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CRC Press eBooks

Second quantization

Bruno Juliá Díaz, Arnau Rios Huguet, Héctor Briongos Merino, Javier Rozalén Sarmiento, Artur Polls Martí

Published Sep 17, 2025• 0 citations
Publisher

Abstract

In non-relativistic quantum mechanics, each particle is characterized by a set of parameters. These are intrinsic to a given particle, and they are the same for a given particle species. Think, for instance, of the mass, the charge, or the spin of an electron. All electrons have the same mass, charge and spin and, from these properties alone, we cannot distinguish two different electrons. Based on these properties, we can divide particles into classes depending on their intrinsic parameters. Particles belonging to the same class are said to be indistinguishable .

DOI
10.1201/9781003191339-2

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In non-relativistic quantum mechanics, each particle is characterized by a set of parameters. These are intrinsic to a given particle, and they are the same for a given particle species. Think, for instance, of the mass, the charge, or the spin of an electron. All electrons have the same mass, charge and spin and, from these properties alone, we cannot distinguish two different electrons. Based on these properties, we can divide particles into classes depending on their intrinsic parameters. Particles belonging to the same class are said to be indistinguishable .
Elementary particles appear in different ways.Particles of the same kind (similar particles) share all state-independent properties like mass, charge, and spin quantum number; there are many similar particles, but they are indistinguishable with respect to a certain class of properties.However, the case might even be stronger: similar particles apparently share all state-dependent properties, like spatial location and spin projection.This threatens the validity of Leibniz's famous Principle of the Identity of Indiscernibles (PII; see Rodriguez-Pereyra (2014) for a historical reconstruction).As it is usually understood, quantum mechanics requires for similar particles to be in permutation invariant states.Take the following state, for example:
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.
In massive free particles the square of Relativistic energy, E’2 = p2c2 + mo2c4/h2 divides into two: relativistic momentum p=m’vg and rest mass mo (where c is the speed of light and h is Planck’s constant}. The rest mass is a function of phase velocity vp; the momentum a function of group velocity vg. In each mass component, mo and (m’-mo) the velocity dependence is exclusive. This consequence of their formulations defines their functions in mass, in internal motion, in intrinsic spin, in superconductivity, in wavefunction collapse, in conservation laws such as momentum, spin, etc.
Details of the many-particle system can also be read from my book titled “Quantum Field Theory: Feynman Path Integrals and Diagrammatic Techniques in Condensed Matter” [ 23 ]. The fundamental difference between classical and quantum mechanics relates the concept of indistinguishability of identical particles where each particle can be equipped with an identifying marker without influencing its behavior in classical mechanics. Besides in classical mechanics, the particle follows its own continuous path in phase space and so it is possible to determine the particular process taking place. So, principally, each particle in a group of identical particles can be identified which is not so in quantum mechanics. It is not possible to mark a particle without influencing its physical state, and worse, if a number of identical particles are brought to the same region in space, their wave functions will rapidly spread out and overlap with one another, thereby soon rendering it impossible to say which particle is where. One of the fundamental assumptions for many-particle systems is therefore that identical particles (i.e., particles characterized by the same quantum numbers such as mass, charge and spin) are, in principle, indistinguishable.
This is a self-contained and hopefully readable account on the method of creation and annihilation operators (also known as the Fock space representation or the "second quantization" formalism) for non-relativistic quantum mechanics of many particles. Assuming knowledge only on conventional quantum mechanics in the wave function formalism, we define the creation and annihilation operators, discuss their properties, and introduce corresponding representations of states and operators of many-particle systems. As the title of the note suggests, we cover most topics treated in sections 6.7 and 6.8 of Feynman's "Statistical Mechanics: A Set of Lectures". As a preliminary, we also carefully discuss the symmetry of wave functions describing indistinguishable particles. We note that all the contents of the present note are completely standard, and the definitions and the derivations presented here have been known to many. Although the style of the present note may be slightly more mathematical than standard physics literatures, we do not try to achieve full mathematical rigor.(Note to experts: In particular we here DERIVE the (anti)commutation relations of the creation and annihilation operators, rather than simply declaring them. In this sense our approach is quite close to that of Feynman's. But we here focus on the action of creation/annihilation operators on general NNN body wave functions, while Feynman makes a heavy use of Slater-determinant-type states from the beginning. We hope that our presentation provides a better perspective on the formalism.)
For a quantum mechanical system of N particles, the Hamiltonian is a sum of the total kinetic energy and the potential energy. In classical physics, it is possible to keep track of individual particles. A system of N identical particles is said to satisfy Bose-Einstein statistics if its wave function is symmetric under the exchange of particles, and these particles are known as bosons. In the nonrelativistic quantum mechanics, this principle must be accepted as an empirical postulate. In the relativistic quantum theory, it can be proven that half-integer-spin particles cannot be bosons and integer-spin particles cannot be fermions. The Pauli exclusion principle plays a critical role in programming physical properties of electrons or more generally fermions.