Diagrammar in classical scalar field theory

Cattaruzza, E.; Gozzi, E.; Neto, Antônio Francisco

doi:10.1016/j.aop.2011.05.009

Cited by 9 publications

(12 citation statements)

References 52 publications

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“…We have seen that the path integral description of the Moyal formalism allows easily to make contact with the path integral description of classical mechanics [22]. It turns out that in this limiting case a supersymmetric extension is very useful and it has been studied thoroughly giving light to many interesting results related to ergodicity, the symplectic geometry of classical mechanics, quantization, and so on [24][25][26][27][28][29][30]. In section 3.1.1 we give a brief overview of the CPI formalism (the reader already familiar with it may skip this section).…”

Section: Super-extended Moyal Formalismmentioning

confidence: 99%

Note on the super-extended Moyal formalism and its BBGKY hierarchy

Pagani

2017

Annals of Physics

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We consider the path integral associated to the Moyal formalism for quantum mechanics extended to contain higher differential forms by means of Grassmann odd fields. After revisiting some properties of the functional integral associated to the (super-extended) Moyal formalism, we give a convenient functional derivation of the BBGKY hierarchy in this framework. In this case the distribution functions depend also on the Grassmann odd fields.Comment: 32 page

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Section: Super-extended Moyal Formalismmentioning

confidence: 99%

Note on the super-extended Moyal formalism and its BBGKY hierarchy

Pagani

2017

Annals of Physics

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“…We will see later how this quantum mechanical treatment of a classical system can be used to deal with the case when it interacts with a quantum mechanical system. The theory has been developed further since 1998 by a number of authors [96,97,98,99,100,101,102,103,104,105,106,107,108,109,110]. They introduce the additional operatorsλ q = −i∂ q andλ p = −i∂ p which can be identified with the operatorsπ and −χ respectively.…”

Section: Classical Mechanicsmentioning

confidence: 99%

The Unfinished Search for Wave-Particle and Classical-Quantum Harmony

Ghose¹

2015

j adv phys

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The main purpose of this paper is to review the progress that has taken place so far in the search for a single unifying principle that harmonizes (i) the wave and particle natures of matter and radiation, both at the quantum and the classical levels, on the one hand and (ii) the classical and quantum theories of matter and radiation on the other hand. In the author's opinion, the Koopman-von Neumann-Sudarshan (KvNS) Hilbert space theory based on complex wave functions underlying particle trajectories in classical phase space, is an important step forward in that direction. To appreciate the similarities and differences between classical and quantum wave functions, it is important first to review the famous paradoxes of quantum theory arising mainly from the dual character of matter and radiation and the mysterious nature of measurements in quantum theory. They have given rise to the suspicion that quantum mechanics is an incomplete theory and to multiple interpretations of quantum mechanics as well as no-go theorems ruling out certain types of hidden variable theories introduced to 'complete' quantum mechanics in some way. It is also important to point out that experimental verifications of the predictions of the double-prism experiment and the observation of average single photon trajectories in weak measurements appear to favour the spacetime picture of particles favoured by Einstein, de Broglie and Bohm over the complementarity approach favoured by Bohr. The KvN theory of classical mechanics provides a clear and beautiful harmony of classical waves and particles. Sudarshan has given an alternative but equivalent formulation that shows that classical mechanics can be regarded as a quantum theory with essentially hidden noncommuting variables. An extension of KvNS theory to classical electrodynamics provides a sound Hilbert space foundation to it and satisfactorily accounts for entanglement and Bell-CHSH-like violations already observed in classical polarization optics. An important new insight that has been obtained through these developments is that entanglement and Bell-like inequality violations are neither unique signatures of quantumness nor of non-locality-they are rather signatures of non-separability. Another new insight is the interpretation of the Wigner function as a KvNS wave function, i.e. a probability amplitude which need not be positive everywhere. This has important implications for simulating certain types of quantum information processes using classical polarization optics. Finally, Sudarshan's proposed solution to the measurement problem using KvNS theory for the measuring apparatus is sketched to show to what extent wave and particles can be harmonized in quantum theory.

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“…For a free particle considered in equations (8,9), V (q) = 0, and the (q, λ p ) representation now exhibits a phase in its KvN wave functions (eqns. (25,26). If one wishes this phase of the wave function to be unobservable, like in the (q, p) representation, one has to invoke a superselection rule, prohibiting superpositions of states to be written.…”

Section: Classical Mechanicsmentioning

confidence: 99%

“…We will show in this paper that the answer lies in the classic works of Koopman [9], von Neumann [10] and later others [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. Sudarshan [26] developed a complete theory of Classical Mechanics (CM) based on Hilbert spaces associated with commuting hermitian operators as observables (hereinafter referred to as the KvNS theory).…”

Section: Introductionmentioning

confidence: 99%

Hilbert space theory of classical electrodynamics

Rajagopal

Ghose

2016

Pramana - J Phys

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Classical electrodynamics is reformulated in terms of wave functions in the classical phase space of electrodynamics, following the Koopman-von Neumann-Sudarshan prescription for classical mechanics on Hilbert spaces sans the superselection rule which prohibits interference effects in classical mechanics. This is accomplished by transforming from a set of commuting observables in one Hilbert space to another set of commuting observables in a larger Hilbert space. This is necessary to clarify the theoretical basis of much recent work on quantum-like features exhibited by classical optics. Furthermore, following Bondar et al (Phys.Rev. A 88, 052108, (2013)), it is pointed out that quantum processes that preserve the positivity or nonpositivity of the Wigner function can be implemented by classical optics. This may be useful in interpreting quantum information processing in terms of classical optics.

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Diagrammar in classical scalar field theory

Abstract: In this paper we work out the Feynman diagrams of a classical scalar field theory

Cited by 9 publications

References 52 publications

Note on the super-extended Moyal formalism and its BBGKY hierarchy

Note on the super-extended Moyal formalism and its BBGKY hierarchy

The Unfinished Search for Wave-Particle and Classical-Quantum Harmony

Hilbert space theory of classical electrodynamics

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