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---
title: "Imaginary time"
date: 2021-11-11
categories:
- Physics
- Quantum mechanics
layout: "concept"
---
Let $\hat{A}_S$ and $\hat{B}_S$ be time-independent in the Schrödinger picture.
Then, in the [Heisenberg picture](/know/concept/heisenberg-picture/),
consider the following expectation value
with respect to thermodynamic equilibium
(as found in [Green's functions](/know/concept/greens-functions/) for example):
$$\begin{aligned}
\expval{\hat{A}_H(t) \hat{B}_H(t')}
&= \frac{1}{Z} \Tr\!\Big( \exp(-\beta \hat{H}_{0,S}(t)) \: \hat{A}_H(t) \: \hat{B}_H(t') \Big)
\end{aligned}$$
Where the "simple" Hamiltonian $\hat{H}_{0,S}$ is time-independent.
Suppose a (maybe time-dependent) "difficult" $\hat{H}_{1,S}$ is added,
so that the total Hamiltonian is $\hat{H}_S = \hat{H}_{0,S} + \hat{H}_{1,S}$.
Then it is easier to consider the expectation value
in the [interaction picture](/know/concept/interaction-picture/):
$$\begin{aligned}
\expval{\hat{A}_H(t) \hat{B}_H(t')}
&= \frac{1}{Z} \Tr\!\Big( \exp(-\beta \hat{H}_S(t)) \: \hat{K}_I(0, t) \hat{A}_I(t) \hat{K}_I(t, t') \hat{B}_I(t') \hat{K}_I(t', 0) \Big)
\end{aligned}$$
Where $\hat{K}_I(t, t_0)$ is the time evolution operator of $\hat{H}_{1,S}$.
In front, we have $\exp(-\beta \hat{H}_S(t))$,
while $\hat{K}_I$ is an exponential of an integral of $\hat{H}_{1,I}$, so we are stuck.
Keep in mind that exponentials of operators
cannot just be factorized, i.e. in general
$\exp(\hat{A} \!+\! \hat{B}) \neq \exp(\hat{A}) \exp(\hat{B})$
To get around this, a useful mathematical trick is
to use an **imaginary time** variable $\tau$ instead of the real time $t$.
Fixing a $t$, we "redefine" the interaction picture along the imaginary axis:
$$\begin{aligned}
\boxed{
\hat{A}_I(\tau)
\equiv \exp\!\bigg(\frac{\tau \hat{H}_{0,S}}{\hbar}\bigg) \: \hat{A}_S \: \exp\!\bigg( \!-\! \frac{\tau \hat{H}_{0,S}}{\hbar}\bigg)
}
\end{aligned}$$
Ironically, $\tau$ is real; the point is that this formula
comes from the real-time definition by replacing $t \to -i \tau$.
The Heisenberg and Schrödinger pictures can be redefined in the same way.
In fact, by substituting $t \to -i \tau$,
all the key results of the interaction picture can be updated,
for example the Schrödinger equation for $\Ket{\psi_S(\tau)}$ becomes:
$$\begin{aligned}
\hbar \dv{}{t}\Ket{\psi_S(\tau)}
= - \hat{H}_S \Ket{\psi_S(\tau)}
\quad \implies \quad
\Ket{\psi_S(\tau)}
= \exp\!\bigg( \!-\! \frac{\tau \hat{H}_S}{\hbar} \bigg) \Ket{\psi_H}
\end{aligned}$$
And the interaction picture's time evolution operator $\hat{K}_I$
turns out to be given by:
$$\begin{aligned}
\boxed{
\hat{K}_I(\tau, \tau_0)
= \mathcal{T} \bigg\{ \exp\!\bigg( \!-\! \frac{1}{\hbar} \int_{\tau_0}^\tau \hat{H}_{1,I}(\tau') \dd{\tau'} \bigg) \bigg\}
}
\end{aligned}$$
Where $\mathcal{T}$ is the
[time-ordered product](/know/concept/time-ordered-product/)
with respect to $\tau$.
This operator works as expected:
$$\begin{aligned}
\Ket{\psi_I(\tau)}
= \hat{K}_I(\tau, \tau_0) \Ket{\psi_I(\tau_0)}
\end{aligned}$$
Where $\Ket{\psi_I(\tau)}$ is related to
the Schrödinger and Heisenberg pictures as follows:
$$\begin{aligned}
\Ket{\psi_I(\tau)}
\equiv \exp\!\bigg(\frac{\tau \hat{H}_{0,S}}{\hbar}\bigg) \Ket{\psi_S(\tau)}
= \exp\!\bigg(\frac{\tau \hat{H}_{0,S}}{\hbar}\bigg) \exp\!\bigg( \!-\! \frac{\tau \hat{H}_S}{\hbar}\bigg) \Ket{\psi_H}
\end{aligned}$$
It is interesting to combine this definition
with the action of time evolution $\hat{K}_I(\tau, \tau_0)$:
$$\begin{aligned}
\Ket{\psi_I(\tau)}
&= \hat{K}_I(\tau, \tau_0) \Ket{\psi_I(\tau_0)}
\\
\exp\!\bigg(\frac{\tau \hat{H}_{0,S}}{\hbar}\bigg) \exp\!\bigg( \!-\! \frac{\tau \hat{H}_S}{\hbar}\bigg) \Ket{\psi_H}
&= \hat{K}_I(\tau, \tau_0) \exp\!\bigg(\frac{\tau_0 \hat{H}_{0,S}}{\hbar}\bigg) \exp\!\bigg( \!-\! \frac{\tau_0 \hat{H}_S}{\hbar}\bigg) \Ket{\psi_H}
\end{aligned}$$
Rearranging this leads to the following useful
alternative expression for $\hat{K}_I(\tau, \tau_0)$:
$$\begin{aligned}
\boxed{
\hat{K}_I(\tau, \tau_0)
= \exp\!\bigg(\frac{\tau \hat{H}_{0,S}}{\hbar}\bigg)
\exp\!\bigg(\!-\! \frac{(\tau \!-\! \tau_0) \hat{H}_{S}}{\hbar}\bigg)
\exp\!\bigg(\!-\! \frac{\tau_0 \hat{H}_{0,S}}{\hbar}\bigg)
}
\end{aligned}$$
Returning to our initial example,
we can set $\tau = \hbar \beta$ and $\tau_0 = 0$,
so $\hat{K}_I(\tau, \tau_0)$ becomes:
$$\begin{aligned}
\hat{K}_I(\hbar \beta, 0)
&= \exp\!\big(\beta \hat{H}_{0,S}\big) \exp\!\big(\!-\! \beta \hat{H}_{S}\big)
\\
\implies \quad
\exp\!\big(\!-\! \beta \hat{H}_{S}\big)
&= \exp\!\big(\!-\! \beta \hat{H}_{0,S}\big) \hat{K}_I(\hbar \beta, 0)
\end{aligned}$$
Using the easily-shown fact that
$\hat{K}_I(\hbar \beta, 0) \hat{K}_I(0, \tau) = \hat{K}_I(\hbar \beta, \tau)$,
we can therefore rewrite the thermodynamic expectation value like so:
$$\begin{aligned}
\expval{\hat{A}_H(\tau) \hat{B}_H(\tau')}
&= \frac{1}{Z} \Tr\!\Big(\! \exp(-\beta \hat{H}_{0,S}) \hat{K}_I(\hbar \beta, \tau)
\hat{A}_I(\tau) \hat{K}_I(\tau, \tau') \hat{B}_I(\tau') \hat{K}_I(\tau', 0) \!\Big)
\end{aligned}$$
We now introduce a time-ordering $\mathcal{T}$,
letting us reorder the (bosonic) $\hat{K}_I$-operators inside,
and thereby reduce the expression considerably:
$$\begin{aligned}
\Expval{\mathcal{T}\Big\{\hat{A}_H \hat{B}_H\Big\}}
&= \frac{1}{Z} \Tr\!\Big( \mathcal{T} \Big\{ \hat{K}_I(\hbar \beta, \tau) \hat{K}_I(\tau, \tau') \hat{K}_I(\tau', 0)
\hat{A}_I(\tau) \hat{B}_I(\tau') \Big\} \exp(-\beta \hat{H}_{0,S}) \Big)
\\
&= \frac{1}{Z} \Tr\!\Big( \mathcal{T}\Big\{ \hat{K}_I(\hbar \beta, 0) \hat{A}_I(\tau) \hat{B}_I(\tau') \Big\} \exp(-\beta \hat{H}_{0,S}) \Big)
\end{aligned}$$
Where $Z = \Tr\!\big(\exp(-\beta \hat{H}_S)\big) = \Tr\!\big(\hat{K}_I(\hbar \beta, 0) \exp(-\beta \hat{H}_{0,S})\big)$.
If we now define $\Expval{}_0$ as the expectation value with respect
to the unperturbed equilibrium involving only $\hat{H}_{0,S}$,
we arrive at the following way of writing this time-ordered expectation:
$$\begin{aligned}
\boxed{
\Expval{\mathcal{T}\Big\{\hat{A}_H \hat{B}_H\Big\}}
= \frac{\Expval{\mathcal{T}\Big\{ \hat{K}_I(\hbar \beta, 0) \hat{A}_I(\tau) \hat{B}_I(\tau') \Big\}}_0}{\Expval{\hat{K}_I(\hbar \beta, 0)}_0}
}
\end{aligned}$$
For another application of imaginary time,
see e.g. the [Matsubara Green's function](/know/concept/matsubara-greens-function/).
## References
1. H. Bruus, K. Flensberg,
*Many-body quantum theory in condensed matter physics*,
2016, Oxford.
|
