9.1 Goldstone theorem and Nambu-Goldstone bosons
4 min read•august 14, 2024
Goldstone's theorem explains how symmetry breaking creates massless particles called . These bosons are crucial in understanding the low-energy behavior of systems with broken symmetries, from particle physics to condensed matter.
The theorem states that for every broken continuous symmetry, a massless boson appears. This concept is key to grasping and its far-reaching implications in various physical phenomena, from to the .
Spontaneous Symmetry Breaking
Concept and Consequences
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- Spontaneous symmetry breaking occurs when the ground state of a system has a lower symmetry than the Lagrangian or Hamiltonian describing the system
- In quantum field theory, spontaneous symmetry breaking happens when the vacuum state is not invariant under a symmetry transformation, even though the Lagrangian is
- Spontaneous symmetry breaking leads to the existence of massless scalar bosons, known as Nambu-Goldstone bosons, which correspond to the broken symmetry generators
- The number of Nambu-Goldstone bosons is equal to the number of broken continuous symmetry generators
Physical Phenomena and Applications
- Spontaneous symmetry breaking plays a crucial role in various physical phenomena
- : Spontaneous magnetization occurs below the Curie temperature, breaking rotational symmetry (iron, nickel)
- Superconductivity: Cooper pairs form a condensate, breaking U(1) gauge symmetry (lead, niobium)
- Higgs mechanism: Spontaneous breaking of electroweak symmetry gives mass to W and Z bosons (Standard Model of particle physics)
- : Quark condensates break chiral symmetry in quantum chromodynamics (QCD)
Goldstone Theorem and Massless Bosons
Theorem Statement and Derivation
- The states that for every spontaneously broken continuous symmetry, there exists a massless scalar boson, known as a Nambu-Goldstone boson
- The derivation of the Goldstone theorem involves considering a Lagrangian with a continuous symmetry and analyzing the consequences of spontaneous symmetry breaking
- The proof relies on the existence of a conserved current associated with the continuous symmetry
- The non-invariance of the vacuum state under the symmetry transformation is a crucial ingredient
Implications and Properties
- The massless nature of Nambu-Goldstone bosons can be understood through the concept of long-range correlations
- The impossibility of having a preferred direction in the vacuum state leads to massless excitations
- The presence of massless Nambu-Goldstone bosons has important implications for the low-energy behavior of the system
- They dominate the low-energy excitation spectrum
- They play a crucial role in the long-range behavior of correlation functions
- The structure of the excitation spectrum is determined by the pattern of symmetry breaking and the number of broken generators
Nambu-Goldstone Bosons in Physical Systems
Particle Physics Examples
- In particle physics, (π mesons) are identified as the Nambu-Goldstone bosons associated with the spontaneous breaking of chiral symmetry in quantum chromodynamics (QCD)
- Pions are the lightest mesons and play a crucial role in the strong interaction
- The small but non-zero masses of pions are attributed to the explicit breaking of chiral symmetry by quark masses
- The Higgs boson in the Standard Model of particle physics is not a Nambu-Goldstone boson but is related to the spontaneous breaking of the electroweak symmetry
- The Higgs mechanism gives mass to the W and Z bosons while preserving gauge invariance
Condensed Matter Examples
- In condensed matter physics, in crystals can be interpreted as Nambu-Goldstone bosons arising from the spontaneous breaking of translational symmetry
- Phonons are collective excitations that represent the vibrations of the crystal lattice (sound waves)
- in ferromagnets are Nambu-Goldstone bosons associated with the spontaneous breaking of rotational symmetry in the magnetic ordering
- Magnons are collective excitations of the spin degrees of freedom in a ferromagnet (spin waves)
- In superfluid helium-4, the phonon and roton excitations can be understood as Nambu-Goldstone bosons resulting from the spontaneous breaking of U(1) symmetry
- The phonon represents the sound wave in the superfluid, while the roton is a collective excitation with a minimum in the dispersion relation
Nambu-Goldstone Bosons in Quantum Field Theory
Theoretical Description
- In quantum field theory, Nambu-Goldstone bosons appear as massless excitations in the spectrum of the theory after spontaneous symmetry breaking
- The presence of Nambu-Goldstone bosons is a consequence of the Goldstone theorem and is essential for maintaining the consistency of the theory
- Nambu-Goldstone bosons are described by that transform non-linearly under the broken symmetry generators
- The non-linear transformation reflects the fact that the vacuum state is not invariant under the symmetry
Interactions and Low-Energy Effective Theory
- The interactions of Nambu-Goldstone bosons with other fields in the theory are dictated by the symmetry structure and the pattern of symmetry breaking
- The low-energy effective theory describing the dynamics of Nambu-Goldstone bosons is known as the
- The non-linear sigma model captures the essential features of the symmetry breaking
- It describes the interactions of Nambu-Goldstone bosons among themselves and with other low-energy degrees of freedom
- In gauge theories, the Nambu-Goldstone bosons associated with the spontaneous breaking of gauge symmetries are "eaten" by the gauge bosons
- This mechanism, known as the Higgs mechanism, gives rise to massive vector bosons (W and Z bosons in the Standard Model)
- The Nambu-Goldstone bosons are absorbed into the longitudinal components of the gauge bosons, providing them with mass
Key Terms to Review (22)
Chiral Symmetry Breaking: Chiral symmetry breaking refers to the phenomenon where a system that exhibits chiral symmetry in its Lagrangian does not have that symmetry in its vacuum state. This breaking is essential in understanding particle masses and interactions, particularly in the context of certain quantum field theories. The consequence of this symmetry breaking often leads to the emergence of massless Goldstone bosons, as outlined in the Goldstone theorem, and plays a critical role in explaining the behavior of quarks and gluons in Quantum Chromodynamics (QCD).
Continuous symmetries: Continuous symmetries refer to transformations that can be smoothly varied without discrete jumps, preserving the physical properties of a system. They are fundamental in understanding the behavior of systems, especially in quantum field theory, where they lead to conservation laws and the emergence of Nambu-Goldstone bosons in cases of spontaneous symmetry breaking.
Ferromagnetism: Ferromagnetism is a phenomenon where certain materials, such as iron, exhibit spontaneous magnetization, meaning they can become permanently magnetized even without an external magnetic field. This behavior is a result of the alignment of magnetic moments of electrons in the material, leading to strong interactions that can break symmetry, and has profound implications in the context of spontaneous symmetry breaking and the Goldstone theorem.
Goldstone Modes: Goldstone modes refer to the massless excitations that arise in a system with spontaneously broken continuous symmetries. When a symmetry is broken, the Goldstone theorem states that there will be corresponding massless scalar particles, known as Nambu-Goldstone bosons, which represent the directions in which the symmetry is broken. These modes play a significant role in various fields, including particle physics and condensed matter systems, highlighting how symmetry breaking can lead to observable physical phenomena.
Goldstone theorem: The Goldstone theorem states that in a quantum field theory, for every continuous symmetry that is spontaneously broken, there exists a corresponding massless scalar particle known as a Nambu-Goldstone boson. This concept is crucial in understanding how symmetries and their breaking lead to the emergence of new particles and phenomena in particle physics.
Group Theory: Group theory is a branch of mathematics that studies algebraic structures known as groups, which consist of a set equipped with an operation that satisfies specific properties like closure, associativity, identity, and invertibility. In the context of physics, especially in particle physics and field theory, group theory helps to analyze symmetries in physical systems and plays a critical role in understanding the behavior of particles and fields.
Higgs mechanism: The Higgs mechanism is a process in particle physics that explains how particles acquire mass through spontaneous symmetry breaking in a quantum field. It introduces a scalar field, known as the Higgs field, which permeates all of space, and through interactions with this field, certain particles gain mass while others remain massless, providing an essential framework for understanding the mass of fundamental particles.
Jeffrey Goldstone: Jeffrey Goldstone is a physicist known for his contributions to theoretical physics, particularly in the formulation of the Goldstone theorem. This theorem is crucial for understanding the behavior of systems that exhibit spontaneous symmetry breaking, leading to the emergence of massless excitations known as Nambu-Goldstone bosons.
Lagrangian Formalism: Lagrangian formalism is a powerful framework in theoretical physics that reformulates classical mechanics using the principle of least action, where the dynamics of a system are described by a function called the Lagrangian. This method is particularly useful in fields like quantum field theory as it elegantly encapsulates symmetries and conservation laws, which are essential for understanding concepts like spontaneous symmetry breaking and the emergence of Nambu-Goldstone bosons.
Magnons: Magnons are quasiparticles that represent collective excitations of spins in a magnetically ordered system, particularly in ferromagnetic materials. They emerge when the spins of atoms in a lattice oscillate out of their equilibrium positions, leading to a quantized excitation of the magnetic order, which can be understood through the lens of the Goldstone theorem and Nambu-Goldstone bosons.
Massless scalar particles: Massless scalar particles are quantum field theory entities that are described by a scalar field and possess no mass, allowing them to travel at the speed of light. These particles play a significant role in various physical theories, particularly in the context of spontaneous symmetry breaking, where their emergence is closely tied to the Goldstone theorem and the concept of Nambu-Goldstone bosons, which are massless excitations resulting from broken symmetries.
Nambu-Goldstone bosons: Nambu-Goldstone bosons are massless scalar particles that arise in quantum field theories as a consequence of spontaneous symmetry breaking. When a system's ground state does not exhibit the same symmetry as the underlying Lagrangian, these bosons emerge as the excitations corresponding to the broken symmetry, reflecting the degrees of freedom associated with the lost symmetry.
Non-linear sigma model: A non-linear sigma model is a type of quantum field theory where fields take values in a non-linear manifold, typically a target space like a sphere or more generally a symmetric space. This model is important for understanding spontaneous symmetry breaking and the dynamics of Nambu-Goldstone bosons, as it effectively captures the low-energy excitations that arise when a continuous symmetry is spontaneously broken.
Phonons: Phonons are quantized modes of vibrations in a crystalline lattice, which can be thought of as the quantum mechanical representation of sound waves in solids. These quasiparticles play a crucial role in understanding thermal and acoustic properties of materials, especially in the context of how they interact with other excitations such as electrons or other bosons. In systems that exhibit spontaneous symmetry breaking, phonons can also be related to the Nambu-Goldstone bosons resulting from such transitions.
Pions: Pions are subatomic particles that act as the exchange particles, or gauge bosons, for the strong nuclear force between nucleons (protons and neutrons). They come in three varieties: positively charged ($ ext{π}^+$), negatively charged ($ ext{π}^-$), and neutral ($ ext{π}^0$). Pions play a crucial role in understanding how spontaneous symmetry breaking leads to the emergence of Nambu-Goldstone bosons, as they can be considered as Goldstone modes resulting from the symmetry breaking in quantum field theories.
Quantum electrodynamics: Quantum electrodynamics (QED) is the quantum field theory that describes how light and matter interact, specifically focusing on the electromagnetic force between charged particles. It unifies the principles of quantum mechanics and special relativity, providing a framework for understanding phenomena like particle interactions, vacuum fluctuations, and the behavior of photons.
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.
Superconductivity: Superconductivity is a phenomenon where a material exhibits zero electrical resistance and expels magnetic fields when cooled below a certain critical temperature. This behavior is significant in understanding phase transitions and broken symmetries, leading to the emergence of Nambu-Goldstone bosons and various condensed matter theories.
Supersymmetry: Supersymmetry is a theoretical framework in particle physics that posits a symmetry relationship between fermions and bosons, suggesting that each particle has a superpartner with different spin characteristics. This concept aims to solve various problems in quantum field theory, such as the hierarchy problem and unification of forces, and plays a crucial role in models like supergravity and string theory.
Vacuum states: Vacuum states are the lowest energy states of a quantum field theory, representing the absence of particles but still possessing physical properties such as fluctuations and virtual particles. They serve as the foundational state from which excitations, like particles and fields, emerge, playing a critical role in phenomena like spontaneous symmetry breaking. This concept connects deeply with symmetry properties and the resulting implications for particle physics, especially in understanding Nambu-Goldstone bosons.
Yoichiro Nambu: Yoichiro Nambu was a Japanese theoretical physicist known for his groundbreaking work on spontaneous symmetry breaking and the development of the Nambu-Goldstone theorem. His contributions have significantly impacted the understanding of particle physics, particularly in relation to how symmetries in quantum field theories lead to the emergence of massless bosons.