In this paper, we show the relation between $sp(4,\mathbb{R})$, the Lie algebra of the symplectic group, and the elements of $Sp(4,\mathbb{R})$. We use this result to obtain some special cases of symplectic matrices relevant to the study of squeezed states. In this regard, we provide some applications in quantum mechanics and analyze the squeezed polymer states obtained from the polymer representation of the symplectic group. Remarkably, the polymer's dispersions are the same as those obtained for the squeezed states in the usual representation.
In this paper, we show the relation between sp(4,R), the Lie algebra of the symplectic group, and the elements of Sp(4,R). We use this result to obtain some special cases of symplectic matrices relevant to the study of squeezed states. In this regard, we provide some applications in quantum mechanics and analyze the squeezed polymer states obtained from the polymer representation of the symplectic group. Remarkably, the polymer's dispersions are the same as those obtained for the squeezed states in the usual representation.
Starting from the symplectic Lie groups Sp (2 n , ) of an n -mode system in classical and quantum mechanics we discuss, in particular, Sp (2, )~ SO (2,1) for a single mode and Sp (4, )~ SO (3,2) with some of its important subgroups such as SU (1,1)~ Sp (2, ) and SU (2). We then look at their applications in quantum optics (squeezing, phase states, beamsplitter and polarization). We explicitly construct the root scheme of the Lie algebra sp (4, ) in two different bases as it is an important means for the investigation of the group properties and the construction of the irreducible representations. Apart from the well known realizations of Sp (2, ) by single-mode and two-mode systems, we discuss some other realizations of this group by fractal combinations of boson operators.
In this paper we show the relation between sp(4,R), the Lie algebra of the symplectic group, and the elements of the symplectic group Sp(4,R). We use this relation to provide a classical analog of the squeeze operator S(ζ). This classical squeeze matrix shares some similarities with the correlation matrix V(2) and its amount of squeezing is half of that in the correlation matrix.
Four-mode squeezed states: two-field quantum systems and the symplectic group Sp(4,R)
Abstract We 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 Sp(4,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.
In this paper, we provide the representation of the symplectic group Sp(2n,R) in polymer quantum mechanics. We derive the propagator of the polymer free particle and the polymer harmonic oscillator without considering a polymer scale. The polymer scale is then introduced to reconcile our results with those expressions for the polymer free particle. The propagator for the polymer harmonic oscillator implies nonunitary evolution.
When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation T of a real algebraic Lie group G. This requires defining an inner product on the Hilbert space mathbbH that carries the representation T. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of T, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (mathfrakFmathbbHD, mathbbHD, mathfrakFmathfrakHD) of Bargmann spaces. Coherent state wave functions -- the elements of mathfrakFmathbbHD and of mathfrakFmathbbHD -- are used to define the inner product on mathbbHD in a way that simplifies the calculation of the overlaps. This inner product and the group action Gamma of G on mathfrakFmathbbHD are used to formulate expressions for the matrix elements of T with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for Gamma relies on matrix factorizations in the complexification of G even though the representation T does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and Gamma action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in mathbbH. The scalar and vector coherent state methods will both be applied to Sp(n) and Sp(n,mathbbR).