Proceedings of International Conference on Physics in ... - KEK

ON THE UNRUH EFFECT ∗
Ralf Schützhold † , Fakultät für Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany
Abstract
After a brief introduction into the Unruh effect and its
generalization to non-uniform (here circular) acceleration,
we discuss prospects for measuring signatures <strong>of</strong> this effect
in strong lasers.
INTRODUCTION
The Unruh effect [1] describes the striking discovery
that an accelerated observer/detector experiences the
Minkowski vacuum as a thermal bath – implying that
the particle concept depends on the inertial state <strong>of</strong> the
observer/detector. After important preparatory works by
Fulling [2] and Davies [3], Unruh [1] realized this phenomenon
while trying to understand how black holes can
evaporate by emitting Hawking radiation [4]. Around the
same time, the mathematical foundation <strong>of</strong> this striking fact
was established by Bisognano & Wichmann [5] by showing
the relation between the Rindler Hamiltonian and thermality
– but apparently without immediately realizing the
broad physical significance.
However, so far this prediction <strong>of</strong> quantum field theory
has eluded a direct experimental verification, see also [6].
There are some observations regarding the imperfect polarizability
<strong>of</strong> electrons in storage rings which are related to
the Sokolov-Ternov effect [7] and can be interpreted as an
indirect verification <strong>of</strong> the Unruh effect generalized to the
case <strong>of</strong> circular acceleration, see, e.g., [8, 9].
In the following, we briefly discuss a recent proposal
[10, 11] for directly observing signatures <strong>of</strong> the Unruh
effect in the form <strong>of</strong> photon pairs created by electrons
which are accelerated in an ultra-strong laser field, see also
[12, 13]. We start with a short introduction into the Unruh
effect for uniform acceleration and discuss the generalization
to circular acceleration, which is relevant for electrons
in storage rings and the Sokolov-Ternov effect.
DETECTOR PLUS FIELD
Let us consider the following action for the field ϕ and
the detector with excitation energy E, coupled with the
coupling strength g (we use = c = 1)
A = Adetector + Afield
∫ [
E
= dτ
2 σz
]
+ gσxϕ (x[τ])
+ 1
∫
2
d 4 x [(∂µϕ)(∂ µ ϕ)] , (1)
∗ Work supported by DFG under grant SCHU 1557/1.
† ralf.schuetzhold@uni-due.de
where σz and σx are the Pauli matrices and τ is the proper
time along the detector trajectory. In the interaction picture,
the transition Hamiltonian reads (with 2σ± = σx ± iσy)
]
ˆϕ (x[τ]) , (2)
ˆHint(τ) = g(τ) [ e iEτ σ+ + e −iEτ σ−
where g(τ) is smooth switching (on and <strong>of</strong>f) function. Initially
(where the interaction is switched <strong>of</strong>f) both, detector
and field, are in their ground state
|Ψin⟩ = |Ψ(τ ↓ −∞)⟩
= |Ψdetector⟩ ⊗ |Ψfield⟩ = |↓⟩ ⊗ |0⟩ . (3)
Assuming small coupling g, we may derive the final state
via perturbation theory
|Ψout⟩ = |Ψ(τ ↑ +∞)⟩
∫
= |Ψin⟩ − i
dτ ˆ Hint(τ) |Ψin⟩ + O(g 2 ) .(4)
This yields the excitation probability <strong>of</strong> the detector
P↑ = ⟨Ψout| ↑⟩ ⟨↑ |Ψout⟩
=
∫ ∫
dτ dτ ′ g(τ) g(τ ′ ) e iE(τ−τ ′ )
×
× ⟨0| ˆ ϕ (x[τ]) ˆ ϕ (x[τ ′ ]) |0⟩ . (5)
For simplicity, we consider the Wightmann function for a
massless scalar field in 3+1 dimensions
⟨0| ˆ ϕ (x) ˆ ϕ (x ′ ) |0⟩ = − 1
(2π) 2
1
(t − t ′ ) 2 − (r − r ′ , (6)
) 2
where the pole structure (at the light-cone) is understood in
such a way that the Fourier transform <strong>of</strong> the Wightmann
function only contains non-negative energies. Roughly
speaking, (t − t ′ ) 2 is replaced by (t − t ′ − iε) 2 with ε ↓ 0.
UNIFORM ACCELERATION
In the case <strong>of</strong> (eternal) uniform acceleration a, the detector
trajectory in terms <strong>of</strong> the proper time τ reads
t[τ] = 1
a
sinh(aτ) , x[τ] = 1
a
cosh(aτ) , y = z = 0 . (7)
Evaluating the two-point function along this trajectory
⟨0| ˆ ϕ (x[τ]) ˆ ϕ (x[τ ′ ]) |0⟩ = − 1
2(2π) 2
a2 cosh(a[τ − τ ′ , (8)
]) − 1
we find a stationary expression which is periodic along the
imaginary τ− = τ − τ ′ -axis and possesses double poles at
this axis
a[τ − τ ′ ] ∈ 2πi N . (9)