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Geometric and Algebraic Approaches to Quantum Theory

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Geometric and Algebraic Approaches to Quantum Theory Geometric and algebraic approaches to quantum theory A. Schwarz Department of Mathematics University of California Davis, CA 95616, USA, schwarz @math.ucdavis.edu Abstract We show how to formulate quantum theory taking as a starting point the set of states (geometric approach). We discuss the equations of motion and the formulas for probabilities of physical quantities in this approach. A heuristic proof of decoherence in our setting is used to justify the formulas for probabili- ties. We show that quantum theory can be obtained from classical theory if we restrict the set of observables. This remark can be used to construct models with any prescribed group of symmetries; one can hope that this construc- tion leads to new interesting models that cannot be build in the conventional framework. The geometric approach can be used to formulate quantum theory in terms of Jordan algebras, generalizing the algebraic approach to quantum theory. The scattering theory can be formulated in geometric approach. 1 Introduction arXiv:2102.09176v3 [quant-ph] 3 Jul 2021 Let us start with some very general considerations. Almost all physical theories are based on the notion of state at the moment t. The set of states will be denoted C0. We can consider mixtures of states: taking states ωi with probabilities pi we obtain the mixed state denoted piωi. Similarly if we have a family of states ω(λ) labeled by elements of a set Λ andP a probability distribution on Λ ( a positive measure µ on Λ obeying µ(Λ) = 1) we can talk about the mixed state Λ ω(λ)dµ. R 1 We assume that the set of states C0 contains also mixed states. Then the set of states is a convex set. We assume that it is a subset of a topological linear space L. The extreme points of C0 are called pure states. We assume that for all convex sets we consider every point is a mixture of extreme points. ( If a set is a convex compact subset of locally convex topological vector space this assumption is a statement of Choquet-Bishop-de Leeuw theorem.) Instead of the set C0 we can work with the corresponding cone C (the set of points of the form αx where α is a positive real number, x ∈ C0). The elements of this cone are called non-normalized states. Two elements x, y ∈ C determine the same normalized state if they are proportional : y = Cx where C is a positive number. An observable specifies a linear functional a on L. This requirement agrees with the definition of mixed state: if ω = piωi then a(ω) is an expectation value of a(ωi). (The non-negative numbers pi obeyingP pi = 1 are considered as probabilities.) We consider deterministic theories.P This means that the state in the moment t = 0 (or in any other moment t0) determines the state in arbitrary moment t. Let us denote by σ(t) the operator transforming the state in the moment t = 0 into the state in the moment t (the evolution operator). The evolution operators constitute a one-parameter family σ(t) of invertible maps σ(t): C0 → C0 , that can be extended to linear maps of L. In other words the operators σ(t) belong to the group U of automorphisms of C0 (to the group of linear bicontinuous maps of L inducing invertible maps of C0 onto itself). In some cases one should impose an additional condition σ(t) ∈V where V is a subgroup of U. The evolution operator satisfies the equation dσ = H(t)σ(t) (1) dt (equation of motion). Here H(t) ∈ Lie(V) is an element of the tangent space to the group V at the unit element (the ”Hamiltonian”). 1 The equation (1) can be regarded as a definition of H(t). However, usually we go in opposite direction: the physical system we consider is specified by the operator H(t) (by the equation of motion) and our goal is to calculate the evolution operator solving the equation of motion. Examples Classical mechanics 1Knowing the topology in L we can define in various ways the topology in V⊂U. This allows us to define the Lie algebra of the group V as the tangent space at the unit element. ( We disregard the subtleties related to the fact that the group V is in general infinite-dimensional.) 2 The cone C consists of positive measures µ on symplectic manifoldM, we assume that µ(M) < ∞ The set C0 consists of probability distributions (normalized positive measures, µ(M) = 1). Observables are functions on symplectic manifold V is the group of symplectomorphisms Lie(V) is the algebra of Hamiltonian vector fields dρ The equation of motion is the Liouville equation dt = {H, ρ} where ρ stands for the density of the measure and {·, ·} denotes the Poisson bracket. Quantum mechanics The set C0 consists of density matrices ( positive trace class operators in real or complex Hilbert space H having unit trace: TrK = 1). Omitting the condition TrK = 1 we obtain the cone C.2 Notice that C is a homogeneous self-dual cone; such cones are closely related to Jordan algebras ( see Section 3). In textbook quantum mechanics it is assumed that that H is a complex Hilbert space. In this case observables are identified with self-adjoint operators. A self- adjoint operator Aˆ specifies a linear functional by the formula K → TrAK.ˆ V = U is isomorphic to the group of invertible isometries. (They are called orthogonal operators if the Hilbert space is real and unitary operators if the Hilbert space is complex.) An operator Vˆ ∈ U acts on C and on C0 by the formula K → VKˆ Vˆ ∗. The tangent space of this group at the unit element can be regarded as the Lie algebra LieU; it consists of bounded operators obeying Aˆ + Aˆ∗ = 0 (skew-adjoint operators) . ( In complex Hilbert space instead of skew-adjoint operator Aˆ we can work with self-adjoint operator Hˆ = iAˆ.) 3 The equation of motion can be written in the form dK = A(t)K = −Aˆ(t)K + KAˆ(t). dt where Aˆ(t) is a family of skew-adjoint operators in H. 2Physicists always work in complex Hilbert space imposing the condition of reality on the states if necessary. We prefer to work in real Hilbert space. 3The group U can be considered as Banach Lie group; the tangent space to it can be defined as the set of tangent vectors to curves (to one-parameter families of operators) that are differentiable with respect to the norm topology. Notice, however, that the families of evolution operators appearing in physics usually do not satisfy this condition (but they can be approximated by differentiable families). For a self-adjoint operator Hˆ in complex Hilbert space the operator Aˆ = −iHˆ can be ˆ ˆ considered as a tangent vector to the one-parameter group of unitary operators eAt = e−iHt. This family is differentiable in norm topology only if the operator Hˆ is bounded. 3 In complex Hilbert space we can write the equation of motion as follows: dK = H(t)K = i(Hˆ (t)K − KHˆ (t)) dt where Hˆ (t) is a family of self-adjoint operators ( here Hˆ (t)= iAˆ(t)). Quantum theory in algebraic approach The starting point is a unital associative algebra A with involution ∗ The set of states C0 is defined as the space of positive normalized linear functionals on A (the functionals obeying ω(A∗A) ≥ 0,ω(1) = 1). Omitting the normalization condition ω(1) = 1 we obtain the definition of the cone of states C. We can define a cone A+ of non-negative elements of A as a convex envelope of elements of the form A∗A; the cone C of states is dual to this cone. V = U denotes the group of involution preserving automorphisms of A (they act naturally on states) To relate the algebraic approach with the Hilbert space formulation we use the GNS (Gelfand-Naimark-Segal) construction: for every state ω there exists pre Hilbert space H, representation A → Aˆ of A in H and a cyclic vector θ ∈ H such that ω(A)= hAθ,ˆ θi ( We say that vector θ is cyclic if every vector x ∈ H can be represented in the form x = Aθˆ where A ∈A. Notice that instead of pre Hilbert space H one can work with its completion, Hilbert space H, then θ is cyclic in weaker sense: the vectors Aθˆ are dense in Hilbert space. ) In the present paper we describe the approach to quantum theory where the pri- mary notion is the convex set of states (geometric approach). We discuss its relations to algebraic approach and to textbook quantum mechanics. We show that a gener- alization of decoherence is correct in geometric approach and use this statement to derive the formulas for probabilities from the first principles. ( To prove decoherence we study the interaction of physical system with random environment that is mod- eled as random adiabatic perturbation of the equations of motion.) We show that quantum theory can be obtained from classical theory if we restrict the set of observ- ables. ( In classical theory a state can be represented as a mixture of pure states in unique way. If not all observables are allowed some states should be identified. This identification permits us to construct any theory from classical theory.) We analyze the relation to Jordan algebras. 4 Finally we discuss the notions of particle and 4The set of self-adjoint elements (observables) is not closed under multiplication, but it is closed 1 with respect to the operation a ◦ b = 2 (ab + ba).
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