Theorem 2 (Darboux) Assume that the Poisson bracket is non-degenerate, i.e.
det θab �= 0. Then for any point p ∈ M there is a local neighborhood p ∈ U ⊂ M2n
with local coordinates q i , pj : U → R which satisfy the classical canonical commutation
relations. In other words,
�
�
0
11 n
(22)
θab =
−11 n 0
Now assume we have a function f on M such that {H, f } = 0. It follows that
f is a constant of motion, and therefore one can reduce the dynamical system to a
lower-dimensional one on the submanifold of Mf where f is constant. Moreover, the
Jacobi identity implies that the Poisson bracket of any two constants of motion is again
a constant of motion,
{H, {f1 , f2 }} = −{f1 , {f2 , H}} − {f2 , {H, f1 }} = 0
(23)
The system is called integrable if one can find n independent constants of motion, since
then one can indeed solve the system completely (using action-angle variables, HamiltonJacobi theory, etc., cf. [5]).
Example: n-Particle system.
A system of n point particles in an external potential is described by the 2n –
dimensional phase space (q i , pj ), i, j = 1, ..., n and Hamiltonian
H=
� 1
p2i + V (q i )
2m
i
i
(24)
Here V (q i ) can describe external as well as interaction potentials. Then the eom are
ṗi = {H, pi } = −
q̇i = {H, qi } =
∂V (q)
,
∂q i
pi
m
(25)
If the potential V has symmetries, then one can obtain conservation laws. Moreover,
these conserved quantities are generators of the underlying symmetry:
8
Symmetries and Noether’s theorem.
Now we assume that the underlying Lagrangian is invariant under a symmetry
q i → qsi = qsi (q)
(26)
i.e. under infinitesimal change of coordinates
δq i =
d i
�
q ≡ qi
ds
(27)
For example, translations in the n-particle system along a vector (ci ) correspond to
qsi = q i + sci , which is a symmetry if V (q i ) is constant along that direction. Then:
Theorem 3 The function
I(p, q) = pi
d i
�
q = pi q i
ds
(28)
is a constant of motion. Moreover, the associated Hamiltonian vector field XI = {I, .}
generates the symmetry, in the sense
δq i = {I, q i }.
(29)
The last statements follow by inspection. The conservation law is seen as follows
�
�
∂L �
d
dq i
dq i
�
I˙ = (ṗi )q i + pi
= i q i + pi
= L=0
dt
∂q
dt
ds
(30)
where the last step follows from the assumed symmetry. These statements constitute
Noethers theorem.
In summary, we have formulated classical (Hamiltonian) mechanics in an algebraic
way in terms of the Poisson algebra (A, {., .}), and a Hamiltonian H ∈ A which determines the dynamics via
f˙ = {H, f }.
(31)
A state is given by a point (p, q) ∈ M, which can be represented by a delta functions
δp,q in A. A measurement of the observable f amounts to evaluating f in the state δp,q ,
resulting in
�
f δp,q dV
(32)
f (p, q) =
M
9
Here dV = n!1 ω n is the canonical (Liouville) volume form on M defined in terms of the
symplectic structure (discussed later).
The above class of Poisson brackets has a general mathematical origin. Consider
classical Lagrangian mechanics on some configuration space C n with local coordinates
q i and Lagrangian L(q i , q̇ i ). Then the ”generalized velocity“ q̇ i is a tangent vector to C,
and the momenta pi = ∂∂L
define a cotangent vector on C, given by pi dq i . Now (q i , pj )
q̇ i
are in fact local coordinates of the cotangent bundle M2n = T ∗ C n . This is the phase
space of the system. It is a mathematical fact that the cotangent bundles always carry
a canonical non-degenerate Poisson structure (in fact a symplectic structure), which is
just the one obtained above. For more details see the excellent book [5] by Arnold.
However, Poisson brackets can also exist on compact manifolds, which are not obtained as above. Their quantization leads to interesting quantum spaces, which we will
study later.
3
3.1
Canonical quantization
Quantum mechanics
In Quantum mechanics, the above setup involving the commutative algebra A of (complexified) observables on a phase space M2n is replaced by a non-commutative algebra
A� of (complexified) observables realized as self-adjoint operators on a Hilbert space H.
Recall first that a Hilbert space H is a (finite-or infinite-dimensional) complete vector
space with a positive definite sesquilinear inner product �, �, i.e.
�ψ � , λ1 ψ1 + λ2 ψ2 � = λ1 �ψ � , ψ1 � + �λ2 ψ2 �,
�ψ, ψ � � = �ψ � , ψ�∗ ,
�ψ, ψ� ≥ 0,
�ψ, ψ� = 0 ⇔ ψ = 0
(33)
We need the notion of the adjoint A† of an operator A on H. It is defined implicitly by
�A† ψ, ψ � � = �ψ, Aψ � �
∀ψ, ψ � ∈ H
(34)
Hence observables are given by A† = A. This implies that the expectation value of any
observable is real,
�ψ, Aψ� ∈ R
10
(35)
This is essential for QM, where the lhs gives the expected result for a measurement of
A in the state ψ ∈ H. In QM one often uses the bra-ket notation, such that
ψ ≡ |ψ�
�ψ , ψ� ≡ �ψ � |ψ�
�
�A† ψ � , ψ� = �ψ � , Aψ� ≡ �ψ � |A|ψ�,
A† = A
(36)
In particular, the operator Pψ = |ψ��ψ| is a projection operator on a normalized vector
ψ ∈ H, �ψ|ψ� = 1.
Technically there are some subtleties - for ”unbounded“ operators one should restrict
the vectors to be in the dense ”domain of definition“ D ⊂ H of the operator. We will
largely ignore these subtleties here - they can be handled in the theory of operators on
Hilbert spaces developed by von Neumann and others.
Moreover, there is an important theorem (the spectral theorem) which states that
any selfadjoint operator on a Hilbert space can be diagonalized, and all eigenvalues are
real. In the simplest case of finite dimensional H, this is simply the statement that any
hermitean matrix A can be diagonalized by a unitary matrix,
A = U diag(λi )U −1 ,
Aψi = λi ψi
λi ∈ R
(37)
or equivalently
A=
n
�
Pi2 = Pi = |ψi ��ψi | = Pi† ,
λ i Pi ,
i=1
λi ∈ R
(38)
This in turn follows easily from the observation that eigenvectors for distinct eigenvalues are mutually orthogonal. The same is true for selfadjoint operators with discrete
spectrum (i.e. only isolated eigenvalues with finite multiplicity), and we can write
�
A=
λ i Pi
Pi2 = Pi = Pi†
(39)
i
In particular, Pi is the observable testing the result λi of observable A.
Now a (pure) state of a quantum mechanical system is given by a normalized vector
ψ ∈ H, up to a phase (or by H/ ∼ where ψ ∼ ψ � = zψ for z ∈ C\{0}). The crucial
difference the classical mechanics is the vector space structure of H. This implies the
superposition principle for states, which is at the heart of QM. This linear structure
means that QM is intrinsically simpler than CM. (recall that states in classical mechanics
are always delta functions on position space). Such a state can also be represented by
a density matrix
ρ = |ψ��ψ|
11
∈ A�
(40)
QM can then be summarized as follows: QM is based on the algebra A� = End(H) of
(complexified) observables i.e. operators on a Hilbert space H, and a Hamilton operator
H ∈ A� which determines the dynamics via
Ḟ =
i
[H, F ].
�
(41)
Real-valued functions f ∈ A correspond to selfadjoint operators F = F † ∈ A� . The set
of possible results for the QM observable F is given by the spectrum of F , which is real.
The expected result of a measurement of F for a system in the state ψ is given by
Eψ (F ) = �ψ, F ψ� = Tr(ρF )
(42)
This is very similar to the above algebraic formulation of CM, noting that the commutator takes the role of the Poisson bracket. Thus to quantize a classical mechanical
system (A, {., .}), we need a way to associate the classical observables with the quantum
mechanical ones.
3.2
Quantization of a classical system
A quantization of some classical system (A, M2n , {, }) as above is given by
1. a noncommuative algebra A� = End(H) of observables on some Hilbert space H,
and
2. a map
Q:
A → A� = End(H)
f �→ F = Q(f )
(43)
Note that A is a commutative algebra while A� is a non-commutative algebra. This
is non-commutativity is at the heart of Quantum mechanics, leading to uncertainty
relations etc.
The map Q is required to obtain an interpretation of the quantum mechanical system
in terms of classical observables, and in particular to replace the classical Hamilton
function H ∈ A with the Quantum mechanical Hamiltonian H = H † .
There are obvious questions:
• what is the appropriate Hilbert space H?
12
• how do we obtain the map Q?
• is Q unique?
These are the questions under investigation in this lecture.
Question 1) is usually simple - all infinite-dimensional (separable) Hilbert spaces
are equivalent. However the question becomes non-trivial for systems described by a
compact phase space (→ second part of the lecture).
The rough answer to the second question is given by Diracs prescription: the Poisson
bracket on M should be replaced by the commutator. More precisely, the following
”axioms“ seem natural:
1. (linearity Q1)
Q(λf + g) = rQ(f ) + Q(g),
λ∈R
(44)
2. (reality Q2)
Q(f )† = Q(f ∗ )
(45)
(then (44) extends to λ ∈ C). In particular, real functions should map to selfadjoint operators.
3. (Q3)
Q(1) = 11
(46)
4. (Lie-homomorphism Q4)
[Q(f ), Q(g)] = −i�Q({f, g})
tentative
(47)
Here � is the Planck constant. This is the crucial step: Note that the commutator
satisfies the same algebraic properties (11) as the Poisson brackets, except that
A� is non-commutative. This is the basis of the correspondence principle.
Finally, we need some ”irreducibility“ requirement:
5. (Irreducibility Q5) if {f1 , ..., fn } is a complete set of observables in A (i.e. the only
function which Poisson-commutes with all of them is ∼ 1), then {Q(f1 ), .., Q(fn )}
is a complete set of observables on End(H) i.e. the only operator which commutes
with all of them is 11 .
It also seems that the map Q should be (essentially) bijective, but sometimes
exceptions arise (e.g. for spin):
13
6. (Q6) Q : A → A� should be an isomorphism of vector spaces, up to subtleties
such as possible non-classical features.
To some extent Q6 is also related to Q5.
Unfortunately, it turns out that (47) is asking too much. This is the content of the
theorem of Groenewold and van Howe, which we will explain later.
lesson: Imposing axioms is dangerous !!
So we have to weaken this requirement, while keeping its essence. The correct
way to proceed is suggested by the correspondence principle, which states that the
quantum system should reduce to the classical system ”for large quantum numbers“, or
– equivalently – in the limit � → 0. Naively, the algebra would become commutative
in this limit, A� → A. However, the Poisson structure must still be incorporated. This
can be achieved by requiring the ”weak quantization requirement“
�
�
i
lim Q−1 [Q(f ), Q(g)] = {f, g}
(48)
�→0 �
In particular, higher-order corrections in � are not specified. Also there are technical
subtleties how precisely to define the limit. This means that such a quantization map
Q is not unique. In general, the quantization of a classical system is not unique.
To achieve uniqueness one can impose Q4 for some given (preferred?) complete set of
observables. In particular in the presence of symmetries, some observables are preferred,
suggesting a preferred quantization.
3.3
Basic example: Quantization of a point particle in R3 .
A classical point particle in R3 is described by its position q i ∈ R3 and its momentum
pi ∈ R3 , with classical canonical CR
{q i , q i } = 0 = {pi , pj },
{q i , pj } = δji
(49)
Note that the Darboux theorem makes this ”assumption“ almost empty. Moreover, they
form a complete set of classical observables.
Now we want to quantize these classical CCR. According to the rule Q4, the operators
Qi = Q(q i ), Pi = Q(pi ) should satisfy
[Qi , Pj ] = i�δji ,
[Qi , Qj ] = 0 = [Pi , Pj ]
14
(50)
These are the (quantum) canonical commutation relations ((q)-CCR), or the Heisenberg
algebra. The additional requirement Q5 tells us to look for irreducible representations
of this algebra.
First, note that finite-dimensional representations of the CCR are impossible (take
the trace).
Now the Stone- von Neumann theorem essentially states that there is a unique irreducible representation of the Heisenberg algebra, given by the usual
Schrödinger quantization: The Hilbert space H is the space of square-integrable functions on R3 ,
�
2
3
d3 qψ ∗ ψ < ∞}
(51)
H = L (R ) = {ψ(q);
and the operators Qi = Q(q i ), Pi = Q(pi ) are defined by
Qi ψ(q) = q i ψ(q)
∂
Pi ψ(q) = −i� i ψ(q)
∂q
(52)
These are self-adjoint operators with spectrum (i.e. the set of eigenvalues) being (−∞, ∞).
Strictly speaking there are no eigenvectors but only ”generalized eigenvectors” or spectral projectors, but this can be handled.
Thus the quantized algebra of observables is given by the operator algebra
A� = End(H)
(53)
and observables corresponding to real-valued functions are (formally) selfadjoint operators A = A† .
Strictly speaking, uniqueness requires slightly stronger assumptions. Consider the
exponentiated (“Weyl”) operators
U (ci ) = eic
jP
j
V (dj ) = eidj Q
,
j
(54)
Recall the Baker-Campbell-Haussdorf formula
1
eA eB = eA+B e 2 [A,B] ,
[A, [A, B]] = 0 = [B, [B, A]].
(55)
As a consequence, the Weyl operators satisfy the Weyl algebra
i
U (ci )V (dj ) = ei�c di V (dj )U (ci )
(56)
If these are “well-behaved” (strongly continuous) operators, then uniqueness follows.
15
An instructive counter-example is given by the quantization of a particle on a circle
S1 ∼
= R/Z: in local coordinates, (ϕ, ∂ϕ ) satisfies the same Heisenberg algebra but in fact
ϕ is not globally defined.
It is important to note that Schrödinger quantization is not some fundamental dogma
which falls from the sky, but is the result of a clear and “almost” unique (taking into
account the translation symmetry) procedure. In particular, it follows (!) that wavefunctions can be realized as square-integrable functions on configuration space.
3.4
The theorem of Groenewold and van Howe
Let P ol(R2n ) be the vector space of polynomials in 2n variables q i , pj . Then the following
fact holds:
Theorem 4 There exists no associative (operator) algebra A� with 11 together with an
irreducible quantization map
Q:
(P ol(R2n ), {., .}) → A�
(57)
such that Q(1) = 11 and
i�Q({f, g}) = [Q(f ), Q(g)]
(58)
irreducible means that if F ∈ A� commutes with the image of Q in A� then F ∼ 11 .
This means that Q4 (together with Q5) is too strong.
We sketch the proof (cf. [Waldmann], [Giulini] etc.) Actually one can drop the
assumption Q(1) = 11 (cf. [Waldmann])
proof
It is enough to consider the one-dimensional case. Assume we had such a Q. Define
Q := Q(q),
P := Q(p)
(59)
Then it follows that Q, P satisfy the CCR
[Q, P ] = i�Q(1) = i�11
(60)
Furthermore, it follows that
[Q, Q(f )] = Q({q, f }) = i�Q(
16
∂f
)
∂q
(61)