Fradkin tensor

Suppose the Hamiltonian of a harmonic oscillator is given by

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

with

• momentum $\backslash vec\; p$,
• mass $m$,
• angular frequency $\backslash omega$, and
• displacement $\backslash vec\; x$,

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

: $F\_\{ij\}\; =\; \backslash frac\{p\_i\; p\_j\}\{2m\}\; +\; \backslash frac\{1\}\{2\}\; m\; \backslash omega^2\; x\_i\; x\_j\; .$

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

• The Fradkin tensor is orthogonal to the angular momentum $\backslash vec\; L\; =\; \backslash vec\; x\; \backslash times\; \backslash vec\; p$:
• : $F\_\{ij\}\; L\_j\; =\; 0$
• contracting the Fradkin tensor with the displacement vector gives the relationship
• : $x\_i\; F\_\{ij\}\; x\_j\; =\; E\backslash vec\; x^2\; -\; \backslash frac\{\backslash vec\; L^2\}\{2m\}$.
• 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
• :

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

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

: $\backslash frac\{\backslash mathrm\; dA\}\{\backslash mathrm\; dt\}\; =\; \backslash \{A,H\backslash \}\; =\; \backslash sum\_k\; \backslash left(\backslash frac\{\backslash partial\; A\}\{\backslash partial\; x\_k\}\; \backslash frac\{\backslash partial\; H\}\{\backslash partial\; p\_k\}\; -\; \backslash frac\{\backslash partial\; A\}\{\backslash partial\; p\_k\}\; \backslash frac\{\backslash partial\; H\}\{\backslash partial\; x\_k\}\backslash right)\; +\; \backslash frac\{\backslash partial\; A\}\{\backslash partial\; t\}$,

so for the Fradkin tensor of the harmonic oscillator,

: $\backslash frac\{\backslash mathrm\; dF\_\{ij\}\}\{\backslash mathrm\; dt\}\; =\; \backslash frac\{1\}\{2\}\; \backslash omega^2\; \backslash sum\_k\; \backslash Big((x\_j\; \backslash delta\_\{ik\}\; +\; x\_i\; \backslash delta\_\{jk\})\; p\_k\; -\; (p\_j\; \backslash delta\_\{ik\}\; +\; p\_i\; \backslash delta\_\{jk\})\; x\_k\; \backslash Big)\; =\; 0\; .$.

The Fradkin tensor is the conserved quantity associated to the transformation

: $x\_i\; \backslash to\; x\_i\text{'}\; =\; x\_i\; +\; \backslash frac\; 12\; \backslash omega^\{-1\}\; \backslash varepsilon\_\{jk\}\; \backslash left(\backslash dot\; x\_j\; \backslash delta\_\{ik\}\; +\; \backslash dot\; x\_k\; \backslash delta\_\{ij\}\backslash right)$

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 }}

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.

Category:Quantum mechanics

Category:Classical mechanics