oneparticlereducible

One-particle reducible, abbreviated as 1PR, is a term used in perturbation theory, quantum field theory, and many-body physics to classify Feynman diagrams or contributions to Green’s functions. A diagram is called one-particle reducible if it can be separated into two disconnected parts by cutting a single internal propagator line. The line corresponds to a single virtual particle that, when removed, leaves two otherwise independent subdiagrams connected only by that propagator.

In contrast, one-particle irreducible (1PI) diagrams cannot be separated by removing a single internal line. The distinction is important because 1PR diagrams can be viewed as iterative insertions of lower-order pieces into propagators or vertices. Consequently, 1PR contributions factorize into a product of simpler amplitudes connected by a propagator, and they often arise from Dyson-like resummations of self-energy or vertex corrections.

In practice, the full Green’s function or S-matrix can be organized into 1PI building blocks, with 1PR diagrams generated by combining these blocks through propagators. This separation underpins the use of effective actions and renormalization, where 1PI diagrams define the primitive, non-redundant parts of the theory, while 1PR diagrams account for repeated insertions and propagator dressing.

Examples of 1PR diagrams include a two-point function where a loop is attached to a propagator line, such that cutting that internal line splits the diagram into two independent pieces. Understanding 1PR versus 1PI helps clarify the structure of perturbative expansions and the ways in which higher-order corrections are assembled.