Second Quantization

SECOND QUANTIZATION LUCA GUIDO MOLINARI In quantum mechanics a particle is objectified by a set of observables, i.e. a set of fundamental operators, such as position, momentum and spin, characterized by certain commutation relations. If they are represented irreducibly (i.e. any operator commuting with them is multiple of the identity), then any observable of the particle is a function of them. The fundamental operators of different particles commute: this translates the concept of a particle being an autonomous entity. However, two or more identical particles are indistinguishable, and the fundamental operators become unphysical. Only observables that are invariant for permutation of particles, i.e. symmetric functions of the fundamental operators, are meaningful. A convenient formalism for the quantum description of identical particles is “sec- ond quantisation”. There are various ways to introduce it. Here, it is a procedure for rewriting physical operators in new basis of operators {ĉ†r , ĉr }∞ r=1 that create and destroy a particle in single-particle states |ri. Their commutation rules take care of the particle statistics. Second-quantized operators are defined in Fock spaces for bosons or fermions, that accomodate any number of particles. The formalism is efficient in describing new quantum phenomena, as decays, absorption and emission processes of particles or excitations such as photons, phonons, etc. The creation and destruction operators with commutation rules (Bose statistics) were introduced by Paul M. A. Dirac (1926, Quantum theory of light radiation and absorption). Pascual Jordan extended the formalism to massive bosons (Jordan and Klein, 1927) and to Fermi statistics, with anti-commutation rules (Jordan and Wigner, 1928). 1. Systems of N identical particles The Hilbert space of N particles is H (N ) = H1 ⊗ · · · ⊗ HN , where Hk is the Hilbert space for the single k-th particle. H (N ) is the closure of the finite linear combinations of factored vectors |u1 , . . . , uN i = |u1 i⊗. . .⊗|uN i, where |uk i belongs to Hk , with the inner product among factored states (1) hv1 , . . . , vN |u1 , . . . , uN i = hv1 |u1 i1 · · · hvN |uN iN where h · | · ik is the inner product in Hk . The tensor product is linear in each term: |u1 , . . . , αui + βvi , . . . , uN i = α|u1 , . . . , ui , . . . , uN i + β|u1 , . . . , vi , . . . , uN i. An operator  on Hk corresponds to an operator Â(k) on H (N ) with action Â(k)|u1 , . . . , uk , . . . uN i = |u1 , . . . , Âuk , . . . , uN i. By construction, operators of dif- ferent particles commute: [Â(k), B̂(k 0 )] = 0. Date: oct.2001, oct. 2019. 1 2 L. G. MOLINARI Example 1.1. A Hilbert space for a spin-less particle is L2 (R3 ). The products f1 (x1 ) · · · fN (xN ) and their linear combinations R form a linear space whose closure is the Hilbert space L2 (R3N ) of functions with dx1 . . . dxN |f (x1 , . . . , xN )|2 < ∞. 1.1. Identical particles. If the N particles are identical, the spaces H1 , ... , HN are copies of the same one-particle Hilbert space H (H (N ) = ⊗N H ). Since the vectors in |u1 , . . . , uN i belong to the same space, it is possible to introduce the permutation operators. If σ is the permutation σ(1, . . . , N ) = (σ1 , . . . , σN ), the corresponding operator on factored states is (2) P̂σ |u1 , . . . uN i = |uσ1 , . . . , uσN i They extend by linearity to unitary operators on the whole space H (N ), and form a representation of the symmetric (non-Abelian) group SN : (3) P̂σ P̂σ0 = P̂σσ0 , P̂σ† = P̂σ−1 An important subclass are the exchange operators. The exchange of particles i and j is P̂ij |u1 . . . ui . . . uj . . . uN i = |u1 . . . uj . . . ui . . . uN i. Since P̂ij2 = I, it follows that P̂ij is self-adjoint with eigenvalues ±1. Any permutation can be factored into exchanges, in various ways. However, the parity of the number of exchanges in any factorization is the same: a permutation is even or odd if the number of exchanges is even or odd1. Let’s introduce the simmetrization operator Ŝ(N ) (or Ŝ(N )+ ) and the antisim- metrization operator Â(N ) (or Ŝ(N )− ) of N particles: 1 X (4) Ŝ(N )± = (±1)σ P̂σ N! σ∈SN (+1)σ = 1 applies to the simmetrization Ŝ(N ), while (−1)σ applies to Â(N ), and is +1 if σ is even, −1 if σ is odd. We omit the simple proofs of (5) P̂σ Ŝ(N ) = Ŝ(N )P̂σ = Ŝ(N ), P̂σ Â(N ) = Â(N )P̂σ = (−1)σ Â(N ) Proposition 1.2. Ŝ(N )± are projection operators: (6) Ŝ(N )†± = Ŝ(N )± , Ŝ(N )2± = Ŝ(N )± Proof. In Ŝ(N )†± = N1 ! σ (±1)σ P̂σ−1 , the sum on σ coincides with the sum on P inverse permutations σ −1 , and σ has the same parity of σ −1 P. This gives self- adjointness. The other property follows from (5): Ŝ(N )2± = N1 ! σ (±1)σ Ŝ(N )± P̂σ = 1 P N! σ Ŝ(N )± = Ŝ(N )± .  Exercise 1.3. Show that A(N )|u1 . . . uN i = 0 if two vectors are equal. PN Exercise 1.4. If |vk i = k=1 Mkj |uj i, k = 1 . . . N , show that (7) A(N )|v1 . . . vN i = (det M )A(N )|u1 . . . uN i 1S N is represented by N × N matrices σij with elements 0 and a single 1 in each column and row, the determinant may be 1 or -1. There are N ! such matrices. The matrix exchanging elements i, j has matrix elements σrs = δrs for r, s 6= i, j, σii = σjj = 0, σij = σji = 1. Its determinant is −1. All factorizations that produce the permutation matrix must give the same value of the determinant.  SECOND QUANTIZATION 3 The projection operators identify two Hilbert (sub)spaces: H (N )+ = Ŝ(N )+ H (N ), H (N )− = Ŝ(N )− H (N ) A vector in H (N )+ is invariant under the action of any exchange operator, while a vector in H (N )− changes sign under the action of any exchange operator. In other words, H (N )+ is the eigenspace with eigenvalue 1 for all exchange operators, while H (N )− is the eigenspace with eigenvalue −1 for all exchange operators. Therefore, H (N )+ and H (N )− are orthogonal subspaces of H (N ). 1.2. Indistinguishability of identical particles. In quantum mechanics iden- tical particles are indistinguishable. This means that the operators associated to observables are symmetric functions of the fundamental 1-particle operators: Ô(1, 2, . . . , N ) = Ô(σ1 , σ2 , . . . , σN ), where for brevity we put i = (xi , pi , si ). There- fore, they commute with all permutation operators: [Ô, P̂σ ] = 0. PN PN Examples: ‘one-particle operators’ Â1 = i=1 A(i), (total momentum p = i=1 pi , PN PN particle density n̂(x) = i=1 δ3 (x − xi )), spin density s(x) = i=1 si δ3 (x − xi ), P ...), two-particle operators Ô2 = i<j o(i, j), with o(i, j) = o(j, i) (the two-particle P interaction energy V̂ = i<j v(xi , xj )). The only subspaces that are left invariant by the action of the observables, and in particular by the time evolution, are H (N )± . This makes them the Hilbert spaces for the physics of N identical particles. The spin-statistics connection states that: H (N )+ is the space for N bosons (integer spin), H (N )− is the space for N fermions (half-integer spin). This is a fundamental result of the relativistic quantum field theory, where a viola- tion would correspond to a violation of causality. 1.3. Slater determinants and permanents. The inner product of two vectors Ŝ(N )± |u1 , . . . uN i and Ŝ(N )± |v1 , . . . vN i is: 1 X hu1 , . . . , uN |Ŝ(N )2± |v1 , . . . , vN i = (±1)σ hu1 , . . . , uN |P̂σ |v1 , . . . , vN i N! σ 1 X = (±1)σ hu1 |vσ1 i . . . huN |vσN i N! σ   hu1 |v1 i . . . hu1 |vN i 1 .. .. (8) = D±    N! . .  huN |v1 i . . . huN |vN i D+ is the permanent2 and D− is the determinant. When the inner product is evaluated with the bra hx1 , m1 ; . . . ; xN , mN |, one gets the permanent or the determinant of a matrix whose elements are functions:   v1 (x1 m1 ) . . . vN (x1 m1 ) (9) D±   .. ..  . .  v1 (xN mN ) . . . vN (xN mN ) The result is a totally symmetric or antisymmetric function. 2The permanent of a matrix is evaluated as a determinant, but omitting the weights ±1. Note that D+ (AB) 6= D+ (A)D+ (B). 4 L. G. MOLINARI Exercise 1.5. The first N eigenfunctions of the 1D harmonic oscillator are 1 1 2 uk (x) = p √ e− 2 x Hk (x) 2k k! π where Hk (x) = 2k xk + . . . are the Hermite polinomials. Show that 1 2 2 Y hx1 . . . xN |Â(N )|u1 . . . uN i = cost. e− 2 (x1 +...+xN ) (xi − xj ) i>j 1.4. The basis of occupation numbers. Given a 1-particle orthonormal com- plete set of vectors |ri, r = 1, 2, 3 . . . the factored vectors |r1 , r2 , . . . , rN i form an orthonormal basis in H (N ). We study their projections Ŝ(N )± |r1 , r2 , . . . rN i. Remark 1.6. Since hri |rj i = δij , by eq.(8) two vectors Ŝ(N )± |r1 , . . . rN i and Ŝ(N )± |r10 , . . . rN 0 i are orthogonal if |r10 , . . . rN 0 i is not a permutation of |r1 , . . . rN i. Bosons. In evaluating the squared norm, the sum on permutations in eq.(8) gives nonzero terms in correspondence of the identity permutation and permuta- tions among vectors that are repeated. Then: n1 !n2 ! . . . n∞ ! kŜ(N )|r1 , r2 , . . . rN ik2 = N! where nr is the number of times that |ri is present in the sequence |r1 , . . . , rN i. Since n1 + n2 + . . . + n∞ = N , most occupation numbers nr are zero. Consider the normalized and orthogonal vectors r N! (10) |r1 , . . . , rN i+ = Ŝ(N )|r1 , . . . rN i, n1 ! . . . n∞ ! where, conventionally, 1-particle basis vectors are in ascending order, r1 ≤ r2 ≤ . . . ≤ rN , to avoid replicas of the same symmetric vector. These vectors form an othonormal basis in the space of N bosons H (N )+ . Fermions. A state |ri cannot appear twice in |r1 , . . . , rN i, therefore nr = 0, 1. In evaluating the norm, only the identity permutation contributes, then kÂ(N )|r1 , r2 , . . . rN ik2 = N1 ! The vectors √ (11) |r1 , r2 , . . . rN i− = N ! Â(N )|r1 , r2 , . . . rN i where r1 < r2 < . . . < rN , form an orthonormal basis in the space of N fermions H (N )− . A vector |r1 , . . . rN i± is proportional to a symmetric or antisymmetric sum on all permutations of |r1 , . . . , rN i. It carries the information that N particles are dis- tribuited with equal probability in the specified 1-particle states. The information is specified by the occupation numbers of each state of the basis. We introduce the equivalent notation: (12) |n1 , n2 , . . . , n∞ i± ≡ |r1 , r2 , . . . rN i± where nr is the occupation number of the single particle state |ri, i.e. the number of times that the vector |ri appears in |r1 , . . . rN i. For bosons nr = 0, 1, 2, . . . , for fermions nr = 0, 1, and n1 + · · · + n∞ = N . For example, if the basis is |1i, |2i, ... the vector for 6 bosons |2, 3, 3, 5, 5, 8i+ can be represented as |0, 1, 2, 0, 2, 0, 0, 1, 0, ...i+ .  SECOND QUANTIZATION 5 The vector for 6 fermions |2, 3, 4, 5, 6, 8i− is equivalent to |0, 1, 1, 1, 1, 1, 0, 1, 0, ...i− . The ortogonality and completeness relations in H (N )± are: 0 0 (13) ± hn1 , n2 , . . . |n1 , n2 , . . .i± = δn1 ,n01 δn2 ,n02 . . . X (14) ± |n1 , n2 , . . .ihn1 , n2 , . . . |± = Iˆ n1 ,n2 ,... n1 +n2 +...=N 2. The Fock space Even though the number of particles might be a constant of the motion, it is advantageous to immerse the Hilbert space of N bosons or fermions in the larger Fock spaces (Vladimir Fock, 1928) for bosons or fermions: F± = |0i ⊕ H (1) ⊕ H (2)± ⊕ . . . ⊕ H (N )± ⊕ . . . |0i is the vacuum state (zero particle), H (1) is the one-particle Hilbert space H , H (N )± is the Hilbert space of N bosons or fermions. If H is separable, the Fock spaces F± are separable. In this construction, vectors with different content of particles are orthogonal. The occupation number vectors |n1 , . . . , n∞ i± with unrestricted sum n1 + · · · + n∞ are a basis for F± . In the larger ambient of Fock space, operators that change the number of par- ticles may be defined. The basic ones are the operators that create or destroy one boson, b̂† , b̂, or the operators that create or destroy one fermion ↠, â. Since many properties can be derived jointly, we’ll also use the single notation ĉ† , ĉ (for bosons they act on F+ , for fermions they act on F− ). 2.1. Creation and destruction operators. The creation operator of a particle in a single particle state |ui is defined through its action on a factored vector: √ (15) ĉ†|ui Ŝ(N )± |u1 , . . . , uN i = N + 1 Ŝ(N + 1)± |u, u1 , . . . , uN i The creation operator simply adds a new particle in state |ui to the existing N particles, and reshuffles the N + 1 states. Therefore ĉ†|ui : H (N )± → H (N + 1)± , N = 0, 1, . . . . In particular, ĉ†|ui |0i = |ui. The action on a generic vector in F± is defined by linearity, once the vector is expanded into factored vectors. The destruction operator of a particle in a state |ui is defined as the adjoint of ĉ†|ui . As such, to find out its action on vectors, we must consider the matrix element between two factored states hv1 , . . . , vN 0 |Ŝ(N 0 )± ĉ|ui Ŝ(N )± |u1 , . . . , uN i. As ĉ|ui adds a particle in the bra-vector, the matrix element is zero if N 0 6= N − 1, meaning that ĉ|ui acts on the ket by removing a particle. Let’s then consider the matrix element hv1 , . . . , vN −1 |Ŝ(N − 1)± ĉ|ui Ŝ(N )± |u1 , . . . , uN i √ = N hu, v1 , . . . , vN −1 |Ŝ(N )± |u1 , . . . , uN i   √ hu|u1 i hu|u2 i . . . hu|uN i N  hv1 |u1 i . . . . . . hv 1 |uN i  = D±   N!  ... ...  hvN −1 |u1 i hvN −1 |uN i 6 L. G. MOLINARI The permanent or determinant is expanded with respect to the first row: √ N NX = (±1)j+1 hu|uj iD± {hvi |uk i}k6=j N ! j=1 N 1 X =√ (±1)j+1 hu|uj ihv1 , . . . , vN −1 |Ŝ(N − 1)± |u1 , . . . , u / j , . . . , uN i N j=1 By the arbitrariness of the bra vector, and being the linear combinations of such vectors a dense set in H (N − 1)± , we obtain the final formula: (16) N 1 X ĉ|ui Ŝ(N )± |u1 , . . . , uN i = √ (±1)j+1 hu|uj iŜ(N − 1)± |u1 . . . , u / j , . . . , uN i N j=1 Destruction operators annihilate the vacuum state: ĉ|ui |0i = 0. From the definition of creation operator on factored states, and being the de- struction operator its adjoint, the following properties of linearity and antilinearity follow: (17) ĉ†α|ui+β|vi = αĉ†|ui + βĉ†|vi , ĉα|ui+β|vi = α∗ ĉ|ui + β ∗ ĉ|vi 2.2. The algebra of (anti)commutation rules. Creation and destruction oper- ators obey a simple and important algebra. Consider the creation of two particles in the single particle states |ui and |vi ĉ†|ui ĉ†|vi Ŝ(N )± |u1 , . . . , uN i = (N + 2)(N + 1)Ŝ(N + 2)± |u, v, u1 , . . . , uN i p If the order is exchanged, the boson state with vectors exhanged remains the same, while for fermions an exchange means a minus sign: ĉ†|vi ĉ†|ui Ŝ(N )± |u1 , . . . , uN i = ± (N + 2)(N + 1)Ŝ(N + 2)± |u, v, u1 , . . . , uN i p By respectively subtracting and summing, the following exact commutation and anti-commutation properties result3: (18) [b̂†|ui , b̂†|vi ] = 0, (bosons) {â†|ui , â†|vi } = 0, (fermions) By taking their adjoint, it follows that destruction operators exactly commute or anticommute: (19) [b̂|ui , b̂|vi ] = 0, (bosons) {â|ui , â|vi } = 0, (fermions) In particular, (â†|ui )2 = 0 and (â|ui )2 = 0 (Pauli principle). 3[A, B] = AB − BA, {A, B} = AB + BA  SECOND QUANTIZATION 7 To obtain the relations among creation and destruction operators, their actions in different order are evaluated: N ĉ†|vi ĉ|ui S(N )± |u1 , . . . , uN i = X (±1)j+1 hu|uj iS(N )± |v, u1 . . . , u / j , . . . , uN i j=1 √ ĉ|ui ĉ†|vi S(N )± |u1 , . . . , uN i = N + 1 c|ui S(N + 1)± |v, u1 , . . . , uN i = N X = hu|viS(N )± |u1 , . . . , uN i ± (±1)j+1 hu|uj iS(N )± |v, u1 , . . . , u / j , . . . , uN i j=1 By comparing the two expressions one concludes that (20) [b̂|ui , b̂†|vi ] = hu|vi (bosons) {â|ui , â†|vi } = hu|vi (fermions) Symmetric or antisymmetric factored states are obtained by acting with creation operators on the vacuum (the state is not normalized, in general): 1 (21) Ŝ(N )± |u1 , u2 , . . . uN i = √ ĉ†|u1 i ĉ†|u2 i . . . ĉ†|uN i |0i N! In the formula bosonic creation operators commute, and may be exchanged. Fermionic ones anticommute, and their exchange may imply a sign change. 3. Canonical (anti)commutation relations In most applications, creation and destruction operators are introduced in asso- ciation with a complete orthonormal basis |ri, r = 1, 2, . . .. For brevity we write ĉr = ĉ|ri and ĉ†r = ĉ†|ri . The algebra of such operators of states. For bosons the canonical commutation relations hold: (22) [b̂r , b̂s ] = 0, [b̂†r , b̂†s ] = 0, [b̂r , b̂†s ] = δrs (CCR) For fermions the canonical anti-commutation relations hold: (23) {âr , âs } = 0, {â†r , â†s } = 0, {âr , â†s } = δrs (CAR) A canonical transformation of creation and destruction operators is a map to an- other set of operators, that preserves the CCR or CAR rules. For example, the exchange of âr with â†r is canonical (particle-hole symmetry). The action of the operators is simple on the occupation number states |n1 , . . . n∞ i± referred to the basis |ri itself. Starting from eqs.(10) and (12), for Bose operators we obtain r N! b̂†r |n1 . . . nr . . . n∞ i = b̂† Ŝ(N )|r1 , r2 , . . . , rN i n1 ! . . . nr ! . . . r s (N + 1)! = Ŝ(N + 1)|r, r1 , r2 , . . . , rN i n1 ! . . . nr ! . . . √ (24) = nr + 1 |n1 . . . nr + 1 . . . n∞ i r N N! 1 X b̂r |n1 . . . nr . . . n∞ i = √ δr,ri Ŝ(N − 1)|r1 . . . , /ri , . . . rN i n1 ! . . . nr ! . . . N i=1 √ (25) = nr |n1 . . . nr − 1 . . . n∞ i 8 L. G. MOLINARI For Fermi operators: √ √ â†r |n1 . . . nr , . . . n∞ i = N ! N + 1Â(N + 1)|r, r1 , r2 . . . , rN i p = (N + 1)!(−1)n1 +...+nr−1 Â(N + 1)|r1 , . . . , r, . . . , rN i ( (−1)n1 +···+nr−1 |n1 . . . nr + 1 . . . n∞ i if nr = 0 (26) = 0 if nr = 1 ( 0 if nr = 0 (27) âr |n1 . . . nr . . . n∞ i = n1 +···+nr−1 (−1) |n1 . . . nr − 1 . . . n∞ i if nr = 1 The factor (−1)n1 +...+nr−1 results from the number of exchanges that bring the vector |r, r1 . . . rN i to the vector with r1 < · · · < r < · · · < rN . The operators n̂r = b̂†r b̂r and n̂r = â†r âr are the occupation numbers of |ri: (28) n̂r |n1 . . . nr . . . n∞ i = nr |n1 . . . nr . . . n∞ i In the basis |ri, the following are normalized occupation number vectors: 1 (29) |n1 , n2 , . . . , n∞ i = √ b̂†n1 · · · b̂†n ∞ |0i ∞ (bosons) n 1 ! . . . n∞ ! 1 (30) |n1 , n2 , . . . , n∞ i = â†n 1 1 · · · â†n ∞ |0i ∞ (fermions) For Fermi statistics ni = 0, 1 and the order of the operators, if changed, may produce a sign. Exercise 3.1. Prove for fermions: the number operators commute, the eigenvalues of n̂r are 0, 1. 3.1. Field operators. The formalism is extended to operators that create and destroy particles in states belonging to a continuum basis. An important example is the single particle basis of position x and spin sz XZ dx|x, mihx, m| = I, hx, m|x0 , m0 i = δmm0 δ3 (x − x0 ) m The operators that create or destroy a particle in the unphysical states |x, mi are named field operators. In place of the symbols ĉ|x,mi e ĉ†|x,mi , it is customary to † use the symbols ψ̂m (x) and ψ̂m (x). For bosons: † 0 0 (31) [ψ̂m (x), ψ̂m 0 (x )] = δ3 (x − x )δm,m0 † (32) [ψ̂m (x), ψ̂m0 (x0 )] = 0, [ψ̂m † (x), ψ̂m 0 0 (x )] = 0 For fermions: † 0 0 (33) {ψ̂m (x), ψ̂m 0 (x )} = δ3 (x − x )δm,m0 † (34) {ψ̂m (x), ψ̂m0 (x0 )} = 0, {ψ̂m † (x), ψ̂m 0 0 (x )} = 0 Given the one-particle state |ui and the function u(x, m) = hx, m|ui, one has the expansions XZ † XZ † (35) ĉ|ui = dx u(x, m) ψ̂m (x), ĉ|ui = dx u(x, m) ψ̂m (x) m m  SECOND QUANTIZATION 9 On the other hand, given a single particle basis |ri: X X † (36) ψ̂m (x) = hx, m|riĉr , ψ̂m (x) = hr|x, miĉ†r r r Let’s introduce the operators that create and destroy a particle in eigenstates of p and sz : p|k, mi = ~k|k, mi and sz |k, mi = ~m|k, mi. In a box of volume V with periodic b.c. the eigenstates form a discrete basis, and we associate to them the canonical operators ĉ†km and ĉkm . Then, for example: Z dx 1 X ik·x (37) ĉkm = √ e−ik·x ψ̂m (x), ψ̂m (x) = √ e ĉkm V V V k In the infinite domain, the basis |k, mi is continuous, and we associate to it the † field operators ψ̂m (k) and ψ̂m (k). it is: Z Z dx dx (38) ψ̂m (k) = e−ik·x ψ̂m (x), ψ̂m (x) = eik·x ψ̂m (k) (2π)3/2 (2π)3/2 4. Second quantization of operators The action of creation operators on the vacuum state gives factored states, that generate the Fock spaces. For this reason creation and destruction operators asso- ciated to a basis, are a basis of operators on F± . The operators associated to the observables of identical particles commute with the projectors S(N )± , and leave the subspaces H (N )± invariant. In this section, they are expanded in creation and destruction operators. The proof of the following identity is left to the reader. If |ui and |vi are any two single-particle states, for any factored state: ĉ†|ui ĉ|vi Ŝ(N )± |u1 , . . . , uN i = X (39) hv|uj iŜ(N )± |u1 , . . . , u, . . . , uN i j=1..N (in the right-hand side, the vector |ui replaces the vector |uj i). The iteration of the formula with new vectors |u0 i and |v 0 i gives: ĉ†|ui ĉ|vi ĉ†|u0 i ĉ|v0 i Ŝ(N )± |u1 , . . . , uN i = hv 0 |uj iĉ†|ui ĉ|vi Ŝ(N )± |u1 , . . . , u0 , . . . , uN i X j=1..N X = hv|u0 i hv 0 |uj iŜ(N )± |u1 , . . . , u, . . . , uN i j=1..N X + hv |uj ihv|uk iŜ(N )± |u1 , . . . , u0 , . . . , u, . . . uN i 0 j6=k In the last line u replaces uk and u0 replaces uj . The term with a single sum is hv|u0 iĉ†|ui ĉ|v0 i Ŝ(N )± |u1 , . . . , uN i. We end up with another useful identity (the generalization is simple to guess). An exchange of operators is done, by means of 10 L. G. MOLINARI (anti)commutation relation hv|u0 i = c|vi c†|u0 i ∓ c†|u0 i c|vi : ĉ†|ui ĉ†|u0 i ĉ|v0 i ĉ|vi Ŝ(N )± |u1 , . . . , uN i = ±(ĉ†|ui ĉ|vi ĉ†|u0 i ĉ|v0 i − hv|u0 iĉ†|ui ĉ|v0 i )Ŝ(N )± |u1 , . . . , uN i X =± hv|uj ihv 0 |uk iŜ(N )± |u1 , . . . , u0 , . . . , u, . . . uN i j6=k X (40) = hv|uj ihv 0 |uk iŜ(N )± |u1 , . . . , u, . . . , u0 , . . . uN i j6=k In the last line, u and u0 replace uj and uk . 4.1. One-particle operators. The operators are the sum of N identical operators PN acting on 1-particle subspaces:  = k=1 A(k); A( · ) is a function of the funda- mental 1-particle operators (e.g. position, momentum and spin). Upon insertion of two resolutions of the identity (not necessarily with same basis): X ÂŜ(N )± |u1 , . . . uN i = Ŝ(N )± |u1 , . . . , Auk , . . . uN i k=1..N X X = Ŝ(N )± hr|A|uk i|u1 , . . . , r, . . . , uN i r k=1..N X X = hr|A|si hs|uk iS(N )± |u1 , . . . , r, . . . , uN i r,s k=1..N The last line is compared with (39) and, since the vectors S(N )± |u1 . . . uN i generate the Fock spaces: X (41)  = ĉ†r hr|A|siĉs r,s If the basis vectors are eigenvectors of the single particle P operator A, with eigen- values ar , the expansion is simple and suggestive:  = r ar n̂r , being n̂r = ĉ†r ĉr the occupation number operator of the state |ri. If the continuous basis of position-spin is used, the expansion reads: XZ (42)  = dxdx0 ψ̂m † (x)hxm|A|x0 m0 iψ̂m0 (x0 ) mm0 For local operators it is hx, m|A|x0 m0 i = δ3 (x − x0 ) Am,m0 (x). A notable example is the particle density operator: N X X † (43) n̂(x) = δ3 (x − xi ) = ψ̂m (x)ψ̂m (x) i=1 m In the basis |k, mi (in a box) the kinetic energy operator is diagonal, while in the basis |x, mi it recalls a 1-particle expectation value N p̂2i X ~2 k 2 † ~2 X X Z † (44) T̂ = = ĉk,m ĉk,m = − dx ψ̂m (x)∇2 ψ̂m (x) i=1 2m 2m 2m m k,m  SECOND QUANTIZATION 11 4.2. Two-particle operators. The operators are the sum of identical two-particle operators: V̂ = 12 i6=j v(i, j) where v(i, j) = v(j, i). The symmetry implies P the exchange symmetry of matrix elements on 2-particle states h1, 2|v|10 , 20 i = h2, 1|v|20 , 10 i. Let us evaluate X V̂ Ŝ(N )± |u1 , . . . , uN i = 12 Ŝ(N )± v(i, j)|u1 , . . . , ui , . . . , uj , . . . , uN i i6=j X X 0 0 = 1 2 hr, s|v|r , s i hr0 |ui ihs0 |uj iS(N )± |u1 , . . . , r, . . . , s, . . . uN i r,s;r 0 ,s0 i6=j After using (40) we obtain the following expression: X (45) V̂ = 12 ĉ†r ĉ†s hr, s|v|r0 , s0 iĉs0 ĉr0 r,s,r 0 ,s0 (note the order of the destruction operators, which is opposite to that of the ket states in the matrix element). If field operators are used: X Z † † (46) V̂ = 12 dx1 . . . dx4 ψ̂m 1 (x1 )ψ̂m 2 (x2 ) mi hx1 m1 , x2 m2 |v|x3 m3 , x4 m4 i ψ̂m4 (x4 )ψ̂m3 (x3 ) The notation can be significantly simplified: Z V̂ = 2 d(1234)ψ̂ † (1)ψ̂ † (2)h1, 2|v|3, 4i ψ̂(4)ψ̂(3) 1 If v depends only on the particles’ positions: hx1 m1 , x2 m2 |v|x3 m3 , x4 m4 i = v(x3 , x4 )hx1 m1 , x2 m2 |x3 m3 , x4 m4 i = = v(x3 , x4 )δ3 (x1 − x3 )δ3 (x2 − x4 )δm1 ,m3 δm2 ,m4 The operator gains a form similar to a first-quantized expectation value: X Z † (47) V̂ = 12 dxdx0 ψ̂m † (x)ψ̂m 0 0 0 0 (x ) v(x, x ) ψ̂m0 (x )ψ̂m (x) m,m0 Remark 4.1. In second-quantized form, operators share some general properties: 1) there is no dependence on the total number N of particles: the operators act on the Fock space, and have the same expression for any N ; 2) if the numbers of creation and destruction operators are the same, they leave the subspaces of N identical particles invariant; 3) they are normally ordered, with destruction operators at the right of creation operators. As a consequence, the expectation value of the operator on the vacuum state (zero particle) is zero. † P Exercise 4.2. Given Ĥ = rs hrs ĉr ĉs , which is the transformation that diago- nalizes the Hamiltonian? Study the Hamiltonian Ĥ = −t r (ĉr ĉr+1 + ĉ†r ĉr−1 ). P † 12 L. G. MOLINARI 5. Symmetries A unitary operator u on one-particle states in H defines a unitary operator on H (N ): ÛN = u ⊗ u ⊗ . . . ⊗ u. ÛN commutes with exchange operators and leaves the subspaces H (N )± invariant. The unitary operator on Fock spaces is Û = I ⊕ u ⊕ Û2 ⊕ . . . ⊕ ÛN ⊕ . . .. Proposition 5.1. A one-particle unitary operator u induces a canonical transfor- mation on creation and destruction operators: (48) Û† ĉ†|vi Û = ĉ†u† |vi , Û† ĉ|vi Û = ĉu† |vi Proof. Û† ĉ†|vi ÛŜ(N )± |v1 , . . . vN i = Û† ĉ†|vi Ŝ(N )± |uv1 , . . . , uvN i √ √ = N + 1Û† Ŝ(N + 1)± |v, uv1 , . . . , uvN i = N + 1Ŝ(N + 1)± |u† v, v1 , . . . , vN i = ĉ†u† |vi Ŝ(N )± |v1 , . . . vN i. The second relation follows by adjunction.  We present the unitary representations of space translations, rotations, dilata- tions and parity on Fock space. They are important in the study of correlators. 5.1. Translations. For one particle, translations are represented by the unitary group u(a) with the following action on the fundamental one-particle operators: u† (a)xi u(a) = xi + ai , u† (a)pi u(a) = pi , and u† (a)si u(a) = si . They imply u(a) = exp(− ~i a · p) and the following transformations of eigenvectors (49) u(a)|x, mi = |x + a, mi, u(a)|k, mi = e−ik·a |k, mi On F± translations are represented by the unitary operators i ~k ĉ†km ĉkm X (50) Û(a) = exp(− a · P), P = ~ k,m with action: (51) Û(a)† ψ̂m † † (x)Û(a) = ψ̂m (x − a), Û(a)† ψ̂m (x)Û(a) = ψ̂m (x − a) (52) Û(a)† ĉ†km Û(a) = eik·a ĉ†km , Û(a)† ĉkm Û(a) = e−ik·a ĉkm . 5.2. Rotations. On the Hilbert space of a particle with spin, space rotations act as unitary operators specified by the vector transformation of the fundamental operators: u(R)† xi u(R) = Rij xj , u(R)† pi u(R) = Rij pj and u(R)† si u(R) = Rij sj . The infinitesimal transformations yield the commutation rules of xi , pi , si with the generator (angular momentum) jk = ijk xi pj + sk . The action on basis vectors is X X u(R)|x, mi = 0 D(R)m0 m |Rx, m0 i, u(R)|k, mi = 0 D(R)m0 m |Rk, m0 i m m where D(R) is a unitary representation of rotations, of dimension 2s + 1, generated by the spin matrices. To the single-particle representation there corresponds a representation on Fock space. The generators are the operators of total angular momentum. The field operators transform as follows: ψ̂ † 0 (R−1 x)D(R)†m0 m X (53) Û(R)† ψ̂m † (x)Û(R) = m0 m X (54) Û(R)† ψ̂m (x)Û(R) = 0 D(R)mm0 ψ̂m0 (R−1 x) m The same relations hold for the field operators in the basis of momentum.  SECOND QUANTIZATION 13 5.3. Dilations. Dilations, or scale trasformations, are here illustrated in the isotropic case. They form a 1-parameter group of unitary transformations u(t) with the fol- lowing action on fundamental operators: (55) u(t)† xi u(t) = et xi , u(t)† pi u(t) = e−t p̂i , u(t)† si u(t) = si and on position and momentum eigenvectors 3 3 u(t)|x, mi = e 2 t |et x, mi u(t)|k, mi = e− 2 t |e−t k, mi 1 With u(t) = exp(−itD), from (55) one obtains the generator D = 2~ (x · p + p · x). The representation on Fock space has action: 3 3 (56) Û(t)† ψ̂m † (x)Û(t) = e− 2 t ψ̂m † (e−t x) Û(t)† ψ̂m (x)Û(t) = e− 2 t ψ̂m (e−t x) 3 3 (57) Û(t)† ψ̂m † † (k)Û(t) = e 2 t ψ̂m (et k) Û(t)† ψ̂m (k)Û(t) = e 2 t ψ̂m (et k) Theorem 5.2 (Virial theorem). Let Ĥ = i Ĥi , where Û(t)† Ĥi Û(t) = eni t Ĥi . If P |Ei is an eigenstate of Ĥ, then X (58) ni hE|Ĥi |Ei = 0. i † † Proof. From Û (t)Ĥ|Ei = E Û (t)|Ei one obtains: i eni t Hi Û† (t)|Ei = E Û† (t)|Ei. P P A derivative in t = 0 gives i ni Ĥi |Ei + iĤ D̂|Ei = iE D̂|Ei. The inner product with |Ei gives the result.  1 P n 2 Example 5.3 (Coulomb Hamiltonian). Ĥ = T̂ + Û , T̂ = 2m i=1 pi , Û = 2 −1 P e i<j |xi −xj | . For a scale transformation the kinetic term has weight nT = −2 while the Coulomb interaction has weight nU = −1. Then hE|T̂ |Ei = − 21 hE|Û |Ei. 5.4. Parity. Parity is the unitary and self-adjoint operator p defined by the prop- erties: pxi p = −xi , ppi p = −pi and psi p = si . They imply p|x, mi = | − x, mi and p|k, mi = | − k, mi. The corrisponding unitary operator on Fock space is (59) Û†p ψ̂m † † (x)Ûp = ψ̂m (−x), Û†p ψ̂m (k)Ûp = ψ̂m (−k) L. G. Molinari: Physics Department, Università degli Studi di Milano and I.N.F.N. sez. di Milano, Via Celoria 16, 20133 Milano Email address: [email protected]