Wintner-Wielandt's theorem

The set of Wintner-Wielandt is a mathematical theorem from the theory of linear operators , a section of the functional analysis , the close ties to the theoretical physics has. In its original version, it goes back to Aurel Wintner (1903–1958) and Helmut Wielandt (1910–2001) and provides information on the extent to which the quantum mechanical basic operators , which are linked to Heisenberg's uncertainty relation, exist as limited operators .

In connection with Wintner-Wielandt's theorem, a number of further investigations were carried out.

Formulation of the sentence

The sentence can be formulated as follows:

Given a normalized vector space and the normalized algebra of the bounded linear operators of , provided with the operator norm . The identity operator by going with designated. ${\displaystyle X}$ ${\displaystyle L(X)}$ ${\displaystyle X}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle X}$ ${\displaystyle \mathbf {1} _{X}}$

For two linear operators and on and a ( real or complex ) scalar , (H ) is understood to mean the following equation ( Heisenberg's commutation relation): ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle X}$ ${\displaystyle a}$

(H)   ${\displaystyle [P,Q]=a\mathbf {1} _{X}}$

Then:

Equation (H) can be fulfilled if and only if is, that is if and only if and are interchangeable . ${\displaystyle a=0}$ ${\displaystyle P}$ ${\displaystyle Q}$

proof

Wintner has provided a proof with the help of spectral theory .

Wielandt gave another and more general, but more easily accessible, proof. Wielandt's proof can be presented as follows:

I: Extension of Heisenberg's exchange relation

Because of this, Heisenberg's exchange relation can be extended to the following identity for each : ${\displaystyle [P,Q]=PQ-QP}$ ${\displaystyle n\in \mathbb {N} }$

(H1)   ${\displaystyle PQ^{n}-Q^{n}P=naQ^{n-1}}$

This results from complete induction :

Induction start :

The induction start for gives (H) itself. ${\displaystyle n=1}$

Induction step : ${\displaystyle n\to n+1}$

${\displaystyle PQ^{n+1}-Q^{n+1}P=(PQ^{n}-Q^{n}P)Q+Q^{n}(PQ-QP)}$  

With the induction hypothesis, the following results are obtained by inserting :

${\displaystyle PQ^{n+1}-Q^{n+1}P=(naQ^{n-1})Q+Q^{n}(a\mathbf {1} _{X})=naQ^{n}+aQ^{n}}$

So it follows:

${\displaystyle PQ^{n+1}-Q^{n+1}P=(n+1)aQ^{n}}$

II: Actual proof of contradiction

The assumption of contradiction is now taken as given. ${\displaystyle a\neq 0}$

Then it follows with (H) that the null operator can not be, and because of (H1) this applies to each and every in the same way. ${\displaystyle Q}$ ${\displaystyle Q^{n}}$ ${\displaystyle Q^{n-1}}$

On the other hand, the following estimate is obtained from (H1) for each : ${\displaystyle n\in \mathbb {N} }$

${\displaystyle n\cdot |a|\cdot \|Q^{n-1}\|=\|PQ^{n}-Q^{n}P\|=\|PQ^{n}+(-1)Q^{n}P\|\leq \|P\|\cdot \|Q^{n}\|+\|Q^{n}\|\cdot \|P\|=2\cdot \|P\|\cdot \|Q^{n}\|}$

So further:

${\displaystyle n\cdot |a|\cdot \|Q^{n-1}\|\leq 2\cdot \|P\|\cdot \|Q^{n-1}Q\|}$

So finally:

${\displaystyle n\cdot |a|\cdot \|Q^{n-1}\|\leq 2\cdot \|P\|\cdot \|Q^{n-1}\|\cdot \|Q\|}$

Now you can divide by and get for each : ${\displaystyle |a|\cdot \|Q^{n-1}\|}$ ${\displaystyle n\in \mathbb {N} }$

(H2)   ${\displaystyle n\leq {\tfrac {2\cdot \|P\|\cdot \|Q\|}{|a|}}}$

With (H2) one arrives at a contradiction, as desired, because the set of natural numbers has no upper bound within the real numbers . ${\displaystyle \mathbb {N} }$

III: Graduation

It must therefore apply. But this means that the zero operator is, which is synonymous with . ${\displaystyle a=0}$ ${\displaystyle [P,Q]}$ ${\displaystyle PQ=QP}$

Connection with the quantum mechanical basic operators

The set of Wintner-Wielandt implies that the quantum mechanical ground operators can not all be limited, so discontinuous need to be. In particular, the Hilbert spaces of quantum mechanics cannot be of finite dimension .

Furthermore, it has been proven that in the case of the validity of (H) the scalar must always be purely imaginary , i.e. without a real part , whereby the prerequisite is that (H) is meaningful at all.

generalization

As the proof shows, the statement of Wintner-Wielandt's theorem is equally valid for every normalized algebra with one element .

literature

Original work

• Arlen Brown and Carl Pearcy: Structure theorem for commutators of operators . In: Bull. Amer. Math. Soc . tape 70 , 1964, pp. 779–780 ( ams.org [PDF; 181 kB ]). MR0167847
• Paul R. Halmos : Commutators of Operators . In: Amer. J. Math . tape 74 , 1952, pp. 237-240 , JSTOR : 2372081 ( MR0045310 ).
• Helmut Wielandt : About the unlimited number of operators in quantum mechanics . In: Math. Ann . tape 121 (1949-1950) , pp. 21 ( MR0030701 ).
• Aurel Wintner : The unboundedness of quantum-mechanical matrices . In: Physical Rev . tape 71 , 1947, pp. 738-739 ( MR0020724 ).

Monographs

• Lothar Collatz : Functional Analysis and Numerical Mathematics . Unchanged reprint of the 1st edition from 1964 (=  The Basic Teachings of Mathematical Sciences in Individual Representations . Volume 120 ). 2nd Edition. Springer-Verlag , Berlin, Heidelberg, New York 1968, ISBN 3-540-04135-4 ( MR0165651 ).
• Paul Halmos: A Hilbert Space Problem Book (=  Graduate Texts in Mathematics . Volume 19 ). Springer-Verlag , Berlin [a. a.] 1982, ISBN 3-540-90090-X ( MR0675952 ).
• Harro Heuser : Functional Analysis . Theory and application (=  mathematical guidelines ). 4th revised edition. Teubner Verlag, Wiesbaden 2006, ISBN 978-3-8351-0026-8 ( MR2380292 ).
• Johann v. Neumann : Mathematical foundations of quantum mechanics. Unchanged reprint of the 1st edition from 1932 (=  The Basic Teachings of the Mathematical Sciences in Individual Representations . Volume 38 ). Springer-Verlag , Berlin [a. a.] 1968, ISBN 3-540-04133-8 ( MR0223138 ).
• Joachim Weidmann : Linear operators in Hilbert spaces. Part 1: Basics (=  mathematical guidelines ). Teubner Verlag, Stuttgart [u. a.] 2000, ISBN 3-519-02236-2 ( MR1887367 ).

Web links

• Cape. 5 of the script for the lecture Mathematical Methods in Quantum Mechanics ( Memento from May 7, 2005 in the Internet Archive ) (SS 2001; PDF; 91 kB), TU Munich

Footnotes and individual references

1. ↑ Wintner: Physical Rev . tape 71 .
2. ↑ ^{a b c} Wielandt: Math. Ann . tape 121 .
3. ↑ ^{a b c} Collatz, pp. 77-79.
4. ↑ ^{a b} Heuser, p. 102.
5. ↑ Halmos, pp. 126-127, 333.
6. ↑ Where is the so-called commutator of the two operators and . ${\displaystyle [P,Q]=PQ-QP}$ ${\displaystyle P}$ ${\displaystyle Q}$
7. ↑ Halmos, p. 333.
8. ↑ Halmos, p. 126, describes the two proofs as two beautiful proofs .
9. ↑ In an operator algebra , for the successive execution of two operators and for reasons of clarity, one often writes instead . ${\displaystyle P\colon X\to X}$ ${\displaystyle Q\colon X\to X}$ ${\displaystyle PQ}$ ${\displaystyle P\circ Q}$
10. ↑ Here is observed. ${\displaystyle Q^{0}=\mathbf {1} _{X}}$
11. ↑ Because after multiplying the two middle terms cancel each other out.
12. ↑ This is shown starting from (H1) with the help of another induction proof .
13. ↑ Read from right to left!
14. ↑ Since is not the null operator, then holds . ${\displaystyle Q^{n-1}}$ ${\displaystyle |a|\cdot \|Q^{n-1}\|\neq 0}$
15. ↑ v. Neumann, p. 123.
16. ↑ Halmos, p. 126.