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RESEARCH PRODUCT

The “Maslov Anomaly” for the Harmonic Oscillator

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Specializing the discussion of the previous section to the harmonic oscillator we have for \(N = 1,\ \eta ^{a} = (p,x),\ a = 1,2,\ \eta ^{1} \equiv p,\ \eta ^{2} \equiv x\) $$\displaystyle{ H(p,x) = \frac{1} {2}\eta ^{a}\eta ^{a} = \frac{1} {2}{\bigl (p^{2} + x^{2}\bigr )}\;. }$$ (30.1) The only conserved quantity is J = H. In the action we need the combination $$\displaystyle{ \frac{1} {2}\eta ^{a}\omega _{ ab}\dot{\eta }^{b} -\mathcal{H}(\eta ) = \frac{1} {2}\eta ^{a}\left [\omega _{ ab} \frac{d} {dt} -{\bigl ( 1 + A(t)\bigr )}\mathrm{1l}_{ab}\right ]\eta ^{b} }$$ (30.2) and $$\displaystyle{ \tilde{M}_{\phantom{a}b}^{a} =\omega ^{ac}\partial _{ c}\partial _{b}(H + AJ\,) ={\bigl ( 1 + A(t)\bigr )}\omega ^{ac}\mathrm{1l}_{ cb} }$$ or, compactly written: $$\displaystyle{ \tilde{M} ={\bigl ( 1 + A(t)\bigr )}\varOmega \;, }$$ (30.3) where we have introduced the notation \(\varOmega \equiv \omega ^{-1} = (\omega ^{ab}).\ A(t)\) is the gauge field for a single U(1) group associated with energy conservation. The fluctuation part of the action is now given by $$\displaystyle{ S_{\text{fl}} = \frac{1} {2}\int _{0}^{T}dt\,\chi ^{a}\omega _{ ab}\left [ \frac{d} {dt} -\tilde{M}\right ]_{\phantom{b}c}^{b}\chi ^{c}\;, }$$ (30.4)