Appendix A Second Quantization The method of second quantization is deeply rooted in relativistic quantum field theory (QFT) which gives a physical meaning of particle indistinguishability and allows for a unification of fields and particles including the effect of interac- tion [72, 227, 228]. However even for non-relativistic systems, second quantiza- tion is very helpful for describing assemblies of identical particles. In particular, it provides mathematical tools that directly account for important symmetries and the (non-)conservation of the system’s particle number. A.1 Symmetry of Many-Body States The quantum dynamics of N identical non-relativistic particles are described by the time-dependent Schrödinger equation (TDSE), ∂ i − Hˆ (N)(t) Ψ (N) = 0, (A.1) ∂t with the Hamiltonian Hˆ (N) being invariant under particle exchange. In the simplest case (neglecting spin degrees of freedom etc.), the wave function is an element of the (N) (N) N-particle Hilbert space H and has the form Ψ = Ψ(r1,...,rN ,t), where the state of each particle i is determined by its position ri relative to some reference. 2 As also the probability density ρ(r1,...,rN ,t)=|Ψ(r1,...,rN ,t)| is invariant under the exchange of any pair of labels (ensuring indistinguishability), the wave function Ψ (N) must obey the symmetry, −iφ Pij Ψ(r1,...,ri,...,rj ,...,rN ,t)= e Ψ(r1,...,rj ,...,ri,...,rN ,t). (A.2) Here, Pij indicates the pairwise permutation operator. In principle, the phase fac- tor e−iφ could be arbitrary. However, we know from experience that at the most only φ = 0 and π are realized1. Thus, we deal with even (odd) permutations, 1 Of course, the operator Pij is Hermitian (and unitary) and thus has real eigenvalues. K. Balzer, M. Bonitz, Nonequilibrium Green’s Functions Approach to 109 Inhomogeneous Systems, Lecture Notes in Physics 867, DOI 10.1007/978-3-642-35082-5, © Springer-Verlag Berlin Heidelberg 2013 110 A Second Quantization PΨ (N) =+Ψ (N) (PΨ (N) =−Ψ (N)). The corresponding particles are called bosons (fermions). A stringent proof of this phase restriction essentially appears in relativistic quantum mechanics and leads to the spin-statistics theorem and, for fermions, to the Pauli exclusion principle. If the total wave function is constructed from a complete set of one-particle states (N) = ····· φi(r) according to Ψ i ...i ai1...iN (t)φi1 (r1) φiN (rN ), the symmetry 1 N ∈ C conditions of Eq. (A.2) directly transfer to the coefficients ai1...iN (t) . Moreover, we can introduce properly (anti-)symmetrized N-particle basis states as tensor prod- ucts. For bosons, the inner product is then a permanent of the matrix {φi(rj )}.For fermions, we obtain Slater determinants, φi (r1)φi (r2) ... φi (rN ) 1 1 1 φi2 (r1)φi2 (r2) ... φi2 (rN ) Φ (r ,...,r ) = . (A.3) i1,...,iN 1 N . . φiN (r1)φiN (r2) ... φiN (rN ) In this basis, any fermionic wave function can be written as, (N) = Ψ bi1...iN (t)Φi1,...,iN (r1,...,rN ), (A.4) i1...iN ∈ C in which the coefficients bi1,...,iN (t) do not need to obey a certain permutation symmetry regarding their indices. With the sum running over all possible Slater determinants, Eq. (A.4) is usually referred to as the configuration interaction (CI) representation of a many-body wave function [36]. On the other hand, if we include only a few determinants, the ansatz is generally called multi-configuration Hartree- Fock (MCHF), e.g., [17]. In the limiting case of only a single Slater determinant, we are back to Eq. (A.3) and the standard (mean-field) Hartree-Fock approach [35]. The central point of the second quantization formalism to be introduced below is that it will automatically keep track of the correct permutation symmetry of the many-body state according to Eq. (A.4). A.2 Occupation Number Representation In the following, we assume that the set of one-particle states {|i=φi(r)} with i = 1, 2, 3,...is complete and orthonormal, i.e., i|j=δij , i|i=1. (A.5) i One possibility of labeling√ the fermionic basis states introduced in Eq. (A.3)isto = !| write Φi1,...,iN (r1,...,rN ) N i1,...,iN , where√ the indices on the r.h.s. form a sequence with i1 <i2 < ···<iN , and the prefactor N! accounts for normalization of |i1,...,iN . As the Slater determinant vanishes when two indices are equal, it is clear that the sequence must be strictly monotonic. For bosons, |i1,...,iN means the respective permanent, and it is i1 ≤ i2 ≤···≤iN . The correct normalization is A.3 Particle Creation and Annihilation in Fock Space 111 √ governed by a factor of n1!n2! ...n∞!, where the integers ni denote the number of particles that occupy the state |i. In both cases, the set of states |i1,...,iN is complete and orthonormal. Another possibility of labeling is to directly resort to occupation numbers (oc- | cupation number representation). Here, any state i1,...,iN is equally described | =|{ } = by the ket vector n1,n2,... n with i ni N, where, for bosons, ni can take any positive integer including zero, and, for fermions, one either has ni = 0or ni = 1. For the inner product of two states, |{n} and |{n }, one then obtains {n}| n = δ . (A.6) ni ,ni i A.3 Particle Creation and Annihilation in Fock Space In the above sections, the particle number N was constant, and we dealt with a wave function involving strictly N coordinates. However, in many physical situations the number of particles can fluctuate, e.g., through induced particle currents or thermal effects. Theoretically, this becomes most natural in the grand canonical ensemble of statistical physics. In quantum statistics, the Fock space H allows for variations in the number of particles [229]. It is defined as the direct sum of all Hilbert spaces with distinct but fixed particle number, ∞ H = H (i) = H (0) ⊕ H (1) ⊕ H (2) ⊕···⊕H (N) ⊕···. (A.7) i=0 An arbitrary state in Fock space then reads, Ψ = Ψ (0) + Ψ (1) + Ψ (2) +···+Ψ (N) +···, (A.8) i.e., it is composed of elements of H (0), H (1), H (2) and so on. H (0) is one- dimensional and consists of the vacuum state |0. Different subspaces of the Fock space with fixed particle number are orthogonal. All bosonic and fermionic states |n1,n2,...=|{n} with arbitrary particle num- ber belong to the Fock space. For this reason, it is useful to define operators which can increase or decrease the number of particles in |{n} by one (producing a dif- ferent state in H ) but let the symmetry of the many-body state unaffected. For fermions, we define such creation (fˆ†) and annihilation operators (fˆ)by, ˆ†| = − s − | + fi n1,n2,...,ni,... ( 1) (1 ni) n1,n2,...,ni 1,... , (A.9) ˆ s fi|n1,n2,...,ni,...=(−1) ni|n1,n2,...,ni − 1,..., = i−1 ˆ | = where s j=1 nj , and fi 0 0 is just a special case of the annihilator action. If the initial state has particle number N, the result of Eq. (A.9) is a properly antisym- 112 A Second Quantization metrized (N ± 1)-particle state2. The built-in antisymmetrization manifests itself in the (equal-time) canonical anticommutation relations3, ˆ ˆ† = fi, fj + δij , (A.10) ˆ ˆ = ˆ† ˆ† = fi, fj + fi , fj + 0, ˆ ˆ ˆ where [ˆa,b]+ =ˆab + baˆ. For bosons, the only difference is that, instead of the anticommutator, we have to take the commutator, [·, ·]−. Some useful relations hold: N N |i ,i ,...,i = fˆ† |0, i ,i ,...,i |=0| fˆ , 1 2 N in 1 2 N in n=1 n=1 ˆ ˆ† ˆ = ˆ ˆ† ˆ† ˆ =− ˆ† fi, fj fk − δij fk, fi , fj fk − δikfj , (A.11) ˆ ˆ† ˆ† ˆ ˆ = ˆ† ˆ ˆ + ˆ† ˆ ˆ fi, fj fk fmfl − δij fk fmfl δikfj flfm, ˆ† ˆ† ˆ† ˆ ˆ =− ˆ† ˆ† ˆ − ˆ† ˆ† ˆ fi , fj fk fmfl − δilfj fk fm δimfk fj fl. Finally, we mention that, if the one-particle states |i change under a unitary transformation, {|i} → {|i}, the creation and annihilation operators transform ac- cording to, ˆ† = | ˆ† ˆ = | ˆ fi i i fi , fi i i fi. (A.12) i i A.4 General Form of Operators The creation and annihilation operators defined in Eq. (A.9) are the basic quanti- ties of the second quantization method. With them, we can reformulate arbitrary operators in quantum mechanics and can redefine how to compute observables and expectation values of Hermitian operators. First, it is easily verified that, ˆ = ˆ† ˆ ρ1,ij fi fj , (A.13) is the one-particle reduced density matrix (1pRDM) operator. Second, the particle H (1) number operator (the identity operator in the Fock subspace ) takes the form ˆ = ˆ† ˆ N i fi fi . In general, an N-particle operator Aˆ(N) can be of S-particle type, i.e., √ 2 ˆ†| = + | + Analogous expressions exist for bosons: bi n1,...,ni ,... ni 1 n1,...,ni 1,... and ˆ √ bi |n1,...,ni ,...= ni |n1,...,ni − 1,.... 3This is a direct consequence of the definitions in (A.9). A.4 General Form of Operators 113 N Aˆ(N) = aˆ(S) ,S≤ N. (A.14) i1...iS i1,...,iS =1 In the second-quantized form, it is rewritten as, ˆ 1 (S) ˆ† ˆ† ˆ ˆ A = i1,...,iS|ˆa i ,...,i f ...f f ...f . (A.15) ! S 1 i1 iS iS i1 S i1...iS ,i1...iS Note that, in contrast to Eq. (A.14) where the sum ranges over the particles, the | | indices i1 to iS in Eq.