Under construction

1. Setup and conventions

Real scalar field with Lagrangian density

$\mathcal{L}[\varphi,\partial_\mu\varphi] = -\frac{1}{2}\varphi(x)(\partial{\cdot} \partial + m^2)\varphi(x)$,

and equation of motion $-(\partial{\cdot} \partial+m^2)\varphi(x)=0$.

Mode expansion

\[\varphi(x) = \varphi^+(x) + \varphi^-(x)\] \[\varphi^+(x) = \int\frac{d^3k}{(2\pi)^{3/2}}\frac{a(\mathbf{k})}{\sqrt{2\omega_k}}\,e^{-\mathrm{i} \mathbf{k\cdot x}}, \qquad \varphi^-(x) = \int\frac{d^3k}{(2\pi)^{3/2}}\frac{a^\dagger(\mathbf{k})}{\sqrt{2\omega_k}}\,e^{+\mathrm{i} \mathbf{k\cdot x}}\]

We denote a density matrix state by $\rho$. Expectation values $\operatorname{tr}(_\,\rho)$ specialise to vacuum expectation values $\braket{0

(_)

0}$ when $\rho = {\ket{0}}!{\bra{0}}$.

2. Operator definitions

Defined directly from the field operators, and trace and state $\rho$.

Name	Definition
Wightman function	$W(x,x’) = \operatorname{tr}(\varphi(x) \varphi(x’)\,\rho)$
Hadamard function	$H(x,x’) = \operatorname{tr}(\lbrace\varphi(x),\varphi(x’)\rbrace\,\rho)$
Pauli–Jordan function	$\mathrm{i} E(x,x’) = \operatorname{tr}([\varphi(x),\varphi(x’)]\,\rho)$

$H$ is symmetric and $E$ is antisymmetric in $x, x’$; see Relations section. $E$ is state-independent (depends only on the classical symplectic structure) and vanishes for spacelike separation (microcausality).

Positive and negative frequency Wightman functions are defined by their Fourier support:

	Definition	Fourier support
$W^+(x,x’)$	$W(x,x’)$	$\tilde{W}^+(k)$ on $k^0 > 0$
$W^-(x,x’)$	$W(x’,x)$	$\tilde{W}^-(k)$ on $k^0 < 0$

Spectral representations

\[\mathrm{i} E(x) = \int\frac{d^4k}{(2\pi)^3}\,e^{-\mathrm{i} \mathbf{k\cdot x}}\,\delta(k{\cdot}k - m^2)\,\operatorname{sgn}(k^0)\]

using $\delta(k{\cdot}k - m^2)\,\theta(\pm k^0) = \delta(k^0 \mp \omega_k)/(2\omega_k)$.

3. Derived propagators

All entries below are Green’s functions: $(\partial{\cdot} \partial + m^2)G = -\delta^{(4)}(x-x’)$.

Retarded and advanced

\[G_R(x,x') = \theta(t-t')\,E(x,x')\] \[G_A(x,x') = -\theta(t'-t)\,E(x,x')\]

$G_R$ has support in the future lightcone of $x’$; $G_A$ in the past lightcone.¹

Time-ordered and anti-time-ordered

\[\mathrm{i} G_F(x,x') = \theta(t-t')\,W^+(x,x') + \theta(t'-t)\,W^-(x,x')\] \[\mathrm{i} G_D(x,x') = \theta(t-t')\,W^-(x,x') + \theta(t'-t)\,W^+(x,x')\]

$G_F$ is the Feynman (time-ordered) propagator; $G_D$ is the Dyson (anti-time-ordered) propagator.

Symmetric combination

\[G_S(x,x') = G_R(x,x') + G_A(x,x')\]

$G_S$ is time-reversal symmetric; also called the principal-part propagator.

4. Relations

Argument exchange

\[G_A(x,x') = G_R(x',x)\]

Commutator function

It’s naturally related to $G_R$ and $G_A$

\[E(x,x') = G_R(x,x') - G_A(x,x')\]

Pauli–Jordan in terms of Wightman functions

\[\mathrm{i} E(x,x') = W^+(x,x') - W^-(x,x')\]

Symmetric and antisymmetric combinations of retarded and advanced propagators

\[G_S(x,x') = \operatorname{sgn}(t-t')\,E(x,x')\]

Wightman decomposition

\[W(x,x') = \frac{H(x,x')}{2} + \mathrm{i}\frac{E(x,x')}{2}\]

Feynman in terms of $H$ and $G_S$

\[\mathrm{i} G_F(x,x') = \frac{H(x,x')}{2} + \mathrm{i}\frac{G_S(x,x')}{2}\]

5. References

Streater, R. F. and Wightman, A. S., PCT, Spin and Statistics, and All That. Princeton University Press, 2000.
Birrell, N. D. and Davies, P. C. W., Quantum Fields in Curved Space. Cambridge University Press, 1984.
Kocic, M. B., Invariant Commutation and Propagation Functions. FK8017 HT15, v1.01, 2016. nLab PDF.

Sign convention differs from Birrell–Davies [2, p. 21] by an overall $-1$ in both $G_R$ and $G_A$. ↩