Antiunitary Research Articles

Chow's theorem for Hilbert Grassmannians as a Wigner-type theorem

Let H be an infinite-dimensional complex Hilbert space. Denote by G ∞ ( H ) the Grassmannian formed by closed subspaces of H whose dimension and codimension both are infinite. We say that X , Y ∈ G ∞ ( H ) are ortho-adjacent if they are compatible and X ∩ Y is a hyperplane in both X, Y. A subset C ⊂ G ∞ ( H ) is called an A-component if for any X , Y ∈ C the intersection X ∩ Y is of the same finite codimension in both X, Y and C is maximal with respect to this property. Let f be a bijective transformation of G ∞ ( H ) preserving the ortho-adjacency relation in both directions. We show that the restriction of f to every A-component is induced by a unitary or anti-unitary operator or it is the composition of the orthocomplementary map and a map induced by a unitary or anti-unitary operator. Note that the restrictions of f to distinct A-components can be related to different operators.

T-violation and the dark sector

It is argued, as a working hypothesis, that “normal” and dark matter interactions can only be [Formula: see text]- and [Formula: see text]-violating. One way to implement this idea is to consider that time reversal in dark matter is implemented, not by an antiunitary operator, but by an unitary operator. It is shown how this occurs naturally in the context of complex spacetime with an extended symmetry group.

Automorphisms of graphs corresponding to conjugacy classes of finite-rank self-adjoint operators

We consider the graph whose vertex set is a conjugacy class C consisting of finite-rank self-adjoint operators on a complex Hilbert space H. The dimension of H is assumed to be not less than 3. In the case when operators from C have two eigenvalues only, we obtain the Grassmann graph formed by k-dimensional subspaces of H, where k is the smallest dimension of eigenspaces. Chow's theorem describes automorphisms of this graph for k>1. Under the assumption that operators from C have more than two eigenvalues we show that every automorphism of the graph is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces with the same dimensions. In contrast to this result, Chow's theorem states that there are graph automorphisms induced by semilinear automorphisms not preserving orthogonality if C is formed by operators with precisely two eigenvalues.

Maps commuting with the λ-Aluthge transform for the semistar Jordan product

Let H and K be two complex separable Hilbert spaces, such that dim ( H ) ≥ 2 and B ( H ) the algebra of bounded linear operators of H on itself. For every A , B ∈ B ( H ) , the semistar Jordan product is denoted by A ▹ B = 1 2 ( AB + B ∗ A ) and for every λ ∈ [ 0 , 1 ] , the λ-Aluthge transform of A is denoted by Δ λ ( A ) . We show that a bijective map Φ : B ( H ) ⟶ B ( K ) satisfies the following condition for some λ ∈ ( 0 , 1 ) , Δ λ ( Φ ( A ) ▹ Φ ( B ) ) = Φ ( Δ λ ( A ▹ B ) ) , forall A , B ∈ B ( H ) , if and only if there exists a unitary or anti-unitary operator U : H ⟶ K , such that Φ ( A ) = UA U ∗ , forall A ∈ B ( H ) .

On a preserver problem concerning Kubo-Ando means of positive operators

We study bijective transformations ϕ of the set of positive operators on a Hilbert space such that given a Kubo-Ando mean σ with certain properties, ‖AσB‖=‖ϕ(A)σϕ(B)‖ holds for every pair A,B of positive operators. It is shown that whenever the Hilbert space is finite dimensional, then ϕ is necessarily conjugation by a fixed unitary or antiunitary operator. Coupled with earlier results, this completely resolves the above preserver problem in the finite dimensional case. Most ideas of the proof carry over to the infinite dimensional setting, thus some partial results are obtained in that case as well.

Construction of maximally-localized Wannier functions using crystal symmetry

In this study, we show two methods concerning constructions of Maximally localized Wannier functions (MLWFs) using crystal symmetry. The first one is to compute input matrices for generating MLWFs only using wavefunctions in the irreducible Brillouin zone (IBZ). This approach reduces the computational costs and the size of intermediate files in proportion to the number of k points in the IBZ. The second method is to calculate the symmetry-adapted Wannier functions that are compatible with so-called frozen window technique, that is, the procedure to disentangle band structures by forcing Bloch states in a certain energy range to be included. We demonstrate how these work in the case of Fe, Co3Sn2S2, Cu, and Nb. These methods are implemented in PW2WANNIER90 code, which interfaces Quantum-ESPRESSO with the Wannier90, and the open-source python library, SymWannier. The code supports both unitary and anti-unitary symmetry operations as well as nonsymmorphic group operations, spin-orbit couplings, and ultrasoft/PAW pseudopotentials. All of these codes are available on GitHub (https://github.com/wannier-utils-dev/symWannier).

Complex-self-adjointness

We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, the latter being not necessarily involutive. We discuss extension theory, refined polar and singular-value decompositions, and antilinear eigenfunction expansions. The study is motivated by physical symmetries in quantum mechanics with non-self-adjoint operators.

Simultaneity and Time Reversal in Quantum Mechanics in Relation to Proper Time

In Newtonian physics, the equation of motion is invariant when the direction of time (t→−t) is flipped. However, in quantum physics, flipping the direction of time changes the sign of the Schrödinger equation. An anti-unitary operator is needed to restore time reversal in quantum physics, but this is at the cost of not having a consistent definition of time reversal applicable to all fundamental theories. On the other hand, a quantum system composed of a pair of entangled particles behaves in such a manner that when the state of one particle is measured, the second particle ‘simultaneously’ acquires a determinate state. A notion of absolute simultaneity seems to be inferred by quantum mechanics, even though it is forbidden by the postulates of relativity. We aim to point out that the above two problems can be overcome if the wavefunction is defined with respect to proper time, which in fact is the real physical time instead of ordinary time.

Spin splitting in monoperiodic systems described by magnetic line groups

In this paper we report the classification of all the 81 magnetic line group families into seven spin splitting prototypes, in analogy to the similar classification previously reported for the 1651 magnetic space groups, 528 magnetic layer groups, and 394 magnetic rod groups. According to this classification, electrically induced (Pekar–Rashba) spin splitting is possible in the antiferromagnetic structures described by magnetic line groups of type I (no anti-unitary operations) and III, both in the presence and in the absence of the space inversion operation. As a specific example, a group theoretical analysis of spin splitting in CoO (8, 8) nanotube is carried out and its predictions are confirmed by ab initio density functional theory calculations.

Commutativity preserving transformations on conjugacy classes of compact self-adjoint operators

Let H be a complex Hilbert space of dimension not less than 3 and let C be a conjugacy class of compact self-adjoint operators on H. Suppose that the dimension of the kernels of operators from C is not less than the dimension of their ranges. In the case when C is formed by operators of finite rank k and dim⁡H=2k, we require that k≥4. We show that every bijective transformation of C preserving the commutativity in both directions is induced by a unitary or anti-unitary operator up to a permutation of eigenspaces of the same dimension.

Automorphisms and Some Geodesic Properties of Ortho-Grassmann Graphs

Let $H$ be a complex Hilbert space. Consider the ortho-Grassmann graph $\Gamma^{\perp}_{k}(H)$ whose vertices are $k$-dimensional subspaces of $H$ (projections of rank $k$) and two subspaces are connected by an edge in this graph if they are compatible and adjacent (the corresponding rank-$k$ projections commute and their difference is an operator of rank $2$). Our main result is the following: if $\dim H\ne 2k$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator; if $\dim H=2k\ge 6$, then every automorphism of $\Gamma^{\perp}_{k}(H)$ is induced by a unitary or anti-unitary operator or it is the composition of such an automorphism and the orthocomplementary map. For the case when $\dim H=2k=4$ the statement fails. To prove this statement we compare geodesics of length two in ortho-Grassmann graphs and characterise compatibility (commutativity) in terms of geodesics in Grassmann and ortho-Grassmann graphs. At the end, we extend this result on generalised ortho-Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators.

Fundamentals of Diatomic Molecular Spectroscopy

The interpretation of optical spectra requires thorough comprehension of quantum mechanics, especially understanding the concept of angular momentum operators. Suppose now that a transformation from laboratory-fixed to molecule-attached coordinates, by invoking the correspondence principle, induces reversed angular momentum operator identities. However, the foundations of quantum mechanics and the mathematical implementation of specific symmetries assert that reversal of motion or time reversal includes complex conjugation as part of anti-unitary operation. Quantum theory contraindicates sign changes of the fundamental angular momentum algebra. Reversed angular momentum sign changes are of a heuristic nature and are actually not needed in analysis of diatomic spectra. This review addresses sustenance of usual angular momentum theory, including presentation of straightforward proofs leading to falsification of the occurrence of reversed angular momentum identities. This review also summarizes aspects of a consistent implementation of quantum mechanics for spectroscopy with selected diatomic molecules of interest in astrophysics and in engineering applications.

The physics and the philosophy of time reversal in standard quantum mechanics

A widespread view in physics holds that the implementation of time reversal in standard quantum mechanics must be given by an anti-unitary operator. In foundations and philosophy of physics, however, there has been some discussion about the conceptual grounds of this orthodoxy, largely relying on either its obviousness or its mathematical-physical virtues. My aim in this paper is to substantively change the traditional structure of the debate by highlighting the philosophical commitments underlying the orthodoxy. I argue that the persuasive force of the orthodoxy can benefit from a relational metaphysics of time and a by-stipulation view on symmetries. Within such philosophical background, I submit, the orthodoxy of time reversal in standard quantum mechanics could find a fertile terrain to lay the groundwork for a more thorough conceptual justification.

Quantum Monte Carlo simulation of generalized Kitaev models

Frustrated spin systems generically suffer from the negative sign problem inherent in Monte Carlo methods. Since the severity of this problem is formulation dependent, optimization strategies can be put forward. We introduce a phase pinning approach in the realm of the auxiliary field quantum Monte Carlo algorithm. If we can find an antiunitary operator that commutes with the one-body Hamiltonian coupled to the auxiliary field, then the phase of the action is pinned to 0 and $\ensuremath{\pi}$. For generalized Kitaev models, we can successfully apply this strategy and observe a remarkable improvement of the average sign. We use this method to study the thermodynamical and dynamical properties of the Kitaev-Heisenberg model down to temperatures corresponding to half of the exchange coupling constant. Our dynamical data reveal finite temperature properties of ordered and spin-liquid phases inherent in this model.

Antiferromagnetism-Induced Spin Splitting in Systems Described by Magnetic Layer Groups

In this paper, we report the classification of all the 528 magnetic layer groups into seven spin splitting prototypes, in analogy to the similar classification previously reported for the 1651 magnetic space groups [Yuan et al., Phys. Rev. Mat. 2021, 5, 014409]. According to this classification, induced (Pekar–Rashba) spin splitting is possible in the antiferromagnetic monolayers described by magnetic layer groups of type I (no anti-unitary operations) and III, both in the presence and in the absence of the space inversion operation. As one specific application, we have studied theoretically the (110) monolayer of antiferromagnetic (AFM) MnO2. Our density functional theory calculations for this structure predicted colossal (on the order of 1 eV) spin splitting in the entire Brillouin zone, similar to the case of the (110) monolayer of AFM MnF2 which was studied by us earlier. This splitting is much larger compared to the corresponding values for the bulk AFM1 structure of β-MnO2, which was rationalized by noting that the (110) monolayer is described by the type I magnetic layer group that does not contain anti-unitary symmetry operations, and therefore, spin splitting is not suppressed anywhere in the entire Brillouin zone, including high-symmetry points and lines.

Generalized Grassmann graphs associated to conjugacy classes of finite-rank self-adjoint operators

Two distinct projections of finite rank m are adjacent if their difference is an operator of rank two or, equivalently, the intersection of their images is (m−1)-dimensional. We extend this adjacency relation on other conjugacy classes of finite-rank self-adjoint operators which leads to a natural generalization of Grassmann graphs. Let C be a conjugacy class formed by finite-rank self-adjoint operators with eigenspaces of dimension greater than 1. Under the assumption that operators from C have at least three eigenvalues we prove that every automorphism of the corresponding generalized Grassmann graph is the composition of an automorphism induced by a unitary or anti-unitary operator and the automorphism obtained from a permutation of eigenspaces with the same dimensions. The case when the operators from C have two eigenvalues only is covered by classical Chow's theorem which says that there are graph automorphisms induced by semilinear automorphisms not preserving orthogonality.

A geometric approach to Wigner‐type theorems

Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.

Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Let V \mathtt {V} be a standard subspace in the complex Hilbert space H \mathcal {H} and G G be a finite dimensional Lie group of unitary and antiunitary operators on H \mathcal {H} containing the modular group ( Δ V i t ) t ∈ R (\Delta _{\mathtt {V}}^{it})_{t \in \mathbb {R}} of V \mathtt {V} and the corresponding modular conjugation J V J_{\mathtt {V}} . We study the semigroup \[ S V = { g ∈ G ∩ U ⁡ ( H ) : g V ⊆ V } S_{\mathtt {V}} = \{ g\in G \cap \operatorname {U}(\mathcal {H})\colon g\mathtt {V} \subseteq \mathtt {V}\} \] and determine its Lie wedge L ⁡ ( S V ) = { x ∈ g : exp ⁡ ( R + x ) ⊆ S V } \operatorname {\textbf {L}}(S_{\mathtt {V}}) = \{ x \in \mathfrak {g} \colon \exp (\mathbb {R}_+ x) \subseteq S_{\mathtt {V}}\} , i.e., the generators of its one-parameter subsemigroups in the Lie algebra g \mathfrak {g} of G G . The semigroup S V S_{\mathtt {V}} is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form G exp ⁡ ( i C ) G \exp (iC) , where C ⊆ g C \subseteq \mathfrak {g} is an Ad ⁡ ( G ) \operatorname {Ad}(G) -invariant closed convex cone. Our main results assert that the Lie wedge L ⁡ ( S V ) \operatorname {\textbf {L}}(S_{\mathtt {V}}) spans a 3 3 -graded Lie subalgebra in which it can be described explicitly in terms of the involution τ \tau of g \mathfrak {g} induced by J V J_{\mathtt {V}} , the generator h ∈ g τ h \in \mathfrak {g}^\tau of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup S V S_{\mathtt {V}} itself.

The structure of maps on the space of all quantum pure states that preserve a fixed quantum angle

Abstract Let $H$ be a Hilbert space and $P(H)$ be the projective space of all quantum pure states. Wigner’s theorem states that every bijection $\phi \colon P(H)\to P(H)$ that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator $U\colon H\to H$. Uhlhorn’s theorem generalizes this result for bijective maps $\phi $ that are only assumed to preserve the quantum angle $\frac{\pi }{2}$ (orthogonality) in both directions. Recently, two papers, written by Li–Plevnik–Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $\alpha $ in both directions, provided that $0 < \alpha \leq \frac{\pi }{4}$ holds. In this paper we solve the remaining structural problem for quantum angles $\alpha $ that satisfy $\frac{\pi }{4} < \alpha < \frac{\pi }{2}$, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner’s.

Dyson\u2019s classification and real division superalgebras

It is well-known that unitary irreducible representations of groups can be usefully classified in a 3-fold classification scheme: Real, Complex, Quaternionic. In 1962 Freeman Dyson pointed out that there is an analogous 10-fold classification of irreducible representations of groups involving both unitary and antiunitary operators. More recently, it was realized that there is also a 10-fold classification scheme involving superdivision algebras. Here we give a careful proof of the equivalence of these two 10-fold ways.