finite-dimensional love

finite-dimensional

Definitions

from Wiktionary, Creative Commons Attribution/Share-Alike License

  • adj. (of a vector space) having a basis consisting of a finite number of elements.

Etymologies

Sorry, no etymologies found.

Examples

  • All finite-dimensional inner product spaces are complete, and I will restrict myself to these.

    Quantum Mechanics

  • If the quantum state evolves in a finite-dimensional Hilbert space, it evolves ergodically through a torus of phases, and will exhibit all of the usual problems of Boltzmann brains and the like (as Dyson, Kleban, and Susskind have emphasized).

    What if Time Really Exists?

  • It would seem at first sight that a lattice formulation could never reproduce this, since in finite volume (e.g. on the 4-torus) the space of lattice spinor fields is finite-dimensional but to have a non-trivial index theory the operators generally must be acting on an infinite-dimensional vectorspace.

    String Theory is Losing the Public Debate

  • Every finite-dimensional vector space is free, being generated by any choice of basis.

    Algebra

  • The essence of duality for finite-dimensional vector spaces resides in its involutary nature along with the reversal of the linear transformations.

    Algebra

  • In the finite-dimensional case, multiplication is realized as the usual matrix product.

    Algebra

  • This result applies more generally to other cases where a macroscopic system (not idealized as finite-dimensional) experiences decoherence due to interaction with its environment (see Donald (1998)).

    Modal Interpretations of Quantum Mechanics

  • However, relying on the (near) ubiquity of decoherence in the macroscopic realm, Bacciagaluppi and Hemmo show that when the apparatus is considered as a finite-dimensional system

    Modal Interpretations of Quantum Mechanics

  • In classical Hamilton-Jacobi theory we also have this equation for the velocity, but there the Hamilton-Jacobi function S can be entirely eliminated and the description in terms of S simplified and reduced to a finite-dimensional description, with basic variables the positions and the (unconstrained) momenta of all the particles, given by Hamilton's or Newton's equations.

    Bohmian Mechanics

  • (If the domain is both compact and discrete, then it is finite, and on a finite-dimensional space all norms are equivalent.)

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