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Second quantization - Scholarpedia. Second quantization is the basic algorithm for the construction of Quantum Mechanics of assemblies of identical particles. It is an essential algorithm in the non- relativistic systems where the number of particles is fixed, however too large for the use of Schr. We present the algorithm clarifying its mathematical content in both non- relativistic and field theory frameworks. Its name, suggesting something beyond quantization, keeps the memory of its invention, aiming at the formulation of a Quantum Theory of Radiation (Dirac PAM, 1. Indeed Dirac established that . 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.
However in the present article we shall first discuss, without any reference to field theory, the algorithm built in 1. Dirac (Dirac PAM, 1. Bosonic particles and extended to Fermions by Jordan and Wigner in 1. Jordan P, Wigner E, 1. 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, 1. Thus one considers systems with a finite.
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. 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, 1. Wightman AS, 1. 99. 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. For many reasons, among which the famous Spin- Statistics(Streater RF, 1.
Bose- Einstein statistics corresponding to the symmetric elements of the tensor product, and Fermi- Dirac statistics corresponding to anti- symmetric tensors. However recently the study of low dimensional phenomena, such as e. In quantum theory, the linear spaces are Hilbert spaces; that is, they have a scalar product, have a countable basis and are topologically complete. In this case, for simplicity and also for practical purposes we use a less elegant definition of the tensor product based on the choice of an orthonormal basis of this space built of \(N\)- tuples of elements of the orthonormal bases of the factor spaces, one element for each factor space. In the case of an assembly of \(N\) identical particles we have to consider the \(N\)- th tensor power of the single- particle space. Thus we start from the choice of an orthonormal basis of this space. In the wave function representation an \(N\)- tuple is identified with the product of the wave functions corresponding to its elements, each depending on the coordinates of one of the \(N\) particles.
It is fairly obvious that this set of product functions is a countable orthonormal basis for the \(N\)- particle wave function space. In this way we find an orthonormal basis for the \(N\)- Boson states whose elements correspond to unordered \(N\)- tuples of elements of the single- particle basis. Obviously the sequence of occupation numbers satisfies the constraint \(\sum. Indeed summing over the permutations of particle coordinates produces \(N!\) terms which form groups of \(\ \prod.
Therefore the normalization factor is \(1/\sqrt. We have already said that, in the case of assemblies of identical particles, the space of observables is a linear space spanned by the symmetric functions of the single- particle dynamical variables. For practical reasons this space is usually decomposed into subspaces spanned by symmetric functions of the dynamical variables of a fixed number (\(p\)) of particles. These are called \(p\)- particle operators. Operators form an algebra, the product of two operators is an operator defined on a suitable domain of vectors. In principle this property allows the construction of the whole algebra starting from a subset, for example the 1- particle operators. Then, using the operator algebra, we shall discuss, in section (6), the generalization of the relations we find to the \(p\)- particle operators .
Since a 1- particle operator acts on a single- particle at a time, it has non- vanishing matrix elements between two states of the occupation number basis only if they differ, at most, in the state occupied by a single- particle, the other \(N- 1\) particles remaining in the same state. This implies that, either the states coincide and we compute an expectation value, or . In Quantum Mechanics (Dirac PAM, 1. More precisely let us assume that the Hamiltonian \(H. We furthermore consider the operator \(X\) as an instantaneous perturbation to \(H. At order \(\ g^2\ \) the perturbation induces transitions between states with different occupation number and the transition probability is equal to \(\ g^2/\hbar^2\ \) times the squared absolute value of the corresponding matrix element of \(X\ .\).
If \(X\) is a single- particle operator and we consider the 1- particle transition between the states \(\mu\neq\rho\ ,\). Therefore the considered matrix element must be proportional to the square root of the product of both numbers. Since for a single- particle state, when \(N. We call it Dirac's factorization. In the case of 1- particle operators the factorization mechanism acts in two steps. In the first step one particle is extracted (annihilated) from an occupied state thus reducing the total occupation number by one. In the second step a particle is introduced (created) into a 1- particle state recovering the original number of particles.
Both steps are carried out by linear operators which are Hermitian conjugate to each other and are respectively called annihilation and creation operators. However these operators do not act into a \(N\)- Boson space. Therefore, in order to realize the algorithm, one has to extend the state space that we denote by\. This state, that we call vacuum state, is denoted by the ket symbol \(. Thus, adding together the vacuum state, the states of the single- particle basis and those of the \(N\)- Boson states bases for any \(N\ ,\) we define the symmetric Fock space (Fock V, 1. The Fock space is a Hilbert space since it is spanned by a countable basis.
Indeed a basic theorem of set theory asserts that the union of countably many countable sets is countable. The elements of this basis are labeled by all the possible sequences of natural numbers with finite sum, that is, in a formula, by \(. It is important to remember here that, in the framework of non- relativistic quantum mechanics of isolated systems of particles, the total mass of the system and hence the number of constituents is strictly conserved (Bargmann V, 1. On the contrary in Quantum Field Theory (Itzykson C, Zuber JB, 1. Fock space gives the natural framework for the construction of a relativistic scattering theory, at least whenever all the particles are massive 6. Its non- vanishing matrix elements between two states in \(.
The operator \(X\) is written as a linear combination of ordered products of creation and annihilation operators, the creation operators lying at the left- hand side of the annihilation ones. This order is called Normal Order (also known as Wick order). For this it is necessary to deepen the study of the annihilation and creation operators. In particular we have to introduce three important relations (A, B, C) which are easily deduced from and equivalent to the above Eq.(2) and Eq.(3). Of course the same commutation relation holds true for the creation operators. This C relation is called the Fock Property (also known as . We analyze this point.
Let us then introduce a second basis of the single- particle space and denote its generic element by \(\varphi. It is clear that any element of the new basis can be written as a, possibly infinite, linear combination of elements of the old one. Assuming, as above, that the vacuum state is independent of the choice of the one particle basis and identifying the single- particle state \(\ \varphi. For this let us introduce a pair of single- particle vector states. However, for the sake of mathematical rigor, we have to consider the fact that creation and annihilation are unbounded operators and hence their commutation algebra suffers from domain problems. For this reason one should better translate the CCR algebraic rules into the Weyl form (Streater RF, 1. This means introducing the vacuum vector state \(.
The second condition replaces the Fock condition, while the first one corresponds to the commutation relations A and B. It is possible to prove a reconstruction theorem stating that Eqs.