Squeezing Bogoliubov transformations on the infinite mode CCR-algebra

Honegger, Reinhard; Rieckers, Alfred

doi:10.1063/1.531656

Cited by 43 publications

(38 citation statements)

References 12 publications

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“…When β(T ) is Hilbert-Schmidt it follows that the exponential factor in (4.52) exists because each of α −1 (T ), σ(T ) and γ(T ) is bounded. This necessary condition is highlighted here because it is known that the symplectic transformation S(T ) is unitarily implementable in the Fock space representation defined by Ω and J if and only if β(T ) is Hilbert-Schmidt [9,6]. * Thus the path integral, normalized using Q 0 and interpreted using the Fredholm determinant, fails to exist if the symplectic transformation generated by the classical Hamiltonian fails to be unitarily implementable in the Fock space representation.…”

Section: Discussionmentioning

confidence: 99%

“…In the case where the Hamiltonian is time-independent, some sufficient conditions for the Fredholm determinant of K to exist (and hence for the exponential factor to exist as well) can be obtained from the results in [10,6]. For example, a relatively simple sufficient condition is that the quadratic form on V + given by B corresponds (via the scalar product on V + ) to a Hilbert-Schmidt operator:…”

Section: Discussionmentioning

confidence: 99%

“…We note that the coherent state path integral result (4.52) for the transition amplitude has been rigorously obtained within the Fock space formalism for a particular class of normal-ordered, time-independent, quadratic Hamiltonians [6].…”

Section: -Parameter Family Of Symplectic Transformations S(t) On V Cmentioning

confidence: 99%

“…In particular, it is well-known that for field theories there exist inequivalent representations of the canonical commutation relations (see, e.g., [8] and references therein). Closely related to this is the fact that many linear canonical transformations -which may include those defining the time evolution of a linear system -cannot be unitarily implemented in the Fock space quantization of a field theory [9,10,6]. This situation is known to occur in a variety of physical settings, e.g., for quantum fields in curved spacetimes [11], for the polarized Gowdy model in General Relativity [12], and for parametrized free field theories in dimensions greater than two [13].…”

Section: Introductionmentioning

confidence: 99%

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Coherent state path integral for linear systems

Torre

2005

Phys. Rev. D

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We present a computation of the coherent state path integral for a generic linear system using "functional methods" (as opposed to discrete time approaches). The Gaussian phase space path integral is formally given by a determinant built from a first-order differential operator with coherent state boundary conditions. We show how this determinant can be expressed in terms of the symplectic transformation generated by the (in general, time-dependent) quadratic Hamiltonian for the system. We briefly discuss the conditions under which the coherent state path integral for a linear system actually exists. A necessary -but not sufficient -condition for existence of the path integral is that the symplectic transformation generated by the Hamiltonian is (unitarily) implementable on the Fock space for the system.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: -Parameter Family Of Symplectic Transformations S(t) On V Cmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Coherent state path integral for linear systems

Torre

2005

Phys. Rev. D

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“…Generally, a canonical transformation U is implementable quantum mechanically as a unitary operator in the Fock representation determined by a complex structure J if and only if the antilinear part of the transformation, given by U J = 1 2 (U + JU J), is a Hilbert-Schmidt operator [30]. This condition implies that the antilinear coefficients of the considered transformation must be square summable.…”

Section: Uniqueness Of the Fock Quantizationmentioning

confidence: 99%

Uniqueness of the Fock quantization of scalar fields and processes with signature change in cosmology

Gomar

Marugán

2014

Phys. Rev. D

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We study scalar fields subject to an equation of the Klein-Gordon type in nonstationary spacetimes, such as those found in cosmology, assuming that all the relevant spatial dependence is contained in the Laplacian. We show that the field description -with a specific canonical pair-and the Fock representation for the quantization of the field are fixed indeed in a unique way (except for unitary transformations that do not affect the physical predictions) if we adopt the combined criterion of (a) imposing the invariance of the vacuum under the group of spatial symmetries of the field equations and (b) requiring a unitary implementation of the dynamics in the quantum theory. Besides, we provide a spacetime interpretation of the field equations as those corresponding to a scalar field in a cosmological spacetime that is conformally ultrastatic. In addition, in the privileged Fock quantization, we investigate the generalization of the evolution of physical states from the hyperbolic dynamical regime to an elliptic regime. In order to do this, we contemplate the possibility of processes with signature change in the spacetime where the field propagates and discuss the behavior of the background geometry when the change happens, proving that the spacetime metric degenerates. Finally, we argue that this kind of signature change leads naturally to a phenomenon of particle creation, with exponential production.

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Unitary quantization of a scalar charged field and Schwinger effect

Garay

Martín-Caro

Martín-Benito

2020

J. High Energ. Phys.

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Quantum field theory in curved spacetimes suffers in general from an infinite ambiguity in the choice of Fock representation and associated vacuum. In cosmological backgrounds, the requirement of a unitary implementation of the field dynamics in the physical Hilbert space of the theory is a good criterion to ameliorate such ambiguity. Indeed, this criterion, together with a unitary implementation of the symmetries of the equations of motion, leads to a unique equivalence class of Fock representations. In this work, we apply the procedure developed for fields in cosmological settings to analyze the quantization of a scalar field in the presence of an external electromagnetic classical field in a flat background. We find a natural Fock representation that admits a unitary implementation of the quantum field dynamics. It automatically allows to define a particle number density at all times in the evolution with the correct asymptotic behavior, when the electric field vanishes. Moreover we show the unitary equivalence of all the quantizations that fulfill our criteria. Although we perform the field quantization in a specific gauge, we also show the equivalence between the procedures taken in different gauges.

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Squeezing Bogoliubov transformations on the infinite mode CCR-algebra

Cited by 43 publications

References 12 publications

Coherent state path integral for linear systems

Coherent state path integral for linear systems

Uniqueness of the Fock quantization of scalar fields and processes with signature change in cosmology

Unitary quantization of a scalar charged field and Schwinger effect

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