Lie product formula

The Lie product formula or Lieche product formula , named after Sophus Lie , is a formula for calculating the value of the exponential function of a sum of two square matrices . Because of later generalizations by Hale Trotter , one also speaks of the Trotter product formula or Lie-Trotter product formula .

The product formula

Let and be two square matrices of equal size above or , say, matrices. Then you can sum them up. Furthermore, one can define the exponential function of square matrices by inserting them into the exponential series , that is ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle n\times n}$ ${\displaystyle A+B}$

${\displaystyle e^{A}:=\sum _{k=0}^{\infty }{\frac {A^{k}}{k!}}}$.

The limit value of the row is formed in the space of the matrices, i.e. component by component. With these data, the Lie product formula applies: ${\displaystyle n\times n}$

${\displaystyle e^{A+B}=\lim _{N\to \infty }\left(e^{A/N}e^{B/N}\right)^{N}}$.

comment

The formula known from analysis for the exponential function of a sum only applies to a limited extent:

${\displaystyle e^{A+B}=e^{A}e^{B}}$   if    , ${\displaystyle AB=BA}$

for if the matrices and interchange, one can use the proof of the formula known from analysis ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle e^{x+y}=e^{x}e^{y}}$   for all    ( stands for or ) ${\displaystyle x,y\in \mathbb {K} }$ ${\displaystyle \mathbb {K} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

just copy. The Lie product formula generalizes this situation, because if and swap, then swap and and you get ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \textstyle e^{A/N}}$ ${\displaystyle \textstyle e^{B/N}}$

${\displaystyle \left(e^{A/N}e^{B/N}\right)^{N}=\left(e^{A/N}\right)^{N}\left(e^{B/N}\right)^{N}=e^{A}e^{B}}$.

application

If a closed subgroup of the general linear group is defined, the associated Lie algebra is defined by ${\displaystyle G}$ ${\displaystyle \mathrm {GL} (n,\mathbb {K} )}$

${\displaystyle L(G):=\{A\in \mathrm {M} _{n}(\mathbb {K} )\mid e^{tA}\in G{\text{ für alle }}t\in \mathbb {R} \}}$.

With this definition it is not even clear that the set so defined is a vector space at all . It is only obvious that with and also is, because it is for everyone exactly when for everyone . In order to show that the addition is also complete, the above Lie product formula is used as follows. ${\displaystyle L(G)}$ ${\displaystyle a\in \mathbb {R} \setminus \{0\}}$ ${\displaystyle A\in L(G)}$ ${\displaystyle aA\in L(G)}$ ${\displaystyle e^{tA}\in G}$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle e^{taA}\in G}$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle L(G)}$

Be . Then by definition it is for everyone . Since there is a group, follows for all . Since is complete, it also contains the limit value for , and according to the Lie product formula, that leads to and therefore also for everyone . But that means by definition . ${\displaystyle A,B\in L(G)}$ ${\displaystyle \textstyle e^{tA/N},e^{tB/N}\in G}$ ${\displaystyle t\in \mathbb {R} ,N\in \mathbb {N} }$ ${\displaystyle G}$ ${\displaystyle \textstyle \left(e^{tA/N}e^{tB/N}\right)^{N}\in G}$ ${\displaystyle t\in \mathbb {R} ,N\in \mathbb {N} }$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle N\to \infty }$ ${\displaystyle \textstyle e^{tA+tB}\in G}$ ${\displaystyle \textstyle e^{t(A+B)}\in G}$ ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle A+B\in L(G)}$

Generalizations

The Lie product formula holds more generally in any Banach algebras with one element . The exponential function of an element of the Banach algebra can again be defined via the exponential series. If and are elements of such a Banach algebra, then: ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle e^{A+B}=\lim _{N\to \infty }\left(e^{A/N}e^{B/N}\right)^{N}}$

In particular, the formula applies to all bounded operators on a Hilbert space. On infinite-dimensional Hilbert spaces, however, certain unbounded operators, the densely-defined, self-adjoint operators with a domain, are of particular interest. For these can by means of the Spectral the unitary operators are formed. Then the following generalization, which goes back to Hale Trotter and is called the Trotter product formula or Lie-Trotter product formula , applies : ${\displaystyle A}$ ${\displaystyle D(A)}$ ${\displaystyle \textstyle e^{\mathrm {i} tA},\,t\in \mathbb {R} }$

Let and be two self-adjoint operators with domains of definition or in a complex Hilbert space, so that on is essentially self-adjoint . Then applies ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle D(A)}$ ${\displaystyle D(B)}$ ${\displaystyle A+B}$ ${\displaystyle D(A)\cap D(B)}$

${\displaystyle e^{\mathrm {i} t(A+B)}=s-\lim _{N\to \infty }\left(e^{\mathrm {i} tA/N}e^{\mathrm {i} tB/N}\right)^{N}}$.

Here, convergence with respect to the strong operator topology means , that is, if both sides of the formula are applied to a fixed vector of the Hilbert space, then norm convergence is present. ${\displaystyle s-\lim }$

Since they are unitary and therefore have the norm 1 as bounded operators , we are dealing with the prototype of a contractive, strongly continuous semigroup on a Banach space . The Lie-Trotter product formula can be generalized to this situation as follows, where the -th semigroup operator denotes the generator : ${\displaystyle \textstyle e^{\mathrm {i} tA}}$ ${\displaystyle \textstyle e^{tA}}$ ${\displaystyle t}$ ${\displaystyle A}$

Be , and producers contractionary, continuous semigroups on a Banach space , and it is for all of a substantial range of . Then applies ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle X}$ ${\displaystyle Cx=Ax+Bx}$ ${\displaystyle x}$ ${\displaystyle C}$

${\displaystyle e^{tC}=s-\lim _{N\to \infty }\left(e^{tA/N}e^{tB/N}\right)^{N}}$   for all    , ${\displaystyle t\in \mathbb {R} _{\geq 0}}$

this means

${\displaystyle e^{tC}x=\lim _{N\to \infty }\left(e^{tA/N}e^{tB/N}\right)^{N}x}$   for everyone    and ${\displaystyle x\in X}$ ${\displaystyle t\in \mathbb {R} _{\geq 0}}$

Individual evidence

1. ^ Brian C. Hall: Lie Groups, Lie Algebras, and Representations , Springer-Verlag 2003, ISBN 0-387-40122-9 , Theorem 2.10
2. ↑ Rajemdra Bathia: Matrix Analysis , Springer-Verlag 1997, ISBN 0-387-94846-5 , Theorem IX.1.3
3. ↑ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , Lemma I.2.13
4. ↑ Joachim Hilgert, Karl-Hermann Neeb: Lie groups and Lie algebras , Vieweg, 1999, ISBN 3-528-06432-3 , proof of Proposition I.3.1
5. ↑ C. Touré, F. Schulz, R. Brits: Some character generating functions on Banach algebras , Theorem 1.6
6. ^ Piotr Soltan: A Primer on Hilbert Space Operators , Birkhäuser-Verlag, ISBN 978-3-319-92060-3 , Theorem 4.17
7. ↑ Barry Simon : Functional Integration and Quantum Physics , Academic Press Inc 1979, ISBN 0-12-644250-9 , Theorem 1.1.
8. ↑ EB Davies: One Parameter Semigroups , Academic Press 1980, ISBN 0-12-206280-9 , Theorem 3.30
9. ↑ HF Trotter: On the product of semi-groups of operators , Proc. Amer. Math. Soc. (1959), Volume 10, pages 545-551