Many-dimensional Hilbert space

The problem of higher dimensional crystallographic groups has been formulated as problem 18 of the famous Hilbert problems. Hilbert in 1900, proposed 23 fairly general problems to stimulate mathematical research. In 1910, this problem was solved by Bieberbach He proved that in any dimension that there were only finitely many groups [20]. Around 60 years later it was found that there are 4783 groups in the four-dimensional space. 

Viewed in terms of quantum computational principles, lipid selectivity is the interaction of molecules embedded in a many-dimensional (Hilbert) state space. In this view, geometry-dependent orbital surfaces are not 3D manifolds but sets of fluctuating hypersurfaces, each with an attached probability value, existing in linear superposition [33,3], The molecular interactions are thus a massively parallel search for a local energy minimum. These features would appear to fulfill the requirements for a quantum computing system. In terms of Atmar s... 

And if this was not enough it emerges that atomic orbitals are described in a many-dimensional Hilbert space which defies visualization since we can only observe objects in three dimensional space. How then can anyone still claim that orbitals have been directly observed And yet this is just what was claimed in Nature magazine and many other journals without... 

The separation surface may be nonlinear in many classification problems, but support vector machines can be extended to handle nonlinear separation surfaces by using feature functions < )(x). The SVM extension to nonlinear datasets is based on mapping the input variables into a feature space of a higher dimension (a Hilbert space of finite or infinite dimension) and then performing a linear classification in that higher dimensional space. For example, consider the set of nonlinearly separable patterns in Figure 28, left. It is... 

The individual unit of classical information is the bit an object that can take either one of two values, say 0 or 1. The corresponding unit of quantum information is the quantum bit or qubit. It describes a state in the simplest possible quantum system [1,2]. The smallest nontrivial Hilbert space is two-dimensional, and we may denote an orthonormal basis for the vector space as 0> and 11 >. A single bit or qubit can represent at most two numbers, but qubits can be put into infinitely many other states by a superposition ... 

The first step toward a practical relativistic many-electron theory in the molecular sciences is the investigation of the two-electron problem in an external field which we meet, for instance, in the helium atom. Salpeter and Bethe derived a relativistic equation for the two-electron bound-state problem [135,170-173] rooted in quantum electrod)mamics, which features two separate times for the two particles. If we assume, however, that an absolute time is a good approximation, we arrive at an equation first considered by Breit [101,174,175]. The Bethe-Salpeter equation as well as the Breit equation hold for a 16-component wave function. From a formal point of view, these 16 components arise when the two four-dimensional one-electron Hilbert spaces are joined by direct multiplication to yield the two-electron Hilbert space. 

In this section we discuss briefly—without any pretense of completeness— further computational approaches to quantum phase transitions. The conceptually simplest method for solving a quantum many-particle problem is (numerically) exact diagonalization. However, as already discussed in the section on Quantum Phase Transitions Computational Challenges, the exponential increase of the Hilbert space dimension with the number of degrees of freedom severely limits the possible system sizes. One can rarely simulate more than a few dozen particles even for simple lattice systems. Systems of this size are too small to study quantum phase transitions (which are a property of the thermodynamic limit of infinite system size) with the exception of, perhaps, certain simple one-dimensional systems. Even in one dimension, however, more powerful methods have largely superceded exact diagonalization. 

In spite of its computational intractability the full configuration interaction method forms the basis of a many-body theory. As stated above, the implementation of the algebraic approximation restricts the domain of the Hamiltonian operator, Jf, to a finite dimensional subspace 8 of the Hilbert space 1). If tP denotes the projector onto the subspace 8, then the Hamiltonian operator, Jf, is replaced by... 

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