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Ordering of the exponential of a quadratic in boson operators. II. Multimode case
1.C. L. Mehta, J. Math. Phys. 18, xxx (1977).
2.N. H. McCoy, Proc. Edinburgh Math. Soc. 3, 118 (1932).
3.R. M. Wilcox, J. Math. Phys. 8, 962 (1967).
4.C. L. Mehta, J. Phys. A (Proc. Phys. Soc. London) 1, 385 (1968).
5.F. A. Berezin, The Method of Second Quantization (Academic, New York, 1966), p. 143. See also Ref. 7.
6.C. L. Mehta, in Progress in Optics, Vol. VIII, edited by E. Wolf (North‐Holland, Amsterdam, 1970), p. 373, Eq. (A. 7). Our Eq. (3.7) is valid only if X is a positive definite matrix (the integral on the left‐hand side would otherwise diverge). However, as in true in the single mode case, one may use the arguments of analytic continuation to justify the final result [Eq. (3.10 or (3.11)], which is valid even when this restriction is relaxed.
7.Normal ordered form (3.10) or (3.11) may also be obtained using a result of R. Balian and E. Brezin [Nuovo Cimento B 64, 37 (1969)]. They have shown by group theoretic arguments that can be expressed as the product where contains only the creation operators, contains only the annihilation operators, and is of the form The normal ordered form of has been obtained in Ref. 4.
8.R. J. Glauber, Phys. Rev. 131, 2766 (1963).
9.See, for example, G. S. Agarwal and E. Wolf, Phys. Rev. D 2, 2161 (1970) or Ref. 4.
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