Starting from the earlier notions of stationary action principles, we show how Schwinger's Quantum Action Principle descended from Dirac's formulation, which independently led Feynman to his path-integral formulation of quantum mechanics. The connection between the two is brought out, and applications are discussed. The Keldysh-Schwinger time-cycle method of extracting matrix elements is described. Part II will discuss the variational formulation of quantum electrodynamics and the development of source theory.
Starting from the earlier notions of stationary action principles, we show how Schwinger's Quantum Action Principle descended from Dirac's formulation, which independently led Feynman to his path-integral formulation of quantum mechanics. The connection between the two is brought out, and applications are discussed. The Keldysh-Schwinger time-cycle method of extracting matrix elements is described. Part II will discuss the variational formulation of quantum electrodynamics and the development of source theory.
Abstract Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory of elementary particle physics, they have also profoundly advanced our understanding of non-relativistic many-particle physics. The Schwinger Action Principle is a quantum-mechanical variational principle that closely parallels the Hamilton Principle of Least Action of classical mechanics, generalizing it to include the role of quantum operators as generalized coordinates and momenta. As such, it unifies all aspects of quantum theory, incorporating Hamilton equations of motion for those operators and the Heisenberg equation, as well as producing the canonical equal-time commutation/anticommutation relations. It yields dynamical coupled field equations for the creation and annihilation operators of the interacting many-body system by variational differentiation of the Hamiltonian with respect to the field operators. Also, equations for the development of matrix elements (underlying Green’s functions) are derived using variations with respect to particle and potential “sources” (and coupling strength). Variational calculus, involving impressed potentials, c-number coordinates and fields, also quantum operator coordinates and fields, is discussed in full detail. Attention is given to the introduction of fermion and boson particle sources and their use in variational calculus.
Through a very careful analysis of Dirac's 1932 paper on the Lagrangian in Quantum Mechanics as well as the second and third editions of his classic book {\it The Principles of Quantum Mechanics}, I show that Dirac's contributions to the birth of the path-integral approach to quantum mechanics is not restricted to just his seminal demonstration of how Lagrangians appear naturally in quantum mechanics, but that Dirac should be credited for creating a path-integral which I call {\it Dirac path-integral} which is far more general than Feynman's while possessing all its desirable features. On top of it, the Dirac path-integral is fully compatible with the inevitable quantisation ambiguities, while the Feynman path-integral can never have that full consistency. In particular, I show that the claim by Feynman that for infinitesimal time intervals, what Dirac thought were analogues were actually proportional can not be correct always. I have also shown the conection between Dirac path-integrals and the Schrödinger equation. In particular, it is shown that each choice of Dirac path-integral yields a {\it quantum Hamiltonian} that is generically different from what the Feynman path-integral gives, and that all of them have the same {\it classical analogue}. Dirac's method of demonstrating the least action principle for classical mechanics generalizes in a most straightforward way to all the generalized path-integrals.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
This book covers the entire span of quantum mechanics whose developments have taken place during the early part of the twentieth century up till the present day. We start with the Rutherford-Bohr model of the atom followed by Schrodinger's wave mechanics with its application to the solution of calculating the energy spectrum of a particle in a box, the harmonic oscillator and finally the hydrogen atom. Heisenberg's matrix mechanics and its duality with Schrodinger's wave mechanics, quantum mechanics in the interaction picture. Dirac's relativistic theory of the electron exhibiting the spin of the electron as a relativistic effect when it interacts with an external electromagnetic field. Feynman's path integral approach to non-relativistic quantum mechanics with is a marvellous intuitive interpretation as a sum over paths and how classical mechanics is obtained from its limit as Planck' constant tends to zero, methods for computing the spectra of the Dirac Hamiltonian in a radial potential, quantum field theory as developed by Feynman, Schwinger, Tomonaga and Dyson for describing the interaction between electrons, positrons, and photons via propagators using both the operator theoretic expansions and Feynman's path integral. We also introduce time independent and time dependent perturbation theory in quantum mechanics with applications to quantum gate design for quantum computers forming a major part of the research conducted by the author's research group, Quantum noise introduced into the Schrodinger and Dirac's equation based on the Hudson-Parthasarathy quantum stochastic calculus in Boson Fock space, scattering theory and wave operators with applications to quantum gate design, some aspects of second quantization like the interpretation of Boson Fock space in terms of harmonic oscillator algebras and the BCS theory of superconductivity, Wigner-Mackey-Frobenius theory of induced representations of a group with applications to Wigner's theory of particle classification, Dirac's equation in a gravitational field and Yang-Mills non-Abelian gauge theories with application to the construction of unified quantum field theories and finally, the more recent theory of super-symmetry which is a Boson-Fermion unification theory. We have discussed the statistics of Boson's, Fermions and Maxwell-Boltzmann based on entropy maximization. The book is written in problem-solution format and it would be of use to physicists and engineers interested respectively in developing unified field theories and in the design of quantum gates. Note: T&F does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.
Feynman's path integral formulation arose from his attempt to incorporate the Lagrangian framework into quantum mechanics, offering what he regarded as a more fundamental perspective than the Hamiltonian approach, particularly in the context of quantum electrodynamics. Inspired by an analogy proposed by P. A. M. Dirac, Feynman showed that Schrödinger's equation could be recovered by promoting this analogy to an equality, up to a constant factor. This insight laid the foundation for the path integral formalism. In this article, we first outline the historical development of dynamical frameworks in physics and then trace Feynman's reasoning as he constructed his formulation, beginning with the double-slit experiment and its interpretation. We subsequently derive the Schrödinger equation from the path integral and demonstrate the conservation of probability. These final sections aim to provide students with a clear link between this elegant yet less commonly presented approach and the more standard methods typically taught in undergraduate courses.
The Feynman path integral plays a central role in quantum mechanics, linking classical action to propagators and relating quantum electrodynamics (QED) to Feynman diagrams. However, the path-integral formulations used in non-relativistic quantum mechanics and in QED are neither unified nor directly connected. This suggests the existence of a missing path integral that bridges relativistic action and the Dirac equation at the single-particle level. In this work, we analyze the consistency and completeness of existing path-integral theories and identify a spinor path integral that fills this gap. Starting from a relativistic action written in spinor form, we construct a spacetime path integral whose kernel reproduces the Dirac Hamiltonian. The resulting formulation provides a direct link between the relativistic classical action and the Dirac equation, and it naturally extends the scalar relativistic path integral developed in our earlier work. Beyond establishing this structural connection, the spinor path integral offers a new way to interpret the origin of classical mechanics for the Dirac equation and suggests a spacetime mechanism for spin and quantum nonlocal correlations. These features indicate that the spinor path integral can serve as a unifying framework for existing path-integral approaches and as a starting point for further investigations into the spacetime structure of quantum mechanics.