A so-called generalized Weyl correspondence is defined among random variables on one side and linear operators in a separable Hilbert space ℋ on the other. Besides such a correspondence, there is a relation among states on ℋ (considered as positive nuclear operators on ℋ) and the distribution functions of the random variables. By adding some new assumptions, several relations are shown. Later, we study two particularly interesting cases. In the first we connect dichotomic random variables with number operators in a Grassmann algebra ℋ, and nuclear operators on ℋ with probability measures in the set of all sequences made up of zero and one. In the second case we relate states between stochastic and quantum electrodynamics.
A so-called generalized Weyl correspondence is defined among random variables on one side and linear operators in a separable Hilbert space ℋ on the other. Besides such a correspondence, there is a relation among states on ℋ (considered as positive nuclear operators on ℋ) and the distribution functions of the random variables. By adding some new assumptions, several relations are shown. Later, we study two particularly interesting cases. In the first we connect dichotomic random variables with number operators in a Grassmann algebra ℋ, and nuclear operators on ℋ with probability measures in the set of all sequences made up of zero and one. In the second case we relate states between stochastic and quantum electrodynamics.
The definition and some general properties of the generalized Weyl correspondence between stochastic processes and operator-valued real functions on a Hilbert space plus a trace class operator are given. The relation between the derivatives of an operator-valued function and the derivative of the corresponding stochastic process are studied. When the operator-valued function is the position (or momentum) in the Heisenberg picture, a condition for the positivity of the joint distribution functions of the corresponding process is given, provided that the evolution Hamiltonian be quadratic in the position and momentum. Finally, the case of an arbitrary Hamiltonian evolution for the position operator is studied and the two-dimensional density functions of the process is related to the Wigner function associated to some state ρ̂, and a necessary condition for the positivity of the densities is given.
In the quantum probability theory the 6-field of random events is replaoed by the lattice of orthogonal projectors in a separable infinite dimensional Hilbert space H.A countably additive function from this lattices to the unit interval constitutes a state, the non-commutative analogue of a probability measure.The Theorem of Gleason [4] asserts that every state is of the form n--tr jtT, where jt runs over all projectors and T is a probability operator on H, i.e. a positive linear operator of unit trace.Conversely, every probability operator determines a state by the Gleason formula.From now onwards let S stand for the set of all states, i.e. all probability operators on H.We shall dienote by the space of-? ell nuclear linear operators acting in H with the norm jjïj|.j= tr{TT*) 1 ^2.Of coarse S is e closed and convex subset Of TyIn quantum theory to every physical quantity or observable th^re corresponds e sslf»adjoint not necessarily bounded linear operator on Hs By 0 ws shall denote the set of all obt ;ry£bles-Given he-0 f tac pi'cb^ivility distribution of A at the etato T is defined for ail Eoi'oi subsets E of the real
Couples are wholes and not wholes, what agrees disagrees, the concordant is discordant. From all things one and from one all things. ( Heraclitus , On the Universe 59.) This chapter introduces the notion of joint (Wigner) density of random variables, for future use in quantum Malliavin calculus. For this we will rely on functional calculus on general Lie algebras, starting with the Heisenberg– Weyl algebra. We also consider some applications to quantum optics and timefrequency analysis. Joint moments of noncommuting random variables The notion of moment ⟨ ϕ , P n ψ ⟩ of order n of P in the state ⟨ ϕ , · ψ ⟩ is well understood as the noncommutative analogue of E [ X n ]. Indeed, in the single variable case, various functional calculi are available for elements of an algebra. In particular, • polynomials can be applied to any element in an algebra; • holomorphic functions can be applied to elements of a Banach algebra provided the function is holomorphic on a neighborhood of the spectrum of the element; • continuous functions can be applied to normal elements of C*-algebras provided they are continuous on the spectrum of the element; • measurable ( i.e. , Borel) functions can be applied to normal elements in a von Neumann algebra ( e.g. , B (h ) ), provided they are measurable on the spectrum of the operator. In addition, the Borel functional calculus can be extended to unbounded normal operators (where normal means that XX ∗ and X ∗ X have the same domain and coincide on this domain). All the above functional calculi work due to a restriction to a commutative subalgebra. In that sense, Weyl calculus is fundamentally different because it applies to functions of noncommuting operators. We now need to focus on the meaning of joint moments. For example, which is the noncommutative analogue of E [ XY ]? A possibility is to choose ⟨ ϕ , PQψ ⟩ as the the noncommutative analogue of E [ XY ], however this would lead to ⟨ ϕ , QPψ ⟩ as the the noncommutative analogue of E [ YX ] and we have E [ XY ] = E [ YX ] while due to noncommutativity. A way to solve this contradiction can be to define the first joint moment as Next, how can we define the analogue of E [ XY 2 ]? Proceeding similarly, a natural extension could be Then what is the correct way to construct E[ X 2 Y 3 ]? Clearly there is a combinatorial pattern which should be made explicit.
In view of the probabilistic quantizer–dequantizer operators introduced, the qubit states (spin-1/2 particle states, two-level atom states) realizing the irreducible representation of the S U ( 2 ) symmetry group are identified with probability distributions (including the conditional ones) of classical-like dichotomic random variables. The dichotomic random variables are spin-1/2 particle projections m = ± 1 / 2 onto three perpendicular directions in the space. The invertible maps of qubit density operators onto fair probability distributions are constructed. In the suggested probability representation of quantum states, the Schrödinger and von Neumann equations for the state vectors and density operators are presented in explicit forms of the linear classical-like kinetic equations for the probability distributions of random variables. The star-product and quantizer–dequantizer formalisms are used to study the qubit properties; such formalisms are discussed for photon tomographic probability distribution and its correspondence to the Heisenberg–Weyl symmetry properties.
The commutator of position and momentum operators is first considered in the coordinate and momentum representations, and then in an abstract Hilbert space. The reader is then introduced to canonical operators with the associated canonical quantization of commutation rules, which consists of maps defined on a symplectic space and taking values in the set of unitary operators on a Hilbert space. If the symplectic space is finite-dimensional, all of its irreducible representations are unitarily equivalent. The Weyl exponentiated form of the commutation relations is described and investigated. One then arrives at the definition of Weyl systems, and it is shown how to recover from them the formulation in terms of self-adjoint operators which satisfy a generalized form of commutation relations. Further topics are the Heisenberg representation for temporal evolution and the generalized uncertainty relations. Unitary operators associated with symplectic linear transformations are then considered, and, within this framework, translations, rotations and the harmonic oscillator are considered. The Weyl programme is then reassessed. Lastly, the basic postulates of modern quantum theory are eventually studied: the probabilistic nature of the predictions of results of measurements; the correspondence between observables and self-adjoint operators on Hilbert spaces; the expected result of measuring an observable quantity; and the evolution of the state vector in systems which are not affected by any external influence. The chapter ends with an outline of rigged Hilbert space formalism and its applications. The commutator in wave mechanics In the first part of this book we studied wave mechanics within the framework of the Schrödinger equation.
In "Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order" [arXiv:1208.4724], it was shown that self-adjoint operators affiliated with a von Neumann algebra N can equivalently be described as certain real-valued functions on the projection lattice P(N) of the algebra, which we call q-observable functions. Here, we show that q-observable functions can be interpreted as generalised quantile functions for quantum observables interpreted as random variables. More generally, when L is a complete meet-semilattice, we show that L-valued cumulative distribution functions and the corresponding L-quantile functions form a Galois connection. An ordinary CDF can be written as an L-CDF composed with a state. For classical probability, one picks L=B(Ω), the complete Boolean algebra of measurable subsets modulo null sets of a measurable space Ω. For quantum probability, one uses L=P(N), the projection lattice of a nonabelian von Neumann algebra N. Moreover, using some constructions from the topos approach to quantum theory, we show that there is a joint sample space for all quantum observables, despite no-go results such as the Kochen-Specker theorem. Specifically, the spectral presheaf Σ of a von Neumann algebra N, which is not a mere set, but a presheaf (i.e., a 'varying set'), plays the role of the sample space. The relevant meet-semilattice L in this case is the complete bi-Heyting algebra of clopen subobjects of Σ. We show that using the spectral presheaf Σ and associated structures, quantum probability can be formulated in a way that is structurally very similar to classical probability.