from the matrix equation:

${\displaystyle \left(\begin{array}{c}\mathrm{\Delta }{\text{OD}}_{{\lambda }_1}\\ \mathrm{\Delta }{\text{OD}}_{{\lambda }_2}\end{array}\right)=\left(\begin{array}{cc}{ϵ}_{{\lambda }_1}^{\text{Hb}}d& {ϵ}_{{\lambda }_1}^{{\text{HbO}}_2}d\\ {ϵ}_{{\lambda }_2}^{\text{Hb}}d& {ϵ}_{{\lambda }_2}^{{\text{HbO}}_2}d\end{array}\right)\left(\begin{array}{c}\mathrm{\Delta }[X{]}^{\text{Hb}}\\ \mathrm{\Delta }[X{]}^{{\text{HbO}}_2}\end{array}\right)}$

Due to their simplicity and cost-effectiveness, CW-fNIRS is by far the most common form of functional NIRS since it is the cheapest to make, applicable with more channels, and ensures a high temporal resolution. However, it does not distinguish between absorption and scattering changes, and cannot measure absolute absorption values: which means that it is only sensitive to relative change in HbO concentration.

Still, the simplicity and cost-effectiveness of CW-based devices prove themselves to be the most favorable for a number of clinical applications: neonatal care, patient monitoring systems, diffuse optical tomography, and so forth. Moreover, thanks to its portability, wireless CW systems have been developed—allowing individuals to be monitored in ambulatory, clinical and sports environments. 

Frequency domain

Frequency domain (FD) system comprises NIR laser sources which provide an amplitude-modulated sinusoid at frequencies near 100 MHz. FD-fNIRS measures attenuation, phase shift and the average path length of light through the tissue.

Changes in the back-scattered signal's amplitude and phase provide a direct measurement of absorption and scattering coefficients of the tissue, thus obviating the need for information about photon path-length; and from the coefficients we determine the changes in the concentration of hemodynamic parameters.

Because of the need for modulated lasers as well as phasic measurements, FD system-based devices are more technically complex (therefore more expensive and much less portable) than CW-based ones. However, the system is capable of providing absolute concentrations of HbO and HbR.

Time domain

Time domain (TD) system introduces a short NIR pulse with a pulse length usually in the order of picoseconds—around 70 ps. Through time-of-flight measurements, photon path-length may be directly observed by dividing resolved time by the speed of light. Information about hemodynamic changes can be found in the attenuation, decay, and time profile of the back-scattered signal. For this photon-counting technology is introduced, which counts 1 photon for every 100 pulses to maintain linearity. TD-fNIRS does have a slow sampling rate as well as a limited number of wavelengths. Because of the need for a photon-counting device, high-speed detection, and high-speed emitters, time-resolved methods are the most expensive and technically complicated.

TD-based devices have the highest depth sensitivity and are capable of presenting most accurate values of baseline hemoglobin concentration and oxygenation.

Diffuse correlation spectroscopy

Diffuse correlation spectroscopy (DCS) is a non-invasive optical imaging technique that utilizes coherent near-infrared light to measure local microvascular cerebral blood flow by quantifying the temporal light intensity fluctuations generated by dynamic scattering of moving red blood cells. This dynamic scattering from moving cells causes the detected intensity to temporally fluctuate. These fluctuations can be quantified by the temporal intensity autocorrelation curve of a single speckle. The decay of the autocorrelation curve is fitted with the solution of the correlation diffusion equation to obtain an index of cerebral blood flow.

Measurement of brain oxyhemoglobin and deoxyhemoglobin concentration changes at high alltitude induced hypoxia with a portable fNIRS device (PortaLite, Artinis Medical Systems)

Application

fNIRS has been successfully implemented as a control signal for brain–computer interface systems. Modern fNIRS systems are combined with virtual or augmented reality in studies on brain-computer interfaces, neurorehabilitation or social perception. fNIRS can be used to monitor musicians' brain activity while playing musical instruments. fNIRS is compatible with some other neuroimaging modalities, including: MRI, EEG, and MEG. fNIRS has also been used to measure the effects of transcranial magnetic stimulation (TMS).

Hypoxia & altitude studies

With our constant need for oxygen, our body has developed multiple mechanisms that detect oxygen levels, which in turn can activate appropriate responses to counter hypoxia and generate a higher oxygen supply. Moreover, understanding the physiological mechanism underlying the bodily response to oxygen deprivation is of major importance and NIRS devices have shown to be a great tool in this field of research.

Mobile and wireless fNIRS and EEG systems synchronized with all-in-one head mounted display (PhotonCap, Cortivision)

Brain mapping

Functional connectivity

fNIRS measurements can be used to calculate a limited degree of functional connectivity. Multi-channel fNIRS measurements create a topographical map of neural activation, whereby temporal correlation between spatially separated events can be analyzed. Functional connectivity is typically assessed in terms correlations between the hemodynamic responses of spatially distinct regions of interest (ROIs). In brain studies, functional connectivity measurements are commonly taken for resting state patient data, as well as data recorded over stimulus paradigms. A study led by Alessandro Crimi team highlighted that the functional connectivity measures obtained with fNIRS measurements are quite different from those obtained via EEG caps.

Cerebral oximetry

NIRS monitoring is helpful in a number of ways. Preterm infants can be monitored reducing cerebral hypoxia and hyperoxia with different patterns of activities. It is an effective aid in Cardiopulmonary bypass, is strongly considered to improve patient outcomes and reduce costs and extended stays.

There are inconclusive results for use of NIRS with patients with traumatic brain injury, so it has been concluded that it should remain a research tool.

Diffuse optical tomography

Diffuse optical tomography is the 3D version of Diffuse optical imaging. Diffuse optical images are obtained using NIRS or fluorescence-based methods. These images can be used to develop a 3D volumetric model which is known as the Diffuse Optical Tomography.

Functional neuroimaging

The use of fNIRS as a functional neuroimaging method relies on the principle of neuro-vascular coupling also known as the haemodynamic response or blood-oxygen-level dependent (BOLD) response. This principle also forms the core of fMRI techniques. Through neuro-vascular coupling, neuronal activity is linked to related changes in localized cerebral blood flow. fNIRS and fMRI are sensitive to similar physiologic changes and are often comparative methods. Studies relating fMRI and fNIRS show highly correlated results in cognitive tasks. fNIRS has several advantages in cost and portability over fMRI, but cannot be used to measure cortical activity more than 4 cm deep due to limitations in light emitter power and has more limited spatial resolution. fNIRS includes the use of diffuse optical tomography (DOT/NIRDOT) for functional purposes. Multiplexing fNIRS channels can allow 2D topographic functional maps of brain activity (e.g. with Hitachi ETG-4000, Artinis Oxymon, NIRx NIRScout, etc.) while using multiple emitter spacings may be used to build 3D tomographic maps.

Hyperscanning

See also: Neural synchrony § Hyperscanning

Hyperscanning involves two or more brains monitored simultaneously to investigate interpersonal (across-brains) neural correlates in various social situations, which proves fNIRS to be a suitable modality for investigating live brain-to-brain social interactions.

Advantages and Limitations

Advantages

The advantages of fNIRS are, among other things: noninvasiveness, low-cost modalities, perfect safety, high temporal resolution, compatibility with other imaging modalities, and multiple hemodynamic biomarkers. People who use medical implants in their brains such as cochlear implant, or metal brain plates are able to wear fNIRs with no risk of device displacement or heating, which can occur in MRI.

Limitations

fNIRs have low brain sensitivity due to it only being able to detect changes on the cortical surface and low spatial resolution, about 1-3 centimeters deep. The signal is sensitive to hair and skin pigment differences, making it difficult to do between-subject designs. Dense or extremely curly hair may prohibit placement of optodes close to the scalp, limiting the ability to use the technique with all individuals. Although this device can be used with brain implant users, the signal over these areas will be disrupted meaning that spatial oversampling must occur to retain spatial resolution of the target area. Alternative strategies must be implemented such as more pressure on optodes, or specific montages in order to compensate for these limitations.

Mathematical physics

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From Wikipedia, the free encyclopedia

Mathematical physics is the development of mathematical methods for use in physics and their applications. A broader definition would include the development of mathematical ideas inspired by physics, known as physical mathematics.

Scope

There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world.

Classical mechanics

Main articles: Lagrangian mechanics and Hamiltonian mechanics

Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead to an understanding of the deep interplay between the notions of symmetry and conserved quantities during the dynamical evolution of mechanical systems, as embodied within the most elementary formulation of Noether's theorem. These approaches and ideas have been extended to other areas of physics, such as statistical mechanics, continuum mechanics, classical field theory, and quantum field theory. Moreover, they have provided multiple examples and ideas in differential geometry (e.g., several notions in symplectic geometry and vector bundles).

Partial differential equations

Main article: Partial differential equations

Within mathematics proper, the theory of partial differential equation, variational calculus, Fourier analysis, potential theory, and vector analysis are perhaps most closely associated with mathematical physics. These fields were developed intensively from the second half of the 18th century (by, for example, D'Alembert, Euler, and Lagrange) until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics.

Quantum theory

Main article: Quantum mechanics

An example of mathematical physics: solutions of the Schrödinger equation for quantum harmonic oscillators (left) with their amplitudes (right).

The theory of atomic spectra (and, later, quantum mechanics) developed almost concurrently with some parts of the mathematical fields of linear algebra, the spectral theory of operators, operator algebras and, more broadly, functional analysis. Nonrelativistic quantum mechanics includes Schrödinger operators, and it has connections to atomic and molecular physics. Quantum information theory is another subspecialty.

Relativity and quantum relativistic theories

Main articles: Theory of relativity and Quantum field theory

The special and general theories of relativity require a rather different type of mathematics. This was group theory, which played an important role in both quantum field theory and differential geometry. This was, however, gradually supplemented by topology and functional analysis in the mathematical description of cosmological as well as quantum field theory phenomena. In the mathematical description of these physical areas, some concepts in homological algebra and category theory are also important.

Statistical mechanics

Main article: Statistical mechanics

Statistical mechanics forms a separate field, which includes the theory of phase transitions. It relies upon the Hamiltonian mechanics (or its quantum version) and it is closely related with the more mathematical ergodic theory and some parts of probability theory. There are increasing interactions between combinatorics and physics, in particular statistical physics.

Usage

Relationship between mathematics and physics

The usage of the term "mathematical physics" is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics are not, in fact, considered parts of mathematical physics, while other closely related fields are. For example, ordinary differential equations and symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian 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".

Mathematical vs. theoretical physics

The term "mathematical physics" is sometimes used to denote research aimed at studying and solving problems in physics or thought experiments within a mathematically rigorous 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 to theoretical physics, 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 and experimental physics, which often requires theoretical physicists (and mathematical physicists in the more general sense) to use heuristic, intuitive, or approximate arguments. Such arguments are not considered rigorous by mathematicians.

Such mathematical physicists primarily expand and elucidate physical theories. 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 of thermodynamics from statistical mechanics are examples. Other examples concern the subtleties involved with synchronisation procedures in special and general relativity (Sagnac effect and Einstein 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 of functional analysis parallel each other in many ways. The mathematical study of quantum mechanics, quantum field theory, and quantum statistical mechanics has motivated results in operator algebras. The attempt to construct a rigorous mathematical formulation of quantum field theory has also brought about some progress in fields such as representation theory.

Prominent mathematical physicists

Before Newton

There is a tradition of mathematical analysis of nature that goes back to the ancient Greeks; examples include Euclid (Optics), Archimedes (On the Equilibrium of Planes, On Floating Bodies), and Ptolemy (Optics, Harmonics). Later, Islamic and Byzantine scholars built on these works, and these ultimately were reintroduced or became available to the West in the 12th century and during the Renaissance.

Kepler's three laws of orbital motion

In the first decade of the 16th century, amateur astronomer Nicolaus Copernicus proposed heliocentrism, and published a treatise on it in 1543. He retained the Ptolemaic idea of epicycles, and merely sought to simplify astronomy by constructing simpler sets of epicyclic orbits. Epicycles consist of circles upon circles. 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 [1571–1630], 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 1623 book The Assayer asserted that the "book of nature is written in mathematics". His 1632 book, about his telescopic observations, supported heliocentrism. Having made use of experimentation, Galileo then refuted geocentric cosmology by refuting Aristotelian physics itself. Galileo's 1638 book Discourse 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 as classical mechanics. By the Galilean law of inertia as well as the principle of Galilean invariance, also called Galilean relativity, for any object experiencing inertia, there is empirical justification for knowing only that it is at relative rest or relative 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 developed analytic geometry, which in time allowed the plotting of locations in 3D space (Cartesian coordinates) and marking their progressions along the flow of time.

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 in Horologium Oscillatorum (1673), and the first to fully mathematize a mechanistic explanation of an unobservable physical phenomenon in Traité de la Lumière (1690). He is thus considered a forerunner of theoretical physics and one of the founders of modern mathematical physics.

Newtonian and post-Newtonian physics

The prevailing framework for science in the 16th and early 17th centuries was one borrowed from ancient 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.

An animation for Newton's shell theorem

By the middle of the 17th century, important concepts such as the fundamental theorem of calculus (proved in 1668 by Scottish mathematician James Gregory) and finding extrema and minima of functions via differentiation using Fermat's theorem (by French mathematician Pierre de Fermat) were already known before Leibniz and Newton. Isaac Newton (1642–1727) developed calculus (although Gottfried Wilhelm Leibniz developed similar concepts outside the context of physics) and Newton's method to solve problems in mathematics and physics. He was extremely successful in his application of calculus and other methods to the study of motion. Newton's theory of motion, culminating in his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) in 1687, 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. The principle of Galilean invariance/relativity was merely implicit in Newton's theory of motion. Having ostensibly reduced the Kepler's laws as well as Galileo's terrestrial laws of motion to a unifying force, Newton achieved great mathematical rigor, but with theoretical laxity.

Standing waves on a string

In the 18th century, the Swiss Daniel Bernoulli (1700–1782) made contributions to fluid dynamics, and vibrating strings. The Swiss Leonhard Euler (1707–1783) did special work in variational calculus, dynamics, fluid dynamics, and other areas. Also notable was the Italian Joseph-Louis Lagrange (1736–1813), 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 (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 physicist Joseph Fourier (1768 – 1830) 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, following mathematicians in France, Germany and England had contributed to mathematical physics. The French Pierre-Simon Laplace (1749–1827) made paramount contributions to mathematical astronomy, potential theory. Siméon Denis Poisson (1781–1840) worked in analytical mechanics and potential theory. In Germany, Carl Friedrich Gauss (1777–1855) made key contributions to the theoretical foundations of electricity, magnetism, mechanics, and fluid dynamics. In England, George Green (1793–1841) published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism in 1828, 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 (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 the luminiferous aether, was accepted. Jean-Augustin Fresnel modeled hypothetical behavior of the aether. The English physicist Michael Faraday introduced the theoretical concept of a field—not action at a distance. Mid-19th century, the Scottish James Clerk Maxwell (1831–1879) 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 Maxwell's field. Later, radiation and then today's known electromagnetic spectrum were found also consequent of this electromagnetic field.

An illustration of Stokes' theorem in the context of electrodynamics

The English physicist Lord Rayleigh [1842–1919] worked on sound. The Irishmen William Rowan Hamilton (1805–1865), George Gabriel Stokes (1819–1903) and Lord Kelvin (1824–1907) 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 mathematician Carl Gustav Jacobi (1804–1851) in particular referring to canonical transformations. The German Hermann von Helmholtz (1821–1894) made substantial contributions in the fields of electromagnetism, waves, fluids, and sound. In the United States, the pioneering work of Josiah Willard Gibbs (1839–1903) became the basis for statistical mechanics. Fundamental theoretical results in this area were achieved by the German Ludwig Boltzmann (1844–1906). Together, these individuals laid the foundations of electromagnetic theory, fluid dynamics, and statistical mechanics.

Relativistic

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 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 1908, Einstein's former mathematics professor Hermann Minkowski, applied linear algebra 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.

In 1887, 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.

Austrian theoretical physicist and philosopher Ernst Mach criticized Newton's postulated absolute space. Mathematician Jules-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. Also in 1905, Albert Einstein published his special 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 to relative space and relative time, whereby length contracts and time dilates along the travel pathway of an object.

Cartesian coordinates arbitrarily used rectilinear coordinates. Carl Friedrich Gauss, inspired by Descartes' work, introduced curved geometry, replacing rectilinear axes by curved ones. Gauss also introduced another key tool of modern physics: curvature. However, Gauss's work was limited to two dimensions; extending it to three or more dimensions introduced a great deal of complexity, and the need for the (not yet invented) tensors. It was Bernhard Riemann who extended curved geometry to N dimensions with differential geometry, now known as Riemannian geometry.

Einstein initially called this "superfluous learnedness", but later used Minkowski spacetime as tangent space in his general theory of relativity, 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. The concept of Newton's gravity: "two masses attract each other" replaced by the geometrical argument: "mass transform curvatures of spacetime and free falling particles with mass move along a geodesic curve in the spacetime". The innovations in geometry were foreshadowed by non-Euclidean geometry and the explorations of curvature by C. F. Gauss. Even massless energy exerts gravitational effect by its mass equivalence locally "curving" the geometry of the four, unified dimensions of space and time.

Quantum mechanics

Another revolutionary development of the 20th century was quantum theory, which emerged from the seminal contributions of Max Planck (1856–1947) (on black-body radiation) and Einstein's work on the photoelectric effect. In 1912, mathematician Henri Poincaré published Sur la théorie des quanta. He introduced the first non-naïve definition of quantization in this paper. The development of early quantum physics followed by a heuristic framework devised by Arnold Sommerfeld (1868–1951) and Niels Bohr (1885–1962), but this was soon replaced by the quantum mechanics developed by Max 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), and Wolfgang Pauli (1900–1958).

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 by mathematicians David Hilbert (1862–1943), Erhard Schmidt (1876–1959) and Frigyes Riesz (1880–1956) in search of generalization of Euclidean space and study of integral equations), 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 (introduced by David Hilbert who investigated quadratic 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 the electron, predicting its magnetic moment and the existence of its antiparticle, the positron.