The Fradkin tensor, or Jauch-Hill-Fradkin tensor, named after Josef-Maria Jauch and Edward Lee Hill [1] and David M. Fradkin, [2] is a conservation law used in the treatment of the isotropic multidimensional harmonic oscillator in classical mechanics. For the treatment of the quantum harmonic oscillator in quantum mechanics, it is replaced by the tensor-valued Fradkin operator.
The Fradkin tensor provides enough conserved quantities to make the oscillator's equations of motion maximally superintegrable. [3] This implies that to determine the trajectory of the system, no differential equations need to be solved, only algebraic ones.
Similarly to the Laplace–Runge–Lenz vector in the Kepler problem, the Fradkin tensor arises from a hidden symmetry of the harmonic oscillator.
Suppose the Hamiltonian of a harmonic oscillator is given by
with
then the Fradkin tensor (up to an arbitrary normalisation) is defined as
In particular, is given by the trace: . The Fradkin Tensor is a thus a symmetric matrix, and for an -dimensional harmonic oscillator has independent entries, for example 5 in 3 dimensions.
In Hamiltonian mechanics, the time evolution of any function defined on phase space is given by
so for the Fradkin tensor of the harmonic oscillator,
The Fradkin tensor is the conserved quantity associated to the transformation
by Noether's theorem. [4]
In quantum mechanics, position and momentum are replaced by the position- and momentum operators and the Poisson brackets by the commutator. As such the Hamiltonian becomes the Hamiltonian operator, angular momentum the angular momentum operator, and the Fradkin tensor the Fradkin operator. All of the above properties continue to hold after making these replacements.
The Fradkin tensor, or Jauch-Hill-Fradkin tensor, named after Josef-Maria Jauch and Edward Lee Hill [1] and David M. Fradkin, [2] is a conservation law used in the treatment of the isotropic multidimensional harmonic oscillator in classical mechanics. For the treatment of the quantum harmonic oscillator in quantum mechanics, it is replaced by the tensor-valued Fradkin operator.
The Fradkin tensor provides enough conserved quantities to make the oscillator's equations of motion maximally superintegrable. [3] This implies that to determine the trajectory of the system, no differential equations need to be solved, only algebraic ones.
Similarly to the Laplace–Runge–Lenz vector in the Kepler problem, the Fradkin tensor arises from a hidden symmetry of the harmonic oscillator.
Suppose the Hamiltonian of a harmonic oscillator is given by
with
then the Fradkin tensor (up to an arbitrary normalisation) is defined as
In particular, is given by the trace: . The Fradkin Tensor is a thus a symmetric matrix, and for an -dimensional harmonic oscillator has independent entries, for example 5 in 3 dimensions.
In Hamiltonian mechanics, the time evolution of any function defined on phase space is given by
so for the Fradkin tensor of the harmonic oscillator,
The Fradkin tensor is the conserved quantity associated to the transformation
by Noether's theorem. [4]
In quantum mechanics, position and momentum are replaced by the position- and momentum operators and the Poisson brackets by the commutator. As such the Hamiltonian becomes the Hamiltonian operator, angular momentum the angular momentum operator, and the Fradkin tensor the Fradkin operator. All of the above properties continue to hold after making these replacements.