Rotate time 90° onto the imaginary axis and quantum time evolution turns into a statistical weight ── temperature was the circumference of that circle
So far we have treated temperature as the slope of a sharing of energy. This episode shows a completely different face ── temperature is the reciprocal of the circumference of a circle formed by time's other axis (imaginary time). It sounds outlandish, but what produces it is a startlingly simple comparison. Quantum time evolution is \(e^{-iHt/\hbar}\); the statistical weight is \(e^{-H/k_BT}\). They look almost identical. For the two to agree we need \(t=-i\hbar/k_BT\) ── a purely imaginary time. And this is not a formal game: it carries the physical content that going once around in imaginary time multiplies you by the Boltzmann factor (the KMS condition), it gives another reading of Episode 3's "freezing," and it is the very origin of the Hawking and Unruh temperatures in the finale. It also settles the homework left in Episode 9 of the sister series "Cosmology That Clicks."
A quantum state's phase rotates with time: \(e^{-iEt/\hbar}\). It only oscillates; the magnitude never changes.
Now twist time onto the imaginary axis. Set \(t=-i\tau\) with \(\tau\) real:
$$e^{-iE t/\hbar}\ \xrightarrow{\ t=-i\tau\ }\ e^{-E\tau/\hbar}$$Oscillation has turned into decay. And the decay \(e^{-E\tau/\hbar}\) looks exactly like Episode 2's Boltzmann factor \(e^{-E/k_BT}\).
In other words ── advancing \(\hbar/k_BT\) in imaginary time is the same thing as coming to thermal equilibrium at temperature \(T\). This length is written \(\beta\hbar\) (with \(\beta=1/k_BT\), the lead of Episode 2).
Moreover, the quantum partition function \(Z=\mathrm{Tr}\,e^{-\beta H}\) has the form of "start and end at the same place" (a trace), so the imaginary-time direction is a circle with its ends joined. A system with a temperature is one in which one direction of time is a circle of circumference \(\beta\hbar\).
The consequence of that rephrasing is the most interesting part.
Take a state of energy \(E\) once around in imaginary time and it comes back multiplied by \(e^{-E\beta\hbar/\hbar}=e^{-E/k_BT}\).
"Being in thermal equilibrium" = "one lap in imaginary time multiplies by the Boltzmann factor." Stated rigorously this is the KMS condition (Kubo–Martin–Schwinger), and in the modern view it is adopted as the definition of temperature. However complicated the interactions, even where no particle number can be defined, "periodicity in imaginary time" can still be written ── this is the most general definition of temperature there is.
Episode 1 had us memorise "room temperature \(k_BT\approx25\) meV." That and 25 fs are two sides of a single fact (\(\hbar/25\,\mathrm{meV}=26\) fs). In energy it is 25 meV; in time it is 25 fs ── one and the same mark for the state called room temperature, tied together by the uncertainty relation \(\Delta E\,\Delta t\sim\hbar\).
| Temperature | k_BT | imaginary-time period ℏ/k_BT | |
|---|---|---|---|
| cosmic microwave background, 2.7 K | 0.23 meV | 2.8 ps | a long circle = cold |
| liquid nitrogen, 77 K | 6.6 meV | 99 fs | |
| room temperature, 300 K | 25 meV | 25 fs | the reference |
| the Sun's surface, 5800 K | 0.50 eV | 1.3 fs | |
| the Sun's core, 1.5×10⁷ K | 1.3 keV | 0.5 as | a very short circle = hot |
| absolute zero | 0 | ∞ | the circle unrolls into a plain straight line |
That last row is suggestive ── absolute zero is the state in which the imaginary-time circle has unrolled to infinite length. Turned around: to have a temperature is for one direction of time to be curled up. Episode 1 said temperature is not a property of matter; by this point it starts to look like a property of spacetime. In the finale that intuition becomes literally true.
The figure below is the complex time plane. Horizontal is real time (ordinary time), vertical is imaginary time \(\tau\). The top and bottom edges of the strip are the same points, curled there into a cylinder ── so the height of the strip is exactly the period \(\beta\hbar\).
Move the temperature slider and the height of the strip changes. Cool it and the strip grows taller (the circle grows longer); heat it and the strip thins (the circle shortens). The shading inside is the decay \(e^{-E\tau/\hbar}\), and the number on the right is the factor picked up in one lap = the Boltzmann factor. Switch the energy and you can read off whether that state gets thermally excited, as a one-lap factor.
① Episode 3's "freezing" becomes a question of fitting on the circle.
If the imaginary-time direction is a circle of circumference \(\beta\hbar\), the frequencies that can live on it are discrete (like standing waves on a string) ── \(\omega_n=2\pi n/\beta\hbar\). These are the Matsubara frequencies. The first is \(\hbar\omega_1=2\pi k_BT\), which at room temperature is 157 meV. A degree of freedom with a level spacing larger than that does not fit on the circle = it cannot be excited thermally ── Episode 3's freezing by \(\hbar\omega/k_BT\), restated as geometry.
② Statistical-mechanics calculations turn into quantum-mechanics calculations.
A \(d\)-dimensional statistical-mechanics problem can be solved as \(d-1\) spatial dimensions plus one (periodic) imaginary-time dimension ── i.e. as a quantum-mechanical problem. Episode 4 of the sister series "Renormalization That Clicks" dealt with the 2D Ising model; that is equivalent to a 1D quantum spin chain ── one dimension gets transferred. This statistical ↔ quantum correspondence is an everyday computational tool, from lattice gauge theory to condensed matter.
③ "Having a temperature" can be stated geometrically.
This is the bridge to the finale. If the imaginary-time period is the temperature, then conversely ── if the imaginary-time direction of spacetime happens to be curled up for some reason of its own, there is a temperature there. That is exactly what happens for an accelerating observer, and for an observer outside a black hole.
Established: that the Wick rotation \(t=-i\tau\) maps the quantum time-evolution operator \(e^{-iHt/\hbar}\) to the statistical weight \(e^{-\beta H}\); that the quantum partition function \(Z=\mathrm{Tr}\,e^{-\beta H}\) can be written as an imaginary-time path integral of period \(\beta\hbar\); that the KMS condition gives a general characterisation of thermal equilibrium states; the Matsubara frequencies \(\omega_n=2\pi n/\beta\hbar\) (for bosons); the correspondence between \(d\)-dimensional classical statistics and \((d-1)+1\)-dimensional quantum systems; and \(\beta\hbar\approx25\) fs at room temperature ── all established physics.
Points to note: (1) Imaginary time is not a "second time that exists." It is a computational construction obtained by analytically continuing the real-time theory; what is observed is always a real-time quantity. After computing in imaginary time you must continue back to real time ── and that reverse continuation is famously ill-conditioned numerically. (2) For fermions the boundary condition in the imaginary-time direction is anti-periodic and the Matsubara frequencies become half-integer (the body treats bosons). (3) The Wick rotation is not always permitted. With time-dependent external fields, out of equilibrium, and in general curved spacetimes, Euclideanisation can be non-unique or simply undefined. (4) The strip height in the figure scales as \(1/T\) but is clipped top and bottom to fit the display (it is not drawn to true scale). (5) "Having a temperature = time being curled up" is a restatement for equilibrium states.
Rotate time 90° onto the imaginary axis (the Wick rotation \(t=-i\tau\)) and quantum phase rotation \(e^{-iEt/\hbar}\) turns into decay \(e^{-E\tau/\hbar}\). Simply comparing that with Episode 2's Boltzmann factor \(e^{-E/k_BT}\) gives \(\tau=\hbar/k_BT=\beta\hbar\). Because the partition function is a trace, the ends of imaginary time join up into a circle of circumference \(\beta\hbar\).
The physical content is "one lap in imaginary time multiplies by the Boltzmann factor" ── made rigorous, this is the KMS condition, which in the modern view is used as the definition of temperature. In orders of magnitude, one lap at room temperature is 25 femtoseconds, the same single mark as Episode 1's "25 meV," tied to it by the uncertainty relation. Absolute zero is the circle unrolled into a line.
Three dividends ── (1) Episode 3's freezing translates into "does it fit on the Matsubara circle?" (the first Matsubara energy at room temperature is 157 meV), (2) \(d\)-dimensional statistics turns into \((d-1)+1\)-dimensional quantum mechanics, and (3) if spacetime itself curls imaginary time up, there is a temperature there ── the bridge to the finale.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen the temperature slider changes the height of the imaginary-time strip (= the period), and the buttons switch the energy so you can read the Boltzmann factor picked up in one lap. "See the answer" opens each solution.