scholarly journals Four-mode squeezed states: two-field quantum systems and the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$

2022 ◽  
Vol 82 (1) ◽  
Author(s):  
Thomas Colas ◽  
Julien Grain ◽  
Vincent Vennin
Keyword(s):  
Phase Space ◽  
Symplectic Group ◽  
Fock Space ◽  
Power Spectra ◽  
Scalar Fields ◽  
Squeezed States ◽  
Alternative Description ◽  
Pedagogical Analysis ◽  
Single Field ◽  
Linear Interactions

AbstractWe construct the four-mode squeezed states and study their physical properties. These states describe two linearly-coupled quantum scalar fields, which makes them physically relevant in various contexts such as cosmology. They are shown to generalise the usual two-mode squeezed states of single-field systems, with additional transfers of quanta between the fields. To build them in the Fock space, we use the symplectic structure of the phase space. For this reason, we first present a pedagogical analysis of the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$ Sp ( 4 , R ) and its Lie algebra, from which we construct the four-mode squeezed states and discuss their structure. We also study the reduced single-field system obtained by tracing out one of the two fields. This procedure being easier in the phase space, it motivates the use of the Wigner function which we introduce as an alternative description of the state. It allows us to discuss environmental effects in the case of linear interactions. In particular, we find that there is always a range of interaction coupling for which decoherence occurs without substantially affecting the power spectra (hence the observables) of the system.

scholarly journals ANGULAR POWER SPECTRUM IN MODULAR INVARIANT INFLATION MODEL

2007 ◽  
Vol 22 (12) ◽  
pp. 2223-2237 ◽  
Author(s):  
MITSUO J. HAYASHI ◽  
SHIRO HIRAI ◽  
TOMOYUKI TAKAMI ◽  
YUSUKE OKAMEI ◽  
KENJI TAKAGI ◽  
...  
Keyword(s):  
Scalar Potential ◽  
Field Potential ◽  
Power Spectra ◽  
Scalar Fields ◽  
Modular Invariant ◽  
Angular Power Spectrum ◽  
Inflation Model ◽  
Density Perturbations ◽  
Cosmological Data ◽  
Single Field

We propose a scalar potential of inflation, motivated by modular invariant supergravity, and compute the angular power spectra of the adiabatic density perturbations that result from this model. The potential consists of three scalar fields, S, Y and T, together with two free parameters. By fitting the parameters to cosmological data at the fixed point T = 1, we find that the potential behaves like the single-field potential of S, which slowly rolls down along the minimized trajectory in Y. We further show that the inflation predictions corresponding to this potential provide a good fit to the recent three-year WMAP data, e.g. the spectral index ns = 0.951. The TT and TE angular power spectra obtained from our model almost completely coincide with the corresponding results obtained from the ΛCDM model. We conclude that our model is considered to be an adequate theory of inflation that explains the present data, although the theoretical basis of this model should be further explicated.

Remarks on the linking theory of shape dynamics

2018 ◽  
Vol 15 (06) ◽  
pp. 1850098
Author(s):  
P. P. Yu
Keyword(s):  
Phase Space ◽  
Vector Bundle ◽  
Short Note ◽  
Extended Phase Space ◽  
Geometric Structures ◽  
Extended Phase ◽  
Shape Dynamics ◽  
Canonical Gravity ◽  
Alternative Description ◽  
Gauge Fixing

This short note is an attempt to bring out the geometric structures in the linking theory of shape dynamics. Symplectic induction is applied to give a natural construction of the extended phase space used in the linking theory as a trivial vector bundle over the original phase space for canonical gravity. The geometry of the gauge fixing for shape dynamics is analyzed with the assistance of the Lichnerowicz–York equation lifted to the extended phase space. An alternative description is provided to show how the same geometry simply derives from symplectic induction.

scholarly journals Proposal of a Computational Approach for Simulating Thermal Bosonic Fields in Phase Space

Physics ◽  
2019 ◽  
Vol 1 (3) ◽  
pp. 402-411 ◽  
Author(s):  
Alessandro Sergi ◽  
Roberto Grimaudo ◽  
Gabriel Hanna ◽  
Antonino Messina
Keyword(s):  
Phase Space ◽  
Thermal Noise ◽  
Fock Space ◽  
Vacuum State ◽  
Thermal Bath ◽  
Wigner Transform ◽  
Thermal Vacuum ◽  
Thermo Field Dynamics ◽  
Augmented Space ◽  
Thermal Vacuum State

When a quantum field is in contact with a thermal bath, the vacuum state of the field may be generalized to a thermal vacuum state, which takes into account the thermal noise. In thermo field dynamics, this is realized by doubling the dimensionality of the Fock space of the system. Interestingly, the representation of thermal noise by means of an augmented space is also found in a distinctly different approach based on the Wigner transform of both the field operators and density matrix, which we pursue here. Specifically, the thermal noise is introduced by augmenting the classical-like Wigner phase space by means of Nosé–Hoover chain thermostats, which can be readily simulated on a computer. In this paper, we illustrate how this may be achieved and discuss how non-equilibrium quantum thermal distributions of the field modes can be numerically simulated.

scholarly journals κ-DEFORMED STATISTICS AND CLASSICAL FOUR-MOMENTUM ADDITION LAW

2008 ◽  
Vol 23 (09) ◽  
pp. 653-665 ◽  
Author(s):  
MARCIN DASZKIEWICZ ◽  
JERZY LUKIERSKI ◽  
MARIUSZ WORONOWICZ
Keyword(s):  
Minkowski Space ◽  
Fock Space ◽  
Scalar Fields ◽  
Cross Product ◽  
Creation And Annihilation Operators ◽  
Multiplication Rule ◽  
Symmetry Properties ◽  
Deformed Algebra ◽  
Free Scalar

We consider κ-deformed relativistic symmetries described algebraically by modified Majid–Ruegg bi-cross-product basis and investigate the quantization of field oscillators for the κ-deformed free scalar fields on κ-Minkowski space. By modification of standard multiplication rule, we postulate the κ-deformed algebra of bosonic creation and annihilation operators. Our algebra permits one to define the n-particle states with classical addition law for the four-momentum in a way which is not in contradiction with the nonsymmetric quantum four-momentum co-product. We introduce κ-deformed Fock space generated by our κ-deformed oscillators which satisfy the standard algebraic relations with modified κ-multiplication rule. We show that such a κ-deformed bosonic Fock space is endowed with the conventional bosonic symmetry properties. Finally we discuss the role of κ-deformed algebra of oscillators in field-theoretic noncommutative framework.

scholarly journals LOCAL EXPONENTS AND INFINITESIMAL GENERATORS OF CANONICAL TRANSFORMATIONS ON BOSON FOCK SPACES

2004 ◽  
Vol 07 (04) ◽  
pp. 547-571 ◽  
Author(s):  
F. HIROSHIMA ◽  
K. R. ITO
Keyword(s):  
Canonical Transformation ◽  
Symplectic Group ◽  
Unitary Operator ◽  
Fock Space ◽  
Adjoint Operator ◽  
The Self ◽  
Canonical Transformations ◽  
Fock Spaces ◽  
Infinitesimal Generators ◽  
Boson Fock Space

A one-parameter symplectic group {etÂ}t∈ℝ derives proper canonical transformations indexed by t on a Boson–Fock space. It has been known that the unitary operator Ut implementing such a proper canonical transformation gives a projective unitary representation of {etÂ}t∈ℝ on the Boson–Fock space and that Ut can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator Δ(Â) and a local exponent [Formula: see text] with a real-valued function τÂ(·) such that [Formula: see text].

scholarly journals Scalar field cosmology — geometry of dynamics

2014 ◽  
Vol 11 (02) ◽  
pp. 1460012 ◽  
Author(s):  
Marek Szydłowski ◽  
Orest Hrycyna ◽  
Aleksander Stachowski
Keyword(s):  
Phase Space ◽  
Scalar Field ◽  
Dynamical Systems ◽  
Potential Function ◽  
Structural Stability ◽  
Saddle Points ◽  
Scalar Fields ◽  
Fine Tuning ◽  
Scalar Field Cosmology ◽  
Structurally Stable

We study the Scalar Field Cosmology (SFC) using the geometric language of the phase space. We define and study an ensemble of dynamical systems as a Banach space with a Sobolev metric. The metric in the ensemble is used to measure a distance between different models. We point out the advantages of visualization of dynamics in the phase space. It is investigated the genericity of some class of models in the context of fine tuning of the form of the potential function in the ensemble of SFC. We also study the symmetries of dynamical systems of SFC by searching for their exact solutions. In this context, we stressed the importance of scaling solutions. It is demonstrated that scaling solutions in the phase space are represented by unstable separatrices of the saddle points. Only critical point itself located on two-dimensional stable submanifold can be identified as scaling solution. We have also found a class of potentials of the scalar fields forced by the symmetry of differential equation describing the evolution of the Universe. A class of potentials forced by scaling (homology) symmetries was given. We point out the role of the notion of a structural stability in the context of the problem of indetermination of the potential form of the SFC. We characterize also the class of potentials which reproduces the ΛCDM model, which is known to be structurally stable. We show that the structural stability issue can be effectively used is selection of the scalar field potential function. This enables us to characterize a structurally stable and therefore a generic class of SFC models. We have found a nonempty and dense subset of structurally stable models. We show that these models possess symmetry of homology.

G-INFLATION: INFLATION WITH THE MOST GENERAL SECOND-ORDER FIELD EQUATIONS

2012 ◽  
Vol 10 ◽  
pp. 77-84
Author(s):  
TSUTOMU KOBAYASHI ◽  
MASAHIDE YAMAGUCHI ◽  
JUN'ICHI YOKOYAMA
Keyword(s):  
Power Spectra ◽  
Second Order ◽  
Stability Criteria ◽  
Cosmological Perturbations ◽  
Field Equations ◽  
Special Cases ◽  
Single Field ◽  
The Stability

In this talk, we have discussed generalized Galileons as a framework to develop the most general single-field inflation models ever, (Generalized) G-inflation, containing previous examples such as k-inflation, extended inflation, and new Higgs inflation as special cases. We have also investigated the background and perturbation evolution in this model, calculating the most general quadratic actions for tensor and scalar cosmological perturbations to give the stability criteria and the power spectra of primordial fluctuations.

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