Relation between time-ordered propagator in condensed matter and Feynman propagator

Question

In particle physics I am used to the Feynman propagator being decomposed into positive and negative frequency Wightman functions. For example, this is the representation used in Eq. (6.2.13) of Weinberg's volume I [1]. In his notation,

\begin{equation} -\mathrm{i} \Delta_{\text{F}}(x) = \theta(x) \Delta_+(x) + \theta(-x) \Delta_+(-x) \end{equation}

This is constructed to have only positive frequency terms at future infinity, and only negative frequency terms at past infinity. In particular, the function $\Delta_+(-x)$ used by Weinberg can be written as the negative frequency Wightman function $\Delta_-(x)$, and

\begin{equation} \Delta_+(x) = \int \frac{\mathrm{d}^3 q}{(2\pi)^3} \frac{1}{2 E_{\mathbf{q}}} \mathrm{e}^{-\mathrm{i} E_{\mathbf{q}} t + \mathrm{i} \mathbf{q} \cdot \mathbf{x}} , \quad \Delta_-(x) = \int \frac{\mathrm{d}^3 q}{(2\pi)^3} \frac{1}{2 E_{\mathbf{q}}} \mathrm{e}^{+\mathrm{i} E_{\mathbf{q}} t -\mathrm{i} \mathbf{q} \cdot \mathbf{x}} \end{equation}

where $E_{\mathbf{q}}$ is the energy for a particle of momentum $\mathbf{q}$.

But when I compare with the formula for the time-ordered propagator given in Altland & Simons [2] or Kamenev [3], the result seems different. Kamenev gives (now in his notation)

\begin{align} \langle \phi^+(t) \bar{\phi}^-(t') \rangle = \mathrm{i} G^<(t,t') & = n_B \mathrm{e}^{-\mathrm{i} \omega_0 (t-t')} \\ \langle \phi^-(t) \bar{\phi}^+(t') \rangle = \mathrm{i} G^>(t,t') & = (n_B + 1) \mathrm{e}^{-\mathrm{i} \omega_0 (t - t')} \\ \langle \phi^+(t) \bar{\phi}^+(t') \rangle = \mathrm{i} G^{\mathbb{T}}(t,t') & = \theta(t-t') \mathrm{i} G^>(t,t') + \theta(t'-t) \mathrm{i} G^<(t,t') \\ \langle \phi^-(t) \bar{\phi}^-(t') \rangle = \mathrm{i} G^{\tilde{\mathbb{T}}}(t,t') & = \theta(t-t') \mathrm{i} G^<(t,t') + \theta(t'-t) \mathrm{i} G^>(t,t') \end{align}

Altland & Simons have something equivalent in their Eq. (11.16). In this notation, if I have understood, $\mathrm{i} G^>$ and $\mathrm{i} G^<$ take the place of the positive and negative frequency Wightman functions.

Then, the $++$ correlation function gives the time-ordered Green's function $G^{\mathbb{T}}$, and the $--$ correlation function gives the anti-time-ordered one $G^{\tilde{\mathbb{T}}}$. But if I go to vacuum where $n_B = 0$, it's clear that, although $G^{\mathbb{T}}$ is built from modes of definite frequency, it has no support at all for $t' > t$. Likewise, $G^{\tilde{\mathbb{T}}}$ has no support for $t > t'$. So whatever $G^{\mathbb{T}}(t,t')$ and $G^{\tilde{\mathbb{T}}}(t,t')$ are, they are apparently not what I would expect for the Feynman and Dyson propagators in vacuum.

Kamenev and Altland & Simons compute their propagators by specifying a density matrix at an initial time. In the standard textbook presentation of the Feynman propagator, the boundary conditions enter in a more opaque way via an explicit computation of the expectation value of the time-ordered product.

The answer I am expecting looks more like the formula for the $++$, $--$, $+-$ and $-+$ correlation functions given by Glavan & Prokopec in their lecture notes on non-equilibrium field theory [4]. Their Eqs (237) and (64) for the $++$ correlation function and Feynman propagator match (as far as I can see) Weinberg's formula for $\Delta_F$ and Kamenev's formula for $\mathrm{i} G^{\mathbb{T}}$. But then, their Wightman functions Eq. (72) don't look like $\mathrm{i} G^>$, $\mathrm{i} G^<$ given by Kamenev and Altland & Simons. Glavan & Prokopec quote (in their notation)

\begin{align} \mathrm{i} \Delta^+(t,t') & \propto (1 + \bar{n}_0) \mathrm{e}^{-\mathrm{i} \omega (t-t')} + \bar{n}_0 \mathrm{e}^{\mathrm{i} \omega (t-t')} \\ \mathrm{i} \Delta^-(t,t') & = \Big[ \mathrm{i} \Delta^+(t,t') \Big]^\ast \end{align}

where I think $\bar{n}_0 \mapsto n_B$ and $\omega \mapsto \omega_0$ in Kamenev's notation. In vacuum, this has the positive/negative frequency behaviour at infinity that I am expecting.

How am I supposed to understand this? Clearly a different boundary condition is being applied to the time ordered propagator. What is different about what Glavan & Prokopec are assuming, compared to Kamenev and Altland & Simons?

• [1] Weinberg, The Quantum Theory of Fields Volume I: Foundations, Cambridge University Press
• [2] Altland & Simons, Condensed Matter Field Theory, 2nd edition, Cambridge University Press
• [3] Kamenev, Field Theory of Non-Equilibrium Systems, 2nd edition, Cambridge University Press
• [4] https://webspace.science.uu.nl/~proko101/LecturenotesNonEquilQFT.pdf

Answer 1

I think you are confusing two different things here. The distinction of positive and negative frequency Weightman functions in Weinberg denoted by $+$ and $-$ have nothing to do with the $\pm$ labels in the book by Kamanev. In the latter case they denote the branch of the closed time contour that is used in the non-equilibrium/Keldysh formalism. The book by Weinberg or e.g. Peskin deal only with equilibrium/vacuum field theory so this more complicated time integration contour does not show up at all.

There is a second distinction, which I unfortunately do not understand 100% myself. Namely that for some reason in CMT the fields are typically defined as $$ \Psi(\textbf{x}) = \sum_\textbf{k} {a}_\textbf{k} e^{i\textbf{x}\cdot\textbf{k}} $$ whereas in high energy physics they are defined as $$ \Psi(\textbf{x}) = \int_\textbf{k}\left( {a}_\textbf{k} e^{i\textbf{x}\cdot\textbf{k}} + {a}^\dagger_\textbf{k} e^{-i\textbf{x}\cdot\textbf{k}}\right) $$ Because of this the CMT fields do not have positive and negative frequency components and the HEP fields do. Why this is the way they are defined, I would like to know myself.