Non-unitary Groups Research Articles

Special Representations of the Iwasawa Subgroups of Simple Lie Groups

In the paper, a family of representations of maximal solvable subgroups of the simple Lie groups O(p, q), U(p, q), and Sp(p, q), where 1 ≤ p ≤ q, is introduced. These subgroups are called the Iwasawa subgroups of the corresponding simple groups. The main property of these representations is the existence of nontrivial 1-cohomology with values in the representations. For groups of rank 1, the representations from this family are unitary; for ranks greater than 1, they are nonunitary. The paper continues a series of our previous papers and serves as an introduction to the theory of nonunitary current groups.

Normalized vacuum states in [formula omitted] supersymmetric Yang–Mills quantum mechanics with any gauge group

We study the question of existence and the number of normalized vacuum states in N=4 super-Yang–Mills quantum mechanics for any gauge group. The mass deformation method is the simplest and clearest one. It allowed us to calculate the number of normalized vacuum states for all gauge groups. For all unitary groups, # vac=1, but for the symplectic groups [starting from Sp(6) ], for the orthogonal groups [starting from SO(8)] and for all the exceptional groups, it is greater than one. We also discuss at length the functional integral method. We calculate the “deficit term” for some non-unitary groups and predict the value of the integral giving the “principal contribution”. The issues like the Born–Oppenheimer procedure to derive the effective theory and the manifestation of the localized vacua in the asymptotic effective wave functions are also discussed.

On the ϵ-ζ 1 transition in molecular oxygen physisorbed on graphite in an external magnetic field

Landau-type theory has been applied to the calculation of the symmetry change at the ϵ-ζ 1 phase transition in an O 2-on-graphite layer placed in an external magnetic field perpendicular to the surface of the substrate. As the high-temperature phase of the system can be described by a unitary or a nonunitary diperiodic magnetic group (C2′/m′2′/m′2/m), its irreducible representations or corepresentations, respectively, were applied in this calculation. It is shown that only one of the three collinear spin structures induced by the representations, i.e. a structure with the spins parallel to the magnetic field direction, is generated by the corepresentations. This seems to indicate that the description of the high-temperature phase of the system in the magnetic field in terms of a nonunitary group and, consequently, its corepresentations is not correct. New experimental studies determining the structure of the ϵ phase in external magnetic field are needed to verify this conclusion.

Construction of grand unified theories

Systematic treatment of the theoretical and aesthetic requirements employed in the construction of unifying groups and their representations is given. Formulating the extensive set of conditions and giving reasons for them we derive a much more compact class of unified models than the one already discussed in the literature. It is shown that most models based on nonunitary groups and those with exotic representations are excluded. On the other hand, the permissible class of unitary unifying models is large but well ordered, thus making them usefull for future applications.

The commensurate to incommensurate phase transitions in dysprosium and terbium crystals

Abstract The commensurate to incommensurate phase transitions at the Neel point in magnetic crystals from the paramagnetic phase of symmetry P63/mmc1′ to the antiferromagnetic phase are discussed on the basis of the Landau theory. Transitions induced by active representations connected with the wave vector k = vb 3, –0.5 ≤ v ≤ 0.5, v ≠ 0, are considered. It is demonstrated that spin structures with spins of constant or varying length can appear in the low-temperature phases. Two types of helical constant-spin-length structures are allowed by the Landau theory. One of these can describe the observed helical structures in Dy and Tb crystals. An example of spin structure with a varying spin length allowed by the Landau theory is given. The high-temperature paramagnetic phase can be described by a unitary or a non-unitary group. A sufficient condition for the equivalence of these two descriptions is derived.

THE FIELD-INDUCED PHASE TRANSITIONS IN FeBr2 CRYSTAL

The field-induced phase transitions in FeBr2-type crystal under hydrostatic pressure have been determined in terms of Landau's theory, with the paramagnetic phase described by a unitary or a non-unitary group. The sets of low-temperature symmetry groups implied by these two descriptions partly overlap.

Symmetry changes at the tricritical point in metamagnets

Abstract Landau-type theory of symmetry changes at the tricritical point of a metamagnet has been formulated and applied to two types of crystals. The high-temperature magnetically ordered phase of a metamagnet can be described by a non-unitary group or by a unitary group. In the calculation of symmetry changes these two descriptions imply the application of corepresentations or representations, respectively. Calculations of symmetry changes in FeCl2-type crystals and in rare-earth crystals with space-group symmetry P63/nunc (D4 6h) have been performed with both methods. In these calculations we deal with case (a) corepresentations for both crystal types. The two sets of low-temperature symmetry groups obtained for the same crystal with the use of Co-representations or representations, respectively, partially overlap, although different symmetry groups also appear. This means that descriptions of a high-temperature magnetically ordered crystal by non-unitary or unitary groups are nonequivalent. With an appropriate alteration of the coefficients in the thermodynamic potential this theory is valid for a line of continuous transitions.

Clebsch-Gordan coefficients for nonunitary groups

It is shown that calculating Clebsch-Gordan coefficients of a nonunitary group can be reduced to formulas containing only representations of the unitary subgroup and additional conditions due to the antiunitary symmetry. This is another example demonstrating that, in applications involving corepresentations of nonunitary groups, the analysis can be made mainly in terms of representations of its unitary part.

On the Decomposition of Direct Products of Irreducible Representations

A lemma concerning irreducible representations contained in the decomposition of a direct product of irreducible representations of simply reducible groups is generalized to arbitrary decomposable unitary and nonunitary groups.

Difficulty with a kinematic concept of unstable particles: the SZ.-Nagy extension and the Matthews-Salam-Zwanziger representation

We discuss the possibility of describing unstable systems, or dissipative systems in general, by vectors in a Hilbert space, evolving in time according to some non-unitary group or semigroup of translations. If the states of the unstable or dissipative system are embedded in a larger Hilbert space containing “decay products” as well, so that the time evolution of the system as a whole becomes unitary, we show that the infinitesimal generator necessarily has all energies from minus to plus infinity in its spectrum. This result supplements and extends the well-known fact that a positive energy spectrum is incompatible with a decay law bounded by a decreasing exponential. As an example of both facts, we discuss Zwanziger's irreducible, nonunitary representation of the Poincare group; and we find its minimal, unitary extension (the Sz.-Nagy construction). The answer provides a mathematically canonical approach to the Matthews-Salam theory of wave functions for unstable, elementary particles, where the spectrum difficulty was already recognized. We speculate on the possibility that the Matthews-Salam-Zwanziger representation might be a strong coupling approximation in the relativistic version of the Wigner-Weisskopf theory, but we have not shown the existence of a physically acceptable model where that is so.

Scattering matrices for the Raman effect in magnetic crystals

The symmetry of a magnetic crystal can be described by one of the black and white Shubnikov point groups containing both unitary and anti-unitary elements. The theory is reviewed that enables a tensor to be symmetry adapted to any one of the irreducible corepresentations of a non-unitary group and it is applied to the determination of scattering matrices for the Raman effect in magnetically ordered crystals. An example is given in detail for the magnetic point group of antiferromagnetic MnF2 and complete tables are presented for all the fifty-eight black and white magnetic point groups.

Matrix Elements in Systems with Nonunitary Symmetry

The Eckart-Wigner theorem is generalized to include nonunitary groups. The proof is based on the connection between corepresentations of a nonunitary group and the representations of its unitary part. All possible cases of the corepresentations have been considered, and general expressions for matrix elements of operators with given symmetry have been obtained. It has been shown that the antiunitary symmetry leads, in general, to additional connections between different matrix elements.

Crystal field theory and the Shubnikov point groups

Abstract Two pieces of theory which have so far remained unconnected, crystal field theory and the theory of corepresentations of non-unitary groups, are brought together here for the study of the splitting of atomic energy levels in a crystalline field with the symmetry of one of the magnetic (Shubnikov) point groups. The cases of the various possible relative strengths of the crystalline field and of spin-orbit coupling are considered. Tables are presented which enable the splitting of any atomic energy level to be obtained very easily in a crystalline field with the symmetry of any one of the 58 magnetic point groups. Examples of the use of these tables are given. A discussion is given of the relevance of Kramers' theorem to the energy levels of electrons in surroundings with the symmetry of any one of the 58 magnetic point groups.

Mass-Splitting Theorem for Translationally Invariant Nonintegrable Lie Algebras

In a previous article, a rigorous theorem prohibiting the derivation of stable baryon mass spectra from unitary representations of Lie groups was generalized to the case of nonunitary group representations. In the present note, the generalization is taken a stage further; namely, it is shown that, subject to some physically mild conditions concerning translational invariance and closure, an analogous theorem can be established for the case of nonintegrable representations of Lie algebras.

Mass-Splitting Theorem for Non-Unitary Group Representations

It is shown that a rigorous theorem on the impossibility of obtaining a stable hadron mass spectrum from unitary representations of Lie groups due to Jost and Segal can be generalized to the case of (nonuniformly) bounded group representations. The only link with unitarity that need be retained is the assumption that the mass-squared operator is self-adjoint.

Double-valued Corepresentations of Magnetic Point Groups

The theory of the corepresentations of non-unitary groups, which was developed by Wigner, is applied here to the determination of the double-valued corepresentations of the magnetic point groups. Complete tables are given for the double-valued corepresentations of each of the 58 magnetic point groups. A comparison is made with previous work by Dimmock and Wheeler.

Short Communications Crystal Field Theory and Magnetic Groups: Spin?Orbit Coupling

Some initial work has recently been done (Cracknell 1966a) on the applicationof the theory of the corepresentations of non-unitary groups to the consideration ofthe splitting of atomic energy levels in a crystalline field possessing the mmetryof one of the magnetic point groups, or Shubnikov point groups. In that work onlythe situation of a strong rystalline field in the complete absence of spin

Ray Representations of Finite Nonunitary Groups

An extension of the ray representation theory is formulated to embrace nonunitary groups. The coray representations are obtained by the ray representations of its unitary subgroup. Theorems of coray representations are stated. The usefulness of the formalism is discussed.

Corepresentations of Magnetic Point Groups

The well-established theory due to Wigner of the corepresentations of non-unitary groups is here applied to the magnetic point groups and the results are tabulated. The equivalence of the theory of the corepresentations of the grey groups to the conventional treatment of time-reversal symmetry in space groups is also considered.

Crystal Field Theory And Magnetic Point Groups

The author's previous work on the application of Wigner's theory of the coreps of non-unitary groups to the Shubnikov groups (magnetic groups) is here considered in relation to crystal field theory. Both the splitting of the energy levels and the symmetry properties of the wave function are considered in magnetic point groups. Examples of 4'mm' and 4m'm' are studied.