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)
and
[P, Q(f )] = Q({p, f }) = −i�Q(
∂f
)
∂p
(62)
It follows that if any element F = Q(f ) ∈ A� commutes with Q and P , then f = const
and therefore F ∼ 11 .
Now consider
[Q(q 2 ), P ] = i�Q({q 2 , p}) = 2i�Q(q) = 2i�Q
(63)
[Q(p2 ), Q] = −2i�P
(64)
and similarly
On the other hand, we have
[Q2 , P ] = 2i�Q
and
[P 2 , Q] = −2i�P
(65)
Thus (Q2 − Q(q 2 )) commutes with both Q and P , which using irreducibility implies
Q(q 2 ) = Q2 + c1 ,
Q(p2 ) = P 2 + c2
(66)
for c1 , c2 ∈ C. Similarly, noting that {q 2 , p2 } = 4pq it follows that
1
1
1
[Q(q 2 ), Q(p2 )] = [Q2 + c1 , P 2 + c2 ] = [Q2 , P 2 ]
i�
i�
i�
= 2(QP + P Q)
4Q(qp) =
(67)
Furthermore we have {qp, p2 } = 2p2 , hence
1
2(P 2 + c2 ) = 2Q(p2 ) = [Q(qp), Q(p2 )]
i�
�
1
1 �1
2
(QP + P Q), P + c2 =
[QP + P Q, P 2 ] = 2P 2
=
i� 2
2i�
(68)
It follows that C2 = 0, and similarly c1 = 0. In summary, we found
Q(q 2 ) = Q2 ,
Q(p2 ) = P 2 ,
1
Q(qp) = (P Q + QP )
2
(69)
Similarly, consider
[Q(q 3 ), P ] = i�Q({q 3 , p}) = 3i�Q(q 2 ) = 3i�Q2
[Q(p3 ), Q] = −3i�P 2
(70)
on the other hand
[Q3 , P ] = 3i�Q2 ,
[P 3 , Q] = −3i�P 2
17
(71)
hence irreducibility implies
Q(q 3 ) = Q3 + c3 ,
Q(p3 ) = P 3 + c4
(72)
Now {q 3 , qp} = 3q 3 implies
3Q(q 3 ) =
1
1
1
[Q(q 3 ), Q(qp)] = [Q3 + c3 , (QP + P Q)] = 3Q3
i�
i�
2
(73)
therefore c3 = 0, and similarly c4 = 0. Thus
Q(q 3 ) = Q3 ,
Q(p3 ) = P 3
(74)
Similarly, {q 3 , p2 } = 6q 2 p implies
6Q(q 2 p) =
1
1
[Q(q 3 ), Q(p2 )] = [Q3 + c3 , P 2 ] = 3(Q2 P + P Q2 )
i�
i�
(75)
and from {p3 , q 2 } = −6p2 q we get
6Q(p2 q) = 3(QP 2 + P 2 Q)
(76)
Finally, noting 9q 2 p2 = {q 3 , p3 } while 3q 2 p2 = {q 2 p, p2 q} gives
1
1
[Q(q 3 ), Q(p3 )] = [Q3 , P 3 ]
i�
i�
(77)
1
1 1
1
[Q(q 2 p), Q(p2 q)] = [ (Q2 P + P Q2 ), (QP 2 + P 2 Q)]
i�
i� 2
2
(78)
9q 2 p2 =
while
3q 2 p2 =
but this is inconsistent since the operator
3
A = [Q3 , P 3 ] − [Q2 P + P Q2 , QP 2 + P 2 Q] �= 0
4
(79)
indeed, using the CCR we get
[Q, [Q, [P, [P, A]]]] = 24�4
This is the contradiction.
(80)
qed
So we have to give up the strong quantization axiom Q4. However, one can “almost” achieve it. One can define a quantization map which maps every (nice) classical
observables one-to-one to QM observables, while “almost” satisfying Q4. Let us start
with polynomial observables:
Note: the irreducibility requirement is essential here. If one gives it up, one can
find “pre-quantizations” which completely respect Q4. However this gives a totally
unphysical quantization, and it means that there are quantum observables which have
no classical counterpart. Incidentally this is the first step in geometric quantization
(before choosing a polarization), which we will not discuss here.
18
3.5
Ordering and Weyl quantization
We want to construct a quantization map which extends the Schrödinger quantization
to polynomials in pi , q i . This can be achieved by agreeing on some ordering prescription
for the QM observables
Q:
P ol(q i , pj ) → A�
qi →
� Qi
pi →
� Pi
q i1 ...q ir pj1 ...pjs �→ e
i
iϕj
Qi1 ...Qir Pj1 ...Pjs
(81)
where j = (j1 , ..., js ) etc., and the phase factors ϕij etc. are determined such that
Q(q i1 ...q ir pj1 ...pjs ) is self-adjoint, (such that Q2 holds: Q(f )† = Q(f ∗ )). Note that the
image of Q really spans the space of all observables A� , More precisely, any polynomial
of degree n in Pi , Qj can be expressed uniquely by the ordered polynomials on the rhs of
(81), since the generators commute up to polynomials of lower order (Poincare-BirkhoffWitt). In particular, the rhs of (81) are precisely the totally symmetrized expressions
in the Qi , Pj : e.g.
1
Q(pq) = Q(qp) = (P Q + QP )
2
1
Q(p2 q) = (P 2 Q + P QP + QP 2 )
3
etc., which are obviously self-adjoint.
(82)
This symmetric “Weyl” quantization is unique if we impose that polynomials in pi , q j
are mapped to selfadjoint differential operators of the same order. However, in principle
there is nothing that forces us to require that.
Weyl quantization is very naturally extended to plane waves on phase space, via
Fourier transform. Denote the 2n coordinates on phase space with xa = (q i , pj ), and
consider the following functions
i
a
uk (x) = ei(ki q +lj pj ) ≡ eika x ,
ka = (ki , lj )
(83)
Expanding this into power series and recalling the symmetrization of Weyl quantization,
we have
i
a
Q(uk (x)) = ei(ki Q +lj Pj ) ≡ eika X̂ = uk (Q(x))
X a = (Qi , Pj )
(84)
So Weyl quantization maps plane waves into “quantum plane waves”! Now we can
expand a general observable f (q, p) into plane waves i.e. write it as Fourier integral
�
a
f (q, p) = d2n k fˆ(k)eika x
(85)
19
and by linearity the Weyl quantization gives
f=
Q:
�
A → A�
a
d k fˆ(k)eika x �→ Q(f ) =
2n
�
a
d2n k fˆ(k)eika X
(86)
This is clearly (essentially) invertible, up to subtleties concerning the domain of definition etc. Furthermore, it follows that
Q(f ∗ ) = Q(f )†
We can define
Δ(x) =
�
(87)
d2n k −ika (xa −X a )
e
(2π)2n
∈ A�
Then we can write the quantization map as
�
Q(f ) = d2n x f (x)Δ(x)
(88)
(89)
The inverse of Weyl quantization can be written down as
Q−1 : A� → A
F �→ f (x) =
1
T r(F Δ(x))
(2π�)n
de-quantization formula
(90)
using the relation
a
T r(eika X ) = (
2π n 2n
) δ (k)
�
(91)
applied to the rhs of (86).
this can be seen formally as follows
�
�
µ
µ
i(aµ Qµ +bµ Pµ )
3
i(aµ Qµ +bµ Pµ )
) = d p�p|e
|p� = d3 peib pµ �p|eiaµ Q |p�
Tr(e
�
µ
= d3 peib pµ �p|p + a��
�
2π
µ
= d3 peib pµ δ 3 (a�) = ( )3 δ 3 (b)δ 3 (a)
�
(92)
using
p xµ
1
i µ�
e
(2π�)3/2
�
TrA = d3 p �p|A|p�,
�x|p� =
20
�p|p� � = δ 3 (p − p� )
(93)
Moreover, we note that the integral over phase space is related to the trace over the
quantized operators,
�
�
�n
d2n xf (x) = 2π� T r(Q(f ))
(94)
This encodes the Bohr-Sommerfeld quantization: each “unit cell” in quantized phase
space has volume 2π�.
To be more rigorous, one can extend this to a map from square-integrable functions
to Hilbert Schmidt operators,
Q:
∼
=
L2 (R2n ) → HS(H)
(95)
or various subs-spaces or sub-algebras. Note that this is a map of vector spaces.
Of course Weyl quantization does not respect the Poisson algebraic structure. Rather,
these quantum plane waves satisfy the Weyl algebra
a
a
a
a
i
eika X̂ eila X̂ = e− 2 ka θ
eika X̂ eila X̂ = e−i ka θ
ab l
ab l
b
b
ei(ka +la )X̂
a
a
eila X̂ eika X̂
a
(96)
where
θ
ab
�
0
= {x , x } = �
−11
a
b
11
0
�
(97)
is the Poisson structure on phase space. In particular
a
i�
eika X̂ = e 2
i
ki li
i
eiki Q eilj Pj
eiki Q eilj Pj = ei� ki li eilj Pj eiki Q
i
(98)
This is the usual exponentiated form of the Heisenberg commutation relations, extended
to all observables.
3.6
Star products and deformation quantization
We want to understand better to which extent this Weyl quantization violates the naive
quantization condition Q4. For this purpose, it is useful to transfer the problem of
quantization into the equivalent problem of defining a quantized product on the classical
observables A. To understand this, we use the fact that Q is a bijective map (up to
21
subtleties). This means that we can “pull back” the noncommutative (operator) algebra
on A� via Q to the usual space A of functions on phase space, interpreted as vector
space (!!). Hence define
f � g := Q−1 (Q(f )Q(g))
Q(f � g) := Q(f )Q(g)
(99)
This defines a new, non-commutative product structure on (A, �), in contrast to the
classical point-wise commutative product. � is clearly non-local. For plane waves, the
formula (96) gives
a
a
i�
eika x � eila x = e− 2
ka θ ab lb i(ka +la )xa
e
(100)
This gives the following integral formula for this star product
�
�
�
�
i
ab �
�
2n ˆ
ikX
2n �
� ik� X
2n
d k ĝ(k )e
= d k d2n k � fˆ(k)ĝ(k � )e− 2 ka θ kb ei(k+k )X
Q(f )Q(g) = d k f (k)e
�
�
i
ab �
�
2n
= d k d2n k � fˆ(k)ĝ(k � − k)e− 2 ka θ kb eik X
�
i
ab �
� a
f � g(x) = d2n kd2n k � fˆ(k)ĝ(k � − k)e− 2 ka θ kb eika x
(101)
Furthermore, � can be written at least formally as follows
(f � g)(x) = f (x) exp(
−
→
−
i� ←
∂ a θab ∂ b )f (x)
2
(102)
(check it for the plane waves). For polynomials, this gives e.g.
xa � x b = x a xb +
i� ab
θ
2
xa � xb − xb � xa = i�θab
(103)
so this is consistent with the basic quantization condition Q4 at least for monomials.
The point of this construction is that
• At O(�0 ) it reduces to the point-wise product, as it must.
• at O(�) it recovers the Poisson bracket of classical mechanics. This basically shows
that the (semi-)classical limit of QM is given by classical mechanics! (However
things are more complicated because also the states |ψ� ∈ H have non-classical
properties)
• it gives a systematic expansion of the noncommutativity of the quantum observables in higher powers of �. These measure the deviation of Q from satisfying the
(naive) strong quantization condition.
22
• it allows to formulate the noncommuative “quantum” geometry, leading to e.g.
physics and field theory on such quantized spaces, very similar to ordinary physics;
only the pointwise product is replaced by the star product.
It turns out that there are lots of different star products and quantization maps Q. The
star product picture is useful because one can classify the inequivalent star products
(and hence quantization Q). One considers two star products �, �� to be equivalent if
there is a map I such that
f �� g = I −1 (I(f ) � I(g)
(104)
or equivalently if they arise from two quantization maps Q� = Q ◦ I. Then one can show
(see e.g. Waldmann) that if the Poisson structure is non-degenerate (i.e. on symplectic
manifolds M), any two star products are equivalent under some topological assumptions
2
on M, if the second deRham-cohomology of M is trivial, HdR
(M) = 0.
In particular, on R2n all such quantizations are equivalent up to such maps I, and
they basically boil down to different (generalized) ordering prescriptions.
Furthermore, this concept allows to quantize also Poisson brackets which are degenerate. It is a recent important result of Kontsevich that this can always be done, at
least formally.
4
Path integral quantization
An entirely different approach to quantization is given by Feynmans path integral. This
is particularly important in quantum field theory, and it can be obtained quite nicely
using the Weyl quantization.
We derive the Feynman path integral representation for quantum mechanics, starting
from Weyl quantization (105),
�
�
2n
Q(f ) = d x f (x)Δ(x) = dn qdn p f (q, p)Δ(q, p)
� n n
d kd l −ik(q−Q)−il(p−P )
e
(105)
Δ(q, p) =
(2π�)n
Where we separate the coordinates xa = (q, p) in phase space, assumed to be 2dimensional for simplicity. Using the basis
Q|q� = q|q�
23
(106)