11.2 Quantum field theory

Quantum Field Theory (QFT) extends quantum mechanics to systems with infinite degrees of freedom. It's crucial for understanding particle physics and forms the basis of the Standard Model. Von Neumann algebras play a key role in QFT's mathematical formulation.

QFT treats fields as fundamental objects, incorporating special relativity and allowing particle creation and annihilation. It uses field operators to create and destroy particles, forming the building blocks for observables and interactions in the theory.

Foundations of QFT

• Quantum Field Theory (QFT) extends quantum mechanics to systems with infinitely many degrees of freedom, crucial for understanding particle physics and condensed matter systems
• QFT provides a framework for describing fundamental interactions in nature, forming the basis for the Standard Model of particle physics
• Von Neumann algebras play a key role in the mathematical formulation of QFT, particularly in the algebraic approach to quantum field theory

Quantum mechanics vs QFT

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• Quantum mechanics deals with discrete particles while QFT treats fields as fundamental objects
• QFT incorporates special relativity, allowing for particle creation and annihilation processes
• Quantum mechanics uses wave functions while QFT employs field operators acting on a vacuum state
• QFT naturally accounts for many-particle systems and interactions, essential for describing high-energy physics phenomena

Field operators

• Field operators create and annihilate particles at specific spacetime points
• Commutation relations for bosonic fields: $[φ(x), π(y)] = iδ(x-y)$
• Anticommutation relations for fermionic fields: $\{ψ(x), ψ^†(y)\} = δ(x-y)$
• Field operators form the building blocks for constructing observables and interaction Hamiltonians in QFT

Hilbert space in QFT

• Fock space serves as the Hilbert space for QFT, accommodating states with varying particle numbers
• Vacuum state |0⟩ represents the ground state of the quantum field
• Creation and annihilation operators act on Fock space to generate multi-particle states
• Unitary representations of the Poincaré group on Fock space ensure Lorentz invariance of the theory

Canonical quantization

• Canonical quantization provides a systematic method for transitioning from classical field theories to quantum field theories
• This approach generalizes the quantization procedure used in quantum mechanics to field theories with infinite degrees of freedom
• Canonical quantization forms the foundation for understanding the operator algebra structure in QFT, connecting to von Neumann algebra theory

Scalar fields

• Klein-Gordon equation describes the dynamics of scalar fields: $(∂_μ∂^μ + m^2)φ(x) = 0$
• Quantization promotes classical fields to operator-valued distributions
• Commutation relations for scalar field operators: $[φ(x), π(y)] = iδ(x-y)$
• Mode expansion of scalar fields in terms of creation and annihilation operators: $φ(x) = ∫ \frac{d^3p}{(2π)^3} \frac{1}{\sqrt{2ω_p}} (a_p e^{-ipx} + a_p^† e^{ipx})$

Fermion fields

• Dirac equation governs the behavior of fermion fields: $(iγ^μ∂_μ - m)ψ(x) = 0$
• Anticommutation relations for fermion field operators: $\{ψ_α(x), ψ_β^†(y)\} = δ_αβ δ(x-y)$
• Spin-statistics theorem connects the spin of particles to their quantization rules
• Fermion fields describe particles with half-integer spin (electrons, quarks)

Gauge fields

• Gauge fields mediate interactions between matter fields (electromagnetic, strong, weak forces)
• Maxwell's equations describe the classical behavior of electromagnetic fields
• Quantization of gauge fields introduces gauge-fixing terms to handle redundant degrees of freedom
• Faddeev-Popov ghosts arise in the path integral formulation to maintain unitarity in non-abelian gauge theories

Path integral formulation

• Path integral formulation provides an alternative approach to quantization, emphasizing the role of classical paths in quantum processes
• This formulation connects quantum field theory to statistical mechanics, enabling powerful computational techniques
• Path integrals form the basis for non-perturbative approaches in QFT, relevant to the study of strongly coupled systems in von Neumann algebras

Feynman diagrams

• Graphical representations of terms in the perturbative expansion of correlation functions
• Vertices represent interactions, while lines correspond to propagators of particles
• Feynman rules associate mathematical expressions to diagram elements
• Loop diagrams account for quantum corrections and renormalization effects

Generating functionals

• Functional derivatives of generating functionals yield correlation functions
• Partition function Z[J] generates all correlation functions: $Z[J] = ∫ Dφ e^{iS[φ] + i∫ J(x)φ(x)d^4x}$
• Connected generating functional W[J] relates to Z[J] via W[J] = -i ln Z[J]
• Effective action Γ[φ] obtained by Legendre transform of W[J]

Effective action

• Incorporates quantum corrections to the classical action
• One-particle irreducible (1PI) diagrams contribute to the effective action
• Effective potential V_eff(φ) determines the vacuum structure of the theory
• Generates equations of motion for the full quantum theory

Renormalization

• Renormalization addresses infinities arising in quantum field theories due to high-energy fluctuations
• This process is crucial for extracting meaningful physical predictions from QFT calculations
• Renormalization group techniques connect to the classification of von Neumann algebras, particularly in the context of scaling limits and critical phenomena

Regularization methods

• Introduce a cutoff to render divergent integrals finite (momentum cutoff, dimensional regularization)
• Pauli-Villars regularization adds heavy auxiliary fields to cancel divergences
• Lattice regularization discretizes spacetime, providing a non-perturbative approach
• Zeta function regularization exploits analytic continuation of divergent sums

Renormalization group

• Describes how coupling constants change with energy scale
• Wilson's approach to renormalization group focuses on effective field theories
• Callan-Symanzik equation governs the scaling behavior of correlation functions
• Fixed points of the renormalization group flow correspond to scale-invariant theories

Beta functions

• Encode the running of coupling constants with energy scale
• Asymptotic freedom in QCD manifests as a negative beta function
• Landau poles indicate potential breakdown of perturbation theory at high energies
• Beta functions determine the ultraviolet behavior of quantum field theories

Symmetries in QFT

• Symmetries play a fundamental role in constraining the structure and dynamics of quantum field theories
• The interplay between symmetries and quantum fields connects to representation theory of von Neumann algebras
• Symmetry principles guide the construction of physically relevant quantum field theories and their operator algebraic formulations

Noether's theorem

• Associates conserved currents and charges with continuous symmetries of the action
• Energy-momentum tensor T^μν conserved for spacetime translation invariance
• Angular momentum tensor J^μνρ conserved for Lorentz invariance
• Charge Q conserved for global U(1) symmetry in quantum electrodynamics

Spontaneous symmetry breaking

• Occurs when the ground state of a system does not respect the symmetries of the Lagrangian
• Goldstone's theorem predicts massless bosons for broken continuous symmetries
• Higgs mechanism generates masses for gauge bosons in spontaneously broken gauge theories
• Anderson-Brout-Englert-Higgs mechanism explains mass generation in the Standard Model

Gauge invariance

• Local symmetry principle underlying fundamental interactions in nature
• Gauge transformations: $ψ(x) → e^{iα(x)}ψ(x), A_μ(x) → A_μ(x) - ∂_μα(x)$
• Covariant derivative D_μ = ∂_μ + ieA_μ ensures gauge-invariant coupling to matter fields
• Non-abelian gauge theories (Yang-Mills theories) describe strong and weak interactions

Operator algebras in QFT

• Operator algebras provide a rigorous mathematical framework for studying the structure of quantum field theories
• This approach connects QFT to von Neumann algebra theory, enabling the application of powerful algebraic techniques
• Algebraic quantum field theory emphasizes the role of local observables and their algebraic relations

Local algebras

• Associate algebras of observables to bounded regions of spacetime
• Isotony: algebras associated with larger regions contain those of smaller regions
• Microcausality: observables in spacelike separated regions commute
• Net of local algebras captures the causal structure of quantum field theory

Haag-Kastler axioms

• Provide a set of axioms for algebraic quantum field theory
• Isotony, locality, covariance, and spectrum condition form the core axioms
• Haag's theorem demonstrates the inequivalence of interacting and free field representations
• Algebraic approach circumvents difficulties associated with the particle picture in curved spacetime

Tomita-Takesaki theory

• Modular theory for von Neumann algebras applied to local algebras in QFT
• Modular automorphism group generates time evolution in thermal states
• Bisognano-Wichmann theorem relates modular operators to Lorentz boosts
• Modular theory provides a deep connection between spacetime symmetries and algebraic structure of QFT

QFT in curved spacetime

• Extends quantum field theory to incorporate effects of gravity on quantum fields
• This area bridges quantum theory and general relativity, providing insights into quantum aspects of gravity
• The algebraic approach to QFT in curved spacetime connects to von Neumann algebra theory through the study of local observable algebras

Hawking radiation

• Predicts thermal emission from black holes due to quantum effects
• Temperature of Hawking radiation proportional to surface gravity of the black hole
• Information paradox arises from the apparent loss of information in black hole evaporation
• Unruh-DeWitt detector model provides an operational approach to detecting Hawking radiation

Unruh effect

• Accelerated observers in flat spacetime perceive the vacuum as a thermal bath
• Unruh temperature proportional to proper acceleration: T = ℏa/(2πck_B)
• Demonstrates the observer-dependence of the particle concept in QFT
• Connects to Hawking radiation through the principle of equivalence

Algebraic approach to QFTCS

• Emphasizes the role of local observable algebras in curved spacetime
• Avoids reliance on global concepts like particles and vacua
• Hadamard states serve as a replacement for the notion of vacuum states
• Allows for a rigorous treatment of quantum fields in general globally hyperbolic spacetimes

Applications to von Neumann algebras

• Quantum field theory provides a rich source of examples and applications for von Neumann algebra theory
• The algebraic structure of QFT naturally leads to the study of operator algebras, particularly type III factors
• Von Neumann algebras offer powerful tools for analyzing the mathematical structure of quantum field theories

Type III factors in QFT

• Local algebras in QFT typically form type III von Neumann factors
• Hyperfinite type III_1 factor emerges in the analysis of thermal states in QFT
• Type III factors reflect the infinite number of degrees of freedom in quantum field theories
• Connes' classification of type III factors provides insights into the structure of local algebras

KMS states

• Kubo-Martin-Schwinger (KMS) condition characterizes thermal equilibrium states in QFT
• KMS states generalize the notion of Gibbs states to infinite systems
• Modular automorphism group of a KMS state generates time evolution
• KMS condition: ⟨A σ_t(B)⟩ = ⟨B σ_{t-iβ}(A)⟩ for observables A and B

Modular theory in QFT

• Tomita-Takesaki theory applied to local algebras in QFT
• Modular operators Δ and J encode information about spacetime symmetries
• Bisognano-Wichmann theorem relates modular operators to Lorentz boosts in Minkowski space
• Modular theory provides a deep connection between algebraic structure and spacetime geometry in QFT

Quantum field theory vs strings

• Compares and contrasts quantum field theory with string theory as approaches to fundamental physics
• This comparison highlights the strengths and limitations of QFT, motivating the exploration of more general mathematical structures
• The relationship between QFT and string theory connects to broader questions in operator algebra theory and mathematical physics

Perturbative vs non-perturbative

• QFT relies heavily on perturbative methods for practical calculations
• String theory incorporates non-perturbative effects more naturally through dualities
• Lattice QFT provides a non-perturbative approach to strongly coupled systems
• Seiberg-Witten theory demonstrates the power of combining perturbative and non-perturbative techniques

Holographic principle

• Proposes that the information content of a region is encoded on its boundary
• AdS/CFT correspondence provides a concrete realization of holography
• Holographic entanglement entropy connects quantum information to geometry
• Tensor networks offer a discrete analog of holography, bridging QFT and quantum information theory

AdS/CFT correspondence

• Relates string theory in anti-de Sitter space to conformal field theory on the boundary
• Provides a tool for studying strongly coupled quantum field theories
• Large N limit of gauge theories connects to classical gravity in the bulk
• Witten diagram technique for computing CFT correlators from bulk interactions