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Science

When Infinity Does Not Count

Vadim Cheianov

Published Jan 8, 2009• 2 citations
Publisher

Abstract

A transformation of variables overcomes fundamental difficulties in formulating a unified theory of one-dimensional quantum fluids.
DOI
10.1126/science.1168389

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In a previous paper, the master-graph formulation of the quantum-statistical theory of quantum fluids was developed. If this formulation is used to calculate the equilibrium properties of quantum fluids, apparent divergences are encountered in the low-temperature limit. In the present paper, we transform this theory by means of a $\ensuremath{\Lambda}$ transformation to overcome these apparent low-temperature divergences. In this transformation, the terms in the theory which gave rise to the apparent low-temperature divergences and which represent the dominant low-temperature contributions are summed explicitly to obtain well-behaved expressions. In addition, a consistent method is developed to obtain the corrections to the dominant low-temperature contributions. Explicit expressions for the $\ensuremath{\Lambda}$-transformed theory are given for the cases of a Bose fluid above the Bose-Einstein condensation temperature and for a Fermi fluid. Finally, the physical implications of the $\ensuremath{\Lambda}$ transformation are discussed.
Second quantization and its diagrams are field theory's working language: the occupation-number formalism—Dirac's 1927 quantized radiation, Jordan and Wigner's anticommuting fermions, Fock's space—that makes variable particle number natural, and the Feynman diagrams—the space-time pictures of 1949 whose algebra Dyson unified—that make perturbation theory drawable. This article presents a narrative review of the primary literature that built that language, from Dirac's 1927 emission and absorption quantization, Jordan and Wigner's 1928 Pauli equivalence, and Fock's 1932 configuration space, through the QED reconstruction of Tomonaga's 1946 invariant formulation, Schwinger's 1948 magnetic moment, Feynman's 1949 theory of positrons and space-time approach, and Dyson's 1949 unification, Feynman's 1962 lecture volume, Mattuck's 1976 guide to the many-body diagrams, Peskin and Schroeder's 1995 introduction, and Coleman's 2015 many-body synthesis. The synthesis is organized around three themes: the occupation-number settlement, in which variable particle number became formalism—bosonic commutators, fermionic anticommutators, Fock's space; the diagrammatic revolution, in which QED's infinities were organized into pictures with rules—and measured; and the many-body expansion, in which the same diagrams described solids, fluids, and superconductors. It is concluded that second quantization's genius is bookkeeping—particles as excitations—and that the diagrams are the bookkeeping's pictures: perturbation theory made visible, its rules the field's grammar.
We give a direct analytic proof that every total Boolean function computable exactly with at most two quantum queries depends on at most six input variables. This bound is sharp. A previous derivation relied on a numerical classification of Boolean functions on at most four variables; to the best of our knowledge, ours is the first unconditional proof. The main difficulty beyond the one-query case is that query-path vectors need not be mutually orthogonal, so interference prevents the scalar weight-counting argument from applying. We overcome this difficulty by establishing a rigid support-and-branch decomposition: after the coordinates in the first-query support are fixed, the second query induces a family of exact one-query branches. This structural description may be of independent interest.