SECOND QUANTIZATION notes by Luca G. Molinari (oct 2001- revised oct 2016) The appropriate formalism for the quantum description of identical particles is second quantisation. There are various equivalent ways to introduce it. In the present approach, it is a procedure to rewrite the operators corresponding to observables from the basis of the fundamental single-particle operators (such as { x i, p i, s i } N i=1) to a new basis of operators {ĉ r, ĉ r } r=1 that create and destroy a particle in single particle states r. Their commutation rules take care of the particle statistics. Second-quantized operators are defined in a broader Hilbert space, the Fock space (Vladimir Fock, 1928), that accomodates any number of particles. The formalism is very advantageous, and becomes necessary to describe absorption or emission processes of particles or excitations such as photons, phonons, magnons, plasmons etc. Creation and destruction operators with commutation rules for Bose statistics were introduced by Paul M. A. Dirac for the quantization of the electromagnetic field (1926, Quantum theory of light radiation and absorption). Jordan extended the formalism to massive bosons (Jordan and Klein, 1927) and then to fermions, whose statistics requires anti-commutation rules (Jordan and Wigner, 1928). 1 Systems of N identical particles 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 their commutation relations. If the relations are represented irreducibly (any operator commuting with them is multiple of the identity), then any other observable of the particle is a function of them, and the fundamental operators are identified up to a unitary transformation. Operators of different particles commute: this translates the concept of a particle being an autonomous entity. It follows that the Hilbert space of N 1
particles is H (N) = H 1 H N, where H k is the Hilbert space for the description of the k-th particle. H (N) is the closure of the finite linear combinations of factored vectors u 1,..., u N = u 1... u N, where u k belongs to H k. The inner product among factored states is: v 1,..., v N u 1,..., u N = v 1 u 1 1 v N u N N (1) where k is the inner product in H k. The tensor product is linear in each term: u 1,..., αu i + βv i,..., u N = α u 1,..., u i,..., u N + β u 1,..., v i,..., u N A single particle operator  on H k corresponds to an operator Â(k) on H (N) with action Â(k) u 1,..., u k,... u N = u 1,..., Âu k,..., u N. Single particle operators of different particles commute: [Â(k), ˆB(k )] = 0. A particle with spin s may be described by a normalised function f in L 2 (R 3 ) 2s+1. f( x, m) 2 d 3 x is the probability that the particle is localised in the volume d 3 x centred in x, with z component of spin equal to m. The products f 1 ( x 1, m 1 ) f N ( x N, m N ) and their linear combinations form a linear space whose closure is the Hilbert space of functions f( x 1 m 1,..., x N m N ) such that d 3 x 1... d 3 x N f( x 1 m 1,..., x N m N ) 2 <. m 1...m N 1.1 Identical particles If the N particles are identical, the spaces H 1,..., H N are copies of the same one-particle Hilbert space H (symbolically H (N) = N H ). Since the vectors that form u 1,... u N belong to the same space, it is possible to introduce the permutation operators. If σ is the permutation σ(1,..., N) = (σ 1,..., σ N ), the corresponding operator on factored states is ˆP σ u 1,... u N = u σ1,..., u σn (2) They extend by linearity to unitary operators on the whole space H (N), and form a representation of the symmetric group S N : ˆP σ ˆPσ = ˆP σσ, ˆP σ = ˆP σ 1 (3) An important subclass are the exchange operators. The exchange of particles i and j is ˆP ij u 1... u i... u j... u N = u 1... u j... u i... u N. Since ˆP ij 2 = I, it follows that ˆP ij is self-adjoint with eigenvalues ±1. 2
Any permutation can be factored into exchanges, in various ways. However, the parity of the number of exchange factors in any factorization is the same: a permutation is even or odd if the number of exchanges is even or odd 1. Let s introduce the simmetrization operator Ŝ(N) (or Ŝ(N) +) and the antisimmetrization operator Â(N) (or Ŝ(N) ) of N particles: Ŝ(N) ± = 1 (±1) σ ˆPσ (4) N! σ S N (+1) σ = 1 applies to the simmetrization, while for the antisimmetrization ( 1) σ is +1 if σ is even, 1 if σ is odd. We omit the simple proofs of ˆP σ Ŝ(N) ± = Ŝ(N) ± ˆP σ = (±1) σ Ŝ(N) ± (5) Proposition 1.1. Ŝ(N) ± are projection operators: Ŝ(N) ± = Ŝ(N) ±, Ŝ(N) 2 ± = Ŝ(N) ± (6) Proof. Ŝ(N) ± = 1 N! σ (±1)σ ˆPσ 1, the sum on σ coincides with the sum on inverse permutations σ 1, and σ has the same parity of σ 1. This gives self-adjointness. The other property follows from (5): Ŝ(N) 2 ± = 1 N! σ (±1)σ Ŝ(N) ± ˆPσ = 1 N! σ Ŝ(N) ± = Ŝ(N) ±. Exercise 1.2. Show that Ŝ(N)Â(N) = 0 (i.e. S(N) +S(N) = 0). Exercise 1.3. Show that A(N) u 1... u N = 0 if two of the N vectors are equal. The two subspaces of projection are the Hilbert 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 If S N is represented by N N matrices σ ij with elements 0 and a single 1 in each column and row, the determinant may be 1 or -1. The exchange matrix among elements ij has matrix elements σ rs = δ rs for r, s i, j, σ ii = σ jj = 0, σ ij = σ ji = 1. Its determinant is 1. All factorizations that produce the matrix must give the same value of the determinant. 3
1.2 Indistinguishability of identical particles In quantum mechanics identical 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(σ 1, σ 2,..., σ N ), where for brevity we put i = ( x i, p i, s i ). Therefore, they commute with permutation operators: [Ô, ˆP σ ] = 0. Examples are the one-particle operators Ô1 = N i=1 o 1(i), (e.g. total momentum P = i p i, particle density ˆn( x) = i δ 3( x x i )), and two-particle operators Ô2 = i<j o 2(i, j), with o 2 (i, j) = o 2 (j, i) (e.g. two-particle interaction ˆV = i<j v( x i, x j )). 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 violation would correspond to a violation of causality. 1.3 Slater determinants and permanents The inner product of two vectors Ŝ(N) ± u 1,... u N and Ŝ(N) ± v 1,... v N is: u 1,..., u N Ŝ(N)2 ± v 1,..., v N = 1 (±1) σ u 1,..., u N N! ˆP σ v 1,..., v N σ = 1 (±1) σ u 1 v σ1... u N v σn N! σ = 1 u 1 v 1... u 1 v N N! D ±.. (7) u N v 1... u N v N D + is the permanent 2 and D is the determinant. When the inner product is evaluated with the bra x 1, m 1 ;... ; x N, m N, one gets the permanent or the determinant of a matrix whose elements are 2 The permanent of a matrix is evaluated as a determinant, but omitting the weights ±1. Note that D + (AB) D + (A)D + (B). 4
functions: D ± v 1 ( x 1 m 1 )... v N ( x 1 m 1 ).. v 1 ( x N m N )... v N ( x N m N ) (8) The result is a totally symmetric or antisymmetric function. Exercise 1.4. In d = 1 consider the first N eigenfunctions of the harmonic oscillator 1 u k (x) = 2k k! 1 π e 2 x2 H k (x) where H k (x) = 2 k x k +... are the Hermite polinomials. Show that x 1... x N Â(N) u 1... u N = cost. e 1 2 (x2 1 +...+x2 N ) i>j(x i x j ) Exercise 1.5. Given N vectors u 1,..., u N and a N N matrix M, construct the vectors v k = M kj u j, k = 1... N. Show that A(N) v 1... v N = (det M)A(N) u 1... u N (9) 1.4 The basis of occupation numbers Given a 1-particle orthonormal (complete) basis of vectors r, r = 1, 2, 3..., the factored vectors r 1, r 2,..., r N form an orthonormal basis in H (N). We study their projections Ŝ(N) ± r 1, r 2,... r N. Since r i r j = δ ij, by eq.(7) two vectors Ŝ(N) ± r 1,... r N and Ŝ(N) ± r 1,... r N are orthogonal if r 1,... r N is not a permutation of r 1,... r N. Bosons. In evaluating the squared norm, the sum on permutations in eq.(7) gives nonzero terms in correspondence of the identity permutation and permutations among vectors that are repeated. Then: Ŝ(N) r 1, r 2,... r N 2 = n 1!n 2!... n! N! where n r is the number of times that the ket r is present in the sequence r 1,..., r N. Since n 1 + n 2 +... + n = N, most occupation numbers n r are zero. Consider the normalized and orthogonal vectors N! r 1,..., r N + = Ŝ(N) r 1,... r N, (10) n 1!... n! 5
where, conventionally, 1-particle basis vectors are in ascending order, r 1 r 2... r N, 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 r cannot appear twice in r 1,..., r N, therefore n r = 0, 1. In evaluating the norm, only the identity permutation contributes, then Â(N) r 1, r 2,... r N 2 = 1 The vectors N! r 1, r 2,... r N = N! Â(N) r 1, r 2,... r N (11) where r 1 < r 2 <... < r N, form an orthonormal basis in the space of N fermions H (N). A vector r 1,... r N ± is proportional to a symmetric or antisymmetric sum on all permutations of r 1,..., r N. It carries the information that N particles are distribuited with equal probability in the specified 1-particle states. The same information is encoded by the occupation numbers. We therefore introduce the useful notation: n 1, n 2,..., n ± r 1, r 2,... r N ± (12) where n r is the occupation number of the single particle state r, i.e. the number of times that the vector r appears in r 1,... r N. For bosons n r = 0, 1, 2,..., for fermions n r = 0, 1, and n 1 + + n = N. The ortogonality and completeness relations in H (N) ± are: ± n 1, n 2,... n 1, n 2,... ± = δ n1,n δ 1 n 2,n... (13) 2 ± n 1, n 2,... n 1, n 2,... ± = Î (14) n 1,n 2,... n 1 +n 2 +...=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 for bosons or fermions: F ± = 0 H (1) H (2) ±... H (N) ±... 0 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. In this construction, vectors with different content of particles are orthogonal. The occupation number vectors n 1,..., n ± without the restriction on n 1 + + n 6
are a basis for F ±. If H is separable, the Fock spaces F ± are separable. In the larger ambient of Fock space, operators that change the number of particles may be defined. The basic ones are the operators that create or destroy one boson, ˆb, ˆb, or the operators that create or destroy one fermion, â, 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 u is defined through its action on a factored vector: ĉ u Ŝ(N) ± u 1,..., u N = N + 1 Ŝ(N + 1) ± u, u 1,..., u N (15) The creation operator simply adds a new particle in state u to the existing N particles, and reshuffles the N + 1 states. Therefore ĉ u : H (N) ± H (N + 1) ±, N = 0, 1,.... In particular, ĉ u 0 = u. 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 u is defined as the adjoint of ĉ u. As such, to find out its action on vectors, we must consider the matrix element between two factored states v 1,..., v N Ŝ(N ) ± ĉ u Ŝ(N) ± u 1,..., u N As ĉ u adds a particle in the bra-vector, the matrix element is zero if N N 1, meaning that ĉ u acts on the ket by removing a particle. Let s then consider the matrix element v 1,..., v N 1 Ŝ(N 1) ±ĉ u Ŝ(N) ± u 1,..., u N = N u, v 1,..., v N 1 Ŝ(N) ± u 1,..., u N u u 1 u u 2... u u N N = N! D v 1 u 1...... v 1 u N ±...... v N 1 u 1 v N 1 u N 7
The permanent or determinant is expanded with respect to the first row: = N N! = 1 N N (±1) j+1 u u j D ± { v i u k } k j j=1 N j=1 (±1) j+1 u u j v 1,..., v N 1 Ŝ(N 1) ± u 1,..., /u j,..., u N By the arbitrariness of the bra vector, and being the linear combinations such vectors a dense set in H (N 1) ±, we obtain the final formula: ĉ u Ŝ(N) ± u 1,..., u N = 1 N N j=1 (±1) j+1 u u j Ŝ(N 1) ± u 1..., /u j,..., u N Destruction operators annihilate the vacuum state: ĉ u 0 = 0. (16) From the definition of creation operator on factored states, and being the destruction operator its adjoint, the following properties of linearity and antilinearity follow: ĉ α u +β v = αĉ u + βĉ v, ĉ α u +β v = α ĉ u + β ĉ v (17) 2.2 The algebra of (anti)commutation rules Creation and destruction operators obey a simple and important algebra. Consider the creation of two particles in the single particle states u and v ĉ u ĉ v Ŝ(N) ± u 1,..., u N = (N + 2)(N + 1)Ŝ(N + 2) ± u, v, u 1,..., u N If the order is exchanged, the boson state with vectors exhanged remains the same, while for fermions an exchange means a minus sign: ĉ v ĉ u Ŝ(N) ± u 1,..., u N = ± (N + 2)(N + 1)Ŝ(N + 2) ± u, v, u 1,..., u N 8
By respectively subtracting and summing, the following exact commutation and anti-commutation properties result 3 : [ˆb u, ˆb v ] = 0, (bosons) (18) {â u, â v } = 0, (fermions) (19) By taking their adjoint, it follows that destruction operators exactly commute or anticommute: [ˆb u, ˆb v ] = 0, (bosons) (20) {â u, â v } = 0, (fermions) (21) In particular, (â u )2 = 0 and (â u ) 2 = 0 (Pauli principle). To obtain the relations among creation and destruction operators, their actions in different order are evaluated: ĉ v ĉ u S(N) ± u 1,..., u N = N (±1) j+1 u u j S(N) ± v, u 1..., /u j,..., u N j=1 ĉ u ĉ v S(N) ± u 1,..., u N = N + 1 c u S(N + 1) ± v, u 1,..., u N = N = u v S(N) ± u 1,..., u N ± (±1) j+1 u u j S(N) ± v, u 1,..., /u j,..., u N j=1 By comparing the two expressions one concludes that [ˆb u, ˆb v ] = u v (bosons) (22) {â u, â v } = u v (fermions) (23) Symmetric or antisymmetric factored states are obtained by acting with creation operators on the vacuum (the state is not normalized, in general): Ŝ(N) ± u 1, u 2,... u N = 1 ĉ u N! 1 ĉ u 2... ĉ u N 0 (24) In the formula bosonic creation operators commute, and may be exchanged. Fermionic ones anticommute, and their exchange may imply a sign change. 3 [A, B] = AB BA, {A, B} = AB + BA 9
2.3 Canonical (anti)-commutation relations In most applications, creation and destruction operators are introduced in association with a complete orthonormal basis r, r = 1, 2,.... For brevity we write ĉ r = ĉ r and ĉ r = ĉ r. The action of such operators is simple on the occupation number states n 1, n 2,... n ± referred to the basis r itself. Starting from eqs.(10) and (12), for Bose operators we obtain N! ˆb r n 1... n r... n = n 1!... n r!...ˆb rŝ(n) r 1, r 2,..., r N (N + 1)! = n 1!... n r!...ŝ(n + 1) r, r 1, r 2,..., r N N! 1 ˆbr n 1... n r... n = n 1!... n r!... N for Fermi operators: = n r + 1 n 1... n r + 1... n (25) N i=1 δ r,ri Ŝ(N 1) r 1..., /r i,... r N = n r n 1... n r 1... n (26) â r n 1... n r,... n = N! N + 1Â(N + 1) r, r 1, r 2..., r N â r n 1... n r... n = = (N + 1)!( 1) n 1+...+n r 1 Â(N + 1) r 1,..., r,..., r N { ( 1) n 1+ +n r 1 n 1... n r + 1... n if n r = 0 = 0 if n r = 1 (27) { 0 if n r = 0 ( 1) n 1+ +n r 1 n 1... n r 1... n if n r = 1 (28) The factor ( 1) n 1+...+n r 1 results from the number of exchanges that bring the vector r, r 1... r N to the vector with r 1 < < r < < r N. The operators ˆn r = ˆb rˆb r and ˆn r = â râ r are the occupation number operators of the state r : ˆn r n 1... n r... n = n r n 1... n r... n (29) The algebra of the creation and destruction operators of states in an orthonormal basis, is very simple. For bosons the canonical commutation relations (CCR) hold: [ˆb r, ˆb s ] = 0, [ˆb r, ˆb s] = 0, [ˆb r, ˆb s] = δ rs (30) 10
For fermions the canonical anti-commutation relations (CAR) hold: {â r, â s } = 0, {â r, â s} = 0, {â r, â s} = δ rs (31) A canonical transformation of creation and destruction operators is a map to another set of operators, that preserves the CCR or CAR rules. For example, the exchange of â r with â r is canonical (particle-hole symmetry). If r is a basis, the following are normalized occupation number vectors: n 1, n 2,..., n = 1 n1!... n!ˆb n 1 n 1 ˆb 0 (32) n 1, n 2,..., n = â n 1 1 â n 0 (33) For Fermi statistics n i = 0, 1 and the order of the operators, if changed, may produce a sign. 2.4 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 s z d 3 x x, m x, m = I, x, m x, m = δ mm δ 3 ( x x ) m The operators that create or destroy a particle in the unphysical states x, m are named field operators. In place of the symbols c x,m e c x,m, it is customary to use the symbols ˆψ m ( x) and ˆψ m( x). For bosons: For fermions: [ ˆψ m ( x), ˆψ m ( x )] = δ 3 ( x x )δ m,m (34) [ ˆψ m ( x), ˆψ m ( x )] = 0, [ ˆψ m( x), ˆψ m ( x )] = 0 (35) { ˆψ m ( x), ˆψ m ( x )} = δ 3 ( x x )δ m,m (36) { ˆψ m ( x), ˆψ m ( x )} = 0, { ˆψ m( x), ˆψ m ( x )} = 0 (37) Given the one-particle state u and the function u( x, m) = x, m u, one has the expansions ĉ u = d 3 x u( x, m) ˆψ m ( x), ĉ u = d 3 x u( x, m) ˆψ m( x) (38) m m 11
On the other hand, given a single particle basis r : ˆψ m ( x) = r x, m r ĉ r, ˆψ m ( x) = r r x, m ĉ r (39) Let s introduce the operators that create and destroy a particle in eigenstates of p and s z : p k, m = k k, m and s z k, m = m k, m. 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 ĉ km. Then, for example: ĉ km = V d 3 x e i k x ˆψm ( x), ˆψm ( x) = 1 V V k e i k xĉ km (40) In the infinite domain, the basis k, m is continuous, and we associate to it the field operators ˆψ m( k) and ˆψ m ( k). it is: ˆψ m ( k) = d 3 x (2π) 3/2 e i k x ˆψm ( x), ˆψm ( x) = 3 Second quantization of operators d 3 x (2π) 3/2 ei k x ˆψm ( k) (41) By acting with creation operators on the vacuum state, one generates factored states that generate the Fock spaces. For this reason creation and destruction operators associated to a basis, are themselves 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 as linear combinations of products of creation and destruction operators. The proof of the following identity is left to the reader. If u and v are any two single-particle states, for any factored state: ĉ u ĉ v Ŝ(N) ± u 1,..., u N = N v u j Ŝ(N) ± u 1,..., u,..., u N (42) j=1 (in the right-hand side, the vector u replaces the vector u j ). The iteration 12
of the formula with new vectors u and v gives: N ĉ u ĉ v ĉ u ĉ v Ŝ(N) ± u 1,..., u N = v u j ĉ u ĉ v Ŝ(N) ± u 1,..., u,..., u N j=1 = v u N v u j Ŝ(N) ± u 1,..., u,..., u N j=1 + j k v u j v u k Ŝ(N) ± u 1,..., u,..., u,... u N In the last line u replaces u k and u replaces u j. The term with a single sum is v u ĉ u ĉ v Ŝ(N) ± u 1,..., u N. We end up with another useful identity (the generalization is simple to guess). An exchange of operators is done, by means of (anti)commutation relation v u = c v c u c u c v : ĉ u ĉ u ĉ v ĉ v Ŝ(N) ± u 1,..., u N = ±(ĉ u ĉ v ĉ u ĉ v v u ĉ u ĉ v )Ŝ(N) ± u 1,..., u N = ± j k v u j v u k Ŝ(N) ± u 1,..., u,..., u,... u N = j k v u j v u k Ŝ(N) ± u 1,..., u,..., u,... u N (43) In the last line, u and u replace u j and u k. 3.1 One-particle operators The operators are the sum of N identical operators acting on 1-particle subspaces: Ô = N k=1 o(k); o( ) is a function of the fundamental 1-particle operators (e.g. position, momentum and spin). Upon insertion of two resolutions of the identity (not necessarily with same basis): ÔŜ(N) ± u 1,... u N = Ŝ(N) ± = Ŝ(N) ± = r,s r r o s N u 1,..., ou k,... u N k=1 N r o u k u 1,..., r,..., u N k=1 N s u k S(N) ± u 1,..., r,..., u N k=1 13
The last line compared with (42) and, since the vectors S(N) ± u 1... u N generate the Fock spaces: Ô = r,s ĉ r r o s ĉ s (44) If the basis vectors are eigenvectors of the single particle operator o, with eigenvalues o r, the expansion is simple and suggestive: Ô = r o rˆn r, being ˆn r = ĉ rĉ r the occupation number operator of the state r. If the continuous basis of position-spin is used, the expansion reads: Ô = mm d 3 xd 3 x ˆψ m ( x) xm o x m ˆψ m ( x ) (45) For local operators it is x, m o x m = δ 3 ( x x ) o m,m ( x). example is the particle density operator: A notable ˆn( x) = N δ 3 ( x x i ) = m i=1 ˆψ m( x) ˆψ m ( x) (46) In the basis k, m the kinetic energy operator is diagonal, while in the basis x, m it recalls a 1-particle expectation value ˆT = N i=1 p 2 i 2m = k,m 2 k 2 2m ĉ k,m ĉ k,m = 2 2m 3.2 Two-particle operators m d 3 x ˆψ m( x) 2 ˆψm ( x) (47) The operators are the sum of identical two-particle operators: ˆV = 1 2 i j v(i, j) where v(i, j) = v(j, i). The symmetry implies the exchange symmetry of matrix elements on 2-particle states 1, 2 v 1, 2 = 2, 1 v 2, 1. Let us evaluate ˆV Ŝ(N) ± u 1,..., u N = 2Ŝ(N) 1 ± v(i, j) u 1,..., u i,..., u j,..., u N = 1 2 i j r, s v r, s r u i s u j S(N) ± u 1,..., r,..., s,... u N r,s;r,s i j After using (43) we obtain the following expression: ˆV = 1 ĉ 2 rĉ s r, s v r, s ĉ s ĉ r (48) r,s;r,s 14
(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: ˆV = 1 d 3 x 2 1... d 3 x 4 ˆψ m1 ( x 1 ) ˆψ m 2 ( x 2 ) (49) m i x 1 m 1, x 2 m 2 v x 3 m 3, x 4 m 4 ˆψ m4 ( x 4 ) ˆψ m3 ( x 3 ) The notation can be significantly simplified: ˆV = 1 d(1234) ˆψ (1) ˆψ (2) 1, 2 v 3, 4 2 ˆψ(4) ˆψ(3) If v depends only on the particle s positions: x 1 m 1, x 2 m 2 v x 3 m 3, x 4 m 4 = v( x 3, x 4 ) x 1 m 1, x 2 m 2 x 3 m 3, x 4 m 4 = = v( x 3, x 4 )δ 3 ( x 1 x 3 )δ 3 ( x 2 x 4 )δ m1,m 3 δ m2,m 4 The operator gains a form similar to the first-quantized expectation value: ˆV = 1 d 3 xd 3 x ˆψ 2 m ( x) ˆψ m ( x ) v( x, x ) ˆψ m ( x ) ˆψ m ( x) (50) m,m Remark 3.1. In their second-quantized expression, the operators share some general properties: 1) the dependence on the total number N of particles is lost: 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. Exercise 3.2. Given Ĥ = rs h rsĉ rĉ s, which is the transformation that diagonalizes the Hamiltonian? In particular, study the Hamiltonian Ĥ = t r (ĉ rĉ r+1 + ĉ rĉ r 1 ) 4 Symmetries A unitary operator u on one-particle states in H defines a unitary operator on H (N): Û N = u u... u. Û N commutes with exchange operators and leaves the subspaces H (N) ± invariant. The unitary operator on Fock spaces is Û = I u Û2... ÛN.... 15
Proposition 4.1. A one-particle unitary operator u induces a canonical transformation on creation and destruction operators: Û ĉ v Û = ĉ u v, Û ĉ v Û = ĉ u v (51) Proof. Û ĉ v ÛŜ(N) ± v 1,... v N = Û ĉ v Ŝ(N) ± uv 1,..., uv N = N + 1Û Ŝ(N +1) ± v, uv 1,..., uv N = N + 1Ŝ(N +1) ± u v, v 1,..., v N = ĉ u v Ŝ(N) ± v 1,... v N. The second relation follows by adjunction. We present the unitary representations of space translations, rotations, dilatations and parity on Fock space. They are important in the study of correlators. 4.1 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)x i u( a) = x i +a i, u ( a)p i u( a) = p i, and u ( a)s i u( a) = s i. They imply u( a) = exp( i a p) and the following transformations of eigenvectors u( a) x, m = x + a, m, u( a) k, m = e i k a k, m (52) On F ± translations are represented by the unitary operators Û( a) = e i a P, P = k ĉ km ĉ km (53) k,m with action: Û( a) ˆψ m ( x)û( a) = ˆψ m( x a), Û( a) ˆψm ( x)û( a) = ˆψ m ( x a) (54) 4.2 Rotations Û( a) ĉ km Û( a) = e i k aĉ km, Û( a) ĉ km Û( a) = e i k aĉ km. (55) 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) x i u(r) = R ij x j, u(r) p i u(r) = R ij p j and u(r) s i u(r) = R ij s j. The infinitesimal transformations yield the commutation rules of x i, p i, s i with the generator (angular momentum) j k = ɛ ijk x i p j + s k. The action on the basis vectors is u(r) x, m = m D(R) m m R x, m, u(r) k, m = m D(R) m m R k, m 16
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: Û(R) ˆψ m ( x)û(r) = m ˆψ m (R 1 x)d(r) m m (56) Û(R) ˆψm ( x)û(r) = m D(R) mm ˆψm (R 1 x) (57) The same relations hold for the field operators in the basis of momentum. 4.3 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 following action on fundamental operators: u(t) x i u(t) = e t x i, u(t) p i u(t) = e tˆp i, u(t) s i u(t) = s i (58) and on position and momentum eigenvectors u(t) x, m = e 3 2 t e t x, m u(t) k, m = e 3 2 t e t k, m With u(t) = exp( itd), from (58) one obtains the generator D = 1 ( x p + 2 p x). The representation on Fock space has action: Û(t) ˆψ m ( x)û(t) = e 3 2 t ˆψ m (e t x) (59) Û(t) ˆψm ( x)û(t) = e 3 2 t ˆψm (e t x) (60) Û(t) ˆψ m ( k)û(t) = e 3 2 t ˆψ m (e t k) (61) Û(t) ˆψm ( k)û(t) = e 3 2 t ˆψm (e t k) (62) Teorema 4.2 (Virial theorem). Let Ĥ = i Ĥi, where Û(t) Ĥ i Û(t) = e nit Ĥ i. If E is an eigenstate of Ĥ, then n i E Ĥi E = 0. (63) i Proof. From Û (t)ĥ E = EÛ (t) E one obtains: i enit H i Û (t) E = EÛ (t) E. A derivative in t = 0 gives i n iĥi E + iĥ ˆD E = ie ˆD E. The inner product with E gives the result. 17
Example 4.3 (Coulomb Hamiltonian). Ĥ = ˆT + Û, ˆT = n i=1 p 2 i 2m, Û = i<j e 2 x i x j For a scale transformation the kinetic term has weight n T = 2 while the Coulomb interaction has weight n U = 1. Therefore E ˆT E = 1 E Û E. 2 4.4 Parity Parity is the unitary and self-adjoint operator p defined by the properties: px i p = x i, pp i p = p i and ps i p = s i. They imply p x, m = x, m and p k, m = k, m. The corrisponding unitary operator on Fock space is Û p ˆψ m( x)ûp = ˆψ m( x), Û p ˆψ m ( k)ûp = ˆψ m ( k) (64) 18