This article needs editing to comply with Wikipedia's Manual of Style. In particular, it has problems with MOS:BBB. Please help improve the content. (February 2024) (Learn how and when to remove this message)

The Schrödinger group is the symmetry group of the free particle Schrödinger equation. Mathematically, the group SL(2,R) acts on the Heisenberg group by outer automorphisms, and the Schrödinger group is the corresponding semidirect product.

Schrödinger algebra

The Schrödinger algebra is the Lie algebra of the Schrödinger group. It is not semi-simple. In one space dimension, it can be obtained as a semi-direct sum of the Lie algebra sl(2,R) and the Heisenberg algebra; similar constructions apply to higher spatial dimensions.

It contains a Galilei algebra with central extension.

${\displaystyle =i\epsilon _{abc}J_{c},\,\!}$

${\displaystyle =i\epsilon _{abc}P_{c},\,\!}$

${\displaystyle =i\epsilon _{abc}K_{c},\,\!}$

${\displaystyle =0,=0,=i\delta _{ab}M,\,\!}$

${\displaystyle H,J_{a}=0,H,P_{a}=0,H,K_{a}=iP_{a}.\,\!}$

where ${\displaystyle J_{a},P_{a},K_{a},H}$ are generators of rotations (angular momentum operator), spatial translations (momentum operator), Galilean boosts and time translation (Hamiltonian) respectively. (Notes: ${\displaystyle i}$ is the imaginary unit, ${\displaystyle i^{2}=-1}$. The specific form of the commutators of the generators of rotation ${\displaystyle J_{a}}$ is the one of three-dimensional space, then ${\displaystyle a,b,c=1,\ldots ,3}$.). The central extension M has an interpretation as non-relativistic mass and corresponds to the symmetry of Schrödinger equation under phase transformation (and to the conservation of probability).

There are two more generators which we shall denote by D and C. They have the following commutation relations:

${\displaystyle H,C=iD,C,D=-2iC,H,D=2iH,\,\!}$

${\displaystyle P_{a},D=iP_{a},K_{i},D=-iK_{a},\,\!}$

${\displaystyle P_{a},C=-iK_{a},K_{a},C=0,\,\!}$

${\displaystyle J_{a},C=J_{a},D=0.\,\!}$

The generators H, C and D form the sl(2,R) algebra.

A more systematic notation allows to cast these generators into the four (infinite) families ${\displaystyle X_{n},Y_{m}^{(j)},M_{n}}$ and ${\displaystyle R_{n}^{(jk)}=-R_{n}^{(kj)}}$, where n ∈ ℤ is an integer and m ∈ ℤ+1/2 is a half-integer and j,k=1,...,d label the spatial direction, in d spatial dimensions. The non-vanishing commutators of the Schrödinger algebra become (euclidean form)

${\displaystyle X_{n},X_{n'}=(n-n')X_{n+n'}}$

${\displaystyle X_{n},Y_{m}^{(j)}=\left({n \over 2}-m\right)Y_{n+m}^{(j)}}$

${\displaystyle X_{n},M_{n'}=-n'M_{n+n'}}$

${\displaystyle X_{n},R_{n'}^{(jk)}=-n'R_{n'}^{(jk)}}$

${\displaystyle Y_m^{(j)},Y_{m^{\prime }}^{}}$Zdroj:https://en.wikipedia.org?pojem=Schrödinger_group
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---