Table of Contents
- Introduction
- Motivation and Context
- Classical Field Theory in Curved Spacetime
- Basics of Curved Spacetime Geometry
- Covariant Derivatives and the Metric
- Scalar Field Quantization
- Canonical Quantization Challenges
- Vacuum Ambiguity in Curved Spacetime
- Bogoliubov Transformations
- Particle Creation by Time-Dependent Backgrounds
- Hawking Radiation
- Unruh Effect
- Renormalization in Curved Spacetime
- Stress-Energy Tensor and Backreaction
- Trace Anomaly
- Hadamard States and Regularization
- Quantum Fields in de Sitter Space
- Quantum Fields in Expanding Universes
- Cosmological Particle Creation
- Inflation and Vacuum Fluctuations
- Entanglement Entropy and Horizons
- Black Hole Backgrounds and Global Structure
- Effective Action and Semiclassical Gravity
- Limitations and Quantum Gravity
- Conclusion
1. Introduction
Quantum field theory in curved spacetime (QFCS) describes how quantum fields behave in a gravitational background. It generalizes flat spacetime QFT to dynamic or static curved geometries — bridging quantum theory with general relativity while gravity remains classical.
2. Motivation and Context
QFCS is essential for:
- Hawking radiation and black hole thermodynamics
- Early universe particle production
- Inflationary cosmology
- Understanding semiclassical effects of quantum matter on classical spacetime
3. Classical Field Theory in Curved Spacetime
Fields are defined over a spacetime with a general metric \( g_{\mu\nu} \). For a scalar field \( \phi \), the Klein–Gordon equation becomes:
\[
(\Box – m^2 – \xi R)\phi = 0
\]
where:
- \( \Box = \nabla^\mu \nabla_\mu \): d’Alembertian
- \( R \): Ricci scalar
- \( \xi \): coupling constant (e.g., \( \xi = 1/6 \) for conformal coupling)
4. Basics of Curved Spacetime Geometry
Spacetime is modeled as a 4-dimensional Lorentzian manifold with:
- Metric \( g_{\mu\nu} \)
- Levi-Civita connection \( \nabla_\mu \)
- Curvature tensors: \( R^\alpha_{\ \beta\mu\nu} \), \( R_{\mu\nu} \), \( R \)
5. Covariant Derivatives and the Metric
Covariant derivatives replace partial derivatives to maintain general covariance. For a vector field \( V^\mu \):
\[
\nabla_\nu V^\mu = \partial_\nu V^\mu + \Gamma^\mu_{\nu\rho} V^\rho
\]
6. Scalar Field Quantization
Field \( \phi(x) \) is promoted to an operator. Mode expansion:
\[
\phi(x) = \sum_i \left( a_i u_i(x) + a_i^\dagger u_i^*(x) \right)
\]
Modes \( u_i(x) \) satisfy the Klein–Gordon equation. However, mode decomposition is observer-dependent in curved spacetime.
7. Canonical Quantization Challenges
In curved spacetime:
- No unique time coordinate
- No preferred vacuum state
- Global hyperbolicity and foliation issues
This leads to vacuum ambiguity.
8. Vacuum Ambiguity in Curved Spacetime
Unlike flat spacetime, there is no unique vacuum. Different observers may define particles differently, leading to effects like:
- Unruh radiation
- Particle creation in expanding universes
9. Bogoliubov Transformations
Relates two sets of mode functions \( \{u_i\}, \{v_j\} \):
\[
v_j = \sum_i \left( \alpha_{ji} u_i + \beta_{ji} u_i^* \right)
\]
The presence of nonzero \( \beta_{ji} \) indicates particle creation.
10. Particle Creation by Time-Dependent Backgrounds
Time-varying backgrounds (e.g., expanding universes) cause mode mixing, leading to particle creation. Important in early universe and inflationary cosmology.
11. Hawking Radiation
In black hole backgrounds, vacuum fluctuations near the horizon lead to thermal radiation at:
\[
T_H = \frac{\hbar \kappa}{2\pi c k_B}
\]
This was first derived using QFCS by Hawking (1974).
12. Unruh Effect
An accelerating observer detects a thermal bath of particles, even in Minkowski vacuum:
\[
T = \frac{\hbar a}{2\pi c k_B}
\]
This shows observer-dependent particle content.
13. Renormalization in Curved Spacetime
Quantum expectation values like \( \langle T_{\mu\nu} \rangle \) diverge. Renormalization involves subtracting singular parts using methods like:
- Point splitting
- Hadamard renormalization
- Adiabatic subtraction
14. Stress-Energy Tensor and Backreaction
The semiclassical Einstein equation:
\[
G_{\mu\nu} = 8\pi G \langle T_{\mu\nu} \rangle
\]
captures the backreaction of quantum fields on the classical geometry.
15. Trace Anomaly
Even if classically \( T^\mu_\mu = 0 \) for conformally invariant fields, quantum corrections give:
\[
\langle T^\mu_\mu \rangle \neq 0
\]
This is the trace anomaly and affects renormalization and dynamics.
16. Hadamard States and Regularization
A physically acceptable quantum state must satisfy the Hadamard condition — local short-distance behavior matching flat spacetime vacuum. This ensures well-defined renormalization.
17. Quantum Fields in de Sitter Space
de Sitter spacetime (constant positive curvature) plays a key role in inflation. The Bunch–Davies vacuum is the preferred state, leading to nearly scale-invariant perturbations.
18. Quantum Fields in Expanding Universes
In FLRW spacetime, quantum fields experience redshifting and mode stretching, with implications for particle creation, cosmological perturbations, and vacuum selection.
19. Cosmological Particle Creation
During rapid expansion, such as inflation, vacuum fluctuations are amplified, producing real particles — a key process in structure formation.
20. Inflation and Vacuum Fluctuations
Inflation stretches quantum fluctuations beyond the Hubble radius. These become classical perturbations that seed the cosmic microwave background (CMB) anisotropies.
21. Entanglement Entropy and Horizons
Event horizons lead to entanglement between inside and outside modes. The reduced density matrix has nonzero entropy:
\[
S_{\text{ent}} = -\text{Tr}(\rho \ln \rho)
\]
This connects quantum fields, thermodynamics, and geometry.
22. Black Hole Backgrounds and Global Structure
Global structure (e.g., horizons, causal boundaries) determines particle content and evolution of quantum fields — essential for phenomena like Hawking radiation.
23. Effective Action and Semiclassical Gravity
Functional methods derive the effective action for quantum fields in curved backgrounds, used to compute vacuum polarization, anomalies, and corrections to Einstein equations.
24. Limitations and Quantum Gravity
QFCS treats gravity classically. Near the Planck scale, backreaction, non-perturbative effects, and spacetime fluctuations require full quantum gravity (e.g., string theory, LQG).
25. Conclusion
Quantum field theory in curved spacetime provides deep insights into black holes, the early universe, and quantum-gravitational effects without needing a full quantum gravity theory. Though limited to semiclassical regimes, it remains an indispensable tool in theoretical physics, bridging relativistic gravitation and quantum field dynamics.
Leave a Reply