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

S = 1
2
dτ dξ d d−1
x⊥ (∂τ φ) 2 − (∂ξφ) 2
−(m 2 φ 2 + (∂⊥φ) 2 ) e 2aξ , (6)
is standard. The expansion <strong>of</strong> the fields in terms <strong>of</strong> the
normal modes (proportional to the Mc Donald functions)
<strong>of</strong> the associated wave equation is given by
dω
φ(τ, ξ, x⊥) ∼ √ d
2ω d−1 k⊥(a(ω, k⊥)e −iωτ+ik⊥x⊥
m⊥ e aξ
, m 2 ⊥ = (m 2 + k 2 ⊥)/a 2 , (7)
+h.c.) Ki ω
a
and the normalization is chosen such that the commutator
<strong>of</strong> creation and annihilation operators a (†) (ω, k⊥) is standard.
The repulsive exponential barrier in uniformly accelerated
frames is <strong>of</strong> similar origin as the centrifugal barrier
in rotating frames. It prevents unlimited propagation <strong>of</strong> the
wave in positive ξ direction. This repulsion accounts for the
fact that a particle moving with arbitrary constant speed in
Minkowski space is seen by the accelerated observer to approach
ξ = −∞ and the speed <strong>of</strong> light for large times τ. In
the accelerated frame, the transverse velocity <strong>of</strong> the particle
vanishes exponentially for large times (∼ exp{−2aτ})
as a result <strong>of</strong> the forever increasing time dilation induced
by the acceleration. Since m 2 ⊥
appears in the action (6) as
a coupling constant <strong>of</strong> the exponential “potential” , the energy
eigenvalues <strong>of</strong> the normal modes do not depend on the
transverse momentum and the mass, though the eigenfunctions
do. This is reminiscent <strong>of</strong> the degeneracy <strong>of</strong> the Landau
levels <strong>of</strong> a particle moving in a constant magnetic field.
The high degeneracy <strong>of</strong> the eigenstates <strong>of</strong> the Hamiltonian,
including the ground state, is due to the inertial force and
has far reaching consequences. It indicates the presence <strong>of</strong>
a symmetry <strong>of</strong> the Rindler space Hamiltonian. In [1] the
invariance <strong>of</strong> the Hamiltonian under scale transformations
valid even in the presence <strong>of</strong> a mass term has been identified
as the source <strong>of</strong> the degeneracy.
The Unruh heat bath
Starting point for establishing the relation between observables
in inertial and accelerated frames is the identity <strong>of</strong> the
(scalar) fields (φ) and ( ˜ φ) in the two frames
φ(τ, ξ, x⊥) = ˜
φ(t, x) . (8)
t,x=t,x(τ,ξ)
Projection <strong>of</strong> this equation onto the Rindler space normal
modes (6) yields the following relation (Bogoliubov transformation)
between the creation and annihilation operators
in the two frames
a(Ω, k⊥) =
a sinh π Ω
a
1
∞
−∞
dk Ω i √ e a
4πωk
βk
· e πΩ
πΩ
2a −
ã(k, k⊥) + e 2a ã † (k, −k⊥)
. (9)
Observations in the accelerated frame are performed in the
Minkowski vacuum |0M 〉 rather than in the Rindler space
vacuum. A fundamental quantity is the number <strong>of</strong> particles
measured in the accelerated frame which, with the help <strong>of</strong>
(9), is found to be
〈0M |a † (Ω, k⊥)a(Ω ′ , k ′ ⊥)|0M 〉
1
=
Ω
e2π a − 1 δ(Ω − Ω′ )δ(k⊥ − k ′ ⊥) . (10)
In the accelerated frame, a thermal distribution <strong>of</strong> (Rindler)
particles is observed with the temperature determined by
the acceleration
T = a
. (11)
2π
For a black hole (cf. Eq. (5)) this temperature agrees with
the black hole temperature
TBH =
1
8πGM .
Although the above derivation makes use <strong>of</strong> the properties
<strong>of</strong> non-interacting fields, relations between observables in
accelerated frames and at finite temperature can be derived
for interacting fields (cf. [9] for a derivation within the path
integral approach). Here we consider the two point function
<strong>of</strong> a self-interacting scalar field and use its invariance
under Lorentz transformations. The relation (8) between
the fields in Rindler and Minkowski space implies that for
arbitrary points in the right Rindler wedge (cf. Fig. 1) the
values <strong>of</strong> the 2-point functions in Minkowski space and in
Rindler space are equal. The two point functions depend
only on (x − x ′ ) 2 . We express the distance in terms <strong>of</strong> the
Rindler space coordinates
(x − x ′ ) 2 = 2ea(ξ+ξ′ )
a2 ′
cosh a(τ − τ ) − cosh η , (12)
with
cosh η = 1 +
and obtain
aξ aξ e − e ′2 2 + a x⊥ − x ′ 2 ⊥
2e a(ξ+ξ′ )
, (13)
D (x − x ′ ) 2 = D(τ − τ ′ , ξ, ξ ′ , x⊥ − x ′ ⊥)
= i〈0M
T φ(τ, ξ, x⊥)φ(τ ′ , ξ ′ , x ′ ⊥) 0M 〉
a(ξ+ξ 2e
= D
′ )
a2 ′
cosh a(τ − τ ) − cosh η
. (14)
After a Wick rotation to imaginary Rindler time,
τ → τE = −iτ ,
the two point function is periodic with period
β = 2π
a ,
and therefore is a finite temperature 2-point function with
T = 1/β, in agreement with the result (11) for noninteracting
fields.