Quantum Black Holes and Global Symmetries Daniel Klaewer - - PowerPoint PPT Presentation

Quantum Black Holes and Global Symmetries

Daniel Klaewer

Max-Planck-Institute for Physics, Munich

Young Scientist Workshop 2017, Schloss Ringberg

Outline

2

1) Quantum fields in curved spacetime 2) The Unruh effect 3) Quantum black holes and Hawking radiation 4) Problems with global symmetries

Quantum Gravity?

3

• Collider physics described by (perturbative) quantum field theory
• Prototype: scalar field theory
• Dictated by unification of quantum mechanics
• With special relativity

S =

• d4x

1 2ηµν∂µφ∂νφ − 1 2m2φ2 − λφ4

• ∼ λ
• Lfree

Lint i∂t|ψ = ˆ H|ψ ηµν =     1 −1 −1 −1     d 2 = µνxµxν = (ct)2 − x2

Quantum Gravity?

4

• General relativity described by highly non-linear, complicated Einstein

field equations for the metric (spacetime geometry)

• Naive treatment as perturbative QFT runs into big problems (non-

renormalizability) Rµν − 1 2Rgµν = 8πG c4 Tµν Gravity has to be quantized - we just don’t know for sure how! R ∼ d(g−1dg) + (g−1dg)2 For this talk we will remain ignorant! ηµν =     1 −1 −1 −1     → gµν =     g00 g01 g02 g03 g10 g11 g12 g13 g20 g21 g22 g23 g30 g31 g32 g33     gµν

Necessity of Quantization

5

Rµν − 1

2Rgµν = 8πG c4 Tµν

• gravity

matter

• Why not just quantize RHS?
• Superposition states: gravitational field is “average” over possible
• utcomes. Upon measurement we have collapse and discontinuous

change of the gravitational field.

• Violates locality, Lorentz invariance, also , which is

inconsistent with the proposed equation. Rµν − 1

2Rgµν = 8πG c4 ⟨ ˆ

Tµν⟩ ∇µ⟨ ˆ Tµν⟩ ̸= 0

Quantum Fields in Curved Spacetime!

6

• Study quantum fields in classical gravity background (e.g. black hole)
• Surprisingly, leads to non-trivial, robust insights about quantum gravity
• Works as long as curvature is not too strong (black hole singularity)
• d4x

1 2ηµν∂µφ∂νφ − 1 2m2φ2 − λφ4

• d4x
• det(g)

1 2gµν∂µφ∂νφ − 1 2m2φ2 − λφ4

The Vacuum State

7

• QFT vacuum: state of lowest energy
• Equivalently killed by annihilation operators for every particle
• Lorentz symmetry: inertial observers agree on vacuum
• In fact only true for inertial observers
• In general relativity: no privileged class of observers!
• Mathematically: creation/annihilation operators of two observers related

by Bogolyubov transformation ˆ H|Ω⟩ = Emin|Ω⟩ ˆ aI|Ω⟩ = 0 The definition of “vacuum” or “particle” in GR is inherently ambiguous ˆ bI =

J

• αIJˆ

aJ + βIJˆ a†

J

• [H, Λµν] = 0

ˆ bI|ΩA =

• J

βIJˆ a†

J|ΩA = 0

The Unruh Effect

8

• Equivalence Principle: gravity is locally equivalent to accelerated frame
• f reference
• For a qualitative picture, we thus consider constantly accelerated
• bserver (constant proper acceleration)
• Calculate expectation value of number density
• perator of the accelerated observer A in

Minkowski vacuum

• This is precisely Planck’s law! A sees radiation!

ct = c2

α sinh(ατ/c)

x = c2

α cosh(ατ/c)

Unruh Effect ⟨ΩM|ˆ nA(E)|ΩM⟩ = 1 exp 2πcE

α

• − 1

T = kB c α 2π

Quantum Black Holes

9

• Classical black hole: nothing can escape from within the horizon
• Hawking showed: vacuum of collapse that of free falling observer
• Observer at infinity has relative acceleration and sees Hawking radiation
• Black holes lose mass and evaporate after all!
• Black hole thermodynamics:

T = c3 8πGkB 1 M dM = TdS S = kB 4πG c3 M 2 = A 4ℓ2

P

Some Ballpark Figures

10

• Both Unruh and Hawking radiation are hard to measure
• Measuring Hawking radiation requires getting close to small black

holes - only option: micro black holes at accelerators

• For a 1K black hole we are looking at
• Unruh radiation is in principle easy to measure but the amount of

acceleration is huge, again for 1K we need M ≃ 10−8M α ≃ 1020 m s2

Global Symmetries and Black Holes

11

• Imagine a world with gravity, matter and a continuous global symmetry

(no gauge symmetry! no associated force!)

• Noether’s theorem guarantees associated charge, e.g. baryon number
• To avoid subtleties, assume single particle species interacting only

through gravity and with charge under such symmetry

• By collapsing of these to a black hole with mass and waiting until

it evaporates to mass we get black holes with arbitrary charge, all of the same mass/energy |p⟩ |q⟩ → e/3|q⟩ |¯ q⟩ → e−/3|¯ q⟩ B = 1 3 (Nq − N¯

q)

p U(1) |p⟩ → ep|p⟩ N N · m m

No Global Symmetries

12

+

• Important: Hawking radiation contains same number of + and - charged

particles, so black hole cannot lose charge

• If we let the black hole completely evaporate, charge is gone!
• We have created a process

which explicitly violates charge conservation

• N · (m, q)

(N · m − ∆E, N · q) (m, N · q) Q = Nq Q = 0

No Global Symmetries: Loophole

13

+

• N · (m, q)

(N · m − ∆E, N · q) (m, N · q)

• Hawking calculation only valid until
• What if evaporation stops and remnant forms?
• No-Hair theorem: black holes with different but same are

indistinguishable from outside

• Since we can construct BH with arbitrary for a fixed and thus

energy, we see that black holes in the theory have infinite microcanonical entropy!

• Leads to various inconsistencies, violates entropy bounds!

MBH ≃ Mp Q M Q M

Bonus: The Stringy Version

14

• We believe string theory is a consistent theory of quantum gravity
• Should rather forbid global symmetries then
• Explicit mechanism: perturbative string theory is described by two

dimensional field theory on the string world sheet

• Introducing a global symmetry on the world-sheet magically gives rise to

a gauge symmetry with associated gauge bosons in the spacetime! S =

• dtdl
• ηMNgab∂aXM∂bXN + · · ·
• XM(t, l) string position

Bonus: The AdS/CFT version

15

• Reminder: AdS/CFT is an isomorphism of two very different theories

1) Quantum gravity in Anti-de-Sitter (AdS) space 2) Conformal field theory (non-gravitational) on the AdS boundary AdS CFT global symmetry in CFT gauge symmetry in AdS global symmetry in AdS Contradiction!

Conclusion

16

• Even if ignorant about the details of quantum gravity, we can gain non-

trivial insights by using usual QFT techniques in curved backgrounds

• Accelerated observers experience Unruh radiation
• A different manifestation of this is Hawking radiation of black holes
• Combining global symmetries with these expectations leads to

paradoxes

• Hence global symmetries are not allowed in quantum gravity and thus

nature!

• String theory seems to obey this!

Thank You