🔬Quantum Field Theory Unit 3 – Free Fields and Symmetries

Key Concepts and Definitions

• Quantum field theory (QFT) framework that combines classical field theory, special relativity, and quantum mechanics to describe subatomic particles and their interactions
• Free field theory subset of QFT that deals with non-interacting fields and serves as a foundation for understanding more complex interacting field theories
• Symmetry transformation that leaves the physical properties of a system unchanged (translation, rotation, gauge)
  • Continuous symmetries described by continuous parameters (rotation angle, phase shift)
  • Discrete symmetries described by discrete parameters (parity, charge conjugation, time reversal)
• Noether's theorem establishes a connection between continuous symmetries and conservation laws in physical systems
• Gauge invariance principle stating that the physical properties of a system remain unchanged under certain local transformations of the fields (U(1), SU(2), SU(3))
• Lagrangian density function that describes the dynamics of a field system and is used to derive equations of motion and conserved quantities
• Hamiltonian density function that represents the energy density of a field system and is related to the Lagrangian density through a Legendre transformation

Free Field Theory Fundamentals

• Free scalar field simplest type of field described by the Klein-Gordon equation, representing spinless particles (Higgs boson)
  • Lagrangian density for a free scalar field: $\mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2$
  • Equation of motion derived from the Lagrangian density using the principle of least action (Euler-Lagrange equation)
• Free spinor field represents fermions with half-integer spin (electrons, quarks) and is described by the Dirac equation
  • Lagrangian density for a free spinor field: $\mathcal{L} = \bar{\psi} (i \gamma^\mu \partial_\mu - m) \psi$
  • Dirac matrices $\gamma^\mu$ satisfy the anticommutation relations $\{\gamma^\mu, \gamma^\nu\} = 2 g^{\mu\nu}$
• Free vector field represents bosons with integer spin (photons, gluons) and is described by the Proca equation
  • Lagrangian density for a free vector field: $\mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \frac{1}{2} m^2 A_\mu A^\mu$
  • Field strength tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ captures the electromagnetic field properties
• Quantization procedure that promotes classical fields to quantum operators acting on a Hilbert space of states (canonical quantization, path integral formulation)
• Fock space representation of the Hilbert space for a multi-particle quantum system, constructed using creation and annihilation operators

Symmetries in Quantum Field Theory

• Poincaré symmetry fundamental symmetry of spacetime in special relativity, consisting of translations, rotations, and boosts
  • Generators of the Poincaré group: momentum ($P^\mu$), angular momentum ($M^{\mu\nu}$)
  • Casimir operators of the Poincaré group: mass ($P_\mu P^\mu$) and spin ($W_\mu W^\mu$, where $W^\mu = \frac{1}{2} \epsilon^{\mu\nu\rho\sigma} P_\nu M_{\rho\sigma}$)
• Internal symmetries transformations that act on the internal degrees of freedom of fields (flavor, color) without affecting spacetime coordinates
  • Examples: U(1) symmetry (electromagnetism), SU(2) symmetry (weak interaction), SU(3) symmetry (strong interaction)
• Spontaneous symmetry breaking phenomenon where the ground state of a system does not exhibit the full symmetry of the Lagrangian (Higgs mechanism)
  • Goldstone theorem states that spontaneous breaking of a continuous global symmetry leads to the appearance of massless scalar particles (Goldstone bosons)
• Discrete symmetries transformations that involve discrete changes in the properties of fields (parity, charge conjugation, time reversal)
  • CPT theorem states that any Lorentz-invariant local QFT must be invariant under the combined transformation of charge conjugation (C), parity (P), and time reversal (T)

Noether's Theorem and Conservation Laws

• Noether's theorem establishes a one-to-one correspondence between continuous symmetries of a Lagrangian and conserved quantities (charges) in the physical system
  • Continuous symmetry transformation that leaves the action $S = \int \mathcal{L} d^4x$ invariant
  • Conserved current $J^\mu$ associated with the continuous symmetry, satisfying the conservation equation $\partial_\mu J^\mu = 0$
  • Conserved charge $Q = \int J^0 d^3x$ that remains constant in time
• Energy-momentum conservation consequence of translational invariance in spacetime
  • Energy-momentum tensor $T^{\mu\nu} = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)} \partial^\nu \phi - g^{\mu\nu} \mathcal{L}$
  • Conservation equation $\partial_\mu T^{\mu\nu} = 0$
• Angular momentum conservation consequence of rotational invariance in spacetime
  • Angular momentum tensor $M^{\mu\nu\rho} = x^\mu T^{\nu\rho} - x^\nu T^{\mu\rho}$
  • Conservation equation $\partial_\mu M^{\mu\nu\rho} = 0$
• Charge conservation consequence of global U(1) symmetry (gauge invariance) in the Lagrangian
  • Noether current $J^\mu = \frac{\partial \mathcal{L}}{\partial(\partial_\mu \psi)} \psi$
  • Conservation equation $\partial_\mu J^\mu = 0$

Gauge Transformations and Invariance

• Gauge transformation local transformation of fields that leaves the physical observables unchanged
  • U(1) gauge transformation for a complex scalar field: $\phi(x) \rightarrow e^{i \alpha(x)} \phi(x)$
  • SU(N) gauge transformation for a multiplet of fields: $\psi(x) \rightarrow e^{i \alpha^a(x) T^a} \psi(x)$, where $T^a$ are the generators of SU(N)
• Gauge invariance requirement that the Lagrangian remains invariant under gauge transformations
  • Gauge-invariant Lagrangian for a complex scalar field coupled to a U(1) gauge field: $\mathcal{L} = (D_\mu \phi)^* (D^\mu \phi) - \frac{1}{4} F_{\mu\nu} F^{\mu\nu}$, where $D_\mu = \partial_\mu - i e A_\mu$ is the covariant derivative
• Gauge fields introduced to maintain gauge invariance in the Lagrangian (vector potential in electromagnetism, gluon fields in quantum chromodynamics)
  • Gauge field transforms under a gauge transformation as $A_\mu(x) \rightarrow A_\mu(x) - \frac{1}{e} \partial_\mu \alpha(x)$
• Gauge fixing procedure that removes the redundancy in the gauge field description by choosing a specific gauge condition (Coulomb gauge, Lorenz gauge)
  • Faddeev-Popov ghost fields introduced to maintain gauge invariance in the quantization procedure for non-Abelian gauge theories

Applications to Particle Physics

• Quantum electrodynamics (QED) QFT describing the electromagnetic interactions between charged particles (electrons, positrons, muons)
  • U(1) gauge symmetry leads to the conservation of electric charge
  • Feynman diagrams used to calculate scattering amplitudes and cross-sections for electromagnetic processes (electron-positron annihilation, Compton scattering)
• Quantum chromodynamics (QCD) QFT describing the strong interactions between quarks and gluons
  • SU(3) gauge symmetry leads to the conservation of color charge
  • Asymptotic freedom property of QCD: the strong coupling constant decreases at high energies, allowing perturbative calculations
  • Confinement property of QCD: quarks and gluons are confined within hadrons and cannot be observed as free particles
• Electroweak theory unified description of electromagnetic and weak interactions based on the SU(2)xU(1) gauge symmetry
  • Spontaneous symmetry breaking through the Higgs mechanism gives rise to the masses of the W and Z bosons
  • Cabibbo-Kobayashi-Maskawa (CKM) matrix describes the mixing of quark flavors in weak interactions
• Standard Model of particle physics theoretical framework that combines QED, QCD, and the electroweak theory to describe the fundamental particles and their interactions
  • Higgs boson scalar particle associated with the Higgs field responsible for the spontaneous symmetry breaking in the electroweak sector
  • Beyond the Standard Model theories (supersymmetry, grand unification) aim to address the limitations of the Standard Model and provide a more comprehensive description of nature

Mathematical Techniques and Tools

• Feynman path integral formulation of QFT that expresses the transition amplitude between initial and final states as a sum over all possible field configurations
  • Path integral for a scalar field: $Z = \int \mathcal{D}\phi e^{i S[\phi]}$, where $S[\phi] = \int \mathcal{L}(\phi, \partial_\mu \phi) d^4x$ is the action
  • Feynman diagrams derived from the path integral by expanding the exponential in powers of the interaction term and representing each term as a diagram
• Renormalization procedure that deals with the infinities arising in the calculation of physical quantities in QFT
  • Regularization methods (dimensional regularization, Pauli-Villars regularization) used to isolate the divergences
  • Renormalization group equations describe the dependence of the renormalized coupling constants on the energy scale
• Perturbation theory approach to solve QFT problems by expanding the quantities of interest in powers of the coupling constant
  • Feynman rules a set of rules for constructing Feynman diagrams and calculating the corresponding amplitudes based on the Lagrangian of the theory
  • Loop diagrams Feynman diagrams containing closed loops, which contribute to higher-order corrections to the perturbative expansion
• Functional methods tools for analyzing the properties of QFTs using functional derivatives and functional integrals
  • Generating functionals $Z[J] = \int \mathcal{D}\phi e^{i S[\phi] + i \int J \phi d^4x}$ used to derive Green's functions and correlation functions
  • Effective action $\Gamma[\phi_c] = -i \ln Z[J] - \int J \phi_c d^4x$ that incorporates the effects of quantum corrections and allows the study of non-perturbative phenomena

Common Challenges and Problem-Solving Strategies

• Divergences and renormalization dealing with the infinities that arise in loop calculations by applying regularization and renormalization techniques
  • Identify the divergent diagrams and isolate the divergences using a suitable regularization scheme
  • Introduce counterterms in the Lagrangian to absorb the divergences and obtain finite, renormalized quantities
  • Verify the renormalizability of the theory by checking that all divergences can be absorbed by a finite number of counterterms
• Gauge fixing and ghost fields choosing an appropriate gauge condition to simplify calculations and introducing ghost fields to maintain gauge invariance in non-Abelian gauge theories
  • Select a gauge fixing condition (Coulomb gauge, Lorenz gauge) that simplifies the equations of motion or the Feynman rules
  • Introduce Faddeev-Popov ghost fields to cancel the unphysical degrees of freedom in the gauge field propagator
  • Use the BRST (Becchi-Rouet-Stora-Tyutin) formalism to ensure the gauge invariance of the quantized theory
• Symmetry breaking and Goldstone bosons analyzing the consequences of spontaneous symmetry breaking and identifying the presence of Goldstone bosons
  • Identify the symmetries of the Lagrangian and the vacuum state of the theory
  • Determine if the vacuum state breaks any of the symmetries and if there are any flat directions in the potential
  • Apply the Goldstone theorem to predict the existence of massless scalar particles (Goldstone bosons) for each broken continuous symmetry
• Feynman diagram calculations organizing and simplifying the calculation of scattering amplitudes and cross-sections using Feynman diagrams
  • Draw all the relevant Feynman diagrams for the process of interest up to the desired order in perturbation theory
  • Apply the Feynman rules to assign factors to each element of the diagrams (propagators, vertices, external lines)
  • Perform the loop integrals using techniques such as Feynman parameters, dimensional regularization, and Passarino-Veltman reduction
  • Evaluate the traces of gamma matrices, simplify the expressions, and calculate the final result for the amplitude or cross-section