The Fradkin tensor, or Jauch-Hill-Fradkin tensor, named after Josef-Maria Jauch and Edward Lee Hill and David M. Fradkin, 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. 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.
Definition.
Suppose the Hamiltonian of a harmonic oscillator is given by
 formula_1
with
then the Fradkin tensor (up to an arbitrary normalisation) is defined as
 formula_6
In particular, formula_7 is given by the trace: formula_8. The Fradkin Tensor is a thus a symmetric matrix, and for an formula_9-dimensional harmonic oscillator has formula_10 independent entries, for example 5 in 3 dimensions.
 where formula_16 is the Poisson bracket, formula_17 is the Kronecker delta, and formula_18 is the Levi-Civita symbol.
Proof of conservation.
In Hamiltonian mechanics, the time evolution of any function formula_19 defined on phase space is given by
 formula_20,
so for the Fradkin tensor of the harmonic oscillator,
 formula_21.
The Fradkin tensor is the conserved quantity associated to the transformation
 formula_22
by Noether's theorem.
Quantum mechanics.
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.