Lagrangian mechanics for fields

An introduction to Lagrangian methods for classical fields in flat spacetime and then in curved spacetime. The Euler-Lagrange equations for Lagrangian densities are obtained, and applied to the wave, Klein-Gordan, Weyl, Dirac, Maxwell and Proca equations. The canonical energy tensor is obtained. Conservation laws and Noether’s theorem are described. An example of the treatment of Interactions is given by presenting the the QED Lagrangian. Finally, covariant Lagrangian methods are described, and the Einstein field eqution is derived from the Einstein-Hilbert action.

Quantum mechanical four-dimensional non-polarizing beamsplitter

Some quantum optics researchers might not realize that classical electromagnetism predicts a $$\mathbf {\pi }$$ π phase shift between S- and P-polarized reflection and might think the reflection coefficients of the transverse modes are independent, or that such a phase shift has no measurable consequences. In this paper, we discuss theoretical grounds to define elements of a 4x4 matrix to represent the beamsplitter, accurately accounting for transverse polarization modes and phase relations between them. We also provide experimental evidence confirming this matrix representation. From a scientific point of view, the paper addresses a non-trivial equivalence between the classical fields Fresnel formalism and the canonical commutation relations of the quantized photonic fields. That the formalism can be readily verified with a simple experiment provides further benefit. The beamsplitter expression derived can be applied in the field of quantum computing.

Theory of Photon Subtraction for Two-Mode Entangled Light Beams

Photon subtraction is useful to produce nonclassical states of light addressed to applications in photonic quantum technologies. After a very accelerated development, this technique makes possible obtaining either single photons or optical cats on demand. However, it lacks theoretical formulation enabling precise predictions for the produced fields. Based on the representation generated by the two-mode SU(2) coherent states, we introduce a model of entangled light beams leading to the subtraction of photons in one of the modes, conditioned to the detection of any photon in the other mode. We show that photon subtraction does not produce nonclassical fields from classical fields. It is also derived a compact expression for the output field from which the calculation of conditional probabilities is straightforward for any input state. Examples include the analysis of squeezed-vacuum and odd-squeezed states. We also show that injecting optical cats into a beam splitter gives rise to entangled states in the Bell representation.

Quantum Mechanical 4-dimensional Non-polarizing Beamsplitter

The effects of a beamsplitter are frequently described mathematically as a matrix acting on a two input ports vector. This might be comprehensive for a scalar field but certainly insufficient in case of photons which are vector fields. In this paper we discuss theoretical grounds to define elements of a 4x4 matrix to more accurately represent the beamsplitter, fully accounting for transverse polarization modes. We also provide experimental evidence confirming this matrix representation. From scientific point of view the paper addresses a non-trivial equivalence between the classical fields Fresnel formalism and the canonical commutation relations of the quantized photonic fields. That the formalism can be readily verified with a simple experiment provides further benefit. The beamsplitter expression derived can be applied in the field of quantum computing.

Berezinskii–Kosterlitz–Thouless phase induced by dissipating quasisolitons

We theoretically study the sound propagation in a two-dimensional weakly interacting uniform Bose gas. Using the classical fields approximation we analyze in detail the properties of density waves generated both in a weak and strong perturbation regimes. While in the former case density excitations can be described in terms of hydrodynamic or collisionless sound, the strong disturbance of the system results in a qualitatively different response. We identify observed structures as quasisolitons and uncover their internal complexity for strong perturbation case. For this regime quasisolitons break into vortex pairs as time progresses, eventually reaching an equilibrium state. We find this state, characterized by only fluctuating in time averaged number of pairs of opposite charge vortices and by appearance of a quasi-long-range order, as the Berezinskii–Kosterlitz–Thouless (BKT) phase.

From management consultancy to management therapy: Diagnostic of managerial disorders and anatomy of management therapy.

In this paper I develop an idea which can provide the ground in order to start the transition from an amatorial/common sense management consultancy practice towards a scientific validated management consultancy practice. Rooted in medicine, psychiatry and psychology, the innovative tool is aiming at moving the management consultancy practice towards a direction already present in the classical fields of science mentioned above. The new more formal, rigorous, systematic and science-based management consultancy practice is called management therapy (MT). The MT is different from management consultancy because it is substantiated in the emerging field of research called Evidence Based Management (EBM). The paper introduces the first two steps in order to favor opening the discussions and the practice of management therapy: (1) developing the analogous/equivalent of Diagnostic and Statistical Manual of Mental Disorders (DSM) in the management field and (2) developing the action/process of MT (the anatomy of the MT).

Differential Forms

In this chapter we present the modern theory of differential forms and see how it applies to the classical fields studied in the previous chapter. We apply the theory to Maxwell fields as well as to Cartan’s formulation of general relativity. A discussion of the generalized Stokes theorem is given.

Classical fields in curved spacetime

This chapter discusses classical fields in an arbitrary Riemann spacetime. General considerations are followed by the formulation of scalar fields with non-minimal coupling. Spontaneous symmetry breaking in curved space is shown to provide the induced gravity action with a cosmological constant. The construction of spinor fields in curved spacetime is based on the notions of group theory from Part I and on the local Lorentz invariance. Massless vector fields (massless vector gauge fields) are described and the interactions between scalar, fermion and gauge fields formulated. A detailed discussion of classical conformal transformations and conformal symmetry for both matter fields and vacuum action is also provided.