He discovered that the so-called Weil representation, previously introduced in quantum mechanics by Irving Segal and Shale, gave a contemporary framework for understanding the classical theory of quadratic forms.

Angular momentum in quantum mechanics differs in many profound respects from angular momentum in classical mechanics.

The classical definition of angular momentum as can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator.

These newer concerns are among the many factors causing researchers to investigate new methods of computing such as the quantum computer, as well as to expand the usage of parallelism and other methods that extend the usefulness of the classical von Neumann model.

Drude's classical model was augmented by Felix Bloch, Arnold Sommerfeld, and independently by Wolfgang Pauli, who used quantum mechanics to describe the motion of a quantum electron in a periodic lattice.

A few theoretical physicists have argued that classical physics is intrinsically incapable of explaining the holistic aspects of consciousness, but that quantum theory provides the missing ingredients.

In other words, it takes no more time to break RSA on a quantum computer ( up to a multiplicative constant ) than to use it legitimately on a classical computer.

Bennett, Bernstein, Brassard, and Vazirani proved in 1996 that a brute-force key search on a quantum computer cannot be faster than roughly 2 < sup > n / 2 </ sup > invocations of the underlying cryptographic algorithm, compared with roughly 2 < sup > n </ sup > in the classical case.

It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta, entities which fit neither the classical idea of particles nor the classical idea of waves.

# The quantum mechanical description of large systems will closely approximate the classical description.

The problem of thinking in terms of classical measurements of a quantum system becomes particularly acute in the field of quantum cosmology, where the quantum system is the universe.

The latter, though more rarely discussed, is interesting, as it is the suitable setting for quantum computation, whereas the former is sufficient for classical logic.

These seemingly contradictory discoveries made it necessary to go beyond classical physics and take the quantum nature of light into account.

Because it demonstrates the fundamental limitation of the observer to predict experimental results, Richard Feynman called it " a phenomenon which is impossible ... to explain in any classical way, and which has in it the heart of quantum mechanics.

* Chaos: classical and quantum.

From a classical perspective, the electromagnetic field can be regarded as a smooth, continuous field, propagated in a wavelike manner ; whereas from the perspective of quantum field theory, the field is seen as quantized, being composed of individual particles.

The study of how charged substances interact is classical electrodynamics, which is accurate insofar as quantum effects can be ignored.

Heisenberg's principle was an attempt to provide a classical explanation of a quantum effect sometimes called non-locality.

It turns out that the usual rules for combining quantum mechanical and classical descriptions violate the principle of locality without violating causality.

The distribution was discovered in the context of classical statistical mechanics by J. W.

Any two black holes that share the same values for these properties, or parameters, are indistinguishable according to classical ( i. e. non-quantum ) mechanics.

These arguments, and a discussion of the distinctions between absolute and relative time, space, place and motion, appear in a Scholium at the very beginning of Newton's work, The Mathematical Principles of Natural Philosophy ( 1687 ), which established the foundations of classical mechanics and introduced his law of universal gravitation, which yielded the first quantitatively adequate dynamical explanation of planetary motion.

In the classical branches of continuum mechanics the development of the theory of stresses is based on non-polar materials.

Even larger molecules are treated by classical mechanics methods that employ what are called molecular mechanics.

Molecular mechanics simulations, for example, use a single classical expression for the energy of a compound, for instance the harmonic oscillator.

More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame.

* The strong cosmic censorship hypothesis asserts that, generically, general relativity is a deterministic theory, in the same sense that classical mechanics is a deterministic theory.

The result of a die roll is determined by the way it is thrown, according to the laws of classical mechanics ; they are made random by uncertainty due to factors like movements in the thrower's hand.

Additionally, the detection of individual photons is observed to be inherently probabilistic, which is inexplicable using classical mechanics.

One of the peculiarities of classical electromagnetism is that it is difficult to reconcile with classical mechanics, but it is compatible with special relativity.

This violates Galilean invariance, a long-standing cornerstone of classical mechanics.

The 900-page book, titled Elementorum physicae mathematicae, written in Latin by Jesuit Father Andrea Caraffa, a professor at the Collegio Romano, covered subjects like mathematics, classical mechanics, astronomy, optics, and acoustics.

Most physicists today believe that quantum mechanics is correct, and that the EPR paradox is a " paradox " only because classical intuitions do not correspond to physical reality.

These are based on classical mechanics and are modified in quantum mechanics and general relativity.

The first step is the realization that classical mechanics and Newton's law of gravity admit of a geometric description.

As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of ( special ) relativistic mechanics.

The geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean.

This treatise ( Della pittura ) was also known in Latin as De Pictura, and it relied in its scientific content on classical optics in determining perspective as a geometric instrument of artistic and architectural representation.

An incomplete and somewhat arbitrary subdivision of model theory is into classical model theory, model theory applied to groups and fields, and geometric model theory.

The result of this synthesis is called geometric model theory in this article ( which is taken to include o-minimality, for example, as well as classical geometric stability theory ).

Architectural interest in Cubism centered on the dissolution and reconstitution of three-dimensional form, using simple geometric shapes, juxtaposed without the illusions of classical perspective.

The Eudoxan planetary model, on which the Aristotelian and Ptolemaic models were based, was the first geometric explanation for the " wandering " of the classical planets.

classical physics, his geometric approaches, called classical unified field theories,

When these sets ( or the motions on them ), are harder to describe than the classical geometric objects, then the attractor is a strange attractor, as described in the section below.

In 1957, Artin wrote a book on geometric algebra an insightful development of the classical groups in a Kleinian context.

The connection of quantum field theories to a physical geometric description is less obvious than the connection between the classical equations ( i. e. non-quantum mechanical descriptions of gravity and electromagnetism ) and geometry.

Buildings increased in geometric complexity, brick and plaster were used in addition to stone in the decoration of important public structures, classical orders were used more freely, mosaics replaced carved decoration, complex domes rested upon massive piers, and windows filtered light through thin sheets of alabaster to softly illuminate interiors.

After 1930, until his death in 1956, Metzinger turned towards a more classical or decorative approach to painting with elements of Surrealism, still concerned with questions of form, volume, dimension, relative position and relationship of figures, along with visible geometric properties of space.

This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems.

" Artificial perspective projection " was the name given by Leonardo da Vinci to what today is called " classical perspective projection " and referred to above as the result of a geometric protocol.

Applications, according to Arnold, are to be seen in symplectic geometry, as the geometric form of classical mechanics.

Much of the work on classical unified field theories consisted of attempts to further extend the general theory of relativity to interpret additional physical phenomena, particularly electromagnetism, within the framework of general covariance, and more specifically as purely geometric objects in the space-time continuum.

His modeling is in a lyrical, classical art-deco manner which effortlessly combines sensuous curves with geometric line patterns.

Geometric transformations are applied to the vertices of polygons, or other geometric objects used as modelling primitives, as part of the first stage in a classical geometry-based graphic image rendering pipeline.

The connection is explained by the geometric model of loop spaces approach to Bott periodicity: there 2-fold / 8-fold periodic embeddings of the classical groups in each other ( corresponding to isomorphism groups of Clifford algebras ), and their successive quotients are symmetric spaces which are homotopy equivalent to the loop spaces of the unitary / orthogonal group.

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory.