A Lecture on Quantum Field Theory with Engineering Applications

Abstract

In this lecture, we shall give a brief survey of the developments leading from classical field theory to quantum field theory starting with the Lagrangian approach to classical mechanics, the Lagrangian approach to classical field theory with examples taken from electrodynamics, general relativity and quantum mechanics. We then introduce the second quantization procedure to classical field theory wherein the field can be regarded as a countable ensemble of particles to which the ordinary rules of particle quantum mechanics is applicable and then state why for aesthetic reasons and conceptual understanding, it is preferable to adopt the operator field-theoretic approach for second quantization based on canonical commutation relations (CCR) for Bosons and canonical anticommutation relations (CAR) for Fermions and in the process discuss how one computes the electron and photon propagators required for making calculations regarding scattering and other processes in the interaction between electrons, positrons and photons. We discuss spontaneous symmetry breaking and approximate symmetry breaking based on non-Abelian gauge theories, i.e. non-commutative generalizations of electrodynamics interacting with the Dirac wave function that is invariant under the Abelian U(1) group. We discuss how spontaneous symmetry breaking leads to the generation of massless Goldstone Bosons and how approximate symmetry breaking on the other hand gives masses to massless particles as it happens in the electroweak theory of Weinberg, Salam and Glashow. We then take the audience to the world of supersymmetry wherein by introducing Fermionic anticommuting variables, all the known fields, namely, the scalar field, the Dirac field, the electromagnetic field, the non-Abelian matter and gauge field and the gravitational field along with their superpartners appear in one big superfield. Using this superfield, Lagrangian densities are constructed that possess all the three kinds of symmetry required, namely, symmetry under the Lorentz group, symmetry under gauge transformations and supersymmetry along with diffeomorphism symmetry if the gravitational field is included. How to construct quantum gates from supersymmetric Lagrangians using the path integral approach of Feynman is discussed by including control superpotentials in the supersymmetric Lagrangian. The use of supersymmetric Lagrangians gives additional degrees of freedom owing to the presence of superpartners of matter and gauge fields and moreover, it unifies Bosons and Fermions into a single Lagrangian. If nature follows supersymmetry, then it is natural to design gates using supersymmetric Lagrangians if one desires to construct gates from physical systems. After this, we move on to quantum communication theory wherein we state the fundamental Cq-communication problem in which classical alphabets are encoded into states and detection operators we decode the input alphabet string from the tensor product state output. The maximum rate at which information can be transmitted reliably over such a channel is derived, namely, the Cq-Shannon capacity theorem. This computation is based on the Von Neumann quantum entropy of a mixed quantum state. Applications of the Cq-communication theory to real-life quantum problems are given involving transmission of information via electromagnetic fields generated by current sources with the receiver being an electron bound to the atomic nucleus interacting with the received electromagnetic field causing its state to change after a finite time to a value dependent on the transmitting current sources. Finally, we touch upon the subject of real-time nonlinear stochastic filtering theory with applications to estimating antenna parameters and medium parameters from noisy measurements of the electromagnetic field at a discrete set of spatial points.