Glossary

12.2 Quantum field theory in curved spacetime

4 min read•august 14, 2024

Quantum field theory in curved spacetime takes our understanding of particles to the next level. It combines quantum mechanics with Einstein's general relativity, exploring how quantum fields behave in warped space near black holes or in the early universe.

This topic introduces mind-bending concepts like and the Unruh effect. These show how the very notion of particles depends on who's observing them, challenging our everyday ideas about reality in extreme gravitational environments.

Quantum Field Theory in Curved Spacetime

Extending QFT to Curved Spacetime

Top images from around the web for Extending QFT to Curved Spacetime

  • Einstein’s Theory of Gravity – University Physics Volume 1 View original

    Is this image relevant?

  • general relativity - Spacetime geometry around two black holes - Physics Stack Exchange View original

    Is this image relevant?

  • Einstein’s Theory of Gravity – University Physics Volume 1 View original

    Is this image relevant?

1 of 3

Top images from around the web for Extending QFT to Curved Spacetime

  • Einstein’s Theory of Gravity – University Physics Volume 1 View original

    Is this image relevant?

  • general relativity - Spacetime geometry around two black holes - Physics Stack Exchange View original

    Is this image relevant?

  • Einstein’s Theory of Gravity – University Physics Volume 1 View original

    Is this image relevant?

1 of 3
  • Quantum field theory (QFT) in flat spacetime is not sufficient to describe the behavior of quantum fields in the presence of strong gravitational fields or curved spacetime geometries
  • General relativity, our best theory of gravity, describes spacetime as a dynamical entity that can be curved by the presence of matter and energy
  • To properly incorporate the effects of gravity on quantum fields, it is necessary to formulate QFT in a way that is compatible with the principles of general relativity
  • The extension of QFT to curved spacetime is crucial for understanding the behavior of quantum fields in the early universe, near black holes, and in other extreme gravitational environments (high-energy astrophysical phenomena, cosmological inflation)

Modifications to QFT in Curved Spacetime

Local Description of Quantum Fields

  • In curved spacetime, the concept of a global inertial frame of reference is no longer valid, requiring a local description of quantum fields
  • The definition of particle states becomes observer-dependent, as the notion of positive and negative frequency modes is not globally well-defined in curved spacetime
  • The creation and annihilation operators for particles must be replaced by local field operators that depend on the spacetime coordinates

Modifications to Commutation Relations and Action

  • The commutation relations for field operators must be modified to account for the spacetime metric, ensuring that the theory remains consistent with the principles of general relativity
  • The action for the quantum field must be generalized to include the coupling of the field to the spacetime curvature, typically through the use of the covariant derivative and the metric tensor
  • procedures need to be adapted to handle the additional divergences that arise due to the coupling of the quantum field to the spacetime curvature

Hawking Radiation and Black Hole Thermodynamics

Hawking Radiation

  • Hawking radiation is a theoretical prediction that arises from the application of QFT to black hole spacetimes
  • showed that, due to near the event horizon, black holes can emit thermal radiation as if they had a temperature inversely proportional to their mass
  • The temperature associated with Hawking radiation is known as the Hawking temperature, given by TH=ℏc38πGMkBT_H = \frac{\hbar c^3}{8\pi G M k_B}, where MM is the mass of the black hole
  • Hawking radiation implies that black holes can evaporate over time, losing mass and shrinking in size, with smaller black holes evaporating faster than larger ones

Black Hole Thermodynamics

  • The discovery of Hawking radiation led to the development of , which draws analogies between the laws of thermodynamics and the properties of black holes
    • The zeroth law of black hole mechanics states that the surface gravity of a stationary black hole is constant over its event horizon, analogous to the zeroth law of thermodynamics
    • The first law of black hole mechanics relates changes in the mass, angular momentum, and charge of a black hole to changes in its area, analogous to the first law of thermodynamics
    • The second law of black hole mechanics states that the area of a black hole's event horizon cannot decrease over time, analogous to the second law of thermodynamics and the increase of entropy
  • The existence of Hawking radiation suggests that black holes have an entropy proportional to the area of their event horizon, known as the Bekenstein-Hawking entropy formula: SBH=kBc3A4ℏGS_{BH} = \frac{k_B c^3 A}{4 \hbar G}, where AA is the area of the event horizon

Unruh Effect and Observer-Dependent Particle States

Unruh Effect

  • The Unruh effect, discovered by , is a prediction of QFT in curved spacetime that demonstrates the observer-dependence of particle states
  • According to the Unruh effect, an accelerating observer in flat spacetime will experience a thermal bath of particles, known as Unruh radiation, with a temperature proportional to their proper acceleration
  • The Unruh temperature is given by TU=ℏa2πckBT_U = \frac{\hbar a}{2\pi c k_B}, where aa is the proper acceleration of the observer

Observer-Dependent Particle States

  • The Unruh effect arises due to the fact that the concept of particle states is not invariant under general coordinate transformations in curved spacetime
  • The , as defined by an inertial observer in flat spacetime, appears as a thermal state to an accelerating observer, leading to the perception of a thermal bath of particles
  • The Unruh effect is closely related to the Hawking radiation phenomenon, as both arise from the application of QFT to non-inertial reference frames or curved spacetime geometries
  • The Unruh effect highlights the importance of the observer's reference frame in defining particle states and demonstrates the inherent ambiguity in the concept of particles in QFT in curved spacetime

Key Terms to Review (19)

Adiabatic evolution: Adiabatic evolution refers to a process in quantum mechanics where a system's parameters change slowly compared to the energy scales of the system, allowing it to remain in its instantaneous ground state. This concept is crucial in understanding how quantum states can evolve without exchanging heat with the environment, particularly in curved spacetime where gravitational effects play a significant role.
Black hole thermodynamics: Black hole thermodynamics is the study of the laws governing the behavior and properties of black holes, drawing parallels between thermodynamic laws and black hole physics. This concept suggests that black holes have entropy, temperature, and emit radiation, known as Hawking radiation, which connects gravitational phenomena to quantum mechanics and thermodynamics.
Cosmic inflation: Cosmic inflation is a theory in cosmology that proposes a rapid exponential expansion of the universe during its first few moments after the Big Bang. This expansion addresses several problems in cosmology, such as the flatness and horizon problems, by suggesting that the universe expanded faster than the speed of light, smoothing out any irregularities and creating a uniform structure we observe today.
David Gross: David Gross is a prominent theoretical physicist best known for his pioneering contributions to the development of string theory and his work in quantum chromodynamics (QCD). He played a crucial role in advancing the understanding of the strong interaction, which is fundamental to explaining how quarks and gluons interact within protons and neutrons. His insights into anomalies, particularly the chiral anomaly, have had significant implications for particle physics and the unification of forces, while his investigations into quantum field theory in curved spacetime have opened new avenues in theoretical physics.
Effective Field Theory: Effective field theory (EFT) is a framework used in quantum field theory that allows physicists to make predictions about physical systems by focusing on low-energy phenomena while ignoring high-energy details. This approach simplifies calculations and is especially useful for dealing with complex interactions by encapsulating the effects of heavy particles and degrees of freedom that are not relevant at the energy scale of interest.
Einstein's Equations: Einstein's equations are a set of ten interrelated differential equations in the theory of general relativity that describe how matter and energy in the universe influence the curvature of spacetime. These equations establish a fundamental relationship between the geometry of spacetime and the distribution of energy and momentum, serving as a cornerstone for understanding gravity in a modern context.
Geometric Phase: Geometric phase, also known as the Berry phase, is a phenomenon in quantum mechanics where a system acquires a phase factor due to its geometric properties when it undergoes adiabatic processes. This phase is not dependent on the dynamics of the system but rather on the path taken in parameter space, highlighting the connection between quantum states and their geometrical representation. The geometric phase becomes particularly important in contexts like quantum field theory in curved spacetime, where the curvature affects the paths that quantum states can take.
Hawking Radiation: Hawking radiation is a theoretical prediction by Stephen Hawking that describes the emission of particles from black holes due to quantum effects near the event horizon. It suggests that black holes are not completely black but can emit radiation, leading to their eventual evaporation over time. This concept bridges quantum mechanics and general relativity, showcasing how quantum field theory operates in curved spacetime, especially in the extreme gravitational fields around black holes.
Particle creation: Particle creation refers to the process in quantum field theory where particles are produced from energy fluctuations, often in high-energy environments. This phenomenon illustrates the principle that energy can be converted into mass, as outlined by Einstein's equation $$E=mc^2$$. The concept is crucial for understanding the dynamics of fields and the interactions that lead to observable particles in both flat and curved spacetime settings.
Perturbation theory: Perturbation theory is a mathematical technique used in quantum mechanics and quantum field theory to approximate the behavior of a system that is subject to small disturbances or interactions. It allows for the calculation of physical quantities by treating the interaction as a small perturbation of a solvable system, providing a powerful method to understand complex systems and their dynamics.
Quantum Fluctuations: Quantum fluctuations are temporary changes in energy levels that occur in a vacuum due to the uncertainty principle, allowing particles to spontaneously appear and disappear. These fluctuations play a fundamental role in various phenomena, influencing particle interactions and the structure of space itself.
Renormalization: Renormalization is a process used in quantum field theory to remove infinities from calculated quantities, leading to meaningful physical predictions. This involves redefining parameters in a theory, such as mass and charge, to absorb these infinities into a finite set of parameters, ensuring that the theory remains predictive and matches experimental results.
Scalar Fields: Scalar fields are mathematical functions that assign a single scalar value to every point in space, often used to describe physical quantities like temperature or energy density. In quantum field theory, scalar fields represent particles with no intrinsic angular momentum, such as the Higgs boson, and play a crucial role in symmetry breaking and the dynamics of quantum fields in curved spacetime.
Spontaneous Symmetry Breaking: Spontaneous symmetry breaking occurs when a system that is symmetric under a certain transformation chooses a specific configuration that does not exhibit that symmetry. This phenomenon is crucial in various fields, leading to the emergence of distinct states and particles, and it helps explain many physical processes, including mass generation and phase transitions.
Stephen Hawking: Stephen Hawking was a renowned theoretical physicist, cosmologist, and author who made significant contributions to our understanding of black holes and the nature of the universe. His work on Hawking radiation provided groundbreaking insights into quantum field theory in curved spacetime, challenging previous notions of black holes as perfect absorbers of matter and energy.
Vacuum State: The vacuum state is the lowest energy state of a quantum field, where no particles are present but fluctuations in the field still exist. It serves as a foundation for understanding particle creation and annihilation processes in quantum field theory, playing a critical role in various aspects such as the second quantization framework, the behavior of fields governed by equations like the Klein-Gordon equation, and the structure of Fock space.
Vector Fields: A vector field is a mathematical construct that assigns a vector to every point in a given space, often used to represent physical quantities that have both magnitude and direction, such as force or velocity. In the context of quantum field theory in curved spacetime, vector fields help describe how quantum particles behave under the influence of gravitational fields and the curvature of spacetime, providing essential tools for understanding particle dynamics in non-flat geometries.
Weyl's invariance: Weyl's invariance refers to the property of a physical theory being invariant under the rescaling of lengths, which is particularly important in the context of quantum field theory in curved spacetime. This concept implies that the theory retains its form under transformations that scale the metric tensor, allowing for consistent formulations of matter fields in varying gravitational backgrounds. Weyl's invariance highlights the interplay between geometry and physics, illustrating how the behavior of quantum fields can change with the curvature of spacetime.
William Unruh: William Unruh is a prominent theoretical physicist known for his contributions to quantum field theory, particularly in the context of curved spacetime. He is best recognized for his discovery of the Unruh effect, which reveals how an accelerating observer perceives a vacuum as a thermal bath of particles, linking acceleration and quantum mechanics in profound ways.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide