6533b7cefe1ef96bd125710c

RESEARCH PRODUCT

Maslov Anomaly and the Morse Index Theorem

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Our starting point is again the phase space integral $$\displaystyle{ \text{e}^{\text{i}\hat{\varGamma }[\tilde{M}]} =\int \mathcal{D}\chi ^{a}\,\text{e}^{\text{i}S_{\text{fl}}[\chi,\tilde{M}]} }$$ (31.1) with periodic boundary conditions χ(0) = χ(T) and $$\displaystyle{ S_{\text{fl}}[\chi,\tilde{M}] = \frac{1} {2}\int _{0}^{T}dt\,\bar{\chi }_{ a}(t)\left [ \frac{\partial } {\partial t} -\tilde{M}(t)\right ]_{\phantom{a}b}^{a}\chi ^{b}(t)\;. }$$ (31.2) Here we have indicated that Sfl and \(\hat{\varGamma }\) depend on ηcl a and A i only through \(\tilde{M}_{\phantom{a}b}^{a}\): $$\displaystyle{ \tilde{M}(t)_{\phantom{a}b}^{a} =\omega ^{ac}\partial _{ c}\partial _{b}\mathcal{H}{\bigl (\eta _{\text{cl}}(t)\bigr )} =\omega ^{ac}\partial _{ c}\partial _{b}{\bigl (H + A_{i}J_{i}\bigr )}{\bigl (\eta _{\text{cl}}(t)\bigr )}\;. }$$ (31.3) We also have used the “dual” \(\bar{\chi }_{a} \equiv \chi ^{b}\omega _{ba}\) in (31.2). We decompose \(\chi ^{a} = (\pi _{i},x_{i}),\ a = 1,2,\,\ldots \,2N\); i = 1, 2…, N. Now, the Morse Index theorem works in configuration space. Therefore we have to convert the phase space path integral (31.1) to a configuration space integral by integrating out the momentum components π i . So let us first write: $$\displaystyle{ S_{\text{fl}} = \frac{1} {2}\int _{0}^{T}dt\left [\chi ^{a}\omega _{ ab}\dot{\chi }^{b} -\chi ^{a}\partial _{ a}\partial _{b}\mathcal{H}{\bigl (\eta _{\text{cl}}(t)\bigr )}\chi ^{b}\right ] }$$ (31.4) and define $$\displaystyle{ Q_{ab}(t):= \partial _{a}\partial _{b}\mathcal{H}{\bigl (\eta _{\mathrm{cl}}(t)\bigr )} =: \left (\begin{array}{*{10}c} Q_{ij}^{\pi \pi }(t) & Q_{ij}^{\pi x}(t) \\ Q_{ij}^{x\pi }(t)Q. }$$ (31.5)