By using a matrix technique, which allows to identify directly the ladder operators, the Penning trap coherent states are derived as eigenstates of the appropriate annihilation operators. These states are compared with the ones obtained through the displacement operator. The associated wave functions and mean values for some relevant operators in these states are also evaluated. It turns out that the Penning trap coherent states minimize the Heisenberg uncertainty relation.

Abstract. A detailed analysis of the BF formulation for general relativity given by Capovilla, Montesinos, Prieto, and Rojas is performed. The action principle of this formulation is written in an equivalent form by doing a transformation of the fields of which the action depends functionally on. The transformed action principle involves two BF terms and the two Lorentz invariants that appear in the original action principle generically. As an application of this formalism, the action principle used by Engle, Pereira, and Rovelli in their spin foam model for gravity is recovered and the coupling of the cosmological constant in such a formulation is obtained.

The action principle of the BF type introduced by Capovilla, Montesinos, Prieto, and Rojas (CMPR) which describes general relativity with Immirzi parameter is modified in order to allow the inclusion of the cosmological constant. The resulting action principle is on the same footing as the original Plebanski action in the sense that the equations of motion coming from the new action principle are equivalent to the Holst action principle plus a cosmological constant without the need of imposing additional restrictions on the fields. We consider this result a relevant step towards the coupling of matter fields to gravity in the framework of the CMPR action principle. The research in quantum gravity led by its two main branches (loop quantum gravity [1] and spin foam models for gravity [2]) has recently motivated the study of the classical descriptions for general relativity and theories related to it, particularly, the formulations of gravity of the BF type. For instance, Cartan's equations in the framework of BF theories are analyzed in Refs. [3,4] while the relationship of general relativity to the Husain-Kuchar model in the framework of BF theory is analyzed in Refs. [5][6][7].As is well know, general relativity was formulated as a constrained BF theory by Plebański in the mid-seventies [8]. The fundamental variables used for the description of the gravitational field are two-form fields, a connection one-form, and some Lagrange multipliers. This framework was extended much later in order to include the coupling of matter fields [9]. These formulations are complex and one has to use reality conditions in order to get real general relativity. There are other formulations for gravity expressed as a constrained BF theory which are real, and were given in Refs. 10, 11 and 12. In particular, the one given in Ref. 12 includes the so-called Immirzi parameter [13][14][15]. Recently, the issue of the introduction of the cosmological constant in the framework of Ref. 12 was studied by Smolin and Speziale [16]. This issue is also the goal of this paper. Our approach is different from the one followed in Ref. 16 and, in our opinion, is close to the original Plebański formulation. So, we are ready to make the introduction of the cosmological constant and, at the end of the paper, we will comment on the relationship between the results of this paper and the ones of Ref. 16.The action principle for pure gravity introduced by Capovilla, Montesinos, Prieto, and Rojas in Ref. 12 (hereafter CMPR) is given bywhere A I J is a Euclidean or Lorentz connection one-form, depending on whether SO(4) or SO(3, 1) is taken as the internal gauge group, and F The variation of the action (1) with respect to the independent fields gives the equations of motionδψ :

Action principles of the BF type for diffeomorphism invariant topological field theories living in n-dimensional spacetime manifolds are presented. Their construction is inspired by Cuesta and Montesinos' recent paper where Cartan's first and second structure equations together with first and second Bianchi identities are treated as the equations of motion for a field theory. In opposition to that paper, the current approach involves also auxiliary fields and holds for arbitrary n-dimensional spacetimes. Dirac's canonical analysis for the actions is detailedly carried out in the generic case and it is shown that these action principles define topological field theories, as mentioned. The current formalism is a generic framework to construct geometric theories with local degrees of freedom by introducing additional constraints on the various fields involved that destroy the topological character of the original theory. The latter idea is implemented in two-dimensional spacetimes where gravity coupled to matter fields is constructed out, which has indeed local excitations. .unam.mx end, four different types of auxiliary fields are introduced. The first one φ I is a (n − 2)-form that will be associated to the basis of vielbeins e I ; the second one φ IJ = −φ JI is also a (n − 2)-form that will be associated to the Lorentz connection ω IJ . The other two auxiliary fields ψ I and ψ IJ = −ψ JI are (n − 3)-forms that will be associated to the "torsion" T I and the "curvature" R IJ , respectively. Note that R I J is not the same as R I J [ω]: R IJ is a set of two-forms while R I J [ω] is the curvature of ω I J (an analog comment applies to T I , see Ref.[3] for more details). In this way, for n-dimensional spacetimes M n with n ≥ 3, the field theories studied in this paper are defined by the action principlewhere ω IJ = −ω JI is a Lorentz (or Euclidean) connection valued in the so(n − 1, 1) or so(n) Lie algebra, e I is a basis of one-forms, T I is a set of n two-forms, R IJ = −R JI is a set of n(n − 1)/2 two-forms. The indices I, J, K, . . . , are raised and lowered with the Minkowski (σ = −1) or Euclidean (σ = +1) metric (η IJ ) = diag(σ, +1, +1, . . . , +1) (see Ref. [4] for the canonical analysis of BF theory with structure group SO(3, 1), Refs. [5,6] for alternative action principles for SO(3, 1) BF theory, and Ref.[7] for the study of its symmetries). The equations of motion that follow from the variation of the action (1) with respect to the independent fields arewhere D is the covariant derivative computed with respect to the connection ω I J . In two-dimensional spacetimes M 2 , on the other hand, only Cartan's first and second structure equations are allowed because there is no room for the first and second Bianchi identities, i.e., the terms involving the ψ's in Eq.(1) are not allowed. Consequently, the natural action principle is given bywhich are totally antisymmetric in the free indices. The constraints (16) behave as constraints (10) do for BF theory, i.e., repeatedly applying the operator D a to Eqs. (16)...

We perform the coupling of the scalar, Maxwell, and Yang-Mills fields as well as the cosmological constant to BF gravity with Immirzi parameter. The proposed action principles employ auxiliary fields in order to keep a polynomial dependence on the B fields. By handling the equations of motion for the B field and for the auxiliary fields, these latter can be expressed in terms of the physical fields and by substituting these expressions into the original action principles we recover the first-order (Holst) and second-order actions for gravity coupled to the physical matter fields. We consider these results a relevant step towards the understanding of the coupling of matter fields to gravity in the theoretical framework of BF theory.Comment: To appear in Phys. Rev D, 9 pages, LaTeX fil