Non-commutative structures and operators
on a Hilbert space
D.Treschev
Steklov Mathematical Institute,
Moscow, Russia.
Luxembourg, September 2009
2
Motivation
Consider the operators of the coordinate and the momentum of a
quantum particle
𝑥
ˆ𝑗 𝜓 = 𝑥𝑗 𝜓(𝑥),
𝑝ˆ𝑗 𝜓 = −𝑖ℏ
∂
𝜓(𝑥),
∂𝑥𝑗
𝑥 ∈ ℝ𝑛 .
They generate the Heisenberg algebra of polynomial (non-commutative)
expressions in 𝑥
ˆ𝑗 , 𝑝ˆ𝑗 with the identities
𝑥
ˆ𝑗 𝑥
ˆ𝑘 − 𝑥
ˆ𝑘 𝑥
ˆ𝑗 = 0,
𝑝ˆ𝑗 𝑝ˆ𝑘 − 𝑝ˆ𝑘 𝑝ˆ𝑗 = 0,
𝑝ˆ𝑗 𝑥
ˆ𝑘 − 𝑥ˆ𝑘 𝑝ˆ𝑗 = −𝑖ℏ𝛿𝑗𝑘 .
3
Any element of the Heisenberg algebra corresponds to a differential
operator with polynomial in 𝑥 coefficients.
Natural problem. To extend the Heisenberg algebra so that its elements
will still correspond to “good” operators on a reasonable function space.
We propose a certain approach to this problem. But first we develop
some formalism, called the non-commutative structures.
4
ˆ +, ⋅) be the free associative algebra over ℂ with 𝑚 generators
Let (𝕆,
𝑧ˆ1 , . . . , 𝑧ˆ𝑚 :
ˆ
𝐹ˆ ∈ 𝕆
⇔
𝐹ˆ =
∑
ˆ
𝑓ˆ𝜁ˆ 𝜁,
𝜁ˆ
where 𝜁ˆ is a word, composed from the letters 𝑧ˆ1 , . . . , 𝑧ˆ𝑚 , and 𝑓ˆ𝜁ˆ ∈ ℂ.
∑
Empty word is identified with 1. Example. 𝐹ˆ = ∞ 𝑘! 𝑧ˆ1 𝑧ˆ𝑘 𝑧ˆ2 .
𝑘=1
2 1
The corresponding commutative object, (𝕆, +, ⋅), is a commutative
associative algebra with generators 𝑧1 , . . . , 𝑧𝑚 .
𝐹 ∈𝕆
⇔
𝐹 =
∑
𝑓𝛼 𝑧 𝛼 ,
𝛼∈ℤ𝑚
+
where 𝛼 is a multi-index, an element of ℤ𝑚
+ = {0, 1, . . .}, and 𝑓𝛼 ∈ ℂ.
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ˆ →𝕆
ˆ and ★ : 𝕆 → 𝕆.
Other structures are the involutions ★ : 𝕆
★
ˆ we have: 𝜁ˆ = 𝑓ˆ 𝑧ˆ𝑗 ⋅ 𝑧ˆ𝑗 .
For any word 𝜁ˆ = 𝑓ˆ𝜁 𝑧ˆ𝑗 ⋅ 𝑧ˆ𝑗 ∈ 𝕆
1
𝑘
𝜁
𝑘
For any monomial 𝑓𝛼 𝑧 𝛼 ∈ 𝕆 we have: (𝑓𝛼 𝑧 𝛼 )★ = 𝑓 𝛼 𝑧 𝛼 .
Then ★ is an anti-isomorphism and ★2 = id.
1
6
ˆ → 𝕆. For any
We define the homomorphism of “averaging” aver : 𝕆
word 𝜁ˆ
ˆ = 𝑧 𝛼,
aver(𝜁)
where 𝑧ˆ𝑗 enters 𝜁ˆ exactly 𝛼𝑗 times.
Example: aver(ˆ
𝑧2 𝑧ˆ12 𝑧ˆ2 𝑧ˆ13 ) = 𝑧15 𝑧22 .
ˆ by linearity.
Then aver is uniquely continued to 𝕆
7
Below we need the derivations
∂1 , . . . , ∂𝑚 ∈ Der(𝕆),
ˆ
∂ˆ1 , . . . , ∂ˆ𝑚 ∈ Der(𝕆).
By definition
∂𝑗 (𝑧𝑘 ) = ∂ˆ𝑗 (ˆ
𝑧𝑘 ) = 𝛿𝑗𝑘 ,
𝑗, 𝑘 = 1, . . . , 𝑚.
Hence, ∂𝑗 = ∂/∂𝑧𝑗 . Obviously,
∂𝑗 aver = aver ∂ˆ𝑗 ,
★ aver = aver ★,
∂𝑗 ★ = ★∂𝑗 ,
∂ˆ𝑗 ★ = ★∂ˆ𝑗 .
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Let 𝐷 ⊂ ℝ𝑚 be a domain. Consider the trivial bundles
𝜋 : 𝐷 × 𝕆 → 𝐷,
ˆ → 𝐷.
𝜋
ˆ :𝐷×𝕆
Sections of these bundles have the form
𝐹 = 𝐹 (𝜉; 𝑧),
𝐹ˆ = 𝐹ˆ (𝜉; 𝑧ˆ),
ˆ
where for any 𝜉 ∈ 𝐷 𝐹 ∈ 𝕆 and 𝐹ˆ ∈ 𝕆.
The sections are said to be horizontal if
(∂𝜉𝑗 − ∂𝑗 )𝐹 = 0,
(∂𝜉𝑗 − ∂ˆ𝑗 )𝐹ˆ = 0.
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ˆ
The algebras of horizontal sections are denoted 𝕆(𝐷), 𝕆(𝐷).
For any 𝜉0 ∈ 𝐷 there are obvious isomorphisms
𝕆(𝐷)∣𝜉=𝜉0 ∼
=𝕆
and
∼ ˆ
ˆ
𝕆(𝐷)∣
𝜉=𝜉0 = 𝕆.
ˆ
Therefore we have: aver : 𝕆(𝐷)∣
𝜉=𝜉0 → 𝕆(𝐷)∣𝜉=𝜉0 . These maps generate
ˆ
the averaging homomorphism aver : 𝕆(𝐷)
→ 𝕆(𝐷).
ˆ
The involutions ★ are naturally extended to anti-isomorphisms of 𝕆(𝐷)
and 𝕆(𝐷).
Examples. 1. The simplest examples (except constants) are
z𝑗 = 𝜉𝑗 + 𝑧𝑗 ∈ 𝕆(ℝ𝑚 ),
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ˆ 𝑚 ).
ẑ𝑗 = 𝜉𝑗 + 𝑧ˆ𝑗 ∈ 𝕆(ℝ
2. Polynomials in z1 , . . . , z𝑚 lie in 𝕆(ℝ𝑚 ). Non-commutative polynomiˆ 𝑚 ).
als in ẑ1 , . . . , ẑ𝑚 lie in 𝕆(ℝ
3. Proposition. 𝐹 ∈ 𝕆(𝐷) iff 𝐹 (𝜉; 𝑧) = 𝑓 (z), where 𝑓 : 𝐷 → ℂ is a
smooth function.
Corollary. 𝕆(𝐷) ∼
= 𝐶 ∞ (𝐷, ℂ).
ˆ
Informally speaking, any 𝐹ˆ ∈ 𝕆(𝐷)
is a “non-commutative smooth
function of ẑ”. Sometimes we will use the notation 𝐹ˆ = 𝐹ˆ (ẑ).
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ˆ
We extend 𝕆(𝐷)
by adding the element 𝑟ˆ which commutes with
everything,
∂𝜉𝑗 𝑟ˆ = ∂𝑗 𝑟ˆ = ∂ˆ𝑗 𝑟ˆ = 0,
𝑟ˆ★ = −ˆ
𝑟.
Physical meaning of 𝑟ˆ is −𝑖ℏ.
ˆ 𝑟ˆ(𝐷). Averaging aver : 𝕆
ˆ 𝑟ˆ(𝐷) →
The extended algebra is denoted 𝕆
𝕆(𝐷) is the same as before, but first one should put 𝑟ˆ = 0.
ˆ 𝑟ˆ(𝐷) generated by the elements
Consider the ideal 𝐽 ∈ 𝕆
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ˆ 𝑟ˆ(𝐷).
𝜑ˆ𝑗𝑘 ∈ 𝕆
ẑ𝑗 ẑ𝑘 − ẑ𝑘 ẑ𝑗 − 𝑟ˆ𝜑ˆ𝑗𝑘 ,
ˆ 𝐽 (𝐷) = 𝕆
ˆ 𝑟ˆ(𝐷)/𝐽. Averaging (again denoted by aver) is defined
We put 𝕆
ˆ 𝐽 so that the following diagram commutes
on 𝕆
ˆ 𝑟ˆ(𝐷)
𝕆
ˆ 𝐽 (𝐷)
𝕆
−→
aver ↘
↙ aver
𝕆(𝐷)
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Definition. If 𝐽 = 𝐽 ★ , then 𝐽 is said to be Hermitian.
ˆ 𝑟ˆ(𝐷) → 𝕆
ˆ 𝐽 (𝐷) is not a divisor
If the image of 𝑟ˆ under the projection 𝕆
ˆ 𝐽 (𝐷) then 𝐽 is said to be Poissonian.
of zero in 𝕆
Below for brevity we denote this image again by 𝑟ˆ.
Proposition. 𝐽 is Poissonian iff for all 𝑗, 𝑘, 𝑙
𝜑ˆ𝑗𝑘 + 𝜑ˆ𝑘𝑗 ∈ 𝐽,
ẑ𝑗 𝜑ˆ𝑘𝑙 + ẑ𝑘 𝜑ˆ𝑙𝑗 + ẑ𝑙 𝜑ˆ𝑗𝑘 ∈ 𝐽.
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Main Example. Suppose that 𝑚 = 2𝑛. We denote
ẑ𝑗 = x̂𝑗 ,
ẑ𝑗+𝑛 = p̂𝑗 ,
𝑗 = 1, . . . , 𝑛,
and generate 𝐽 by the elements
x̂𝑗 x̂𝑘 − x̂𝑘 x̂𝑗 ,
p̂𝑗 p̂𝑘 − p̂𝑘 p̂𝑗 ,
p̂𝑗 x̂𝑘 − x̂𝑘 p̂𝑗 − 𝑟ˆ𝛿𝑗𝑘 ,
x̂𝑗 p̂𝑘 − p̂𝑘 x̂𝑗 + 𝑟ˆ𝛿𝑗𝑘 .
ˆ 𝐽 (ℝ2𝑛 )
𝐽 is Poissonian and Hermitian. The corresponding algebra 𝕆
is denoted ℍ̂(ℝ2𝑛 ). Subalgebra of polynomial elements in ℍ̂(ℝ2𝑛 ) is
isomorphic to the Heisenberg algebra.
Any Poissonian ideal generates a quantum bracket [ , ]:
ˆ =
[𝐹ˆ , 𝐺]
)
1( ˆ ˆ
ˆ 𝐹ˆ
𝐹𝐺 − 𝐺
𝑟ˆ
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ˆ∈𝕆
ˆ𝐽.
for any 𝐹ˆ , 𝐺
ˆ 𝐽 because 𝐹ˆ 𝐺
ˆ−
Important: the right-hand side of this equation lies in 𝕆
ˆ
ˆ
𝐺𝐹 is divisible by 𝑟ˆ.
Then the corresponding classical Poisson bracket { , } is uniquely
determined from the commutative diagram
[,]
ˆ 𝐽 (𝐷) × 𝕆
ˆ 𝐽 (𝐷) −−−
ˆ 𝐽 (𝐷)
𝕆
−→ 𝕆
⏐
⏐
⏐
⏐aver
aver × avery
y
𝕆(𝐷) × 𝕆(𝐷)
{,}
−−−−→ 𝕆(𝐷)
ˆ 𝐽 (𝐷).
If 𝐽 is Hermitian, ★ is naturally defined as an anti-isomorphism of 𝕆
16
If we regard 𝐷 as a coordinate domain, these algebras can be defined
over a manifold. They are called non-commutative structures.
This language is analogous to the language of star-products in deformation quantization. There are strong relations between this approach to
deformation quantization and the traditional one, but we will not discuss
them now.
We concentrate on the algebra ℍ̂(ℝ2𝑛 ) and try to interpret its elements
as operators on 𝐿2 (ℝ𝑛 ).
17
Recall the standard conventions:
x̂𝑗 𝜓 = 𝑥𝑗 𝜓(𝑥),
p̂𝑗 𝜓 = −𝑖ℏ
∂
𝜓(𝑥),
∂𝑥𝑗
𝑟ˆ𝜓 = −𝑖ℏ𝜓,
𝜓 : ℝ𝑛 → ℂ.
Consider 𝐹ˆ ∈ ℍ(ℝ2𝑛 ). How it is possible to associate with 𝐹ˆ an
operator on 𝐿2 (ℝ𝑛 ) ? If 𝐹ˆ is not polynomial in p̂, power expansions do
not help because we have to deal with a “differential operator of an infinite
order”.
Main idea. For any 𝜁 ∈ ℝ𝑚
𝑒−𝑖𝜁 p̂ 𝜓(𝑥) = 𝑒−ℏ
a quite innocent operator.
∑
𝜁𝑗 ∂/∂𝑥𝑗
𝜓(𝑥) = 𝜓(𝑥 − ℏ𝜁),
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Expansion in exponents 𝑒−𝑖𝜁 p̂ means the Fourier expansion in the
variable p̂. The most convenient space for harmonic analysis on ℝ𝑚 is
the Schwartz space 𝒮. We define
∞
{
∑
𝒮𝑟ˆ = 𝑓 (𝜂, 𝜁, 𝑟ˆ) =
𝑓𝑘 (𝜂, 𝜁)ˆ
𝑟𝑘 ,
𝑘=0
}
𝑓𝑘 ∈ 𝒮 ,
𝜂, 𝜁 ∈ ℝ𝑛 ,
the series is formal. The structure of a topological space is introduced in
a standard way.
19
Fourier transform:
𝒮𝑟ˆ ∋ 𝑓 7→ 𝑓ˆ = ℱ (𝑓 ) =
1
(2𝜋)𝑛/2
∫
𝑓 (x̂, 𝜁, 𝑟ˆ)𝑒−𝑖𝜁 p̂ 𝑑𝜁.
ℝ𝑛
We denote 𝒮ˆ𝑟ˆ = ℱ (𝒮𝑟ˆ). Proposition. 𝒮ˆ𝑟ˆ ⊂ ℍ̂(ℝ2𝑛 ) is a subalgebra.
Inverse Fourier transform:
𝒮ˆ𝑟ˆ ∋ 𝑓ˆ 7→ 𝑓 = ℱ −1 (𝑓ˆ) =
1
(2𝜋)𝑛/2
∫
𝑓ˆ(x̂, p̂)𝑒𝑖𝜁 p̂ 𝑑𝑝,
ℝ𝑛
where as usual, p̂ = 𝑝 + 𝑝ˆ.
Proposition. ℱ −1 ℱ = id𝒮𝑟ˆ and ℱ ℱ −1 = id𝒮ˆ𝑟ˆ .
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2𝑛
ˆ
The derivations ∂𝑗 still exist in ℍ(ℝ ). We use for them the notation
ˆ
∂𝑥ˆ𝑗 and ∂ˆ𝑝ˆ𝑗 .
Proposition. For any 𝑓 ∈ 𝒮𝑟ˆ and 𝑓ˆ = ℱ (𝑓 )
∂ˆ𝑝ˆ𝑗 𝑓ˆ = ℱ (−𝑖𝜁𝑗 𝑓 ),
𝑖𝑓ˆp̂𝑗 = ℱ (∂𝑓 /∂𝜁𝑗 ).
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For any 𝑓 = 𝑓 (𝜂, 𝜁, 𝑟ˆ), 𝑔 = 𝑔(𝜂, 𝜁, 𝑟ˆ) ∈ 𝒮𝑟ˆ and 𝑓ˆ = 𝑓ˆ(x̂, p̂), 𝑔ˆ =
𝑔ˆ(x̂, p̂) ∈ 𝒮ˆ𝑟ˆ we define the convolutions
∫
1
𝑓 ∗𝑔 =
𝑓 (x̂, 𝜁 − 𝜌, 𝑟ˆ) 𝑔(x̂ + 𝑖ˆ
𝑟𝜁, 𝜌, 𝑟ˆ) 𝑑𝜌,
(2𝜋)𝑛/2 ℝ𝑛
∫
𝑓ˆ ∗ 𝑔ˆ =
𝑔ˆ(x̂, 𝑃 ) 𝑓ˆ(x̂, p̂ − 𝑃 ) 𝑑𝑃.
ℝ𝑛
Proposition. Suppose that 𝑓ˆ = ℱ (𝑓 ) and 𝑔ˆ = ℱ (𝑔). Then
𝑓ˆ ∗ 𝑔ˆ = ℱ (𝑓 𝑔),
𝑓ˆ𝑔ˆ = ℱ (𝑓 ∗ 𝑔).
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Representation of 𝒮ˆ𝑟ˆ in 𝐿2 (ℝ𝑛 ).
For any 𝑓ˆ ∈ 𝒮ˆ𝑟ˆ we have:
𝑓ˆ =
1
(2𝜋)𝑛/2
∫
𝑓 (x̂, 𝜁, 𝑟ˆ)𝑒−𝑖𝜁 p̂ 𝑑𝜁.
ℝ𝑛
Therefore for 𝜓 ∈ 𝐿2 (ℝ𝑛 )
𝑓ˆ𝜓 =
1
(2𝜋)𝑛/2
∫
ℝ𝑛
𝑓 (𝑥, 𝜁, −𝑖ℏ)𝜓(𝑥 − ℏ𝜁) 𝑑𝜁.
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For any 𝑓, 𝑔 ∈ 𝒮𝑟ˆ and 𝑓ˆ, 𝑔ˆ ∈ 𝒮ˆ𝑟ˆ we define the scalar products
∫
⟨𝑓, 𝑔⟩ =
𝑓 (𝜂, 𝜁, 𝑟ˆ) 𝑔 ★ (𝜂, 𝜁, 𝑟ˆ) 𝑑𝜂𝑑𝜁,
𝑛
∫ℝ
ˆ
⟨𝑓 , 𝑔ˆ⟩ =
𝑓ˆ(x̂, p̂) 𝑔ˆ★ (x̂, p̂) 𝑑𝑥𝑑𝑝.
ℝ𝑛
Proposition. Suppose that 𝑓ˆ = ℱ (𝑓 ) and 𝑔ˆ = ℱ (𝑔). Then
⟨𝑓ˆ, 𝑔ˆ⟩ = ⟨𝑓, 𝑔⟩.
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By using these scalar products, we can consider the spaces of distributions 𝒮𝑟ˆ′ and 𝒮ˆ𝑟ˆ′ , corresponding to the spaces of “test functions” 𝒮𝑟ˆ and 𝒮ˆ𝑟ˆ.
The space 𝒮ˆ𝑟ˆ′ contains the algebra of non-commutative polynomials in
x̂ and p̂ (the Heisenberg algebra.)
Representation of 𝒮ˆ𝑟ˆ′ in 𝐿2 (ℝ𝑛 ) is given by the same formula.
Note that 𝒮ˆ𝑟ˆ′ ∕⊂ ℍ(ℝ2𝑛 ) and ℍ(ℝ2𝑛 ) ∕⊂ 𝒮ˆ𝑟ˆ′ .
25
A possible application
ˆ ∈ ℍ̂(ℝ2𝑛 ) is
Suppose that a quantum system with Hamiltonian 𝐻
integrable in the sense that there exist sufficiently regular independent
𝐹ˆ1 , . . . , 𝐹ˆ𝑛 ∈ ℍ̂(ℝ2𝑛 ) such that
ˆ = 0.
[𝐹ˆ𝑗 , 𝐹ˆ𝑘 ] = [𝐹ˆ𝑗 , 𝐻]
In particular, the corresponding classical Hamiltonian system, obtained by
means of the averaging homomorphism, is Liouville integrable.
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If the classical common integral levels are compact, there exist actionangle variables.
It turns out that there exist non-commutative action-angle variables
ˆ
𝐼, 𝜑,
ˆ i.e.,
ˆ 𝑖ˆ
(a) 𝐼ˆ = ℳ(𝐹ˆ , 𝑖ˆ
𝑟 ), 𝐹ˆ = 𝑊 (𝐼,
𝑟).
(b) 𝜑ˆ are angular (defined mod2𝜋).
(c) [𝐼ˆ𝑗 , 𝐼ˆ𝑘 ] = [𝜑ˆ𝑗 , 𝜑ˆ𝑘 ] = 0, [𝐼ˆ𝑗 , 𝜑ˆ𝑘 ] = 𝛿𝑗𝑘 .
27
Existence of such non-commutative action-angle variables has been
proven so far locally [T2007]: 𝐼ˆ𝑗 , 𝜑ˆ𝑗 ∈ ℍ̂(𝐷), where 𝐷 is a neighborhood of
any classical Liouville torus. The corresponding global result is probably
much more complicated, but under reasonable assumptions does not look
hopeless.
28
Suppose that 𝐼ˆ𝑗 , 𝜑ˆ𝑗 ∈ ℍ̂(ℝ2𝑛 ).
Suppose moreover that 𝐼ˆ𝑗 , 𝑒𝑖𝜑ˆ𝑗 correspond to some good (pseudodifferential) operators.
Then for any eigenfunction 𝜓 ∕≡ 0, 𝐹ˆ𝑗 𝜓 = 𝜆𝑗 𝜓 we have:
𝐹ˆ𝑗 𝜓𝑘 = Λ𝑗𝑘 𝜓𝑘 ,
(
)
Λ𝑗𝑘 = 𝑊𝑗 ℳ(𝜆, ℏ) + ℏ𝑘, ℏ .
𝜓𝑘 = 𝑒𝑖𝑘𝜑ˆ 𝜓,
𝑘 ∈ ℤ.
Hence, having one eigenfunction, we obtain many (infinitely many?
all?) eigenfunctions and eigenvalues.