Second quantization

Post-publication activity

Curator: Carlo Maria Becchi

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 Schrödinger’s wave function representation, and in the relativistic case, field theory, where the number of degrees of freedom is infinite since particles are produced and absorbed. We present the algorithm clarifying its mathematical content in both non-relativistic and field theory frameworks.

History

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.

The Mathematical aspects of the Quantum Mechanics of Identical Particles

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.

It follows that, studying many identical particle systems, one has to select a particular symmetry class of the wave functions; this is often called selection of statistics. For many reasons, among which the famous Spin-Statistics (Streater RF, 1975) relation, the two most important statistics are 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.g. the Fractional Quantum Hall Effect (Stormer HL, Tsui DC, Gossard AC, 1999), has revealed the role of intermediate statistics. Following Dirac we shall begin with Bose-Einstein statistics.

The Bose-Einstein Statistics, the State Space

Mathematically the tensor product of $$N$$ linear spaces is a set of equivalence classes of elements of the Cartesian product of the same spaces, that is the space of $$N$$-tuples of vectors each belonging to a different factor space 1. 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. We denote by $$\psi_\mu\ ,$$ with $$\mu$$ a positive integer, the generic element of the chosen basis; this is always possible since the basis is a countable set. 2.

Having chosen the single-particle basis we define an orthonormal basis of the $$N$$-th tensor power of this space to consist of all (ordered) $$N$$-tuples of elements of this basis. 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. The chosen basis identifies a single element in each equivalence class of the $$N$$-th Cartesian power and hence allows a construction of the tensor product space which, according to the equivalence criterion, does not depend on the basis.

Next we build a basis for the $$N$$-Boson space by symmetrizing the elements of the $$N$$-particle basis with respect to the permutations of particle coordinates and normalizing the results. 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. While the ordered $$N$$-tuples naturally correspond to ordered sets of possibly repeated positive integers, unordered $$N$$-tuples are better identified giving for each state of the single-particle basis, and hence for each positive integer $$\nu\ ,$$ the number $$N_\nu$$ of times the $$N_\nu$$-th state appears in the $$N$$-tuple. $$N_\nu$$ is called the occupation number. Obviously the sequence of occupation numbers satisfies the constraint $$\sum_{\nu=1}^\infty N_\nu=N\ \ .$$ The normalization factor of the symmetrized products of single-particle states is uniquely determined by the occupation numbers. Indeed summing over the permutations of particle coordinates produces $$N!$$ terms which form groups of $$\ \prod_{\nu=1}^\infty N_\nu!\$$ identical terms (Notice that $$0!=1!=1$$). Therefore the normalization factor is $$1/\sqrt{ N!\prod_{\nu=1}^\infty N_\nu!}\ .$$ In conclusion we identify the state space of an assembly of $$N$$ through the occupation number basis whose generic element is identified in Dirac's ket form by$|\{N_\nu\}\rangle\ .$

Identical Particle Operators

Given the reference ortho-normal basis, in order to complete the mathematical apparatus of Quantum Mechanics one has to compute the matrix elements of relevant operators among which are those corresponding to observables. 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 3. 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. For this reason we concentrate on the study of matrix elements of a generic single-particle operators, that is on the operator: $X=\sum_{i=1}^N x_i,$ where $$x_i$$ acts only on the $$i$$-th particle.

Our purpose is to relate the matrix elements of $$X$$ to those of the operator $$x_i$$ between two elements of the chosen basis of the $$i$$-th particle Hilbert space. Then, using the operator algebra, we shall discuss, in section (6), the generalization of the relations we find to the $$p$$-particle operators .

Avoiding heavy combinatorial analysis we deduce the wanted relations for the 1-particle operators using the simplest possible arguments which were used for the first time by Einstein in his study of the Black Body Radiation (Einstein A, 1917). 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 "bra" and "ket" states correspond to occupation number sequences $$N'_\nu$$ and $$N_\nu$$ which are related by$N'_\nu = N_\nu+\delta_{\mu\nu} -\delta_{\rho\nu}\ ,$ for some single-particle states $$\mu\neq\rho\ .$$

Concerning the expectation value in a $$N$$-particle state, since the operator $$X$$ is the sum of single-particle contributions, its expectation value coincides with the sum of the expectation values of its one particle component $$x_i$$ in all the occupied 1-particle states, each expectation value multiplied by the corresponding occupation number. Denoting the matrix element of $$x_i$$ between the single-particle states $$\mu$$ and $$\nu$$ by $$x_{\mu,\nu}\ ,$$ we have the formula: $\tag{1} \langle \{N_\nu\}|X|\{N_\nu\}\rangle=\sum_\nu x_{\nu\nu}N_\nu\ .$

We now consider the non-diagonal matrix elements. In Quantum Mechanics (Dirac PAM, 1958) the squared absolute value of non-diagonal matrix elements of operators, in much the same way as scalar products, have a probabilistic interpretation. More precisely let us assume that the Hamiltonian $$H_0$$ of the Bosonic assembly is diagonal in the chosen occupation number basis. We furthermore consider the operator $$X$$ as an instantaneous perturbation to $$H_0\ ,$$ that is, we consider the dynamics corresponding to the perturbed Hamiltonian $$\ H=H_0+g\delta( t)X\ ,$$ where $$\ g\$$ is a "small" parameter and $$\ \delta( t)\$$ Dirac's delta function. 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 $$\mu\neq\rho\ ,$$ this matrix element is: $\langle \{N_\nu+\delta_{\mu\nu} -\delta_{\rho\nu}\}|X|\{N_\nu\}\rangle\ .$ Since we are considering a single-particle operator and hence independent single-particle dynamics, the mentioned probability is expected to be proportional to the occupation number in the initial state$N_\rho$ if the particle passes from $$\rho$$ to $$\mu$$ and $$N_\mu+1$$ for the reverse transition. 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 $$N_\rho=1$$ and $$N_\mu=0\ ,$$ the matrix element coincides with $$x_{\mu\rho}\ ,$$ we conclude that: $\langle \{N_\nu+\delta_{\mu\nu} -\delta_{\rho\nu}\}|X|\{N_\nu\}\rangle=\sqrt{ N_\rho (N_\mu+1)} x_{\mu\rho}\$4

The Fock space for Bosons

The idea of the second quantization algorithm is based on a particular factorization mechanism for the action of $$p$$-particle operators. 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${\mathcal H}^{(N)}_S$ to the direct sum of Bosonic spaces with particle number ranging from zero to infinity.

The direct sum of linear spaces being defined as the linear space spanned by vectors belonging to one of the spaces, we define the extended Bosonic state space introducing a basis of this space which is the union of the occupation number bases of the $$N$$-Boson spaces for all values of $$N\ .$$ This however is not enough since we have also to introduce a state with occupation number identically zero. This state, that we call vacuum state, is denoted by the ket symbol $$|\{0\}\rangle$$ and is understood not to depend on the particular choice of the single-particle state basis. 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, 1932) $${\mathcal H}_S\ .$$ This space is given by the formula${\mathcal H}_S={\mathcal H}^{(0)}\bigoplus{\mathcal H}^{(1)}\bigoplus_{N=2}^\infty{\mathcal H}^{(N)}_S\ ,$ where $${\mathcal H}^{(1)}$$ is the 1-particle Hilbert space and $${\mathcal H}^{(0)}$$ is the space spanned by the vacuum vector. 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 $$|\{N_\nu\}\rangle$$ with $$\sum_{\nu=1}^\infty N_\nu<\infty$$5. 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, 1954), therefore the vector sum of spaces with different number of particles is physically meaningless, it has a pure algorithmic role. On the contrary in Quantum Field Theory (Itzykson C, Zuber JB, 1980), the field operators do not commute with particle numbers and hence the Fock space gives the natural framework for the construction of a relativistic scattering theory, at least whenever all the particles are massive 6.

Second Quantization of the Bose-Einstein Assemblies

Given the Fock space and considering the matrix elements of 1-particle operators computed above, in order to carry the operator factorization out, we introduce the annihilation operator in the single-particle state $$\mu$$ defining its action on the Fock basis states according to (Landau LD, Lifshitz EM, 1958): $\tag{2} A_\mu |\{N_\nu\}\rangle=\sqrt{N_\mu}|\{N_\nu-\delta_{\mu,\nu}\}\rangle\ .$

One also defines the Hermitian conjugate of the annihilation operator, which is identified with the creation operator, by: $\tag{3} A^{\dagger}_\mu |\{N_\nu \}\rangle =\sqrt{N_\mu+1}|\{N_\nu+\delta_{\mu,\nu}\}\rangle\ .$

Then it is easy to verify that the operator$N_{\nu}\equiv A^{\dagger }_\nu A_\nu$ is diagonal in the Fock basis with eigenvalues coinciding with the $$\nu$$ state occupation number. Furthermore one sees that the operator: $\tag{4} X\equiv\sum_{\mu,\rho=1}^\infty x_{\mu,\rho} A^{\dagger}_\mu A_\rho\ ,$

commutes with the particle total number $$N$$ and hence maps every $${\mathcal H}^{(N)}_S$$ in itself. Its non-vanishing matrix elements between two states in $${\mathcal H}^{(N)}_S$$ coincide with those computed in section (4) 7.

It is apparent from this formula what we mean by Dirac's factorization. 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).

Before discussing the extension of the creation-annihilation factorization to more than one-particle operators we must understand how the above construction depends on the single-particle basis. 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).

A - The first relation states that two arbitrarily chosen annihilation operators commute$\ [A_\mu,A_\rho]=0\$ since they act independently on the occupation numbers of different single-particle states. Of course the same commutation relation holds true for the creation operators.

B - The second relation (Landau LD, Lifshitz EM, 1958), which follows from Eq.(2) and Eq.(3), states that annihilation and creation operators satisfy a set of Canonical Commutation Relations (CCR) (Streater RF, 1975) $\ \ [A_\mu, A^{\dagger}_\nu]=\delta_{\mu,\nu} \ ,$ where the identity operator is understood in the right-hand side.

C - The third relation, which is valid for any annihilation operator, that is for any $$\mu\ ,$$ is$A_\mu |\{0\}\rangle=0\ ,$ where $$0$$ is the null element of the vector space, here in particular of the Fock space. This C relation is called the Fock Property (also known as "vacuum property") since it crucially identifies the Fock representation of the CCR algebra defined by Eq.(2) and Eq.(3).

Indeed, using the above three properties one verifies without much effort that:

• the set of states built by the repeated action of creation operators on the vacuum state, more precisely the set of states:

$| \Psi\{N_\nu \}\rangle \equiv\prod_{\nu=1}^\infty {(A^{\dagger}_\nu)^{N_\nu}\over \sqrt{N_\nu!}} |\{0\}\rangle$ is an orthonormal set.

• the action of a creation operator on $$|\Psi\{N_\nu \}\rangle$$ coincides with that shown in Eq.(3) if one identifies this state with $$|\{N_\nu \}\rangle \ .$$
• an annihilation operator satisfies Eq.(2).

Therefore, identifying the states $$|\Psi\{N_\nu \}\rangle$$ with the elements of the occupation number basis, we prove that properties A, B and C are equivalent to the Fock representation of the CCR algebra.

Now the dependence of the representation on the chosen one-particle basis $$\psi_\nu$$ is directly related to that of properties A, B and C. We analyze this point. Let us then introduce a second basis of the single-particle space and denote its generic element by $$\varphi_\nu$$ with again $$\nu$$ a positive integer. 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. That is$\ \varphi_\mu=\sum_{\nu=1}^{\infty}\langle\phi_\nu|\varphi_\mu\rangle \phi_\nu\equiv \sum_{\nu=1}^{\infty} U_{\mu,\nu}\phi_\nu \ .$ Here $$\ U_{\mu,\nu}\$$ are elements of a possibly infinite, unitary matrix. Assuming, as above, that the vacuum state is independent of the choice of the one particle basis and identifying the single-particle state $$\ \varphi_\mu\$$ with second quantization vector $$\ \hat A^{\dagger}_\mu|\{0\}\rangle\ ,$$ we have the relation$\ \hat A^{\dagger}_\mu|\{0\}\rangle=\sum_{\nu=1}^{\infty} U_{\mu,\nu} A^{\dagger}_\nu|\{0\}\rangle\ ,$ which is automatically satisfied if$\quad\hat A^{\dagger}_\mu =\sum_{\nu=1}^{\infty} U_{\mu,\nu} A^{\dagger}_\nu\ ,$ whose Hermitian conjugate relation is$\quad\hat A_\mu =\sum_{\nu=1}^{\infty} U^*_{\mu,\nu} A_\nu\ .$ From this relation it is immediate to prove that properties A and C hold true for the new annihilation operators $$\hat A\ .$$ Concerning property B one has$\quad [\hat A_\mu, \hat A^{\dagger}_\nu]= \sum_{\rho,\sigma=1}^\infty U^*_{\mu,\rho} U_{\nu,\sigma}\delta_{\rho,\sigma}=\delta_{\mu,\nu}\ ,$ where we have taken into account the unitarity of the matrix $$U\ .$$ Thus we have verified that a change of the reference single-particle basis corresponds to an automorphism of the CCR algebra which leaves the vacuum state invariant.

It is natural at this point to look for a formulation of properties A, B and C without any reference to a one-particle space basis. For this let us introduce a pair of single-particle vector states: $|f\rangle=\sum_{\nu=1}^{\infty} f_\nu |\psi_\nu\rangle\quad {\rm and}\quad |g\rangle=\sum_{\nu=1}^{\infty} g_\nu |\psi_\nu\rangle\ ,$ and the corresponding creation operators: $\tag{5} A^{\dagger}_f=\sum_{\nu=1}^{\infty} f_\nu A^{\dagger}_\nu\ ,\quad{\rm and }\ A^{\dagger}_g=\sum_{\nu=1}^{\infty} g_\nu A^{\dagger}_\nu\ .$

Computing the Hermitian conjugate operators we find the annihilation operators$\quad A_f=\sum_{\nu=1}^{\infty} f^*_\nu A_\nu\ ,\quad{\rm and }\quad A_g=\sum_{\nu=1}^{\infty} g^*_\nu A_\nu\ .$

Given the scalar product $$\langle f|g\rangle$$ one has: $\tag{6} [A_f,A_g]=0\quad\ ,\quad [A_f,A^{\dagger}_g]= \langle f|g\rangle \quad\ ,\quad A_f|\{0\}\rangle= A_g|\{0\}\rangle=0\ .$

These equations give a basis independent form of properties A, B and C and hence characterize the Fock representation of the CCR algebra. 8

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, 1975). This means introducing the vacuum vector state $$|\{0\}\rangle$$ and, for any single-particle vector state $$f\ ,$$ the unitary operators: $\tag{7} W(f)\equiv \exp({A^{\dagger}_f-A_f\over\sqrt{2}})\ ,$

with the following conditions: $\tag{8} W(f)W(g)=\exp({ \langle g|f\rangle-\langle f|g\rangle\over 4})W(f+g)\ ,$

and $\tag{9} \langle \{0\}|W(f) |\{0\}\rangle=\exp(-\langle f|f\rangle/4)\ ,$

which can be deduced from Eq. (6) using the well known Baker–Campbell–Hausdorff formula . 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. (8) and (9) imply Eq. (7) and hence a Fock representation of the $$W$$ operator algebra 9.

It remains to be seen how Dirac's factorization is generalized to $$p$$-particle operators with $$p$$ larger than one. We shall study the case $$p=2$$ from which we shall deduce the general rule. The idea is to avoid heavy combinatorics following the strategy mentioned in section (4) and using the operator algebra. Thus we consider the product of two generic one-particle operators $$X=\sum_{i=1}^Nx_i$$ and $$Y=\sum_{i=1}^Ny_i \ .$$ acting on the $$N$$-Boson assembly. The operator product $$XY$$ is not anymore a single-particle operator, indeed it can be decomposed according to $\tag{10} \ XY=\sum_{i=1}^Nx_iy_i+\sum_{i>j=1}^N(x_iy_j+y_ix_j)\ ,$

where the first term in the right-hand side is the single-particle operator which acts on the $$i$$-th particle space as the product $$x_iy_i\ ,$$ while the second term has the structure of a two-particle operator since it has the form $$\sum_{i>j=1}^No_{i,j}\$$ with $$\ o_{i,j}=o_{j,i}\ .$$ Now we consider the same operator product in the Fock space.

We have$X\equiv\sum_{\mu,\rho=1}^\infty x_{\mu,\rho} A^{\dagger}_\mu A_\rho\equiv ( A^{\dagger}xA)\ ,$ and $$Y\equiv\sum_{\mu,\rho=1}^\infty y_{\mu,\rho} A^{\dagger}_\mu A_\rho\equiv ( A^{\dagger}yA) \ ,$$ therefore the operator product has the form $$XY=(A^{\dagger}xA)(A^{\dagger}yA)$$ which is not normal ordered but can be normal ordered using the commutation rule B $\tag{11} \ XY=\sum_{\mu,\nu=1}^\infty [\sum_{ \rho=1}^\infty x_{\mu,\rho}y_{\rho,\nu}] A^{\dagger}_\mu A_\nu +\sum_{\mu,\rho,\nu,\sigma=1}^\infty x_{\mu,\sigma}y_{\nu,\rho}A^{\dagger}_\mu A^{\dagger}_\nu A_\rho A_\sigma\ .$

In this normal ordered form the first term is clearly a 1-particle operator since it contains one creation and one annihilation operator. Furthermore the matrix of coefficients is the product of the matrices of coefficients appearing in the second quantization form of $$X$$ and $$Y\ .$$ Therefore this first term corresponds to the 1-particle part of the operator product $$XY$$ shown in Eq.(10).

It follows that the second term in the normal ordered product in Eq.(11) corresponds to the 2-particle part of the operator product. Comparing Eq.(10) with Eq.(11) we see that the coefficient of the normal ordered product of the pair of creation $$\ A^{\dagger}_\mu A^{\dagger}_\nu\$$ and the pair of annihilation operators $$\ A_\rho A_\sigma\$$ is related to a matrix element of the 2-particle operator $$\ o_{i,j}\ \ ,$$ which in the present case is $$\ x_iy_j+y_ix_j\ .$$ More precisely the coefficient is half of the matrix element of the operator between two elements of the second Cartesian power of the single-particle space. These are$\langle\psi_\mu\times\psi_\nu|$ and $$|\psi_\sigma\times\psi_\rho\rangle$$ where the first wave function in each pair refers to the $$i$$-th particle and the second to the $$j$$-th.

Now, using the fact that the matrix elements of any 2-particle operator $$o_{i,j}$$ can be approximated by linear combinations of matrix elements of products like $$x_iy_j$$ in much the same way as a function of two variables $$F(x,y)$$ can be expanded in series of products of functions of the single variables $$f_n(x)f_m(y)\ ,$$ 10 we see that the above result can be extended to any 2-particle operator. For example it can be applied to the pair interaction operator corresponding to the two-particle potential $$\ v(r)\ \ .$$ Choosing the plane wave single-particle basis2, the interaction energy operator is: $V={1\over 2 \Omega}\sum_{\vec p,\vec k,\vec q}\tilde v (q)A^{\dagger}_{\vec p+\vec q}A^{\dagger}_{\vec k-\vec q}A _{\vec k}A_{\vec p }\ ,$ where $$\ \tilde v\$$ is the Fourier transform of the potential.

The analysis of the two-particle case also indicates a rule for the second quantized construction of a generic $$p$$-particle operator. A $$p$$-particle operator decomposes into a linear combination of normal ordered products of $$p$$ creation ($$A^{\dagger}_{\mu_1}\cdot\cdot A^{\dagger}_{\mu_p}$$) and $$p$$ annihilation operators ($$A _{\nu_1}\cdot\cdot A _{\nu_p}$$) divided by $$p!\ .$$ The coefficient of a generic term is the matrix element of the first quantized operator between the $$p$$-particle state corresponding to the wave function product $$\psi^*_{\mu_1}(\xi_1)\cdot\cdot \psi^*_{\mu_p}(\xi_p)$$ and that corresponding to the wave function product $$\psi_{\nu_1}(\xi_1)\cdot\cdot \psi_{\nu_p}(\xi_p) \ .$$

We stop here the analysis of the structure of the observables in a non-relativistic second quantized theory and we turn our attention toward the relations between second quantization and the quantization of free fields.

Quantization of the Free Bose-Einstein Fields

We limit our analysis to neutral free fields among which those of major physical interest are the field of elastic deformations of an ideal continuous medium, the electromagnetic field and the relativistic scalar field. The basic common property of these systems is the quadratic and local nature of their energy. That is, their energy is equal to the space integral of an energy density function which is the sum of a kinetic term proportional to a positive definite quadratic form in the first time derivative of the field components and of a positive semi-definite potential part with quadratic terms in the fields and in their gradients. In the electromagnetic case fields are replaced by potential components. Avoiding technical complications we consider a field with a single real component $$\phi (\vec r,t)$$ whose energy is given by$\quad E={1\over 2}\int d\vec r\ [\dot \phi^2 +a\phi^2+b |\vec\nabla\phi|^2]\quad\ ,$ where $$\dot\phi$$ is the field time derivative. Requiring energy positivity one has $$a>0$$ and $$b>0\ .$$ We require strict energy positivity in order to avoid difficulties with the so-called zero modes11, in the present case a space independent solution of the field equations linearly varying in time. Of course with strict energy positivity one also avoids infrared problems in the infinite volume limit. We assume a finite cubic volume $$\Omega=L^3$$ and periodic boundary conditions for the field. This allows a Fourier decomposition of the field in the form$\quad \phi(\vec r, t)={1\over\sqrt{\Omega}}\sum_{\vec k}\varphi_{\vec k}(t)e^{i\vec k\cdot\vec r}\ ,$ where $$\vec k$$ is the wave number. In terms of the Fourier components the energy assumes the canonical form: $\ E={1\over 2}\sum_{\vec k}[|\dot\varphi_{\vec k}|^2+(a +bk^2)| \varphi_{\vec k}|^2] ={1\over 2}\sum_{\vec k}[|\dot\varphi_{\vec k}|^2+\omega^2(k)| \varphi_{\vec k}|^2]\ .$ This appears as the energy of an infinite assembly of independent harmonic oscillators, Normal Modes, whose displacements correspond to the real and imaginary parts of the Fourier amplitude $$\varphi_{\vec k}$$ for all wave numbers 12. In summary the field system appears as a composite system with an infinite number of components.

Quantizing this system we meet two major difficulties.

The first difficulty is essentially related to the Uncertainty Principle (Dirac PAM, 1958) which implies that the energy of a harmonic oscillator must be larger than or equal to its zero-point value, which is half of the product of the Planck constant and the oscillator frequency ($$h\nu/2$$). Of course we are defining zero energy state the classical state of an oscillator at rest. In the case of many independent oscillators the ground state energy is the sum of the zero-point energies of all the oscillators. For the above described field this sum diverges. There are two different solution to this paradox.

• In the case of elastic deformations in a real solid material the zero point energy is not infinite since, in fact, solids are not continuous and the number of normal modes is three times that of the atoms in the solid lattice. This zero point energy gives a significant contribution to the binding energy of the solid crystal.
• In the case of relativistic fields, such as the electromagnetic and the scalar fields, the number of normal modes is, to our present knowledge, infinite. However there is no compelling reason to compare the quantum with the classical situation. If we consider the quantum field oscillations in the vacuum, that is in open space without boundaries, the theory and its ground state must be Lorentz invariant. Lorentz's invariance of the ground state requires that its energy and momentum should vanish. Therefore one should not take into account the normal mode zero-point energies13.

In conclusion the zero-point energy problem is solved by subtracting this contribution from the Hamiltonian. Applying Dirac's procedure, that is introducing for any wave number the momentum variable $$\pi_{\vec k}$$ conjugate to $$\varphi_{\vec k}$$ and the operator $$A_{\vec k} =(\pi_{\vec k}-i\omega( k)\varphi_{\vec k})/\sqrt{2\hbar \omega(k)}$$ and its Hermitian conjugate, one gets$\ H=\sum_{\vec k}\hbar\omega(k) A^{\dagger}_{\vec k} A_{\vec k}\ ,$ with the canonical commutation relations $\ [A_{\vec k} A^{\dagger}_{\vec k'}]=\delta_{\vec k,\vec k'}\ .$ Since our theory is translation invariant we also have the total momentum operator$\quad\vec P=\sum_{\vec k}\hbar\vec k A^{\dagger}_{\vec k} A_{\vec k}\ .$

Now we come to the second difficulty: The free field has an infinite number of components and hence its vector state space should be identified with the infinite tensor product of the Hilbert states associated with all the normal modes, that is with $$\bigotimes_{\vec k}{\mathcal H}_{\vec k}\ .$$ This is not a Hilbert space, as it has not got a countable basis.

In order to make this point clear we identify an orthonormal basis in each $${\mathcal H}_{\vec k}$$ choosing the set of eigenstates of the Hamiltonian $$H_{\vec k}=\hbar \omega(k)A^{\dagger}_{\vec k}A_{\vec k}$$ that we denote by $$\{|N_{\vec k}\rangle\}$$ with the condition$A^{\dagger}_{\vec k} A_{\vec k}|N_{\vec k}\rangle=N_{\vec k}|N_{\vec k}\rangle\ .$ As we have seen, we can build an orthonormal set of states in the infinite tensor product $$\bigotimes_{\vec k}{\mathcal H}_{\vec k}$$ whose elements are identified selecting a vector in each $${\mathcal H}_{\vec k}\ .$$ The elements of this set correspond to sequences of natural numbers, the quantum excitation number of each mode. We denote the vector corresponding to the sequence $$\{N_{\vec k}\}$$ in the set by $$|\{N_{\vec k}\}\rangle\ .$$ The infinite tensor product should be identified with (the completion of) the linear span of the above set which however is uncountable. The linear span of the above set decomposes into the union of an uncountable set of Hilbert spaces and hence it is not clear which is the Hilbert space corresponding to our quantized system.

However this is clearly a problem that we can solve on the basis of a physical argument. Indeed we are only interested in the set of finite energy states. This has a precise meaning since we know the Hamiltonian of our system. The selected set of states corresponds to the linear span of those among the above defined $$|\{N_{\vec k}\}\rangle$$ states for which $$\sum_{\vec k}\omega(k)N_{\vec k}$$ is finite. This is a subset of the linear span of the states for which $$\sum_{\vec k}N_{\vec k}$$ is finite and hence belongs to the Fock space generated by the action of the creation operators $$A^{\dagger}_{\vec q}$$ on the ground state $$|\{0\}\rangle\ ,$$ this state being identified with the vacuum state. Furthermore the Fock space is the completion of the finite energy state space since this space is dense in the Fock space in its Hilbert space norm, see e.g. (Haag R, 1992).

In conclusion we quantize the free field in a Fock space interpreting its quantum states as superposition of states of an assembly of Bosons with momentum $$\hbar\vec k$$ and energy $$\hbar\omega (k)\ .$$ The excitation number of the $$\vec k$$-th normal mode is identified with the number of quanta (e. g. photons or phonons) with momentum $$\hbar\vec k\ .$$ The finite energy quantization criterion is also crucial in the Haag-Ruelle (Haag R, 1992) construction of a relativistic quantum theory of scattering.

Notice that in free field quantization one has observables which do not appear in non-relativistic second quantization. Indeed the space averages of the quantum field $$\ \phi_f=\int d\vec r \phi (\vec r)f(\vec r)\ ,$$ defined by smearing the field with suitably regular functions $$f$$ with compact support, are observables. On the contrary the field is a operator valued distribution and hence it is not a proper observable. The smeared field appears as a linear combination of the Fourier components $$\varphi_{\vec k}$$ and hence of the operators $$A_{\vec k}+A^{\dagger} _{-\vec k}$$ which do not commute with the particle number operator $$\sum_{\vec k}A^{\dagger}_{\vec k}A_{\vec k}\ .$$ This field-particle number uncertainty is a general property of quantum field theory for which the $$N$$-particle sectors of the state space are not anymore super-selection sectors and hence, at least in the neutral field case, the whole Fock space acquires a physical meaning losing its purely algorithmic role.

Second Quantization of the Fermi-Dirac Assemblies

Fermions are identical particles with anti-symmetric wave functions. In three space dimensions particles with half-integer spin are Fermions. The construction of the second quantization algorithm for Fermions follows the same line of the Boson case, therefore we shall keep the same notation whenever possible and repeat the same construction putting into evidence the main differences between the two statistics. First of all we choose a basis of single-particle Hilbert space whose elements are denoted by $$\psi_\mu$$ in much the same way as for Bosons. Then we build a basis of the $$N$$-Fermion Hilbert space considering the anti-symmetrized $$N$$-tuples of elements of the single-particle basis. An explicit way of constructing an element of this basis consists:

• first in selecting an $$N$$-tuple of single-particle wave functions, each dependent on the coordinates $$\xi_i$$ of a different particle, let it be $$\psi_{\nu_1}(\xi_1)\cdot\cdot\psi_{\nu_N}(\xi_N)\ ,$$
• then, considering the $$N\times N$$ matrix whose elements are the single-particle wave functions $$\psi_{\nu_i}(\xi_j)$$ where the row is identified by the index $$\nu_i$$ and the column by the particle coordinate $$\xi_j\ ,$$
• finally, computing the determinant of this matrix, called Slater Determinant , divided by the normalization factor $$\sqrt{N!}\ .$$

It is apparent that the result vanishes whenever at least two wave function indices coincide. This is Pauli Exclusion Principle (Dirac PAM, 1958) preventing two identical Fermions from occupying the same state. It is also apparent that the set of normalized Slater determinants is an ortho-normal set of functions of the particle coordinates. Notice that the sign of the determinant depends on how the elements of the single-particle basis are ordered in a sequence. We shall see that physically this sign and in general the choice of the single-particle basis are irrelevant, but one must take the sign carefully into account in the calculations.

Next step of the construction is the evaluation of the operator matrix elements within states of the chosen basis which are labeled by sequences of occupation numbers $$N_\nu$$ ($$N_\nu$$ being either $$0$$ or $$1$$). We continue using for Fermions the same notations as for Bosons. Once again we begin with the single-particle operators whose non-vanishing matrix elements are, either the diagonal ones, or those between "bra" and "ket" states corresponding to occupation number sequences $$N'_\nu$$ and $$N_\nu$$ related by$N'_\nu = N_\nu+\delta_{\mu\nu} -\delta_{\rho\nu}\ ,$ for some single-particle states $$\mu\neq\rho\ .$$

The diagonal matrix element corresponding to the state with occupation numbers $$\{N_\nu\}$$ is the sum of the expectation values of the single-particle operator in the occupied states, indeed this just follows from the additivity of single-particle contributions. They are given by Eq.(1).

The generic non-diagonal matrix element of a single-particle operator is associated with the transition of a particle from the single-particle state $$\rho$$ to the state $$\mu$$ in much the same way as it is in the Boson case. It is an obvious consequence of the exclusion principle that this matrix element vanishes unless $$N_{\rho}=1$$ and $$N_{\mu}=0\ .$$ Therefore the matrix element must be proportional to $$N_{\rho}(1-N_{\mu})\ .$$ Furthermore, due to the identification of the occupation number state wave functions with Slater determinants whose rows and lines obey a fixed ordering criterion, the transition of one particle from the state $$\rho$$ to the state $$\mu$$ might lead to a determinant with wrongly ordered rows. Indeed, in general, replacing the elements $$\psi_{\rho}(\xi_j)$$ with $$\psi_{\mu}(\xi_j)$$ into the Slater matrix violates the chosen order rule which can be recovered exchanging as many pairs of rows as there are occupied states lying in the ordered sequence between $$\rho$$ and $$\mu\ .$$ Let the number of these occupied states be $$\Sigma_{\rho,\mu}\ ,$$ the state vector obtained moving the $$i$$-th particle from the occupied state $$\rho$$ to $$\mu$$ is equal to the state of the basis with occupation number $$N'_\nu = N_\nu+\delta_{\mu\nu} -\delta_{\rho\nu}\ ,$$ multiplied by $$(-1)^{\Sigma_{\rho,\mu}}\ .$$ Combining the above comments with the condition that the matrix element should coincide with $$\ x_{\mu,\rho}\ ,$$ if $$\ \sum_{\nu}N_{\nu}=1\ ,$$ we find that in general the matrix element is: $\langle \{N_\nu+\delta_{\mu\nu} -\delta_{\rho\nu}\}|X|\{N_\nu\}\rangle=N_\rho (1-N_\mu)(-1)^{\Sigma_{\rho,\mu}}x_{\mu\rho}\ .$ Next we introduce the Fermionic Fock space and the creation and annihilation operators with the purpose of factorizing the one particle operator $$X$$ into a linear combination of normal ordered products. The Fermionic Fock space is built exactly in the same way as the Bosonic one. In the general situation we have a mixed Bosonic and Fermionic Fock space with a unique vacuum state.

Then we introduce the annihilation operator in the one-particle state $$\mu\ :$$ $\tag{12} A_\mu |\{N_\nu\}\rangle=N_\mu (-1)^{\Sigma_{\mu}}|\{N_\nu-\delta_{\mu,\nu}\}\rangle\ ,$

where $$\Sigma_{\mu}\equiv\sum_{\nu<\mu}N_{\nu}$$ is the number of occupied states preceding the state $$\mu$$ in the chosen order. Since $$A_\mu$$ empties the state $$\mu\ ,$$ its Hermitian conjugate, the creation operator $$A^{\dagger}_\mu\ ,$$ fills it. Thus we have $\tag{13} A^{\dagger}_\mu |\{N_\nu\}\rangle=(1-N_\mu) (-1)^{\Sigma_{\mu}}|\{N_\nu+\delta_{\mu,\nu}\}\rangle\ .$

Now it is easy to verify that in the Fock space the single-particle operator $$X$$ satisfies Eq.(4) with however Bosonic creation and annihilation operators replaced by Fermionic ones.

Next question is how the algebraic properties of the Bosonic creation and annihilation operators change in the Fermionic case. As a matter of fact the most relevant difference is that, while the action of Bosonic operators $$A_{\mu}$$ ($$\ A_{\mu}^{\dagger}\$$) only depends on the single-particle state occupation number $$N_{\mu}$$ and hence the operators corresponding to different single-particle states commute, this is not true for the Fermionic operators. Indeed the sign factors appearing in Eq.(12) and Eq.(13) depend on the occupation numbers $$N_{\nu}$$ for $$\nu<\mu\ .$$ Consequently the order in which two Fermionic creation/annihilation operators corresponding to different single-particle states act is relevant since to the sign factor changes from $$+1$$ to $$-1$$ changing the order. This implies that Fermionic creation/annihilation operators corresponding to different single-particle states anti-commute. If instead we consider products of Fermionic creation/annihilation operators corresponding to the same single-particle state, it is clear that the square creation $$\ A_{\mu}^2\$$ and hence $$\ (A_{\mu}^{\dagger})^2\$$ vanish, since they should change the occupation number by two contrasting with Pauli's principle. It remains to consider the products creation times annihilation operators in the two possible orders. These products, acting on a base state, give back the original state multiplied by a factor which is $$\ N_{\mu}^2\equiv N_{\mu}\$$ for $$\ A_{\mu}^{\dagger}A_{\mu}\$$ and $$\ (1-N_{\mu})^2\equiv 1-N_{\mu}\$$ for $$\ A_{\mu}A_{\mu}^{\dagger}\ .$$ In conclusion we get the following equations: $A_\mu A_\nu +A_\nu A_\mu \equiv\{A_\mu ,A_\nu \}=0\quad {\rm and} \quad A_\mu A^{\dagger}_\nu +A^{\dagger}_\nu A_\mu \equiv\{A_\mu ,A^{\dagger}_\nu \}=\delta_{\mu,\nu}\ ,$ which define the algebra of Canonical Anti-Commutation Relations (CAR) (Streater RF, 1975). Combining these equations with the Fock property, that is, all annihilation operators acting on the vacuum state give the trivial element of the vector state space, one completes the Fermionic version of the conditions identifying the Fock representation in the Bosonic case. In particular a generic occupation number state being defined by $|\{N_\nu\}\rangle\equiv\prod^{(ord)}_{\nu}(A^{\dagger}_\nu)^{N_{\nu}}|\{0\}\rangle\ ,$ one recovers Eq's.(12) and (13). Notice the order prescription in the product.

It remains the question of the dependence of the algebra on the chosen ordered single-particle state basis. This question has exactly the same answer of the Bosonic case. That is: the Fock representation is identified, up to unitary equivalence, by the set of anti-commutation relations strictly analogous to the commutation relations given in Eq.(6). Furthermore there is no need of a Weyl form of the Fermionic algebra since in the Fermi-Dirac case creation and annihilation are bounded operators.

In Quantum Field Theory Fermions appear associated with the normal modes of theories whose energy in not bounded from below, such as the well known Dirac electron theory (Dirac PAM, 1958). In this situation the choice of Fermi-Dirac statistics is needed in order to make the ground state of the theory stable due to Pauli's exclusion principle. Indeed in the ground state of these models all the normal modes under a certain level (the zero-energy Fermi level) are occupied while those above this level are empty. This corresponds to the picture of the Fermi Sea (Itzykson C, Zuber JB, 1980).

There are situations in quantum field theory and in the case of non-relativistic non-isolated systems in which the Hamiltonian is highly simplified by linear transformations mixing together creation and annihilation operators while preserving CCR (CAR) algebra. These transformations are called Valatin-Bogoliubov transformations ( Bogoliubov NN, Tolmachev VV, Shirkov DV, 1959). Contrary to the automorphisms described in section (6) they do not leave the vacuum invariant. In some cases there is a new vacuum which is a Fock state and hence the transformations are implemented by an unitary operator, in other cases this is not true and the related representations are not unitarily equivalent. If the Hamiltonian is (approximately) diagonalized, that is, reduced to a function of occupation numbers, following Landau (Landau LD, 1957), one interprets the states generated by the new creation operators on the new ground state as Quasi Particle states. The most important applications concern BCS theory of super-conductivity, that of super-fluidity and the description of squeezed states in Quantum Optics (Mandel L, Wolf E, 1995).

Other Statistics

We have noticed in the introduction that, since the first studies on quantum statistics, people discussed the possibility of statistics differing from the two principal choices studied above. In the framework of the second quantization algorithm one has many possible choices among which, following for example the scheme introduced by Green in 1953 (Green HS, 1992), that of changing the algebraic relations involving pairs of creation/annihilation operators (either CCR or CAR) into different relations involving triple or larger products of these operators.

More recently the discovery of the Fractional Quantum Hall Effect (Stormer HL, Tsui DC, Gossard AC, 1999) has attracted the attention of theoreticians on the properties of interacting systems of electrons bound to a surface in a magnetic field orthogonal to the same surface. As shown by Laughlin this system has elementary excitations, Laughlin vortices, which are strictly two-dimensional objects, carrying fractional charge, spin and a quantum of magnetic flux $$h/e\ .$$ These vortices should behave as anyons ( Wilczek F, 1982), i.e. particles with quantum intermediate statistics.

In order to understand this point let us consider the wave function of a pair of vortices in a plane. This is a function of the center of mass position of the system and of the relative distance. Let us assume that the wave function vanishes with the relative distance since the vortices cannot overlap. The system being two-dimensional, both variables are two component vectors that we identify with complex numbers. Let $$Z$$ correspond to the center of mass position and $$z$$ to the relative distance. The wave function has the form $$\Psi(Z,z)$$ and vanishes in the $$z$$-origin. Now suppose that each vortex carries an elementary magnetic flux $$h/e$$ and a fractional charge $$e/n\ ,$$ the wave function must be polydrome with a branch point at $$\ z=0\ \ .$$ Indeed a transformation $$\ z\to\exp(i\phi)z\$$ corresponds to a rotation of one vortex around the other . Due to the Aharonov Bohm effect, this rotation should induce into the wave function a phase factor $$\ \exp(i\phi)\$$ with $$\ \phi=h/e\times e/(n\hbar)=2\pi/n\ \ .$$ If instead we consider an exchange of the vertices, this corresponds to a change of sign $$\ z \to - z\ \ ,$$ and hence to a phase factor $$\ \exp(i\pi/n)\$$ in the wave function.

The corresponding effect in a Fermionic second quantized theory amounts to replacing of the sign factor appearing in Eq.(13) with a phase factor, e.g. replacing $$(-1)^{\Sigma_n}$$ with $$\exp (\pm i\phi \Sigma_n)$$ and the opposite phase in Eq.(12). In the Bosonic case one should introduce the same factors into Eq.'s (2) and (3) . With this choice a particle exchange into the wave function corresponds to a phase factor $$\exp ( i\phi )\ .$$

Notice that the introduction of the above factors in the algebraic rules of second quantization requires a physical order of single-particle states. In fact this order naturally appears since the wave function of a system of Laughlin vortices should depend on the path along which every vortex has been introduced into the system and in particular on the point (rather one should speak of direction) where the path crosses the border. It is clear that the border of the system is topologically equivalent to a circle which, in order to have a unique identification of the crossing directions, should be broken into a segment, thus establishing, up to cyclic permutations, a "physical" order of single-particle states.

Notes

1 ^   The equivalence criterion reduces the multi-linearity of the Cartesian product to the linearity of the tensor product.
2 ^a b   Notice however that it is often not convenient to label the elements of the basis by positive integers. Consider for example a perfect gas of spinless particles in a cubic box $$\Omega=L^3$$ that we want to represent in terms of particle momentum states. This requires translation invariance and hence periodic boundary conditions for the wave functions, that is, the value of the wave function on a face of the volume coincides with that on the corresponding point on the opposite face. In other words we must identify the coordinate manifold with a torus. In this situation it is possible to choose the single-particle basis of plane wave functions $$\exp(i\ k\cdot\vec r)/\sqrt{\Omega}$$ where $$\vec k=2\pi \vec n/L$$ is the wave number vector and $$\vec n$$ has integer Cartesian components. This is, of course, a countable orthonormal basis for the 1-particle state space.
3 ^   For example the kinetic energy of an assembly of $$N$$ non-relativistic identical particles with mass $$m\ :$$ $$T=\sum_{i=1}^Np_i^2/(2m)\ ,$$ is a 1-particle operator in much the same way as the total momentum of the assembly, the particle density and many others. On the contrary the pair interaction potential $$V=\sum_{i>j=1}^Nv(|\vec r_i-\vec r_j|)$$ is a 2-particle operator. A generic $$p$$-particle operator has the form$X^{(p)}=\sum_{i_1>\cdot\cdot>i_p=1}^N x_{i_1,\cdot\cdot,i_p}$ where $$x_{i_1,\cdot\cdot,i_p}$$ acts only on the particles $$i_1,\cdot\cdot,i_p$$ and is invariant under $$p$$-particle permutations.
4 ^   Notice that the $$\sqrt{N_{\mu}+1}$$ factor, when $$\mu$$ is the final state, accounts for the stimulated emission processes which are the origin of optical amplification (Mandel L Wolf E , 1995).
5 ^   Notice that e.g. the state $$|\{1\}\rangle$$ does not belong to the Fock space since it does not belong to any $${\mathcal H}^{(N)}_S\ .$$
6 ^   However it turns out that in the presence of mass-less particles, in particular in Quantum Electrodynamics, one has to deal with states with unbounded total occupation number. This Infra-Red Problem (Itzykson C Zuber JB, 1980) appears when one goes to the infinite volume limit and demands great care in the definition of transition probabilities.
7 ^   Notice that in general Eq.(4) contains a series whose convergence needs a short comment: the creation and annihilation operators are unbounded operators due to the $$\sqrt{N}$$ factor in Eq.(2), therefore one should not consider the uniform operator topology. On the contrary the analysis which has led us to this formula shows that the series should converge after the selection of a matrix element of $$X\ ,$$ that is, in the weak operator topology.
8 ^   In many text-books the possibility of changing the single-particle basis is extended to generalized bases, such as, e.g. in the case of spinless particles, the generalized basis of localized states $$|\vec r\rangle\ .$$ In this way one defines the Wave Function Operator$\ \Phi (\vec r)\equiv \sum_{\mu}\psi_{\mu}(\vec r)A_{\mu}\equiv \sum_{\mu}\langle \vec r|\psi_{\mu}\rangle A_{\mu}\ ,$ which is an annihilation operator valued distribution. Indeed one has the commutation relation$[\Phi (\vec r),\Phi^{\dagger} (\vec r')] =\delta(\vec r-\vec r')$ whose right-hand side is Dirac's distribution. From a different point of view the restriction to smooth test functions of the annihilation operator $$A_f$$ given in Eq.(5) is a linear, annihilation operator valued, functional defining the above mentioned distribution. The formal advantage of introducing the wave function operator is the direct translation of the Schrödinger’s representation operators into the second quantized form. For example the kinetic energy operator is given by$\hbar^2\int d\vec r \Phi^{\dagger} (\vec r)\nabla^2\Phi (\vec r)/(2m)\ ,$ and the particle density operator in the point $$\vec R$$ is$\Phi^{\dagger} (\vec R)\Phi (\vec R)\ .\ :$

9 ^   We limit ourselves to a short comment: in Weyl’s formulation the Fock space is identified with the linear span of the coherent states which are the vector states generated by the action of the unitary operators $$W$$ on the vacuum state. The scalar products between coherent states are computed using Eq.(9).

10 ^   Notice that strictly speaking one should discuss the meaning of "approximating" mentioned above showing that this holds true in the weak operator topology. However in the present case this point is not particularly relevant since, as a matter of fact, our final formula has a purely combinatorial content.
11 ^   Normal modes with zero frequency are called zero modes.
12 ^   Contrary to what appears at first sight there is a single oscillator for each wave number since $$\phi$$ is a real field and hence its Fourier components corresponding to opposite wave numbers are complex conjugate to one-another.
13 ^   However if one introduces boundaries, for example considering the electromagnetic oscillations between two conducting surfaces, one induces a variation of the normal modes and hence of their zero-point energies and this variation should be and is in fact measurable. This is the Casimir Effect (Itzykson C Zuber JB, 1980).

References

• Dirac, P A M (1958). The principles of Quantum Mechanics. Fourth Edition. Claredon Press, Oxford.
• Landau, L D and Lifshitz, E M (1958). Quantum Mechanics. Non-Relativistic Theory. Course of Theoretical Physics Vol.3 Pergamon Press, London.
• Haag, R (1992). Local Quantum Physics: Fields, Particles, Algebras. Springer-Verlag, Berlin. ISBN 3540610499
• Itzykson, C and Zuber, J B (1980). Quantum Field Theory. Dover Publications, Berlin. ISBN 0486445682
• Bogoliubov , N N; Tolmachev , V V and Shirkov, D V (1959). A new method in the theory of superconductivity. Consultants Bureau, New York.
• Mandel, L and Wolf, E (1995). Optical Coherence and Quantum Optics. Cambridge University Press, Cambridge.
• Dirac, P A M (1927). The Quantum Theory of Emission and Absorbtion of Radiation. Proc. Roy. Soc. [A] 114: 243. doi:10.1098/rspa.1927.0039.
• Jordan, P and Wigner, E (1928). Uber Paulisches Äquivalenzverbot. Zeit. f. Phys. 47: 631.
• Einstein, A (1917). Quantum Theorie des Strahlung. Physikalische Zeitscrift XVIII: 121.
• Green, H. S. (1953). A Generalized Method of Field Quantization. Phys. Rev. 90: 270.
• Stormer, H L; Tsui, D C and Gossard, A C (1999). The Fractional Quantum Hall Effect. Rev. Mod. Phys. 71: S298. doi:10.1103/revmodphys.71.s298.
• Fock, V (1932). Konfigurationsraum und zweite Quantelung. Zeit. f. Phys. 75: 622. doi:10.1007/bf01344458.
• Streater, R F (1975). An outline of axiomatic relativistic quantum field theory. Report Prog, Phys. 48: 771. doi:10.1088/0034-4885/38/7/001.
• Bargmann, V (1954). On Unitary Ray Representations of Continuous Groups. Ann. of Math. 59: 1. doi:10.2307/1969831.
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Internal references

• Jean Zinn-Justin and Riccardo Guida (2008) Gauge invariance. Scholarpedia, 3(12):8287.

• A short, however clear presentation of Second Quantization is given in Landau L D Lifshitz E M (1958) Quantum Mechanics. Non-Relativistic Theory. Course of Theoretical Physics Vol.3. Pergamon Press (London)
• see also Umezawa H Matsumoto H Tachiki M (1982) Thermo field dynamics and condensed states North-Holland Pub. Co. (Amsterdam)
• the mathematical aspects of second quantization are discussed in Friedrichs K. O. (1953) Mathematical Aspects of the Quantum Theory of Fields Interscience Publishers Inc (New York)
• and in Bogoliubov N N Logunov A A Todorov I T (1975) Introduction to axiomatic quantum field theory , (Mathematical physics monograph series; 18). Benjamin (Reading)
• and in Streater R F (1975) An outline of axiomatic relativistic quantum field theory Report Prog, Phys. Vol. 48 p.771
• The more important applications to many particle physics can be found in Nozières, P. (1964) Theory of interacting Fermi systems W.A. Benjamin (New York)
• and in Fetter A L. Walecka J D (1971) Quantum theory of many-particle systems | McGraw-Hill (San Francisco)
• about the discovery of anyons see Biedenharn L Lieb E Simon B Wilczek F (1990) The ancestry of the anyon | Physics Today (L AUG 90)