Scientists / Open questions

Kenneth G. Wilson

Renormalization group; critical phenomena

1982 Nobel Prize in Physics

24 questions

Cyclic renormalization-group flows of the Glazek-Wilson quantum Hamiltonian and the prevalence of limit cycles
Constructive realization of Wilson's 1971 differential renormalization group for the Ising model below four dimensions: analytic flow functions, fixed-point convergence, and the critical scaling limit
Asymptotic structure of the $O(n>1)$ scaling equation of state near the coexistence curve below $T_c$ : Goldstone singularities beyond the leading term
Completing the similarity renormalization group for Hamiltonians: explicit flow construction, counterterm fixing, convergence, and infrared (small-x) behavior for light-front QCD
Rigorous error control, internal-basis selection, and extensions of Hilbert-space basis set reduction (HS-BSR)
Causal molecular and biochemical characterization of the chemically induced chlorophyll-deficient mutant panel of Mimulus cardinalis (Erythranthe cardinalis)
Continuum confining limit of Wilson's lattice gauge theory and the fate of its strong-coupling expansion
Rigorous foundations of Wilson's $\epsilon$ expansion for $O(n)$ critical exponents: large-order behavior, Borel summability, and nonperturbative validity of the matching construction
Axiomatic foundations of the Wilson-Zimmermann operator product expansion: conditions on a Wightman quantum field theory that guarantee its short-distance hypotheses
Extending the transverse renormalization program to the full light-front QCD Hamiltonian
Modification of the operator-product expansion $T(\psi(x)\bar\psi(y))$ under general gauge transformations in the Thirring model
Systematically convergent weak-coupling expansion of similarity-renormalized Hamiltonians at the bound-state formation scale in asymptotically free theories
Parameter selection for real-space renormalization-group transformations in finite-lattice approximations
Beyond first order: the fixed-point structure and eigenvalue spectrum of the Bell-Wilson nonlinear renormalization group for the Gaussian model
Coupling-coherent perturbative renormalization with infinitely many relevant and marginal operators: well-posedness, nonperturbative reliability, and the light-front QCD Hamiltonian
Completing the renormalization-group approach for electronic structure: coarse-graining of bound-state atoms, core-electron integration, and low-temperature auxiliary-field sampling without sign-problem bias (White-Wilkins-Wilson, 1986)
Wilson's 1975 challenge: practical numerical renormalization-group schemes for turbulence, the molecular bond, and strongly coupled quantum field theory, and a systematic procedure for constructing renormalization-group transformations
Open problems in light-front Tamm-Dancoff field theory: nonperturbative renormalization beyond the lowest truncation and effective Hamiltonians converging to QCD
Genericity of universality features in renormalization-group limit cycles: marginal operators in discrete cycles, tunability of irrelevant operators, and multiplicity of geometric spectra
Nonperturbative renormalization-group structure of strongly coupled relativistic field theories: strong-coupling fixed points, mass-deformation eigenvalues, and limit cycles
Completing Wilson's 1970 rigorous model of coupling-constant renormalization: operator renormalization, regularity of the fixed-point analysis, and extension to continuum relativistic field theory
Completing and characterizing the Metropolis ('Monte Carlo') code optimizer: convergence and expected time, safe-verification precision, address resolution, resource-bounded tracking, and the guidance-versus-robustness trade-off
Beyond the second-order valence approximation: a systematically improvable, first-principles light-front Hamiltonian QCD calculation of heavy-quarkonium spectra
Finite-N status of Wilson's fixed points below four dimensions: the four-fermion–Yukawa equivalence and summation of the $\epsilon$ expansion

Back to open problems

Cyclic renormalization-group flows of the Glazek-Wilson quantum Hamiltonian and the prevalence of limit cycles

Constructive realization of Wilson's 1971 differential renormalization group for the Ising model below four dimensions: analytic flow functions, fixed-point convergence, and the critical scaling limit

Asymptotic structure of the O(n>1)O(n>1)O(n>1) scaling equation of state near the coexistence curve below TcT_cTc​: Goldstone singularities beyond the leading term

Completing the similarity renormalization group for Hamiltonians: explicit flow construction, counterterm fixing, convergence, and infrared (small-x) behavior for light-front QCD

Rigorous error control, internal-basis selection, and extensions of Hilbert-space basis set reduction (HS-BSR)

Causal molecular and biochemical characterization of the chemically induced chlorophyll-deficient mutant panel of Mimulus cardinalis (Erythranthe cardinalis)

Continuum confining limit of Wilson's lattice gauge theory and the fate of its strong-coupling expansion

Rigorous foundations of Wilson's ϵ\epsilonϵ expansion for O(n)O(n)O(n) critical exponents: large-order behavior, Borel summability, and nonperturbative validity of the matching construction

Axiomatic foundations of the Wilson-Zimmermann operator product expansion: conditions on a Wightman quantum field theory that guarantee its short-distance hypotheses

Extending the transverse renormalization program to the full light-front QCD Hamiltonian

Modification of the operator-product expansion T(ψ(x)ψˉ(y))T(\psi(x)\bar\psi(y))T(ψ(x)ψˉ​(y)) under general gauge transformations in the Thirring model

Systematically convergent weak-coupling expansion of similarity-renormalized Hamiltonians at the bound-state formation scale in asymptotically free theories

Parameter selection for real-space renormalization-group transformations in finite-lattice approximations

Beyond first order: the fixed-point structure and eigenvalue spectrum of the Bell-Wilson nonlinear renormalization group for the Gaussian model

Coupling-coherent perturbative renormalization with infinitely many relevant and marginal operators: well-posedness, nonperturbative reliability, and the light-front QCD Hamiltonian

Completing the renormalization-group approach for electronic structure: coarse-graining of bound-state atoms, core-electron integration, and low-temperature auxiliary-field sampling without sign-problem bias (White-Wilkins-Wilson, 1986)

Wilson's 1975 challenge: practical numerical renormalization-group schemes for turbulence, the molecular bond, and strongly coupled quantum field theory, and a systematic procedure for constructing renormalization-group transformations

Open problems in light-front Tamm-Dancoff field theory: nonperturbative renormalization beyond the lowest truncation and effective Hamiltonians converging to QCD

Genericity of universality features in renormalization-group limit cycles: marginal operators in discrete cycles, tunability of irrelevant operators, and multiplicity of geometric spectra

Nonperturbative renormalization-group structure of strongly coupled relativistic field theories: strong-coupling fixed points, mass-deformation eigenvalues, and limit cycles

Completing Wilson's 1970 rigorous model of coupling-constant renormalization: operator renormalization, regularity of the fixed-point analysis, and extension to continuum relativistic field theory

Completing and characterizing the Metropolis ('Monte Carlo') code optimizer: convergence and expected time, safe-verification precision, address resolution, resource-bounded tracking, and the guidance-versus-robustness trade-off

Beyond the second-order valence approximation: a systematically improvable, first-principles light-front Hamiltonian QCD calculation of heavy-quarkonium spectra

Finite-N status of Wilson's fixed points below four dimensions: the four-fermion–Yukawa equivalence and summation of the ϵ\epsilonϵ expansion

Asymptotic structure of the $O(n>1)$ scaling equation of state near the coexistence curve below $T_c$ : Goldstone singularities beyond the leading term

Rigorous foundations of Wilson's $\epsilon$ expansion for $O(n)$ critical exponents: large-order behavior, Borel summability, and nonperturbative validity of the matching construction

Modification of the operator-product expansion $T(\psi(x)\bar\psi(y))$ under general gauge transformations in the Thirring model

Finite-N status of Wilson's fixed points below four dimensions: the four-fermion–Yukawa equivalence and summation of the $\epsilon$ expansion