What is Hilbert: Definition and 302 Discussions

David Hilbert (; German: [ˈdaːvɪt ˈhɪlbɐt]; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory).
Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century.Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic.

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  1. S

    I How to check if a matrix is Hilbert space and unitary?

    I have a matrix, [ a, ib; -1 1] where a and b are constants. I have to represent and analyse this matrix in a Hilbert space: I take the space C^2 of this matrix is Hilbert space. Is it sufficient to generate the inner product: <x,y> = a*ib -1 and obtain the norm by: \begin{equation}...
  2. S

    I How to study an ODE in matrix form in a Hilbert space?

    Hello, I have derived the matrix form of one ODE, and found a complex matrix, whose phase portrait is a spiral source. The matrix indicates further that the ODE has diffeomorphic flow and requires stringent initial conditions. I have thought about including limits for the matrix, however the...
  3. S

    I Generating a Hilbert space representation of a wavefunction

    Hello, I Have a particle with wavefunction Psi(x) = e^ix and would like to find its Hilbert space representation for a period of 0-2pi. Which steps should I follow? Thanks!
  4. D

    I Einstein Hilbert action

    My question is why is the lagrangian density term in the action is equal to ricci scaler for gravitational field
  5. M

    A Can the convolution operator be diagonalized using the Fourier transform?

    Hi there, I am also familiar with Hilbert spaces and Functional Analysis and I find your question very interesting. I agree that the Fourier transform is a powerful tool for analyzing LTI systems and diagonalizing the convolution operator. As for your question about whether the same can be...
  6. P

    I Hilbert space in Everettian QM

    Is it assumed that Hilbert space is an infinite manifold that the non-collapsing wave function occupies in Everettian QM? Thank you.
  7. D

    To find the energy eigenvalues in the 3D Hilbert space

    A fictitious system having three degenerate angular momentum states with ##\ell=1## is described by the Hamiltonian \hat H=\alpha (\hat L^2_++\hat L^2_-) where ##\alpha## is some positive constant. How to find the energy eigenvalues of ##\hat H##?
  8. B

    I Understanding Hilbert Subspace for Two-Particle Entangled Systems

    I read that if we construct an observable on a two-particle entangled system like the "center of mass" observable, this observable does not pick out a single state of the two-particle system. It only picks out a subspace of the full Hilbert space of all possible states--the subspace that...
  9. D

    A What Hilbert space for a spinless particle?

    I'm looking for a rigorous mathematical description of the quantum mechanical space state of, for instance, a particle with no internal states. At university we were told that it the Hilbert state of wave functions. They gave us no particular restrictions on these functions, such as continuity...
  10. bhobba

    A Rigged Hilbert Spaces In Quantum Mechanics

    In discussing stuff in another thread I used the standard Dirac notion expanding a state in position eigenvectors namely |u> = ∫f(x) |x>. By definition f(x) is the wave-function. I omitted the dx which is my bad but the following question was posed which I think deserved a complete answer. It...
  11. M

    I Understanding Abstract Kets & Hilbert Space

    Some posts in another thread lead me to a search which ended when I read the following "kets such as ##|\psi\rangle## are elements of abstract Hilbert Space". That lead me to this paper. http://www.phy.ohiou.edu/~elster/lectures/qm1_1p2.pdf "The abstract Hilbert space ##l^2## is given by a...
  12. Gean Martins

    I Phase Space and two dimensional Hilbert Space

    I always had this doubt,but i guess i never asked someone. What's the main difference between the Classical phase space, and the two dimensional Hilbert Space ?
  13. A

    How to interpret the infinity of Hilbert Space?

    This is basically just a comprehension question, but what makes elements of the Hilbert space exist in infinite dimensions? I understand that the number of base vectors to write out an element, like a wavefunction, are infinite: \begin{equation*} \psi(x) = \int c_s u_s (x) ds = \sum_k^{\infty}...
  14. C

    Hilbert Dynamics in choosing position in BM/MWI

    In General Relativity. Gravity is caused by curvature of spacetime. In MWI and Bohmian Mechanics.. the position observable and position preferred basis is chosen. There must be a non-zero energy or some kind of dynamics that would lock the particular Hilbert space vectors into those special...
  15. Demystifier

    I Why is Hilbert not the last universalist?

    It is often said that Poincare was the last universalist, i.e. the last mathematician who understood more-or-less all mathematics of his time. But Hilbert's knowledge of math was also quite universal, and he came slightly after Poincare. So why was Hilbert not the last universalist? What branch...
  16. O

    B Does Cutting an Object Affect Its Hilbert Space or Quantum State?

    When you cut an object with a knife.. say a sausage. Does it's Hilbert Space or Quantum State split into two too? Or is it like in a holographic film.. in which even after cutting it, all the original image is in each of the cut portion?
  17. H

    I Why do Hydrogen bound states span the Hilbert space?

    As the title says, why does the set of hydrogen bound states form an orthonormal basis? This is clearly not true in general since some potentials (such as the finite square well and reversed gaussian) only admit a finite number of bound states.
  18. H

    MATLAB Trying to compute Hilbert transform numerically

    I know the result: \widehat{\mathscr{H}(f)}(k)=-i\sgn (k)\hat{f}(k) I want to use this to compute the Hilbert transform. I have written code for Fourier transform,inverse Fourier transform and that the Hilbert transform. My code is the following: function y=ft(x,f,k) n=length(k); %See now long...
  19. I

    I Interpretation of direct product of Hilbert spaces

    Dear all, I know how to interpret a vector, inner product etcetera in one Hilbert space. However, I can not get my head around how the direct product of two (or more) Hilbert spaces can be interpreted. For instance, the Hilbert space ##W## of a larger system is spanned by the direct product of...
  20. hilbert2

    A Approximating a QF with finite-dimensional Hilbert space

    Is it possible to approximately calculate the dynamics of a "phi-fourth" interacting Klein-Gordon field by using a finite dimensional Hilbert state space where the possible values of momentum are limited to a discrete set ##-p_{max},-\frac{N-1}{N}p_{max},-\frac{N-2}{N}p_{max}...
  21. mertcan

    A Einstein Hilbert action integral

    hi, when I see the einstein hilbert action I really started to be curious about that equation $$S=\int{\sqrt{g}d^4xR}$$. How is this action derived?? Is there a any proof using action integral involving the lagrangian density ? If there is not a derivation from lagrangian action, What is the...
  22. Muthumanimaran

    I Representing Mixed States in Hilbert Space

    Why cannot we represent mixed states with a ray in a Hilbert space like a Pure state. I know Mixed states corresponds to statistical mixture of pure states, If we are able to represent Pure state as a ray in Hilbert space, why we can't represent mixed states as ray or superposition of rays in...
  23. F

    I Why are Hilbert spaces used in quantum mechanics?

    In classical mechanics we use a 6n-dimensional phase space, itself a vector space, to describe the state of a given system at anyone point in time, with the evolution of the state of a system being described in terms of a trajectory through the corresponding phase space. However, in quantum...
  24. R

    What Is the Hilbert Transform of the Sinc Function?

    Homework Statement Show that the Hilbert transform of ##\frac{\sin(at)}{at}## is given by $$\frac{\sin^2(at/2)}{at/2}.$$ Homework Equations The analytic signal of a function is given by ##f_a(t) = 2 \int^\infty_0 F(\nu) \exp(j2 \pi \nu t) \ d\nu,## where ##F(\nu)## is the Fourier transform...
  25. R

    Hilbert Transform Homework: Show Envelope is |m(t)|

    Homework Statement For a real, band-limited function ##m(t)## and ##\nu_v > \nu_m,## show that the Hilbert transform of $$h(t) = m(t) cos(2\pi \nu_c t)$$ is $$\hat{h}(t) = m(t) sin(2 \pi \nu_c t),$$ and therefore the envelope of ##h(t)## is ##|m(t)|.## Homework Equations Analytic signal...
  26. KarminValso1724

    B Has the existence of Hilbert Space been proven 100%?

    Or is it only theoretical.
  27. Perturbative

    I General Relativity within the confines of a Hilbert Space

    Introduction If Quantum Mechanics is more fundamental than General Relativity as most Physicists believe, and Quantum Mechanics is described using Hilbert Spaces wouldn't finding a compatible version of General Relativity that operates within the confines of a Hilbert Space be of utmost...
  28. B

    Vectors, Hilbert Spaces, and Tensor Products

    If I ever say anything incorrect, please promptly correct me! The state of a system in classical mechanics is specified by point in phase space, the point giving us the position and velocity at a given instance. Could we rephrase it by saying a vector in phase space specifies the system? If...
  29. DuckAmuck

    Can a cross product be defined between two Hilbert space vectors?

    The equivalent of a dot product in Hilbert space is: \langle f | g \rangle = \int f(x) g(x) dx And you can find the angle between functions/vectors f and g via: \theta = arccos\left( \frac{\langle f | g \rangle}{\sqrt{\langle f|f \rangle \langle g|g \rangle}} \right) So is it possible to...
  30. J

    I Hilbert Space vs Quantum Vacuum

    If Hilbert space is just a mathematical tool like a column for an accountant and doesn't have factual existence. How about the quantum vacuum. Isn't it quantum vacuum is just another tool? Is it like Hilbert space or does the quantum vacuum have more factual existence? If the quantum vacuum is...
  31. J

    Does Hilbert Space Include Constants of Nature Beyond Basic Quantum Information?

    Does Hilbert Space contain the fine structure constant or store the values of other constants of nature or their information or does it only contain the position, momentum basis information of particles?
  32. F

    M,N is subset of Hilbert space, show M+N is closed

    Homework Statement [/B] Let M, N be a subset of a Hilbert space and be two closed linear subspaces. Assume that (u,v)=0, for all u in M and v in N. Prove that M+N is closed. Homework Equations I believe that (u,v)=0 is an inner product space The Attempt at a Solution This is a problem from...
  33. kostoglotov

    Verifying the Fourier Series is in Hilbert Space

    The text does it thusly: imgur link: http://i.imgur.com/Xj2z1Cr.jpg But, before I got to here, I attempted it in a different way and want to know if it is still valid. Check that f^{*}f is finite, by checking that it converges. f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x +...
  34. M

    Superposition of Hilbert space of qutrit states

    Homework Statement Given a orthonormal basis of the hilbert space of qutrit states: H = span (|0>, |1>, |2>) write in abstract notation and also a chosen consistent matrix representation, the states a) An equiprobable quantum superposition of the three elements of the basis b) An...
  35. N

    Depending on interpretation of QM, can Hilbert space be....

    Depending on interpretation of QM, can hilbert space be considered just as real as space time? In MWI the wave function is real, but still lies in hilbert space, so would hilbert space be considered a real space according to this interpretation?
  36. DrClaude

    Quantum Hilbert Hotel: Endless Possibilities

    Researchers have created a quantum version of the Hilbert hotel: http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.115.160505
  37. R

    How Many Basis Vectors in Hilbert Space?

    What is the dimensionality, N, of the Hilbert space (i.e., how many basis vectors does it need)? To be honest I am entirely lost on this question. I've heard of Hilbert space being both finite and infinite so I'm not sure as to a solid answer for this question. Does the Hilbert space need 4...
  38. ognik

    MHB Exploring Hilbert Space: What Makes It Confusing?

    I see I am not the only one finds Hilbert confusing - because all it's properties seem so familiar. I have gathered together what I could find, please comment? A Hilbert space is a vector space that: Has an inner product: • Inner product of a pair of elements in the space must be equal to...
  39. Andre' Quanta

    Separability of Hilbert Spaces

    Why we require the separability of Hilbert spaces in Quantum Mechanics?
  40. D

    State Vectors as elements of Hilbert Space

    The state, ##| S\rangle##, say, of a system is represented as a vector in a Hilbert space. ##\psi (x, t)## is the representation of the state vector in the position eigenbasis; ##\psi (p, t)## in the momentum eigenbasis et cetera. That is, ##\psi (x, t) = \langle x|S\rangle##; ##\psi (p, t) =...
  41. T

    The Joy of X: Solving the Hilbert Hotel Question with Steven Strogatz

    In "The Joy of X" Steven Strogatz discuss in a chapter on the Hilbert Hotel,a hotel with an infinite number of rooms, the problem of assigning rooms when an infinite number of buses arrive, and each bus has an infinite number of passengers. "There is always room at the Hilbert Hotel" says the...
  42. W

    Commutation between operators of different Hilbert spaces

    Hi! If I have understood things correctly, in a multi-electron atom you have that the spin operator ##S## commutes with the orbital angular momentum operator ##L##. However, as these operators act on wavefunctions living in different Hilbert spaces, how is it possible to even calculate the...
  43. pellman

    How does this statement follow? (adjoints on Hilbert spaces)

    If A is an operator on a Hilbert space H and A* is its adjoint, then . That is, the orthogonal complement of the range of A is the same subspace as the kernel of its adjoint. Then the author I am reading says it follows that the statements "The range of A is a dense subspace of H" and "A* is...
  44. pellman

    Equality of two elements of a hilbert space defined?

    Given x,y elements of a hilbert space H, how do we conclude that x = y? Since there is an inner product, we can say that x = y only if (x,z) = (y,z) for all z in H. But is there a definition of equality which does not depend on the inner product? A hilbert space is a special instance of...
  45. B

    Are Hilbert spaces still necessary?

    I have never been happy with the fact a single quantum state could be encoded by an infinite number of vectors |\phi\rangle. Choosing a unit vector limits this overabundance but you have still an infinity of (physically equivalent) possibilities left. I later realized that the projector...
  46. A. Neumaier

    When are isomorphic Hilbert spaces physically different?

    In quantum mechanics, a Hilbert space always means (in mathematical terms) a Hilbert space together with a distinguished irreducible unitary representation of a given Lie algebra of preferred observables on a common dense domain. Two Hilbert spaces are considered (physically) different if this...
  47. Logic Cloud

    Can Quantum Mechanics Have 'Elegant' Trajectories in Hilbert Space?

    State-space trajectories in classical mechanics can be used to nicely represent the time evolution of a given system. In the case of the harmonic oscillator, for instance, we get ellipses. How does this situation carry over to quantum mechanics? Can the time evolution of, say, the quantum...
  48. P

    Non-Commutative Hilbert Nullstellensatz?

    Hi, In my very naive understanding of algebraic geometry, I get the impression that it's written in language of commutative algebra and the main theorem (at least at the basic level) is Hilbert's Nullstellensatz. I'm curious if there's an analog of the Nullstellensatz for non-commutative...
  49. S

    Is this function in Hilbert space?

    Homework Statement (a) For what range of ##\nu## is the function ##f(x) = x^{\nu}## in Hilbert space, on the interval ##(0,1)##. Assume ##\nu## is real, but not necessarily positive. (b) For the specific case ##\nu = \frac{1}{2}##, is ##f(x)## in Hilbert space? What about ##xf(x)##? What...
  50. S

    Double Orthogonal Closed Subspace Inner Product => Hilbert

    Let X be an Inner Product Space. If for every closed subspace M, M^{\perp \perp} = M, then X is a Hilbert Space (It's complete). Hint: Use the following map: T : X \longrightarrow \overset{\sim}{X}: T(y)=(x,y)=f(x) where (x,y) is the inner product of X. Relevant equations: S^{\perp} is always...
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