Quantum fields in curved spacetimes

Quantum fields in curved spacetimes Martin Scholtz February 5th Charles University in Prague

Outline Quantum field theory in the flat spacetime Curved spacetimes and Bogoljubov transformation Unruh effect Hawking effect

QFT in curved spacetimes

Quantum field theory in the flat spacetime • Lagrangian (density) of the real Klein-Gordon field L=

1 1 (∂µ φ)(∂ µ φ) − m2 φ2 , 2 2

+ m2 φ = 0

• Momentum π=

∂L ˙ = φ, ∂ φ˙

φ˙ ≡ ∂t φ

• Hamiltonian (density) H = π φ˙ − L =

1 ~ 2 1 1 2 π + (∇φ) + m2 φ2 2 2 2

Martin Scholtz | Charles University in Prague

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QFT in curved spacetimes • Klein-Gordon scalar product

Z (φ1 , φ2 ) = −i

(φ1 ∂t φ∗2 − φ∗2 ∂t φ1 ) d~ x

Σ

• Positive and negative frequency modes f˙k = −i ω fk ,

f˙∗ = i ω fk∗

• Plane waves, kµ = (ω, ~k), ω 2 + ~k2 = m2 fk (x) =

e−i kµ x

p

µ

,

2(2π)3 ω

kµ xµ = ω t − ~k · ~ x

are normalized by (fk , fk0 ) = δ(~k − ~k0 ),

(fk , fk∗0 ) = 0,


(fk∗ , fk∗0 ) = −δ(~k − ~k0 )

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QFT in curved spacetimes • Canonical quantization [φ(t, ~ x), φ(t, ~ x0 )] = 0,

[φ(t, ~ x), π(t, ~ x0 )] = iδ(~ x−~ x0 ),

[φ(t, ~ x), π(t, ~ x0 )] = 0

• Expansion in positive/negative modes φ(x) =

Z h

i

a~k f~k (x) + a~† f~∗ (x) d~k k

k

• Creation/annihilation operators [a~k , a~k0 ] = 0,

[a~k , a~† 0 ] = δ(~k − ~k0 ) k

• Number of particles with momentum ~k N~k = a~† a~k k


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QFT in curved spacetimes • Vacuum state a~k |0i = 0 • State with n1 , n2 , . . . particle with momenta ~k1 , ~k2 , . . . |n1 , n2 , . . .i = √

1 n1 ! n2 ! . . .

a~†

n1

k1

a~†

n2

k2

. . . |0i

• Standard relations a~k |n1 , n2 , . . .i = j

a~† |n1 , n2 , . . .i = kj

√

nj |n1 , n2 , . . . nj − 1, . . .i

p

nj + 1 |n1 , n2 , . . . nj + 1, . . .i

N~k |n1 , n2 , . . .i = nj |n1 , n2 , . . .i j


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QFT in curved spacetimes • This analysis holds in a chosen inertial frame • Consider boosted frame t0 = γ (t − ~v · ~ x) ,

~ x0 = γ (~ x − ~v t)

• Then ∂t0 f~k = −i ω~0 f~k k

where ω~0 = γ(ω − ~v · ~k) ≥ 0 k

• Hence, f~k is positive-frequency mode for any inertial observer • In particular, the vacuum state |0i is invariant


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Curved spacetimes and Bogoljubov transformation • General metric gab without Poincaré group • No preferred time coordinate, except for – proper time of the observer – timelike Killing vector field in stationary spacetime • Lagrangian of the scalar field L=

1 µν 1 g (∇µ φ)(∇ν φ) − m2 φ2 2 2

• Klein-Gordon scalar product

Z (φ1 , φ2 ) = −i

(φ1 ∇µ φ∗2 − φ∗2 ∇µ φ1 ) dΣµ

Σ


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• Positive/negative frequency modes £K fk∗ = i ω fk∗

£K fk = −i ω fk ,

• In the Minkowski spacetime – We choose K = ∂t – Any other choice K 0 = ∂t0 gives rise to the same vacuum state • No preferred choice of K in curved spacetime • Suppose we have found two complete sets {fi } and {gi } (fi , fj ) = δij

(gi , gj ) = δij


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• Bogoljubov transformation

X

gi =

αij fi + βij fj∗ ,

fi =

j

X

α∗ji gj − βji gj∗

j

• Field expansion φ=

X

ai fi + a†i fi∗ =

i

X

bi gi + b†i gi∗

i

• Bogoljubov transformation of the operators ai =

X

∗ † αji bj + βji bj ,

bi =

j


X

∗ † α∗ij aj + βij aj

j

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QFT in curved spacetimes • Two different vacua

ai 0f

• In 0f

bi |0g i = 0

=0

vacuum, the number of particles observed by the observer with bi modes is

0f | b†i bi | 0f

=

X

|βij |2

j

• Summary – There is no preferred choice of the time coordinate – Each observer has different set of modes adapted to his frame – The number of particles depends on the observer


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Unruh effect • Particle detection by an accelerated observer in the flat spacetime • Rindler spacetime = part of Minkowski spacetime • Minkowski (inertial) observer – Time coordinate t – Positive/negative frequency modes fk = √

1 4πω

e−i ω t+i k x

fk∗ = √

1 4πω

ei ω t−i k x

– Field expansion

Z φ=

ak fk + a† fk∗ dk


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QFT in curved spacetimes • Rindler (uniformly accelerated) observer

U

– Time coordinate η – Positive/negative frequency modes

=

(2) gk

√1 4πω

e−i ω η+i k ξ

=

√1 4πω

ei ω η+i k ξ

st.

∂η

in L

0

L

in R

con

ξ = const.

(1) gk

η=

∂η

in L

R

in R

0

D – Field expansion φ=

Z

(1)

(1)

b k gk

(2)

(2)

+ bk gk

(1)†

+ bk

(1)∗

gk


(2)†

+ bk

(2)∗

gk

dk

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• Normalization of the modes

(m)

gk

(n)

, gk 0

= δmn δ(k − k0 ),

(m)

gk

(n)∗

, gk 0

=0

• Two vacua Minkowski: ak |0M i = 0,

(m)

• Extension of gk (1)

gk

=

(1)

(2)

Rindler: bk |0R i = bk |0R i = 0

modes

ai ω/a (x − t)i ω/a 4π ω


(2)

gk

=

a−i ω/a (−x − t)i ω/a 4π ω

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• Normalized modes in entire spacetimes (1)

hk

(2)

hk

(m) (n) (hk , hk0 )

=

1

p

2 sinh

=

2 sinh

eω π/(2a) gk

eω π/(2a) gk

ωπ a

1

p

ωπ a

(2)∗

(1)∗

(1)

+ e−ω π/(2a) g−k

(2)

+ e−ω π/(2a) g−k

= δmn δ(k − k0 )

• Field expansion φ=

Z h

(1)

(1)

(2)

(2)

(1)†

ck hk + ck hk + ck


(1)∗

hk

(2)†

+ ck

(2)∗

hk

i

dk

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QFT in curved spacetimes (m)

• Operators ck

(m)

and ak

(m)

ak |0M i = ck

have the same (Minkowski) vacuum

|0M i = 0

• It is easier to find the Bogoljubov transformation to Rindler modes (1)

bk

(2)

bk

=

1

p

2 sinh

=

1

p

2 sinh

eω π/2a ck + e−ω π/2a c−k

eω π/2a ck + e−ω π/2a c−k

ωπ a

ωπ a

(1)

(2)†

(2)

(1)†

• Unruh effect – number of particles registered by Rindler observer in vacuum |0iM

D

(1)† (1) bk

0M | bk

| 0M

E

=

1 δ(0) e2πω/a − 1


→

T =

a~ 2 π kB

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Hawking effect • Analogy – Rindler spacetime – Killing horizon with surface gravity a – Unruh temperature T =

a 2π

– Schwarzschild spacetime – Killing horizon with surface gravity κ – Hawking temperature T =

κ 2π

– Preferred modes are chosen w.r.t. I ± • Both Unruh and Hawking effect can be understood via virtual particles creation


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QFT in curved spacetimes • Charged,spinning black hole with different fields allowed h0M | nk | 0M i =

Γ(ω) e2π(ω−µ)/a ± 1

– Fermionic/bosonic particles ± – Γ – grey body factor (backscattering) – µ–chemical potential (tendency to Schwarzschild) • Exact relations for T and S T =

κ 2π

S=

A 4

recall δM =

1 κ dA + ΩH δJH 8π


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