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Spectral functions and Green’s functions: A critique

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Abstract

The various relations linking the spectral functions and the Green’s functionse.g.,J AB (ε)=[ImG R AB (ε)]/[1+γe −βε], are usually established via Lehmann’s approach. We demonstrate here how the same can be deduced elegantly through their respective superoperator representations. There are contexts in which the said relationships may not be strictly true and have to be modified through acδ(ε) term (c, a constant) e.g.,J AB (ε)=[ImG R AB (ε)]/[1+γe −βε]+cδ(ε). The origin and unambiguous analysis of such situations are presented. A proper bilinear form for the causal Green’s function is derived.

The ‘displaced harmonic oscillator’ is analysed in detail, to illustrate the issues raised above.

The superoperator representation of the imaginary time Green functions is also given, in passing.

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References

  • Abrikasov A A, Gorkov L P, Dzyaloshinski I E 1963Methods of quantum field theory in statistical physics (Englewood Cliffs, NJ: Prentice-Hall)

    Google Scholar 

  • Banwell C N and Primas H 1963Mol. Phys. 6 225

    Article  CAS  Google Scholar 

  • Bonch-Bruevich V L and Tyablikov S V 1962The Green function method in statistical mechanics (Amsterdam: North-Holland)

    Google Scholar 

  • Bowen S P 1975J. Math. Phys. (N.Y.) 16 620

    Article  Google Scholar 

  • Dalgaard E 1977J. Phys. B10 147

    Google Scholar 

  • Dalgaard E and Simons J 1977J. Phys. B10 2767

    Google Scholar 

  • Economou E N 1979Green’s functions in quantum physics (Berlin: Springer Verlag)

    Google Scholar 

  • Felter A L and Walecka J D 1971Quantum theory of many-particle systems (New York: Mc-Graw Hill)

    Google Scholar 

  • Fong F K 1975Theory of molecular relaxation (New York: Wiley-Interscience) Chap. 4, p. 61

    Google Scholar 

  • Haken H 1976Quantum field theory of solids (Amsterdam: North-Holland) Chap. 2, p. 13

    Google Scholar 

  • Jorgensen P and Simons J 1981Second quantization-based methods in quantum chemistry (New York: Academic Press)

    Google Scholar 

  • Linderberg J and Ohrn V 1973Propagators in quantum chemistry (New York: Academic Press)

    Google Scholar 

  • Lowdin P O 1982Int. J. Quantum Chem. S16 485

    Google Scholar 

  • Lehmann H 1954Nuovo Cimento 11 342

    Article  Google Scholar 

  • Mahan G D 1981Many-particle physics (New York: Plenum Press)

    Google Scholar 

  • Mattuck R D 1976A guide to Feynman diagrams in the many-body problem (New York: Mc-Graw Hill)

    Google Scholar 

  • Mc-Weeny R and Pickup B T 1980Rep. Prog. Phys. 43 1065

    Article  Google Scholar 

  • Newton R G 1966Scattering theory of waves and particles (New York: Mc-Graw Hill)

    Google Scholar 

  • Paul R 1982Field theoretical methods in chemical physics (Amsterdam: Elsevier)

    Google Scholar 

  • Ter Haar D 1962Fluctuation relaxation and resonance in magnetic systems (London: Oliver and Boyd)

    Google Scholar 

  • Thouless D J 1972The quantum mechanics of many-body systems (New York: Academic Press)

    Google Scholar 

  • Weiner B 1983J. Math. Phys. (N.Y.) 24 1791

    Article  CAS  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Inorganic and Physical Chemistry, Indian Institute of Science, 560 012, Banglore, India

    A. K. Mishra & S. K. Rangarajan

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  1. A. K. Mishra
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  2. S. K. Rangarajan
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Dedicated to Professor K S G Doss on his eightieth birthday.

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Mishra, A.K., Rangarajan, S.K. Spectral functions and Green’s functions: A critique. Proc. Indian Acad. Sci. (Chem. Sci.) 97, 307–331 (1986). https://doi.org/10.1007/BF02849197

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