Path Integrals as a Calculational Framework for Many-Body Physics

Luca Salasnich

Chris L. Lin, CARLOS R. ORDÓÑEZ

In this paper we calculate the scale anomaly for a quantum field theoretic 2D-nonrelativistic Bose gas with contact interactions using Fujikawa's method, both in vacuum and in many-body systems. The use of path integrals for these problems is novel and motivated by a recently developed path-integral framework for addressing questions about scaling in these systems. A natural class of regulators is found that produces the correct value of the anomaly traditionally calculated via other methods, e.g., diagrammatically via the $\ensuremath{\beta}$ function.

John W. Negele, Henri Orland

This chapter describes the range of techniques available for the stochastic evaluation of path integrals and how this flexibility may be effectively exploited to incorporate one's understanding of the underlying physics in the method. It provides precise calculations of physical observables which are exact within controllable systematic and stochastic sampling errors. A special problem frequently arising in stochastic solutions of many-body problems is sampling of the inverse of a matrix. The principal intellectual challenge is to exploit the freedom and to utilize physical insight to incorporate as much of the essential physics of the problem as possible into the stochastic algorithm. It is natural to ask whether any of the functional integral representations of the many-body evolution operator using fields are preferable to the particle coordinate path integral for stochastic calculations. One alternative to evaluate the evolution operator for spin systems is to use the Trotter formula for other path integrals.

Horacio Grinberg, Julio Marañón

ming Cheng

Quantum mechanics presents two seemingly distinct theoretical pictures: the unitary evolution of state vectors in the Schrödinger picture, and the summation over all possible paths in Feynman's path integral formulation. This article systematically argues that the equivalence between these two pictures is a rigorous mathematical fact rather than a choice between competing physical mechanisms. We analyze the meaning of this equivalence at three levels—physical, mathematical, and philosophical—and explore its implications for the generation of entanglement, many-body systems, and quantum field theory. The complete mathematical proof of the equivalence, from unitary evolution to the path integral, is provided in the appendix, along with a detailed construction of quantum entanglement from the path integral framework. Building on this analysis, we distill a broader philosophical thesis: every physical theory, no matter how successful, is a representation of the natural world rather than the world itself. Physical theories are like maps to a territory—they can be extraordinarily precise, yet they are never the territory. The Schrödinger equation, the path integral, and Heisenberg's matrix mechanics are three different maps of the same "quantum territory." Asking whether the wave function "really exists" or whether a particle "really travels along all paths" commits a category mistake—conflating mathematical entities internal to a theory with ontological reality beyond it. The history of physics—from Newtonian gravity to general relativity, from thermodynamics to statistical mechanics, from field-based to action-at-a-distance formulations of electrodynamics—repeatedly teaches this lesson: ontological claims are relative to a given theoretical framework and carry no absolute significance beyond it. The principle of "representational freedom" is thus proposed: when multiple mathematically equivalent formulations of a physical theory yield strictly identical predictions for all observables, no experiment can adjudicate among them ontologically. Choosing one formulation as a calculational tool is a pragmatic decision, not an ontological verdict. The ultimate conclusion is that the path integral is simply the integral representation of the time-evolution operator in the position basis; neither picture enjoys ontological primacy. To claim that "the path sum really happens" or that "the state vector really evolves" is to cross the same boundary. True scientific wisdom lies in knowing where the boundary between representation and reality lies—and then, with ontological humility, continuing to explore the deeper structures that lie beyond it.