Fradkin tensor

The Fradkin tensor, or Jauch-Hill-Fradkin tensor, named after Josef-Maria Jauch and Edward Lee Hill{{cite journal |last1=Jauch |first1=Josef-Maria |last2=Hill |first2=Edward Lee |title=On the Problem of Degeneracy in Quantum Mechanics |journal=Physical Review |date=1 April 1940 |volume=57 |issue=7 |pages=641–645 |doi=10.1103/PhysRev.57.641|bibcode=1940PhRv...57..641J |url=https://archive-ouverte.unige.ch/unige:162207 }} and David M. Fradkin,{{cite journal |last1=Fradkin |first1=David M. |title=Existence of the Dynamic Symmetries O_{4} and SU_{3} for All Classical Central Potential Problems |journal=Progress of Theoretical Physics |date=1 May 1967 |volume=37 |issue=5 |pages=798–812 |doi=10.1143/PTP.37.798}} 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.{{cite journal |last1=Miller |first1=W. |last2=Post |first2=S. |last3=Winternitz |first3=P. |title=Classical and quantum superintegrability with applications |journal=J. Phys. A: Math. Theor. |date=2013 |volume=46 |issue=42 |page=423001 |doi=10.1088/1751-8113/46/42/423001|arxiv=1309.2694 |bibcode=2013JPhA...46P3001M }} 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

: H = \frac{\vec p^2}{2m} + \frac{1}{2} m \omega^2 \vec x^2

with

then the Fradkin tensor (up to an arbitrary normalisation) is defined as

: F_{ij} = \frac{p_i p_j}{2m} + \frac{1}{2} m \omega^2 x_i x_j .

In particular, H is given by the trace: H = \operatorname{Tr}(F). The Fradkin Tensor is a thus a symmetric matrix, and for an n-dimensional harmonic oscillator has \tfrac{n(n+1)}{2} - 1 independent entries, for example 5 in 3 dimensions.

Properties

  • The Fradkin tensor is orthogonal to the angular momentum \vec L = \vec x \times \vec p:
  • : F_{ij} L_j = 0
  • contracting the Fradkin tensor with the displacement vector gives the relationship
  • : x_i F_{ij} x_j = E\vec x^2 - \frac{\vec L^2}{2m}.
  • The 5 independent components of the Fradkin tensor and the 3 components of angular momentum give the 8 generators of SU(3), the three-dimensional special unitary group in 3 dimensions, with the relationships
  • : \begin{align} \{L_i, L_j\} &= \varepsilon_{ijk} L_k \\

\{L_i, F_{jk}\} &= \varepsilon_{ijn} F_{nk} + \varepsilon_{ikn} F_{jn} \\

\{F_{ij},F_{kl}\} &= \frac{\omega^2}{4} \left(\delta_{ik} \varepsilon_{jln} + \delta_{il}\varepsilon_{jkn} + \delta_{jk} \varepsilon_{iln} + \delta_{jl} \varepsilon_{ikn}\right) L_n\,,\end{align}

: where \{\cdot,\cdot\} is the Poisson bracket, \delta is the Kronecker delta, and \varepsilon is the Levi-Civita symbol.

Proof of conservation

In Hamiltonian mechanics, the time evolution of any function A defined on phase space is given by

: \frac{\mathrm dA}{\mathrm dt} = \{A,H\} = \sum_k \left(\frac{\partial A}{\partial x_k} \frac{\partial H}{\partial p_k} - \frac{\partial A}{\partial p_k} \frac{\partial H}{\partial x_k}\right) + \frac{\partial A}{\partial t},

so for the Fradkin tensor of the harmonic oscillator,

: \frac{\mathrm dF_{ij}}{\mathrm dt} = \frac{1}{2} \omega^2 \sum_k \Big((x_j \delta_{ik} + x_i \delta_{jk}) p_k - (p_j \delta_{ik} + p_i \delta_{jk}) x_k \Big) = 0 ..

The Fradkin tensor is the conserved quantity associated to the transformation

: x_i \to x_i' = x_i + \frac 12 \omega^{-1} \varepsilon_{jk} \left(\dot x_j \delta_{ik} + \dot x_k \delta_{ij}\right)

by Noether's theorem.{{cite journal |last1=Lévy-Leblond |first1=Jean-Marc |title=Conservation Laws for Gauge-Variant Lagrangians in Classical Mechanics |journal=American Journal of Physics |date=1 May 1971 |volume=39 |issue=5 |pages=502–506 |doi=10.1119/1.1986202|bibcode=1971AmJPh..39..502L }}

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.

References