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7
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34047183478
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A. Garg, Lecture Notes at the Boulder School for Condensed Matter and Materials Physics, Boulder, CO, 30 June-25 July 2003 (unpublished) (http://research.yale.edu/boulder/Boulder-2003/reading/garg_lecture_2.pdf).
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A. Garg, Lecture Notes at the Boulder School for Condensed Matter and Materials Physics, Boulder, CO, 30 June-25 July 2003 (unpublished) (http://research.yale.edu/boulder/Boulder-2003/reading/garg_lecture_2.pdf).
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8
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34047142421
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For one particle, the fluctuation integral can be found by the procedure in Refs. 5 and 7: (i) discretizing the path integral into M time slices, say, (ii) formal integration of the resulting Gaussian integral in terms of the determinant of the appropriate quadratic form, (iii) evaluation of this determinant [which is of order 2(M-1)] by finding a recursion relation for its successive diagonal subdeterminants, and (iv) converting this recursion relation into a differential equation which is identical to the Jacobi accessory equation. This approach fails for N particles, and we must resort to the integration by time slices.
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For one particle, the fluctuation integral can be found by the procedure in Refs. 5 and 7: (i) discretizing the path integral into M time slices, say, (ii) formal integration of the resulting Gaussian integral in terms of the determinant of the appropriate quadratic form, (iii) evaluation of this determinant [which is of order 2(M-1)] by finding a recursion relation for its successive diagonal subdeterminants, and (iv) converting this recursion relation into a differential equation which is identical to the Jacobi accessory equation. This approach fails for N particles, and we must resort to the integration by time slices.
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9
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34047125608
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In discussions with colleagues we have found that this complexification is often a source of confusion, so it is worthwhile to comment further on it. It is true that Eq, 2.4) only makes sense when z and z̄ are true complex conjugates (with
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j are complex.
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10
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34047190617
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In the one-particle case, one can also obtain a first-order recursion relation for det Gj, which turns out to be of the continued-fraction type. The standard substitution det Gj=aj/aj-1 turns this into a three-term linear recursion relation for aj, which goes directly into the Jacobi-like equation (2.46) when Δ → 0. In the N-particle case, however, this direct procedure fails, and this is why we examine the differential equation for Gud in the text
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ud in the text.
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11
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0003491619
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McGraw-Hill, New York, NY, Sec. 1.6
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C. M. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, NY, 1978), Sec. 1.6.
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(1978)
Advanced Mathematical Methods for Scientists and Engineers
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Bender, C.M.1
Orszag, S.2
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12
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34047132518
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The corresponding conditions on Y are Y(0)=1, Ẏ(0)=iA(0), which are precisely the ones that must be imposed on the Jacobi equation.
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The corresponding conditions on Y are Y(0)=1, Ẏ(0)=iA(0), which are precisely the ones that must be imposed on the Jacobi equation.
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