Reference
Concise definitions of core QFT concepts, with the distinctions needed to use them correctly.
Core definitions for introductory and graduate QFT; specialized terms are explained in the linked Library treatments.
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The Legendre transform of the connected generating functional with respect to its source. Functional derivatives of this action generate one-particle-irreducible vertices.
Applies to: The stated field variables, source convention, and domain of the Legendre transform.
Full explanation: The 1PI Effective Action and Mean-Field Equations
The change of a renormalized coupling with renormalization scale while the bare theory is held fixed. It is a vector field on the chosen coupling coordinates.
Applies to: A specified renormalization scheme and set of coupling coordinates.
Full explanation: Beta Functions, Running Masses, and Field Anomalous Dimensions
A cumulant obtained by subtracting products of lower-order expectation values from a full correlation function. The logarithm of a normalized source functional generates these cumulants.
Applies to: A source functional and ordering prescription held fixed.
The expectation value of a product of field or observable insertions in a specified state, with any time ordering or contour prescription stated explicitly.
Applies to: Smeared insertions or distributions with a declared extension at coincident points.
A description using the relevant low-energy degrees of freedom and a systematic expansion that retains every contribution required at a stated accuracy.
Applies to: Specified observables, scales, symmetries, power counting, and truncation order.
Full explanation: Effective Field Theory as a Controlled Expansion
An equivalence between field configurations or descriptions representing the same physical configuration. Physical quantities must be insensitive to the redundant description.
Applies to: Gauge transformations treated as redundancies under the chosen boundary conditions.
A consistent transformation of physical states and observables that preserves the dynamics and observable relations of a theory. An internal symmetry acts without moving spacetime points.
Applies to: The symmetry action and its realization in the chosen state must be specified.
An operator with a nonzero vacuum-to-one-particle matrix element for a specified stable particle species. Its correlation functions can reveal that particle's pole.
Applies to: A theory with a stable one-particle sector and a chosen vacuum.
Full explanation: Fields, Observables, and Interpolating Operators
Observables localized in spacelike-separated regions commute. For homogeneous graded-local fields the corresponding graded commutator vanishes.
Applies to: Local relativistic QFT with a declared localization and grading.
Full explanation: Spacelike Compatibility and Local Observables
A two-point kernel whose meaning depends on the specified state, ordering, and boundary conditions. In perturbative vacuum calculations the scalar Feynman propagator is the time-ordered two-point function.
Applies to: The word must be accompanied by a definition: Feynman, retarded, advanced, or another specified kernel.
Full explanation: Scalar Propagators, Ordered Correlators, and Sources
An operator-valued distribution: pairing a field with a suitable test function gives an operator on a specified domain. Its values at sharp spacetime points need not exist as operators.
Applies to: Continuum quantum fields in a chosen representation.
Full explanation: Quantum Fields as Operator-Valued Distributions
A modification or prescription that makes otherwise singular expressions well defined during a calculation, such as a lattice spacing, a momentum cutoff, or dimensional regularization.
Applies to: The regulated theory or perturbative prescription must be specified.
Fixing parameters and operator definitions through stated conditions so that predictions can be expressed in terms of measured or otherwise specified inputs, with regulator dependence controlled at the claimed order.
Applies to: A declared scheme, regulator, observable, and perturbative or nonperturbative construction.
Full explanation: Renormalized Perturbation Theory and Counterterm Rules
A symmetry of the dynamics is not preserved by the selected state or phase. An order parameter can distinguish symmetry-related phases when an appropriate thermodynamic limit exists.
Applies to: The symmetry, state, order parameter, and order of source and volume limits are specified.
A normalized positive assignment of expectation values to the observable algebra. A Hilbert-space vector or density matrix represents a state only after a representation has been chosen.
Applies to: Algebraic and Hilbert-space descriptions of quantum states.
A state with the ground-state or spacetime-invariance and spectrum properties appropriate to the specified system. In Minkowski QFT the standard vacuum is Poincaré invariant and has positive-energy excitations.
Applies to: Vacuum states in relativistic field theory.
