Spectrum generating algebra

dynamical symmetry (in quantum mechanics)

An enveloping algebra of a Lie algebra (or other group-like algebraic structure, cf. also Universal enveloping algebra) under which the Hilbert space $ {\mathcal H} $ of the quantum system forms a (unitary) irreducible representation. Typically, the Hamiltonian $ H $ of the quantum system coincides with the action of a non-central element of the spectrum generating algebra.

The concept should not be confused with the notion of a conserved symmetry algebra $ G $, which is the enveloping algebra of a Lie algebra (or other group-like structure) acting on the Hilbert space of the quantum system and commuting with the Hamiltonian. In the ideal case, the Hamiltonian is the action of a central element (a Casimir or Casimir element) of $ G $. In quantum mechanics, the eigenvalues of the Hamiltonian are the energy levels of the quantum system and hence these, and their corresponding eigenspaces, can be obtained by a decomposition (assuming this is possible) into irreducible representations of $ G $. The values of the Casimir in the occurring irreducibles are the energy levels. A spectrum generating algebra typically extends the symmetry to a larger algebra, the additional generators of which map from one energy eigenspace to another. The relevant transition operators can typically be expressed in terms of the action of these additional generators, hence transition amplitudes can be computed by group theory as the matrix elements in the representation of the spectrum generating algebra consisting of the whole Hilbert space.

The concept has been used particularly in molecular and nuclear physics. In molecular physics the various vibrational and rotational modes of the molecule are directly observable through the spectrum of the Hamiltonian. The formalism of quantum mechanics is used as a language for building an effective model of the molecule to fit the spectrum (rather than as a fundamental process). In this context it is useful to look for a chain of subalgebras

$$ \textrm{ spectrum generating algebra } = G _ {0} \supset G _ {1} \supset G _ {2} \supset \dots $$

and successively refine the Hilbert space by pulling back to, and decomposing into irreducibles of, each of the $ G _ {n} $ in the chain. One decomposes $ {\mathcal H} $ into irreducibles of $ G _ {1} $, and each of these into irreducibles of $ G _ {2} $, etc. In this way, $ {\mathcal H} $ is typically decomposed into a basis of one-dimensional irreducibles of the last subalgebra in the chain, labelled uniquely by the chain of successive irreducible representations to which a basis element belongs. The labels are called quantum numbers and are usually expressed through the values of the Casimirs $ \{ C _ {n, \alpha } \} $ of the successive subalgebras $ G _ {n} $. Moreover, the Hamiltonian is typically taken in the form

$$ H = \mu _ {0} + \sum _ {n > 0, \alpha } \mu _ {n, \alpha } C _ {n, \alpha } $$

for some coefficients $ \mu _ {0} , \mu _ {n, \alpha } $, and hence has a diagonal form in this basis. Thus, the form of the spectrum of energy eigenvalues is again determined by group theory. A special case is when only one $ n $, say $ n = N $, contributes to the Hamiltonian, in which case $ G = G _ {N} $ is a conserved symmetry. All the $ G _ {n} $, $ n < N $, can be called spectrum generating algebras, although, by definition, only $ G _ {0} $ is a spectrum generating algebra in the strict sense of acting irreducibly.

A simple example is provided by the rigid rotator. This is a model of a di-atomic (dumbbell-shaped) molecule where only the rotation of the dumbbell about its centre of mass is considered (vibrational modes along the dumbbell are ignored). The Hilbert space and Hamiltonian are:

$$ {\mathcal H} = \oplus _ {j} V _ {j} , $$

$$ H = ( 2I ) ^ {- 1 } \sum _ {i = 1 } ^ { 3 } J _ {i} ^ {2} , $$

where $ V _ {j} $ is the $ 2j + 1 $- dimensional representation of the Lie algebra $ { \mathop{\rm so} } _ {3} $, and $ J _ {i} $ are operators representing the generators of $ { \mathop{\rm so} } _ {3} $ with the relations

$$ [ J _ {i} ,J _ {j} ] = i \sum _ { k } \epsilon _ {ijk } J _ {k} , $$

where $ \epsilon _ {ijk } $ is totally anti-symmetric with $ \epsilon _ {123 } = 1 $. The constant $ I $ is the moment of inertia of the molecule. The $ J _ {i} $ correspond to the angular momentum. The subalgebra $ { \mathop{\rm so} } _ {2} $ corresponding to $ J _ {3} $ is chosen to further label a quantum state according to the eigenvalue of $ J _ {3} $. On the other hand, the entire Hilbert space $ {\mathcal H} $ is a standard irreducible representation space for $ { \mathop{\rm so} } ( 3,1 ) $, the enveloping algebra of which may be taken as the spectrum generating algebra. Actually, for the present interpretation, one should rather take a contraction limit of $ { \mathop{\rm so} } ( 3,1 ) $ to the semi-direct sum Lie algebra $ e ( 3 ) = \mathbf R ^ {3} \lcs { \mathop{\rm so} } _ {3} $. Then the additional generators are represented by operators $ x _ {i} $ obeying

$$ [ x _ {i} ,x _ {j} ] = 0, [ J _ {i} ,x _ {j} ] = i \sum _ { k } \epsilon _ {ijk } x _ {k} . $$

The reduction chain at the level of the underlying Lie algebras is then $ e ( 3 ) \supset { \mathop{\rm so} } _ {3} \supset { \mathop{\rm so} } _ {2} $. The energy spectrum in this model is

$$ \left \{ { { \frac{j ( j + 1 ) }{2I } } } : {j \in \mathbf N } \right \} , $$

as the set of eigenvalues of $ H $, with eigenspaces $ V _ {j} $. One has $ { \mathop{\rm so} } _ {3} $ as conserved symmetry, while the operators $ x _ {i} $ do not commute with $ H $ and hence mix the $ V _ {j} $.

More interesting "vibron modelvibron models" include the vibrational modes as well, and have spectrum generating algebra $ { \mathop{\rm u} } _ {4} $, the Lie algebra of $ { \mathop{\rm U} } _ {4} $. The reduction chain $ { \mathop{\rm u} } _ {4} \supset { \mathop{\rm so} } _ {4} \supset { \mathop{\rm so} } _ {3} \supset { \mathop{\rm so} } _ {2} $ can be used to model the hydrogen molecule (cf. also Hydrogen-like atom). A similar "interacting boson modelinteracting boson model" in nuclear physics has spectrum generating algebra given by $ { \mathop{\rm u} } _ {6} $ and a typical reduction chain $ { \mathop{\rm u} } _ {6} \supset { \mathop{\rm su} } _ {3} \supset { \mathop{\rm so} } _ {3} \supset { \mathop{\rm so} } _ {2} $. Also, a version of the above rotator with spectrum generating algebra $ { \mathop{\rm so} } ( 3,1 ) $ can be considered one of the origins of string theory.

A more fundamental point of view is to consider the underlying algebra of observables $ {\mathcal A} $ of the quantum system. The spectrum generating algebra and its subalgebras may be realized in $ {\mathcal A} $ itself by means of an algebra homomorphism

$$ \pi : {\textrm{ spectrum generating algebra } } \rightarrow {\mathcal A} . $$

This induces an inner "adjoint" action of the spectrum generating algebra and its subalgebras on $ {\mathcal A} $. The formalism can also be extended to allow these as outer automorphism of $ {\mathcal A} $. Moreover, any concrete irreducible representation of the system on a Hilbert space can then be pulled back to one of the spectrum generating algebra, and should be irreducible as such (if not, then the proposed spectrum generating algebra needs, strictly speaking, to be enlarged). A sufficient condition for this is that $ \pi $ be surjective.

The simplest example here is obtained by taking $ {\mathcal A} $ the quantum harmonic oscillator generated by $ a,a ^ \dag $ with relations $ [ a,a ^ \dag ] = 1 $. Its spectrum generating algebra is the enveloping algebra of the Lie superalgebra $ { \mathop{\rm osp} } ( 1 \mid 2 ) $( cf. also Superalgebra) defined by generators $ J _ {i} $, $ V _ \pm $ of degrees $ 0 $, $ 1 $, respectively, and Lie superbracket

$$ [ J _ {3} ,V _ \pm ] = \pm { \frac{1}{2} } V _ \pm , $$

$$ \{ V _ {+} ,V _ {-} \} = - { \frac{1}{2} } J _ {3} , $$

$$ [ J _ \pm ,V _ \mps ] = \pm V _ \pm , $$

$$ [ J _ \pm ,V _ \pm ] = 0, \{ V _ \pm ,V _ \pm \} = \pm { \frac{1}{2} } J _ \pm , $$

along with the Lie bracket of $ { \mathop{\rm so} } ( 2,1 ) $. Here, $ J _ \pm $ are the usual linear combinations of $ \mps { \frac{( J _ {1} \pm i J _ {2} ) }{\sqrt 2 } } $. The required homomorphism is defined by

$$ \pi ( V _ {+} ) = { \frac{a ^ \dag }{\sqrt 8 } } , \quad \pi ( V _ {-} ) = - { \frac{a}{\sqrt 8 } } . $$

The usual Fock representation $ {\mathcal H} $ of $ {\mathcal A} $ with basis $ \{ | n \rangle = { {( a ^ \dag ) ^ {n} } / {\sqrt {n! } } } | 0 \rangle \} $( cf. also Fock space) pulls back to an irreducible representation of $ { \mathop{\rm osp} } ( 1 \mid 2 ) $, the metaplectic one. The reduction chain is $ { \mathop{\rm osp} } ( 1 \mid 2 ) \supset { \mathop{\rm so} } ( 2,1 ) \supset { \mathop{\rm so} } _ {2} $, where $ {\mathcal H} $ decomposes into two irreducibles of $ { \mathop{\rm so} } ( 2,1 ) $, namely those containing states of odd and even $ n $. Here, $ \pi ( 2J _ {3} ) = a ^ \dag a + {1 / 2 } $ is the Hamiltonian.

Similar ideas can be used to solve the radial equation for the hydrogen atom. After separating out the angular dependence, the Schrödinger equation becomes the eigenvalue problem $ H \psi ( r ) = \lambda \psi ( r ) $ for the operator

$$ H = - { \frac{1}{2} } \nabla ^ {2} + { \frac{a}{2 r ^ {2} } } - { \frac{b}{r} } , $$

$$ \nabla = { \frac \partial {\partial r } } + { \frac{1}{r} } , $$

where $ a $, $ b $ are parameters. In the algebra generated by $ \nabla $ and $ r $, $ r ^ {- 1 } $, there is the realization

$$ \pi ( 2i J _ {1} ) = r \nabla ^ {2} - { \frac{a}{r} } + r, $$

$$ \pi ( 2J _ {3} ) = - r \nabla ^ {2} + { \frac{a}{r} } + r, $$

$$ \pi ( J _ {2} ) = r \nabla. $$

This yields an irreducible representation of $ { \mathop{\rm so} } ( 2,1 ) $ on a suitable Hilbert space of functions. On the other hand,

$$ \pi ( ( 1 - 2 \lambda ) J _ {3} - ( 1 + 2 \lambda ) i J _ {1} - 2b ) = 2r ( H - \lambda ) , $$

with the result that the problem can be solved by group theory without recourse to the particular differential operators. This is the modern version of Pauli's original algebraic solution of the hydrogen atom. It may be considered an origin of the concept of spectrum generating algebra.

As well as the enveloping algebras of (typically non-compact) Lie algebras and Lie superalgebras, one may also use infinite-dimensional Kac–Moody Lie algebras (cf. Kac–Moody algebra) and quantum groups as spectrum generating algebras.

References

[a1]	F. Iachello, A. Arima, "The interacting boson model" , Cambridge Univ. Press (1987)
[a2]	M. de Crombrugghe, V. Rittenberg, "Supersymmetric quantum mechanics" Ann. Phys. , 151 (1983) pp. 99–126
[a3]	A. Bohm, Y. Ne'eman, A.O. Barut, "Dynamical groups and spectrum generating algebras" , I , World Sci. (1988)