Download UNRUH EFFECT Outline Introduction Accelerated motion (1)

UNRUH EFFECT
Marc Wagner
[email protected]
http://theorie3.physik.uni-erlangen.de/∼mcwagner
8th August 2005
Outline
• Introduction
• Accelerated motion in Minkowski spacetime
• Coordinates of an accelerated frame
• Rindler spacetime
• Massless scalar field in Minkowski and in Rindler spacetime
• Lightcone coordinates
• Bogolyubov transformation (relation between creation and annihilation operators in Minkowski and in Rindler spacetime)
• Particle numbers of the Minkowski vacuum in an accelerated
frame
• Unruh temperature
Marc Wagner, “Unruh effect”, 8th August 2005
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Introduction
Accelerated motion (1)
Literature
Three different frames
• V. F. Mukhanov and S. Winitzki, Introduction to Quantum
Fields in Classical Backgrounds, 2005.
www.theorie.physik.uni-muenchen.de/∼serge/T6/
What is the Unruh effect?
• Vacuum state in Minkowski spacetime: state where no particles are present; lowest energy eigenstate.
• Laboratory frame (inertial frame; coordinates xµ = (t, x)):
the usual inertial frame.
• Accelerated frame or proper frame (non inertial frame; coordinates (τ, ξ)): the frame where the accelerated observer is at
rest.
• Comoving frames (inertial frames; coordinates x′µ = (t′, x′)):
frames where the accelerated observer is momentarily at rest.
• Unruh effect: an accelerated observer moving through the
Minkowski vacuum detects particles (Minkowski vacuum and
vacuum in the frame of the accelerated observer are different
quantum states).
• In this talk: observer moves in 1+1 dimensions with constant
acceleration.
Marc Wagner, “Unruh effect”, 8th August 2005
Marc Wagner, “Unruh effect”, 8th August 2005
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Accelerated motion (2)
Accelerated motion (3)
Constant acceleration (i)
Constant acceleration (ii)
• Constant acceleration: constant four acceleration in the comoving frames.
• Any inertial frame:
d
d
(uµ uµ ) = 2uµ uµ = 2uµ aµ.
dτ
dτ
0 =
(1)
• Comoving frame (at that time when the accelerated observer
is at rest):
u′µ = (1, 0)
dt′ d
d
= u′0 ′
dτ dt′
dt
d2 ′
d
d
′0 d ′
A =
u
=
x
=
x
u′0 v ′ =
2
′
dτ
dτ
dt
dτ
d ′
d ′0 ′
′0
u v +u
v
=
=
dτ
dτ
2 d ′
= a′0v ′ + u′0
= a′ .
v
dt′
u′0 =
uµ uµ = 1
→
• A is the ordinary three acceleration a′ in the comoving frame
(at the time when the accelerated observer is at rest):
a′µ = (0, A).
→
(2)
dt′
dτ
→
d
dτ
=
(3)
(4)
• Constant three acceleration in the laboratory frame (or any
inertial frame) is impossible. a = dv/dt = constant would
imply that the observer can move faster than the speed of
light.
Marc Wagner, “Unruh effect”, 8th August 2005
Marc Wagner, “Unruh effect”, 8th August 2005
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Accelerated motion (4)
Accelerated frame (coordinates) (1)
Trajectory of the accelerated observer
• Two equations:
2
2
u0 − u1
= uµ uµ = 1
d 0
u
dτ
2
−
d 1
u
dτ
2
= aµ a
(5)
µ
a′µ a′µ
=
′2
= −a . (6)
• Solution:
u
0
′
= cosh(a τ ) ,
u
1
′
= sinh(a τ ).
(7)
• Trajectory (initial conditions xµ (0) = (0, 1/a′), uµ (0) = (1, 0),
e. g. at t = τ = 0 the particle is at rest at x = 1/a′):
t =
1
sinh(a′τ ) ,
a′
x =
1
cosh(a′τ ).
a′
(8)
• To describe quantum fields in the accelerated frame and to
compare them with quantum fields in the laboratory frame we
need coordinates (τ, ξ) in the accelerated frame and transformation laws t = t(τ, ξ) and x = x(τ, ξ).
• τ = proper time of the observer (or anybody moving along the
trajectory ξ = 0).
• ξ = spatial distance from the observer at ξ = 0.
• Consider a measuring stick of length ξ0 in the accelerated
frame. In the current comoving frame it is represented by the
four vector s′µ = (0, ξ0) (the measuring stick is momentarily
at rest in the current comoving frame).
• Four vector of the measuring stick in the laboratory system:
1
1 v
0
sµ = √
=
ξ0
1 − v2 v 1
1 0 1 u ξ0
0
u u
.
(9)
=
=
u0ξ0
u1 u0
ξ0
Marc Wagner, “Unruh effect”, 8th August 2005
Marc Wagner, “Unruh effect”, 8th August 2005
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Accelerated frame (coordinates) (2)
• The far end of the measuring stick has proper coordinates
(τ, ξ0). From that, (8) and (9) the transformation law between
laboratory coordinates (t, x) and proper coordinates (τ, ξ) can
be derived:
1 + a′ ξ
1
sinh(a′τ )
t = ′ sinh(a′τ ) + u1ξ =
a
a′
x =
′
1+aξ
1
cosh(a′τ ) + u0ξ =
cosh(a′τ ).
a′
a′
• Inverse transformation law:
x+t
1
, τ ∈ (−∞, ∞)
ln
τ =
2a′
x−t
p
1
ξ =
x2 − t2 − ′ , ξ ∈ [−1/a′, ∞).
a
(10)
Rindler spacetime (1)
• (10) and (11):
dt
dt
dτ + dξ =
dτ
dξ
= (1 + a′ ξ) cosh(a′τ )dτ + sinh(a′ τ )dξ
(14)
dx
dx
dτ + dξ =
dτ
dξ
= (1 + a′ ξ) sinh(a′τ )dτ + cosh(a′ τ )dξ.
(15)
dt =
dx =
(11)
• Rindler spacetime:
(12)
ds2 = dt2 − dx2 = (1 + a′ ξ)2dτ 2 − dξ 2.
(13)
Marc Wagner, “Unruh effect”, 8th August 2005
Marc Wagner, “Unruh effect”, 8th August 2005
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Rindler spacetime (2)
Rindler spacetime (3)
Conformally flat Rindler spacetime (i)
Conformally flat Rindler spacetime (ii)
• Quantising fields in conformally flat spacetime in 1+1 dimensions is as easy as quantising fields in Minkowski spacetime.
• To get a conformally flat metric we need a coordinate trans˜ with
formation ξ = ξ(ξ)
˜
dξ = (1 + a′ξ)dξ.
(17)
• Separation of variables yields
Z
1
1
ξ̃ =
dξ
= ′ ln(1 + a′ξ) ,
1 + a′ ξ
a
ξ˜ ∈ (−∞, ∞).
• Transformation law between laboratory coordinates (t, x) and
˜
conformally flat Rindler coordinates (τ, ξ):
′˜
t =
ea ξ
sinh(a′τ )
a′
x =
ea ξ
cosh(a′τ ).
a′
(20)
′˜
(21)
(18)
• This is a rescaling of the spatial coordinate ξ. ξ˜ is not the
spatial distance but parameterises the spatial distance ξ.
• Conformally flat Rindler spacetime:
′˜
ds2 = e2a ξ dτ 2 − dξ˜2 .
(16)
(19)
Marc Wagner, “Unruh effect”, 8th August 2005
Marc Wagner, “Unruh effect”, 8th August 2005
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Massless scalar field (1)
• Action of a minimally coupled massless scalar field:
Z
√ 1
S[φ] =
d2x −g g µν (∂µ φ)(∂ν φ).
2
• Laboratory frame (Minkowski spacetime):
Z
1
S[φ] =
dt dx
(∂t φ)2 − (∂xφ)2 .
2
Massless scalar field (2)
Quantisation in Minkowski spacetime
(22)
(23)
• Accelerated frame (conformally flat Rindler spacetime;
√
′˜
′˜
′˜
−g = e2a ξ , g µν = diag(e−2a ξ , e−2a ξ )):
Z
2 1
.
(24)
S[φ] =
dτ dξ˜
(∂τ φ)2 − ∂ξ˜φ
2
• In 1+1 dimensions minimal coupling is equivalent to conformal coupling. Therefore the action in conformally flat Rindler
spacetime is identical to the action in Minkowski spacetime.
Quantising the field φ in conformally flat Rindler spacetime is
therefore as easy as in Minkowski spacetime.
• Field operator in laboratory coordinates (a(k) = annihilation
operators, a†(k) = creation operators):
φ(t, x) = Z
1
1
dk p
= √
2π
2|k|
e−i|k|t+ikx a(k) + ei|k|t−ikx a†(k) .
• Minkowski vacuum:
a(k)|0M i = 0.
(26)
• Expectation values of certain operators, e. g. H (energy), P
(momentum), T µν (energy momentum tensor, i. e. energy
and momentum density), allow a physical interpretation of
a-particle states.
• Example: a†(k)|0M i represents a particle with definite momentum k.
Marc Wagner, “Unruh effect”, 8th August 2005
Marc Wagner, “Unruh effect”, 8th August 2005
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Massless scalar field (3)
Massless scalar field (4)
Quantisation in Rindler spacetime
• Field operator in conformally flat Rindler coordinates (b(k) =
annihilation operators, b†(k) = creation operators):
˜ =
φ(τ, ξ)
Z
1
1
dk p
= √
2π
2|k|
˜
˜
e−i|k|τ +ikξ b(k) + ei|k|τ −ikξ b†(k) .
(27)
• The field operators represent the same quantum field, i. e.
˜
φ(t, x) = φ(τ, ξ).
• The creation and annihilation operators a(k), a†(k) and b(k),
b†(k) are different, i. e. they create or annihilate different field
excitations.
• Therefore the Minkowski vacuum |0M i and the Rindler vacuum |0Ri are different quantum states.
• Rindler vacuum:
b(k)|0Ri = 0.
(25)
(28)
• Physical interpretation of b-particle states: analogous to physical interpretation of a-particle states (the action which is identical in both cases determines the physical meaning of all quantum states).
Marc Wagner, “Unruh effect”, 8th August 2005
Marc Wagner, “Unruh effect”, 8th August 2005
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Lightcone coordinates
Bogolyubov transformation (1)
• Get the relation between a(k), a† (k) and b(k), b†(k) by comparing the left and right hand side of
˜
φ(t, x) = φ(τ, ξ).
(29)
• Lightcone coordinates will simplify this procedure:
u = t−x ,
v = t+x
(30)
ũ = τ − ξ˜ ,
˜
ṽ = τ + ξ.
(31)
• Relation between (u, v) and (ũ, ṽ):
u = t−x =
1 ′˜
= − ′ ea (ξ−τ )
a
v = t+x =
=
1 a′(ξ+τ
˜ )
e
a′
′˜
′˜
ea ξ
ea ξ
sinh(a′ τ ) − ′ cosh(a′τ ) =
′
a
a
1
′
= − ′ e−a ũ
a
(32)
φ(u, v) = Z
1
1
= √
dk p
2π
2|k|
e−i|k|t+ikx a(k) + ei|k|t−ikx a†(k) =
Z ∞
1 −iω(t−x)
1
dω √
e
a(ω) + eiω(t−x) a†(ω) +
= √
2π 0
2ω
Z 0
1
1
√
dω √
−2ω
2π −∞
eiω(t+x)a(ω) + e−iω(t+x) a†(ω) =
Z ∞
1 −iωu
1
dω √
= √
e
a(ω) + eiωua†(ω) +
2π 0
2ω
e−iωv a(−ω) + eiωv a†(−ω) =
= A(u) + B(v).
′˜
′˜
ea ξ
ea ξ
sinh(a′τ ) + ′ cosh(a′τ ) =
′
a
a
1 ′
= ′ ea ṽ
a
• Field operator in (u, v)-coordinates:
(33)
(34)
• Advantage: φ(u, v) now is a sum of a u-dependent part and a
v-dependent part.
• Lightcone coordinates do not mix: u = u(ũ), v = v(ṽ).
Marc Wagner, “Unruh effect”, 8th August 2005
Marc Wagner, “Unruh effect”, 8th August 2005
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Bogolyubov transformation (2)
Bogolyubov transformation (3)
• Field operator in (ũ, ṽ)-coordinates:
φ(ũ, ṽ) =
Z ∞
1
1 −iΩũ
e
b(Ω) + eiΩũ b†(Ω) +
= √
dΩ √
2π 0
2Ω
e−iΩṽ b(−Ω) + eiΩṽ b†(−Ω)
=
= P (ũ) + Q(ṽ).
(35)
• The u-dependent parts of (34) and (35) must be equal:
Z ∞
1
1 −iωu(ũ)
√
e
a(ω) + eiωu(ũ)a†(ω) =
dω √
2π 0
Z ∞2ω
1
1
= √
dΩ √
e−iΩũ b(Ω) + eiΩũ b†(Ω) .
(36)
2π 0
2Ω
• Right hand side:
Z
Z ∞
1
1
1
√
dũ eiΩũ √
dΩ′ √
′
2π 2π 0
2Ω
−iΩ′ ũ
′
iΩ′ ũ †
′
e
b(Ω ) + e b (Ω ) =
Z ∞
Z
1 1
=
dΩ′ √
dũ eiΩũ
′ 2π
2Ω
0
′
′
e−iΩ ũ b(Ω′) + eiΩ ũb†(Ω′) =
Z ∞
1
dΩ′ √
=
δ(Ω − Ω′)b(Ω′) + δ(Ω + Ω′)b†(Ω′) =
′
2Ω√
0
b(Ω)/
√ 2Ω for Ω > 0 .
=
(38)
b†(−Ω)/ −2Ω for Ω < 0
• Can be solved for b(Ω) and b†(Ω) by performing a Fourier
transformation on both sides:
Z
1
√
dũ eiΩũ . . . .
(37)
2π
Marc Wagner, “Unruh effect”, 8th August 2005
Marc Wagner, “Unruh effect”, 8th August 2005
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Bogolyubov transformation (4)
• Left hand side:
Z
Z ∞
1
1
1
√
dũ eiΩũ √
dω √
2π 2π 0
2ω
−iωu(ũ)
iωu(ũ) †
e
a(ω) + e
a (ω) =
Z
Z ∞
1
1
a(ω) dũ eiΩũ−iωu(ũ)+
=
dω √
2π
2ω
0
Z
1
a†(ω) dũ eiΩũ+iωu(ũ)
=
2π
Z ∞
1
=
dω √
0 Z 2ω
1
1
′
a(ω) dũ
exp iΩũ + iω ′ e−a ũ +
2π
a
Z
1 −a′ũ
1
†
.
exp iΩũ − iω ′ e
a (ω) dũ
2π
a
Bogolyubov transformation (5)
• Result (Ω > 0):
Z ∞
b(Ω) =
dω αω,Ω a(ω) + βω,Ωa†(ω)
αωΩ
βωΩ
(39)
r
1
1 −a′ũ
exp
iΩũ
+
iω
e
=
2π
a′
r
Z
|Ω|
1
1
′
=
exp iΩũ − iω ′ e−a ũ .
dũ
ω
2π
a
|Ω|
ω
Z
dũ
• Transformations like (40) and (43) which relate two different sets of creation and annihilation operators are called Bogolyubov transformations. The coefficients (41) and (42) are
called Bogolyubov coefficients.
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Particle numbers
Unruh temperature
• Expectation value of the number of b-particles with “momentum” Ω in the Minkowski-vacuum (Ω > 0):
h0M |b†(Ω)b(Ω)|0M i =
Z ∞
∗
∗
= h0M |
dω αω,Ω
a†(ω) + βω,Ω
a(ω)
0
Z ∞
dω ′ αω′,Ω a(ω ′) + βω′,Ωa† (ω ′ ) |0M i =
Z0 ∞
Z ∞
∗
=
dω
dω ′ βω,Ω
βω′,Ω h0M |a(ω)a†(ω ′ )|0M i =
0
Z0 ∞
=
dω |βω,Ω|2 .
(44)
• Comparing (46) with the Bose distribution
n(Ω) =
1
e|Ω|/T − 1
(47)
yields the Unruh temperature
T =
a
.
2π
(48)
• Conclusion: An accelerated observer moving through the
Minkowski vacuum has the impression of moving through a
thermal bath of b-particles with temperature T .
0
• The integral on the right hand side of (44) can be solved
(Mukhanov et al., page 112 and 113):
(45)
• An analogous calculation for Ω < 0 can be carried out. The
result for arbitrary Ω is
h0M |b†(Ω)b(Ω)|0M i =
1
.
e2π|Ω|/a − 1
(42)
0
Marc Wagner, “Unruh effect”, 8th August 2005
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.
e2πΩ/a − 1
(41)
• Result (Ω < 0; follows from an analogous calculation with the
v-dependent parts of (34) and (35)):
Z ∞
b(Ω) =
dω αω,−Ω a(−ω) + βω,−Ωa† (−ω) .
(43)
Marc Wagner, “Unruh effect”, 8th August 2005
h0M |b†(Ω)b(Ω)|0M i =
(40)
0
(46)
Marc Wagner, “Unruh effect”, 8th August 2005
Marc Wagner, “Unruh effect”, 8th August 2005
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