On quantum field theory — I: explicit solution of Dyson’s equation in electrodynamics without use of feynman graphs

Caianiello, E. R.

doi:10.1007/bf02781659

Cited by 137 publications

(66 citation statements)

References 3 publications

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“…Bosons are one of the two basic types of particle in the universe; they include photons and the carriers of nuclear forces. It has been known since work by Caianiello [14] in 1953 (if not earlier) that the amplitudes for n-boson processes can be written as the permanents of n × n matrices. Meanwhile, Valiant [65] proved in 1979 that the permanent is #P-complete.…”

Section: Related Workmentioning

confidence: 99%

“…The model that we are going to define involves a quantum system of n identical photons 14 and m modes (intuitively, places that a photon can be in). We will usually be interested in the case where n ≤ m ≤ poly (n), though the model makes sense for arbitrary n and m. 15 Each computational basis state of this system has the form |S = |s 1 , .…”

Section: Physical Definitionmentioning

confidence: 99%

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The computational complexity of linear optics

Aaronson

Архипов

2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

859

1,578

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We give new evidence that quantum computers-moreover, rudimentary quantum computers built entirely out of linear-optical elements-cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions.Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P #P = BPP NP , and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation.Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the Permanent-of-Gaussians Conjecture, which says that it is #P-hard to approximate the permanent of a matrix A of independent N (0, 1) Gaussian entries, with high probability over A; and the Permanent Anti-Concentration Conjecture, which says that |Per (A)| ≥ √ n!/ poly (n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.

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Section: Related Workmentioning

confidence: 99%

Section: Physical Definitionmentioning

confidence: 99%

The computational complexity of linear optics

Aaronson

Архипов

2011

Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing

859

1,578

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“…It has been known since work by Caianiello [15] in 1953 (if not earlier) that the amplitudes for n-boson processes can be written as the permanents of n × n matrices. Meanwhile, Valiant [71] proved in 1979 that the permanent is #P-complete.…”

Section: Related Workmentioning

confidence: 99%

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Aaronson¹,

2013

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“…apparently prevent the convergence of the series. While in the case of bosonic models such factorial is really there, in the case of fermionic systems cancellations between Feynman diagrams (ultimately related to the fermionic anticommutativity properties) have the effect that the real final bound is much better and convergence of the series for W (0) n can be finally achieved [40]. Indeed the fermionic expectation can be written as a determinant…”

Section: Exact Renormalization Group Analysismentioning

confidence: 99%

Luttinger Model and Luttinger Liquids

Mastropietro

2013

Series on Directions in Condensed Matter Physics

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The Luttinger model owes its solvability to a number of peculiar features, like its linear relativistic dispersion relation, which are absent in more realistic fermionic systems. Nevertheless according to the Luttinger liquid conjecture a number of relations between exponents and other physical quantities, which are valid in the Luttinger model, are believed to be true in a wide class of systems, including tight binding or jellium one dimensional fermionic systems. Recently a rigorous proof of several Luttinger liquid relations in non solvable models has been achieved; it is based on exact Renormalization Group and methods coming from Constructive Quantum Field Theory and its main steps will be reviewed below. IntroductionThe crucial observation that the low energy excitations of one-dimensional metals are well described in terms of 1 + 1 massless Dirac fermions dates back to Tomonaga [1] and it was one of the earliest examples of emerging Quantum Field Theory (QFT) description in condensed matter physics; an idea which, later on, found several applications including the recent one to graphene (which is described by Dirac fermions in d = 2 + 1). The QFT description of 1D metals in [1] is based on several unproven assumptions and, in order to separate approximations from exact results, Luttinger [2] proposed a model, nowadays called Luttinger model, incorporating the Tomonaga approximations in it; the quasi-particle excitations are described in terms of two kinds of spinless fermions with linear dispersion relation, and a Dirac sea is introduced filling the infinite states with negative energy. The Luttinger model looks quite similar to the Thirring model [3] proposed a little earlier, as it was noted by Luttinger himself; both models describe massless Dirac fermions in d = 1 + 1 with an interaction quartic in the fermionic fields. There are however important differences: while in the Thirring model the quartic current-current interaction is local, the

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On quantum field theory — I: explicit solution of Dyson’s equation in electrodynamics without use of feynman graphs

Cited by 137 publications

References 3 publications

The computational complexity of linear optics

The computational complexity of linear optics

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Luttinger Model and Luttinger Liquids

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