📐 QFT Equation Reference
A comprehensive, searchable database of all fundamental equations in Quantum Field Theory. Find formulas by name, browse by category (Classical Fields, Canonical Quantization, Path Integrals, Gauge Theories, Renormalization), and jump directly to detailed explanations.
📐QFT Equation Reference
Euler-Lagrange Equation
Classical Field Theory∂μ(∂ℒ/∂(∂μφ)) - ∂ℒ/∂φ = 0Equation of motion for a field from the Lagrangian density
Klein-Gordon Equation
Classical Field Theory(∂μ∂μ + m²)φ = 0Relativistic wave equation for spin-0 fields
Dirac Equation
Classical Field Theory(iγμ∂μ - m)ψ = 0Relativistic wave equation for spin-1/2 fermions
Noether's Theorem
Classical Field Theory∂μjμ = 0Conserved current from continuous symmetry
Energy-Momentum Tensor
Classical Field TheoryTμν = ∂ℒ/∂(∂μφ) ∂νφ - gμνℒStress-energy tensor from translational symmetry
Canonical Commutation Relations
Canonical Quantization[φ(x), π(y)] = iδ³(x-y)Equal-time commutator for scalar fields
Field Mode Expansion
Canonical Quantizationφ(x) = ∫d³k/(2π)³ 1/√(2ωₖ) [aₖe^(-ikx) + aₖ†e^(ikx)]Expansion in creation/annihilation operators
Ladder Operator Commutators
Canonical Quantization[aₖ, aₚ†] = (2π)³δ³(k-p)Commutation relations in Fock space
Feynman Propagator
Canonical QuantizationDF(x-y) = ⟨0|T{φ(x)φ(y)}|0⟩Time-ordered vacuum expectation value
Propagator (Momentum Space)
Canonical QuantizationDF(p) = i/(p² - m² + iε)Feynman propagator in momentum space
Path Integral Formula
Path IntegralsZ = ∫𝒟φ e^(iS[φ])Partition function as functional integral
Generating Functional
Path IntegralsZ[J] = ∫𝒟φ e^(i∫d⁴x(ℒ + Jφ))Functional that generates correlation functions
Wick's Theorem
Path IntegralsT{φ₁...φₙ} = :φ₁...φₙ: + all contractionsRelates time-ordered to normal-ordered products
S-Matrix
Interacting TheoriesS = T exp(-i∫₋∞^∞ dt H_int(t))Scattering operator in interaction picture
LSZ Reduction Formula
Interacting Theories⟨f|S|i⟩ = (i∫d⁴x e^(ipx)(□+m²))^n ⟨0|T{φ...φ}|0⟩Relates S-matrix to correlation functions
Differential Cross Section
Interacting Theoriesdσ/dΩ = (1/(64π²s))|𝓜|²2→2 scattering cross section
Gauge Transformation
Gauge TheoriesAμ → Aμ - ∂μαU(1) gauge transformation for electromagnetic field
Covariant Derivative
Gauge TheoriesDμ = ∂μ - ieAμGauge-covariant derivative for QED
Yang-Mills Field Strength
Gauge TheoriesFμν^a = ∂μAν^a - ∂νAμ^a + gf^abc Aμ^b Aν^cNon-abelian field strength tensor
QCD Lagrangian
Gauge Theoriesℒ_QCD = -¼Fμν^a F^aμν + ψ̄(iγμDμ - m)ψQuantum Chromodynamics Lagrangian
Beta Function
Renormalizationβ(g) = μ dg/dμRunning of coupling constant with scale
Callan-Symanzik Equation
Renormalization[μ∂/∂μ + β(g)∂/∂g + nγ]G^(n) = 0RG equation for correlation functions
QED Beta Function
Renormalizationβ(α) = α²/(3π)Running of QED coupling (1-loop)
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