• 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)
  • 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)
  • 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)
  • 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)
  • 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)
  • 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)
  • 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)
  • 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)
  • 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)