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.
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.
Let Φ : 𝔖(M1) → 𝔖(M2) be a bijection (not assumed affine nor continuous) between the sets of normal states of two quantum systems, modelled on the self-adjoint parts of von Neumann algebras M1 and M2, respectively. This paper concerns with the situation when Φ preserves (or partially preserves) one of the following three notions of “transition probability” on the normal state spaces: the transition probability PU introduced by Uhlmann [Rep. Math. Phys. 9, 273-279 (1976)], the transition probability PR introduced by Raggio [Lett. Math. Phys. 6, 233-236 (1982)], and an “asymmetric transition probability” P0 (as introduced in this article). It is shown that the two systems are isomorphic, i.e., M1 and M2 are Jordan ∗-isomorphic, if Φ preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, in the sense that for any normal states μ and ν, we have PΦ(μ),Φ(ν)=0 if and only if P(μ, ν) = 0, where P stands for PU (respectively, PR or P0). Furthermore, as an extension of Wigner’s theorem, it is shown that there is a Jordan ∗-isomorphism Θ : M2 → M1 satisfying Φ = Θ∗|𝔖(M1) if and only if Φ preserves the “asymmetric transition probability.” This is also equivalent to Φ preserving the Raggio transition probability. Consequently, if Φ preserves the Raggio transition probability, it will preserve the Uhlmann transition probability as well. As another application, the sets of normal states equipped with either the usual metric, the Bures metric or “the metric induced by the self-dual cone,” are complete Jordan ∗-invariants for the underlying von Neumann algebras.
What are the minimal requirements for a physically reasonable mathematical model for quantum theory or a potential extension? In addressing this question, rather than relying on the generalized probabilistic theories or other usual approaches, we propose a reset based on the following fundamental features: transition probabilities, which are typical of quantum theory, and continuous reversible dynamical processes (e.g., represented by a Lie group), which play the same crucial role in classical as well as in quantum physics. In doing so, we get back to the very elementary transition probability space framework originally introduced by Bogdan Mielnik in 1968 and combine it with the continuous reversibility condition. The primary class of valid spaces arises from the irreducible atomic JBW algebras, which encompass the irreducible atomic von Neumann algebras and the Jordan matrix algebras. The von Neumann algebras represent the model of common quantum theory. Beyond these, there is a non-standard valid space, where the exceptional Lie group E6 acts transitively. While E6 has been proposed as a candidate for internal symmetries in particle physics, we highlight that this model deviates from standard quantum theory in critical ways. Specifically, it fails to guarantee the existence of post-measurement states in all situations. Though the above postulates are powerful, they still allow for some physically meaningless models. A further feature of quantum theory (the correspondence between the pure states and the minimal projections) leads us to an additional requirement that rules out these models. However, a complete classification of the valid models remains one of the open issues, pointed out in the paper, which is intended for the mathematical and physical experts, inspiring them to tackle these problems.
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).
As earlier conjectured by several authors and much later established by Solèr (relying on partial results by Piron, Maeda–Maeda and other authors), from the lattice theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. Stückelberg provided some physical, but not mathematically rigorous, reasons for ruling out the real Hilbert space formulation, assuming that any formulation should encompass a statement of Heisenberg principle. Focusing on this issue from another — in our opinion, deeper — viewpoint, we argue that there is a general fundamental reason why elementary quantum systems are not described in real Hilbert spaces. It is their basic symmetry group. In the first 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 real 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 real one, all selfadjoint operators represent observables in accordance with Solèr’s thesis, and the standard quantum version of Noether theorem may be formulated. In the second part of this work, 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. This complex structure reveals a nice interplay of Poincaré symmetry and the classification of the commutant of irreducible real von Neumann algebras.
The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert space that leads to a reduction. This structure is given by the irreducible representations of the group, and in general it can be identified with an operator algebra (a.k.a. C*-algebra or von Neumann algebra). The primary focus of this thesis is the extension of the framework of reductions from symmetries to operator algebras, and its applications in finite-dimensional quantum mechanics. Finding the irreducible representations structure is the principal problem when working with operator algebras. We will therefore review the representation theory of finite-dimensional operator algebras and elucidate this problem with the help of two novel concepts: minimal isometries and bipartition tables. One of the main technical results that we present is the Scattering Algorithm for analytical derivations of the irreducible representations structure of operator algebras. For applications, we will introduce a symmetry-agnostic approach to the reduction of dynamics where we circumvent the non-trivial task of identifying symmetries, and directly reduce the dynamics generated by a Hamiltonian. We will also consider quantum state reductions that arise from operational constraints, such as the partial trace or the twirl map, and study how operational constraints lead to decoherence. Apart from that, we will extend the idea of reduction beyond operator algebras to operator systems, and formulate a quantum notion of coarse-graining that so far only existed in classical probability theory. We will also characterize how the uncertainty principle transitions to the classical regime under coarse-grained measurements and discuss the implications in a finite-dimensional setting.
We describe the symmetry group of the stabilizer polytope for any number n of systems and any prime local dimension d. In the qubit case, the symmetry group coincides with the linear and anti-linear Clifford operations. In the case of qudits, the structure is somewhat richer: For n = 1, it is a wreath product of permutations of bases and permutations of the elements within each basis. For n > 1, the symmetries are given by affine symplectic similitudes. These are the affine maps that preserve the symplectic form of the underlying discrete phase space up to a non-zero multiplier. We phrase these results with respect to a number of a priori different notions of “symmetry,” including Kadison symmetries (bijections that are compatible with convex combinations), Wigner symmetries (bijections that preserve inner products), and symmetries realized by an action on Hilbert space. Going beyond stabilizer states, we extend an observation of Heinrich and Gross [Quantum 3, 132 (2019)] and show that the symmetries of fairly general sets of Hermitian operators are constrained by certain moments. In particular: the symmetries of a set that behaves like a 3-design preserve Jordan products and are therefore realized by conjugation with unitaries or anti-unitaries. (The structure constants of the Jordan algebra are encoded in an order-three tensor, which we connect to the third moments of a design). This generalizes Kadison’s formulation of the classic Wigner theorem on quantum mechanical symmetries.