Quartic quantum theory: an extension of the standard quantum mechanics (original) (raw)
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The paper presents a new methodology how to extend the well-known quantum model [2] with (2N − 1) free parameters (moduli and phases) of wave probabilistic functions ψ(A i) assigned into events A i , i ∈ {1, 2,. .. , N } to N •(N +1) 2 free parameters necessary for full N-dimensional representation of complex system. Our approach generally enables to include additional functions applied on events A i , i ∈ {1, 2,. .. , N }. In the paper, we will demonstrate this mathematical instrument on additional wave probabilistic functions ψ(A k ∩ A m ∩ • • • ∩ A n) connected with macroscopic events' intersections A k ∩ A m ∩ • • • ∩ A n where k, m,. .. , n ∈ {1, 2,. .. , N }.
Physical Review A, 2009
We show that several classes of mixed quantum states in finite-dimensional Hilbert spaces which can be characterized as being, in some respect, 'most classical' can be described and analyzed in a unified way. Among the states we consider are separable states of distinguishable particles, uncorrelated states of indistinguishable fermions and bosons, as well as mixed spin states decomposable into probabilistic mixtures of pure coherent states. The latter were the subject of the recent paper by Giraud et. al. [1], who showed that in the lowest-dimensional, nontrivial case of spin 1, each such state can be decomposed into a mixture of eight pure states. Using our method we prove that in fact four pure states always suffice.
A Royal Road to Quantum Theory (or Thereabouts), Extended Abstract
Electronic Proceedings in Theoretical Computer Science, 2017
A representation of finite-dimensional probabilistic models in terms of formally real Jordan algebras is obtained, in a strikingly easy way, from simple assumptions. This provides a framework in which real, complex and quaternionic quantum mechanics can be treated on an equal footing, and allows some (but not too much) room for other alternatives. This is based on earlier work (arXiv:1206:2897), but the development here is further simplified, and also extended in several ways. I also discuss the possibilities for organizing probabilistic models, subject to the assumptions discussed here, into symmetric monoidal categories, showing that such a category will automatically have a dagger-compact structure. (Recent joint work with Howard Barnum and Matthew Graydon (arXiv:1507.06278) exhibits several categories of this kind.
Principles of a Second Quantum Mechanics - The 4th and definitive version
2015
A qualitative but formalized representation of microstates is first established quite independently of the quantum mechanical mathematical formalism, exclusively under epistemological-operational-methodological constraints. Then, using this representation as a reference-and-imbedding-structure, the foundations of an intelligible reconstruction of the Hilbert-Dirac formulation of Quantum Mechanics is developed. Inside this reconstruction the measurement problem as well as the other major problems raised by the quantum mechanical formalism, dissolve. This is the 4th and definitive version appearing in arXiv:1310:1728v3 and in arXiv:1506.00431
A physical system S is represented by a (finite-dimensional) Hilbert space H over C, that is, a complex vector space with inner product ·|· : H × H → C. Each (pure) state of S is represented as a (normalized) vector in H, denoted |ψ .
Journal of Geometry and Physics, 2001
The manifold of pure quantum states can be regarded as a complex projective space endowed with the unitary-invariant Riemannian geometry of Fubini and Study. According to the principles of geometric quantum mechanics, the detailed physical characteristics of a given quantum system can be represented by specific geometrical features that are selected and preferentially identified in this complex manifold. In particular, any specific feature of projective geometry gives rise to a physically realisable characteristic in quantum mechanics. Here we construct a number of examples of such geometrical features as they arise in the state spaces for spin-1 2 , spin-1, and spin-3 2 systems, and for pairs of spin-1 2 systems. A study is undertaken on the geometry of entangled states, and a natural measure is assigned to the degree of entanglement of a given state for a general multi-particle system. The properties of this measure are analysed in detail for the entangled states of a pair of spin-1 2 particles, thus enabling us to determine the structure of the space of maximally entangled states. With the specification of a quantum Hamiltonian, the resulting Schrödinger trajectories induce an isometry of the Fubini-Study manifold. For a generic quantum evolution, the corresponding Killing trajectory is quasiergodic on a toroidal subspace of the energy surface. When the dynamical trajectory is lifted orthogonally to Hilbert space, it induces a geometric phase shift on the wave function. The uncertainty of an observable in a given state is the length of the gradient vector of the level surface of the expectation of the observable in that state, a fact that allows us to calculate higher order corrections to the Heisenberg relations. A general mixed state is determined by a probability density function on the state space, for which the associated first moment is the density matrix. The advantage of the idea of a general state is in its applicability in various attempts to go beyond the standard quantum theory, some of which admit a natural phase-space characterisation.
A new look on quantum mechanic
2020
Many physics scientists feel uneasy with the Copenhagen quantum interpretation and try to forward alternatives, but none of them solve the problems satisfactorily. There are new approaches worth to mention: • The Randell-Sundrum model- 3D branes in a 4D universe, where we are on one brane. It is similar to the MWI, where instead of many duplicated worlds, there are many 3D branes in a 4D universe. These branes are always there. They do not emerge in a collapse of the wave function, like in MWI interpretation. Therefore it solves the measurement problem. • Polchinski and Strassler theorem – Like the Randell-Sundrum model, they base the theory on particles' interaction in the "strong force" and explain the particle-wave duality. And other things. Here is a suggestion for a possible way to shed light on how our universe functions by assuming that we are in a 4D universe (or more), contrary to the Randell-Sundrum model avoiding the problems in that model. This model also fits the Peres model eliminating quantum problems.
Extreme Dimensionality Reduction with Quantum Modeling
Physical Review Letters, 2020
Effective and efficient forecasting relies on identification of the relevant information contained in past observations-the predictive features-and isolating it from the rest. When the future of a process bears a strong dependence on its behaviour far into the past, there are many such features to store, necessitating complex models with extensive memories. Here, we highlight a family of stochastic processes whose minimal classical models must devote unboundedly many bits to tracking the past. For this family, we identify quantum models of equal accuracy that can store all relevant information within a single two-dimensional quantum system (qubit). This represents the ultimate limit of quantum compression and highlights an immense practical advantage of quantum technologies for the forecasting and simulation of complex systems.
2008
The following publications involve content that is directly relevant to the topic of this thesis. For more information. P. van Loock, C. Weedbrook, and M. Gu. Building gaussian cluster states by linear optics. Phys. Rev. A, 76(3):032321, 2007. S. Sridharan, M. Gu, and M. James. Gate complexity using dynamic programming. Phys. Rev. A, 78(5):052327, 2008. Nicolas C. Menicucci, Peter van Loock, Mile Gu, C. Weedbrook, T. Ralph, and M. Nielsen. Universal quantum computation with continuous-variable cluster states. Phys.
Finite-dimensional quantum mechanics of a particle. II
International Journal of Theoretical Physics, 1982
The finite-dimensional quantum mechanics (FDQM) based on Weyrs form of Heisenberg's canonical commutation relations, developed for the case of onedimensional space, is extended to three-dimensional space. This FDQM is applicable to the physics of particles confined to move within finite regions of space and is significantly different from the current quantum mechanics in the case of atomic and subatomic particles only when the region of confinement is extremely small--of the order of nuclear or even smaller dimensions. The configuration space of such a particle has a quantized eigenstructure with a characteristic dependence on its rest mass and dimension of the region of confinement, and the current SchriSdinger-Heisenberg formalism of quantum mechanics becomes an asymptotic approximation of this FDQM. As an example a spherical harmonic oscillator with a particular radius of confinement is considered.