**mechanics**- Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. (springer.com)
- This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. (springer.com)
- and the path-integral approach to quantum mechanics. (springer.com)
- But why do this, especially as there is no apparent causal link between quantum mechanics and the conscious mind? (slashdot.org)
- While I was away last week on vacation, it seems that Springer has published my book on quantum mechanics and representation theory (previously discussed in various blog posts ). (columbia.edu)
- Quantum mechanics is when God cheats . (principiadiscordia.com)
- Yes, that is a summary of how quantum mechanics relates to the reality we perceive. (principiadiscordia.com)
- When the dimensionality of the wave packet is negligible you get the special case of quantum mechanics known as classical mechanics. (principiadiscordia.com)
- This touches on interpretational issues of quantum mechanics, cosmological observations and properties of quantum gravity. (perimeterinstitute.ca)
- The nature and behavior of matter and energy at that level is sometimes referred to as quantum physics and quantum mechanics. (techtarget.com)
- The old quantum theory is a collection of results from the years 1900-1925 which predate modern quantum mechanics. (wikipedia.org)
- The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. (wikipedia.org)
- The theory is now understood as the semi-classical approximation to modern quantum mechanics. (wikipedia.org)
- Heisenberg went on to reformulate all of quantum theory in terms of a version of these transition matrices, creating matrix mechanics. (wikipedia.org)
- The system obeys classical mechanics except that not every motion is allowed, only those motions which obey the old quantum condition: ∮ H ( p , q ) = E p i d q i = n i h {\displaystyle \oint \limits _{H(p,q)=E}p_{i}\,dq_{i}=n_{i}h} where the p i {\displaystyle p_{i}} are the momenta of the system and the q i {\displaystyle q_{i}} are the corresponding coordinates. (wikipedia.org)
- It studies complexity classes defined using quantum computers and quantum information which are computational models based on quantum mechanics. (wikipedia.org)
- Quantum tic tac toe: not a quantum game in the sense above, but a pedagogical tool based on metaphors for quantum mechanics Quantum pseudo-telepathy Quantum refereed game Jan Sładkowski Jens Eisert Simon C. Benjamin and Patrick M. Hayden (13 August 2001), "Multiplayer quantum games", Physical Review A, 64 (3): 030301, arXiv:quant-ph/0007038 , Bibcode:2001PhRvA. (wikipedia.org)
- It is a set of notions and mathematical tools that combines classical fields, special relativity, and quantum mechanics, and, when combined with the cluster decomposition principle, it may be the only way to do so, while retaining the ideas of quantum point particles and locality. (wikipedia.org)
- Even though QFT is an unavoidable consequence of the reconciliation of quantum mechanics with special relativity (Weinberg (1995)), historically, it emerged in the 1920s with the quantization of the electromagnetic field (the quantization being based on an analogy of the eigenmode expansion of a vibrating string with fixed endpoints). (wikipedia.org)
- However, quantum mechanics as formulated by Dirac, Heisenberg, and Schrödinger in 1926-27 started from atomic spectra and did not focus much on problems of radiation. (wikipedia.org)
- As soon as the conceptual framework of quantum mechanics was developed, a small group of theoreticians tried to extend quantum methods to electromagnetic fields. (wikipedia.org)
- In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. (wikipedia.org)
- Time evolution, quantum transitions, and symmetry transformations in Quantum mechanics may thus be viewed as the systematic theory of abstract, generalized rotations in this space of quantum state vectors. (wikipedia.org)
- The Principles of Quantum Mechanics, Cambridge University Press Prugovecki, Eduard (2006). (wikipedia.org)
- Quantum Mechanics in Hilbert Space, Second Edition (Dover Books on Physics) Paperback. (wikipedia.org)
- The partition function in quantum field theory is a special case of the mathematical partition function, and is related to the statistical partition function in statistical mechanics. (wikipedia.org)
- J=0}} The generating functional is the quantum field theory analog of the partition function in statistical mechanics: it tells us everything we could possibly want to know about a system. (wikipedia.org)
- Unlike the partition function in statistical mechanics, the partition function in quantum field theory contains an extra factor of i in front of the action, making the integrand complex, not real. (wikipedia.org)
- This need to put together relativity and quantum mechanics was the second major motivation in the development of quantum field theory. (wikipedia.org)
- A further boost for quantum field theory came with the discovery of the Dirac equation, which was originally formulated and interpreted as a single-particle equation analogous to the Schrödinger equation, but unlike the Schrödinger equation, the Dirac equation satisfies both the Lorentz invariance, that is, the requirements of special relativity, and the rules of quantum mechanics. (wikipedia.org)
- The integral of interest is (for an example of an application see Relation between Schrödinger's equation and the path integral formulation of quantum mechanics) ∫ − ∞ ∞ exp ( 1 2 i a x 2 + i J x ) d x . {\displaystyle \int _{-\infty }^{\infty }\exp \left({1 \over 2}iax^{2}+iJx\right)dx. (wikipedia.org)

**physics**- British physicist Roger Penrose recently argued "that we will need to invoke 'new physics and exotic biological structures': rewriting quantum theory to make sense of consciousness. (slashdot.org)
- Quantum theory is the theoretical basis of modern physics that explains the nature and behavior of matter and energy on the atomic and subatomic level. (techtarget.com)
- Planck won the Nobel Prize in Physics for his theory in 1918, but developments by various scientists over a thirty-year period all contributed to the modern understanding of quantum theory. (techtarget.com)
- Quantum Theory and the Schism in Physics. (wikipedia.org)
- This theory is based on the physics of information much like quantum computing. (wikipedia.org)
- In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics. (wikipedia.org)
- In quantum physics, in order to quantize a gauge theory, like for example Yang-Mills theory, Chern-Simons theory or the BF model, one method is to perform a gauge fixing. (wikipedia.org)
- In condensed-matter physics, topological quantum field theories are the low-energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states. (wikipedia.org)
- Topological field theories are not very interesting on the flat Minkowski spacetime used in particle physics. (wikipedia.org)
- In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum theory is mathematically compatible with special relativity. (wikipedia.org)
- Today one of the most important open problems, both in theoretical physics and in mathematics, is to establish similar results for gauge theory in the realistic case d = 4. (wikipedia.org)
- There is a firm belief in the physics community that the gauge theory of Yang and Mills can lead to a tractable theory, but new ideas and new methods will be required to actually establish this, and this could take many years. (wikipedia.org)
- Constructive Quantum Field Theory (pdf) review article from Mathematical Physics 2000 p. 111. (wikipedia.org)
- The Haag-Kastler axiomatic framework for quantum field theory, introduced by Haag and Kastler (1964), is an application to local quantum physics of C*-algebra theory. (wikipedia.org)
- Indeed, the viewpoint of local quantum physics is in particular suitable to generalize the renormalization procedure to the theory of quantum fields developed on curved backgrounds. (wikipedia.org)
- In theoretical physics, thermal quantum field theory (thermal field theory for short) or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature. (wikipedia.org)
- In mathematical physics, noncommutative quantum field theory (or quantum field theory on noncommutative spacetime) is an application of noncommutative mathematics to the spacetime of quantum field theory that is an outgrowth of noncommutative geometry and index theory in which the coordinate functions are noncommutative. (wikipedia.org)
- One of the novel features of noncommutative field theories is the UV/IR mixing phenomenon in which the physics at high energies affects the physics at low energies which does not occur in quantum field theories in which the coordinates commute. (wikipedia.org)
- In particle physics, the history of quantum field theory starts with its creation by Paul Dirac, when he attempted to quantize the electromagnetic field in the late 1920s. (wikipedia.org)
- By the late 1970s, these efforts successfully utilized gauge theory in the strong nuclear force and weak nuclear force, producing the modern standard model of particle physics. (wikipedia.org)
- In elementary particle physics and mathematical physics, in particular in effective field theory, a form factor is a function that encapsulates the properties of a certain particle interaction without including all of the underlying physics, but instead, providing the momentum dependence of suitable matrix elements. (wikipedia.org)
- It is further measured experimentally in confirmation or specification of a theory-see experimental particle physics. (wikipedia.org)

**field theory**- It is now believed, primarily due to the continued failures of quantization of general relativity, to be only a very good low-energy approximation, i.e. an effective field theory, to some more fundamental theory. (wikipedia.org)
- In quantum field theory, quantum mechanical interactions among particles are described by interaction terms among the corresponding underlying quantum fields. (wikipedia.org)
- A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. (wikipedia.org)
- Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory. (wikipedia.org)
- In a topological field theory, the correlation functions do not depend on the metric of spacetime. (wikipedia.org)
- Axiomatic quantum field theory is a mathematical discipline which aims to describe quantum field theory in terms of rigorous axioms. (wikipedia.org)
- Attempts to put quantum field theory on a basis of completely defined concepts have involved most branches of mathematics, including functional analysis, differential equations, probability theory, representation theory, geometry, and topology. (wikipedia.org)
- Along with work of their students, coworkers, and others, constructive field theory resulted in giving a mathematical foundation and exact interpretation to what previously was only a set of recipes, also in the case d (wikipedia.org)
- The traditional basis of constructive quantum field theory is the set of Wightman axioms. (wikipedia.org)
- They also fall in the related framework introduced by Haag and Kastler, called algebraic quantum field theory. (wikipedia.org)
- Because of this it is also known as algebraic quantum field theory (AQFT). (wikipedia.org)
- More recently, the approach has been further implemented to include an algebraic version of quantum field theory in curved spacetime. (wikipedia.org)
- This allows one to perform calculations with the same tools as in ordinary quantum field theory, such as functional integrals and Feynman diagrams, but with compact Euclidean time. (wikipedia.org)
- In this Euclidean field theory, real-time observables can be retrieved by analytic continuation. (wikipedia.org)
- Quantum Field Theory and Critical Phenomena. (wikipedia.org)
- The implication is that a quantum field theory on noncommutative spacetime can be interpreted as a low energy limit of the theory of open strings. (wikipedia.org)
- In quantum field theory, the partition function Z [ J ] {\displaystyle Z[J]} is the generating functional of all correlation functions, generalizing the characteristic function of probability theory. (wikipedia.org)
- The generating functional is the holy grail of any particular field theory: if you have an exact closed-form expression for Z [ J ] {\displaystyle Z[J]} for a particular theory, you have solved it completely. (wikipedia.org)
- This i points to a deep connection between quantum field theory and the statistical theory of fields. (wikipedia.org)
- The study of quantum field theory is still flourishing, as are applications of its methods to many physical problems. (wikipedia.org)
- Quantum field theory originated in the 1920s from the problem of creating a quantum mechanical theory of the electromagnetic field. (wikipedia.org)
- This theory assumed that no electric charges or currents were present and today would be called a free field theory. (wikipedia.org)
- This quantum field theory could be used to model important processes such as the emission of a photon by an electron dropping into a quantum state of lower energy, a process in which the number of particles changes-one atom in the initial state becomes an atom plus a photon in the final state. (wikipedia.org)
- It is now understood that the ability to describe such processes is one of the most important features of quantum field theory. (wikipedia.org)
- In it, fermion species nonconservation was shown to follow from second quantization: creation and annihilation of fermions came to the fore and quantum field theory was seen to describe particle decays. (wikipedia.org)
- An Introduction to Quantum Field Theory. (wikipedia.org)
- Condensed Matter Field Theory. (wikipedia.org)
- Schroeder, Daniel V. and Michael Peskin, An Introduction to Quantum Field Theory. (wikipedia.org)
- There are common integrals in quantum field theory that appear repeatedly. (wikipedia.org)
- The first integral, with broad application outside of quantum field theory, is the gaussian integral. (wikipedia.org)
- This integral is also known as the Hubbard-Stratonovich transformation used in field theory. (wikipedia.org)

**perturbation theory**- Our group's research is focused on understanding unique properties and behavior of materials and interfaces, using first principles quantum mechanical calculations based mostly on density functional theory and many-body perturbation theory. (weizmann.ac.il)
- These interactions are conveniently visualized by Feynman diagrams, which are a formal tool of relativistically covariant perturbation theory, serving to evaluate particle processes. (wikipedia.org)
- Perturbation theory is an important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the Schrödinger equation for Hamiltonians of even moderate complexity. (wikipedia.org)
- Using perturbation theory, we can use the known solutions of these simple Hamiltonians to generate solutions for a range of more complicated systems. (wikipedia.org)
- Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem. (wikipedia.org)
- This instability shows up as a broadening of the energy spectrum lines, which perturbation theory fails to reproduce entirely. (wikipedia.org)
- The expressions produced by perturbation theory are not exact, but they can lead to accurate results as long as the expansion parameter, say α, is very small. (wikipedia.org)
- Under some circumstances, perturbation theory is an invalid approach to take. (wikipedia.org)
- clarification needed] Perturbation theory also fails to describe states that are not generated adiabatically from the "free model", including bound states and various collective phenomena such as solitons. (wikipedia.org)
- Perturbation theory can only detect solutions "close" to the unperturbed solution, even if there are other solutions for which the perturbative expansion is not valid. (wikipedia.org)
- As well as Feynman diagrams and perturbation theory, other techniques such as dispersion relations and the finite temperature analog of Cutkosky rules can also be used in the real time formulation. (wikipedia.org)

**electrodynamics**- Here Dirac coined the name "quantum electrodynamics" (QED) for the part of QFT that was developed first. (wikipedia.org)
- In the theory of quantum electrodynamics (QED), in which the electron-photon interaction is treated perturbatively, the calculation of the electron's magnetic moment has been found to agree with experiment to eleven decimal places. (wikipedia.org)
- Major advances in the theory were made in the 1950s, and led to the introduction of quantum electrodynamics (QED). (wikipedia.org)
- The first reasonably complete theory of quantum electrodynamics, which included both the electromagnetic field and electrically charged matter (specifically, electrons) as quantum mechanical objects, was created by Paul Dirac in 1927. (wikipedia.org)

**Hilbert**- A 348 (2006) 147-152, Classical and Quantum Contents of Solvable Game Theory on Hilbert Space Piotrowski, E. W. (wikipedia.org)
- This "transformation" idea refers to the changes a quantum state undergoes in the course of time, whereby its vector "moves" between "positions" or "orientations" in its Hilbert space. (wikipedia.org)
- While the terminology is reminiscent of rotations of vectors in ordinary space, the Hilbert space of a quantum object is more general, and holds its entire quantum state. (wikipedia.org)
- These axioms attempt to describe QFTs on flat Minkowski spacetime by regarding quantum fields as operator-valued distributions acting on a Hilbert space. (wikipedia.org)
- The i arises from the fact that the partition function in QFT calculates quantum-mechanical probability amplitudes between states, which take on values in a complex projective space (complex Hilbert space, but the emphasis is placed on the word projective, because the probability amplitudes are still normalized to one). (wikipedia.org)

**Heisenberg**- Kramers gave a prescription for calculating transition probabilities between quantum states in terms of Fourier components of the motion, ideas which were extended in collaboration with Werner Heisenberg to a semiclassical matrix-like description of atomic transition probabilities. (wikipedia.org)
- The Physical Principles of the Quantum Theory (German: Physikalische Prinzipien der Quantentheorie) by Nobel laureate (1932) Werner Heisenberg and subsequently translated by Carl Eckart and Frank C. Hoyt. (wikipedia.org)
- In the book, after briefly discussing various theories, including quantum theory, Heisenberg discusses the basis for the fundamental concepts of quantum theory. (wikipedia.org)
- Heisenberg was the first to suggest extending noncommutativity to the coordinates as a possible way of removing the infinite quantities appearing in field theories before the renormalization procedure was developed and had gained acceptance. (wikipedia.org)

**algebraic**- Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. (wikipedia.org)

**mechanical**- The shortest summary of quantum mechanical reality I know of would be this: The dimensionality of the wave packet is not always negligible, although it often is. (principiadiscordia.com)
- Sommerfeld's model was much closer to the modern quantum mechanical picture than Bohr's. (wikipedia.org)
- In 1926 Erwin Schrödinger found a completely quantum mechanical wave-equation, which reproduced all the successes of the old quantum theory without ambiguities and inconsistencies. (wikipedia.org)
- See also list of functional analysis topics, list of Lie group topics, list of quantum-mechanical systems with analytical solutions. (wikipedia.org)
- Dirac supplied a systematic procedure for transferring the characteristic quantum phenomenon of discreteness of physical quantities from the quantum-mechanical treatment of particles to a corresponding treatment of fields. (wikipedia.org)
- The methods of QFT could be applied to derive equations resulting from the quantum-mechanical (field-like) treatment of particles, e.g. the Dirac equation, the Klein-Gordon equation and the Maxwell equations. (wikipedia.org)
- For example, by adding a perturbative electric potential to the quantum mechanical model of the hydrogen atom, tiny shifts in the spectral lines of hydrogen caused by the presence of an electric field (the Stark effect) can be calculated. (wikipedia.org)

**harmonic oscillator**- Employing the theory of the quantum harmonic oscillator, Dirac gave a theoretical description of how photons appear in the quantization of the electromagnetic radiation field. (wikipedia.org)
- The Hamiltonians to which we know exact solutions, such as the hydrogen atom, the quantum harmonic oscillator and the particle in a box, are too idealized to adequately describe most systems. (wikipedia.org)

**Wikipedia**- This is a list of mathematical topics in quantum theory, by Wikipedia page. (wikipedia.org)

**spacetime**- The action (which determines the path integral) is S = ∫ M B F {\displaystyle S=\int _{M}BF} The spacetime metric does not appear anywhere in the theory, so the theory is explicitly topologically invariant. (wikipedia.org)
- One discovery of the theory, that can be related in non-technical terms, is that the dimension d of the spacetime involved is crucial. (wikipedia.org)

**1927**- The term transformation theory refers to a procedure and a "picture" used by P. A. M. Dirac in his early formulation of quantum theory, from around 1927. (wikipedia.org)

**Gears of**- Tecmo Koei have been nice enough to release new screenshots of Gears of War… I mean Quantum Theory for our viewing pleasure. (playstationlifestyle.net)

**theoretical**- Quantum complexity theory is a part of computational complexity theory in theoretical computer science. (wikipedia.org)
- In fact most topics in the early development of quantum theory (the so-called old quantum theory, 1900-25) were related to the interaction of radiation and matter and thus should be treated by quantum field theoretical methods. (wikipedia.org)
- Theoretical physicists had given these rules the name "renormalization," but most physicists had been skeptical about whether they could be turned into a mathematical theory. (wikipedia.org)

**2000**- A 272 (2000) 291-303, A Quantum Approach To Static Games of Complete Information T. Cheon, I. Tsutsui, Phys. (wikipedia.org)

**Feynman**- In QED and other quantum field theories, special calculation techniques known as Feynman diagrams are used to systematically sum the power series terms. (wikipedia.org)
- In the free theory, it is simply the Feynman propagator (for n=2). (wikipedia.org)

**axioms**- The first set of axioms for quantum field theories, known as the Wightman axioms, were proposed by Arthur Wightman in the early 1950s. (wikipedia.org)

**electromagnetic**- Classical or quantum electromagnetic theory? (physicsforums.com)
- Weak, strong, and electromagnetic forces of nature are believed to have their natural description in terms of quantum fields. (wikipedia.org)
- The idea was that not only the quanta of the electromagnetic field, photons, but also other particles might emerge and disappear as a result of their interaction with other particles. (wikipedia.org)
- It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate Einstein's relativity theory, which had grown out of the study of classical electromagnetism. (wikipedia.org)

**principles**- Einstein, followed by Debye, applied quantum principles to the motion of atoms, explaining the specific heat anomaly. (wikipedia.org)

**photons**- Bose and Einstein gave the correct quantum statistics for photons. (wikipedia.org)
- Since photons have rest mass zero, and correspondingly travel in the vacuum at the speed c, a non-relativistic theory such as ordinary QM cannot give even an approximate description. (wikipedia.org)

**Dirac**- Paul Dirac later proved in 1926 that both methods can be obtained from a more general method called transformation theory. (wikipedia.org)
- This work was performed first by Dirac himself with the invention of hole theory in 1930 and by Wendell Furry, Robert Oppenheimer, Vladimir Fock, and others. (wikipedia.org)

**procedure**- The main tool of the old quantum theory was the Bohr-Sommerfeld quantization condition, a procedure for selecting out certain states of a classical system as allowed states: the system can then only exist in one of the allowed states and not in any other state. (wikipedia.org)

**superposition**- Since we do not know, the cat is both dead and alive, according to quantum law - in a superposition of states. (techtarget.com)
- It differs from classical game theory in three primary ways: Superposed initial states, Quantum entanglement of initial states, Superposition of strategies to be used on the initial states. (wikipedia.org)
- In the quantum version of the game, the bit is replaced by the qubit, which is a quantum superposition of two or more base states. (wikipedia.org)

**1999**- 83 (1999) 3077-3080, Quantum Games and Quantum Strategies N. F. Johnson, Playing a Quantum Game with a Corrupted Source L. Marinatto, T. Weber, Phys. (wikipedia.org)

**entanglement**- If you've done any reading on quantum theory -- particularly the double-slit experiment and entanglement -- you've probably realized the same thing as Albert "Spooky" Einstein: That sh*t is craaazy. (comicsalliance.com)

**1900**- In 1900, physicist Max Planck presented his quantum theory to the German Physical Society. (techtarget.com)
- In 1900, Planck made the assumption that energy was made of individual units, or quanta. (techtarget.com)
- The old quantum theory was sparked by the 1900 work of Max Planck on the emission and absorption of light, and began in earnest after the work of Albert Einstein on the specific heats of solids. (wikipedia.org)

**gauge**- An alternative approach using lattice approximations is covered in (Wick rotated) lattice gauge theory. (wikipedia.org)
- It has been generalized to theories with gauge invariance and was a central tool in the study of a conjectured deconfining phase transition of Yang-Mills theory. (wikipedia.org)

**invariance**- In the next few years Arnold Sommerfeld extended the quantum rule to arbitrary integrable systems making use of the principle of adiabatic invariance of the quantum numbers introduced by Lorentz and Einstein. (wikipedia.org)
- Relativistic invariance can however be retained in the sense of twisted Poincaré invariance of the theory. (wikipedia.org)

**interpretation**- The two major interpretations of quantum theory's implications for the nature of reality are the Copenhagen interpretation and the many-worlds theory. (techtarget.com)
- Niels Bohr proposed the Copenhagen interpretation of quantum theory, which asserts that a particle is whatever it is measured to be (for example, a wave or a particle), but that it cannot be assumed to have specific properties, or even to exist, until it is measured. (techtarget.com)
- The second interpretation of quantum theory is the many-worlds (or multiverse theory. (techtarget.com)

**oscillators**- The ideas of QM were thus extended to systems having an infinite number of degrees of freedom, so an infinite array of quantum oscillators. (wikipedia.org)

**probability theory**- Osterwalder and Schrader showed that there is an equivalent problem in mathematical probability theory. (wikipedia.org)

**Planck**- Planck wrote a mathematical equation involving a figure to represent these individual units of energy, which he called quanta . (techtarget.com)
- Planck assumed there was a theory yet to emerge from the discovery of quanta, but, in fact, their very existence implied a completely new and fundamental understanding of the laws of nature. (techtarget.com)

**Einstein's**- These are but a few of the extraordinary consequences of Einstein's theory of relativity. (audible.com)
- This theory became known as the uncertainty principle , which prompted Albert Einstein's famous comment, 'God does not play dice. (techtarget.com)

**multiplayer**- In addition to single player campaign, Quantum Theory also supports four modes of online multiplayer, complete with voice chat for real-time communication. (worthplaying.com)
- Introducing quantum information into multiplayer games allows a new type of "equilibrium strategy" which is not found in traditional games. (wikipedia.org)

**Lorentz**- Pascual Jordan and Wolfgang Pauli showed in 1928 that quantum fields could be made to behave in the way predicted by special relativity during coordinate transformations (specifically, they showed that the field commutators were Lorentz invariant). (wikipedia.org)

**behavior**- This has been shown to be a useful tool in studying the behavior of quantum field theories at finite temperature. (wikipedia.org)

**coordinates**- In order for the old quantum condition to make sense, the classical motion must be separable, meaning that there are separate coordinates q i {\displaystyle q_{i}} in terms of which the motion is periodic. (wikipedia.org)

**quantization**- The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization. (springer.com)
- Sommerfeld made a crucial contribution by quantizing the z-component of the angular momentum, which in the old quantum era was called space quantization (Richtungsquantelung). (wikipedia.org)

**fundamental theory**- Since quantum theory is the most fundamental theory of matter that is currently available, it is a legitimate question to ask whether quantum theory can help us to understand consciousness. (hindawi.com)

**endpoints**- They argued in the context of string theory that the coordinate functions of the endpoints of open strings constrained to a D-brane in the presence of a constant Neveu-Schwarz B-field-equivalent to a constant magnetic field on the brane-would satisfy the noncommutative algebra set out above. (wikipedia.org)

**Consciousness**- Does Quantum Theory Explain Consciousness? (slashdot.org)

**known**- Most of the known topological field theories are defined on spacetimes of dimension less than five. (wikipedia.org)
- The known topological field theories fall into two general classes: Schwarz-type TQFTs and Witten-type TQFTs. (wikipedia.org)
- It is known that a quantum field is inherently hard to handle using conventional mathematical techniques like explicit estimates. (wikipedia.org)

**sometimes referred**- Witten TQFTs are also sometimes referred to as cohomological field theories. (wikipedia.org)

**displaystyle**- The quantum numbers n i {\displaystyle n_{i}} are integers and the integral is taken over one period of the motion at constant energy (as described by the Hamiltonian). (wikipedia.org)

**fields**- Quantum fluctuations of matter fields are supposed to have provided the initial seeds of all the structure of the current universe, and quantum gravity is assumed to have been essential in the earliest stages. (perimeterinstitute.ca)
- The existence theorems for quantum fields can be expected to be very difficult to find, if indeed they are possible at all. (wikipedia.org)

**aims**- One of the main aims of quantum complexity theory is to find out where these classes lie with respect to classical complexity classes such as P, NP, PP, PSPACE and other complexity classes. (wikipedia.org)

**gravity**- Quantum gravity is believed to be background-independent (in some suitable sense), and TQFTs provide examples of background independent quantum field theories. (wikipedia.org)