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arXiv:2101.00670 · Analysis and Mathematical Physics

Representation of symmetry transformations on the sets of tripotents of spin and Cartan factors

Yaakov Friedman, Antonio M. Peralta

Published Jan 20, 2022• 6 citations
arXivHTMLPDF

Abstract

Abstract There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states $${\mathbf {P}}$$ P —i.e., the set of one-dimensional projections on a complex Hilbert spaceH– and the orthomodular lattice $${\mathbf {L}}$$ L of closed subspaces ofH. These six groups are isomorphic when the dimension ofHis $$\ge 3$$ ≥3 . The latter hypothesis is absolutely necessary in this identification. For example, the automorphisms group of all bijections preserving orthogonality and the order on $${\mathbf {L}}$$ L identifies with the bijections on $${\mathbf {P}}$$ P preserving transition probabilities only if dim $$(H)\ge 3$$ (H)≥3 . Despite of the difficulties caused by $$M_2({\mathbb {C}})$$ M2(C) , rank two algebras are used for quantum mechanics description of the spin state of spin- $$\frac{1}{2}$$ 12 particles. However, there is a counterexample for Uhlhorn’s version of Wigner’s theorem for such state space. In this note we prove that in order that the description of the spin will be relativistic, it is not enough to preserve the projection lattice equipped with its natural partial order and orthogonality, but we also need to preserve the partial order set of all tripotents and orthogonality among them (a set which strictly enlarges the lattice of projections). Concretely, letMandNbe two atomic JBW $$^*$$ ∗ -triples not containing rank–one Cartan factors, and let $${\mathcal {U}} (M)$$ U(M) and $${\mathcal {U}} (N)$$ U(N) denote the set of all tripotents inMandN, respectively. We show that each bijection $$\Phi : {\mathcal {U}} (M)\rightarrow {\mathcal {U}} (N)$$ Φ:U(M)→U(N) , preserving the partial ordering in both directions, orthogonality in one direction and satisfying some mild continuity hypothesis can be extended to a real linear triple automorphism. This, in particular, extends a result of Molnár to the wider setting of atomic JBW $$^*$$ ∗ -triples not containing rank–one Cartan factors, and provides new models to present quantum behavior.

DOI
10.1007/s13324-021-00644-8

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Abstract There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states P{\mathbf {P}}P P —i.e., the set of one-dimensional projections on a complex Hilbert spaceH– and the orthomodular lattice L{\mathbf {L}}L L of closed subspaces ofH. These six groups are isomorphic when the dimension ofHis ≥3\ge 3≥3 ≥3 . The latter hypothesis is absolutely necessary in this identification. For example, the automorphisms group of all bijections preserving orthogonality and the order on L{\mathbf {L}}L L identifies with the bijections on P{\mathbf {P}}P P preserving transition probabilities only if dim (H)≥3(H)\ge 3(H)≥3 (H)≥3 . Despite of the difficulties caused by M2(C)M_2({\mathbb {C}})M2​(C) M2(C) , rank two algebras are used for quantum mechanics description of the spin state of spin- 12\frac{1}{2}21​ 12 particles. However, there is a counterexample for Uhlhorn’s version of Wigner’s theorem for such state space. In this note we prove that in order that the description of the spin will be relativistic, it is not enough to preserve the projection lattice equipped with its natural partial order and orthogonality, but we also need to preserve the partial order set of all tripotents and orthogonality among them (a set which strictly enlarges the lattice of projections). Concretely, letMandNbe two atomic JBW ∗^*∗ ∗ -triples not containing rank–one Cartan factors, and let U(M){\mathcal {U}} (M)U(M) U(M) and U(N){\mathcal {U}} (N)U(N) U(N) denote the set of all tripotents inMandN, respectively. We show that each bijection Φ:U(M)→U(N)\Phi : {\mathcal {U}} (M)\rightarrow {\mathcal {U}} (N)Φ:U(M)→U(N) Φ:U(M)→U(N) , preserving the partial ordering in both directions, orthogonality in one direction and satisfying some mild continuity hypothesis can be extended to a real linear triple automorphism. This, in particular, extends a result of Molnár to the wider setting of atomic JBW ∗^*∗ ∗ -triples not containing rank–one Cartan factors, and provides new models to present quantum behavior.
Symmetry in quantum mechanicsRoughly speaking, a symmetry of some mathematical object is an invertible transformation that leaves all relevant structure as it is.Thus a symmetry of a set is just a bijection (as sets have no further structure, whence invertibility is the only demand on a symmetry), a symmetry of a topological space is a homeomorphism, a symmetry of a Banach space is a linear isometric isomorphism, and, crucially important for this chapter, a symmetry of a Hilbert space H is a unitary operator, i.e., a linear map u : H → H satisfying one and hence all of the following equivalent conditions:• u is invertible and preserves the inner product, i.e., uϕ, uψ = ϕ, ψ (ϕ, ψ ∈ H).The discussion of symmetries in quantum physics is based on the above idea, but the mathematically obvious choices need not be the physically relevant ones.Even in elementary quantum mechanics, where A = B(H), i.e., the C*-algebra of all bounded operators on some Hilbert space H, the concept of a symmetry is already diverse.The main structures whose symmetries we shall study in this chapter are:1.The normal pure state space P 1 (H), i.e., the set of one-dimensional projections on H, with transition probability τ : P 1 (H) × P 1 (H) → [0, 1] defined by (2.44).2. The normal state space D(H), i.e. the convex set of density operators ρ on H. The self-adjoint operators B(H)sa on H, seen as a Jordan algebra (see below).4. The effects E (H) = [0, 1] B(H) , seen as a convex partially ordered set (poset). 5.The projections P(H) on H, seen as an orthocomplemented lattice.6.The unital commutative C*-subalgebras C (B(H)) of B(H), seen as a poset.Each of these structures comes with its own notion of a symmetry, but the main point of this chapter will be to show these notions are equivalent, corresponding in all cases to either unitary or-surprisingly-anti-unitary operators, both merely defined up to a phase.The latter subtlety will open the world of projective unitary group representation to quantum mechanics (without which the existence of spin-1 2 particles such as electrons, and therewith also of ourselves, would be impossible).
Abstract Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an α∈(0,π2)\alpha \in (0,\tfrac{\pi }{2} )α∈(0,2π​). The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle α\alphaα (or Fubini–Study distance α\alphaα) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when H=C2H=\mathbb{C}^2H=C2 we do this for every α\alphaα, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for α≤π4\alpha \leq \tfrac{\pi }{4}α≤4π​. Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability cos⁡2α\cos ^2\alphacos2α in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when H=C2H=\mathbb{C}^2H=C2 and α=π4\alpha = \tfrac{\pi }{4}α=4π​ where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the α=π2\alpha = \tfrac{\pi }{2}α=2π​ case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.
In the usual formulation of quantum mechanics, groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex Hilbert space, in accordance with Wigner's theorem. In the phase-space formulation, they have real, true unitary representations in the space of square-integrable functions on phase space. Each such phase-space representation is a Weyl–Wigner product of the corresponding Hilbert space representation with its contragredient, and these can be recovered by 'factorizing' the Weyl–Wigner product. However, not every real, unitary representation on phase space corresponds to a group of automorphisms, so not every such representation is in the form of a Weyl–Wigner product and can be factorized. The conditions under which this is possible are examined. Examples are presented.
Orthogonality preserving transformations on the set of nnn-dimensional subspaces of a Hilbert space
We characterize bijective transformations on the set of all nnn-dimensional subspaces of a Hilbert space that preserve orthogonality in both directions. This extends Uhlhorn's improvement of Wigner's classical theorem on symmetry transformations.
Five conceptually distinct notions of symmetry in quantum theory are studied in the algebraic setting where a quantum system is characterized by a von Neumann algebra of observables and the set of normal states on the algebra. It is shown that all five symmetry notions are closely related and that the glue binding them together is the concept of a Jordan ∗-automorphism. For factor algebras a Jordan ∗-automorphism reduces either to an ∗-automorphism or a ∗-anti-automorphism. If the algebra is put in standard form then a ∗-automorphism is always unitarily implementable, whereas a ∗-anti-automorphism is always anti-unitarily implementable. However, there is no guarantee that a general von Neumann algebra admits ∗-anti-automorphisms or, if it does, that it admits order two (or involutory) ∗-anti-automorphisms). For non-factor algebras there can be genuine Jordan ∗-automorphisms that are neither ∗-automorphisms nor ∗-anti-automorphisms, and implementation is possible only through partial isometries. These developments enable generalized versions of Wigner's theorem on the implementation of transition probability preserving symmetries for von Neumann algebras. This review is largely an exercise in connecting the dots in existing mathematics and physics literature. But in the service of the philosophy of physics it is an exercise worth doing since the practitioners in this field seem largely unaware of or unappreciative of this literature and how it fits together to yield a multifaceted but unified picture of quantum symmetries. Along the way various interpretations issues worthy of further discussion are flagged.
As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.