CANONICAL COMMUTATION RELATIONS, BOGOLIUBOV TRANSFORMATIONS QUADRATIC HAMILTONIANS Jan Derezi´ nski Based partly on joint work with Christian G´ erard
PLAN 1. CANONICAL COMMUTATION RELATIONS 2. BOGOLIUBOV TRANSFORMATIONS AND QUADRATIC HAMILTONIANS IN FOCK REPRESENTATION 3. EXAMPLE: SCALAR FIELD WITH POSITION DEPENDENT MASS
CANONICAL COMMUTATION RELATIONS Let ( Y , ω ) be a real vector space equipped with an antisymmetric form. We will usually assume that ω is symplectic, which means that if it is nondegenerate. We will denote by Sp ( Y ) the group of linear transformations preserving ω .
Heuristically, we are interested in a linear map Y ∋ y �→ ˆ φ ( y ) with values in self-adjoint operators such that the Heisenberg com- mutation relations hold: [ˆ φ ( y ) , ˆ φ ( y ′ )] = i y · ωy ′ This is unfortunately a non-rigorous statement, since typically such ˆ φ ( y ) are unbounded. It is however possible to give a rigorous formulation of the above idea.
A regular representations of the canonical commutation relations or a regular CCR representation over ( Y , ω ) on a Hilbert space H is a map Y ∋ y �→ ˆ φ ( y ) with values in self-adjoint operators on H such that φ ( y ′ ) = e − i e iˆ φ ( y ) e iˆ 2 y · ωy ′ e iˆ φ ( y + y ′ ) , ˆ φ ( ty ) = tφ ( y ) , t ∈ R ˆ φ ( y ) are called field operators. It is easy to show that they depend linearly on y and satisfy the Heisenberg commutation relations on appropriate domains.
Consider a regular CCR representation Y ∋ y �→ ˆ φ ( y ) . (1) Let R ∈ Sp ( Y ). Then Y ∋ y �→ ˆ φ ( Ry ) (2) is also a regular CCR representation. We say that (2) has been obtained from (1) by a Bogoliubov transformation.
One can ask whether there exists a unitary U such that φ ( y ) U ∗ = ˆ U ˆ φ ( Ry ) , y ∈ Y . Such a U is called a Bogoliubov implementer. If Y = R 2 d is finite dimensional, then it is possible to characterize all Bogoliubov implementers. They are products of operators of the form e i ˆ H , where ˆ H is a Bogoliubov Hamiltonian ˆ � b ij ˆ φ i ˆ H = φ j + c.
Let us describe two basic constructions of CCR representations in the symplectic case: 1. the Schr¨ odinger representation, 2. the Fock representation Strictly speaking, the former works only for a finite number of degrees of freedom. The latter works for any dimension of Y .
Consider the Hilbert space L 2 ( R d ). Let φ i denote the i th coor- dinate of R d . Let ˆ φ i denote the operator of multiplication by the π i the momentum operator 1 variable φ i on and ˆ i ∂ φ i . Then, R d ⊕ R d ∋ ( η, q ) �→ η · ˆ φ + q · ˆ (3) π is an irreducible regular CCR representation on L 2 ( R d ). (3) is called the Schr¨ odinger representation over the symplectic space R d ⊕ R d .
Let ( Y , ω ) be a finite dimensional symplectic space. Clearly, Y is always equivalent to R d ⊕ R d with the natural symplectic form. The Stone–von Neumann Theorem says that all irreducible reg- ular CCR representations over Y are unitarily equivalent to the Schr¨ odinger representation.
Let Z be a complex Hilbert space. Consider the bosonic Fock space Γ s ( Z ). We use the standard notation for creation/annihilation a ∗ ( z ), ˆ operators ˆ a ( z ), z ∈ Z . We equip Z with the symplectic form z · ωz ′ := Im( z | z ′ ) . The following regular CCR representation is called the Fock repre- sentation. φ ( z ) := 1 Z ∋ z �→ ˆ a ∗ ( z ) + ˆ � � √ ˆ a ( z ) . 2
BOGOLIUBOV TRANSFORMATIONS AND QUADRATIC HAMILTONIANS IN FOCK REPRESENTATION For simplicity, we will assume that the one-particle space is finite dimensional: Z = C m . Operators on C m are identified with m × m matrices. If h = [ h ij ] is a matrix, then h , h ∗ and h # will denote its complex conjugate, hermitian conjugate and transpose.
We are interested in operators on the bosonic Fock space Γ s ( C m ). a ∗ a i , ˆ ˆ j will denote the standard annihilation and creation operators on Γ s ( C m ), where ˆ a ∗ i is the Hermitian conjugate of ˆ a i , a ∗ a ∗ [ˆ a i , ˆ a j ] = [ˆ i , ˆ j ] = 0 , a ∗ [ˆ a i , ˆ j ] = δ ij .
It is convenient to consider the doubled Hilbert space C m ⊕ C m equipped with the complex conjugation J ( z 1 , z 2 ) = ( z 2 , z 1 ) . (4) Operators that commute with J have the form � � p q R = (5) , q p and will be called J -real.
We also introduce the charge form � � 1 l 0 S = (6) . 0 − 1 l We say that a J -real operator � � p q R = (7) . q p is symplectic if R ∗ SR = S.
Here are the equivalent conditions p ∗ p − q # q = 1 l , p ∗ q − q # p = 0 , pp ∗ − qq ∗ = 1 l , pq # − qp # = 0 . We denote by Sp ( R 2 m ) the group of all symplectic transformations.
Note that pp ∗ ≥ 1 p ∗ p ≥ 1 l , l . Hence p − 1 and p ∗− 1 are well defined, and we can set d 1 := q # ( p # ) − 1 , d 2 := qp − 1 . Note that d # 1 = d 1 , d 2 = d # 2 . One has the following factorization: ( p ∗ ) − 1 0 � � � � � � 1 l d 2 1 l 0 R = (8) . 0 1 l 0 d 1 1 l p
In the present context, U is a (Bogoliubov) implementer of a symplectic transformation R if i U ∗ = p ij ˆ a ∗ a ∗ U ˆ j + q ij ˆ a j , a i U ∗ = q ij ˆ a ∗ U ˆ j + p ij ˆ a j . Every symplectic transformation has an implementer, unique up to a choice of a phase factor.
We will need a compact notation for double annihilators/creators: if d = [ d ij ] is a symmetric matrix, then � a ∗ ( d ) = a ∗ a ∗ ˆ d ij ˆ i ˆ j , ij � ˆ a ( d ) = d ij ˆ a i ˆ a j , ij
We have the following canonical choices: the natural implementer U nat R , and a pair of metaplectic implementers ± U met R : := | det pp ∗ | − 1 4 e − 1 1 a ∗ ( d 2 ) Γ U nat 2 ˆ ( p ∗ ) − 1 � 2 ˆ a ( d 1 ) , � e R := ± (det p ∗ ) − 1 2 e − 1 1 a ∗ ( d 2 ) Γ ± U met 2 ˆ ( p ∗ ) − 1 � 2 ˆ a ( d 1 ) . � e R Bogoliubov implementers fom a group called sometimes the c -metaplectic group Mp c ( R 2 m ). Metaplectic Bogoliubov imple- menters form a subgroup of Mp c ( R 2 m ) called the metaplectic group Mp ( R 2 m ). We have an obvious homomorphism Mp c ( R 2 m ) ∋ U �→ R ∈ Sp ( R 2 m ).
Various homomorphisms related to the metaplectic group can be described by the following diagram 1 1 1 ↓ ↓ ↓ 1 → → U (1) → U (1) → 1 Z 2 ↓ ↓ ↓ 1 → Mp ( R 2 m ) → Mp c ( R 2 m ) → U (1) → 1 ↓ ↓ ↓ 1 → Sp ( R 2 m ) → Sp ( R 2 m ) → 1 ↓ ↓ 1 1
Of special importance are positive symplectic transformations. They satisfy p = p ∗ , p > 0 , q = q # . (9) For such transformations d 1 = d 2 will be simply denoted by d := q ( p # ) − 1 For positive symplectic transformations the natural implementer coincides with one of the metaplectic implementers: := det p − 1 2 e − 1 1 a ∗ ( d ) Γ U nat 2 ˆ p − 1 � 2 ˆ a ( d ) . � e R
By a quadratic classical Hamiltonians, we will mean an expression of the form � � � h ij a ∗ g ij a ∗ i a ∗ H = 2 i a j + j + g ij a i a j , where a i , a ∗ j are classical (commuting) variables such that a ∗ i is the complex conjugate of a i and the following Poisson bracket relations hold: { a i , a j } = { a ∗ i , a ∗ j } = 0 , { a i , a ∗ j } = − i δ ij . We will assume that h = h ∗ , g = g # .
Classical Hamiltonians can be identified with self-adjoint J -real operators on the doubled space: � � h g H = , g h We also introduce � � h g B := SH = . − g − h
By a quantization of H we will mean an operator on the bosonic Fock space Γ s ( C m ) of the form � � � a ∗ a ∗ a ∗ g ij ˆ i ˆ j + g ij ˆ a i ˆ a j + 2 h ij ˆ i ˆ a j + c, where c is an arbitrary real constant.
Two quantizations of H are especially useful: the Weyl quanti- H w and the normally ordered (or Wick) quantization ˆ zation ˆ H n : H w := ˆ � � � � a ∗ a ∗ a ∗ a ∗ g ij ˆ i ˆ j + g ij ˆ a i ˆ a j + h ij ˆ i ˆ a j + h ij ˆ a j ˆ i , H n := ˆ � � � a ∗ a ∗ a ∗ g ij ˆ i ˆ j + g ij ˆ a i ˆ a j + 2 h ij ˆ i ˆ a j . The two quantizations obviously differ by a constant: H w = ˆ H n + Tr h. ˆ H w ∈ Mp ( R 2 m ). For any quadratic Hamiltonian H , we have e i t ˆ
Theorem Suppose that H > 0. 1. B has real nonzero eigenvalues. 2. sgn( B ) is symplectic. 3. K := S sgn B is symplectic and has positive eigenvalues. 1 4. Using the positive square root, define R := K 2 . Then R is symplectic and diagonalizes H . That means, for some h 1 , � � 0 h 1 R ∗− 1 HR − 1 = . 0 h # 1
Here is an alternative exppression for K : 2 � − 1 1 1 1 1 2 � 2 H K = H H 2 SHSH 2 . On the quantum level, if R diagonalizes H , then the correspond- ing unitary Bogoliubov implementers U remove double annihila- tors/creators from ˆ H : H w U ∗ = 2 h 1 ,ij ˆ U ˆ a ∗ a j + E w , i ˆ H n U ∗ = 2 h 1 ,ij ˆ U ˆ a ∗ a j + E n , i ˆ where E w , resp. E n is the infimum of ˆ H w , resp. of ˆ H n .
We can compute the infimum of the Bogoliubov Hamiltonians The simplest expression is valid for the Weyl quantization, which we present in various equivalent forms: √ H w = 1 E w := inf ˆ B 2 2Tr = 1 � 1 1 2Tr H 2 SHSH 2 B 2 � d τ = Tr B 2 + τ 2 2 π � 1 h 2 − gg ∗ � 2 − hg + gh # = 1 2Tr g ∗ h − h # g ∗ h # 2 − g ∗ g
Recommend
More recommend
