Coherent state path integral for linear systems

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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“…Further considerations have been pursued in the literature rather sparsely and in very specific contexts, see e.g., Refs. [16,[24][25][26].…”

Section: Introductionmentioning

confidence: 99%

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

2017

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“…[22]. There it was studied the solution of the coherent-state evolution kernel for quadratic Hamiltonians of the form H = j,k A j k z * j (t)z k (t) + 1 2 B j k z j (t)z k (t) + 1 2 B * j k z * j (t)z * k (t) .…”

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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Inequivalent representations in the functional integral framework

Blasone

Jizba

Smaldone

2017

J. Phys.: Conf. Ser.

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Abstract. An important feature of Quantum Field Theory is the existence of unitarily inequivalent representations of canonical commutation relations. When one works with the functional integral formalism, it is not clear, however, how this feature emerges. By following the seminal work of M. Swanson on canonical transformations in phase-space path integral, we generalize his approach to coherent-state functional integrals which in turn will lead to a simplified formalism which makes the appearance of the inequivalent representations more transparent. IntroductionSince the early days of its formulation, it was realized that Quantum Field Theory (QFT) is not simply the extension of quantum mechanics (QM) to relativistic systems [1]. There is indeed more: QFT deals with systems with an infinite number of degrees of freedom and thus the Stonevon Neumann uniqueness theorem [2], stating the unitary equivalence of the unitary irreducible representations of the canonical commutation relations (CCR), does not apply. Consequently there exist in QFT infinitely many unitarily inequivalent representations of the field algebra or, in other words, for a given dynamics, there is an infinite number of (inequivalent) Hilbert spaces.A related, and often not sufficiently appreciated, feature of QFT is that the interacting (Heisenberg) fields do not have a unique representation in terms of the asymptotic fields, i.e. fields which directly act on the Fock space. In fact, the functional relation between the asymptotic fields and the Heisenberg fields is known as Haag's map and represents the so called weak operatorial relation, i.e. a relation valid only for expectation values over states belonging to the Hilbert space of the asymptotic fields. All this is of course well known and indeed important phenomena like the spontaneous symmetry breaking mechanism [3,4] or the Hawking black hole radiation [5,6] and quantization of dissipative systems [7], are possible in QFT only because of the existence of inequivalent representations.In the last two decades the fundamental role of inequivalent representations has been recognized in the problem of quantization of mixed particles: in such case, a simple canonical transformation (rotation) of fields with different masses has a dramatic effect on the structure of the Hilbert space, leading to the orthogonality of the vacua for fields with definite flavor

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Coherent state path integral for linear systems

Cited by 3 publications

References 21 publications

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

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

Functional integrals and inequivalent representations in Quantum Field Theory

Inequivalent representations in the functional integral framework

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