Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Fradkin tensor
open-in-new
<h2 id="definition">Definition</h2> <p>Suppose the <a href="/facts/Hamiltonian_function/2ReerlNA">Hamiltonian</a> of a harmonic oscillator is given by </p> H = p → 2 2 m + 1 2 m ω 2 x → 2 {\displaystyle H={\frac {{\vec {p}}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}{\vec {x}}^{2}} <p>with </p> <ul><li><a href="/facts/Momentum/TTkGv5eY">momentum</a> p → {\displaystyle {\vec {p}}} ,</li> <li><a href="/facts/Mass/HHdc8XmF">mass</a> m {\displaystyle m} ,</li> <li><a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> ω {\displaystyle \omega } , and</li> <li><a href="/facts/Displacement_(physics)/jkmEfS3C">displacement</a> x → {\displaystyle {\vec {x}}} ,</li></ul> <p>then the Fradkin tensor (up to an arbitrary normalisation) is defined as </p> F i j = p i p j 2 m + 1 2 m ω 2 x i x j . {\displaystyle F_{ij}={\frac {p_{i}p_{j}}{2m}}+{\frac {1}{2}}m\omega ^{2}x_{i}x_{j}.} <p>In particular, H {\displaystyle H} is given by the <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a>: H = Tr ( F ) {\displaystyle H=\operatorname {Tr} (F)} . The Fradkin Tensor is a thus a symmetric matrix, and for an n {\displaystyle n} -dimensional harmonic oscillator has n ( n + 1 ) 2 − 1 {\displaystyle {\tfrac {n(n+1)}{2}}-1} independent entries, for example 5 in 3 dimensions. </p> <h2 id="properties">Properties</h2> <ul><li>The Fradkin tensor is orthogonal to the <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a> L → = x → × p → {\displaystyle {\vec {L}}={\vec {x}}\times {\vec {p}}} : F i j L j = 0 {\displaystyle F_{ij}L_{j}=0} </li> <li>contracting the Fradkin tensor with the displacement vector gives the relationship x i F i j x j = E x → 2 − L → 2 2 m {\displaystyle x_{i}F_{ij}x_{j}=E{\vec {x}}^{2}-{\frac {{\vec {L}}^{2}}{2m}}} .</li> <li>The 5 independent components of the Fradkin tensor and the 3 components of angular momentum give the 8 generators of S U ( 3 ) {\displaystyle SU(3)} , the three-dimensional <a href="/facts/Special_unitary_group/Qgy2orcu">special unitary group</a> in 3 dimensions, with the relationships { L i , L j } = ε i j k L k { L i , F j k } = ε i j n F n k + ε i k n F j n { F i j , F k l } = ω 2 4 ( δ i k ε j l n + δ i l ε j k n + δ j k ε i l n + δ j l ε i k n ) L n , {\displaystyle {\begin{aligned}\{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{aligned}}} </li></ul> where { ⋅ , ⋅ } {\displaystyle \{\cdot ,\cdot \}} is the <a href="/facts/Poisson_bracket/7WUH7VXV">Poisson bracket</a>, δ {\displaystyle \delta } is the <a href="/facts/Kronecker_delta/gcGAy0bm">Kronecker delta</a>, and ε {\displaystyle \varepsilon } is the <a href="/facts/Levi-Civita_symbol/WBKUA3Tw">Levi-Civita symbol</a>. <h2 id="proof-of-conservation">Proof of conservation</h2> <p>In Hamiltonian mechanics, the time evolution of any function A {\displaystyle A} defined on <a href="/facts/Phase_space/qPe4Fq57">phase space</a> is given by </p> d A d t = { A , H } = ∑ k ( ∂ A ∂ x k ∂ H ∂ p k − ∂ A ∂ p k ∂ H ∂ x k ) + ∂ A ∂ t {\displaystyle {\frac {\mathrm {d} A}{\mathrm {d} t}}=\{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}}} , <p>so for the Fradkin tensor of the harmonic oscillator, </p> d F i j d t = 1 2 ω 2 ∑ k ( ( x j δ i k + x i δ j k ) p k − ( p j δ i k + p i δ j k ) x k ) = 0. {\displaystyle {\frac {\mathrm {d} F_{ij}}{\mathrm {d} t}}={\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.} . <p>The Fradkin tensor is the conserved quantity associated to the transformation </p> x i → x i ′ = x i + 1 2 ω − 1 ε j k ( x ˙ j δ i k + x ˙ k δ i j ) {\displaystyle x_{i}\to x_{i}'=x_{i}+{\frac {1}{2}}\omega ^{-1}\varepsilon _{jk}\left({\dot {x}}_{j}\delta _{ik}+{\dot {x}}_{k}\delta _{ij}\right)} <p>by <a href="/facts/Noether%2527s_theorem/VFjqrEvz">Noether's theorem</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="quantum-mechanics">Quantum mechanics</h2> <p>In quantum mechanics, position and momentum are replaced by the <a href="/facts/Position_operator/zQUM4fIk">position-</a> and <a href="/facts/Momentum_operator/nPx4UL3k">momentum operators</a> and the Poisson brackets by the <a href="/facts/Commutator/eYxzFmFY">commutator</a>. As such the Hamiltonian becomes the <a href="/facts/Hamiltonian_operator/5XwK2HmD">Hamiltonian operator</a>, angular momentum the <a href="/facts/Angular_momentum_operator/d55sXn5u">angular momentum operator</a>, and the Fradkin tensor the Fradkin operator. All of the above properties continue to hold after making these replacements. </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Jauch, Josef-Maria; Hill, Edward Lee (1 April 1940). "On the Problem of Degeneracy in Quantum Mechanics". Physical Review. 57 (7): 641–645. Bibcode:1940PhRv...57..641J. doi:10.1103/PhysRev.57.641. <a href="https://archive-ouverte.unige.ch/unige:162207" target="_blank">https://archive-ouverte.unige.ch/unige:162207</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Fradkin, David M. (1 May 1967). "Existence of the Dynamic Symmetries O 4 {\displaystyle O_{4}} and S U 3 {\displaystyle SU_{3}} for All Classical Central Potential Problems". Progress of Theoretical Physics. 37 (5): 798–812. doi:10.1143/PTP.37.798. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Miller, W.; Post, S.; Winternitz, P. (2013). "Classical and quantum superintegrability with applications". J. Phys. A: Math. Theor. 46 (42): 423001. arXiv:1309.2694. Bibcode:2013JPhA...46P3001M. doi:10.1088/1751-8113/46/42/423001. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Lévy-Leblond, Jean-Marc (1 May 1971). "Conservation Laws for Gauge-Variant Lagrangians in Classical Mechanics". American Journal of Physics. 39 (5): 502–506. Bibcode:1971AmJPh..39..502L. doi:10.1119/1.1986202. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>