Two Different Squeeze Transformations

Lorentz boosts are squeeze transformations. While these transformations are similar to those in squeezed states of light, they are fundamentally different from both physical and mathematical points of view. The difference is illustrated in terms of two coupled harmonic oscillators, and in terms of the covariant harmonic oscillator formalism. The word “squeezed state” is relatively new and was developed in quantum optics, and was invented to describe a set of two photon coherent states [1]. However, the geometrical concept of squeeze or squeeze transformations has been with us for many years. As far as the present authors can see, the earliest paper on squeeze transformations was published by Dirac in 1949 [2], in which he showed that Lorentz boosts are squeeze transformations. In this report, we show that Dirac’s Lorentz squeeze is different from the squeeze transformations in the squeezed state of light. The question then is how different they are. In order to answer this question, we shall use a system of two coupled harmonic oscillators. Let us look at a phase-space description of one simple harmonic oscillator. Its orbit in phase space is an ellipse. This ellipse can be canonically transformed into a circle. The ellipse can also be rotated in phase space by canonical transformation. This combined operation is dictated by a threeparameter group Sp(2) or the two-dimensional symplectic group. The group 1 Sp(2) is locally isomorphic to SU(1, 1), O(2, 1), and SL(2, r), and is applicable to many branches of physics. Its most recent application was to single-mode squeezed states of light [1, 3]. Let us next consider a system of two coupled oscillators. For this system, our prejudice is that the system can be decoupled by a coordinate rotation. This is not true, and the diagonalization requires a squeeze transformation in addition to the rotation applicable to two coordinate variables [3, 4]. This is also a transformation of the symplectic group Sp(2). If we combine the Sp(2) symmetry of mode coupling and the Sp(2) symmetry in phase space, the resulting symmetry is that of the (3 + 2)dimensional Lorentz group [5]. Indeed, it has been shown that this is the symmetry of two-mode squeezed states [6, 7]. It is known that the (3 + 2)dimensional Lorentz group is locally isomorphic to Sp(4) which is the group of linear canonical transformations in the four-dimensional phase space for two coupled oscillators. These canonical transformations can be translated into unitary transformations in quantum mechanics [7]. In addition, for the two-mode problem, there is another Sp(2) transformation resulting from the relative size of the two phase spaces. In classical mechanics, there are no restrictions on the area of phase space within the elliptic orbit in phase space of a single harmonic oscillator. In quantum mechanics, however, the minimum phase-space size is dictated by the uncertainty relation. For this reason, we have to adjust the size of phase space before making a transition to quantum mechanics. This adds another Sp(2) symmetry to the coupled oscillator system [8]. However, the transformations of this Sp(2) group are not necessarily canonical, and there does not appear to be a straightforward way to translate this symmetry group into the present formulation of quantum mechanics. We shall return to this problem later in this report. If we combine this additional Sp(2) group with the above-mentioned O(3, 2), the total symmetry of the two-oscillator system becomes that of the group O(3, 3), which is the Lorentz group with three spatial and three time coordinates. This was a rather unexpected result and its mathematical details have been published recently by the present authors [8]. This O(3, 3) group has fifteen parameters and is isomorphic to SL(4, r). It has six Sp(4)-like subgroups and many Sp(2) like subgroups. Let us consider a system of two coupled harmonic oscillators. The La2 grangian for this system is L = 1 2 { m1ẋ 2 1 +m2ẋ 2 2 − A′x21 +B′x22 + C ′x1x2 } , (1) with A′ > 0, B′ > 0, 4A′B′ − C ′2 > 0. (2) Then the traditional wisdom from textbooks on classical mechanics is to diagonalize the system by solving the eigenvalue equation