Quantum theory Quantum field theory on curved spacetime Theorem

Unruh effect

Builds on Quantum field theory in curved spacetime Quantum field theory Bogoliubov transformation

Statement

Let $\phi$ be the free real scalar quantum field of mass $m \ge 0$ on Minkowski space $\R^{1,3}$ , with coordinates $(t, x, y, z)$ . Use units with $\hbar = c = k_B = 1$ , and let $\omega_0$ be the vacuum state of $\phi$ . Let $W = \set{(t, x, y, z) : x > \abs{t}}$ be the right Rindler wedge. For $s \in \R$ , let $\Lambda_s$ be the Lorentz boost $(t, x) \mapsto (t \cosh s + x \sinh s,\ t \sinh s + x \cosh s)$ , which maps $W$ to itself. Let $\omega_W$ be the restriction of $\omega_0$ to the algebra of observables localized in $W$ . Then $\omega_W$ is a thermal (KMS) state for the time evolution given by the boosts $\Lambda_s$ , at inverse temperature $2\pi$ in the parameter $s$ . Consider an observer in $W$ with constant proper acceleration $a > 0$ ; the proper time $\tau$ is related to the boost parameter by $s = a\tau$ . Consequently, this observer sees the vacuum as a thermal state at temperature $T = \frac{a}{2\pi}, \qquad\text{that is,}\qquad T = \frac{\hbar a}{2\pi c k_B}$ in ordinary units.

Remarks

One proof expands the field in modes adapted to the uniformly accelerated observers and relates them to the Minkowski modes by a Bogoliubov transformation. Written in terms of the accelerated modes, the Minkowski vacuum has the occupation numbers $1/(e^{2\pi\omega/a} - 1)$ for each frequency $\omega$ . This is the Planck distribution at temperature $a/(2\pi)$ . In algebraic quantum field theory, the statement follows from the Bisognano–Wichmann theorem. This theorem identifies the modular group of the wedge algebra with respect to the vacuum with the boosts. Example: for $a = 1\ \mathrm{m/s^2}$ , the temperature is about $4 \times 10^{-21}$ K, far too small to measure directly. The result does not say that the field contains real particles for all observers. The number of particles depends on the observer, which is the point of quantum field theory in curved spacetime.

Going deeper

For a student

The Unruh effect is the flat-space model of Hawking radiation. Near the horizon of a large black hole, a static observer is like a uniformly accelerated observer in Minkowski space. Its proof uses the two main tools of quantum field theory in curved spacetime, Bogoliubov transformations between mode expansions and the algebraic notion of a KMS state. The Rindler wedge $W$ has a horizon, the light cone $x = \abs{t}$ , and the accelerated observer cannot see the region beyond it. Tracing out that hidden region turns the pure vacuum into a mixed thermal state. In the same way, a pure entangled state of two systems restricts to a mixed state of one of them.

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Bogoliubov transformation concept
Quantum field theory concept
Quantum field theory in curved spacetime concept
Unruh effect statement

Tag	`HM260RP02P`
URI	`m2://HM260RP02P`
Kind	statement, theorem
Writing	W0 stub
MSC	81T20, 83C47
Status	This copy of the site has no build data. Run `m2 status HM260RP02P`.
About	Quantum field theory in curved spacetime
Uses	Quantum field theory in curved spacetime, Quantum field theory, Bogoliubov transformation
Same as	Wikidata Q1568269 Wikipedia Unruh effect
Changed	3 November 2026