Quantum Field Theory – Derivation of Schwinger and Hadamard Functions in Birrell’s and Davies’ Book

analyticitygreens-functionspropagatorqft-in-curved-spacetimequantum-field-theory

In Birrell's and Davies' book on "Quantum Fields in Curved Space", and in particularly in Chapter 2.7, the authors claim that from the expression
$$\mathcal{G}(x,x')=
\int\frac{d^nk}{(2\pi)^n}
\frac{e^{i\vec{k}\cdot(\vec{x}-\vec{x}')-ik^0(t-t')}}
{\big[(k^0)^2-|\vec{k}|^2-m^2\big]}\tag{2.76}$$
one can obtain all of the Green functions discussed later on.

For the case of the Feynman propagator, the retarded and the advanced propagators, it is fairly understandable that from the expression above, one can shift the locations of the poles accordingly, perform the $\int dk^0$ integral using contour integration methods and specifically the Cauchy residue theorem, and obtain the relevant expressions. For the case of Pauli-Jordan or Schwinger function
$$iG(x,x')=\langle0|[\phi(x),\phi(x')]|0\rangle,\tag{2.65}$$
the case of Hadamard's elementary function
$$G^{(1)}(x,x')=\langle0|\{\phi(x),\phi(x')\}|0\rangle\tag{2.66}$$
and the cases of the Wightman functions
$$G^+(x,x')=\langle0|\phi(x)\phi(x')|0\rangle \tag{2.68}$$ or
$$G^-(x,x')=\langle0|\phi(x')\phi(x)|0\rangle,\tag{2.68}$$ it is not obvious how the integral over $k^0$ is related to the integrals over the contours depicted in the Figure (Fig. 3 of the book).

Or to phrase my question a littile bit better, how am I supposed to obtain from the expression that yields all the possible Green functions, an integral over the contours depicted in Fig. 3, such that I can perform the aforesaid integrals and obtain the relevant expressions for the Pauli-Jordan or Schwinger function, the Hadamard's elementary function and the Wightman functions?

P.S.#1: Not related to Physical Interpretation for Schwinger and Hadamard functions

Best Answer

  1. That the $\pm$ Wightman functions (2.68) correspond to closed contours encircling the $\pm$ poles, respectively, is e.g. derived in eq. (2.50) of Peskin & Schroeder.

  2. The Pauli-Jordan/Schwinger and Hadamard functions are just some linear combinations of the Wightman functions, cf. Fig. 3.

  3. The above integration contours are closed.

  4. In contrast, the Feynman, advanced & retarded propagators have open integration coutours from $-\infty$ to $+\infty$ near the real axis, as OP essentially already mentions.

Related Question