Introduction to Quantization
Jan Dereziński
Dept. of Math. Methods in Phys.,
Faculty of Physics, University of Warsaw
ul. Pasteura 5, 02-093 Warszawa, Poland
email [email protected]
March 7, 2017
1
Introduction
1.1
Basic classical mechanics
Basic classical mechanics takes place in phase space Rd ⊕ Rd . The variables are the positions
xi , i = 1, . . . , d, and the momenta pi , i = 1, . . . , d. Real-valued functions on Rd ⊕ Rd are
called observables. (For example, positions and momenta are observables). The space of
observables is equipped with the (commutative) product bc and with the Poisson bracket
{b, c} = ∂xi b∂pi c − ∂pi b∂xi c .
(We use the summation convention of summing wrt repeated indices). Thus in particular
{xi , xj } = {pi , pj } = 0,
{xi , pj } = δji .
(1.1)
The dynamics is given by a real function on Rd ⊕ Rd called the (classical) Hamiltonian
H(x, p). The equations of motion are
dx(t)
dt
dp(t)
dt
=
∂p H(x(t), p(t)),
=
−∂x H(x(t), p(t)).
We treat x(t), p(t) as the functions of the initial conditions
x(0) = x,
p(0) = p.
More generally, the evolution of an observable b(x, p) is given by
d
b(x(t), p(t)) = {b, H}(x(t), p(t)).
dt
1
The dynamics preserves the product (this is obvious) and the Poisson bracket:
bc(x(t), p(t)) = b(x(t), p(t))c(x(t), p(t)),
{b, c}(x(t), p(t)) = {b(x(t), p(t)), c(x(t), p(t))},
Examples of classical Hamitonians:
particle in electrostatic and
1
(p − A(x))2 + V (x),
2m
1 ij
pi g (x)pj ,
2
1 2 ω2 2
p +
x ,
2
2
1
1
(p1 − Bx2 )2 + (p2 + Bx1 )2 ,
2
2
1 ij
1
a pi pj + bij xi pj + cij xi xj .
2
2
magnetic potentials
particle in curved space
harmonic oscillator
particle in constant magnetic field
general quadratic Hamiltonian
1.2
Basic quantum mechanics
Let ~ be a positive parameter, typically small.
Basic quantum mechanics takes place in the Hilbert space L2 (Rd ). Self-adjoint operators
on L2 (Rd ) are called observables. For a pair of such operators A, B we have their product
AB. (Note that we disregard the issues that arise with unbounded operators for which the
product is problematic). From the product one can derive their commutative Jordan product
1
2 (AB + BA) and their commutator [A, B]. The dynamics is given by a self-adjoint operator
H called the Hamiltonian. On the level of the Hilbert space the evolution equation is
i~
dΨ
= HΨ(t), Ψ(0) = Ψ,
dt
it
so that Ψ(t) = e ~ H Ψ. On the level of observables,
~
it
dA(t)
= i[H, A(t)], A(t0 ) = A,
dt
it
so that A(t) = e ~ H Ae− ~ H . The dynamics preserves the product:
it
it
it
it
it
it
e ~ H ABe− ~ H = e ~ H Ae− ~ H e ~ H Be− ~ H .
We have distinguished observables: the positions x̂i , i = 1, . . . , n, and the momenta p̂i :=
~
i
i ∂x , i = 1, . . . , n. They satisfy
[x̂i , x̂j ] = [p̂i , p̂j ] = 0,
2
[x̂i , p̂j ] = i~δji .
Examples of quantum Hamiltonians
particle in electrostatic and
1
(p̂ − A(x̂))2 + V (x̂),
2m
1 −1/4
g
(x̂)p̂i g ij (x̂)g 1/2 (x̂)p̂j g −1/4 (x̂),
2
1 2 ω2 2
p̂ +
x̂ ,
2
2
1
1
(p̂1 − B x̂2 )2 + (p̂2 + B x̂1 )2 ,
2
2
1 ij
1
i
a p̂i p̂j + bj x̂i p̂j + cij x̂i x̂j .
2
2
magnetic potentials
particle in curved space
harmonic oscillator
particle in constant magnetic field
general quadratic Hamiltonian
1.3
Concept of quantization
Quantization usually means a linear transformation with good properties, which to a function
on phase space b : R2d → C associates an operator Op• (b) acting on the Hilbert space
L2 (Rd ). (The superscript • stands for a possible decoration indicating the type of a given
quantization).
(Sometimes we will write Op• (b(x, p)) for Op• (b) – x, p play the role of coordinate
functions on the phase space and not its concrete points).
Here are desirable properties of a quantization:
(1) Op• (1) = 1l, Op• (xi ) = x̂i , Op• (pj ) = p̂j .
(2) 21 Op• (b)Op• (c) + Op• (c)Op• (b) ≈ Op• (bc).
(3) [ Op• (b), Op• (c)] ≈ i~Op• ({b, c}).
(4) [ Op• (b), Op• (c)] = i~Op• ({b, c}) if b is a 1st order polynomial.
The function b will be called the symbol (or dequantization) of the operator B.
Note that (1) implies that (3) and (4) is true at least if b is a 1st order polynomial.
1.4
The role of the Planck constant
Recall that the position operator x̂i , is the multiplication by xi and the momentum operator
is p̂i := ~i ∂xi . Thus we treat the position x̂ as the distinguished physical observable. The
momentum is scaled. This is the usual convention in physics.
Let Op• (b) stand for the quantization with ~ = 1. The quantization with any ~ will be
denoted by Op•~ (b). Note that we have the relationship
Op•~ (b) = Op• (b~ ),
b~ (x, p) = b(x, ~p).
However this convention breaks the symplectic invariance of the phase space. In some
situations it is more natural to use the Planck constant
√ differently and to use the position
operator
x̃
which
is
the
multiplication
operator
by
~xi , and the momentum operator
i
√
~
p̃ := i ∂xi . Note that they satisfy the usual commutation relations
[x̃i , p̃j ] = i~δij .
3
The corresponding quantization of a function b is
•
f ~ (b) := Op• (b̃~ ),
Op
so that
√
√
b̃~ (x, p) = b( ~x, ~p).
•
(1.2)
•
f ~ (xi ) = x̃i ,
Op
f ~ (pi ) = p̃i .
Op
The advantage of (1.2) is that positions and momenta are treated on equal footing. This
approach is typical when we consider coherent states.
Of course, both approaches are unitary equivalent. Indeed, introduce the scaling
τλ Φ(x) = λ−d/2 Φ λ−1/2 x .
Then
•
f ~ (b~ ) = τ~1/2 Op• (b~ )τ~−1/2 .
Op
1.5
Aspects of quantization
Quantization has many aspects in contemporary mathematics and physics.
1. Fundamental formalism
– used to define a quantum theory from a classical theory;
– underlying the emergence of classical physics from quantum physics (Weyl-WignerMoyal, Wentzel-Kramers-Brillouin).
2. Technical parametrization
– of operators used in PDE’s (Maslov, 4 volumes of Hörmander);
– of observables in quantum optics (Nobel prize for Glauber);
– signal encoding.
3. Subject of mathematical research
– geometric quantization;
– deformation quantization (Fields medal for Kontsevich!);
4. Harmonic analysis
– on the Heisenberg group;
– special approach for more general Lie groups and symmetric spaces.
We will not discuss (3), where the starting point is a symplectic or even a Poison manifold.
We will concentrate on (1) and (2), where the starting point is a (linear) symplectic space,
or sometimes a cotangent bundle.
A seperate subject is quantization of systems with an infinite number of degrees of
freedom, as in QFT, where it is even nontrivial to quantize linear dynamics.
4
2
Preliminary concepts
2.1
Fourier tranformation
We adopt the following definition of the Fourier transform.
Z
Ff (ξ) := e−iξx f (x)dx.
The inverse Fourier transform is given by
F
−1
−d
Z
g(x) = (2π)
eixξ g(ξ)dξ.
Formally, F −1 F = 1l can be expressed as
Z
(2π)−d ei(x−y)ξ dξ = δ(x − y).
Hence
(2π)−d
Z Z
eixξ dξdx = 1.
Proposition 2.1. The Fourier transform of e−itpx is (2π)−d t−d e
iηξ
t
.
Proof.
Z
e−itpx−ipη−ixξ dpdx
=
e
iηξ
t
Z Z
η
ξ
e−it(p+ t )(x+ t ) dpdx.
2
Suppose the variable has a generic name, say x. Then we set
Dx :=
1
∂x .
i
Clearly,
eitDx Φ(x) = Φ(x + t).
2.2
Semiclassical Fourier transformation
If we use the parameter ~, it is natural to use the semiclassical Fourier tranformation
Z
i
F~ f (p) := e− ~ px f (x)dx.
Its inverse is given by
F~−1 g(x) = (2π~)−d
5
Z
i
e ~ xp g(p)dp.
Proposition 2.2. The semiclassical Fourier transformation swaps the position and momentum:
F~−1 x̂F~
=
p̂,
F~−1 p̂F~
=
−x̂.
Proof.
Z
Z
i
i
1
dx
dke ~ px xe− ~ kx Ψ(k)
d
(2π~)
Z
Z
i
i
1
dx dk~i∂k e ~ px e− ~ kx Ψ(k)
=
(2π~)d
Z
Z
i
i
1
dx
dke ~ px e− ~ kx ~i∂k Ψ(k)
=−
(2π~)d
F~−1 x̂F~ Ψ (p) =
F~−1 p̂F~ Ψ
Z
Z
i
i
1
~
(x) =
dx dke ~ px ∂p e− ~ py Ψ(y)
(2π~)d
i
Z
Z
i
i
1
=−
dx
dke ~ px ye− ~ py Ψ(y) =
(2π~)d
(2.3)
(2.4)
=
p̂Ψ(p).
(2.5)
(2.6)
−x̂Ψ(x).
(2.7)
2Hence, for Borel functions
F~−1 f (x̂)F~
= f (p̂),
(2.8)
F~−1 g(p̂)F~
= g(−x̂).
(2.9)
(2.9) can be rewritten as
Z Z
i
1
e ~ (x−y)p g(p)Ψ(y)dydp
d
(2π~)
Z 1
y − x
=
ĝ
Ψ(y)dy.
(2π~)d
~
g(p̂)Ψ)(x) =
2.3
(2.10)
(2.11)
Gaussian integrals
Suppose that ν is positive definite. Then
Z
−1
1
d
1 1
e− 2 x·νx+η·x dx = (2π) 2 (det ν)− 2 e 2 η·ν η .
(2.12)
1
In particular, if f (x) = e− 2 x·νx , its Fourier transform is
−1
d
1
1
fˆ(ξ) = (2π) 2 (det ν)− 2 e− 2 ξ·ν ξ .
6
(2.13)
Consequently,
1
e 2 ∂x ·ν∂x Ψ(x)
=
=
=
1
e− 2 p̂·ν p̂ Ψ(x)
(2.14)
1
d
Z
1
e− 2 (x−y)·ν
(2π)− 2 (det ν)− 2
R − 1 (x−y)·ν −1 (x−y)
e 2
Ψ(y)dy
.
R − 1 y·ν −1 y
dy
e 2
−1
(x−y)
Ψ(y)dy
(2.15)
As a consequence, we obtain the following identity
Z
−1
1
1
(det 2πν)− 2
Ψ(x)e− 2 x·ν x dx
R
−1
1
Ψ(x)e− 2 x·ν x dx
.
=
R − 1 x·ν −1 x
e 2
dx
1
e 2 ∂x ·ν∂x Ψ(0)
=
(2.16)
For example,
e
i~∂x ∂p
−d
Z
i
e ~ (x−y)(p−w) Φ(y, w)dydw.
Φ(x, p) = (2π~)
(2.17)
As an exercise let us check (2.14) for polynomial Ψ. By diagonalizing ν and then changing
the variables, we can reduce ourselves to 1 dimension. Now
1
1
(2π)− 2
Z
1
∞
X
n!
xn−2k ;
2k (n − 2k)!k!
k=0
Z
1
1 2
= (2π)− 2
e− 2 y (y + x)n dy
2
e 2 ∂x xn
=
2
e− 2 (x−y) y n dy
=
∞
X
1
xn−m (2π)− 2
m=0
n!
(n − m)!m!
(2.18)
(2.19)
Z
1
2
e− 2 y y m dy.
Now
1
(2π)− 2
Z
1
(2π)− 2
1
2
e− 2 y y 2k+1 dy
Z
1 2
e− 2 y y 2k dy
=
0,
=
(2k)!
.
2k k!
Indeed, the lhs of (2.20) is
Z ∞
1 k− 12 1 1 2
1
1
1
π − 2 2k
e− 2 y
y2
d y 2 = π − 2 2k Γ k +
,
2
2
2
0
which using
1 (2k)!
1
Γ k+
= π 2 2k
2
2 k!
equals the rhs of (2.20).
Therefore, (2.18)=(2.19).
7
(2.20)
2.4
Gaussian integrals in complex variables
Consider the space Rd ⊕ Rd with the generic variables x, p. It is often natural to identify it
with the complex space Cd by introducing the variables
ai
=
2−1/2 (xi + ipi ),
a∗i
=
2−1/2 (xi − ipi ),
so that
xi = 2−1/2 (a + a∗ ), pi = −i2−1/2 (a − a∗ ).
−d
The Lebesgue measure dxdp will be denoted i
da∗j ∧ daj =
(2.21)
∗
da da. To justify this notation note that
1
dx − idp ∧ dx + idp = idx ∧ dp.
2
Let β be a Hermitian positive quadratic form. We then have
Z
−1
∗
∗
(2πi)−d e−a ·βa+w1 ·a +w2 ·a da∗ da = (det β)−1 ew1 ·β w2 ,
e∂a∗ ·β∂a Φ(a∗ , a)
=
=
(2.22)
Z
∗
∗
−1
(2πi)−d (det β)−1 e−(a −b )·β (a−b) Φ(b∗ , b)db∗ db
R −(a∗ −b∗ )·β −1 (a−b)
e
Φ(b∗ , b)db∗ db
R
.
∗
∗
−1
e−(a −b )·β (a−b) db∗ db
(2.23)
As a consequence, we obtain the following identity
∂a∗ ·β∂a
e
Φ(0, 0)
−d
Z
(2πi) (det β)
Φ(a∗ , a)e−a·β
R
−1 ∗
Φ(a∗ , a)e−a·β a da∗ da
R
.
e−a·β −1 a∗ da∗ da
=
=
−1
−1 ∗
a
da∗ da
(2.24)
As an exercise let us check (2.23) for polynomial Φ. By diagonalizing β and then changing
the variables, we can reduce ourselves to 1 (complex) dimension. Now
∞
X
n!m!
a∗(n−k) am−k ;
(2.25)
(n − k)!(m − k)!k!
k=0
Z
Z
∗
−1
−(a∗ −b∗ )(a−b) ∗n m ∗
−1
(2πi)
e
b b db db = (2πi)
e−b b (a∗ + b∗ )n (a + b)m db∗ db (2.26)
e∂a∗ ∂a a∗n am
=
∞
X
=
a∗(n−k) a(m−l)
k,l=0
n!m!
k!(n − k)!l!(m − l)!
Z
∗
e−b b b∗k bl db∗ db.
Now
(2πi)−1
Z
∗
e−b b b∗k bl db∗ db = k!δkl .
8
(2.27)
Indeed, if we use the polar coordinates with b∗ b = 12 r2 , the lhs of (2.27) becomes
Z ∞
1 21 (k+l)
1 2
e− 2 r
r2
(2π)−1
ei(n−m)φ rdrdφ
2
Z ∞ 0
1 k 1 1 2
= δkl
e− 2 r
r2 d r2
2
2
0
which equals the rhs of (2.27).
Therefore, (2.25)=(2.26).
2.5
Integral kernel of an operator
Every linear operator A on Cn can be represented by a matrix [Aji ].
One would like to generalize this concept to infinite dimensional spaces (say, Hilbert
spaces) and continuous variables instead of a discrete variables i, j. Suppose that a given
vector space is represented, say, as L2 (X), where X is a certain space with a measure.
One often uses the representation of an operator A in terms of its integral kernel X × X 3
(x, y) 7→ A(x, y), so that
Z
AΨ(x) =
A(x, y)Ψ(y)dy.
Note that strictly speaking A(·, ·) does not have to be a function. E.g. in the case X = Rd it
could be a distribution, hence one often says the distributional kernel instead of the integral
kernel. Sometimes A(·, ·) is ill-defined anyway. Below we will describe some situations where
there is a good mathematical theory of integral/distributional kernels.
At least formally, we have
Z
AB(x, y) = A(x, z)B(z, y)dz,
A∗ (x, y) = A(y, x).
2.6
Tempered distributions
The space of Schwartz functions on Rn is defined as
R
S(Rn ) := Ψ ∈ C ∞ (Rn ) : |xα ∇βx Ψ(x)|2 dx < ∞,
Remark 2.3. The definition (2.28) is equivalent to
S(Rn ) = Ψ ∈ C ∞ (Rn ) : |xα ∇βx Ψ(x)| ≤ cα,β ,
α, β ∈ Nn .
α, β ∈ Nn .
(2.28)
(2.29)
S 0 (Rn ) denotes the space of continuous functionals on S(Rn ), ie. S(Rn ) 3 Ψ 7→ hT |Ψi ∈
C belongs to S 0 iff there exists N such that
X Z
21
|hT |Ψi| ≤
|xα ∇βx Ψ(x)|2 dx .
|α|+|β|<N
9
We will use the integral notation
Z
hT |Ψi =
T (x)Ψ(x)dx.
The Fourier transformation is a continuous map from S 0 into itself. We have continuous
inclusions
S(Rn ) ⊂ L2 (Rn ) ⊂ S 0 (Rn ).
Here are some examples of elements of S 0 (R):
Z
δ(t)Φ(t)dt := Φ(0),
Z
Z
(t ± i0)λ Φ(t)dt := lim (t ± i)λ Φ(t)dt.
&0
Theorem 2.4 (The Schwartz kernel theorem). B is a continuous linear transformation
from S(Rd ) to S 0 (Rd ) iff there exists a distribution B(·, ·) ∈ S 0 (Rd ⊕ Rd ) such that
Z
(Ψ|BΦ) = Ψ(x)B(x, y)Φ(y)dxdy, Ψ, Φ ∈ S(Rd ).
Note that ⇐ is obvious. The distribution B(·, ·) ∈ S 0 (Rd ⊕Rd ) is called the distributional
kernel of the transformation B. All bounded operators on L2 (Rd ) satisfy the Schwartz kernel
theorem.
Examples:
(1) e−ixy is the kernel of the Fourier transformation
(2) δ(x − y) is the kernel of identity.
(3) ∂x δ(x − y) is the kernel of ∂x . .
2.7
Hilbert-Schmidt operators
We say that an operator B is Hilbert-Schmidt if
X
X
∞ > TrB ∗ B =
(ei |B ∗ Bei ) =
(Bei |Bei ),
i∈I
i∈I
where {ei }i∈I is an arbitrary basis and the RHS does not depend on the choice of the basis.
Hilbert-Schmidt are bounded.
Proposition 2.5. Suppose that H = L2 (X) for some measure space X. The following
conditions are equivalent
(1) B is Hilbert-Schmidt.
(2) The distributional kernel of B is L2 (X × X).
Moreover, if B, C are Hilbert-Schmidt, then
Z
TrB ∗ C = B(x, y)C(x, y)dxdy.
10
2.8
Trace class operators
B is trace class if
X √
√
∞ > Tr B ∗ B =
(ei | B ∗ Bei ).
i∈I
If B is trace class, then we can define its trace:
X
TrB :=
(ei |Bei ).
i∈I
where again {ei }i∈I is an arbitrary basis and the RHS does not depend on the choice of the
basis.
Trace class operators are Hilbert-Schmidt:
B ∗ B = (B ∗ B)1/4 (B ∗ B)1/2 (B ∗ B)1/4 ≤ (B ∗ B)1/4 kBk(B ∗ B)1/4 = kBk(B ∗ B)1/2 .
Hence
√
TrB ∗ B ≤ kBkTr B ∗ B.
Consider a trace class operator C and a bounded operator B. On the formal level we
have the formula
Z
TrBC = B(y, x)C(x, y)dxdy.
(2.30)
In particular by setting B = 1l, we obtain formally
Z
TrC = C(x, x)dx.
3
3.1
x, p- and Weyl-Wigner quantizations
x, p-quantization
Suppose we look for a linear transformation that to a function b on phase space associates
an operator Op• (b) such that
Op• (f (x)) = f (x̂), Op• (g(p)) = g(p̂).
The so-called x, p-quantization, often used in the PDE community, is determined by the
additional condition
Opx,p (f (x)g(p)) = f (x̂)g(p̂).
Note that
f (x̂)g(p̂) Ψ(x) = (2π~)−d
Z
Z
dp
i(x−y)p
~
Ψ(y).
(3.31)
Hence we can generalize (3.31) for a general function on the phase space b
Z
Z
i(x−y)p
Opx,p (b)Ψ (x) = (2π~)−d dp dyb(x, p)e ~ Ψ(y).
(3.32)
11
dyf (x)g(p)e
In the PDE-community one writes
Opx,p (b) = b(x, ~D).
We also have the closely related p, x-quantization, which satisfies
Opp,x (f (x)g(p)) = g(p̂)f (x̂).
It is given by the formula
p,x
Op
(b)Ψ (x) = (2π~)−d
Z
Z
dp
dyb(y, p)e
i(x−y)p
~
Ψ(y).
Thus the kernel of the operator as x, p- and p, x-quantization is given by:
Z
i(x−y)p
Opx,p (bx,p ) = B, B(x, y) = (2π~)−d dpbx,p (x, p)e ~ ,
Z
i(x−y)p
Opp,x (bp,x ) = B, B(x, y) = (2π~)−d dpbp,x (y, p)e ~ .
Proposition 3.1. We can compute the symbol from the kernel: If (3.34), then
Z
izp
bx,p (x, p) = B(x, x − z)e− ~ dz.
Proof.
B(x, x − z) = (2π~)
−d
Z
e
izp
~
(3.33)
(3.34)
(3.35)
(3.36)
bx,p (x, p)dp.
We apply the Fourier transform. 2
∗
Proposition 3.2. Opx,p (b) = Opp,x (b).
Proposition 3.3. We can go from x, p- to p, x-quantization: If (3.34) and (3.35) hold, then
e−i~Dx Dp bx,p (x, p) = bp,x (x, p).
Proof.
Z
bx,p (x, p)
i
=
B(x, x − z)e− ~ zp dz
Z Z
i
(2π~)−d
bp,x (x − z, w)e ~ z(w−p) dzdw
Z Z
i
(2π~)−d
bp,x (y, w)e ~ (x−y)(w−p) dydw
=
e−i~Dx Dp bp,x (x, p).
=
=
2
Therefore, formally,
Opx,p (b) = Opp,x (b) + O(~).
12
(3.37)
Proposition 3.4. We have the following formula for the symbol of the product: If
Opx,p (b)Opx,p (c) = Opx,p (d),
then
(3.38)
d(x, p) = e−i~Dp1 Dx2 b(x1 , p1 )c(x2 , p2 ) x := x = x , .
1
2
p := p1 = p2 .
Proof.
Opx,p (b)Opx,p (c)(x, y)
Z Z Z
i
b(x, p1 )c(x2 , p2 )e ~ (x−x2 )p1 +(x2 −y)p2 dp1 dx2 dp2
= (2π~)−2d
Z
i
−d
= (2π~)
dp2 e ~ (x−y)p2
Z Z
i
×(2π~)−d
b(x, p1 )c(x2 , p2 )e ~ (x2 −x)(p2 −p1 ) dp1 dx2
Z
i
−d
= (2π~)
dp2 e ~ (x−y)p2 e−i~Dp1 Dx2 b(x1 , p1 )c(x2 , p2 ) x := x = x , .
1
2
p := p1 = p2 .
2
Besides d given by (3.38), let
Opx,p (c)Opx,p (b) = Opx,p (d1 ),
(3.39)
Note that formally
d(x, p)
= b(x, p)c(x, p) − i~∂p b(x, p)∂x c(x, p) + O(~2 ),
d1 (x, p)
= b(x, p)c(x, p) − i~∂x b(x, p)∂p c(x, p) + O(~2 ).
Therefore,
Opx,p (b)Opx,p (c)
=
Opx,p (bc) + O(~),
[Opx,p (b), Opx,p (c)]
=
i~Opx,p ({b, c}) + O(~2 ).
Obviously,
3.2
[x̂, Opx,p (b)]
=
i~Opx,p (∂p b) = i~Opx,p ({x, b}),
[p̂, Opx,p (b)]
= −i~Opx,p (∂x b) = i~Opx,p ({p, b}).
Weyl-Wigner quantization
The definition of the Weyl-Wigner quantization looks like a compromise between the x, p
and p, x-quantizations:
Z
Z
x + y i(x−y)p
Op(b)Ψ (x) = (2π~)−d dp dyb
, p e ~ Ψ(y).
(3.40)
2
13
In the PDE-community it is usually called the Weyl quantization and denoted by
Op(b) = bw (x, ~D).
If Op(b) = B, the kernel of B is given by:
Z
x + y i(x−y)p
−d
,p e ~ .
B(x, y) = (2π~)
dpb
2
Proposition 3.5. We can compute the symbol from the kernel:
Z z
z − izp
b(x, p) = B x + , x −
e ~ dz.
2
2
Proof.
z
z
B(x + , x − ) = (2π~)−d
2
2
Z
e
izp
~
(3.41)
b(x, p)dp.
We apply the Fourier transform. 2
Usually, b is called in the PDE community the Weyl symbol and in the quantum physics
community the Wigner function.
1 2
d
Let P0 be the orthogonal projection onto the normalized vector π − 4 e− 2 x . The integral
kernel of P0 equals
d
1 2
1 2
P0 (x, y) = π − 2 e− 2 x − 2 y .
Its various symbols equal
x, p-symbol:
p, x-symbol:
Weyl-Wigner symbol:
− 21 p2 −ix·p
,
d
2
− 12 x2 − 21 p2 +ix·p
,
d
2
− 12 x2 − 21 p2
d
1
2
2 2 e− 2 x
2 e
2 e
.
Proposition 3.6. We can go from the x, p- to the Weyl quantization:
Opx,p (bx,p ) = Op(b), then
if
i
e 2 ~Dx Dp b(x, p) = bx,p (x, p).
(3.42)
Consequently,
bx,p = b + O(~).
3.3
Star product
Proposition 3.7. We have the following formula for the symbol of the product:
if
Op(b)Op(c) = Op(d), then
i
d(x, p) = e 2 ~(Dp1 Dx2 −Dx1 Dp2 ) b(x1 , p1 )c(x2 , p2 ) x := x = x , .
1
2
p := p1 = p2 .
14
(3.43)
Consequently,
1
Op(b)Op(c) + Op(c)Op(b) = Op(bc) + O(~2 ),
2
[Op(b), Op(c)] = i~Op({b, c}) + O(~3 ),
if suppb ∩ suppc = ∅,
then
Op(b)Op(c)
O(~∞ ).
=
(3.44)
(3.45)
(3.46)
Often one denotes d in (3.43) by b ∗ c and calls the star product or the Moyal product of
b and c.
Proposition 3.8. If h is a polynomial of degree ≤ 1, then
1
Op(h)Op(b) + Op(b)Op(h)
2
=
Op(bh),
Proof. Consider for instance h = x.
i
e 2 ~(Dp1 Dx2 −Dx1 Dp2 ) x1 b(x2 , p2 ) x := x = x ,
1
2
p := p1 = p2 .
i
e 2 ~(Dp1 Dx2 −Dx1 Dp2 ) b(x1 , p1 )x2 x := x = x ,
1
2
= xb(x, p) +
i~
∂p b(x, p),
2
= xb(x, p) −
i~
∂p b(x, p).
2
p := p1 = p2 .
2
Consequently,
2
p̂ − A(x̂)
Op aij xi xj + 2bij xi pj + cij pi pj
=
2 Op p − A(x)
,
=
aij x̂i x̂j + bij x̂i p̂j + bij p̂j x̂i + cij p̂i p̂j .
Proposition 3.9. Let h be a polynomial of degree ≤ 2. Then
(1)
[Op(h), Op(b)] = i~Op({h, b}).
(3.47)
(2) Let x(t), p(t) solve the Hamilton equations with the Hamiltonian h. Then the affine
symplectic transformation
rt x(0), p(0) = x(t), p(t)
satisfies
it
it
e ~ Op(h) Op(b)e− ~ Op(h) = Op(b ◦ rt ).
15
Proof.
i
e 2 ~(Dp1 Dx2 −Dx1 Dp2 ) h(x1 , p1 )b(x2 , p2 ) x := x = x ,
1
2
p := p1 = p2 .
i~
= h(x, p)b(x, p) + (Dp h(x, p)Dx b(x, p) − Dp h(x, p)Dx b(x, p) ,
2
(i~)2
(Dp1 Dx2 − Dx1 Dp2 )2 h(x1 , p1 )b(x2 , p2 ) x := x = x ,
+
1
2
8
p := p1 = p2 .
When we swap h and b, we obtain the same three terms except that the second has the
opposite sign. This proves (1).
To prove (2) note that
d
b ◦ rt = {h, b ◦ rt }
dt
[Op(h), Op(b ◦ rt )] = i~Op({h, b ◦ rt }).
it
it
Now, e− ~ Op(h) Op(b ◦ rt )e ~ Op(h) = Op(b) and
t=0
=
it
d −it Op(h)
Op(b ◦ rt )e ~ Op(h)
e ~
dt
i
it
−it
d
e ~ Op(h) − [Op(h), Op(b ◦ rt )] + Op
b ◦ rt e ~ Op(h) = 0
~
dt
2
3.4
Weyl operators
Proposition 3.10 (Baker-Campbell-Hausdorff formula). Suppose that
[A, B], A = [A, B], B = 0.
Then
1
eA+B = eA eB e− 2 [A,B] .
Proof. We will show that for any t ∈ R
1 2
et(A+B) = etA etB e− 2 t
[A,B]
.
First, using the Lie formula, we obtain
etA Be−tA
=
∞ n
X
t
adnA (B)
n!
n=0
=
B + t[A, B].
16
(3.48)
Now
d tA tB − 1 t2 [A,B]
e e e 2
dt
1 2
AetA etB e− 2 t
=
[A,B]
1 2
+etA BetB e− 2 t
[A,B]
1 2
−etA etB t[A, B]e− 2 t
1 2
(A + B)etA etB e− 2 t
=
[A,B]
[A,B]
.
Besides, (3.48) is true for t = 0. 2
Let ξ = (ξ1 , . . . , ξd ), η = (η1 , . . . , ηd ) ∈ Rd . Clearly,
[ξi x̂i , ηj p̂j ] = i~ξi ηi .
Therefore,
i~
e− 2 ξi ηi ei(ξi x̂i +ηi p̂i )
= e−i~ξi ηi eiηi p̂i eiξi x̂i .
eiξi x̂i eiηi p̂i
=
The operators ei(ξi x̂i +ηi p̂i ) are sometimes called Weyl operators. They satisfy the relations
that involve the symplectic form:
0
0
0
0
0
0
i~
ei(ξi x̂i +ηi p̂i ) ei(ξi x̂i +ηi p̂i ) = e− 2 (ξi ηi −ηi ξi ) ei (ξi +ξi )x̂i +(ηi +ηi )p̂i
(3.49)
They translate the position and momentum:
i
i
i
~ (−y p̂+wx̂)
i
~ (y p̂−wx̂)
e ~ (−yp̂+wx̂) x̂e ~ (yp̂−wx̂)
e
3.5
p̂e
=
x̂ − y,
=
p̂ − w.
Weyl-Wigner quantization in terms of Weyl operators
Note that
i
i
ei(ξi x̂i +ηi p̂i ) = e 2 ξi x̂i eiηi p̂i e 2 ξi x̂i .
Hence the integral kernel of e
i(ξi x̂i +ηi p̂i )
(2π~)−d
Z
is
1
1
i
dpei( 2 ξi xi +ηi pi + 2 ξi yi )+ ~ (xi −yi )pi .
Therefore,
Op(ei(ξi xi +ηi pi ) ) = ei(ξi x̂i +ηi p̂i ) .
More generally, if f is a function on R, then
=
(2π)−1
Z
f (ξi x̂i + ηi p̂i )
=
(2π)−1
Z
f (ξi xi + ηi pi )
17
Ff (t)ei(ξi x̂i +ηi p̂i )t dt,
Ff (t)ei(ξi xi +ηi pi )t dt.
(3.50)
Therefore,
Op f (ξi xi + ηi pi ) = f (ξi x̂i + ηi p̂i ).
Every function b on Rd ⊕ Rd can be written in terms of its Fourier transform:
Z Z Z Z
b(x, p) = (2π)−2d
ei(xi −yi )ξi +i(pi −wi )ηi b(y, w)dydwdξdη.
(3.51)
(3.52)
This leads to
Op(b) = (2π)−2d
Z Z Z Z
ei(x̂i −yi )ξi +i(p̂i −wi )ηi b(y, w)dydwdξdη,
(3.53)
which can be treated as an alternative definition of the Weyl-Wigner quantization.
3.6
Classical and quantum mechanics over a symplectic vector space
A symplectic form is a nondegenerate antisymmetric form. A vector space equipped with a
symplectic form is called a symplectic vector space. It has to have an even dimension.
Let R2d be an even dimensional vector space. Let φj , j = 1, . . . , 2d denote the canonical
coordinates in R2d . Let ω = [ωij ] be a symplectic form. We will denote by [ω ij ] the inverse
of [ωij ].
With every symplectic space we have a classical system. Indeed, we can define the
Poisson bracket for functions b, c on R2d :
{b, c} = ∂φi bω ij ∂φj c.
Proposition 3.11. In every symplectic space we can choose a basis φi = xi , φd+i = pi so
that the Poisson bracket has the form (1.1), that is
ωi,i+d = 1,
ωi+d,i = −1.
We also have a quantum system with operators φ̂j , j = 1, . . . , 2d satisfying
[φ̂j , φ̂k ] = iω jk 1l.
(3.54)
We say that a linear transformation r = [rij ] is symplectic if it preserves the symplectic
form. Explicitly, r# ωr = ω, or
rip ωpq rjq = ωij
The set of such transformations is denoted Sp(R2d ). It is a Lie group.
We say that a linear transformation b = [bji ] is infinitesimally symplectic if it infinitesimally preserves the symplectic form. In other words, 1l + b is for small approximately
symplectic. Explicitly, b# ω + ωb = 0, or
bpi ωpj + ωip bpj = 0.
The set of such transformations is denoted sp(R2d ). It is a Lie algebra.
18
Proposition 3.12. b is an infinitesimally symplectic transformaton iff c = ω −1 b is symmetric, so that bij = ω ik ckj . Then
H=
1
1
cjk φj φk , Ĥ = Op(H) = cjk φ̂j φ̂k
2
2
is the corresponding classical and quantum Hamiltonian. Let [rij (t)] be the corresponding dynamics, which is a 1-parameter group in Sp(R2d ). Then the classical and quantum dynamics
generated by this Hamiltonian are given by the flow r(t):
φj (t) = rkj (t)φk (0),
3.7
φ̂j (t) = rkj (t)φ̂k (0),
Weyl quantization for a symplectic vector space
Let φ̂i satisfy the relations (3.54). For ζ = (ζ1 , . . . , ζ2d ) ∈ R2d set
X
φ̂·ζ :=
φ̂i ζi , W (ζ) := eiζ·φ̂ .
i
Then
i
W (ζ)W (θ) = e− 2 ζ·ωθ W (ζ + θ).
For a function b on R2d we can define its Weyl-Wigner quantization:
Z Z
i
−2d
Op(b) := (2π)
ei(φ̂i −ψi )·ζ b(ψ)dψdζ.
Note that
Op eiφ·ζ = eiφ̂·ζ .
(3.55)
More generally, for any Borel function f
Op f (φ·ζ) = f (φ̂·ζ).
(3.56)
1 X
φ̂ζσ(i) · · · φ̂ζσ(n) .
Op φζ1 · · · φζn =
n!
(3.57)
Op (φζ)n = (φ̂ζ)n .
(3.58)
Proposition 3.13.
σ∈Sn
Proof. We have
This is a special case of (3.57), it is also seen directly from (3.55) by expanding into a power
series. Let t1 , . . . , tn ∈ R. By (3.58),
Op (t1 φ·ζ1 + · · · + tn φ·ζn )n = (t1 φ̂·ζ1 + · · · + tn φ̂·ζn )n .
(3.59)
The coefficient at t1 · · · tn on both sides is
Op(n!φ·ζ1 · · · φ·ζn ) =
X
σ∈Sn
2
19
φ̂·ζσ−1 (1) · · · φ̂·ζσ−1 (n) .
3.8
Positivity
Clearly,
Op(b)∗ = Op(b).
Therefore, b is real iff Op(b) is Hermitian. What about positivity? We will see that there is
no implication in either direction between the positivity of b and of Op(b).
We have
(x̂ − ip̂)(x̂ + ip̂) = x̂2 + p̂2 − ~ ≥ 0.
Therefore
Op(x2 + p2 − ~) ≥ 0,
2
(3.60)
2
even though x + p − ~ is not everywhere positive.
The converse is more complicated. Consider the generator of dilations
A :=
i
1
(x̂p̂ + p̂x̂) = x̂p̂ − = Op(xp).
2
2
Its name comes from the 1-parameter group it generates:
eitA Φ(x) = et/2 Φ(et x).
Note that sp A = R. Indeed, A preserves the direct decomposition L2 (R) = L2 (0, ∞) ⊕
L2 (−∞, 0). We will show that the spectrum of A restricted to each of these subspaces
is R. Consider the unitary operator U : L2 (0, ∞) → L2 (R) given by U Φ(s) = es/2 Φ(es )
with the inverse U ∗ Ψ(x) = x−1/2 Ψ(log x). Then U ∗ p̂U = A. But sp p̂ = R. Therefore,
sp A2 = (sp A)2 = [0, ∞[.
We have A2 = Op(b), where
i~
b(x, p) = e 2 (Dp1 Dx2 −Dx1 Dp2 ) x1 p1 x2 p2 x := x = x ,
1
2
p := p1 = p2 .
2
~
2Dp1 Dx2 Dx1 Dp2 x1 p1 x2 p2 x := x = x ,
= x2 p 2 +
1
2
4·2
p := p1 = p2 ,
= x2 p 2 +
~2
.
4
Hence
Op(x2 p2 ) = A2 −
~2
.
4
Therefore Op(x2 p2 ) is not a positive operator even though its symbol is positive
3.9
Parity operator
Define the parity operator
IΨ(x) = Ψ(−x).
20
(3.61)
More generally, set
i
i
I(y,w) := e ~ (−yp̂+wx̂) Ie ~ (yp̂−wx̂) .
(3.62)
Clearly,
2i
I(y,w) Ψ(x) = e ~ w·(x−y) Ψ(2y − x).
Let δ(y,w) denote the delta function at (y, w) ∈ Rd ⊕ Rd .
Proposition 3.14.
Op (π~)d δ(0,0) = I.
(3.63)
Op (π~)d δ(y,w) = I(y,w) .
(3.64)
More generally,
Proof.
d
Op (π~) δ(0,0) (x, y)
−d
Z
=
2
=
2−d δ
δ
x + y
2
x + y
2
i
, ξ e ~ (x−y)·ξ dξ
= δ(x + y).
To see the last step we substitute y2 = ỹ below and evaluate the delta function:
Z Z x + y
x
δ
Φ(y)dy = δ
+ ỹ Φ(2ỹ)2d dỹ = 2d Φ(−x).
2
2
(3.65)
2
Theorem 3.15. Let Op(b) = B.
(1) If b ∈ L1 (Rd ⊕ Rd ), then B is a compact operator. In terms of an absolutely norm
convergent integral, we can write
Z
B = (π~)−d I(x,p) b(x, p)dxdp.
(3.66)
Hence,
kBk ≤ (π~)−d kbk1 .
(3.67)
(2) If B is trace class, then b is continuous, vanishes at infinity and
b(x, p) = 2d TrI(x,p) B.
Hence
|b(x, p)| ≤ 2d Tr|B|.
Proof. Obviously,
Z
b=
b(x, p)δx,p dxdp.
21
(3.68)
Hence
Z
Op(b)
=
=
b(x, p)Op(δx,p )dxdp
Z
−d
(π~)
b(x, p)I(x,p) dxdp.
Next,
Z
b(x, p)
=
δ(x,p) (y, w)b(y, w)dydw
=
(2π~)d TrOp(δ(x,p) )Op(b)
=
2d TrI(x,p) Op(b).
2
3.10
Special classes of symbols
Proposition 3.16. The following conditions are equivalent
(1) B is continuous from S to S 0 .
(2) The x, p-symbol of B is Schwartz.
(3) The p, x-symbol of B is Schwartz.
(4) The Weyl-Wigner symbol of B is Schwartz.
Proof. By the Schwartz kernel theorem (1) is equivalent to B having the kernel in S 0 . The
formulas () involve only partial Fourier transforms and some constant coefficients. 2
Proposition 3.17. The following conditions are equivalent
(1) B is Hilbert-Schmidt.
(2) The x, p-symbol of B is L2 .
(3) The p, x-symbol of B is L2 .
(4) The Weyl-Wigner symbol of B is L2 .
Moreover, if b, c are L2 , then
TrOpx,p (b)∗ Opx,p (c) = TrOp(b)∗ Op(c) = (2π~)−d
Z
b(x, p)c(x, p)dxdp.
(3.69)
One often uses (3.69) when B = Op(b) is, say, bounded and describes an observable, and
C = Op(c) is trace class, and describes a density matrix. Setting b(x, p) = 1 we formally
obtain
Z
x,p
−d
TrOp(c) = TrOp (c) = (2π~)
c(x, p)dxdp.
22
With the x, p-quantization we can compute the so-called marginals involving the position
and momentum:
Z
Trf (x̂)Opx,p (c) = (2π~)−d f (x)c(x, p)dxdp,
Z
Trg(p̂)Opx,p (c) = (2π~)−d g(p)c(x, p)dxdp.
With the Weyl-Wigner quantization we have much more possibilities: eg. for any α
Z
Trf (cos αx̂ + sin αp̂)Op(c) = (2π~)−d f (cos αx + sin αp)c(x, p)dxdp.
4
4.1
Coherent states
General coherent states
Fix a normalized vector Ψ ∈ L2 (Rd ). The family of coherent vectors associated with the
vector Ψ is defined by
i
Ψ(y,w) := e ~ (−yp̂+wx̂) Ψ,
(y, w) ∈ Rd ⊕ Rd .
The orthogonal projection onto Ψ(y,w) , called the coherent state, will be denoted
i
i
P(y,w) := |Ψ(y,w) )(Ψ(y,w) | = e ~ (−yp̂+wx̂) |Ψ)(Ψ|e ~ (yp̂−wx̂) .
It is natural to assume that
Ψ|x̂Ψ = 0,
Ψ|p̂Ψ = 0.
This assumption implies that
Ψ(y,w) |x̂Ψ(y,w) = y,
Ψ(y,w) |p̂Ψ(y,w) = w.
Note however that we will not use the above assumption in this section.
Explicitly,
i
=
e ~ (w·x− 2 y·w) Ψ(x − y),
P(y,w) (x1 , x2 )
=
Ψ(x1 − y)Ψ(x2 − y)e ~ (x1 −x2 )·w .
i
Theorem 4.1.
−d
(2π~)
Proof.
1
Ψ(y,w) (x)
Z
P(y,w) dydw = 1l.
Let Φ ∈ L2 (Rd ). Then
Z Z
(Φ|P(y,w) Φ)dydw
Z Z Z Z
i
=
Φ(x1 )Ψ(x1 − y)Ψ(x2 − y)e ~ (x1 −x2 )·w Φ(x2 )dx1 dx2 dydw
Z Z
= (2π~)d
Φ(x)Ψ(x − y)Ψ(x − y)Φ(x)dxdy = (2π~)d kΦk2 kΨk2 .
23
(4.70)
4.2
Contravariant quantization
Let b be a function on te phase space. We define its contravariant quantization by
Z
Opct (b) := (2π~)−d P(x,p) b(x, p)dxdp.
(4.71)
If B = Opct (b), then b is called the contravariant symbol of B.
We have
R
(1) |TrOpct (b)| ≤ (2π~)−d |b(x, p)|dxdp;
(2) kOpct (b)k ≤ supx,p |b(x, p)|;
(3) Opct (1) = 1l;
(4) Opct (b)∗ = Opct (b).
(5) Let b ≥ 0. Then Opct (b) ≥ 0.
4.3
Covariant quantization
The covariant quantization is the operation dual to the contravariant quantization. Strictly
speaking, the operation that has a natural definition and good properties is not the covariant
quantization but the covariant symbol of an operator.
Let B ∈ B(H). Then we define its covariant symbol by
b(x, p) := TrP(x,p) B = Ψ(x,p) |BΨ(x,p) .
B is then called the covariant quantization of b and is denoted by
Opcv (b) = B.
(1) Opcv (1) = 1l,
(2) Opcv (b)∗ = Opcv (b).
(3) kOpcv (b)k ≥ supx,p |b(x, p)|.
(4) Let Opcv (b) ≥ 0. Then bcv ≥ 0.
R
(5) TrOpcv (b) = (2π~)−d b(x, p)dxdp.
4.4
Connections between various quantizations
Let us compute various symbols of P(y,w) :
covariant symbol(x, p)
= |(Ψ|Ψ(y−x,w−p) |2 ,
Weyl symbol(x, p)
=
2d (Ψ(y−x,w−p) |IΨ(y−x,w−p) ),
contravariant symbol(x, p)
=
(2π~)d δ(x − y)δ(p − w).
Let us now show how to pass between the covariant, Weyl-Wigner and contravariant
quantization. Note that there is a preferred direction: from contravariant to Weyl, and then
from Weyl-Wigner to covariant. Going back is less natural.
24
Let
Opct (bct ) = Op(b) = Opcv (bcv ).
Then
−d
b(x, p)
=
(π~)
bcv (x, p)
=
(π~)−d
bcv (x, p)
=
Z
bct (y, w)(Ψ(y−x,w−p) |IΨ(y−x,w−p) )dydw,
Z
(2π~)−d
b(y, w)(Ψ(−y+x,−w+p) |IΨ(−y+x,−w+p) )dydw,
Z
2
bct (y, w)(Ψ|Ψ(y−x,w−p) ) dydw.
We have
TrOpcv (a)Opct (b) = (2π~)−d
Z
a(x, p)b(x, p)dxdp.
(4.72)
Indeed, let A = Opcv (a). Then the lhs of (4.72) is
Z
TrA(2π~)−d b(x, p)|Ψ(x,p) )(Ψ(x,p) |dxdp
Z
−d
= (2π~)
(Ψ(x,p) |A|Ψ(x,p) )b(x, p)dxdp,
which is the rhs of (4.72).
4.5
Gaussian coherent vectors
Consider the normalized gaussian vector scaled appropriately with the Planck constant
1
d
2
Ω(x) = (π~)− 4 e− 2~ x .
(4.73)
The corresponding coherent vectors are equal to
d
i
i
1
2
Ω(y,w) (x) = (π~)− 4 e ~ w·x− 2~ y·w− 2~ (x−y) .
(4.74)
In the literature, when one speaks about coherent states, one has usually in mind (4.74).
They are also called Gaussian or Glauber’s coherent states.
Z Z
2
2
2
2
1
~
1
b(x, p) =
bct (y, w)(π~)−d e− ~ (x−y) − ~ (p−w) dydw, b = e− 4 (Dx +Dp ) bct ;
Z Z
2
2
2
2
1
1
~
cv
b (x, p) =
b(y, w)(π~)−d e− ~ (x−y) − ~ (p−w) dydw, bcv = e− 4 (Dx +Dp ) b;
Z Z
2
2
2
2
1
1
~
bcv (x, p) =
bct (y, w)(2π~)−d e− 2~ (x−y) − 2~ (p−w) dydw, bcv = e− 2 (Dx +Dp ) bct .
In the case of Gaussian states, there are several alternative names of the covariant and
contravariant symbol of an operator:
(1) For contravariant symbol:
25
(i) anti-Wick symbol,
(ii) Glauber-Sudarshan function,
(iii) P-function;
(2) For covariant symbol:
(i) Wick symbol,
(ii) Husimi or Husimi-Kano function,
(iii) Q-function.
We will use the terms Wick/anti-Wick quantization/symbol.
One can distinguish 5 most natural quantizations. Their respective relations are nicely
described by the following diagram, called sometimes the Berezin diagram:
anti-Wick
quantization
 ~ 2 2
 − 4 (Dx +Dp )
ye
p, x
quantization
i~
e 2 Dx ·Dp
−→
Weyl-Wigner
quantization
 ~ 2 2
 − 4 (Dx +Dp )
ye
i~
e 2 Dx ·Dp
−→
x, p
quantization
Wick
quantization
4.6
Creation and annihilation operator
Set
ai
=
(2~)−1/2 (xi + ipi ),
a∗i
=
(2~)−1/2 (xi − ipi ).
We have
i
{ai , a∗j } = − δij .
~
~1/2
~1/2
xi = 1/2 (ai + a∗i ), pi = 1/2 (ai − a∗i ).
(4.75)
2
i2
In this way, the classical phase space Rd ⊕ Rd has been identified with the complex space
Cd . The Lebesgue measure has also a complex notation:
~d ∗
da da = dxdp.
id
To justify the notation (4.76) we write in terms of differential forms:
da∗j ∧ daj =
1
dx − idp ∧ dx + idp = i~−1 dx ∧ dp.
2~
26
(4.76)
On the quantum side we introduce the operators
âi
=
(2~)−1/2 (x̂i + ip̂i ),
â∗i
=
(2~)−1/2 (x̂i − ip̂i ).
We have
[âi , â∗j ] = δij .
~1/2
~1/2
∗
(â
+
â
),
p̂
=
(âi − â∗i ).
i
i
i
21/2
i21/2
Let b, b∗ be classical variables obtained from y, w, as above. Note that
(4.77)
x̂i =
i
e ~ (−yp̂+wx̂) = e(−b
∗
â+bâ∗ )
.
(4.78)
We have
i
i
i
e ~ (−yp̂+wx̂) x̂ = (x̂ + y)e ~ (−yp̂+wx̂) ,
e(−b
∗
∗
∗
â+bâ ) ∗
â = (â∗ + b∗ )e(−b
∗
â+bâ )
i
e ~ (−yp̂+wx̂) p̂ = (p̂ + w)e ~ (−yp̂+wx̂) ,
∗
e(−b
,
∗
â+bâ )
∗
â = (â + b)e(−b
∗
â+bâ )
.
(4.79)
(4.80)
Recall that in the real notation we had coherent vectors
i
Ωy,w := e ~ (−yp̂+wx̂) Ω.
(4.81)
In the complex notation they become
Ωb := e(−b
x2
∗
â+bâ∗ )
Ω.
(4.82)
1
Using Ω(x) = e− 2~ and âi = (2~)− 2 (x̂i + ~∂xi ) we obtain
âi Ω = 0.
This justifies the name “annihilation operators” for âi . More generally, by (4.80),
âj Ωb = bj Ωb ,
Note that the identity (4.70) can be rewritten as
Z
−d
1l = (2πi)
|Ωa )(Ωa |da∗ da.
4.7
Quantization by an ordering prescription
Consider a polynomial function on the phase space:
X
w(x, p) =
wα,β xα pβ .
α,β
27
(4.83)
It is easy to describe the x, p and p, x quantizations of w in terms of ordering the positions
and momenta:
X
Opx,p (w) =
wα,β x̂α p̂β ,
α,β
Opp,x (w)
X
=
wα,β p̂β x̂α .
α,β
The Weyl quantization amounts to the full symmetrization of x̂i and p̂j , as described in
(3.57).
We can also rewrite the polynomial (4.83) in terms of ai , a∗i by inserting (4.75). Thus
we obtain
X
w(x, p) =
w̃γ,δ a∗γ aδ =: w̃(a∗ , a).
(4.84)
γ,δ
Then we can introduce the Wick quantization
X
∗
Opa ,a (w) =
w̃γ,δ â∗γ âδ
(4.85)
γ,δ
and the anti-Wick quantization
∗
Opa,a (w) =
X
w̃γ,δ âδ â∗γ .
(4.86)
γ,δ
Theorem 4.2. (1) The Wick quantization coincides with the covariant quantization for
Gaussian coherent states.
(2) The anti-Wick quantization coincides with the contravariant quantization for Gaussian
coherent states.
Proof. (1)
∗
(Ω(x,p) |Opa
,a
(w)Ω(x,p) )
Ωa |
=
X
w̃γ,δ â∗γ âδ Ωa )
γ,δ
X
=
w̃γ,δ a∗γ aδ
γ,δ
= w(x, p).
(2)
a,a∗
Op
(w)
=
X
−d
δ
w̃γ,δ â (2πi)
Z
|Ωa )(Ωa |da∗ daâ∗γ
γ,δ
=
(2πi)−d
XZ
w̃γ,δ aδ a∗γ |Ωa )(Ωa |da∗ da
γ,δ
=
(2π~)−d
Z
w(x, p)|Ω(x,p) )(Ω(x,p) |dxdp.
28
2
The Wick quantization is widely used, especially for systems with an infinite number of
degrees of freedom. Note the identity
∗
Ω|Opa
4.8
,a
(w)Ω) = w̃(0, 0).
(4.87)
Connection between the Wick and anti-Wick quantization
As described in equation (4.83), there are two natural ways to write the symbol of the Wick
(or anti-Wick) quantization. We can either write it in terms of x, p, or in terms of a∗ , a. In
the latter notation we decorate the symbol with a tilde.
Let
∗
∗
∗
∗
Opa,a (wa,a ) = Opa ,a (wa ,a ).
Then
∗
wa
∗
w̃a
,a
,a
(x, p)
(a∗ , a)
~
2
∗
2
e 2 (∂x +∂p ) wa,a (x, p)
Z Z
1
∗
− 2~
(x−y)2 +(p−w)2
−d
= (2π~)
e
wa,a (y, w)dydw,
=
(4.88)
∗
e∂a∗ ∂a w̃a,a (a∗ , a)
Z Z
∗
∗
∗
= (2πi)−d
e−(a −b )(a−b) w̃a,a (b∗ , b)db∗ db..
=
(4.89)
(4.88) was proven before. To see that (4.89) and (4.88) are equivalent we note that
∂a
=
∂a∗
=
~1/2
(∂x + i∂p ),
21/2
~1/2
(∂x − i∂p ),
21/2
hence
~ 2
(∂ + ∂p2 ).
2 x
One can also see (4.89) directly. To this end it is enough to consider a∗n am (a and a∗ are
now single variables). To perform Wick ordering we need to make all possible contractions.
Each contraction involves a pair of two elements: one from {1, . . . , n} and the other from
{1, . . . , m}. The number of possible k-fold contractions is
∂a∗ ∂a =
m!
1
n!
m!
n!
k! =
.
k!(n − k)! k!(m − k)!
k! (n − k)! (m − k)!
But
e∂a∗ ∂a a∗n am =
∞
X
1
n!
m!
a∗(n−k)
am−k .
k! (n − k)!
(m − k)!
k=0
29
4.9
Wick symbol of a product
Suppose that
∗
Opa
,a
(w) = Opa
Then
w̃(a∗ , a) = e
∂a2 ∂a∗
1
∗
,a
∗
(w2 )Opa
,a
(w1 ).
w̃2 (a∗2 , a2 )w̃1 (a∗1 , a1 )
a = a2 = a1 .
(4.90)
(Clearly a = a2 = a1 implies a∗ = a∗2 = a∗1 ). This follows essentially by the same argument
as the one used to show (4.89). Using (2.14), one can rewrite (4.90) as an integral:
Z Z
db∗ db
w̃(a∗ , a) =
.
(4.91)
w̃2 (a∗ , b)w̃1 (b∗ , a)
(2πi)d
We also have a version of (4.90) for an arbitrary number of factors. Let
Opa
∗
,a
∗
(w) = Opa
,a
(wn ) · · · Opa
∗
,a
(w1 ).
Then
∂ak ∂a∗j w̃n (a∗n , an ) · · · w̃1 (a∗1 , a1 )
a = an = · · · = a1 .
∗
∗
Ω|Opa ,a (wn ) · · · Opa ,a (w1 )Ω
X
exp
∂ak ∂a∗j w̃n (a∗n , an ) · · · w̃1 (a∗1 , a1 )
0 = an = · · · = a1 .
w̃(a∗ , a) = exp
X
k>j
(4.92)
Consequenly,
=
k>j
We can rewrite (4.92) as an integral:
w̃(a∗ , a)
(4.93)
Z
Z
n−1
n−1
X
Y dbj+1 db∗j
.
(bj+1 − bj )(b∗j − a∗ ) w̃n (b∗n , bn ) · · · w̃1 (b∗1 , b1 )
= · · · exp
(2πi)d
j=1
j=1
Indeed, introduce new variables
cn := an − an−1 ,
cn−1 = an−1 − an−2 , . . . ,
c1 = a1 .
Then
∂cn = ∂an ,
∂cn−1 = ∂an + ∂an−1 , . . . ,
∂c1 = ∂an + · · · + ∂a1 .
Therefore,
X
k>j
∂ak ∂a∗j =
n−1
X
j=1
30
∂cj+1 ∂a∗j .
Hence
exp
X
∂ak ∂a∗j Ψ(a∗n , an , . . . , a∗1 , a1 )
(4.94)
k>j
Z
=
Z
···
exp
n−1
n−1
X
Y dbj+1 db∗j
(bj+1 − bj − aj+1 + aj )(b∗j − a∗j ) Ψ(b∗n , bn , . . . , b∗1 , b1 )
(2πi)d
j=1
j=1
Finally, we insert in (4.94) an = · · · = a1 = a, a∗n = · · · = a∗1 = a∗ .
5
5.1
Pseudodifferential calculus with uniform symbol classes
Gaussian dynamics on uniform symbol classes
We will denote by S(Rn ) the space of b ∈ C ∞ (Rn ) such that
|∂xα b| ≤ Cα .
Note that S(Rn ) has the structure of a Frechet space.
i
Theorem 5.1. Let ν be a quadratic form. Then e 2 DνD is bounded on S(Rn ). Then, for ν
in a bounded set of quadratic forms, there exist C and m such that
i
sup |e 2 DνD b(x)| ≤ C sup |∂β b(x)|.
(5.95)
|β|≤m
i
Consequently, e 2 DνD is bounded on S(Rn ).
5.2
The boundedness of quantized uniform symbols
Let us set ~ = 1.
We will denote by S(Rd ⊕ Rd ) the space of b ∈ C ∞ (Rd ⊕ Rd ) such that
|∂xα ∂pβ b| ≤ Cα,β .
Theorem 5.2 (The Calderon-Vaillancourt Theorem). If b ∈ S(Rd ⊕ Rd ), then Opx,p (b) is
bounded and
X
kOpx,p (b)k ≤ C(n)
sup |∂xα ∂pβ b(x, p)|.
|α|+|β|≤N (n)
There exists also a version of the Calderon-Vaillancourt Theorem with the Weyl quantization replacing the x, p-quantization.
We will write adB (A) := [B, A].
Theorem 5.3. The following conditions are equivalent:
(1)
β
2
d
adα
x̂ adp̂ B ∈ B(L (R )), α, β.
31
(2) B = Opx,p (b), b ∈ S(Rd ⊕ Rd ),
(3) B = Op(b), b ∈ S(Rd ⊕ Rd ).
The implication (1)⇒(2) is called the Beals Criterion. The Calderon-Vaillancourt Theorem implies (1)⇐(2). Let us denote by Ψ the set of operators described in Theorem 5.3.
Theorem 5.4. (1) Ψ is an algebra.
(2) Let B ∈ Ψ be boundedly invertible. Then B −1 ∈ Ψ.
(3) Let f be a function holomorphic on sp B, where B ∈ Ψ. Then f (B) ∈ Ψ.
(4) Let f be a function smooth on sp B, where B = B ∗ ∈ Ψ. Then f (B) ∈ Ψ.
5.3
Semiclassical calculus
We go back to ~. We will write Op~ for the quantization depending on ~.
Theorem 5.5. Let a, b ∈ S(Rd ⊕ Rd ). Then there exist c0 , . . . , cn ∈ S and ~ 7→ r~ ∈
S(Rd ⊕ Rd ) such that
Op~ (a)Op~ (b) =
n
X
~2j Op~ (c2j ) + ~2n+2 Op~ (r~ ),
j=0
α β k ∂x ∂p ∂~ r~ ≤ Cα,β,k .
Besides,
c0 = ab, c1 =
i
{a, b}.
2
If in addition a or b is 0 on an open set Θ ⊂ Rd ⊕ Rd , then so are c0 , . . . , cn .
For a complex function b let ran(b) denote the closure of the image of b.
Theorem 5.6. Let b ∈ S and 0 6∈ ran(b). Then for small enough ~ the operator Op~ (b) is
invertible and there exist c0 , c2 , . . . , c2n ∈ S and ~ 7→ r~ ∈ S such that
Op~ (b)−1 =
n
X
~2j Op~ (c2j ) + ~2n+2 Op~ (r~ ),
j=0
α β k ∂x ∂p ∂~ r~ ≤ Cα,β,k .
Besides,
c0 = b−1 .
As a corollary of the above
theorem, for any neighborhood of ran(b) there exists ~0 such
that, for |~| ≤ ~0 , sp Op~ (b) is contained in this neighborhood.
32
Theorem 5.7. Let b ∈ S and f be a function holomorphic on a neighborhood
of the image
of b. Then for small enough ~ the function f is defined on sp Op~ (b) and there exist
c0 , c2 , . . . , c2n ∈ S and ~ 7→ r~ ∈ S such that
n
X
f Op~ (b) =
~2j Op~ (c2j ) + ~2n+2 Op~ (r~ ),
j=0
α β k ∂x ∂p ∂~ r~ ≤ Cα,β,k .
Besides,
c0 = f ◦ b.
If b is real and smooth, then the same conclusion holds.
Theorem 5.8. Let c ∈ S(Rd ⊕ Rd ⊕ Rd ). Then the operator B with the kernel
Z
i
−d
B(x, y) = (2π~)
c(x, p, y)e ~ (x−y)p dp
belongs to Ψ and equals Op(b), where
i~
b(x, p) = e 2 Dp (−Dx +Dy ) c(x, p, y)
.
x=y
Consequently,
b(x, p) = c(x, p, x) +
i~
∂x c(x, p, y) − ∂y c(x, p, y) + O(~2 ).
2
x=y
(5.96)
Proof. We compute:
b(x, p) = (2π~)−d
Z
i
z
z
e ~ z(w−p) c x + , w, x − dzdw,
2
2
then we apply (2.17). 2
5.4
Inequalities
∗
Theorem 5.9. (1) Let b ∈ S and Op(b) = Opa
O(~) in S
∗
∗
,a
∗
(ba
,a
∗
). Then ba
,a
∈ S and b − ba
∗
,a
=
∗
∗
(2) Let ba,a ∈ S and Opa,a (ba,a ) = Op(b). Then b ∈ S and ba,a − b = O(~) in S.
Proof. We use
ba
∗
,a
2
~
2
= e 4 (∂x +∂p ) b,
b=e
2
2
~
4 (∂x +∂p )
2
2
ba,a ,
and the obvious mapping properties of e 4 (∂x +∂p ) . 2
~
33
(5.97)
∗
(5.98)
Theorem 5.10 (Sharp Gaarding Inequality). Let b ∈ S be positive. Then
Op(b) ≥ −C~.
Proof. Let b0 be the Wick symbol of Op(b), that is,
∗
Opa,a (b0 ) = Op(b).
(5.99)
Then, by Thm 5.9 (1), we have b0 ∈ S and b0 − b = O(~) in S. Besides,
∗
Op(b0 ) = Opa
,a
(b).
Now
Op(b) = Op(b0 ) + Op(b − b0 ) = Opa
(5.100)
∗
,a
(b) + O(~).
(5.101)
The first term on the right of (5.101) is positive, because it is the anti-Wick quantization of
a positive symbol. 2
Theorem 5.11 (Fefferman-Phong Inequality). Let b ∈ S be positive. Then
Op~ (b) ≥ −C~2 .
We will not give a complete proof. We will only note that the inequality follows by basic
calculus if we assume that
k
X
c2j
b=
j=1
for real cj ∈ S. We note also that the Sharp Gaarding inequality is true for matrix valued
symbols, with the same proof. This is not the case of the Fefferman-Phong Inequality.
5.5
Semiclassical asymptotics of the dynamics
Theorem 5.12 (Egorov Theorem). Let h be the sum of a polynomial of second order and
a S(Rd ⊕ Rd ) function.
(1) Let x(t), p(t) solve the Hamilton equations with the Hamiltonian h and the initial conditions x(0), p(0). Then
γt (x(0), p(0)) = x(t), p(t)
defines a symplectic (in general, nonlinear) transformation which preserves S(Rd ⊕Rd ).
(2) Let b ∈ S(Rd ⊕ Rd ). Then there exist bt,2j ∈ S(Rd ⊕ Rd ), j = 0, 1, . . . , such that for
|t| ≤ t0
n
X
it
it
e ~ Op(h) Op(b)e− ~ Op(h) −
Op ~2j bt,2j = O(~2n+2 ).
(5.102)
j=0
Moreover,
bt,0 (x, p) = b γt−1 (x, p)
and suppbt,2j ⊂ γt suppb, j = 0, 1, . . . .
34
(5.103)
Proof. We have
n
X
=
j=0
n
X
~2j
it
d − it Op(h)
Op bt,2j e ~ Op(h)
e ~
dt
~2j e
−it
~ Op(h)
j=0
−
it
i
d
Op(h), Op bt,2j + Op
bt,2j e ~ Op(h)
~
dt
(5.104)
(5.105)
This gives us a sequence of equations for bt,2j of the form
where rt,2j
d
bt,0 = {bt,0 , h},
dt
d
bt,2 = {bt,2 , h} + rt,2 ,
dt
......
d
bt,2j = {bt,2j , h} + rt,2j ,
dt
involve bt,0 , . . . , bt,2j−2 . The initial conditions are
(5.106)
(5.107)
(5.108)
(5.109)
bt,0 = b,
(5.110)
bt,2 = 0,
(5.111)
......
(5.112)
bt,2j = 0.
(5.113)
(5.106) is solved by (5.103). (5.109) is solved by
Z t
bt,2j (x, p) =
rt−s,2j γs−1 (x, p) ds.
(5.114)
0
The error term in (5.105) is O(~2n+2 ). Integrating (5.105) from 0 to t we obtain the
estimate (5.102). 2
5.6
Frequency set
Let ~ 7→ ψ~ ∈ L2 (X ). Let (x0 , p0 ) ∈ Rd ⊕ Rd .
Theorem 5.13. The following conditions are equivlent:
(1) There exists b ∈ Cc∞ (Rd ⊕ Rd ) such that b(x0 , p0 ) 6= 0 and
kOp~ (b)ψ~ k = O(~∞ ).
(2) There exists a neighborhood U of (x0 , p0 ) such that for all c ∈ Cc∞ (U)
kOp~ (c)ψ~ k = O(~∞ ).
The set of points in Rd ⊕ Rd that do not satisfy the conditions of Theorem 7.7 is called
the frequency set of ~ 7→ ψ~ and denoted F S(ψ~ ).
Note that we can replace the Weyl quantization by the x, p or p, x quantization in the
definition of the frequency set.
35
5.7
Properties of the frequency set
Theorem 5.14. Let a ∈ S. Then
F S (Op~ (a)ψ~ ) = supp(a) ∩ F S(ψ~ ).
Theorem 5.15. Let h ∈ S + Pol≤2 be real. Let t 7→ γt be the Hamiltonian flow generated
by h. Then
F S(eitOp~ (h) ψ~ ) = γt (F S(ψ~ )) .
Theorem 5.16. Let
i
ψ~ (x) = a(x)e ~ S(x) .
Then
F S(ψ~ ) = {x ∈ suppa, p = ∇S(x)}.
Proof. We apply the nonstationary method. Let p 6= ∂x S(x) on the support of b ∈
d
Cc∞ (Rd ⊕ Rd ). Let (2π~)− 2 F~ denote the unitary semiclassical Fourier transformation.
Then
Z
i
i
p,x
−d
−d
2
2
(2π~) F~ Op~ (b)ψ~ (p) = (2π~)
e− ~ xp b(p, x)a(x)e ~ S(x) dx. (5.115)
Let
T := (p − ∂x S(x))−2 (p − ∂x S(x))∂x .
Let
T # = ∂ˆx (p − ∂x S(x))−2 (p − ∂x S(x))
be the transpose of T . Clearly,
i
i
−i~T e ~ (S(x)−xp) = e ~ (S(x)−xp) .
(5.116)
Therefore, (5.115) equals
d
Z
d
Z
(−i~)n (2π~)− 2
=
(−i~)n (2π~)− 2
i
b(p, x)a(x)T n e ~ (S(x)−xp) dx
i
(5.117)
d
e ~ (S(x)−xp) T #n b(p, x)a(x)dx = O(~n− 2 ).
(5.118)
2
5.8
Algebras with a filtration/gradation
Let S be an algebra.
We say that it is an algebra with filtration {S m : m ∈ Z− } iff S m are linear subspaces
T
0
0
0
of S such that S = S 0 , S m ⊂ S m , m ≤ m0 and S m ·S m ⊂ S m+m . We write S −∞ := S m .
m
Clearly, S −∞ is an ideal and so are S m .
36
0
⊕ S (m) such that S (m) ·S (m ) ⊂
P (m0 )
0
S (m+m ) . Clearly, a gradation induces a filtration by taking S m :=
S
.
We say that S is an algebra with a gradation if S =
m∈Z−
m0 ≤m
Instead of Z+ we can take any ordered group, eg. Z orP
R, then the definition slightly
changes: the full algebra is denoted S ∞ and satisfies S ∞ =
S m . S −∞ is an ideal in S ∞ .
m
S m for m ≥ 0 are ideals in the algebra S 0 .
5.9
Algebra of semiclassical operators
We say that ~ 7→ b~ is a admissible semiclassical symbol if for any n there exist b0 , . . . , bn ∈ S
and ~ 7→ r~ ∈ S such that for any n
b=
n
X
~j bj + ~n+1 r~
j=0
α β k ∂x ∂p ∂~ r~ ≤ Cα,β,k .
Note that the sequence b0 , b1 , . . . is uniquely defined (does not depend on n).
Let Θ ⊂ Rd ⊕ Rd be closed. We say that b~ is O(~∞ ) outside Θ if b0 , b1 , · · · = 0 outside
Θ.
Let Ssc denote the space of admissible semiclassical symbols and Ψsc the set of their
semiclassical quantizations. We write Ssc (Θ) for the space of symbols that vanish outside Θ
and Ψsc (Θ) for their quantizations.
Clearly, Ψsc is an algebra with gradation given by Ψ−m
:= ~m Ψsc . It is closed wrt
sc
operations described in Theorems. Ψsc (Θ) are ideals in Ψsc .
5.10
Principal and subprincipal symbols
P∞
Recall that by s(A~ ) = a~ (x, ξ) we denote the symbol of A~ . We have a~ ' m=0 ~j a−j .
We call a0 the principal and a−1 the subprincipal symbol. Let us introduce the extended
principal sumbol:
sep (A) := a0 + ~a−1 .
P∞ k
x,p
If A = Op (b) and b ' k=0 ~ b−k , then the principal symbol is b0 and the subprincipal
symbol is b−1 + i~
2 ∂x ∂ξ b0 .
Theorem 5.17. Let A ∈ Ψsc and A0 ∈ Ψsc . Then
A~ A0~ ∈ Ψsc
1
sep
[A~ , A0~ ]+
2
[A~ , A0~ ] ∈ ~Ψsc
sep ~−1 [A~ , A0~ ]
and
=
(5.119)
sep (A~ )sep (A0~ ) (mod ~2 ),
(5.120)
{sep (A~ ), sep (A0~ )} (mod ~2 ).
(5.122)
and
=
(5.121)
37
5.11
Algebra of formal semiclassical operators
We also have the space of formal semiclassical symbols, which are formal power series
b=
n
X
~j bj
j=0
f
with coefficients, say, in S. We denote this space by Ssc
and their quantizations by Ψfsc .
Theorem 5.18. We have the isomorphism
Ψsc /Ψ−∞
= Ψfsc .
sc
5.12
More about star product
The star product can be written in the following asymmetric form:
b ∗ c(x, p)
=
=
~
b x − Dp , p +
2
~
b x − Dp , p +
2
~ Dx c(x, p)
2
~ Dx c(x, p).
2
(5.123)
(5.124)
Note that the operators x − ~2 Dp and p + ~2 Dx commute. Thus we can understand the rhs
of(5.123) as a function of two commuting operators. Alternatively, we can understand it as
the PDE notation for the x, p quantization. Alternatively, (5.124) gives us an interpretation
in terms of the Weyl quantization.
To see this we start from
i
b ∗ c(x, p) = e 2 ~(Dp1 Dx2 −Dx1 Dp2 ) b(x1 , p1 )c(x2 , p2 ) x := x = x , .
(5.125)
1
2
p := p1 = p2 .
i
We treat b(x, p) as the operator of multiplication. We move e 2 ~(Dp1 Dx2 −Dx1 Dp2 ) to the right
obtaining
~
~
b x1 − Dp2 , p1 + Dx2 c(x2 , p2 ) x := x = x , ,
1
2
2
2
p := p1 = p2 .
which equals the LHS of (5.125).
38
Note the following consequences:
f Op(h) Op(b)
! !
~
~ = Op f h x − Dp , p + Dx b ,
2
2
f Op(h)
=
Op f
! !
1 ~
~ 1 ~
~ h x − Dp , p + Dx + h x + Dp , p − Dx 1 ,
2
2
2
2
2
2
i
i
e ~ Op(h) Op(b)e− ~ Op(h)
! !
i
~ ~ ~
~
= Op exp
h x − Dp , p + Dx − h x + Dp , p − Dx b .
~
2
2
2
2
We have similar formulas for the product in the x, p-quantization. Let
Opx,p (a) = Opx,p (b)Opx,p (c).
Then
a(x, p)
6
6.1
= b x, p + ~Dx c(x, p)
= c x + ~Dp , p b(x, p).
Spectral asymptotics
Functional calculus and Weyl asymptotics
It follows from (3.44) that
Op(b)n = Op(bn ) + O(~2 ).
Hence for polynomial functions
f Op(b) = Op(f ◦ b) + O(~2 ).
(6.126)
One can expect (6.126) to be true for a larger class of nice functions. Consequently,
Trf Op(b) = Tr Op(f ◦ b) + O(~2 )
Z
= (2π~)−d f b(x, p) dxdp + O(~−d+2 ).
(6.127)
In particular, we can try to use f = 1l]−∞,µ] . It is too optimistic to expect
1l]−∞,µ] (Op(h)) = Op 1l]−∞,µ] (h) + O(~2 ).
(6.128)
After all the step function is not nice – it is not even continuous. If there is a gap in the
spectrum around µ, one can try to smooth it out. Therefore, there is a hope for some weaker
error term instead of O(~2 ).
39
6.2
Weyl asymptotics
For a bounded from below self-adjoint operator H set
Nµ (H) := #{eigenvalues of H counted with multiplicity ≤ µ}
= Tr1l]−∞,µ] (H).
(6.129)
(6.130)
If (6.128) were true, then we would have
Nµ Op(h)
=
(2π~)−d
Z
dxdp + O(~−d+2 ).
(6.131)
h(x,p)≤µ
Asymptotics of the form (6.131) are called the Weyl asymptotics. In practice the error
term O(~−d+2 ) may be too optimistic and one gets something worse (but hopefully at least
o(~−d )).
We will show that if V is continuous potential with V − µ > 0 outside a compact set
then
Z
d
(6.132)
Nµ (−~2 ∆ + V (x)) ' (2π~)−d cd
|V (x) − µ|−2 dx + o(~−d ).
V (x)≤µ
This is (6.131) for H = p2 + V (x) with a rather weak error estimate.
We will prove this without using the Weyl calculus. Here are the tools that we will use:
A ≤ B ⇒ Nµ (A) ≥ Nµ (B),
Nµ (A ⊕ B) = Nµ (A) + Nµ (B).
To simplify we will assume that d = 1.
Lemma 6.1. Let ∆[0,L],D , resp. ∆[0,L],N denote the Dirichlet, resp. Neumann Laplacian
on [0, L]. For α ∈ R let [α] denote the largest integer ≤ α, θ(α) the Heavyside function and
|µ|+ := µθ(µ). Then
1/2
Nµ − ~2 ∆[0,L],D
=
[L(π~)−1 |µ|+ ],
Nµ − ~2 ∆[0,L],N
=
[L(π~)−1 |µ|+ ] + θ(µ).
1/2
Proof. The eigenfunctions and the spectrum of ∆[0,L],D , resp. ∆[0,L],N are
πnx
,
L
πnx
cos
,
L
sin
~2 π 2 n2
,
L2
~2 π 2 n2
,
L2
n = 1, 2, . . . ;
n = 0, 1, 2, . . . .
1/2
Thus the last eigenvalue has the number n = [L(~π)−1 |µ|+ ]. 2
Divide R into intervals
h
i
IN,j := (j − 1/2)N −1 , (j + 1/2)N −1 .
40
Put at the borders of the intervals Neumann/Dirichlet boundary conditions. The Neumann
conditions lower the expectation value and the Dirichlet conditions increase them. Set
V N,j
=
sup{V (x) : x ∈ IN,j },
V N,j
=
inf{V (x) : x ∈ IN,j }.
We have
⊕
− ~2 ∆IN,j ,N + V N,j
− ~2 ∆IN,j ,D + V N,j .
j∈Z
≤ −~2 ∆ + V (x) ≤
⊕
j∈Z
Hence,
X
Nµ − ~2 ∆IN,j ,N + V N,j
j∈Z
≥ Nµ
− ~2 ∆ + V (x)
≥
X
Nµ − ~2 ∆IN,j ,D + V N,j .
j∈Z
Therefore,
X
1/2
N −1 (~π)−1 |V N,j − µ|− +
≥ Nµ − ~2 ∆ + V (x)
≥
X
θ µ − V N,j
j∈Z
j∈Z
X
N −1 (~π)−1 |V N,j −
1/2
µ|− .
j∈Z
Using the fact that |V − µ|− has a compact support, we can estimate
X
θ µ − V N,j
≤ N C.
j∈Z
By properties of Riemann sums we can find N such that for N ≥ N
Z
X
1/2
1/2
−1
N |V N,j − µ|− − |V (x) − µ|− dx < /3,
(6.133)
j∈Z
Z
X
1/2
1/2
−1
N |V N,j − µ|− − |V (x) − µ|− dx <
/3.
(6.134)
j∈Z
Therefore,
Z
1
1/2
2
|V (x) − µ|− dx
Nµ − ~ ∆ + V (x) −
~π
<
2
CN
+
.
~π3
π
(6.135)
Hence the right hand side of (6.135) is o(~−1 ). This proves (6.132)
If we assume that V is
to be C0 −1 . Then we
√ differentiable, then N can be assumed
−1
can optimize and set = ~. This allows us to replace o(~ ) by O(~−1/2 ).
41
6.3
Energy of many fermion systems
Consider fermions with the 1-particle space is spanned by an orthonormal basis Φ1 , Φ2 , . . . .
The n-particle fermionic space is spanned by Slater determinants
1
Ψi1 ,...,in := √ Φi1 ∧ · · · ∧ Φin ,
n!
i1 < · · · < in .
Suppose that we have noninteracting fermions with the 1-particle Hamiltonian H. Then
the Hamiltonian on the N -particle space will be denoted by dΓn (H).
Suppose that E1 < E2 < . . . are the eigenvalues of H in the ascending order and
Φ1 , Φ2 , . . . are the corresponding normalized eigenvectors. This means that the full Hamiltonian dΓn (H) acts on Slater determinants as
dΓn (H)Ψi1 ,...,in = (Ei1 + · · · + Ei1 )Ψi1 ,...,in .
For simplicity we assume that eigenvalues are nondegenerate. Then the ground state of
the system is the Slater determinant
1
Ψ := √ Φ1 ∧ · · · ∧ Φn .
n!
(6.136)
The ground state energy is E1 + · · · + En .
In practice it is often more convenient as the basic parameter to use the chemical potential
µ instead of the number of particles n. Then the 1-particle density matrix of the ground
state is given by 1l]−∞,µ] (H), where we find µ from the relation
Tr1l]−∞,µ] (H) = n.
If B is a 1-particle observable, then its expectation value in an n-fermionic state Ψ is
given by
ΨdΓn (B)Ψ = TrBγΨ
where γΨ is the so-called reduced 1-particle density matrix. Note that 0 ≤ γΨ ≤ 1l and
TrγΨ = n. The reduced 1-particle density matrix of the Slater determinant (6.136) Ψ is the
projection onto the space spanned by Φ1 , . . . , Φn .
Suppose that the 1-particle space is L2 (Rd ). Then the 1-particle reduced density matrix
can be represented by its kernel γΨ (x, y). Expliccitly,
Z
Z
(6.137)
γΨ (x, y) = dx2 · · · dxn Ψ(x, x2 , · · · , xn )Ψ(y, x2 , · · · , xn ).
We are particularly interested in expectation values of the position. For position independent observables we do not need to know the full reduced density matrix, but only the
density:
Z
TrγΨ f (x̂) = ρΨ (x)f (x),
42
where
ρΨ (x) := γΨ (x, x).
Note that
Z
ρΨ (x)dx = n.
(6.138)
If γ = Op(g), then
TrγΨ f (x̂) = (2π~)−d
Z Z
g(x, p)f (x)dxdp.
Hence
ρΨ (x) = (2π~)−d g(x, p)dp.
Suppose now that the 1-particle Hamiltonian is H = Op(h). Remember that then the
symbol of 1l]−∞,µ] (H) is approximately given by
1l]−∞,µ] h(x, p) .
Hence, the reduced 1-particle density matrix of the ground state is γ = 1l]−∞,µ] (H). The
corresponding density is
Z
Z
−d
−d
ρ(x) ≈ (2π~)
1l]−∞,µ] h(x, p) dp = (2π~)
dp.
h(x,p)≤µ
Let cd rd be the volume of the ball of radius r. If h(x, p) = p2 + v(x), then
Z
ρ(x) ≈ (2π~)−d
dp
p2 +v(x)≤µ
d
=
(2π~)−d cd |v(x) − µ|−2 .
Let us compute the kinetic energy
Z Z
−d
2
Trp̂ 1l]−∞,µ] (H) ≈ (2π~)
p2 dxdp
p2 +v(x)<µ
=
−d
Z
(2π~)
Z
dx
dcd |p|d+1 d|p|
1
2
|p|<|v(x)−µ|−
Z
d+2
dcd
|v(x) − µ|−2
d+2
Z
d+2
d
−2/d
−d
≈ (2π~)
c
ρ d (x)dx.
d+2 d
=
(2π~)−d
dx
Thus if we know that ρ is the density of a ground state of a Schrödinger Hamiltonian, then
we expect that the kinetic energy is given by the functional
Z
d+2
d −2/d
Ekin (ρ) := (2π~)−d
cd
ρ d (x)dx.
(6.139)
d+2
43
Clearly, the potential energy of a state with density ρ in the potential V is given by
Z
Epot (ρ) := V (x)ρ(x)dx.
(6.140)
Suppose in addition that the fermions interact with the potential W . That is, we consider
the Hamiltonian
N
X
X
p2i + V (xi ) +
W (xi − xj )
(6.141)
i=1
1≤i<j≤N
on the N -particle antisymmetric space ∧N L2 (Rd ).
Then we can expect by classical arguments that for a state Ψ
Z Z
X
Ψ|
W (xi − xj )Ψ '
W (x − y)ρΨ (x)ρΨ (y)dxdy.
1≤i<j≤N
This suggests to introduce the interaction functiona
Z Z
Eint (ρ) :=
W (x − y)ρ(x)ρ(y)dxdy.
(6.142)
The Thomas-Fermi functional is given by the sum of (6.139), (6.140) and ((6.142):
ETF (ρ) := Ekin (ρ) + Epot (ρ) + Eint (ρ)
Z
d+2
d −2/d
cd
ρ d (x)dx
= (2π~)−d
d+2
Z
Z Z
+ V (x)ρ(x)dx +
W (x − y)ρ(x)ρ(y)dxdy.
(6.143)
(6.144)
We expect that
Z
n
inf ETF (ρ) | ρ ≥ 0,
ρ(x)dx = n
o
(6.145)
approximates the ground state energy of (6.141).
7
Standard pseudodifferential calculus on Rd
7.1
Classes of symbols
We will often write X = Rd .
m
Let m ∈ N. We define Spol
(T# Rd ) to be the set of functions of the form
a(x, ξ) =
X
aβ (x)ξ β ,
(7.146)
|β|≤m
where
|∂xα aβ | ≤ cα,β .
44
(7.147)
Let m ∈ R. We define S m (T# Rd ) to be the set of functions a ∈ C ∞ (T# Rd ) such that
for any
|∂xα ∂ξβ a(x, ξ)| ≤ cα,β hξim−|β| .
(7.148)
m
We set Sph
(T# Rd ) to be the set of functions a ∈ S m (T# Rd ) such that for any n there
exist functions am−k , k = 0, . . . , n, homogeneous of degree m − k such that
|∂xα ∂ξβ am−k (x, ξ)| ≤
n
X
α β
am−k (x, ξ) ≤
∂x ∂ξ a(x, ξ) −
cα,β |ξ|m−k−|β| , |ξ| > 1,
cα,β,n |ξ|m−k−n−1 . |ξ| > 1.
k=0
P∞
We then write a ' k=0 am−k , where am−k are uniquely determined.
We introduce also
m
S −∞ := ∩ S m = ∩ Sph
,
m
S
Clearly, for m ∈ N,
7.2
∞
m
:= ∪ S ,
m
Spol
m
⊂
m
Sph
.
∞
Sph
m
:=
m
∪ Sph
,
m
For any m ∈ R,
∞
m
Spol
:= ∪ Spol
.
m
m
Sph
m
⊂S .
Classes of pseudodifferential operators
m
Let A be an operator from S(Rd ) to S 0 (Rd ). We will say that A ∈ Ψm , Ψm
ph , Ψpol iff
A = Op(a)
m
m
for a ∈ S m , Sph
, Spol
. (Note, that instead of the Weyl quantization we could use the KohnNirenberg quantization as well, obtaining the same class of operators).
For k ∈ R, the kth Sobolev space is defined as
L2,k (Rd ) := (1 − ∆)−k/2 L2 (Rd ).
We also write
L2,∞ := ∩ L2,k (Rd ),
k
L2,−∞ := ∪ L2,k (Rd ).
k
The Calderon-Vaillancourt Theorem implies immediately the following theorem:
Theorem 7.1. For any k, m ∈ R, A ∈ Ψm extends to a bounded operator
A : L2,k (Rd ) → L2,k−m (Rd ),
and also to a continuous operator on L2,∞ and L2,−∞ . A ∈ Ψ−∞ maps L2,−∞ to L2,∞ .
45
7.3
Extended principal symbol
(m−2)
We will write sep (A) for s(A)
) and call it the extended principal symbol of a.
P∞(modS
If A = Op(a) and a ' k=0 am−k , then am is called the principal symbol and am−1 the
subprincipal symbol of a. We
then identify aep with am + am−1 .
Pcan
∞
If A = Opx,p (b) and b ' k=0 bm−k , then the principal symbol is bm and the subprincipal
symbol is bm−1 + 2i ∂x ∂ξ bm .
0
Theorem 7.2. Let A ∈ Ψm and A0 ∈ Ψm . Then
0
AA0 ∈ Ψm+m
1
sep [A, A0 ]+
2
0
0
[A, A ] ∈ Ψm+m −1
0
sep ([A, A ])
7.4
and
=
(7.149)
0
sep (A)sep (A0 )(mod S m+m −2 )
(7.150)
and
=
(7.151)
0
{sep (A), sep (A )}(mod S
m+m0 −3
).
(7.152)
Diffeomorphism invariance
Let F : Rd → Rd be a diffeomorphism that moves only a bounded part of Rd . We can define
its prolongation to the cotangent bundle T# Rd , denoted T# F and the corresponding acttion
on S(Rd ), denoted F# .
m
m
Theorem 7.3. The spaces Spol
(T# Rd ), Sph
(T# Rd ), S m (T# Rd ) are invariant wrt T# F .
The operators F# are bounded invertible on spaces L2,m .
The algebras Ψpol (Rd ), Ψph (Rd ) and Ψ(Rd ) are invariant wrt F# .
7.5
Ellipticity
Let m ≥ 0. We say that b ∈ S m (Rd ) is elliptic if b is real and for some r, c0 > 0
b(x, ξ) ≥ c0 |ξ|m , |ξ| > r.
Theorem 7.4. Let p ∈ R. Let b ∈ S m (Rd ) be elliptic and z − b(x, ξ) invertible.
(1) (−z + b(x, ξ))p is well defined and belongs to S mp (T # Rd ).
(2) Op(b) is essentially selfadjoint and there exists c such that for z ∈ C\[c, ∞[
z 6∈ sp Op(b), (z − Op(b))m ∈ Ψmp (Rd ).
7.6
Asymptotics of the dynamics
1
Theorem 7.5 (Egorov Theorem). Let h ∈ Sph
be real and elliptic.
(1) Let x(t), ξ(t) solve the Hamilton equations with the Hamiltonian h and the initial conditions x(0), ξ(0). Then
γt (x(0), ξ(0)) = x(t), ξ(t)
defines a symplectic transformation homogeneous in ξ.
46
m
(2) Let b ∈ Sph
. Then there exist bt,m−2j ∈ S m−2j , j = 0, 1, . . . , and Rt,m−2n−2 uniformly
in B(L2,k , L2,k−2n−2 )). such that for |t| ≤ t0
eitOp(h) Op(b)e−itOp(h) =
n
X
Op bt,m−2j + Rt,m−2n−2 .
(7.153)
j=0
Moreover,
bt,m (x, ξ) = b γt (x, ξ)
(7.154)
and suppbt,m−2j ⊂ γt−1 suppb, j = 0, 1, . . . .
We will say that a ∈ S m has a homogeneous principal symbol if there exists am ∈ S m
homogeneous of degree m in ξ and m1 < m such that a − am ∈ S m1 for |ξ| > 1.
7.7
Singular support
Let f ∈ Dc0 (Rd ). It belongs to Cc∞ (Rd ) iff
|fˆ(ξ)| ≤ cn hξi−n ,
n ∈ N.
(7.155)
Let f ∈ D0 (Rd ) and x0 ∈ Rd . Clearly, f is C ∞ in a neighborhood of x0 iff
c (ξ)| ≤ cn hξi−n ,
∃χ∈Cc∞ (Rd ) χ(x0 ) 6= 0, |χf
n ∈ N..
(7.156)
This is equivalent to saying that
∃U a neighborhood of
∞
∀
x0 χ∈Cc (U )
c (ξ)| ≤ cn hξi−n ,
|χf
n ∈ N.
(7.157)
The complement of points where f is C ∞ is called the singular support of f .
7.8
Preservation of the singular support
Theorem 7.6. Let f ∈ L2,−∞ and A ∈ Ψ∞ . Then Af is C ∞ away from suppf .
Proof. Let f ∈ L2,k and A ∈ Ψm . Let χ, χ1 , ∈ C ∞ have all bounded derivatives. Let χ = 1
on suppf and χ1 = 0 on suppχ. Let
χ1 (x)Af = χ1 (x)Aχ(x)n f = χ1 (x)adnχ (A)
(7.158)
But adnχ (A) ∈ Ψm−n . Thus (7.158) belongs to L2,k−m+n . 2
7.9
Wave front
We equip X ⊕ (X # \{0}) with an action of R+ as follows:
(x, ξ) 7→ (x, tξ).
47
(7.159)
We say that a subset of X ⊕ (X # \{0}) is conical iff it is invariant with respect to this action.
Conical subsets can be identified with X ⊕ (X # \{0})/R+ .
Let (x0 , ξ0 ∈ X ⊕ X # \{0}. We say that f is smooth in a conical neighborhood of (x0 , ξ0 )
iff there exists χ ∈ Cc∞ (X ) and a conical neighborhood U of ξ ∈ U such that
ˆ (ξ)| ≤ cn hξi−n , n ∈ N.
|χf
(7.160)
The complement in X ⊕ (X # \{0})/R+ . of points where f is smooth is called the wave
front set of f and denoted W F (f ).
∞
P
m
m
We will write Sph
(Θ) for the set of a ∈ Sph
with a '
am−k such that suppam−k ⊂ Θ.
k=0
The following theorem gives two possible alternative definitions of the wave front set.
Theorem 7.7. Let u ∈ L−∞ . The following conditions are equivalent:
m
(1) There exists b ∈ Sph
such that bm (x0 , ξ0 ) 6= 0 and
Op(b)u ∈ L2,∞ .
m
(2) There exists a conical neighborhood U of (x0 , ξ0 ) such that for all c ∈ Sph
(U)
Op(c)u ∈ L2,∞ .
7.10
Properties of the wave front set
Example 7.8. Let Y be a k-dimensional submanifold with a k-form β. Then the distribution
Z
hF |ψi :=
φβ
Y
has the wave front set in the conormal bundle to Y:
W F (F ) ⊂ N # Y := {(x, ξ) : x ∈ Y, hξ|vi = 0, v ∈ TY}.
Example 7.9. For X = R,
W F ((x + i0)−1 ) = {(0, ξ) : ξ > 0}.
Example 7.10. Let H be a homogeneous function of degree 1 smooth away from the origin
and v ∈ C ∞ ,
|∂ξβ v(ξ)| ≤ cβ hξim−|β|
Then
Z
eixξ−iH(ξ) v(ξ)dξ = u(x)
satisfies
W F (u) = {(∇ξ H(ξ), ξ) : ξ ∈ suppv}.
48
Theorem 7.11. Let u ∈ L2,−∞ and a '
∞
P
k=0
m
am−k ∈ Sph
. Then
∞
W F (Op(a)u) = W F (u) ∩ ∩ suppam−k .
k=0
1
Theorem 7.12 (Theorem about propagation of singularities). Let h ∈ Sph
be real and
elliptic. Then
W F (eitOp(h) u) = γt (W F (u)) .
8
Path integrals
In this section ~ = 1 and we do not put hats on p and x. We will be not very precise
concerning the limits – often lim may mean the strong limit.
8.1
Evolution
Suppose that we have a family of operators t 7→ B(t) depending on a real variable. Typically,
we will assume that B(t) are generators of 1-parameter groups (eg. i times a self-adjoint
operator). Under certain conditions on the continuity that we will not discuss there exists
a unique operator function that in appropriate sense satisfies
d
U (t+ , t− )
dt+
U (t, t)
= B(t+ )U (t+ , t− ),
=
1l.
It also satisfies
d
U (t+ , t− ) =
dt−
U (t2 , t1 )U (t1 , t0 ) =
−U (t+ , t− )B(t− ),
U (t2 , t0 ).
If B(t) are bounded then
U (t+ , t− ) =
∞
X
Z
Z
B(tn ) · · · B(t1 )dtn · · · dt1 .
...
n=0 t >t >···>t >t
+
n
1
−
We will write
Z
!
t+
Texp
B(t)dt
:= U (t+ , t− ).
t−
In particular, if B(t) = B does not depend on time, then U (t+ , t− ) = e(t+ −t− )B .
In what follows we will restrict ourselves to the case t− = 0 and t+ = t and we will
consider
Z
t
U (t) := Texp
B(s)ds .
0
49
(8.161)
Note that the whole evolution can be retrieved from (8.161) by
U (t+ , t− ) = U (t+ )U (t− )−1 .
We have
n
Y
U (t) = lim
n→∞
t
jt
e n B( n ) .
(8.162)
j=1
(In multiple products we will assume that the factors are ordered from the right to the left).
Now suppose that F (s, u) is an operator function such that uniformly in s
euB(s) − F (s, u)
=
o(u),
kF (s, u)k
≤
C.
Then
U (t) = lim
n
Y
n→∞
F
j=1
jt t ,
.
n n
(8.163)
Indeed,
n
Y
e
jt
t
n B( n )
−
j=1
=
F
j=1
n
n
X
Y
k=1 j=k+1
= no(n−1 )
n
Y
jt t ,
n n
!
kt t k−1
jt t t kt
Y t jt
F
,
e n B( n ) − F
,
e n B( n )
n n
n n
j=1
→
0.
n→∞
Example 8.1. (1) F (s, u) = 1l + uB(s). Thus
U (t) = lim
!
t jt 1l + B
.
n
n
n
Y
n→∞
j=1
Strictly speaking, this works only if B(t) is uniformly bounded.
In particular,
etB = lim
n→∞
1l +
t n
B .
n
−1
(2) F (s, u) = 1l − uB(s)
. Then
U (t) := lim
n→∞
n
Y
t jt 1l − B
n
n
j=1
This should work also if B(t) is unbounded.
In particular,
etB = lim
n→∞
50
1l −
t −n
B
.
n
!−1
.
(3) Suppose that B(t) = A(t)+C(t), where both A(t) and C(t) are generators of semigroups.
Set F (s, u) = euA(t) euC(t) . Thus
U (t) = lim
n
Y
n→∞
jt
t
t
jt
e n A( n ) e n C( n ) .
(8.164)
j=1
In particular, we obtain the Lie-Trotter formula
t t n
et(A+C) = lim e n A e n C .
n→∞
8.2
Scattering operator
We will usually assume that the dynamics is generated by iH(t) where H(t) is a self-adjoint
operator. Often,
H(t) = H0 + V (t),
where H0 is a fixed self-adjoint operator. The evolution in the interaction picture is
Z t+
H(t)dt e−it− H0 .
S(t+ , t− ) := eit+ H0 Texp − i
t−
The scattering operator is defined as
S
:=
lim
t+ ,−t− →∞
S(t+ , t− ).
Introduce the Hamiltonian in the interaction picture
HInt (t) := eitH0 V (t)e−itH0 .
Note that
∂t+ S(t+ , t− )
=
−iHInt (t+ )S(t+ , t− ),
∂t− S(t+ , t− )
=
iS(t+ , t− )HInt (t+ ),
S(t, t)
=
1l.
Therefore,
S(t+ , t− )
=
Z
Texp − i
=
Z
Texp − i
t+
HInt (t)dt ,
t−
∞
S
HInt (t)dt
−∞
8.3
Bound state energy
Suppose that Φ0 and E0 , resp. Φ and E are eigenvectors and eigenvalues of H0 , resp H, so
that
H0 Φ0 = E0 Φ0 , HΦ = EΦ.
51
We assume that Φ, E are small perturbations of Φ0 , E0 when the coupling constant λ is
small enough.
The following heuristic formulas can be sometimes rigorously proven:
E − E0
=
lim (2i)−1
t→±∞
d
log(Φ0 |e−itH0 ei2tH e−itH0 Φ0 ).
dt
(8.165)
To see why we can expect (8.165) to be true, we write
(Φ0 |e−itH0 ei2tH e−itH0 Φ0 ) = |(Φ0 |Φ)|2 ei2t(E−E0 ) + C(t).
Then, if we can argue that for large t the term C(t) does not play a role, we obtain (8.165).
8.4
Path integrals for Schrödinger operators
We consider
1 2
p + V (t, x),
2
1
H(t) := Op(h(t)) = − ∆ + V (t, x),
2
Z t
U (t) := Texp −i
H(s)ds .
h(t, x, p)
:=
(8.166)
0
We have
2
i
i
e− 2 t∆ (x, y) = (2πit)−d/2 e 2t (x−y) .
From
U (t)
=
lim
n→∞
n
Y
t
jt
t
e−i n V ( n ,x) ei 2n ∆
j=1
we obtain
Z
U (t, x, y)
=
=
n→∞
lim
n Y
2πit − d2
Z
dxn−1 · · ·
lim
dx1
n
j=1
Z
2πit − dn
2
n→∞
× exp
n
e
in(xj−1 −xj )2
2t
Z
dxn−1 · · ·
dx1
!
n
jt
it X n2 (xj−1 − xj )2
−V
, xj
y=x , .
0
n j=1
2t2
n
x = xn .
Heuristically, this is written as
Z
Z t
L s, x(s), ẋ(s) ds Dx,y x(·) ,
U (t, x, y) = exp i
0
52
y=x ,
0
x = xn .
t
−i n
V ( jt
n ,xj )
where
L(s, x, ẋ) :=
1 2
ẋ − V (s, x)
2
is the Lagrangian and
2πit − dn
t
(n − 1)t 2
· · · dx
Dx,y (x(·) := lim
dx
n→∞
n
n
n
(8.167)
is some kind of a limit of the Lebesgue measure on paths [0, t] 3 s 7→ x(s) such that x(0) = y
and end up at x(t) = x.
8.5
Example–the harmonic oscillator
Let
H=
1 2 1 2
p + x .
2
2
It is well-known that for t ∈]0, π[,
1
e−itH (x, y) = (2πi sin t)− 2 exp
−(x2 + y 2 ) cos t + 2xy 2i sin t
.
(8.168)
(8.168) is called the Mehler formula.
We will derive (8.168) from the path integral formalism. We will use the explicit formula
for the free dynamics with H0 = 12 p2 :
1
e−itH0 (x, y) = (2πit)− 2 exp
−(x − y)2 2it
.
(8.169)
For t ∈]0, π[, there exists a unique trajectory for H starting from y and ending at x. Similarly
(with no restriction on time) there exists a unique trajectory for H0 :
cos(s − 2t )
sin(s − 2t )
(x
+
y)
+
(x − y),
cos 2t
sin 2t
(t − s)
s
.
x0,cl (s) = x + y
t
t
xcl (s) =
(8.170)
(8.171)
Now we set x(s) = xcl (s) + z(s) and obtain
Z
0
t
Z
t
1 2
ẋ (s) − x2 (s) ds
2
0
Z t
Z t
=
L xcl (s), ẋcl (s) ds +
L z(s), ż(s) ds
0
0
Z t
(x2 + y 2 ) cos t − 2xy
1 2
=
+
ż (s) − z 2 (s) ds.
2 sin t
0 2
L x(s), ẋ(s) ds =
53
(8.172)
(8.173)
(8.174)
Similarly, setting x(s) = x0,cl (s) + z(s) we obtain
Z
t
L0 x(s), ẋ(s) ds =
0
Z
t
0
1 2
ẋ (s)ds
2
(8.175)
(8.176)
2
=
(x − y)
+
2t
t
Z
0
1 2
ż (s)ds.
2
(8.177)
Therefore,
R
t
exp
i
L
x(s),
ẋ(s)
ds Dx,y x(·)
0
e
(x, y)
R
=R
t
e−itH0 (x, y)
exp i 0 L0 x(s), ẋ(s) ds Dx,y x(·)
2 2
R
Rt
cos t−2xy
exp i (x +y 2)sin
+ i 0 12 ż 2 (s) − z 2 (s) ds D0,0 z(·)
t
=
R
Rt 1
2
2 (s)ds D
exp i (x−y)
+
i
ż
0,0 z(·)
2t
0 2
2 2
! 21
(x +y ) cos t−2xy
i
exp
i
(−∆)
2 sin t
= det i 2
2
(x−y)
exp i
2 (−∆ − 1)
−itH
R
(8.178)
(8.179)
(8.180)
2t
Here −∆ denotes the minus
o the Dirichlet boundary conditions on the interval
n 2 2 Laplacian with
[0, t]. Its spectrum is π t2k | k = 1, 2, . . . . Therefore, at least formally,
det
i
2 (−∆)
i
2 (−∆
− 1)
!
1
=
det 1l + ∆−1
=
∞ Y
1−
k=1
(8.181)
t2 t
=
.
2
2
π k
sin t
(8.182)
Now (8.169) implies (8.168).
8.6
Path integrals for Schrödinger operators with the imaginary
time
Let us repeat the same computation for the evolution generated by
−H(t) = − − ∆ + V (t, x) .
We add the superscript E for “Euclidean”:
Z t
U E (t) := Texp −
H(s)ds
=
0
54
lim
n→∞
n
Y
j=1
t
jt
t
e− n V ( n ,x) e n ∆ .
Using
2
1
1
e 2 t∆ (x, y) = (2πt)−d/2 e− 2t (x−y) .
we obtain
E
U (t, x, y)
=
Z
2πt − dn
2
lim
n→∞
× exp
Z
dxn−1 · · ·
n
dx1
!
n
jt
t X −n2 (xj − xj−1 )2
−V
, xj
y=x , .
0
n j=1
2t2
n
x = xn .
Heuristically, this is written as
Z
Z t
E
E
LE s, x(s), ẋ(s) ds Dx,y
x(·) ,
U (t, x, y) = exp −
0
where
LE (s, x, ẋ) :=
1 2
ẋ + V s, x
2
is the “Euclidean Lagrangian” and
2πt − dn
(n − 1)t t
2
E
Dx,y
(x(·) := lim
dx
· · · dx
n→∞
n
n
n
is similar to (8.167).
8.7
Wiener measure
dWy x(·) = exp −
Z
t
0
1 2 E
ẋ (s) dsDx(t),y
x(·) dx(t)
2
can be interpreted as a measure on paths, functions [0, t] 3 s 7→ x(s) such that x(0) = y—the
Wiener measure.
Let us fix tn > · · · > t1 > 0, and F is a function on the space of paths depending only
on x(tn ), . . . , x(t1 ) (such a function is called a cylinder function). Thus
F x(·) = Ftn ,...,t1 x(tn ), . . . , x(t1 ) .
Then we set
Z
dWy x(·) F x(·)
Z
=
Ftn ,...,t1 (xn , . . . , x1
(8.183)
e
−
(xn −xn−1 )2
2(tn −tn−1 )
e
−
(x2 −x1 )2
2(t2 −t1 )
e−
(x1 −y)2
2t1
d dxn · · ·
d dx2
d dx1 .
2π(tn − tn−1 ) 2
2π(t2 − t1 ) 2
2πt1 2
We easily check the correctness of the definition on all cylinder functions. Then we extend
the measure to a larger space of paths–there are various possibilities.
55
We can use the Wiener measure to (rigorously) express the integral kernel of U E (t). Let
Φ, Ψ ∈ L2 (Rd ). Then the so-called Feynman-Katz formula says
(Φ|U E (t)Ψ)
Z
Z
Z t
V (s, x(s) ds .
= dx(0) dWx(0) x(·) Φ x(t) Ψ x(0) exp −
(8.184)
0
Theorem 8.2. Let t, t1 , t2 > 0. Then
Z
x(t)dW0 x(·) = 0,
Z
xi (t2 )xj (t1 )dW0 x(·) = δij min(t2 , t1 ),
Z
2
x(t2 ) − x(t1 ) dW0 x(·) = d|t2 − t1 |.
(8.185)
(8.186)
(8.187)
Proof. Let us prove (8.186). Let t2 > t1 . Then
Z
x(t2 )x(t1 )dW0 x(·) =
Z Z
x2
Z
=
e
−
x2
1
(x2 −x1 )2
2(t2 −t1 )
d
(2π(t2 − t1 )) 2
x1
e− 2t1
d
(2πt1 ) 2
dx1 dx2
(8.188)
x2
1
x21
e− 2t1
dx1
d
(2πt1 ) 2
=
t1 .
(8.189)
2
Recall the formula (2.16)
e
1
2 ∂x ·ν∂x
Ψ(0)
=
− 21
(det 2πν)
Z
1
Ψ(x)e− 2 x·ν
−1
x
dx,
(8.190)
which says that for Gaussian measures you can “integrate by differentiating”. The Wiener
measure is Gaussian, and in this case (8.190) has the form
Z
1
dW0 x(·) F x(·) = exp
∂x(s2 ) min(s2 , s1 )∂x(s1 ) F x(·) .
(8.191)
2
Indeed, the operator whose quadratic form appears in the Wiener measure is the Laplacian
on [0, t], which is Dirichlet at 0 and Neumann at t. Now the operator with the integral
kernel min(t2 , t1 ) is the inverse of this Laplacian.
8.8
General Hamiltonians – Weyl quantization
Let [0, t] 3 s 7→ h(s, x, p) ∈ R be a time dependent classical Hamiltonian. Set
H(s) := Op h(s)
and U (t) as in (8.201).
56
Lemma 8.3.
e−iuOp
h(s)
− Op e−iuh(s) = O(u3 ).
(8.192)
Proof. Let us drop the reference to s in h(s). We have
d iuOp(h)
e
Op e−iuh
du
Now
ieiuOp(h) Op(h)Op e−iuh − Op he−iuh .
=
(8.193)
i
Op(h)Op(e−iuh ) = Op he−iuh + Op {h, e−iuh } + O(u2 ).
2
(8.194)
The second termon the right of (8.194) is zero. Therefore, (8.193) is O(u2 ). Clearly,
eiuOp(h) Op e−iuh u=0 = 1l. Integrating O(u2 ) from 0 we obtain O(u3 ). 2
Thus we can use F (s, u) := Op e−iuh(s) in (8.163), so that
U (t)
=
lim
n→∞
n
Y
t
−i n
h
Op e
jt
n
j=1
Thus
Z
U (t, x, y)
=
lim
n→∞
Z Y
n
exp
− it
nh
jt xj +xj−1
n ,
2
, pj + i(xj − xj−1 )pj
j=1
n
Y
dpj (2π)d y = x0 ,
j=1
j=1
x = xn .

Z
Z
n X
(xj −xj−1 )pj
it

· · · exp n
lim
−h
t
×
=
n−1
Y
···
dxj
n→∞
n−1
Y
n
j=0
n
Y
dpj .
×
dxj
(2π)d y = x0 ,
j=1
j=1
x = xn .
(8.195)

jt xj +xj−1
n ,
2
, pj

(8.196)
Heuristically, this is written as follows:
Z t
Z
U (t, x, y) =
Dx,y (x(·)) D (p(·)) exp i
(ẋ(s)p(s) − h(s, x(s), p(s)) ds ,
0
where [0, t] 3 s 7→ (x(s), p(s)) is an arbitrary phase space trajectory with x(0) = y, x(t) = x
and the “measure on the phase space paths” is
n−1
n dp (j − 1 ) t
Y jt Y
2 n
Dx,y (x(·)) = lim
dx
, D p(·) =
.
n→∞
n
(2π)d
j=1
j=1
57
8.9
Hamiltonians quadratic in momenta I
Assume in addition that
h(t, x, p) =
1
(p − A(t, x))2 + V (t, x).
2
(8.197)
Then
Op h(t)
1
(pi − Ai (t, x))2 + V (t, x).
2
=
Introduce
v = p − A(t, x).
The Lagrangian for (8.199) is
L(t, x, v) =
1 2
v + vA(t, x) − V (t, x).
2
Consider the phase space path integral (8.196). The exponent depends quadratically on
p. Therefore, we can integrate it out, obtaining a configuration space path integral. More
precisely, first we make the change of variables
xj +xj−1
vj = pj − A jt
,
n ,
2
and then we do the integration wrt v:
Z
U (t, x, y)
=
Z
···
lim
n→∞
exp
it
n
j −xj−1 )
t
n
j=1
1
− vj2 − V
2
jt xj +xj−1
n ,
2
Z
lim
n→∞
! n−1
Y
dxj
j=1

=
n X
(x
Z
···
exp  itn
n
X
j=1
d
×(2π itn )−n 2
L
xj +xj−1
vj + A( jt
)
n ,
2
n
Y
dvj (2π)d y = x0 ,
j=1
x = xn

jt xj + xj−1 (xj − xj−1 ) 
,
,
t
n
2
n
n−1
Y
dxj y = x , .
0
j=1
x = xn
(8.198)
Heuristically, this is written as
Z
U (t, x, y)
=
Z t
Dx,y (x(·)) exp i
L(s, x(s), ẋ(s))ds ,
0
where [0, t] 3 s 7→ x(s) is a configuration space trajectory with x(0) = y, x(t) = x and the
formal “measure on the configuration space paths” is the same as in (8.167)
58
8.10
Hamiltonians quadratic in momenta II
Suppose, more generally, that
h(t, x, p) =
1
(pi − Ai (t, x))g ij (t, x)(pj − Aj (t, x)) + V (t, x).
2
(8.199)
Then
Op h(t)
=
1
(pi − Ai (t, x))g ij (t, x)(pj − Aj (t, x)) + V (t, x)
2
1X
−
∂ i ∂ j g ij (t, x).
4 ij x x
(For brevity, [g ij ] will be denoted g −1 and [gij ] is denoted g)
Introduce
v = g −1 (t, x) p − A(t, x)
The Lagrangian for (8.199) is
L(t, x, v) =
1 i
v gij (t, x)v j + v j Aj (t, x) − V (t, x).
2
Consider the phase space path integral (8.196). The exponent depends quadratically on
p. Therefore, we can integrate it out, obtaining a configuration space path integral. More
precisely, first we do the integration wrt p(·):
Z
U (t, x, y)
=
Dx,y x(·) D p(·) exp i
Z t
ẋ(s)p(s)
0
=
=
8.11
!
−1
1
− p(s) − A(s, x(s) g
s, x(s) p(s) − A s, x(s) − V s, x(s) ds
2
Z
Z t
1
ẋ(s)g s, x(s) ẋ(s) + ẋ(s)A(s, x(s) − V s, x(s) ds
Dx,y x(·) exp i
2
0
!
Z t
1
+
Trg s, x(s) ds
2 0
Z t Z
1
(8.200)
Dx,y (x(·)) exp
iL(s, x(s), ẋ(s)) + Trg s, x(s) ds .
2
0
Semiclassical path integration
Let us repeat the most important formulas in the presence of a Planck constant ~.
Z
i t
U (t) := Texp −
H(s)ds .
(8.201)
~ 0
59
Z
U (t, x, y)
=
−1
Dx,y (x(·)) D ~
Z t
i
(ẋ(s)p(s) − h(s, x(s), p(s)) ds ,
p(·) exp
~ 0
We assume in addition that the Hamiltonian has the form (8.199), and we set
√
x(s) = xcl (s) + ~z(s),
where xcl is the classical solution such that xcl (0) = y and xcl (t) = x.
Z t
Z
1
i
−
−d
Dx,y ~ 2 x(·) exp
U (t, x, y) = ~ 2
L(s, x(s), ẋ(s))ds
~ 0
Z t
d
i
L(s, xcl (s), ẋcl (s))ds
= ~− 2 exp
~ 0
Z
Z
i t 2
× D0,0 (z(·)) exp
∂x(s) L s, xcl (s), xcl (s) z(s)z(s)
2 0
+2∂x(s) ∂ẋ(s) L s, xcl (s), xcl (s) z(s)ż(s)
!
√ 2
+∂ẋ(s) L s, xcl (s), xcl (s) ż(s)ż(s) + O( ~) ds
=~
8.12
−d
2
" Rt
#!− 12
Rt
2
∂
L
s,
x
(s),
x
(s)
∂
∂
L
s,
x
(s),
x
(s)
1
cl
cl
cl
cl
x(s)
ẋ(s)
0R t 2
R t 0 x(s)
det
2π 0 ∂x(s) ∂ẋ(s) L s, xcl (s), xcl (s)
∂
L s, xcl (s), xcl (s)
0 ẋ(s)
Z t
√ i
× exp
L(s, xcl (s), ẋcl (s))ds 1 + O( ~) .
~ 0
General Hamiltonians – Wick quantization
Let [0, t] 3 s 7→ h(s, a∗ , a) ∈ R be a time dependent classical Hamiltonian expressed in terms
of the complex coordinates. Set
∗
H(t) := Opa ,a h(t)
and U (t) as in (8.201). (We drop the tilde from h̃ and ũ, as compared with the notation of
(4.83).)
Following Lemma 8.3 we prove that
a∗ ,a
∗
h(s)
e−iuOp
− Opa ,a e−iuh(s) = O(u2 ).
(8.202)
∗
Thus we can use F (s, u) := Opa
Opa
∗
,a
,a
e−iuh(s) in (8.163), so that
u(t) := U (t)
=
lim
n→∞
60
n
Y
j=1
∗
Opa
,a
t
e−i n h
jt
n
.
Thus, by (4.92),
u(t, a∗ , a)
=
lim exp
n→∞
X
∂ak ∂a∗j
n
Y
exp − itn h
jt
n
, a∗j , aj
j=1
k>j
a = an = · · · = a1 .
Heuristically, this can be rewritten as
Z t
Z t
∗
ds+
ds− θ(s+ − s− )∂a∗ (s+ ) ∂a(s− )
u(t, a , a) = exp
0
0
Z t
∗
.
h (s, a (s), a(s)) ds × exp −i
.
(8.203)
a=a(s),t>s>0
0
Alternatively, we can use the integral formula (4.93), and rewrite (8.203) as
∗
Z
Z
···
u(t, a , a) =
exp
n−1
X
(b∗j − a∗ )(bj+1 − bj )
(8.204)
j=1
×
n
Y
exp − itn h
j=1
jt
n
, b∗j , bj
Y dbj+1 db∗j
n−1
.
(2πi)d
j=1
Heuristically, it can be rewritten as
u(t, a∗ , a)
R
R
t
D b(·) exp 0 b∗ (s) − a∗ ∂s b(s) − ih (s, b∗ (s), b(s)) ds
=
.
Rt
R
D b(·) exp 0 b∗ (s)∂s b(s)ds
(8.205)
Here, D b(·) is some kind of a “measure” on the complex trajectories.
Let us describe another derivation of (8.205), which starts from (8.203). Consider the
operator θ on L2 ([0, t]) with the integral kernel θ(s+ − s− ) Note that
∂s+ θ(s+ − s− ) = δ(s+ − s− ).
Besides, θf (0) = 0. Therefore, ∂s θ = 1l. Thus θ is the inverse (“the Green’s function”) of
the operator ∂s with the boundary condition f (0) = 0. It is an unbounded operator with
empty resolvent. It is not antiselfadjoint – its adjoint is ∂x with the boundary condition
f (t) = 0. The corresponding sesquilinear form can be written as
Z
t
a∗ (s)∂s a(s)ds.
0
Using (2.23), (8.203) can be rewritten formally as (8.205).
61
8.13
Vacuum expectation value
In particular, we have the following expression for the vacuum expectation value:
Ω|U (t)Ω
R
R
t
D a(·) exp 0 (a∗ (s)∂s a(s) − ih (s, a∗ (s), a(s))) ds
=
.
Rt
R
D a(·) exp 0 a∗ (s)∂s a(s)ds
(8.206)
For f, g ∈ Cd we will write
a∗ (f ) = ai fi ,
a(g) = ai g i .
One often tries to express everything in terms of vacuum expectation values. To this end
introduce functions
[0, t] 3 s 7→ F (s), G(s) ∈ Cd ,
and a (typically, nonphysical) Hamiltonian
H(s) + a∗ F (s) + a G(s) .
The vacuum expectation value for this Hamiltonian is called the generating function:
! !
Z t
∗
Z(F, G) = Ω|Texp − i
H(s) + a F (s) + a G(s) ds Ω .
0
Note that we can retrieve full information about U (t) from Z(F, G) by differentiation.
Indeed let
Fi (s) = fi δ(s − t), Gi (s) = gi δ(s), fi , gi ∈ Cd .
Then
m
F1 · · · Fn G1 · · · Gm ∂Fn ∂G
Z(F, G)
F =0,G=0
n−m
∗
∗
∗
= i
a (f1 ) · · · a (fn )Ω|U (t)a (g1 ) · · · a∗ (gm )Ω
To see this, assume for simplicity that
F1 (s) = · · · = Fn (s) = f δ(s − t),
and approximate the delta function:
(
1/ 0 < s < ;
δ(s) ≈
,
0
< s < t;
G1 (s) = · · · = Gm (s) = gδ(s),
(
δ(s − t) ≈
0
1/
0 < s < t − ;
.
t − < s < t;
Using these approximations, we can write
∗
Z(sF, uG) = lim Ω|e−is a(f ) U (t)e−iu a (g) Ω
→0
∗
∗
=
eisa (f ) Ω|U (t)e−iua (g) Ω .
62
Now
m
F1 · · · F1 G1 · · · G1 ∂Fn ∂G
Z(F, G)
F =0,G=0
∗
n m
isa (f )
−iua∗ (g)
= ∂s ∂u e
Ω|U (t)e
Ω s=0, u=0
∗
n
∗
m
n−m
a (f1 ) Ω|U (t)a (g1 ) Ω .
= i
8.14
Scattering operator for Wick quantized Hamiltonians
Assume now that the Hamiltonian is defined for all times and is split into a time-independent
quadratic part and a perturbation:
h(t, a∗ , a) = a∗ εa + λq(t, a∗ , a).
Set
H0
=
Opa
∗
,a
(a∗ εa) = â∗ εâ =
Q(t)
=
Opa
∗
,a
(q(t)),
X
â∗i εi âi
i
so that H(t) = H0 + λQ(t). The scattering operator is
Z ∞
S = Texp − i
HInt (t)dt ,
−∞
where the interaction Hamiltonian is
HInt (t)
∗
Setting S = Opa
=
λeitH0 Q(t)e−itH0
=
λOpa
∗
,a
q(t, eitε a∗ , e−itε a) .
,a
(s), we can write
Z ∞
Z ∞
∗
s(a , a) = exp
dt+
dt− θ(t+ − t− )∂a(t+ ) ∂a∗ (t− )
−∞
−∞
Z ∞
iεt ∗
−iεt
× exp −iλ
q t, e a (t), e
a(t) dt a∗ = a∗ (t),
−∞
a = a(t), t ∈ R
Z ∞
Z ∞
iε(t+ −t− )
= exp
dt+
dt− e
θ(t+ − t− )∂a(t+ ) ∂a∗ (t− )
−∞
−∞
Z ∞
∗
× exp −iλ
q (t, a (t), a(t)) dt eitε a∗ = a∗ (t),
−∞
R
=
itε
(8.207)
e a = a(t), t ∈ R
R
∞
D b(·) exp −∞ b∗ (t) − eiεt a∗ (∂t + iε) b(t) − e−iεt a − iλq (t, b∗ (t), b(t)) dt
.
R
R∞ D b(·) exp −∞ b∗ (t) − eiεt a∗ (∂t + iε) b(t) − e−iεt a
63
In the firtst step we made the substitution
a(t) = e−itε aInt (t),
a∗ (t) = eitε a∗Int (t),
subsequently dropping the subscript Int. Then the differential operator was represented
as a convolution involving Green’s function of the operator ∂t + iε that has the kernel
eiε(t+ −t− ) θ(t+ − t− ).
8.15
Friedrichs diagrams
Suppose that the perturbation is written as the linear combination of monomials:
+
∗
q(a , a) =
X
k+ ,k−
+
X
|α+ | = k + ,
|α− | = k −
−
wαk +,k
,α−
α+ !α− !
+
−
a∗α aα .
−
With every term wk ,k we associate a “vertex” – a dot with k − incoming legs (to the right)
and k + outgoing legs (to the left). A Friedrichs diagram is an ordered sequence of n vertices
and a family of lines. Each leg belongs to an internal or external lines. Internal lines join an
outgoing leg with an incoming leg that belongs to a later vertex. External incoming lines are
attached to an incoming line. Such a line is extended to the right of the diagram. External
outgoing lines are attached to outgoing legs – they are extended to the left of the diagram.
We will introduce several kinds of evaluations of a Friedrichs diagram. In this subsection
we introduce the time-dependent evaluation. Suppose B is a Friedrichs diagram with vertices
numbered by n, . . . , 1. For any sequence of times tn > · · · > t1 the corresponding evaluation
of B is computed as follows.
+ −
Each line (external and internal) has its index. For each vertex we put wk ,k , considered
+ −
as a tensor with the indices corresponding to its lines. We put wk ,k for each vertex and
eiε(tj+ −tj− ) for every internal line joining the j − th and j + th vertex. At incoming external
line we put e−iεtj− a, at each outgoing external line we put eiεtj+ a∗ . We multiply all these
factors. Thus we obtain a polynomial B(tn , . . . , t1 ; a∗ , a).
Each diagram comes with its symmetry factor. It is the product of several factorials.
If there are mj + ,j − lines joining the vertices j − and j + , it involves mj + ,j − !. If there are
mj − /mj + incoming/outgoing external lines from a vertex j, it involves also mj − !/mj + !. The
complete symmetry factor will be denoted by B!
We can interpret B(tn , . . . , t1 ; a∗ , a) as a product of operators. For each line we introduce
a Hilbert space isomorphic to the one-particle space of the system. We have n + 1 time
intervals
t > tn , . . . , tj+1 > t > tj , . . . , t1 > t.
For each of these intervals we have a collection of lines that are “open” in this interval. (This
should be obvious from the diagram). Within each of these intervals we consider the tensor
product of the spaces corresponding to each of the lines that are open in this interval.
+ −
wk ,k (t) is interpreted as an operator from a k − -particle space to a k + -particle space.
If it is on the jth place in the diagram, it will be denoted Wj (tj ). 1ljB will denote the
64
time
Figure 1: Various Friedrichs vertices
identity on the tensor product of spaces corresponding to the lines that pass the jth vertex.
At the left/right end we put symmetrizators corresponding to external outgoing/incoming
−
lines, denoted Θ+
B / ΘB . Between each two consecutive vertices j + 1 and j we put the free
dynamics for time tj+1 − tj , which, by the abuse of notation, will be denoted e−i(tj+1 −tj )H0 ,
and where H0 is the sum of ε for each line. For the final/initial interval we put eitn H0 /
e−it1 H0 . Thus the evaluation of B interpreted as an operator can be written as
B(tn , . . . , t1 )
so that
=
itn H0
Θ+
(Wn (tn ) ⊗ 1lnB ) e−i(tn −tn−1 )H0 · · ·
Be
×e−i(t2 −t1 )H0 W1 (t1 ) ⊗ 1l1B e−it1 H0 Θ−
B,
+
+
B(tn , . . . , t1 ; a∗ , a) = a∗α B α
65
,α−
−
(tn , . . . , t1 )aα .
t5
t
t4
t
3
t
2
1
time
Figure 2: A disconnected Friedrichs diagram
We have the identities
=
s(a∗ , a)
∞
X
X
n=0 all
=
exp
(−iλ)
n
Z
diag.
∞
X
B(tn , . . . , t1 ; a∗ , a)
dtn · · · dt1
B!
Z
···
(8.208)
tn >···>t1
X
n=0 con.
Z
(−iλ)n
B(tn , . . . , t1 ; a∗ , a)
dtn · · · dt1
B!
Z
···
diag.
!
log (Ω|SΩ) = s(0, 0)
Z
Z
∞
X
X
B(tn , . . . , t1 )
n
=
dtn · · · dt1 ,
(−iλ)
···
B!
n=0
t
>···>t
con. diag.
n
1
no ext. lines
=
s(a∗ , a)
(Ω|SΩ)
∞
X
X
n=0
=
exp
Z
(−iλ)n
linked diag.
∞
X
n=0
X
(8.209)
tn >···>t1
Z
B(tn , . . . , t1 ; a∗ , a)
···
dtn · · · dt1
66
B!
(8.210)
(8.211)
tn >···>t1
n
(−iλ)
con. linked
diag.
Z
Z
···
tn >···>t1
!
B(tn , . . . , t1 ; a∗ , a)
dtn · · · dt1 .
B!
(8.212)
In (8.208) we sum over all diagrams. To derive (8.208) we use (8.207):
=
s(a∗ , a)
Z
∞
X
1
m!
m=0
∞
X
∞
∞
Z
dt− eiε(t+ −t− ) θ(t+ − t− )∂a(t+ ) ∂a∗ (t− )
dt+
−∞
m
−∞
Z
Z
q (tn , a∗ (tn ), a(tn )) · · · q (t1 , a∗ (t1 ), a(t1 )) dtn · · · dt1 e−itε a∗ = a∗ (t),
n=0
tn >···>t1
eitε a = a(t), t ∈ R

m
Z
Z X
∞
∞
X
X
1 
=
(−iλ)n · · ·
eiε(tj+ −tj− ) ∂a(tj+ ) ∂a∗ (tj− ) 
m!
n=0
n≥j + >j − ≥1
tn >···>t1 m=0
×q (tn , a∗ (tn ), a(tn )) · · · q (t1 , a∗ (t1 ), a(t1 )) dtn · · · dt1 e−itj ε a∗ = a∗ (t ),
×
(−iλ)n
···
j
=
∞
X
(−iλ)n
n=0
Z
Z
···
X
tn >···>t1 all
Y
diag.
1
mj + ,j − !
eitj ε a = a(tj ), n ≥ j ≥ 1.
mj+ ,j−
eiε(tj+ −tj− ) ∂a(tj+ ) ∂a∗ (tj− )
×q (tn , a (tn ), a(tn )) · · · q (t1 , a∗ (t1 ), a(t1 )) dtn · · · dt1 e−itj ε a∗ = a∗ (t ),
j
eitj ε a = a(tj ), n ≥ j ≥ 1.
∗
In the sum above, for each pair of times tj − and tj + with j + > j − we first choose mj + ,j − =
0, 1, . . . , then sum over all such possibilities. Clearly, this can be represented by a diagram.
To determine the combinatorial factor, assume for simplicity that we have one degree of
mj
akj −m (t )
a(t )kj
, then we obtain (kj −m)!j .
freedom. Then if we have a factor kjj ! which is hit by ∂a(t)
This explains the combinatorial factor on a incoming external lines. Similarly for a outgoing
external line. For internal lines the combinatorial factor comes from mj + ,j − !.
In (8.209) we sum over all connected diagrams. This follows from (8.208) by an easy
combinatorial argument and is often called the Linked Cluster Theorem.
In (8.210) we sum over all connected diagrams without external lines. In (8.211) we sum
over all linked diagrams, that is, diagrams whose each connected component has at least one
external line. In (8.212) we sum over all connected diagrams without external lines. Clearly,
(8.210), (8.211) and (8.212) follow from (8.209).
8.16
Scattering operator for time-independent perturbations
+
−
+
−
Let us now assume that the monomials wk ,k (t) = wk ,k do not depend on time.
For diagrams with no external legs the integrals
Z
Z
· · · B(tn , . . . , t1 ; a∗ , a)dtn · · · dt1
(8.213)
tn >···>t1
are then infinite or zero. We will come back to these diagrams in the next subsection.
67
Consider a linked diagram. One can expect that then (8.213) is finite. Let us compute
its value.
For ξ ∈ R introduce heuristic expressions
Z +∞
−1
,
(8.214)
eiu(H0 −ξ) du = i H0 − (ξ + i0)
0
Z
0
eiu(H0 −ξ) du =
−1
,
−i H0 − (ξ − i0)
(8.215)
2πδ(H0 − ξ).
(8.216)
−∞
Z
eit(H0 −ξ) dt =
If B is a connected linked diagram, we introduce its evaluation for the scattering amplitude
at energy E as follows. Each line (external and internal) has its index. For each vertex we
+ −
put wk ,k , considered as a tensor with the indices corresponding to its lines. Between each
consecutive vertices we insert (ξ + i0 − εj1 − · · · − εjp )−1 , where each εj corresponds to each
line between these vertices. From the right we multiply by δ(ξ − εj1 − · · · − εjp )−1 aj1 . . . ajp ,
where all ε correspond to incoming external lines. From the left we multiply by δ(ξ −
εj1 − · · · − εjp )−1 a∗j1 . . . a∗jp , where all ε correspond to incoming external lines. We obtain a
polynomial denoted Bsc (ξ; a∗ , a).
Theorem 8.4. For every connected linked diagram B the corresponding integral (8.213)
equals
Z
−2πi Bsc (ξ; a∗ , a)dξ.
Proof. We compute the integrand using the operator interpretation of B(tn , . . . , t1 ):
B(tn , . . . , t1 )
itn H0
Θ+
(Wn ⊗ 1lnB ) e−i(tn −tn−1 )H0 · · ·
Be
×e−i(t2 −t1 )H0 W1 ⊗ 1l1B e−it1 H0 Θ−
B
Z
n
−iun (H0 −ξ)
=
δ(H0 − ξ)dξΘ+
···
B (Wn ⊗ 1lB ) e
×e−iu2 (H0 −ξ) W1 ⊗ 1l1B e−it1 (H0 −ξ) Θ−
B,
=
where we substituted
un := tn − tn−1 , . . . , u2 := t2 − t1 .
and used
Z
1l =
δ(H0 − ξ)dξ.
68
Now
Z
Z
B(tn , . . . , t1 )dtn · · · dt1
···
tn >···>t1
Z ∞
Z
=
=
∞
dun · · ·
dξ
×e
Z
0
−iu2 (H0 −ξ)
n−1
2π(−i)
Z
Z
∞
n
−iun (H0 −ξ)
− ξ)Θ+
···
B (Wn ⊗ 1lB ) e
du1
0
W1 ⊗
dt1 δ(H0
−∞
1l1B e−it1 (H0 −ξ) Θ−
B
n
dξδ(H0 − ξ)Θ+
B (Wn ⊗ 1lB ) H0 − (ξ − i0)
−1
···
−1
× H0 − (ξ − i0)
W1 ⊗ 1l1B δ(H0 − ξ)Θ−
B,
2
The sum of linked diagrams will be called the linked scattering operator. Formally, we
have
S
(Ω|SΩ)
∗
slink (a , a)
=
:=
∗
Slink := Opa ,a (slink ),
∞
X
exp
n=0
=
exp −2iπ
X
Z
n
···
(−iλ)
con. linked
diag.
∞
X
n=0
Z
B(tn , . . . , t1 )
dtn · · · dt1
B!
!
tn >···>t1
X
n
λ
con. linked
diag.
Z
!
Bsc (ξ)
dξ .
B!
Note that, at least diagramwise
e−itH0 ei2tH e−itH0
.
t→∞ (Ω|e−itH0 ei2tH e−itH0 Ω)
Slink = lim
(8.217)
We can make (8.217) more general, and possibly somewhat more satisfactory as follows.
We introduce a temporal switching function R 3 t 7→ χ(t) that decays fast ±∞ and χ(0) = 1.
We then replace the time independent perturbation Q by Q (t) := χ(t/)Q. Let us denote
the corresponding scattering operator by S . Then the linked scattering operator formally
is
S
Slink = lim
,
&0 (Ω|S Ω)
One often makes the choice
χ(t/) = e−|t|/ ,
(8.218)
which goes back to Gell-Mann–Low.
Note that Slink commutes with H0 . More precisely, each diagram commutes with H0 .
69
If ε > 0, we can make a heuristic argument why Slink is unitary. It is proportional to a
unitary as a formal power series. All diagrams with only outgoing legs have zero stationary
evaluation because of the conservation of the energy. Therefore, Slink Ω = Ω. Hence Slink is
unitary.
8.17
Energy shift
We still consider a time independent perturbation. Let B be a connected diagram with no
external lines. Its evaluation is invariant wrt translations in time:
B(tn , . . . , t1 ) = B(tn + s, . . . , t1 + s).
Therefore,
Z
Z
···
B(tn , . . . , t1 )dtn · · · dt1
tn >···>t1
Z
=
Z
Z
···
dt1
B(t1 + un , . . . , t1 + u2 , t1 )dun · · · du2
un >···>u2 >0
Z
=
Z
Z
···
dt1
B(un , . . . , u2 , 0)dun · · · du2 .
(8.219)
un >···>u2 >0
This is infinite if i nonzero. However, if we do not integrate wrt t1 , we typically obtain a
finite expression in (8.219). We will see below that it can be used to compute the energy
shift.
For a connected diagram with no external lines its evaluation for the energy shift is
defined as follows. Each line (external and internal) has its index. For each vertex we put
+ −
wk ,k , considered as a tensor with the indices corresponding to its lines. Between each
consecutive vertices we insert −(εj1 + · · · + εjp )−1 , where each εj corresponds to each line
between these vertices. We obtain a number denoted Bes . (es stands for the energy shift).
Let E denote the ground state energy of H, that is E := inf sp H. We assume that we
can use the heuristic formula for the energy shift
E
=
lim (2i)−1
t→−∞
d
log(Ω|e−itH0 ei2tH e−itH0 Ω),
dt
(8.220)
which follows from (8.165) if we note that E0 = 0. Then we can derive the following
diagrammatic expansion for the energy E:
Theorem 8.5 (Goldstone theorem).
E
=
∞
X
n=0
X
con. diag.
no ext. lines
70
λn
Bes
.
B!
Figure 3: Goldstone diagram
Proof.
Applying (8.210), we get
=
log(Ω|e−itH0 ei2tH e−itH0 Ω)
∞
X
X
(−iλ)n
n=0
con. diag.
no ext. lines
Z
Z
···
B(tn , . . . , t1 )
dtn · · · dt1 .
B!
0>tn >···>t1 >t
So
(2i)−1
=
∞
X
n=0
d
log(Ω|e−itH0 ei2tH e−itH0 Ω)
dt
Z
Z
X
i(−iλ)n
···
con. diag.
no ext. lines
B(tn , . . . , t2 , t)
dtn · · · dt2 .
B!
0>tn >···>t2 >t
Now introduce
u2 := t − t2 , . . . , un := tn−1 − tn .
71
Then u2 , . . . , un ≤ 0, t ≤ u2 + · · · + un ≤ 0, and
B(tn , . . . , t2 , t)
= Wn e−i(tn −tn−1 )H0 Wn−1 ⊗ 1ln−1
···
B
−i(t2 −t)H0
2
× W2 ⊗ 1lB e
W1
n−1
iun H0
= Wn e
Wn−1 ⊗ 1lB
···
iu2 H0
2
W1 .
× W2 ⊗ 1lB e
Then we replace t by −∞ and evaluate the integral using the heuristic relation (8.215).
9
9.1
Method of characteristics
Manifolds
Let X be a manifold and x ∈ X . Tx X , resp. T#
x X will denote the tangent, resp. cotangent
space at x. TX , resp. T# X will denote the tangent, resp. cotangent bundle over X .
Suppose that x = (xi ) are coordinates on X . Then we have a natural basis in TX denoted
∂xi and a natural basis in T# X , denoted dxi . Thus every vector field can be written as v =
v(x)∂x = v i (x)∂xi and every differential 1-form can be written as α = α(x)dx = αi (x)dxi .
We will use the following notation: ∂ˆx is the operator ∂x that acts on everything on the
right. ∂x acts only on the function immediately to the right. Thus the Leibniz rule can be
written as
∂ˆx f (x)g(x) = ∂x f (x)g(x) + f (x)∂x g(x).
(9.221)
ˆ
There are situations when we could use both kinds of notation: ∂x and ∂x , as in the last
term of (9.221). In such a case we make a choice based on esthetic reasons.
9.2
1st order differential equations
Let v(t, x)∂x be a vector field and f (t, x) a function, both time-dependent. Consider the
equation
∂t + v(t, x)∂x + f (t, x) Ψ(t, x) = 0,
Ψ(0, x)
To solve it one finds first the solution of
∂t x(t, y) =
x(0, y)
=
=
Ψ(x).
v(t, x(t, y))
y;
Let x 7→ y(t, x) be the inverse function.
Proposition 9.1. Set
Z
F (t, y) :=
t
f s, x(s, y) ds.
0
Then
Ψ(t, x) := e−F
t,y(t,x)
is the solution of (9.222).
72
Ψ y(t, x)
(9.222)
(9.223)
Proof. Set
Φ(t, y) := Ψ t, x(t, y) .
(9.224)
We have
∂t Φ(t, y)
=
=
∂t + ∂t x(t, y)∂x Ψ t, x(t, y)
∂t + v t, x(t, y) ∂x Ψ t, x(t, y) .
Hence (9.222) can be rewritten as
∂t + f t, x(t, y) Φ(t, y)
=
0,
Φ(0, y)
=
Ψ(y).
(9.225)
(9.225) is solved by
Φ(t, y) := e−F (t,y) Ψ(y).
2
Consider now a vector field v(x)∂x and a function f (x), both time-independent. Consider
the equation
v(x)∂x + f (x) Ψ(x) = 0.
(9.226)
Again, first one finds solutions of
∂t x(t) = v x(t) .
(9.227)
Then we try to find a manifold Z in X of codimension 1 that crosses each curve given by a
solution of (9.226) exactly once. If the field is everywhere nonzero, this should be possible
at least locally. Then we can define a family of solutions of (9.226) denoted x(t, z), z ∈ Z,
satisfying the boundary conditions
x(0, z) = z, z ∈ Z.
This gives a local parametrization
R × Z 3 (t, z) 7→ x(t, z) ∈ X .
Let x 7→ t(x), z(x) be the inverse function.
Proposition 9.2. Set
t
Z
f x(s, z) ds.
F (t, z) :=
0
Then
Ψ(t, x) := e
−F t(x),z(x)
Ψ z(x)
is the solution of (9.222).
Proof. Set Φ(t, z) := Ψ x(t, z) . Then
∂t Φ(t, z)
= ∂t x(t, z)∂x Ψ x(t, z)
= v x(t, z) ∂x Ψ x(t, z) .
73
(9.228)
Hence we can rewrite (9.226) together with the boundary conditions as
∂t + f x(t, z) Φ(t, z) = 0,
Φ(0, z)
=
Ψ(z).
(9.229)
(9.229) is solved by
Φ(t, z) := e−F (t,z) Ψ(z).
2
9.3
1st order differential equations with a divergence term
Fort a vector field v(x)∂x we define
divv(x) = ∂xi v i (x).
Note that divv(x) depends on the coordinates.
Consider a time dependent vector field v(t, x)∂x and the equation
(∂t + v(t, x)∂x + αdivv(t, x)) Ψ(t, x)
=
0,
Ψ(0, x)
=
Ψ(x),
(9.230)
Proposition 9.3. (9.230) is solved by
Ψ(t, x) := det ∂x y(t, x)
α
Ψ(y(t, x)).
(9.231)
Proof. We introduce Φ(t, y) as in (9.224) and rewrite (9.230) as
∂t + αdivv t, x(t, y) Φ(t, y) = 0,
Φ(0, y)
=
Ψ(y)
(9.232)
We have the following identity for the determinant of a matrix valued function t 7→ A(t):
∂t det A(t) = Tr ∂t A(t)A(t)−1 det A(t).
(9.233)
Therefore,
−α
∂t det ∂y x(t, y)
−α
= −αdiv∂t x(t, y) det ∂y x(t, y)
−α
= −αdivv t, x(t, y) det ∂y x(t, y)
.
Therefore, (9.232) is solved by
−α
Φ(t, y) := det ∂y x(t, y)
Ψ(y).
2
Consider again a time independent vector field v(x)∂x and the equation
(v(x)∂x + αdivv(x)) Ψ(x)
=
0.
(9.234)
We introduce the a hypersurface Z and solutions x(t, z), as described before Prop. 9.2.
74
Proposition 9.4. Set
w(x) := ∂z x t(x), z(x) .
Then the solution of (9.234) which on Z equals Ψ(z) is
−α
Ψ(x) := det[v(x), w(x)]
Ψ z(x) .
(9.235)
Note that if X is one-dimensional, so that we can locally identify it with R and v is a
−α
.
number, (9.235) becomes Ψ(x) = C v(x)
9.4
α-densities on a vector space
Let α > 0. We say that f : (Rd )d → R is an α-density, if
hf |av1 , . . . , avd i = | det a|α hf |v1 , . . . , vd i,
(9.236)
for any linear transformation a on Rd and v1 , . . . , vd ∈ Rd .
If X is a manifold, then by an α-density we understand a function on X 3 x 7→ Ψ(x)
where Ψ(x) is an α-density on Tx X .
Clearly, given coordinates x = (xi ) on X , using the basis ∂xi in TX , we can identify an
α-density Ψ with the function
x 7→ hΨ|∂x1 , . . . , ∂xd i(x),
(9.237)
which, by abuse of notation will be also denoted Ψ(x). If we use some other coordinates
x0 = x0i , then we obtain another function x0 7→ Ψ0 (x0 ). We have the transformation property
α
Ψ(x) = |∂x x0 | Ψ0 (x0 ).
(9.238)
A good mnemotechnic way to denote an α-density is to write Ψ(x)|dx|α . Note that 0densities are usual functions, 1-densities, or simply densities are measures. p1 -densities raised
to the pth power give a density, and so one can invariantly define their Lp -norm:
Z
Z 1 p
p
=
|Ψ(x)|p dx = kφkpp .
(9.239)
Ψ(x)|dx| Proposition 9.5. If v(x)∂x is a vector field, the operator
v(x)∂x + αdivv(x)
(9.240)
is invariantly defined on α-densities.
Proof. In fact, suppose we consider some other coordinates x0 . In the new coordinates
the vector field v(x)∂x becomes v 0 (x0 )∂x0 = (∂x x0 )v(x(x0 ))∂x0 . We will denote div0 v 0 the
divergence in the new coordinates. We need to show that if
Φ = | det ∂x x0 |α Φ0 ,
Ψ = | det ∂x x0 |α Ψ0 ,
then
v(x)∂x + αdivv(x) Φ = Ψ
75
is equivalent to
v 0 (x0 )∂x0 + αdiv0 v 0 (x0 ) Φ0 = Ψ0 .
We have
divv 0
=
=
v ∂ˆx | det ∂x x0 |α
=
αv k
∂xj ∂ˆ ∂x0i k
v
∂x0i ∂xj ∂xk
∂v j
∂xj ∂ 2 x0i k
+
v
j
∂x
∂x0i ∂xj ∂xk
∂ 2 x0i ∂xj
| det ∂x x0 |α + | det ∂x x0 |α v ∂ˆx .
∂xj ∂xk ∂x0i
Therefore,
v(x)∂ˆx + αdivv(x) | det ∂x x0 |α Φ0
= | det ∂x x0 |α v 0 (x0 )∂ˆx0 + αdiv0 v 0 (x0 ) Φ0 .
2
Note that (9.231) can be written as an α-density:
Ψ(t, x)|dx|α := | det ∂x y(t, x)|α Ψ(y(t, x))|dx|α
(9.241)
Also (9.235) is naturally an α-density.
10
10.1
Hamiltonian mechanics
Symplectic manifolds
Let Y be a manifold equipped with a 2-form ω ∈ ∧2 T# Y. We say that it is a symplectic
manifold iff ω is nondegenerate at every point and dω = 0.
Let (Y1 , ω1 ), (Y2 , ω2 ) be symplectic manifolds. A diffeomorphism ρ : Y1 → Y2 is called a
symplectic transformation if ρ# ω2 = ω1 .
In what follows (Y, ω) is a symplectic manifold. We will often treat ω as a linear map
from T Y to T # Y. Therefore, the action of ω on vector fields u, w will be written in at least
two ways
hω|u, wi = hu|ωwi = ωij ui wj .
The inverse of ω as a map T Y → T # Y will be denoted ω −1 . It can be treated as a
section of ∧2 T X . The action of ω −1 on 1-forms η, ξ can be written in at least two ways
hω −1 |η, ξi = hη|ω −1 ξi = ω ij ηi ξj .
If H is a function on Y, then we define its Hamiltonian field ω −1 dH. We will often
consider a time dependent Hamiltonian H(t, y) and the corresponding dynamic defined by
the Hamilton equations
∂t y(t) = ω −1 dy H(t, y(t)).
(10.242)
76
Proposition 10.1. Flows generated by Hamilton equations are symplectic
If F, G are functions on Y, then we define their Poisson bracket
{F, G} := hω −1 |dF, dGi.
Proposition 10.2. {·, ·} is a bilinear antisymmetric operation satisfying the Jacobi identity
{F, {G, H}} + {H, {F, G}} + {G, {H, F }} = 0
(10.243)
and the Leibnitz identity
{F, GH} = {F, G}H + G{F, H}.
(10.244)
Proposition 10.3. Let t 7→ y(t) be a trajectory of a Hamiltonian H(t, y). Let F (t, y) be an
observable. Then
d
F t, y(t) = ∂t F t, y(t) + {H, F } t, y(t) .
dt
In particular,
d
H t, y(t) = ∂t H t, y(t) .
dt
10.2
Symplectic vector space
The most obvious example of a symplectic manifold is a symplectic vector
space. As we discussed before, it has the form Rd ⊕ Rd with variables (x, p) = (xi ), (pj ) and the symplectic
form
ω = dpi ∧ dxi .
(10.245)
The Hamilton equations read
∂t x =
∂p H(t, x, p),
∂t p
−∂x H(t, x, p).
=
(10.246)
The Poisson bracket is
{F, G} = ∂xi F ∂pi G − ∂pi F ∂xi G.
(10.247)
Note that Prop 10.1 and 10.2 are easy in a symplectic vector space. To show that ω is
invariant under the Hamiltonian flow we compute
d
ω
dt
=
=
=
d
dp(t) ∧ dx(t)
dt
−d∂x H x(t), p(t) ∧ dx(t) + dp(t) ∧ d∂p H x(t), p(t)
−∂p ∂x H x(t), p(t) dp(t) ∧ dx(t) + dp(t) ∧ ∂x ∂p H x(t), p(t) dx(t) = 0
Proposition 10.4. The dimension of a symplectic manifold is always even. For any symplectic manifold Y of dimension 2d locally there exists a symplectomorphism onto an open
subset of Rd ⊕ Rd .
Now (10.4) implies Prop. 10.1. Similarly, to see Prop. 10.2 we first check the Jacobi and
Leibniz identity for (10.247).
77
10.3
The cotangent bundle
Let X be a manifold. We consider the cotangent bundle T# X . It is equipped with the
canonical projection π : T# X → X .
We can always cover X with open sets equipped with charts. A chart on U ⊂ X allows
us to identify U with an open subset of Rd through coordinates x = (xi )∈ Rd . T # U can be
identified with U × Rd , where we use the coordinates (x, p) = (xi ), (pj ) .
T# X is equipped with the tautological 1-form
X
θ=
pi dxi ,
(10.248)
i
(also called Liouville or Poincaré 1-form), which does not depend on the choice of coordinates. The corresponding symplectic form, called the canonical symplectic form is
X
ω = dθ =
dpi ∧ dxi .
(10.249)
i
Thus locally we can apply the formalism of symplectic vector spaces. In particular, the
Hamilton equations have the form (10.246) and the Poisson bracket (10.247).
10.4
Lagrangian manifolds
Let Y be a symplectic manifold. Let L be a submanifold of Y and iL : L → Y be its
embedding in Y. Then we say that L is isotropic iff i#
L ω = 0. We say that it is Lagrangian
if it is isotropic and of dimension d (which is the maximal possible dimesion for an isotropic
manifold). We say that L is coisotropic if the dimension of the null space of i#
L ω is maximal
possible, that is, 2d − dim L.
Theorem 10.5. Let E ∈ R. Let L be a Lagrangian manifold contained in the level set
H −1 (E) := {y ∈ Y : H(y) = E}.
Then ω −1 dH is tangent to L.
Proof. Let y ∈ Y and v ∈ Ty L. Then since L is contained in a level set of H, we have
0 = hdH|vi = −hω −1 dH|ωvi.
(10.250)
By maximality of Ty L as an isotropic subspace of Ty Y, we obtain that ω −1 dH ∈ Ty L. 2
Clearly, symplectic transformations map Lagrangian manifolds onto Lagrangian manifolds.
10.5
Lagrangian manifolds in a cotangent bundle
Proposition 10.6. Let U be an open subset of X and consider a function U 3 x 7→ S(x) ∈
R. Then
x, dS(x) : x ∈ U
(10.251)
is a Lagrangian submanifold of T # X .
78
Proof. Tangent space of (10.251) at the point (xi , ∂xj S(x)dxj ) is spanned by
vi = ∂xi , ∂xi ∂xj S(x)∂pj
Now
hω|vi , vk i =
X
∂xi ∂xj S(x) −
i,j
X
∂xk ∂xj S(x) = 0.
k,j
2
U 3 S(x) is called a generating function of the Lagrangian manifold (10.251). If U is
connected, it is uniquely defined up to an additive constant.
Suppose that L is a connected and simply connected Lagrangian submanifold. Fix
(x0 , p0 ) ∈ L. For any (x, p) ∈ L, let γ(x,p) be a path contained in L joining (x0 , p0 ) with
(x, p).
Z
T x, p :=
θ.
γ(x,p)
#
#
Using that di#
L θ = iL dθ = iL ω = 0 and the Stokes Theorem we see that the integral does
not depend on the path. We have
dT = i#
(10.252)
L θ.
If π is injective we will say that L is projectable on the base. Then we can use U := π(L)
L
to parametrize L:
U 3 x 7→ x, p(x) ∈ L.
We then define
S(x) := T x, p(x) .
We have
∂xi S(x)dxi = dS(x) = dT x, p(x) = pi dxi .
Hence x 7→ S(x) is the unique generating functon of L satisfying S x(z0 ) = 0.
Both x 7→ S(x) and L 3 (x, p) 7→ T (x, p) will be called generating functions of the
Lagrangian manifold L. To distinguish between them we may add that the former is viewed
as a function on the base and the latter is viewed as a function on L.
We can generalize the construction of T to more general Lagrangian
manifolds. We
cov
consider the universal covering Lcov → L with the base point at
x
,
p
0
0
. Recall that L
is defined as the set of homotopy classes of curves from (x0 , p0 to x, p ∈ L contained in
L. On Lcov we define the real function
Z
Lcov 3 [γ] 7→ T ([γ]) :=
θ.
(10.253)
γ
Exactly as above we see that (10.253) does not depend on the choice of γ and that (10.252)
is true.
79
10.6
Generating function of a symplectic transformations
Let (Yi , ωi ) be symplectic manifolds. We can than consider the symplectic manifold Y2 × Y1
with the symplectic form ω1 − ω2 . Let R be the graph of a diffeomorphism ρ, that is
R := (ρ(y), y) ∈ Y2 × Y1 .
(10.254)
Clearly, ρ is symplectic iff R is a Lagrangian manifold.
Assume that Yi = T# Xi . We can identify Y2 × Y1 with T# (X2 × X1 ).
Let T# X1 3 (x1 , ξ1 ) 7→ (x2 , ξ2 ) ∈ T# X2 be a symplectic transformation. A function
X2 × X1 3 (x2 , x1 ) 7→ S(x2 , x1 ).
(10.255)
is called a generating function of the transformation ρ if it satisfies
ξ2 = −∇x2 S(x2 , x1 ),
ξ1 = ∇x1 S(x2 , x1 ).
(10.256)
Note that if assume that the graph of ρ is projectable onto X2 × X1 , then we can find a
generating function.
10.7
The Legendre transformation
Let X = Rd be a vector space. Consider the symplectic vector space X ⊕ X # = Rd ⊕ Rd
with the generic variables (v, p). It can be viewed as a cotangent bundle in two ways – we
can treat either X or X # as the base. Correspondingly, to describe any Lagrangian manifold
L in X ⊕ X # we can try to use a generating function on X or on X # . To pass from one
description to the other one uses the Legendre transformation, which is described in this
subsection.
Let U be a convex set of X . Let
U 3 v 7→ S(v) ∈ R
(10.257)
be a strictly convex C 2 -function. By strict convexity we mean that for distinct v1 , v2 ∈ U,
v1 6= v2 , 0 < τ < 1,
τ S(v1 ) + (1 − τ )S(v2 ) > S τ v1 + (1 − τ )v2 .
Then
U 3 v 7→ p(v) := ∂v S(v) ∈ X #
(10.258)
is an injective function. Let Ũ be the image of (10.258). It is a convex set, because it is the
image of a convex set by a convex function. One can define the function
Ũ 3 p 7→ v(p) ∈ U
inverse to (10.258). The Legendre transform of S is defined as
S̃(p) := pv(p) − S v(p) .
80
Theorem 10.7. (1) ∂p S̃(p) = v(p).
−1
(2) ∂p2 S̃(p) = ∂p v(p) = ∂v2 S v(p)
. Hence S̃ is convex.
˜
(3) S̃(v)
= S(v).
Proof. (1)
∂p S̃(p) = v(p) + p∂p v(p) − ∂v S v(p) ∂p v(p) = v(p).
(2)
−1
−1
.
= ∂v2 S v(p)
∂p2 S̃(p) = ∂p v(p) = ∂v p v(p)
(3)
˜
S̃(v)
= vp(v) − p(v)v p(v) + S v p(v) = S(p).
2
Thus the same Lagrangian manifold has two descriptions:
v, dS(v) : v ∈ U
=
dS̃(p), p : p ∈ Ũ .
Examples.
(1) U = Rd , S(v) = 12 vmv,
Ũ = Rd , S̃(p) = 21 pm−1 p,
√
1 − v2 .
(2) U = {v ∈ Rd : |v|
<
1},
S(v)
=
−m
p
d
2
2
Ũ = R , S̃(p) = p + m ,
(3) U = R, S(v) = ev ,
Ũ =]0, ∞[, S̃(p) = p log p − p.
Note that we sometimes apply the Legendre transformation to non-convex functions. For
instance, in the first example m can be any nondegenerate matrix.
Proposition 10.8. Suppose that S depends on an additional parameter α. Then we have
the identity
∂α S α, v(α, p) = −∂α S̃(α, p).
(10.259)
Proof. Indeed,
∂α S̃(α, p)
= ∂α pv(α, p) − S α, v(α, p)
= p∂α v(α, p) − ∂α S(α, v(α, p) − ∂v S(α, v(α, p) ∂α v(α, p)
= −∂α S α, v(α, p) .
2
81
10.8
The extended symplectic manifold
Let Y be a symplectic manifold. We introduce the extended symplectic manifold as
T# R × Y = R × R × Y,
where its coordinates have generic names (t, τ, y). Here t has the meaning of time, τ of the
energy. For the symplectic form we choose
σ := −dτ ∧ dt + ω.
Let R × Y 3 (t, y) 7→ H(t, y) be a time dependent function on Y. Let ρt be the flow
generated by the Hamiltonian H(t), that is
ρt y(0) = y(t),
(10.260)
where y(t) solves
∂t y(t) = ω −1 dy H t, y(t) .
(10.261)
Set
G(t, τ, y) := H(t, y) − τ.
It will be convenient to introduce the projection
T# R × Y 3 (t, τ, y) 7→ κ(t, τ, y) := (t, y) ∈ R × Y,
that involves forgetting the variable τ . Note that κ restricted to
G−1 (0) := {(t, τ, y) : G(t, τ, y) = 0}
(10.262)
is a bijection onto R × Y. Its inverse will be denoted by κ−1 , so that
κ−1 (t, y) = t, H(t, y), y .
Proposition 10.9. Let L be a Lagrangian manifold in Y. The set
M := {(t, τ, y) : y ∈ ρt (L), τ = H(t, y), t ∈ R}
(10.263)
satisfies the following properties:
(1) M is a Lagrangian manifold in T# R × Y;
(2) M is contained in G−1 (0)
(3) we have
κ(M) ∩ {0}×Y = {0} × L;
(4) every point in κ(M) is connected to (10.264) by a curve contained in κ(M).
Besides, conditions (1)-(4) determine M uniquely.
82
(10.264)
Proof. Let (t0 , τ0 , y0 ) ∈ M. Let v be tangent to ρt0 (L) at y0 . Then
hdy H(t0 , y0 )|vi∂τ + v
(10.265)
is tangent to M. Vectors of the form (10.265) are symplectically orthogonal to one another,
because ρt0 (L) is Lagrangian. The curve t 7→ t, H t, y(t) , y(t) is contained in M. Hence the following vector is
tangent to M:
∂t + ∂t H(t0 , y0 )∂τ + ω −1 dy H(t0 , y0 ).
(10.266)
The symplectic form applied to (10.265) and (10.266) is
−hdy H(t0 , y0 )|vi + hω|v, ω −1 dy H(t0 , y0 )i = 0.
(10.267)
(10.265) and (10.266) span the tangent space of M. Hence (1) is true (M is Lagrangian).
(2), (3) and (4) are obvious.
Let us show the uniqueness of M satisfying (1), (2), (3) and (4). Let M be a Lagrangian
submanifold contained in G−1 (0) and (t0 , τ0 , y0 ) ∈ M. By Thm 10.5, the vector
σ −1 dG(t0 , τ0 , y0 )
= σ −1 ∂t H(t0 , y0 )dt + dy H(t0 , y0 ) − dτ
(10.268)
is tangent to M. But (10.268) coincides with (10.266). Hence
∂t + ω −1 dy H(t0 , y0 ).
(10.269)
is tangent to κ(M). This means that κ(M) is invariant for the Hamiltonian flow generated
by H(t, y). Consequently,
[
κ(M) ⊃
{t} × ρt (L).
(10.270)
t∈R
κ(M) cannot be larger than the rhs of (10.270), because then condition (4) would be violated.
2
10.9
Time-dependent Hamilton-Jacobi equations
Let R × T# X 3 (t, x, p) 7→ H(t, x, p) be a time-dependent Hamilonian on T# X . Let X ⊃
U 3 x 7→ S(x) be a given function. The time-dependent Hamilton-Jacobi equation equipped
with initial conditions reads
∂t S(t, x) − H t, x, ∂x S(t, x) = 0,
S(0, x)
=
S(x).
(10.271) can be reinterpreted in more geometric terms as follows: Set
G(t, τ, x, p) := τ − H(t, x, p).
83
(10.271)
Consider a Lagrangian manifold L in Y. We want to find a Lagrangian manifold M in
G−1 (0) := {(t, τ, x, p) ∈ T# R × T# X : τ − H(t, x, p) = 0}
(10.272)
such that
κ(M) ∩ {0}×T# X = {0}×L.
Here, as in the previous subsection,
κ(t, τ, x, p) := (t, x, p).
We will also use its inverse
κ−1 (t, x, p) := t, H(t, x, p), x, p .
The relationship between the two formulations is as follows. Assume that L is a generating function of L. Then the function (t, x) 7→ S(t, x) that appears in (10.271) is the
generating function of M, which for t = 0 coincides with x 7→ S(x).
Note that the geometic formulation is superior to the traditional one, because it does
not have a problem with caustics.
The Hamilton-Jacobi equations can be solved as follows. Let R 3 t 7→ x(t, y), p(t, y) ∈
T# X be the solution of the Hamilton equation with the initial conditions on the Lagrangian
manifold L:
x(0, y), p(0, y) = y, ∂y S(y) .
Then
n
o
M = κ−1 t, x(t, y), p(t, y) : (t, y) ∈ R × U .
Let us find the generating function of M. We will use s as an alternate name for the time
variable. The tautological 1-form of T# R × T# X is
−τ ds + pdx.
Fix a point y0 ∈ U. Then the generating function of M satisfying
T κ−1 0, y0 , p(0, y0 ) = S(y0 )
is given by
−1
T κ
t, x(t, y), p(t, y) = S(y0 ) +
Z
(pdx − τ ds),
γ
where γ is a curve in M joining
κ−1 0, y0 , p(0, y0 )
with
−1
κ
(10.273)
t, x(t, y), p(t, y) .
(10.274)
We can take γ as the union of two disjoint segments: γ = γ1 ∪ γ2 . γ1 is a curve in (10.272)
with the time variable equal to zero ending at
κ−1 0, y, p(0, y) .
(10.275)
84
Clearly, since ds = 0 along γ1 , we have
Z
Z
S(y0 ) +
(pdx − τ ds) = S(y0 ) +
γ1
pdx = S(y).
(10.276)
γ1
γ2 starts at (10.275), ends at (10.274), and is given by the Hamiltonian flow. More precisely,
γ2 is
[0, t] 3 s 7→ κ−1 s, x(s, y), p(s, y) .
We have
Z
(pdx − τ ds) =
Z t
p(s, y)∂s x(s, y) − H s, x(s, y), p(s, y)
ds.
(10.277)
0
γ2
Putting together (10.276) and (10.277) we obtain the formula for the generating function of
M viewed as a function on M:
T (t, y)
=
S(y)
Z t
+
p(s, y)∂s x(s, y) − H s, x(s, y), p(s, y) ds.
(10.278)
0
If we can invert y 7→ x(t, y) and obtain the function x 7→ y(t, x), then we have a generating
of M viewed as a function on the base:
S(t, x) = T t, y(t, x) .
(10.279)
10.10
The Lagrangian formalism
Given a time-dependent Hamiltonian H(t, x, p) set
v := ∂p H(t, x, p).
Suppose that we can express p in terms of t, x, v. We define then the Lagrangian
L(t, x, v) := p(t, x, v)v − H t, x, p(t, x, v)
naturally defined on TX . Thus we perform the Legendre transformation wrt p, keeping t, x
as parameters. Note that p = ∂v L(t, x, v) and ∂x H(t, x, p) = −∂x L(t, x, v). The Hamilton
equations are equivalent to the Euler-Lagrange equations:
d
x(t) = v(t),
dt
d
∂v L t, x(t), v(t) = ∂x L x(t), v(t) .
dt
Using the Lagrangian, the generating function (10.282) can be rewritten as
Z t
T (t, y) = S(y) +
L s, x(s, y), ẋ(s, y) ds.
0
85
(10.280)
(10.281)
Lagrangians often have quadratic dependence on velocities:
L(x, v) =
1 −1
vg (x)v + vA(x) − V (x).
2
(10.282)
The momentum and the velocity are related as
p = g −1 (x)v + A(x),
v = g(x) p − A(x) .
(10.283)
The corresponding Hamiltonian depends quadratically on the momenta:
H(x, p) =
10.11
1
p − A(x) g(x) p − A(x) + V (x).
2
(10.284)
Action integral
In this subsection, which is independent of Subsect. 10.9, we will rederive the formula for
the generating function of the Hamiltonian flow constructed (10.282). Unlike in Subsect.
10.9, we will use the Lagrangian formalism.
Let [0, t] 3 s 7→ x(s, α), v(s, α) ∈ TX be a family of trajectories, parametrized by an
auxiliary variable α. We define the action along these trajectories
Z t
I(t, α) :=
L(x(s, α), v(s, α))ds.
(10.285)
0
Theorem 10.10.
∂α I(t, α) = p(x(t, α), v(t, α))∂α x(t, α) − p(x(0, α), v(0, α))∂α x(0, α).
(10.286)
Proof.
Z
∂α I(t, α)
=
t
∂x L(x(s, α), ẋ(s, α))∂α x(s, α)ds
0
=
Z t
+
∂ẋ L(x(s, α), ẋ(s, α))∂α ẋ(s, α)ds
0
Z t
d
∂x L(x(s, α), ẋ(s, α)) − ∂ẋ L(x(s, α), ẋ(s, α)) ∂α x(s, α)ds
ds
0
s=t
+p(x(s, α), v(s, α))∂α x(s, α) .
s=0
2
Theorem 10.11. Let U be an open subset in X . For y ∈ U define a family of trajectories
x(t, y), p(t, y) solving the Hamilton equation and satisfying the intial conditions
x(0, y) = y,
p(0, y) = ∂y S(y).
(10.287)
Let I(t, y) be the action along these trajectories defied as in (10.285). We suppose that we
can invert the y 7→ x(t, y) obtaining the function x 7→ y(t, x). Then
S(t, x) := I t, y(t, x) + S(y(t, x))
(10.288)
86
is the solution of (10.271), and
∂x S(t, x) = p t, y(t, x) .
(10.289)
Proof. We have
∂y I(t, y) + S(y)
= p(t, y)∂y x(t, y) − p(0, y)∂y x(0, y) + ∂y S(y)
= p(t, y)∂y x(t, y).
(10.290)
= ∂x I t, y(t, x) + S y(t, x)
(10.291)
= p(t, y)∂y x(t, y)∂x y(t, x) = p(t, y).
(10.292)
Hence,
∂x S(t, x)
Now
L x(t, y), ẋ(t, y)
=
=
∂t I(t, y) = ∂t I(t, y) + S(y)
∂t S t, x(t, y) + ∂x S t, x(t, y) ẋ(t, y).
(10.293)
(10.294)
Therefore,
∂t S t, x(t, y)
=
=
L x(t, y), ẋ(t, y) − p(t, y)ẋ(t, y)
−H x(t, y), p(t, y) .
(10.295)
(10.296)
2
10.12
Completely integrable systems
Let Y be a symplectic manifold of dimension 2d. We say that functions F1 and F2 on Y are
in involution if {F1 , F2 } = 0.
Let F1 , . . . , Fm be functions on Y and c1 , . . . , cm ∈ R. Define
−1
L := F1−1 (c1 ) ∩ · · · ∩ Fm
(cm ).
(10.297)
dF1 ∧ · · · ∧ dFm 6= 0
(10.298)
We assume that
on L. Then L is a manifold of dimension 2d − m.
Proposition 10.12. Suppose that F1 , . . . , Fm are in involution and satisfy (10.298). Then
m ≤ d and L is coisotropic. If m = d, then L is Lagrangian.
Proof. We have
hdFi |ω −1 dFj i = {Fi , Fj } = 0.
Hence ω −1 dFj is tangent to L.
hω|ω −1 dFi , ω −1 dFj i = hω −1 dFi |dFj i = −{Fi , Fj } = 0.
87
Hence the tangent space of L contains an m-dimensional subspace on which ω is zero. In
the case of a 2d − m dimensional manifold this means that L is coisotropic. 2
If H is a single function on Y, we say that it is completely integrable if we can find a
family of functions in involution F1 , . . . , Fd satisfying (10.298) on Y such that H = Fd .
Note that for completely integrable H it is easy to find Lagrangian manifolds contained
in level sets of H – one just takes the sets of the form (10.297).
11
11.1
Quantizing symplectic transformations
Linear symplectic transformations
Let ρ ∈ L(Rd ⊕ Rd ). Write ρ as a 2×2 matrix and introduce a symplectic form:
a b
0 −1l
ρ=
, ω :=
.
c d
1l 0
ρ ∈ Sp(Rd ⊕ Rd ) iff
(11.299)
ρ# ωρ = ω,
which means
a# d − c# b = 1l, c# a = a# c, d# b = b# d.
(11.300)
If
x̂0i
= aij x̂j + bij p̂j ,
p̂0i
= cij x̂j + dji p̂j ,
(11.301)
then x̂0 , p̂0 satisfy the same commutation relations as x̂, p̂. We define M pc (Rd ⊕ Rd ) to be
the set of U ∈ U (L2 (Rd )) such that there exists a matrix ρ such that
U x̂i U ∗
=
x̂0i ,
U p̂i U ∗
= p̂0i .
We will say that U implements ρ. Obviously, ρ has to be symplectic, M pc (Rd ⊕ Rd ) is a
group and the map U 7→ ρ is a homomorphism.
11.2
Metaplectic group
If χ is a quadratic polynomial on Rd ⊕ Rd , then clearly eitOp(χ) ∈ M pc (Rd ⊕ Rd ) and
implements the symplectic flow given by the Hamiltonian χ. We will denote the group
generated by such maps by M p(Rd ⊕ Rd ). Every symplectic transformation is implemented
by exactly two elements of M p.
88
11.3
Generating function of a symplectic transformation
Let ρ be as above with b invertible. We then have the factorization
a b
1l 0
0
b
1l 0
ρ=
=
,
c d
e 1l
−b#−1 0
f 1l
where
(11.302)
e = db−1 = b#−1 d# ,
f = b−1 a = a# b#−1 .
are symmetric. Define
X × X 3 (x1 , x2 ) 7→ S(x1 , x2 )
Then
a
c
b
d
:=
x1
ξ1
1
1
x1 ·f x1 − x1 ·b−1 x2 + x2 ·ex2 .
2
2
=
x2
ξ2
(11.303)
iff
∇x1 S(x1 , x2 ) = −ξ1 ,
∇x2 S(x1 , x2 ) = ξ2 .
(11.304)
The function S(x1 , x2 ) is called a generating function of the symplectic transformation ρ.
It is easy to check that the operators ±Uρ ∈ M p(X # ⊕ X ) implementing ρ have the
integral kernel equal to
i
dp
±Uρ (x1 , x2 ) = ±(2πi~)− 2 − det ∇x1 ∇x2 S e− ~ S(x1 ,x2 ) .
11.4
Harmonic oscillator
As an example, we cosider the 1-dimensional harmonic oscillator with ~ = 1. Let χ(x, ξ) :=
1 2
1 2
1 2
1 2
−tOp(χ)
equals
2 ξ + 2 x . Then Op(χ) = 2 D + 2 x . The Weyl-Wigner symbol of e
w(t, x, ξ) = (ch 2t )−1 exp(−(x2 + ξ 2 )th 2t ).
(11.305)
Its integral kernel is given by
1
1
W (t, x, y) = π − 2 (sht)− 2 exp
−(x2 + y 2 )cht + 2xy
2sht
.
e−itOp(χ) has the Weyl-Wigner symbol
w(it, x, ξ) = (cos 2t )−1 exp −i (x2 + ξ 2 )tg 2t
(11.306)
and the integral kernel
W (it, x, y) = π
− 21
t
− 12 − iπ
− iπ
4
2 [π]
| sin t|
e
e
89
exp
−(x2 + y 2 ) cos t + 2xy
2i sin t
.
Above, [c] denotes the integral part of c.
We have W (it + 2iπ, x, y) = −W (it, x, y). Note the special cases
W (0, x, y)
=
δ(x − y),
W ( iπ
2 , x, y)
=
(2π)− 2 e− 4 e−ixy ,
W (iπ, x, y)
=
e− 2 δ(x + y),
W ( i3π
2 , x, y)
=
(2π)− 2 e−
1
iπ
iπ
1
i3π
4
eixy .
1
Corollary 11.1. (1) The operator with
kernel ±(2πi)− 2 e−ixy belongs to the metaplectic
0 −1
group and implements
.
1 0
(2) The
operator
with kernel ±iδ(x + y) belongs to the metaplectic group and implements
−1 0
.
0 −1
11.5
The stationary phase method
For a quadratic form B, inert B will denote the inertia of B, that is n+ − n− , where n± is
the number of positive/negative terms of B in the diagonal form.
Theorem 11.2. Let a be smooth function on X and S a function on suppa. Let x0 be a
critical point of S, that is it satisfies
∂x S(x0 ) = 0.
(For simplicity we assume that it is the only one on suppa). Then for small ~,
Z
2
d
π
i
d
i
e ~ S(x) a(x)dx ' (2π~)− 2 ei 4 inert ∂x S(x0 ) e ~ S(x0 ) a(x0 ) + O(~− 2 +1 ).
(11.307)
Proof. The left hand side of (11.2) is approximated by
Z
2
i
i
e ~ S(x0 )+ 2~ (x−x0 )∂x S(x0 )(x−x0 ) a(x0 )dx,
(11.308)
which equals the right hand side of (11.2). 2
11.6
Semiclassical FIO’s
Suppose that X2 × X1 3 (x2 , x1 ) 7→ a(x2 , x1 ), is a function called an amplitude. Let suppa 3
(x2 , x1 ) 7→ S(x2 , x1 ) be another function, which we calle a phase. We define the Fourier
integral operator with amplitude a and phase S to be the operator from Cc∞ (X1 ) to C ∞ (X2 )
with the integral kernel
1
d
i
FIO(a, S)(x2 , x1 ) = (2π~)− 2 (∇x2 ∇x1 S(x2 , x1 )) 2 e ~ S(x2 ,x1 )
90
(11.309)
We treat FIO(a, S) as a quantization of the symplectic transformation with the generating
function S. Suppose that we can solve
∇x S(x̃, x) = p
(11.310)
FIO~ (a2 , S)∗ FIO~ (a1 , S) = Op~ (b) + O(~),
(11.311)
b(x, p) = a2 (x̃(x, p), x)a1 (x̃(x, p), x).
(11.312)
obtaining (x, p) 7→ x̃(x, p). Then
where
In particular, Fourier integral operators with amplitude 1 are asymptotically unitary.
Indeed
=
=
=
FIO~ (a2 , S)∗ FIO~ (a1 , S)(x2 , x1 )
Z
p
p
i
i
dx ∂x ∂x2 S(x, x2 ) ∂x ∂x1 S(x, x1 ) a(x2 , x)a1 (x, x1 )e− ~ S(x,x2 )+ ~ S(x,x1 )
Z
i
dxb(x2 , x, x1 )e ~ p(x2 ,x,x1 )(x2 −x1 )
Z
i
dp∂p x(x2 , p, x1 )b x2 , x(x2 , p, x1 ), x1 e ~ p(x2 −x1 ) ,
where
b(x2 , x, x1 )
p(x2 , x, x1 )
=
p
p
∂x ∂x2 S(x, x2 ) ∂x ∂x1 S(x, x1 ) a(x2 , x)a1 (x, x1 ),
Z
1
∂x S(τ x2 + (1 − τ )x1 )dτ.
=
0
11.7
Composition of FIO’s
Suppose that
X × X1 3 (x, x1 ) 7→ S1 (x, x1 ), X2 × X 3 (x2 , x) 7→ S2 (x2 , x)
(11.313)
are two functions. Given x2 , x1 , we look for x(x2 , x1 ) satisfying
∇x S2 (x2 , x(x2 , x1 )) + ∇x S1 (x(x2 , x1 ), x1 ) = 0.
(11.314)
Suppose such x(x2 , x1 ) exists and is unique. Then we define
S(x2 , x1 ) := S2 (x2 , x(x2 , x1 )) + S1 (x(x2 , x1 ), x1 )
(11.315)
Suppose S1 is a generating function of a symplectic map ρ1 : T# X1 → T# X and S2 is a
generating function of a symplectic map ρ : T# X → T# X2 . Then S is a generating function
of ρ2 ◦ ρ1 .
91
Proposition 11.3.
∇x2 ∇x1 S(x2 , x1 )
=
−∇x2 ∇x S2 (x2 , x(x2 , x1 ))
(11.316)
−1
(2)
× ∇(2)
x S2 (x2 , x(x2 , x1 )) + ∇x S1 (x(x2 , x1 ), x1 )
×∇x ∇x1 S1 (x(x2 , x1 ), x1 ).
Proof. Differentiating (11.314) we obtain
(2)
(∇x2 x)(x1 , x2 ) ∇(2)
S
(x
,
x(x
,
x
))
+
∇
S
(x(x
,
x
),
x
)
2 2
2
1
1
2
1
1
x
x
+∇x ∇x2 S2 (x2 , x(x2 , x1 ))
=
0.
(11.317)
Differentiating (11.315) we obtain
∇x1 S(x1 , x2 )
=
∇x1 S1 (x(x1 , x2 ), x1 ),
∇x2 ∇x1 S(x1 , x2 )
=
(∇x2 x)(x1 , x2 )∇x ∇x1 S1 (x(x1 , x2 ), x1 ).
(11.318)
Then we use (11.317) and (11.318). 2
In addition to two phases S1 , S2 , let
X2 × X 3 (x2 , x) 7→ a2 (x2 , x),
X × X1 3 (x, x1 ) 7→ a1 (x, x1 )
(11.319)
be two amplitudes. Then we define the composite amplitude as
a(x2 , x1 ) := a2 (x2 , x(x2 , x1 ))a1 (x(x2 , x1 ), x1 ).
(11.320)
FIO~ (a2 , S2 )FIO~ (a1 , S1 ) = FIO~ (a, S) + O(~).
(11.321)
Theorem 11.4.
12
12.1
WKB method
Lagrangian distributions
Consider a quadratic form
1
1
xSx := xi Sij xj ,
2
2
and a function on Rd
i
e 2~ xSx .
(12.322)
(12.323)
Clearly, we have the identity
i
p̂i − Sij x̂j e 2~ xSx = 0,
i = 1, . . . , d.
One can say that the phase space support of (12.323) is concentrated on
{(x, p) : pi − Sij xj = 0, i = 1, . . . , d},
92
(12.324)
which is a Lagrangian subspace of Rd ⊕ Rd .
Let us generalize (12.323). Let L be an arbitrary Lagrangian subspace of Rd ⊕ Rd . Let
an
L be the set of linear functionals on Rd ⊕ Rd such that
L=
∩
φ∈Lan
Kerφ.
Every functional in Lan has the form
φ(ξ, η) = ξj xj + η j pj .
The corresponding operator on L2 (Rd ) will be decorated by a hat:
φ̂(ξ, η) = ξij x̂j + ηij p̂j .
We say that f ∈ S 0 (Rd ) is a Lagrangian distribution associated with the subspace L iff
φ(ξ, η) ∈ Lan .
φ̂(ξ, η)f = 0,
In the generic case, the intersection of L and 0 ⊕ Rd is (0, 0). We then say that the
Lagrangian subspace is projectable onto the configuration space. Then one can find a generating function of the distribution L of the form (12.322). Lagrangian distributions associated
with L are then multiples of (12.323).
The opposite case is L = 0 ⊕ Rd . Lan is then spanned by xi , i = 1, . . . , d. The corresponding Lagrangian distributions are multiples of δ(x)
12.2
Semiclassical Fourier transform of Lagrangian distributions
Consider now the semiclassical Fourier transformation, which is an operator F~ on L2 (Rd )
given by the kernel
i
(12.325)
F~ (p, x) := e− ~ xp .
Note that for all ~, (2π~)−d/2 F~ is unitary – it will be called the unitary semiclassical
Fourier transformation. Multiplied by ±id it is an element of the metaplectic group.
Consider the Lagrangian distribution
i
e 2~ xSx ,
(12.326)
with an invertible S. Then its is easy to see that the image of (12.326) under (2π~)−d/2 F~
is
−1
i
id/2 (det S −1 )1/2 e− 2~ pS p .
More generally, we can check that the semiclassical Fourier transformation in all or only a
part of the variables preserves the set of Lagrangian distributions.
93
12.3
The time dependent WKB approximation for Hamiltonians
In this subsection we describe the WKB approximation for the time-dependent Schrödinger
equation and Hamiltonians quadratic in the momenta. For simplicity we will restrict ourselves to stationary Hamiltonians – one could generalize this subsection to time-dependent
Hamiltonians.
Consider the classical Hamiltonian
H(x, p) =
1
(p − A(x))g(x)(p − A(x)) + V (x)
2
(12.327)
with the corresponding Lagrangian
L(x, v) =
1 −1
vg (x)v + vA(x) − V (x).
2
(12.328)
We quantize the Hamiltonian in the naive way:
H~ :=
1
(−i~∂ˆ − A(x))g(x)(−i~∂ˆ − A(x)) + V (x).
2
(12.329)
We look for solutions of
i~∂t Ψ~ (t, x)
=
H~ Ψ~ (t, x).
(12.330)
Ψ~ (t, x)
e ~ S(t,x) a~ (t, x),
(12.331)
We make an ansatz
Ψ~ (0, x)
=
=
i
e
i
~ S(x)
a(x).
(12.332)
i
where a(x), S(x) are given functions. We multiply the Schrödinger equation by e− ~ S(t,x)
obtaining
i~∂ˆt − ∂t S(t, x) a~ (t, x)
(12.333)
1 −1 ˆ
=
(i ~∂x + ∂x S(t, x) − A(x))g(x)(i−1 ~∂ˆx + ∂x S(t, x) − A(x)) + V (x) a~ (t, x).
2
To make the zeroth order in ~ part of (12.333) vanish we demand that
−∂t S(t, x)
=
1
(∂x S(t, x) − A(x))g(x)(∂x S(t, x) − A(x)) + V (x).
2
(12.334)
This is the Hamilton-Jacobi equation for the Hamiltonian H. Together with the initial
conditions (12.334) can be rewritten as
−∂t S(t, x) = H x, ∂x S(x) ,
(12.335)
S(0, x)
= S(x),
94
Recall that (12.335) is solved as follows. First we need to solve the equations of motion:
ẋ(t, y) = ∂p H x(t, y), p(t, y) ,
ṗ(t, y) = −∂x H x(t, y), p(t, y) ,
x(0, y)
=
y,
p(0, y)
=
∂y S(y).
We can do it in the Lagrangian formalism. We replace the variable p by v:
v(t, x) = ∂p H x, ∂x S(t, x) .
Then
ẋ(t, y)
= v(t, y),
v̇(t, y)
= ∂x L x(t, y), v(t, y) ,
x(0, y)
= y,
v(0, y)
= ∂p H y, ∂y S(y) .
Then
S t, x(t, y) = S(y) +
Z
t
L x(s, y), v(s, y) ds
0
defines the solution of (12.335) with the initial condition (12.332), provided that we can
invert y 7→ x(t, y).
We have also the equation for the amplitude:
i~
1
ˆ
ˆ
ˆ
(12.336)
∂t + (v(t, x)∂x + ∂x v(t, x)) a~ (t, x) = ∂ˆx g(x)∂ˆx a~ (t, x).
2
2
Note that for any function b
1
1
∂ˆt + (v(t, x)∂ˆx + ∂ˆx v(t, x)) (det ∂x y(t, x)) 2 b(y(t, x)) = 0
2
(12.337)
Thus setting
i
1
Ψcl (t, x) := (det ∂x y(t, x)) 2 a y(t, x) e ~ S(t,x) .
(12.338)
We solve the Schrödinger equation modulo O(~), taking into account the initial condition:
i~∂t Ψcl (t, x)
=
H~ Ψcl (t, x) + O(~2 ),
Ψcl (0, x)
=
e ~ S(x) a(x)
i
We can improve on Ψcl by setting
1
Ψ~ (t, x) := (det ∂x y(t, x)) 2
∞
X
n=0
95
i
~n bn (t, y(t, x))e ~ S(t,x) ,
(12.339)
where
∂t bn+1
b0 (y) = a(y),
1
−1
t, y(t, x) = i~ (det ∂x y(t, x)) 2 ∂ˆx g(x)∂ˆx (det ∂x y(t, x)) 2 bn t, y(t, x) .
(The 0th order yields Ψcl (t, x)). If caustics develop after some time we can use the prescription of Subsection 12.10 to pass them.
12.4
Stationary WKB metod
The WKB method can be used to compute eigenfunctions of Hamiltonians. Let H and H~
be as in (12.327) and (12.329). We would like to solve
H~ Ψ~ = EΨ~ .
We make the ansatz
i
Ψ~ (x) := e ~ S(x) a~ (x).
i
We multiply the Schrödinger equation by e− ~ S(x) obtaining
Ea~ (x)
(12.340)
1 −1 ˆ
=
(i ~∂x + ∂x S(x) − A(x))g(x)(i−1 ~∂ˆx + ∂x S(x) − A(x)) + V (x) a~ (x).
2
To make the zeroth order in ~ part of (12.340) vanish we demand that
E
=
1
(∂x S(x) − A(x))g(x)(∂x S(x) − A(x)) + V (x),
2
which is the stationary version of theHamilton-Jacobi equation, called sometimes the eikonal
equation. Set v(x) = ∂p H x, ∂x S(x) . We have the equation for the amplitude
1
i~
v(x)∂ˆx + ∂ˆx v(x) a~ (x) = ∂ˆx g(x)∂ˆx a~ (x).
2
2
We set
a~ (x) :=
∞
X
~n an (x).
(12.341)
(12.342)
n=0
Now (12.341) can be rewritten as
1
v(x)∂ˆx + ∂ˆx v(x) a0 (x) = 0,
2
1
v(x)∂ˆx + ∂ˆx v(x) an+1 (x) = i~∂ˆx g(x)∂ˆx an (x).
2
In dimension 1 we can solve (12.343) obtaining
1
a0 (x) = |v(x)|− 2 .
96
(12.343)
This leads to an improved ansatz
− 21
Ψ~ (x) = |v(x)|
∞
X
i
~n bn (x)e ~ S(x)
n=0
We obtain the chain of equations
b0 (x) = 1,
1
1
∂t bn+1 x = i~|v(x)| 2 ∂ˆx g(x)∂ˆx |v(x)|− 2 bn x).
Thus the leading approximation is
1
i
Ψ0 (x) := |v(x)|− 2 e ~ S(x) .
(12.344)
In the case of quadratic Hamiltonians we can solve for v(x) and S(x):
q
v(x) = g(x)−1 2 E − V (x) ,
q
∂x S(x) = g(x)−1 2 E − V (x) + A(x).
12.5
Conjugating quantization with a WKB phase
Lemma 12.1. The operator B~ with the kernel
Z
i
−d
2
(x − y)p dp
(2π~)
b(x, y)p exp
~
(12.345)
equals
~ ˆ
∂x b(x, x) + b(x, x)∂ˆx + i~ (∂x b(x, y) − ∂y b(x, y)) .
2i
y=x
(12.346)
Proof. We apply Theorem 5.8. 2
Theorem 12.2. Let S, h be smooth functions. Then
i
i
e− ~ S(x) Op~ (G)e ~ S(x)
= G(x, ∂x S(x))
(12.347)
~ ˆ
+
∂x ∂p G x, ∂x S(x) + ∂p G x, ∂x S(x) ∂ˆx + O(~2 ).
2i
97
Proof. The integral kernel of the left-hand side equals
Z i
x+y −d
2
(2π~)
G
, p exp
− S(x) + S(y) + (x − y)p dp
2
~
Z Z 1
d
i
x
+
y
−2
= (2π~)
, p exp (x − y) −
∂S τ x + (1 − τ )y dτ + p dp
G
2
~
0
Z Z 1
i
d
x
+
y
= (2π~)− 2
∂S τ x + (1 − τ )y dτ exp
G
,p +
(x − y)p dp
2
~
0
Z Z 1
i
d
x
+
y
∂S τ x + (1 − τ )y dτ exp
= (2π~)− 2
G
,
(x − y)p dp
(12.348)
2
~
0
Z
Z 1
i
d
x+y
∂S(τ x + (1 − τ )y)dτ exp
+ (2π~)− 2
p∂p G
,
(x − y)p dp(12.349)
2
~
0
Z Z 1
d
+ (2π~)− 2
dσ(1 − σ)pp
0
Z 1
i
x+y
, σp +
(x − y)p dp. (12.350)
×∂p ∂p G
∂S(τ x + (1 − τ )y)dτ exp
2
~
0
We have
(12.348)
=
(12.349)
=
(12.348)
=
G(x, ∂x S(x)),
~ ˆ
∂x ∂p G(x, ∂x S(x)) + ∂p G(x, ∂x S(x))∂ˆx ,
2i
O(~2 ),
where we used Lemma 12.1 to compute the second term. 2
12.6
WKB approximation for general Hamiltonians
The WKB approximation is not restricted to quadratic Hamiltonians. Using Theorem 12.2
we easily see that the WKB method works for general Hamiltonians.
One can actually unify the time-dependent and stationary WKB method into one setup.
Consider a function H on Rd ⊕ Rd having the interpretation of the Hamiltonian. We are
interested in the two basic equations of quantum mechanics:
(1) The time-dependent Schrödinger equation:
(i~∂t − Op~ (H)) Φ~ (t, x) = 0.
(12.351)
(2) The stationary Schrödinger equation:
(Op~ (H) − E) Φ~ (x) = 0
98
(12.352)
They can be written as
Op~ (G)Φ~ (x) = 0,
(12.353)
where
(1) for (12.351), instead of the variable x actually we have t, x ∈ R × Rd , instead of p we
have τ, p ∈ R × Rd and
G(x, t, p, τ ) = τ − H(x, p).
(2) for (12.351),
G(x, p) = H(x, p) − E.
In order to solve (12.353) modulo O(~) we make an ansatz
i
Φ~ (x) = e ~ S(x) a~ (x).
i
We insert Φ~ into (12.353), we multiply by e− ~ S(x) , we set
v(x) := ∂p G(x, p),
and by (12.347) we obtain
i
e− ~ S(x) Op~ (G)Φ~
= G x, ∂x S(x) a~ (x)
~ ˆ
+
∂x v(x) + v(x)∂ˆx a~ (x)
2i
+O(~2 ).
Thus we obtain the Hamilton-Jacobi equation
G(x, ∂x S(x)) = 0
and the transport equation
1 ˆ
∂x v(x) + v(x)∂ˆx a~ (x) = O(~).
2
If we choose any solution of
1 ˆ
∂x v(x) + v(x)∂ˆx a0 = 0
2
and set
i
Φcl (x) := e ~ S(x) a0 (x)
then we obtain an approximate solution:
Op~ (G)Φcl (x) = O(~).
99
12.7
WKB functions. The naive approach
Distributions associated with a Lagrangian subspaces have a natural generalization to Lagrangian manifolds in a cotangent bundle.
Let X be a manifold and L a Lagrangian manifold in T# X . First assume that L is
projectable onto U ⊂ X and U 3 x 7→ S(x) is a generating function of L. Then
i
U 3 x 7→ a(x)e ~ S(x)
(12.354)
is a function that semiclassically is concentrated in L.
Suppose now that L is not necessarily
projectable. Then we can consider its covering
Lcov parametrized by z 7→ x(z), p(z) ∈ Lcov . Let T be a generating function of L viewed
as a function on Lcov . We would like to think of (12.354) as derived from a half-density on
the Lagrangian manifold
i
z 7→ b x(z), p(z) |dz|1/2 e ~ T (x(z),p(z)) .
(12.355)
where b is a nice function on Lcov .
If a piece of Lcov is projectable over U ⊂ X , then we can express (12.355) in terms of x:
1/2 i
(12.356)
U 3 x 7→ b x, p(z(x)) det ∂x z(x) e ~ T x,p(z(x)) |dx|1/2 .
(12.356) is actually not quite correct – there is a problem along the caustics, which should
be corrected by the so-called Maslov index.
12.8
Semiclassical Fourier transform of WKB functions
Let us apply (2π~)−d/2 F~ to a function given by the WKB ansatz:
i
Ψ~ (x) := a(x)e ~ S(x) .
Thus we consider
(2π~)−d/2
Z
i
a(x)e ~
S(x)−xp
(12.357)
dx.
We apply the stationary phase method. Given p we define x(p) by
∂x (S(x(p)) − x(p)p) = ∂x S(x(p)) − p = 0.
We assume that we can invert this function obtaining p 7→ x(p). Note that
−1
∂p x(p) = ∂x2 S x(p)
,
so locally it is possible if ∂x2 S is invertible. Let p 7→ S̃(p) denote the Legendre transform of
x 7→ S(x), that is
S̃(p) = px(p) − S x(p) .
100
Then by the stationary phase method
(2π~)−d/2 F~ Ψ~ (p)
=
e
=
e
2 S(x(p))
iπinert ∂x
4
2 S̃(p)
iπinert ∂p
4
i
|∂x2 S x(p) |−1/2 e− ~ S̃(p) a(x(p)) + O(~)
i
|∂p2 S̃(p)|1/2 e− ~ S̃(p) a(x(p)) + O(~).
One can make this formula more symmetric by replacing Ψ~ with
i
Φ~ (x) := |∂x2 S(x)|1/4 a(x)e ~ S(x) .
(12.358)
Then
(2π~)−d/2 F~ Φ~ (p)
12.9
=
e
2 S̃(p)
iπinert ∂p
4
i
|∂p2 S̃(p)|1/4 e− ~ S̃(p) a(x(p)) + O(~).
WKB functions in a neighborhood of a fold
Let us consider R × R and the Lagrangian manifold given by x = −p2 . Note that it is
not projectable in the x coordinates. It is however projectable in the p coordinates. Its
3
generating function in the p coordinates is p 7→ p3 .
We consider a function given in the p variables by the WKB ansatz
3
i p
3
1
(2π~)− 2 F~ Ψ~ (p) = e ~
Then
1
Ψ~ (x) = (2π~)− 2
Z
i
e~(
p3
3
b(p).
+xp)
b(p)dp.
(12.359)
(12.360)
√
The stationary phase method gives for x < 0, p(x) = ± −x. Thus, for x < 0,
Ψ~ (x) '
3
√
iπ
i2
1
e 4 − ~3 (−x) 2 (−x)− 4 b(− −x)
3
√
iπ
i2
1
+e− 4 + ~3 (−x) 2 (−x)− 4 b( −x).
(12.361)
iπ
Thus we see that the phase jumps by e 2 .
For x > 0 the non-stationary method gives Ψ~ (x) ' O(~∞ ). If b is analytic, we can
apply the steepest descent method to obtain
3
√
2
1
Ψ~ (x) ' e− ~3 x 2 x− 4 b(i x)
(12.362)
Note that the stationary phase and steepest descent method are poor in a close vicinity
of the fold – they give a singular behavior, even though in reality the function is continuous.
It can be approximated by replacing b(p) with b(0) in terms of the Airy function
Z ∞
i 2
1
e 3 p +ipx dp.
Ai(x) :=
2π −∞
In fact,
Ψ~ (x) ≈ b(0)(2π)1/2 ~−1/6 Ai(~−2/3 x).
101
12.10
Caustics and the Maslov correction
Let us go back to the construction described in Subsection 12.7. Recall that we had problems
with the WKB approximation near a point where the Lagrangian manifold is not projectable.
There can be various behaviors of L near such point, but it is enough to assume that we have
a simple fold. We can then represent locally the manifold as X = R × X⊥ with coordinates
(x1 , x⊥ ). The corresponding coordinates on the cotangent bundle T# X = R × R × T# X⊥
are (x1 , p1 , x⊥ , p⊥ ).
Suppose that we have a Lagrangian manifold that locally can be parametrized by (p1 , x⊥ )
with a generating function (p1 , x⊥ ) 7→ T (p1 , x⊥ ), but is not projectable on X . More precisely,
we assume that it projects to the left of x1 = 0, where it has a fold. Thus it has two sheets
given by
{x = (x1 , x⊥ ) : x1 ≤ 0} 3 x 7→ p± (x).
By applying the Legendre transformation in x1 we obtain two generating functions
{(x1 , x⊥ ) : x1 ≤ 0} 3 x 7→ S ± (x).
Suppose that we start from a function given by
i
Φ~ (p1 , x⊥ ) = e ~ T (p1 ,x⊥ )+iα b(p1 , x⊥ ),
where α is a certain phase. If we apply the unitary semiclassical Fourier transformation wrt
the variable p1 we obtain
Ψ~ (x)
=
21
−
b p−
1 (x), x⊥ det ∂x1 p1 (x)
21
π
i +
+
+ O(~).
+e ~ S (x1 ,x⊥ )+iα+i 4 b p+
1 (x), x⊥ det ∂x1 p1 (x)
i
e~S
−
(x1 ,x⊥ )+iα−i π
4
(12.363)
(12.364)
π
Thus the naive ansatz is corrected by the factor of ei 2 .
In the case of a general Lagrangian manifold, we can slightly deform it so that we can
reach each point by passing caustics only through simple folds.
12.11
Global problems of the WKB method
Let us return to the setup of Subsection 12.6. Note that the WKB method gives only a
local solution. To find a global solution we need to look for a Lagrangian manifold L in
G−1 (0). Suppose we found such a manifold. We divide it into projectable patches Li such
that π(Li ) = Ui . For each of these patches on Ui we can write the WKB ansatz
i
e ~ S(x) a(x).
Then we try to sew them together using the Maslov conditon.
This might work in the time dependent case. In fact, we can choose a WKB ansatz
corresponding to a projectable Lagrangian manifold at time t = 0, with a well defined
generating function. For small times typically the evolved Lagrangian manifold will stay
projectable and the WKB method will work well. Then caustics may form – we can then
102
consider the generating function viewed as a (univalued) function on the Lagrangian manifold
and use the Maslov prescription.
When we apply the WKB method in more than 1 dimension for the stationary Schrödinger
equation, problems are more serious. First, it is not obvious that we will find a Lagrangian
manifold. Even if we find it, it is typically not simply connected. In principle we should use
its universal covering. Thus above a single x we can have contributions from various sheets
of Lcov – typically, infinitely many of them. They may cause “destructive interference”.
12.12
Bohr–Sommerfeld conditions
The stationary WKB method works well in the special case of X = R. Typically, a
Lagrangian manifold coincides in this case with a connected component of the level set
{(x, p) ∈ R × R : H(x, p) = E}. The transport equation has a univalued solution. L is
topologically a circle, and it is the boundary of a region D, which is topologically a disc.
(This
equips
L with an orientation). The function T after going around L increases by
R
R
θ = D ω. Suppose that L crosses caustics only at simple folds, n+ of them in the “posL
itive” direction and n− in the “negative” direction. Clearly, n+ − n− = 2. (In fact, in a
typical case, such as that of a circle, we have n+ = 2, n− = 0). Then when we come back
to the initial point the WKB solution changes by
i
e~
R
D
ω−iπ
.
(12.365)
If (12.365) is different from 1, then going around we obtain contributions to WKB that
intefere destructively. Thus (12.365) has to be 1. This leads to the condition
Z
1
ω − π = 2πn, n ∈ Z,
(12.366)
~ D
or
1
2π
Z
1
ω =~ n+
2
D
,
(12.367)
which is the famous Bohr-Sommerfeld condition.
Contents
1 Introduction
1.1 Basic classical mechanics . . . .
1.2 Basic quantum mechanics . . .
1.3 Concept of quantization . . . .
1.4 The role of the Planck constant
1.5 Aspects of quantization . . . .
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1
1
2
3
3
4
2 Preliminary concepts
2.1 Fourier tranformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Semiclassical Fourier transformation . . . . . . . . . . . . . . . . . . . . . . .
2.3 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.4
2.5
2.6
2.7
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space
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11
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4 Coherent states
4.1 General coherent states . . . . . . . . . . . . . . . . . . .
4.2 Contravariant quantization . . . . . . . . . . . . . . . . .
4.3 Covariant quantization . . . . . . . . . . . . . . . . . . . .
4.4 Connections between various quantizations . . . . . . . .
4.5 Gaussian coherent vectors . . . . . . . . . . . . . . . . . .
4.6 Creation and annihilation operator . . . . . . . . . . . . .
4.7 Quantization by an ordering prescription . . . . . . . . . .
4.8 Connection between the Wick and anti-Wick quantization
4.9 Wick symbol of a product . . . . . . . . . . . . . . . . . .
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23
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29
30
5 Pseudodifferential calculus with uniform symbol classes
5.1 Gaussian dynamics on uniform symbol classes . . . . . . .
5.2 The boundedness of quantized uniform symbols . . . . . .
5.3 Semiclassical calculus . . . . . . . . . . . . . . . . . . . . .
5.4 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Semiclassical asymptotics of the dynamics . . . . . . . . .
5.6 Frequency set . . . . . . . . . . . . . . . . . . . . . . . . .
5.7 Properties of the frequency set . . . . . . . . . . . . . . .
5.8 Algebras with a filtration/gradation . . . . . . . . . . . .
5.9 Algebra of semiclassical operators . . . . . . . . . . . . . .
5.10 Principal and subprincipal symbols . . . . . . . . . . . . .
5.11 Algebra of formal semiclassical operators . . . . . . . . . .
5.12 More about star product . . . . . . . . . . . . . . . . . . .
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31
31
31
32
33
34
35
36
36
37
37
38
38
3 x, p3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
Gaussian integrals in complex
Integral kernel of an operator
Tempered distributions . . . .
Hilbert-Schmidt operators . .
Trace class operators . . . . .
variables
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and Weyl-Wigner quantizations
x, p-quantization . . . . . . . . . . . . . . . . . . . . . . .
Weyl-Wigner quantization . . . . . . . . . . . . . . . . . .
Star product . . . . . . . . . . . . . . . . . . . . . . . . .
Weyl operators . . . . . . . . . . . . . . . . . . . . . . . .
Weyl-Wigner quantization in terms of Weyl operators . .
Classical and quantum mechanics over a symplectic vector
Weyl quantization for a symplectic vector space . . . . . .
Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . .
Parity operator . . . . . . . . . . . . . . . . . . . . . . . .
Special classes of symbols . . . . . . . . . . . . . . . . . .
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6 Spectral asymptotics
39
6.1 Functional calculus and Weyl asymptotics . . . . . . . . . . . . . . . . . . . . 39
6.2 Weyl asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.3 Energy of many fermion systems . . . . . . . . . . . . . . . . . . . . . . . . . 42
7 Standard pseudodifferential calculus on Rd
7.1 Classes of symbols . . . . . . . . . . . . . .
7.2 Classes of pseudodifferential operators . . .
7.3 Extended principal symbol . . . . . . . . . .
7.4 Diffeomorphism invariance . . . . . . . . . .
7.5 Ellipticity . . . . . . . . . . . . . . . . . . .
7.6 Asymptotics of the dynamics . . . . . . . .
7.7 Singular support . . . . . . . . . . . . . . .
7.8 Preservation of the singular support . . . .
7.9 Wave front . . . . . . . . . . . . . . . . . .
7.10 Properties of the wave front set . . . . . . .
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8 Path integrals
8.1 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Scattering operator . . . . . . . . . . . . . . . . . . . . . . .
8.3 Bound state energy . . . . . . . . . . . . . . . . . . . . . . .
8.4 Path integrals for Schrödinger operators . . . . . . . . . . .
8.5 Example–the harmonic oscillator . . . . . . . . . . . . . . .
8.6 Path integrals for Schrödinger operators with the imaginary
8.7 Wiener measure . . . . . . . . . . . . . . . . . . . . . . . . .
8.8 General Hamiltonians – Weyl quantization . . . . . . . . . .
8.9 Hamiltonians quadratic in momenta I . . . . . . . . . . . .
8.10 Hamiltonians quadratic in momenta II . . . . . . . . . . . .
8.11 Semiclassical path integration . . . . . . . . . . . . . . . . .
8.12 General Hamiltonians – Wick quantization . . . . . . . . . .
8.13 Vacuum expectation value . . . . . . . . . . . . . . . . . . .
8.14 Scattering operator for Wick quantized Hamiltonians . . . .
8.15 Friedrichs diagrams . . . . . . . . . . . . . . . . . . . . . . .
8.16 Scattering operator for time-independent perturbations . .
8.17 Energy shift . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9 Method of characteristics
9.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 1st order differential equations . . . . . . . . . . . . .
9.3 1st order differential equations with a divergence term
9.4 α-densities on a vector space . . . . . . . . . . . . . .
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72
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10 Hamiltonian mechanics
10.1 Symplectic manifolds . . . . . . . . . . . . . . . . . .
10.2 Symplectic vector space . . . . . . . . . . . . . . . .
10.3 The cotangent bundle . . . . . . . . . . . . . . . . .
10.4 Lagrangian manifolds . . . . . . . . . . . . . . . . .
10.5 Lagrangian manifolds in a cotangent bundle . . . . .
10.6 Generating function of a symplectic transformations
10.7 The Legendre transformation . . . . . . . . . . . . .
10.8 The extended symplectic manifold . . . . . . . . . .
10.9 Time-dependent Hamilton-Jacobi equations . . . . .
10.10The Lagrangian formalism . . . . . . . . . . . . . . .
10.11Action integral . . . . . . . . . . . . . . . . . . . . .
10.12Completely integrable systems . . . . . . . . . . . .
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11 Quantizing symplectic transformations
11.1 Linear symplectic transformations . . . . . . . . .
11.2 Metaplectic group . . . . . . . . . . . . . . . . . .
11.3 Generating function of a symplectic transformation
11.4 Harmonic oscillator . . . . . . . . . . . . . . . . . .
11.5 The stationary phase method . . . . . . . . . . . .
11.6 Semiclassical FIO’s . . . . . . . . . . . . . . . . . .
11.7 Composition of FIO’s . . . . . . . . . . . . . . . .
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88
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12 WKB method
12.1 Lagrangian distributions . . . . . . . . . . . . . . . . . . . .
12.2 Semiclassical Fourier transform of Lagrangian distributions
12.3 The time dependent WKB approximation for Hamiltonians
12.4 Stationary WKB metod . . . . . . . . . . . . . . . . . . . .
12.5 Conjugating quantization with a WKB phase . . . . . . . .
12.6 WKB approximation for general Hamiltonians . . . . . . . .
12.7 WKB functions. The naive approach . . . . . . . . . . . . .
12.8 Semiclassical Fourier transform of WKB functions . . . . .
12.9 WKB functions in a neighborhood of a fold . . . . . . . . .
12.10Caustics and the Maslov correction . . . . . . . . . . . . . .
12.11Global problems of the WKB method . . . . . . . . . . . .
12.12Bohr–Sommerfeld conditions . . . . . . . . . . . . . . . . .
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92
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