Inequivalent representations in the functional integral framework

Blasone, Massimo; Jizba, Petr; Smaldone, L. A.

doi:10.1088/1742-6596/804/1/012006

Cited by 3 publications

(2 citation statements)

References 22 publications

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“…The properly chosen measure for this integration d d−1 N o,i is Ω i . Note that this adjustment of the measure is not unusual, since it is known that canonical transformations, similar to those generated by N µ o,i , can result in a change of the path integral measure [14][15][16].…”

Section: E Path Integralmentioning

confidence: 93%

A hidden constraint on the Hamiltonian formulation of relativistic worldlines

Koch

Muñoz

2019

Eur. Phys. J. C

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Gauge theories with general covariance are particularly reluctant to quantization. We discuss the example of the Hamiltonian formulation of the relativistic point particle that, despite its apparent simplicity, is of crucial importance since a number of point particle systems can be cast into this form on a higher dimensional Rindler background, as recently pointed out by Hojman. It is shown that this system can be equipped with a hidden local, symmetry generating, constraint which on the one hand does not bother the classical evolution and on the other hand simplifies the realization of the path integral quantization. Even though the positive impact of the hidden symmetry is more evident in the Lagrangian version of the theory, it is still present through the suggested Hamiltonian constraint.

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Section: E Path Integralmentioning

confidence: 93%

A hidden constraint on the Hamiltonian formulation of relativistic worldlines

Koch

Muñoz

2019

Eur. Phys. J. C

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“…Beside new results, we also collect and generalize some older ones, which are scattered in an incoherent fashion over a series of articles. Furthermore, we also wish to promote the concept of inequivalent representations in QFT which is not yet sufficiently well known among the path-integral practitioners [30].…”

Section: Introductionmentioning

confidence: 99%

Functional integrals and inequivalent representations in Quantum Field Theory

2017

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We discuss canonical transformations in Quantum Field Theory in the framework of the functional-integral approach. In contrast with ordinary Quantum Mechanics, canonical transformations in Quantum Field Theory are mathematically more subtle due to the existence of unitarily inequivalent representations of canonical commutation relations. When one works with functional integrals, it is not immediately clear how this algebraic feature manifests itself in the formalism. Here we attack this issue by considering the canonical transformations in the context of coherent-state functional integrals. Specifically, in the case of linear canonical transformations, we derive the general functional-integral representations for both transition amplitude and partition function phrased in terms of new canonical variables. By means of this, we show how in the infinite-volume limit the canonical transformations induce a transition from one representation of canonical commutation relations to another one and under what conditions the representations are unitarily inequivalent. We also consider the partition function and derive the energy gap between statistical systems described in two different representations which, among others, allows to establish a connection with continuous phase transitions. We illustrate the inner workings of the outlined mechanism by discussing two prototypical systems: the van Hove model and the Bogoliubov model of weakly interacting Bose gas.

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Generalized generating functional for mixed-representation Green's functions: A quantum-mechanical approach

2017

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When one tries to take into account the non-trivial vacuum structure of Quantum Field Theory, the standard functional-integral tools such as generating functionals or transitional amplitudes, are often quite inadequate for such purposes. Here we propose a generalized generating functional for Green's functions which allows to easily distinguish among a continuous set of vacua that are mutually connected via unitary canonical transformations. In order to keep our discussion as simple as possible, we limit ourselves to Quantum Mechanics where the generating functional of Green's functions is constructed by means of phase-space path integrals. The quantum-mechanical setting allows to accentuate the main logical steps involved without embarking on technical complications such as renormalization or inequivalent representations that should otherwise be addressed in the full-fledged Quantum Field Theory. We illustrate the inner workings of the generating functional obtained by discussing Green's functions among vacua that are mutually connected via translations and dilatations. Salient issues, including connection with Quantum Field Theory, vacuum-to-vacuum transition amplitudes and perturbation expansion in the vacuum parameter are also briefly discussed.

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Inequivalent representations in the functional integral framework

Cited by 3 publications

References 22 publications

A hidden constraint on the Hamiltonian formulation of relativistic worldlines

A hidden constraint on the Hamiltonian formulation of relativistic worldlines

Functional integrals and inequivalent representations in Quantum Field Theory

Generalized generating functional for mixed-representation Green's functions: A quantum-mechanical approach

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