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

Abstract
Quantum fields and static interactions in accelerated frames
Frieder Lenz
Institute for Theoretical Physics III, University <strong>of</strong> Erlangen-Nürnberg
91058 Erlangen, Germany
Properties <strong>of</strong> quantum fields in Rindler space or equivalently
in accelerated frames are explored. Consequences <strong>of</strong>
the inertial forces for the kinematics and the dynamics <strong>of</strong>
scalar particles and photons are discussed and results <strong>of</strong> investigations
<strong>of</strong> interaction energies generated by scalar and
vector particle exchange in Rindler space are presented.
INTRODUCTION
Quantum fields observed in accelerated and in inertial
frames are related to each other by a coordinate transformation.
Nevertheless their properties differ significantly
which is due to the existence <strong>of</strong> a horizon in accelerated
frames (Rindler space). In the following I will present the
results <strong>of</strong> studies [1, 2] <strong>of</strong> both kinematical and dynamical
properties <strong>of</strong> quantum fields. The kinematics <strong>of</strong> noninteracting
quantum fields in Rindler space [3, 4, 5] and
their relation to fields in Minkowski space together with the
interpretation in terms <strong>of</strong> finite temperature quantum fields
[6] have been the subject <strong>of</strong> many investigations [7]. Here
the focus will be on the peculiar properties <strong>of</strong> the spectrum
<strong>of</strong> the particles in Rindler space and their signatures in the
Unruh radiation <strong>of</strong> photons in comparison to blackbody radiation.
The impact <strong>of</strong> the unusual kinematics in accelerated
frames on the dynamics will be demonstrated in the
discussion <strong>of</strong> static interactions.
KINEMATICS
A uniformly accelerated observer (acceleration a) at fixed
coordinates transverse to the acceleration x⊥ moves along
a hyperbola [8]
x 2 − t 2 = 1
a 2 , x⊥ = 0 . (1)
Quantum fields as seen by accelerated observers are most
conveniently described in terms <strong>of</strong> the coordinates obtained
by transformation to the rest frame t, x, x⊥ → τ, ξ, x⊥
t(τ, ξ) = 1
a eaξ sinh aτ , x(τ, ξ) = 1
a eaξ cosh aτ . (2)
By construction, ξ = 0 corresponds to the hyperbolic motion
(1). More generally, a particle at rest in the observers
system at ξ = ξ0 =const. corresponds to the uniformly
accelerated motion in Minkowski space with acceleration
a exp{−aξ0}. Trajectories <strong>of</strong> uniformly accelerated particles
for different values <strong>of</strong> ξ0 are shown in Fig. 1 together
with the lines τ =const. .
II
t
10
5
+
III x I
10 5 5 10
5
10
IV
→
ξ = −τ = −∞
→
←
ξ = const.
← τ = const.
ξ = τ = −∞
Figure 1: Kinematics <strong>of</strong> uniform acceleration
The coordinate transformation (2) is not one-to-one. The
coordinates −∞ < τ, ξ < ∞ cover only one quarter <strong>of</strong> the
Minkowski space, the right ”Rindler wedge”R+ (region I)
R± = x µ |t| ≤ ±x . (3)
Upon reversion <strong>of</strong> the sign <strong>of</strong> x in Eq. (2) the left Rindler
wedge R− (region III) is covered by a corresponding
parametrization. As illustrated in the Figure, the boundaries
x = |t| corresponding to ξ = ±τ = −∞ form an
event horizon. The light cones shown in the Figure indicate
that in region I signals can be transmitted to region II
but not received from it. Signals received from region IV
appear to have originated from the horizon ξ = τ = −∞.
The space-time defined by the coordinate transformation
(2) is called Rindler space and its metric is given by
ds 2 = gµν(ξ)dx µ dx ν = e 2aξ (dτ 2 − dξ 2 ) − dx 2 ⊥ . (4)
The Rindler metric derives its importance from the fact that
essentially any static metric which possesses a horizon can
be approximated near the horizon by the Rindler metric.
This is the case for instance for the Schwarzschild metric.
Acceleration and Schwarzschild radius, or black hole mass,
are related by
a = 1 1
= . (5)
2R 4GM
QUANTUM FIELDS IN RINDLER SPACES
Quantization <strong>of</strong> scalar fields
Quantization <strong>of</strong> <strong>of</strong> a non-interacting scalar field in Rindler
space with the action