Second Quantization Pdf

[Fritzi Vanderweel]

Secondquantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac,[1] and were later developed, most notably, by Pascual Jordan[2] and Vladimir Fock.[3][4]In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles.[5] The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory.

This exchange symmetry property imposes a constraint on the many-body wave function. Each time a particle is added or removed from the many-body system, the wave function must be properly symmetrized or anti-symmetrized to satisfy the symmetry constraint. In the first quantization formalism, this constraint is guaranteed by representing the wave function as linear combination of permanents (for bosons) or determinants (for fermions) of single-particle states. In the second quantization formalism, the issue of symmetrization is automatically taken care of by the creation and annihilation operators, such that its notation can be much simpler.

In the second quantization language, instead of asking "each particle on which state", one asks "How many particles are there in each state?". Because this description does not refer to the labeling of particles, it contains no redundant information, and hence leads to a precise and simpler description of the quantum many-body state. In this approach, the many-body state is represented in the occupation number basis, and the basis state is labeled by the set of occupation numbers, denoted

Note that besides providing a more efficient language, Fock space allows for a variable number of particles. As a Hilbert space, it is isomorphic to the sum of the n-particle bosonic or fermionic tensor spaces described in the previous section, including a one-dimensional zero-particle space C.

The creation and annihilation operators are introduced to add or remove a particle from the many-body system. These operators lie at the core of the second quantization formalism, bridging the gap between the first- and the second-quantized states. Applying the creation (annihilation) operator to a first-quantized many-body wave function will insert (delete) a single-particle state from the wave function in a symmetrized way depending on the particle statistics. On the other hand, all the second-quantized Fock states can be constructed by applying the creation operators to the vacuum state repeatedly.

The creation and annihilation operators (for bosons) are originally constructed in the context of the quantum harmonic oscillator as the raising and lowering operators, which are then generalized to the field operators in the quantum field theory.[7] They are fundamental to the quantum many-body theory, in the sense that every many-body operator (including the Hamiltonian of the many-body system and all the physical observables) can be expressed in terms of them.

These two equations can be considered as the defining properties of boson creation and annihilation operators in the second-quantization formalism. The complicated symmetrization of the underlying first-quantized wave function is automatically taken care of by the creation and annihilation operators (when acting on the first-quantized wave function), so that the complexity is not revealed on the second-quantized level, and the second-quantization formulae are simple and neat.

These commutation relations can be considered as the algebraic definition of the boson creation and annihilation operators. The fact that the boson many-body wave function is symmetric under particle exchange is also manifested by the commutation of the boson operators.

The raising and lowering operators of the quantum harmonic oscillator also satisfy the same set of commutation relations, implying that the bosons can be interpreted as the energy quanta (phonons) of an oscillator. The position and momentum operators of a Harmonic oscillator (or a collection of Harmonic oscillating modes) are given by Hermitian combinations of phonon creation and annihilation operators,

This idea is generalized in the quantum field theory, which considers each mode of the matter field as an oscillator subject to quantum fluctuations, and the bosons are treated as the excitations (or energy quanta) of the field.

It is particularly instructive to view the results of creation and annihilation operators on states of two (or more) fermions, because they demonstrate the effects of exchange. A few illustrative operations are given in the example below. The complete algebra for creation and annihilation operators on a two-fermion state can be found in Quantum Photonics.[8]

These anti-commutation relations can be considered as the algebraic definition of the fermion creation and annihilation operators. The fact that the fermion many-body wave function is anti-symmetric under particle exchange is also manifested by the anti-commutation of the fermion operators.

where i , j \displaystyle i,j labels any Majorana fermion operators on equal footing (regardless their origin from Re or Im combination of complex fermion operators c α \displaystyle c_\alpha ). The anticommutation relation indicates that Majorana fermion operators generates a Clifford algebra, which can be systematically represented as Pauli operators in the many-body Hilbert space.

The term "second quantization", introduced by Jordan,[11] is a misnomer that has persisted for historical reasons. At the origin of quantum field theory, it was inappositely thought that the Dirac equation described a relativistic wavefunction (hence the obsolete "Dirac sea" interpretation), rather than a classical spinor field which, when quantized (like the scalar field), yielded a fermionic quantum field (vs. a bosonic quantum field).

One is not quantizing "again", as the term "second" might suggest; the field that is being quantized is not a Schrdinger wave function that was produced as the result of quantizing a particle, but is a classical field (such as the electromagnetic field or Dirac spinor field), essentially an assembly of coupled oscillators, that was not previously quantized. One is merely quantizing each oscillator in this assembly, shifting from a semiclassical treatment of the system to a fully quantum-mechanical one.

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The second quantization of scalar and neutrino fields in the Kerr metric is discussed, and an instability to spontaneous emission is found in the scalar and neutrino cases. The dependence of these results on assumptions about the vacuum state is discussed, as is the semiclassical origin of this creation.

Second quantization was introduced by Dirac as an algorithm for the construction of Quantum Mechanics of assemblies of identical particles (Dirac PAM, 1958). Its name, suggesting something beyond quantization, keeps the memory of its invention, aiming at the formulation of a Quantum Theory of Radiation (Dirac PAM, 1927). Indeed Dirac established that "an Einstein-Bose assembly is dynamically equivalent to a set of harmonic oscillators" (at least whenever to each particle of the assembly corresponds a finite set of independent states). Clearly this equivalence is directly related to the idea of photon introduced long before by Einstein in his interpretation of the photo-electric effect and the name refers to field quantization. Indeed the electromagnetic radiation is known to be equivalent to a set of harmonic oscillators whose excited states, after Einstein, should be interpreted as states of assemblies of photons. The reason why a mathematical algorithm suitable for the study of assemblies of identical particles has a key role in field quantization will be clarified in section (7) where we shall see that field variables and the number of associate quanta, e.g. photons, do not commute and hence Quantum Field Theory (Itzykson C, Zuber JB, 1980) has to deal with assemblies of identical particles. However in the present article we shall first discuss, without any reference to field theory, the algorithm built in 1927 by Dirac (Dirac PAM, 1927) for, even non-relativistic, Bosonic particles and extended to Fermions by Jordan and Wigner in 1928 (Jordan P, Wigner E, 1928). In the case of isolated systems of non-relativistic particles the total number of particles is fixed, and hence finite, since it cannot be changed by interactions (Bargmann V, 1954). Thus one considers systems with a finite number of degrees of freedom. However in the cases of interest this number is of the order of Avogadro's number and one aims at avoiding wave functions and operators depending on too many variables and hence at simplifying calculations. Once we have presented the algorithm we shall relate it to field quantization. Further extensions of the algorithm related to the constructions of intermediate statistics, neither Bosonic nor Fermionic, were introduced more recently.

One of the basic principles of Quantum Mechanics (Dirac PAM, 1958) relates compositeness to the tensor product of Hilbert spaces. The state space of an assembly of systems is identified with the tensor product of the state spaces of each system. This implicitly defines the action of operators. In the case of \(N\) identical particles the \(N\)-tensor power of the single-particle state space decomposes into distinguished super-selection sectors(Wick GC Wightman AS Wigner EP, 1952) (Wightman AS, 1995) orthogonal Hilbert subspaces such that no transition is possible between states belonging to different sectors. This is due to the fact that the observables of an assembly of \(N\) identical particles are permutation invariant, symmetric, functions of the single-particle dynamical variables and hence have vanishing matrix elements between states of the assembly belonging to different permutation symmetry classes of tensors. These classes distinguish the different super-selection sectors of the Hilbert space.

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