3.THE ALGEBRA OF CANONICAL COMMUTATION RELATIONS
3.1 Fock spaces
In classical mechanics a point system is characterized by its coordinates Qi(t) and impulsion Pi(t), i = 1 ...n. In the Hamiltonian description of motion equa- tions there exists a fundamental function H(P,Q) of motion, which describes the system and satisfies Euler-Lagrange equations: ∂H ∂H = Q˙ i , = P˙i . ∂Pi ∂Qi − If f(P,Q) is a functional of the trajectory, we then have the evolution equation df ∂f ∂P ∂f ∂Q = + dt ∂P ∂t ∂Q ∂t or else df = f,H dt { } where g,f denote the Poisson bracket of f by h: { } ∂g ∂h ∂g ∂h g, h = . { } ∂P ∂Q − ∂Q ∂P In particular we have P ,P = Q ,Q =0 { i j} { i j} P ,Q = δ . { i j} ij It happens that it is not exactly the definitions of the Pi and Qi which is important, but the relations above. Indeed, a change of coordinates P 0(P,Q), Q0(P,Q) will give rise to the same motion equations if and only if P 0 and Q0 satisfy the relations above. In quantum mechanics it is essentially the same situation. We have a self- adjoint operator H (the Hamiltonian) which describes all the evolution of the system via the Schr¨odinger equation d i¯h ψ(t) = H ψ(t) . dt There are also self-adjoint operators Qi, Pi which represent the position and the impulsion of the system and which evolve following itH itH Qi(t) = e Qie itH itH Pi(t) = e− Pie . Thus any observable A defined from P and Q satisfies the evolution equation d i A(t) = [A(t),H] dt −¯h where [ , ] denotes the commutator. · · 1 But the operators Pi, Qi satisfy the relation
[Pi,Pj]=[Qi,Qj]=0
[Qi,Pj] = i¯hδij I .
Once again, it is not the choice of the representations of Pi and Qi which is important, it is the relations above. It is called commutation relation. In quantum field theory we have an infinite number of degrees of freedom. The operators position and impulsion are indexed by IR3 (for example): we have a field of operators and the relations [P (x),P (y)]= [Q(x),Q(y)]=0 [Q(x),P (y)] = i¯hδ(x y)I . 1 − 1 If one puts a(x) = (Q(x) + iP (x)) and a∗(x) = (Q(x) iP (x)) then a(x) √2 √2 − and a∗(x) are mutually adjoint and satisfy the canonical commutation relations (CCR)
[a(x),a(y)]=[a∗(x),a∗(y)]=0
[a(x),a∗(y)]=¯hδ(x y)I . − Actually it happens that these equations are valid only for a particular family of particles: the bosons (photons, mesons, gravitons,...). There is another family of particles: the fermions (electrons, muons, neutrinos, protons, neutrons, baryons,...) for which the correct relations are the canonical anticommutation relations (CAR)
b(x),b(y) = b∗(x),b∗(y) =0 { } { } b(x),b∗(y) =¯hδ(x y)I { } − where A,B = AB + BC is the anticommutator of operators. A{ natural} problem, which has given rise to a huge literature, is to find concrete realisations of these relations. Let us see the simplest example: find two self-adjoint operators P and Q such that QP PQ = i¯hI . − In a certain sense there is only one solution. This solution is realized on L2(IR) by d Q = x (multiplication by x) and P = i¯h dx . It is the Schr¨odinger representation of the CCR. But in full generality this problem is not well-posed. We need to be able to define the operators PQ and QP on good common domains. One can construct pathological counter-examples (Reed-Simon). The problem is well-posed if we transform it in terms of bounded operators. i(xP yQ) Let Wx,y = e− − and Wz = Wx,y when z = x + iy C. We then have the Weyl commutation relations ∈ i z,z0 WzWz0 = e− =h iWz+z0 . Posed in these terms the problem has only one solution: the symmetric Fock space (Stone-von Neumann theorem). The anticommutation relations as they are written with b(x) and b∗(x) have a more direct solution for b(x) and b∗(x) have to be bounded. We will come back to that later.
2 The importance of Fock space comes from the fact they give an easy realization of the CCR and CAR. They are also a natural tool for quantum field theory, second quantization... (all sorts of physical important notions that we will not develop here). The physical ideal around Fock spaces is the following. If is the Hilbert space describing a system of one particle, then describes aH system H ⊗n H consisting of two particles of the same type. The space ⊗ = , n-fold, nH H⊗···⊗H describes n such particles. Finally the space n IN ⊗ describes a system where there can be any number of such particles which⊕ ∈ canH disappear (annihilate) or be created. But depending on the type of particles (bosons or fermions) we deal with, n there are some symmetries which force to look at certain subspaces of n ⊗ . We did not aim to describe the physics behind Fock spaces (we are not able⊕ H to), but we just wanted to motivate them. Let us come back to mathematics.
Let be a complex Hilbert space. For any integer n 1 put H ≥ n ⊗ = H H⊗···⊗H the n-fold tensor product of . That is, the Hilbert space obtained after completion H of the pre-Hilbert space of finite linear combinations of elements of the form u1 u , with the scalar product ⊗ ···⊗ n u u , v v = u , v u , v . h 1 ⊗···⊗ n 1 ⊗···⊗ ni h 1 1i···h n ni For u ,...,u we define the symmetric tensor product 1 n ∈ H 1 u u = u u , 1 ◦···◦ n n! σ(1) ⊗···⊗ σ(n) σ Sn X∈ where Sn is the group of permutations of 1, 2 ...n , and the antisymmetric tensor product { } 1 u u = ε u u , 1 ∧···∧ n n! σ σ(1) ⊗···⊗ σ(n) σ Sn X∈ where εσ is the signature of the permutation σ. n The closed subspace of ⊗ generated by the u1 un (resp. u1 un) is n n H ◦···◦ ∧···∧ denoted ◦ (resp. ∧ ). It is called the n-fold symmetric (resp. antisymmetric) tensor productH of .H H n Sometimes, when the notation is clear, one denotes by n the space ⊗ , n n H H ◦ or ∧ , and one calls it the n-th chaos of . In any of the three cases we put H H H = C . H0 The element 1 C = 0 plays an important role. One denotes it by 1l (usually by Ω in the literature)∈ H and one calls it the vacuum vector. If one computes 1 u u , v v = ε ε u , v u , v h 1 ∧···∧ n 1 ∧···∧ ni (n!)2 σ τ h σ(1) τ(1)i···h σ(n) τ(n)i σ,τ Sn X∈ one finds 1 det( u , v ) . n! h i ji ij 3 n In order to remove the n! factor we put a scalar product on ∧ which is n H different from the one induced by ⊗ , namely: H u1 un, v1 vn = det( ui, vj )ij . h ∧···∧ ∧···∧ i∧ h i This way, we have 2 2 u1 un = n! u1 un . k ∧···∧ k∧ k ∧···∧ k⊗ n In the same way, on ◦ we put H u1 un, v1 vn = per( ui, vj )ij , h ◦···◦ ◦···◦ i◦ h i where per denotes the permanent of the matrix (that is, the determinant without the minus signs). This way we get 2 2 u1 un = n! u1 un . k ◦···◦ k◦ k ◦···◦ k⊗ We call free (or full) Fock space over the space H ∞ n Γ ( ) = ⊗ . f H H n=0 M We call symmetric (or bosonic) Fock space over the space H ∞ n Γ ( ) = ◦ . s H H n=0 M We call antisymmetric (or fermionic) Fock space over the space H ∞ n Γ ( ) = ∧ . a H H n=0 M It is understood that in the definition of Γf ( ), Γs( ) and Γa( ) each of the n n n H H H spaces ⊗ , ◦ or ∧ is equipped with its own scalar product , , , H H H h· ·i⊗ h· ·i◦ or , . In other words, the elements of Γf ( ) (resp. Γs( ), Γa( )) are those h· ·i∧ n H n n H H series f = n IN fn such that fn ⊗ (resp. ◦ , ∧ ) for all n and ∈ ∈ H H H 2 2 P f = fn < k k k kε ∞ n IN X∈ for ε = (resp. , ). If one⊗ want to◦ ∧ write everything in terms of the usual tensor norm, we simply n have that an element f = n IN fn is in Γs( ) (resp. Γa( )) if fn ◦ (resp. n ∈ H H ∈ H ∧ ) for all n and H P 2 2 f = n! fn < . k k k k⊗ ∞ n IN ∈ X 2 The simplest case is obtained by taking = C, this gives Γs(C) = ` (IN). If m H is of finite dimension n then ∧ = 0 for m>n and thus Γa( ) is of finite H n H H dimension 2 ; this is never the case for Γs( ). In physics, one usually consider bosonicH or fermionic Fock spaces over = L2(IR3). H 2 + In quantum probability it is the space Γs(L (IR )) which is important for quantum stochastic calculus (we will meet this space during the second semester).
4 We now only consider symmetric Fock spaces Γs( ). For a u one notes that u u = u Hu. The coherent vector (or exponential vector∈ H ) associated to u◦···◦is ⊗···⊗ u n ε(u) = ⊗ n! n IN X∈ so that u,v ε(u), ε(v) = eh i h i in Γ ( ). s H Proposition 3.1 – The vector space of finite linear combinations of coherent E vectors, is dense in Γs( ). Every finite familyH of coherent vectors is linearly independent. Proof Let us prove the independence. Let u ...u . The set 1 n ∈ H E = u ; u, u = u, u , i,j ∈ H h ii 6 h ji for i = j, is open and dense in . Thus the set E is non empty. Thus 6 H i,j i,j there exists a v such that the θj = v, uj are two by two different. Now, if n ∈ H h i T i=1 αi ε(ui) = 0 this implies that n n P zθi 0 = ε(zv), αiε(ui) = αie i=1 i=1 X X for all z C. Thus the αi all vanish and the family ε(u1) ...ε(un) is free. In order∈ to show the density, we first notice that the set u u, u is total in Γ ( ) for { ◦···◦ ∈ H} s H n u u = (ε u + + ε u )◦ . 1 ◦···◦ n 1 1 ··· n n εi= 1 n X± n d But u◦ = n ε(tu) . This gives the result. dt t=0
Corollary 3.2 – If S is dense subset, then the space (S) generated by the ε(u), u S, is dense in⊂Γ H( ). E ∈ s H Proof
We have 2 2 2 u e v u,v ε(u) ε(v) = ek k + e k k 2 eh i . k − k − < Thus the mapping u ε(u) is continuous. We now conclude easily from Proposi- tion 3.1. 7→
Theorem 3.3 – Let 1, 2 be two Hilbert spaces. Then there exists a unique unitary isomorphismH H U : Γ ( ) Γ ( ) Γ ( ) s H1 ⊕ H2 −→ s H1 ⊗ s H2 ε(u v) ε(u) ε(v). ⊕ 7−→ ⊗ 5 Proof
The space ( i) is dense in Γs( i), i =1, 2, and ε(u) ε(v) ; u 1, v is total in ΓE (H ) Γ ( ). Furthermore,H we have{ ⊗ ∈ H ∈ H2} s H1 ⊗ s H2 u v,u0 v0 ε(u v), ε(u0 v0) = eh ⊕ ⊕ i h ⊕ ⊕ i u,u0 + v,v0 = eh i h i u,u0 v,v0 = eh ieh i
= ε(u), ε(u0) ε(v), ε(v0) h ih i = ε(u) ε(v), ε(u0) ε(v0) . h ⊗ ⊗ i Thus the mapping U is isometric. One concludes easily.
An example we will follow all along this chapter: the space Γs(C). It is equal to `2(IN) but it can be advantageously interpreted as L2(IR). Indeed, let U 2 be the mapping from Γs(C) to L (IR) which maps ε(z) to the function fz(x) = 2 2 1/4 zx s /2 x /4 (2π)− e − − . It is easy to see that U extends to a unitary isomorphism. We will come back to this example later.
There is an interesting characterization of the space Γ ( ) which says roughly s H that Γs( ) is the exponential of . Idea which is already confirmed by Theorem 3.3. H H
Theorem 3.4 – Let be a separable Hilbert space. If K is a Hilbert space such that there exists a mappingH λ : K H −→ u λ(u) 7−→ satisfying u,v i) λ(u), λ(v) = eh i for all u, v ii) hλ(u) ; u i is total in K. ∈ H { ∈ H} Then there exists a unique unitary isomorphism U : K Γ ( ) −→ s H λ(u) ε(u). 7−→
Proof Clearly U is isometric and maps a dense subspace onto a dense subspace.
It is useful to stop a moment in order to describe Γs( ) when is of the form L2(E, , m). We are going to see that if (E, , m)H is a measured,H non atomic, σ-finite,E separable measured space then Γ(L2E(E, , m)) can be written as L2( , , µ) for an explicit measured space ( , , µ). E P EP P EP 6 2 n 2 n n n If = L (E, , m), then ⊗ interprets naturally as L (E , ⊗ , m⊗ ) and n H E 2 n Hn n E ◦ interprets as Lsym(E , ⊗ , m⊗ ) the space of symmetric, square integrable functionsH on En. E If f(x1 ...xn) is a n-variable symmetric function on E then
f(x1,...,xn) = f(xσ(1),...,xσ(n)) for all σ Sn. If the xi are two by two different we thus can see f as a function on the set∈ x ,...,x . But as m is non atomic, almost all the (x ...x ) En { 1 n} 1 n ∈ satisfy xi = xj once i = j. An element of Γs( ) is of the form f = n IN fn 6 6 H ∈ where each fn is a function on the n-element subsets of E. Thus f can be seen as a function on the finite subsets of E. P More rigorously, let be the set of finite subsets of E. Then = n IN n P P ∪ ∈ P where 0 = and n is the set of n-elements subsets of E. Let fn be an element 2 P n {∅}n Pn of L (E , ⊗ , m⊗ ), we define f on by sym E P f(σ)=0 if σ and σ = n , f( x ,...,x ) = f (x ,...,x ) otherwise.∈ P | | { 1 n} n 1 n Let be the smallest σ-field on which makes all these functions measurable on E.P P P n Let ∆n E be the set of (x1 ...xn) such that xi = xj once i = j. By the non-atomicity⊂ of m, we have m(En ∆n)=0. For F 6 we put 6 \ ∈EP ∞ 1 µ(F ) = 1l (F ) + 1lF n (x1,...,xn)dm(x1) dm(xn) . ∅ n! ∩P ··· n=1 ∆n X Z For example, if E = IR with the Lebesgue structure, then n can be identified n P n to the increasing simplex Σn = x1 <
Theorem 3.4 – The mapping πu ε(u) extends to a unitary isomorphism from 2 2 7−→ L ( , , µ) onto Γs(L (E, , m)). P EP E Proof u,v Clearly πu,πv = eh i = ε(u), ε(v) . The set of functions πu is total in L2( , , µ).h One concludesi easily.h i P EP 3.2 Creation and annihilation operators We now come back to general symmetric and antisymmetric Fock spaces Γ ( ) and Γ ( ). s H a H 7 For u we define the following operators: ∈ H n (n+1) a∗(u): ◦ ◦ u H u −→u u H u 1 ◦···◦ n 7−→ ◦ 1 ◦···◦ n n (n+1) b∗(u): ∧ ∧ u H u −→u u H u 1 ∧···∧ n 7−→ ∧ 1 ∧···∧ n n (n 1) a(u): ◦ ◦ − u H u −→n u, u u H u u 1 ◦···◦ n 7−→ i=1h ii 1 ◦···◦ i ◦···◦ n n P (n 1) b(u): ◦ ◦ − b u H u −→n ( 1)i u, u Hu u u . 1 ∧···∧ n 7−→ i=1 − h ii 1 ∧···∧ i ∧···∧ n These operators are respectivelyP called bosonic creation operator, fermionic creation operator, bosonic annihilation operator and fermionic annihilationb opera- tor. Notice that a∗(u) and b∗(u) depend linearly on u, whereas a(u) and b(u) depend antilinearly on u. Actually, one often finds in the literature notations with
“bras” and kets”: a∗u , b∗u , a u , b u . h | h | Note that | i | i
a∗(u)1l= b∗(u)1l= u a(u)1l= b(u)1l= 0 . All the operators above extend to the space Γf ( )(resp. Γf ( )) of finite sums s H a H of chaos that is those f = n IN fn Γs( ) (resp. Γa( )) such that only a ∈ ∈ H H finite number of fn do not vanish. This subspace is dense in the corresponding P Fock space. It is included in the domain of the operators a∗(u), b∗(u), a(u), b(u) (defined as operators on Γs( )(resp. Γa( ))), and it is stable under their action. On this space we have the followingH relations:H
a∗(u)f,g = f,a(u)g h i h i [a(u),a(v)]=[a∗(u),a∗(v)]=0
[a(u),a∗(v)] = u, v I h i b∗(u)f,g = f,b(u)g h i h i b(u),b(v) = b∗(u),b∗(v) =0 { } { } b(u),b∗(v) = u, v I { } h i In other words, when restricted to Γf ( ) (resp. Γf ( )) the operators a(u) s H a H and a∗(u) (resp. b(u) and b∗(u)) are mutually adjoint and they satisfy the CCR (resp. CAR).
Proposition 3.6 – For all u we have 2 ∈ H i) b∗(u) =0, ii) b(u) = b∗(u) = u . k k k k k k 8 Proof
The anticommutation relation b∗(u),b∗(u) = 0 means 2b∗(u)b∗(u) = 0, this gives i). { } We have
b∗(u)b(u)b∗(u)b(u) = b∗(u) b(u),b∗(u) b(u) 2 { } = u b∗(u)b(u) . k k Thus 4 2 b(u) = b∗(u)b(u)b∗(u)b(u) = u b∗(u)b(u) k k k k k k k k = u 2 b(u) 2 . k k k k As the operator b(u) is null if and only if u = 0 we easily deduce that b(u) = u . k k k k The identity i) expresses the so-called Pauli exclusion principle: “One cannot have together two fermionic particles in the same state”. The bosonic case is less simple for the operators a∗(u) and a(u) are never n (n 1) bounded. Indeed, we have a(u)v◦ = n u, v v◦ − , thus the coherent vectors are in the domain of a(u) and h i a(u)ε(v) = u, v ε(v) . h i In particular 2 v /2 sup a(u)h sup a(u)e−k k ε(v) h =1 k k ≥ v k k k k ∈H = sup u, v = + . v |h i| ∞ ∈H Thus a(u) is not bounded. The action of a∗(u) can be also be made explicit. Indeed, we have
n d n a∗(u)v◦ = u v v = (u + εv)◦ . ◦ ◦···◦ dε ε=0
Thus ε(v) is in the domain of a∗(u) and
d a∗(u)ε(v) = ε(u + εv) . dε ε=0 The operators a(u) and a∗(u) are thus closable (they have a densely defined ad- joint). We extend them by closure, while keeping the same notations a(u), a∗(u).
Proposition 3.7 – We have a∗(u) = a(u)∗.
Proof f On Γ ( ) we have f,a(u)g = a∗(u)f,g . We extend this relation to f s H h i h i ∈ Dom a∗(u). The mapping g f,a(u)g is thus continuous and f Dom a(u)∗. 7→ h i ∈ We have proved that a∗(u) a(u)∗. ⊂ Conversely, if f Dom a(u)∗ and if h = a(u)∗f. We decompose f and h in chaoses: f = f and∈ h = h . We have f,a(u)g = h, g for all g Γf ( ). n n n n h i h i ∈ s H P P 9 n Thus, taking g ◦ we get fn 1,a(u)g = hn,g that is, a∗(u)fn 1,g = ∈ H h − i h i h 2 − i hn,g . This shows that hn = a∗(u)fn 1 This way a∗(u)fn is finite, f h i − n k k belongs to Dom a∗(u) and a∗(u)f = a(u)∗f. P 2 3 n In physics, the space is often L (IR ). An element hn of ◦ is thus a symmetric function of n variablesH on IR3. With our definitions we haveH
a(f)hn (x1 ...xn 1) = hn(x1 ...xn 1,x)f(x) dx − − Z and n+1
a∗(f)hn (x1 ...xn+1) = hn(x1 ... xˆi ...xn+1)f(xi) . i=1 X But in the physic literature one often use creation and annihilation operators indexed by the points of IR3, instead of the elements of L2(IR3). One can find there a(x) and a∗(x) formally defined by
a(f) = f(x)a(x) dx Z a∗(f) = f(x)a∗(x) dx Z with
a(x)hn (x1 ...xn 1) = hn(x1 ...xn 1,x) − − n+1 a∗(x)h (x ...x ) = δ(x x )h (x ... xˆ ...x ) . n 1 n+1 − i n 1 i n+1 i=1 X If one comes back to our example Γ (C) L2(IR), we have the creation and s ' annihilation operators a∗(z), a(z), z C. They are actually determined by two ∈ operators a∗ = a∗(1) and a = a(1). They operate on coherent vectors by d a ε(z) = z(ε(z), a∗ε(z) = ε(z + ε) . ∈ dε ε=0 On L2(IR) this gives
d x af (x) = zf (x) = + f (x) z z dx 2 z d x d a∗fs(x) = fz+ε(x) =(x z)fz(x) = fz(x) . dε ε=0 − 2 − dx The operators Q = a + a∗ and P = i(a a∗) are thus respectively represented by d 2 − the operator x and 2i dx on L (IR). That is, the Schr¨odinger representation of the CCR (withh ¯ = 2).
To conclude in this section, note that the operator Q = a+a∗ is an observable, in the physical sens. If we are given a state on L2(IR), for example the vaccum state 1l,then the observable Q has a natural probability law. This law isthe one which describes the probabilistic behaviour of the observable Q if one tries to
10 measure it in the state 1l. One can also see this law in the following way: the mapping t < 1l , eitQ1l > satisfies the Bochner criterion and is thus the Fourier transform of7→ some probability measure µ. From the postulats of quantm mechanics, the n-th moment of this law are given by < 1l , Qn1l >. Passing by the L2(IR) interpretation, this quantity equals
2 n 1 n x /2
If one is given an operator A from an Hilbert space to another , it is H K possible to rise, in a natural way, this operator into an operator Γ(A) from Γs( ) to Γ (K) (and in a similar way from Γ ( ) to Γ (K)) by putting H s a H a Γ(A)(u u ) = Au Au . 1 ◦···◦ n 1 ◦···◦ n One easily sees that Γ(A)ε(u) = ε(Au) . This operator Γ(A) is called the second quantization of A. One must be careful that even if A is bounded operator, Γ(A) is not bounded in general. Indeed, if A > 1 then Γ(A) is not bounded. But on easily sees that k k Γ(AB)=Γ(A)Γ(B) and Γ(A∗)=Γ(A)∗.
In particular if A is unitary, then so is Γ(A). Even more, if (Ut)t IR is a ∈ strongly continuous one parameter group of unitary operators then so is (Γ(Ut)t IR). itH ∈ In other words, if Ut = e for some self-adjoint operator H, then Γ(Ut) = itH0 e for some self-adjoint operator H0. The operator H0 is denoted Λ(H) (or sometimes dΓ(H) in the litterature) and called the differential second quantization of H. One easily checks that n Λ(H) u u = u Hu u 1 ◦···◦ n 1 ◦···◦ i ◦···◦ n Xi=1 and Λ(H)1l= 0. In particular, if H = I we have Λ(I) u u = n u u . 1 ◦···◦ n 1 ◦···◦ n This operator is called number operator.
11 Proposition 3.7 – We have
Λ(H)ε(u) = a∗(Hu)ε(u) .
Proof n We have Λ(H)u◦ = n(Hu) u u. Thus ◦ ◦···◦ n (n 1) (n 1) u◦ u◦ − u◦ − Λ(H) =(Hu) = a∗(Hu) . n! ◦ (n 1)! (n 1)! − −
Proposition 3.8 – For all u , we have ∈ H Λ( u u ) = a∗u a u . | ih | | i h |
Proof Indeed, we have
Λ( u u )ε(v) = a∗( u, v u)ε(v) | ih | h i = u, v a∗(u)ε(v) h i = a∗u a u ε(v) . | i h |
Coming back to L2(IR), there is only one differential second quantization:
Λ(I)=Λ= a∗a . We obtain x d x d Λ = + 2 − dx 2 dx x2 d2 1 = 4 − dx2 − 2 that is 1 x2 d2 Λ + = 2 4 − dx2 the Hamiltonian of the one dimensional harmonic oscillator. Note that Λ is self-adjoint and its law in the vacuum state is just the Dirac mass in 0, for Λ1l= 0.
3.4 Weyl operators
Let be a Hilbert space. Let G be the group of displacements of that is, H H G = (U, u) ; U ( ), u , ∈ U H ∈ H where ( ) is the group of unitary operators on . This group acts on by U H H H (U, u)h = Uh + u .
12 The composition law of G is thus (U, u)(V, v)=(UV,Uv + u) and in particular 1 (U, u)− =(U ∗, U ∗u) . − For every α =(U, u) G one defines the Weyl operator Wα on Γs( ) by ∈ 2 H u /2 u,Uv Wα ε(v) = e−k k −h iε(Uv + u) . In particular i u,Uv WαWβ = e− =h iWαβ for all α = (U, u), β = (V, v) in G. These are called the Weyl commutation relations.
Proposition 3.8 – The Weyl operators Wα are unitary. Proof We have 2 u Uk,u u,U` Wαε(k), Wαε(`) = e−k k −h i−h i ε(Uk + u), ε(U` + u) h i 2 h i u Uk,u u,U` Uk+u,U`+u = e−k k −h i−h ieh i Uk,U` k,` = eh i = eh i = ε(k), ε(`) . h i Thus Wα extends to an isometry. But we furthermore have i u, UU ∗ u WαWα−1 = e− =h − iWαα−1 2 i ( u ) = e− = −k k W(I,0) = I .
Thus Wα is invertible. The mapping : G (Γ( )) −→ U H α W 7−→ α is a unitary projective representation of G.
If one considers the group G of (U, u, t), U ( ), u and t IR with ∈ U H ∈ H ∈ (U, u, t)(V,v,s)=(UV,Uv + u, t + s + u, Uv ) ; e =h i we obtain the so-called Heisenberg group of . The mapping (U, u, t) W eit H 7−→ (U,u) is thus a unitary representation of G.
Conversely, if W(U,u,t) is a unitarye representation of the Heisenberg group of we then have H W(U,u,t) = W(U,u,0)W(I,0,t) and W(I,0,t) = W(I,0,s)W(I,0,t+s) .
13 This means that itH W(U,u,t) = W(U,u,0)e for some self-adjoint operator H, and the W(U,u,0) satisfy the Weyl commutation relations.
If we come back to our Weyl operators W(U,u) one easily sees that
W(U,u) = W(I,u)W(U,0) .
By definition W(U,0)ε(k) = ε(Uk) and thus
W(U,0) = Γ(U) .
Finally, write Wu for W(I,u). Then i u,v WuWv = e− =h iWu+v . These relations are often also called Weyl commutation relations. As a consequence (W(I+tu))t IR is a unitary group; it is strongly continuous (exercise). ∈ Proposition 3.9 – We have 1 ∗ it i (a(u) a (u)) W(I,tu) = e − .
Proof
1 d 1 d t2 2 2 u t u,k W(I,tu)ε(k) = e− k k − h iε(k + tu) i dt t=0 i dt t=0 1 1 d = u, k ε(k) + ε(k + tu) − i h i i dt t=0 1 = ( a(u) + a∗(u))ε(k ) . i −
Coming back to our example on L2(IR), the Weyl operators are defined by 2 |z| 0 2 zz Wzε(z0) = e− − ε(z + z0) . These operators are very helpfull for computing the law of some observables.
2 Proposition 3.10 – The observable 1/i (za∗ za) follows a law (0, z ) in the vaccum state. − N | | Proof We have t(za∗ za) < 1l , e − 1l > = < 1l , Witz1l >
= < (0) , Witz (0) > E E 2 2 t z /2 = < (0) , (itz) >e− | | E2 2 E t z /2 = e− | | .
14 Proposition 3.11 – The observable Λ+αI follows the law δα in the vaccum state.
Proof Indeed, < 1l , (Λ + αI)n1l > = αn< 1l , 1l > = αn.
Let us now compute the law, in the vaccum state, of the observable Λ + za∗ za + z 2 I, sum of the two previous observables. − | | Lemma 3.12 – 2 itΛ it(Λ+za∗ za+ z I) W ze Wz = e − | | . −
Proof 2 It suffices to show that W zΛWz = Λ + za∗ za + z I. We have − − | | < (z1) , W zΛWz (z2) > = < aWz (z1) , aWz (z2) > E − E E E = < (z1 + z) (z1 + z) , (z2 + z) (z2 + z) > 2 E E × z zz2 z1z e−| | − − × =(z z + z z + z z + z 2)ez1z2 . 1 2 1 2 | | 2 Proposition 3.13 – The law of the observable Λ + za∗ za + z I in the vaccum − | | state is the Poisson law ( z 2). P | | Proof We have 2 it(Λ+za∗ za+ z I) itΛ < 1l , e − | | 1l > = < Wz1l , e Wz1l > 2 z itΛ = e−| | < (z) , e (z) > 2 E2 it E z z e = e−| | e| | 2 it z (e 1) = e| | − .
Comments: This result may seem very surprising: the sum of a Gaussian variable and a deterministic one, gives a Poisson distribution! Of course such a phenomena cannot be realised with usual random variables. What does this mean? Actually it is one of the manifestation of the fact that the random phe- nomena attached to quantum mechanics cannot be modelized by usual probability theory.
There has been many attemps to give a probabilistic model to the stochastic phenomena of quantum mechanics, such as the “hidden variable theory” which tried to give a model of the randomness in measurement by saying that many
15 parameters of the systems are unknown to us (hidden variables) since the begining and that their effects appear at the measurement and give this uncertainty, this randomness. But it has been proved by Bell, that if one tries to attach random variables behind each observable and try to find (complicated) rules that explain the princi- ples of quantum mechanics, then this reaches an impossibility. Indeed, taking the spin of a particle in three well-choosen directions, one obtains three Bernoulli vari- ables, but their correlations cannot be obtained by any triplet of classical Bernoulli variables. What then can we do to express the probabilistic effects of measurement in a probabilistic language? Actually, one does need to look very far away. Quantum mechanics in itself, in its axioms, contains the germ of a new probability theory. Indeed, now accept to consider a probability space to be a couple ( , Ψ), where is a Hilbert space and Ψ is a normalized vector of ; instead of the usualH (Ω, ,PH). Accept to consider a random variable to be a self-adjointH operator A on , insteadF of a measurable function X : Ω IR. Accept that the probability distributionH of A under the state Ψ is the one described→ above: E < Ψ , 1l (A)Ψ >. 7−→ E Then what do we obtain ? Actually this probability theory, as stated here, is equivalent to the usual one when considering a single random variable. Indeed, a classical probabilistic situation (Ω, ,P,X) is easily seen to be also a quantum one ( , Ψ,A) by putting 2 F H = L (Ω, ,P ), Ψ = 1l and A = X the operator of multiplication by X. The (quantum)H F distribution of A is thenM the same as the (classical) distribution of X. Conversely, given a quantum triplet ( , Ψ,A), then by the spectral theorem the operator A can be represented as a multiplicationH operator on some measured space (by diagonalization). Where does the difference lie? When considering two non commuting observ- ables on (for example P,Q on L2(IR)), then each of them is a classical random variable,H but on its own probability space (they cannot be diagonalized simultane- ously). We have put together two classical random variables which have nothing to do together, in a same context. We have not stick the together by declaring them independant, there is a dependency ([Q,P ] = iI) which has some consequences (uncertainty principle for example) which cannot be expressed in classical terms. This is to say that what quantum mechanics teaches us is that the observables, under measurement, behave as true random variables, but each one with its own random, and that their interdependency cannot be expressed in a simpler way than the axioms of quantum mechanics. It is then not a surprise that adding two observables which do not commute we obtain a distribution which has nothing to do with the convolution of their respective distributions. This is the example above. One also obtains a surprising one with P 2 + Q2 (exercise).
16 3.5 The CCR algebra
We now denote by W (f) the Weyl operator W , f . (I,f) ∈ H Theorem 3.14 – Let be any (algebraic) subspace of a Hilbert space . There K H exists a C∗-algebra, denoted CCR( ) of operators on Γ( ), unique up to isomor- phism, generated by nonzero elementsK W (f), f , suchH that ∈K W (f)∗ = W ( f) for all f − i
Proof
The existence of a C∗-algebra satisfying the two conditions is obvious. It suffices to consider the C∗-algebra generated by the Weyl operators W (f),f of Γ( ). ∈K H We now give the proof of uniqueness but it can omited by the reader as it makes use of tools that are not pertinent for us. Put b(f,g) = exp( i
U (W R)(g)U ∗ = I R (g). i i × i i ⊗ b If i denotes the C∗-algebra generated by (Wi R)(g); g , then there exists a B-isomorphism τ from to such that{ × ∈ K} ∗ B1 B2 τ((W R)(g))=(W R)(g). 1 × 2 × Thus it would be sufficient now to find -isomorphisms τ from to such that ∗ i Ui Bi τ (W (g))=(W R)(g). i i i × Now set W = Wi. It suffices to show that n n
λi(W R)(fi) = λiW (fi) × i=1 i=1 X X
17 for all λi C, fi . The representation∈ ∈K W R is, via Fourier transform on `2( ), unitary equiv- × K alent to the representation W R on `2( , ) defined by × K H ((W R)(g)Ψ)(χ) = W (g)χ(g)Ψ(χ), χ × ∈ K and hence b b n b n b λ (W R)(f ) = sup λ χ(f )W (f ) . i × i i i i i=1 χ i=1 X ∈K X 2 The set of characters χg on of the form χg(f) = b(f,g) is dense in for it { } K K is a subgroup with annihilator zero. b Note that χg(f)W (f) = W (g)W (f)W (g)∗, thus b n n λ (W R)(f ) = sup λ χ (f )W (f ) i × i i g i i i=1 g i=1 X ∈K X n
= sup W (g) λiW (fi)W (g)∗ g i=1 ∈K X n
= λiW (fi) .
i=1 X
Proposition 3.15 – Let be a subspace of . It follows that CCR( ) = CCR( ) if and only if K= ⊂. H H K H K H Proof If = , then consider the representation of CCR( ) on `2( ) defined by K 6 H H H (W (g)F )(f) = b(f,g)F (f + g) with the same notation as in the proof of uniqueness in the above theorem. If g then ∈H\K n W (g) λ W (g ) F (f) = b(f,g) F (f + g) + − i i i=1 ! ! X n λ b(f,g g)F (f + g ) . − i i − i i=1 ! If F is supported by then X K n
W (g) λiW (gi) F F − ! ≥ || || i=1 X for the vector f F ( f + g) is orthogonal to each of the vectors f b(f,gi g)F (f + g i). 7→ 7→ − Therefore− inf W (g) A 1 A CCR( ) || − || ≥ ∈ K and hence W (g) CCR( ). 6∈ K 18