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The Basic Formalism of Field Theory
Published in A.N. Vasiliev, Patricia A. Millard, Functional Methods in Quantum Field Theory and Statistical Physics, 2019
A.N. Vasiliev, Patricia A. Millard
In going to the quantum theory the coordinates and momenta are realized as linear Hermitian operators q^i and p^i acting in some Hilbert space and satisfying the canonical commutation relations q^sp^m−p^mq^s=iδsm. The quantum analog of a classical observable F(p, q) is the operator F(p^,q^); in particular, the quantum energy operator, the Hamiltonian H, is by definition ℋ(p^,q^). These rules for going from the classical theory to the quantum theory are the well known recipe for canonical quantization.
On fractional uncertainty: a dyadic approach
Published in Applicable Analysis, 2021
Hugo Aimar, Pablo Bolcatto, Ivana Gómez
Let us denote by the wave function describing the non-relativistic state of a quantum system at time t. The evolution in time of the system described by ψ is governed by the Schrödinger equation with H the Hamiltonian operator determined by the system and where h is the Planck constant. The Hamiltonian operator H takes into account the total energy of the system. The canonical quantization of the classical description of the energy in terms of the kinetic and potential energies, , where we substitute the momentum by the vector operator , gives the most classical form of the Schrödinger equation for a particle of mass m, where the kinetic energy is essentially the Laplace operator . When V =0 we say that we have a free system. The Laplacian is the Euler–Lagrange operator corresponding to the energy quadratic form associated to the energy bilinear form In the classical book of Landkof [5], a huge family of energy bilinear and quadratic forms is introduced and the corresponding potential theory is developed. In particular the family (0<s<1) from which can be regarded as a limit case. Precisely, the energy bilinear form on on real valued functions ϕ and η is given by and the associated energy quadratic form The Euler–Lagrange operator associated to is the fractional Laplacian , which at least in the principal value sense, is given by Schrödinger equations with Hamiltonians involving fractional Laplacians have been considered before, see for example [1–4]. We aim to use the energy bilinear forms approach described before, in order to prove some uncertainty relations that sound natural to the context of finite s-energy forms for small s. To precise the above considerations, let us briefly review the uncertainty inequality for the Heisenberg Principle in one space dimension. At this point we have to mention that there is an extensive literature on uncertainty inequalities including Balian–Low's theorem and many other extensions and points of view. We only refer here to the classical survey [6] and to [7]. Let us also refer to [8,9] for related results on fractals and some metric measure spaces.