Classical Physic

As an example, take Leibnizian spacetime,¹⁹ whose structure consists of all and only the following: a notion of absolute or observer-independent simultaneity; a temporal metric (giving the lapse of time between non-simultaneous events); and a Euclidean spatial metric (giving the spatial distance between events lying on a given plane of absolute simultaneity). In a coordinate system (x^α, t), α = 1, 2, 3 adapted to this structure, the spacetime symmetries are

where $R_{\beta }^{\alpha }(t)$ is an orthogonal time dependent matrix and the a^α(t) are arbitrary smooth functions of t. Clearly, the symmetries (1) contain determinism killing symmetries.

Two different strategies for saving determinism in the face of the above construction can be tried. They correspond to radically different attitudes towards the ontology of spacetime. The first strategy is to beef up the structure of the background spacetime. Adding a standard of rotation kills the time dependence in $R_{\beta }^{\alpha }(t)$, producing what is called Maxwellian spacetime. But since the a^α(t) are still arbitrary functions of t there remain determinism killing symmetries. Adding a standard of inertial or straight line motion linearizes the a^α(t) to v^αt + c^α, where the v^α and c^α are constants, producing neo-Newtonian spacetime²⁰ whose symmetries are given by the familiar Galilean transformations

The point can be illustrated with the help of a humble example of particle mechanics constructed within the Maxwellian spacetime introduced in the preceding subsection. An appropriate Lagrangian invariant under the symmetries of this spacetime is given by

This Lagrangian is singular in the sense that Hessian matrix ${\partial }^{2}L/\partial {\dot{\text{x}}}_i\partial {\dot{\text{x}}}_j$ does not have an inverse. The Euler-Lagrange equations are

Applying the formalism to our toy case of particle mechanics in Maxwellian spacetime, the canonical momenta are:

These momenta are not independent but must satisfy three primary constraints, which require the vanishing of the x, y, and z-components of the total momentum:

The technical elaboration of the constraint formalism is complicated, but one should not lose sight of the fact that the desire to save determinism is the motivation driving the enterprise. Here is a relevant passage from [Henneaux and Teitelboim, 1992], one of the standard references on constrained Hamiltonian systems:

The presence of arbitrary functions … in the total Hamiltonian tells us that not all the q's and p's [the configuration variables and their canonical momenta] are observable [i.e. genuine physical magnitudes]. In other words, although the physical state is uniquely defined once a set of q's and p's is given, the converse is not true — i.e., there is more than one set of values of the canonical variables representing a given physical state. To see how this conclusion comes about, we note that if we are given an initial set of canonical variables at the time t₁ and thereby completely define the physical state at that time, we expect the equations of motion to fully determine the physical state at other times. Thus, by definition, any ambiguity in the value of the canonical variables at t₂ ≠ t₁ should be a physically irrelevant ambiguity. [pp. 16–17]

In addition to this standard reaction to the apparent failure of determinism in the above examples, two others are possible. The first heterodoxy takes the apparent violation of determinism to be genuine. This amounts to (a) treating what the constraint formalism counts as gauge dependent quantities as genuine physical magnitudes, and (b) denying that these magnitudes are governed by laws which, when conjoined with the laws already in play, restore determinism. The second heterodoxy accepts the orthodox conclusion that the apparent failure of determinism is merely apparent; but it departs from orthodoxy by accepting (a), and it departs from the first heterodoxy by denying (b) and, accordingly, postulates the existence of additional laws that restore determinism. Instances that superficially conform to part (a) of the two heterodoxies are easy to construct from examples found in physics texts where the initial value problem is solved by supplementing the equations of motion, stated in terms of gauge-dependent variables, with a gauge condition that fixes a unique solution. For instance, Maxwell's equations written in terms of electromagnetic potentials do not determine a unique solution corresponding to the initial values of the potentials and their time derivatives. Imposing the Lorentz gauge condition converts Maxwell's equations to second order hyperbolic partial differential equations (pdes) that do admit an initial value formulation (see Section 4.2).²⁶ Similar examples can be concocted in general relativity theory where orthodoxy treats the metric potentials as gauge variables (see Section 6.2). In these examples orthodoxy is aiming to get at the values of the gauge independent variables via a choice of gauge. If this aim is not kept clearly in mind, the procedure creates the illusion that gauge-dependent variables have physical significance. It is exactly this illusion that the two heterodoxies take as real. The second heterodoxy amounts to taking the gauge conditions not as matters of calculational convenience but as additional physical laws. I know of no historical examples where this heterodoxy has led to fruitful developments in physics.

Newtonian gravitational theory can be construed as a field theory. The gravitational force is given by F_grav = −Δ φ, where the gravitational potential φ satisfies the Poisson equation

with ρ being the mass density. If φ is a solution to Poisson's equation, then so is φ′ = φ + g (x)f (t) where g (x) is a linear function of the spatial variables and f (t) is an arbitrary function of t. Choose f so that f (t) = 0 for t ≤ 0 but f (t) > 0 for t > 0. The extra gravitational force, proportional to f (t), that a test particle experiences in the primed solution after t = 0 is undetermined by anything in the past.

In the Newtonian setting the field equations that naturally arise are elliptic (e.g. the Poisson equation) or parabolic, and neither type supports determinism-without-crutches. An example of the latter type of equation is the classical heat equation

where Φ is the temperature variable and k is the coefficient of heat conductivity.²⁹ Solutions to (8) can cease to exist after a finite time because the temperature “blows up.” Uniqueness also fails since, using the fact that the heat equation propagates heat arbitrarily fast, it is possible to construct surprise solutions Φ_s with the properties that (i) Φ_s is infinitely differentiable, and (ii) Φ_s(x, t) = 0 for all t ≤ 0 but Φ_s(x, t) ≠ 0 for t > 0 (see [John, 1982, Sec. 7.1]). Because (8) is linear, if Φ is a solution then so is Φ′ = Φ + Φ_s. And since Φ and Φ′ agree for all t ≠ 0 but differ for t > 0, the existence of the surprise solutions completely wrecks determinism.

The Navier-Stokes equations for an incompressible fluid moving in ${ℝ}^N,N=2,3$, read

where u (x, t) = (u¹, u², …, u^N) is the velocity of the fluid, p (x, t) is the pressure, v = const. ≥ 0 is the coefficient of viscosity, and $D/dt:=\partial /\partial t+\sum _{j=1}^N{u}^j\partial /\partial x^j$ is the convective derivative (see Foias at al. 2001 for a comprehensive survey). If the fluid is subject to an external force, an extra term has to be added to the right hand side of (9a). The Euler equations are the special case where u = 0. The initial value problem for (9a-b) is posed by giving the initial data

where u₀ (x) is a smooth (C^∞) divergence-free vector field, and is solved by smooth functions $\text{u},p\in C^{\infty }({ℝ}^{N}x[0,\infty ))$ satisfying (9)-(10). For physically reasonable solutions it is required both that u₀(x) should not grow too large as |x| → ∞ and that the energy of the fluid is bounded for all time:

When v = 0 the energy is conserved, whereas for v > 0 it dissipates.

Consider a single particle of mass m moving on the real line $ℝ$ in a potential V (x), x ∈ℝ. The standard existence and uniqueness theorems for the initial value problem of odes can be used to show that the Newtonian equation of motion

has a locally (in time) unique solution if the force function F (x) satisfies a Lipschitz condition.³¹ An example of a potential that violates the Lipschitz condition at the origin is $-\frac{9}{2}|x{|}^{4/3}$. For the initial data $x(0)=0=\dot{x}(0)$ there are multiple solutions of (12): x (t) ≡ 0, x (t) = t³, and x (t) = −t³, where m has been set to unity for convenience. In addition, there are also solutions x (t) where x (t) = 0 for t < k and ±(t – k)³ for t ≥ k, where k is any positive constant. That such force functions do not turn up in realistic physical situations is an indication that Nature has some respect for determinism. In QM it turns out that Nature can respect determinism while accommodating some of the non-Lipschitz potentials that would wreck Newtonian determinism (see Section 5.2).

Laplace's vision of a deterministic universe makes reference to an “intelligence” (which commentators have dubbed ‘Laplace's Demon’):

We ought to regard the present state of the universe as the effect of its antecedent state and as the cause of the state that is to follow. An intelligence knowing all of the forces acting in nature at a given instant, as well as the momentary positions of all things in the universe, would be able to comprehend in one single formula the motions of the largest bodies as well as the lightest atoms in the world, provided that its intellect were sufficiently powerful to subject all data to analysis; to it nothing would be uncertain, the future as well as the past would be present to its eyes.³⁶

Jacques Hadamard, who made seminal contributions to the Cauchy or initial value problem for pdes, took the terminology of “well-posed” (a.k.a. “properly posed”) quite literally. For he took it as a criterion for the proper mathematical description of a physical system that the equations of motion admit an initial value formulation in which the solution depends continuously on the initial data (see [Hadamard, 1923, 32]). However, the standard Courant-Hilbert reference work, Methods of Mathematical Physics, opines that

“properly posed” problems are by far not the only ones which appropriately reflect real phenomena. So far, unfortunately, little mathematical progress has been made in the important task of solving or even identifying such problems that are not “properly posed” but still are important and motivated by realistic situations. [1962, Vol. 2, 230].

1.

Mathematically, decoherence boils down to the idea of adding one more link to the von Neumann chain (see Subsection 2.5) beyond S + A (i.e. the system and the apparatus). Conceptually, however, there is a major difference between decoherence and older approaches that took such a step: whereas previously (e.g., in the hands of von Neumann, London and Bauer, Wigner, etc.)³¹⁰ the chain converged towards the observer, in decoherence it diverges away from the observer. Namely, the third and final link is now taken to be the environment (taken in a fairly literal sense in agreement with the intuitive meaning of the word). In particular, in realistic models the environment is treated as an infinite system (necessitating the limit N → ∞), which has the consequence that (in simple models where the pointer has discrete spectrum) the post-measurement state Σ_n c_nΨ_n ⊗ ϕ_n ⊗ χ_n (in which the χ_n are mutually orthogonal) is only reached in the limit t → ∞. However, as already mentioned in Subsection 6.6, infinite time is only needed mathematically in order to make terms of the type ~ exp −γt (with γ > 0) zero rather than just very small: in many models the inner products (χ_n, χ_m) are actually negligible for n ≠ m within surprisingly short time scales.³¹¹

If only in view of the need for limits of the type N → ∞ (for the environment) and t → ∞, in our opinion decoherence is best linked to stance 1 of the Introduction: its goal is to explain the approximate appearance of the classical world from quantum mechanics seen as a universally valid theory. However, decoherence has been claimed to support almost any opinion on the foundations of quantum mechanics; cf. [Bacciagaluppi, 2004] and [Schlosshauer, 2004] for a critical overview and also see Point 3 below.

2.

Originally, decoherence entered the scene as a proposed solution to the measurement problem (in the precise form stated at the end of Subsection 2.5). For the restriction of the state Σ_n c_nΨ_n ⊗ ϕ_n ⊗ χ_n to S + A (i.e. its trace over the degrees of freedom of the environment) is mixed in the limit t → ∞, which means that the quantum-mechanical interference between the states Ψ_n⊗ϕ_n for different values of n has become ‘delocalized’ to the environment, and accordingly is irrelevant if the latter is not observed (i.e. omitted from the description). Unfortunately, the application of the ignorance interpretation of the mixed post-measurement state of S + A is illegal even from the point of view of stance 1 of the Introduction. The ignorance interpretation is only valid if the environment is kept within the description and is classical (in having a commutative C*-algebra of observables). The latter assumption [Primas, 1983], however, makes the decoherence solution to the measurement problem circular.³¹²

In fact, as quite rightly pointed out by Bacciagaluppi [2004], decoherence actually aggravates the measurement problem. Where previously this problem was believed to be man-made and relevant only to rather unusual laboratory situations (important as these might be for the foundations of physics), it has now become clear that “measurement” of a quantum system by the environment (instead of by an experimental physicist) happens everywhere and all the time: hence it remains even more miraculous than before that there is a single outcome after each such measurement. Thus decoherence as such does not provide a solution to the measurement problem [Leggett, 2002];³¹³ Adler, 2003; Joos and Zeh, 2003], but is in actual fact parasitic on such a solution.

3.

There have been various responses to this insight. The dominant one has been to combine decoherence with some interpretation of quantum mechanics: decoherence then finds a home, while conversely the interpretation in question is usually enhanced by decoherence. In this context, the most popular of these has been the many-worlds interpretation, which, after decades of obscurity and derision, suddenly started to be greeted with a flourish of trumpets in the wake of the popularity of decoherence. See, for example, [Saunders, 1993; 1995; Joos et al., 2003] and [Zurek, 2003]. In quantum cosmology circles, the consistent histories approach has been a popular partner to decoherence, often in combination with many worlds; see below. The importance of decoherence in the modal interpretation has been emphasized by Dieks [1989b] and Bene and Dieks [2002], and practically all authors on decoherence find the opportunity to pay some lip-service to Bohr in one way or another. See [Bacciagaluppi, 2004] and [Schlosshauer, 2004] for a critical assessment of all these combinations.

In our opinion, none of the established interpretations of quantum mechanics will do the job, leaving room for genuinely new ideas. One such idea is the return of the environment: instead of “tracing it out”, as in the original setting of decoherence theory, the environment should not be ignored! The essence of measurement has now been recognized to be the redundancy of the outcome (or “record“) of the measurement in the environment. It is this very redundancy of information about the underlying quantum object that “objectifies” it, in that the information becomes accessible to a large number of observers without necessarily disturbing the object³¹⁴ [Zurek, 2003; Ollivier et al. 2004; Blume-Kohout and Zurek, 2004; 2005]. This insight (called “Quantum Darwinism“) has given rise to the “existential” interpretation of quantum mechanics due to Zurek [2003].

4.

Another response to the failure of decoherence (and indeed all other approaches) to solve the measurement problem (in the sense of failing to win a general consensus) has been of a somewhat more pessimistic (or, some would say, pragmatic) kind: all attempts to explain the quantum world are given up, yielding to the point of view that ‘the appropriate aim of physics at the fundamental level then becomes the representation and manipulation of information’ [Bub, 2004]. Here ‘measuring instruments ultimately remain black boxes at some level’, and one concludes that all efforts to understand measurement (or, for that matter, EPR-correlations) are futile and pointless.³¹⁵

5.

Night thoughts of a quantum physicist, then?³¹⁶ Not quite. Turning vice into virtue: rather than solving the measurement problem, the true significance of the decoherence program is that it gives conditions under which there is no measurement problem! Namely, foregoing an explanation of the transition from the state Σ_n c_nΨ_n ⊗ ϕ_n ⊗ χ_n of S + A + ε to a single one of the states Ψ_n ⊗ ϕ_n of S + A, at the heart of decoherence is the claim that each of the latter states is robust against coupling to the environment (provided the Hamiltonian is such that Ψ_n ⊗ ϕ_n tensored with some initial state I_ε of the environment indeed evolves into Ψ_n ⊗ ϕ_n ⊗ χ_n, as assumed so far). This implies that each state Ψ_n ⊗ ϕ_n remains pure after coupling to the environment and subsequent restriction to the original system plus apparatus, so that at the end of the day the environment has had no influence on it. In other words, the real point of decoherence is the phenomenon of einselection (for environment-induced superselection), where a state is ‘einselected’ precisely when (given some interaction Hamiltonian) it possesses the stability property just mentioned. The claim, then, is that einselected states are often classical, or at least that classical states (in the sense mentioned at the beginning of this section) are classical precisely because they are robust against coupling to the environment. Provided this scenario indeed gives rise to the classical world (which remains to be shown in detail), it gives a dynamical explanation of it. But even short of having achieved this goal, the importance of the notion of einselection cannot be overstated; in our opinion, it is the most important and powerful idea in quantum theory since entanglement (which einselection, of course, attempts to undo!).

6.

The measurement problem, and the associated distinction between system and apparatus on the one hand and environment on the other, can now be omitted from decoherence theory. Continuing the discussion in Subsection 3.4, the goal of decoherence should simply be to find the robust or einselected states of a object $\mathcal{O}$ coupled to an environment ε, as well as the induced dynamics thereof (given the time-evolution of $\mathcal{O}$ + ε). This search, however, must include the correct identification of the object $\mathcal{O}$ within the total $\mathcal{S}$ + ε, namely as a subsystem that actually has such robust states. Thus the Copenhagen idea that the Heisenberg cut between object and apparatus be movable (cf. Subsection 3.2) will not, in general, extend to the “Primas-Zurek” cut between object and environment. In traditional physics terminology, the problem is to find the right “dressing” of a quantum system so as to make at least some of its states robust against coupling to its environment [Amann and Primas, 1997; Brun and Hartle, 1999; Omnès, 2002]. In other words: What is a system? To mark this change in perspective, we now change notation from $\mathcal{Q}$ (for “object”) to $\mathcal{S}$ (for “system”). Various tools for the solution of this problem within the decoherence program have now been developed — with increasing refinement and also increasing reliance on concepts from information theory [Zurek, 2003] — but the right setting for it seems the formalism of consistent histories, see below.

7.

Various dynamical regimes haven been unearthed, each of which leads to a different class of robust states [Joos et al., 2003; Zurek, 2003; Schlosshauer, 2004]. Here H_S is the system Hamiltonian, H_I is the interaction Hamiltonian between system and environment, and H_ε is the environment Hamiltonian. As stated, no reference to measurement, object or apparatus need be made here.

•

In the regime H_S << H_I, for suitable Hamiltonians the robust states are the traditional pointer states of quantum measurement theory. This regime conforms to von Neumann's [1932] idea that quantum measurements be almost instantaneous. If, moreover, H_ε << H_I as well — with or without a measurement context — then the decoherence mechanism turns out to be universal in being independent of the details of ε and H_ε [Strunz et al., 2003).

•

If H_S Ã H_I, then (at least in models of quantum Brownian motion) the robust states are coherent states (either of the traditional Schrödinger type, or of a more general nature as defined in Subsection 5.1); see [Zurek et al., 1993] and [Zurek, 2003]. This case is, of course, of supreme importance for the physical relevance of the results quoted in our Section 5 above, and — if only for this reason — decoherence theory would benefit from more interaction with mathematically rigorous results on quantum stochastic analysis.³¹⁷

•

Finally, if H${\hspace{0.17em}}_{\mathcal{S}}$ >> H_I, then the robust states turn out to be eigenstates of the system Hamiltonian H${\hspace{0.17em}}_{\mathcal{S}}$ [Paz and Zurek, 1999; Ollivier et al., 2004]. In view of our discussion of such states in Subsections 5.5 and 5.6, this shows that robust states are not necessarily classical. It should be mentioned that in this context decoherence theory largely coincides with standard atomic physics, in which the atom is taken to be the system $\mathcal{S}$ and the radiation field plays the role of the environment ε; see [Gustafson and Sigal, 2003] for a mathematically minded introductory treatment and [Bach et al., 1998; 1999] for a full (mathematical) meal.

8.

Further to the above clarification of the role of energy eigenstates, decoherence also has had important things to say about quantum chaos [Zurek, 2003; Joos et al., 2003]. Referring to our discussion of wave packet revival in Subsection 2.4, we have seen that in atomic physics wave packets do not behave classically on long time scales. Perhaps surprisingly, this is even true for certain chaotic macroscopic systems: cf. the case of Hyperion mentioned in the Introduction and at the end of Subsection 5.2. Decoherence now replaces the underlying superposition by a classical probability distribution, which reflects the chaotic nature of the limiting classical dynamics. Once again, the transition from the pertinent pure state of system plus environment to a single observed system state remains clouded in mystery. But granted this transition, decoherence sheds new light on classical chaos and circumvents at least the most flagrant clashes with observation.³¹⁸

9.

Robustness and einselection form the state side or Schrödinger picture of decoherence. Of course, there should also be a corresponding observable side or Heisenberg picture of decoherence. But the transition between the two pictures is more subtle than in the quantum mechanics of closed systems. In the Schrödinger picture, the whole point of einselection is that most pure states simply disappear from the scene. This may be beautifully visualized on the example of a two-level system with Hilbert space $\mathcal{H}$${\hspace{0.17em}}_{\mathcal{S}}$ = ℂ² [Zurek, 2003]. If ↑ and ↓ (cf. (33)) happen to be the robust vector states of the system after coupling to an appropriate environment, and if we identify the corresponding density matrices with the north-pole (0, 0, 1) ∈ B³ and the south-pole (0, 0, −1) ∈ B³, respectively (cf. (3)), then following decoherence all other states move towards the axis connecting the north- and south poles (i.e. the intersection of the z-axis with B³) as t → ∞. In the Heisenberg picture, this disappearance of all pure states except two corresponds to the reduction of the full algebra of observables M₂(ℂ) of the system to its diagonal (and hence commutative) subalgebra ℂ ⊕ ℂ in the same limit. For it is only the latter algebra that contains enough elements to distinguish ↑ and ↓ without containing observables detecting interference terms between these pure states.

10.

To understand this in a more abstract and general way, we recall the mathematical relationship between pure states and observables [Landsman, 1998]. The passage from a C*-algebra $\mathcal{A}$ of observables of a given system to its pure states is well known: as a set, the pure state space $\mathcal{P}$($\mathcal{A}$) is the extreme boundary of the total state space $\mathcal{S}$($\mathcal{A}$) (cf. footnote 259). In order to reconstruct $\mathcal{A}$ from $\mathcal{P}$($\mathcal{A}$), the latter needs to be equipped with the structure of a transition probability space (see Subsection 6.3) through (27). Each element $\mathcal{A}$ ∈ $\mathcal{A}$ defines a function  on $\mathcal{P}$($\mathcal{A}$) by Â(ω) = ω(A). Now, in the simple case that $\mathcal{A}$ is finite-dimensional (and hence a direct sum of matrix algebras), one can show that each function  is a finite linear combination of the form $\mathcal{A}$ = Σ_i pωi, where ω_i ∈ $\mathcal{P}$($\mathcal{A}$) and the elementary functions p_ρ on $\mathcal{P}$($\mathcal{A}$) are defined by p_ρ(σ) = p (ρ, σ). Conversely, each such linear combination defines a function $\mathcal{A}$ for some $\mathcal{A}$ ∈ $\mathcal{A}$. Thus the elements of $\mathcal{A}$ (seen as functions on the pure state space $\mathcal{P}$($\mathcal{A}$)) are just the transition probabilities and linear combinations thereof. The algebraic structure of $\mathcal{A}$ may then be reconstructed from the structure of $\mathcal{P}$($\mathcal{A}$) as a Poisson space with a transition probability (cf. Subsection 6.5). In this sense $\mathcal{P}$($\mathcal{A}$) uniquely determines the algebra of observables of which it is the pure state space. For example, the space consisting of two points with classical transition probabilities (31) leads to the commutative algebra $\mathcal{A}$ = ℂ ⊕ ℂ, whereas the unit two-sphere in ℝ³ with transition probabilities (32) yields $\mathcal{A}$ = M₂(ℂ).

This reconstruction procedure may be generalized to arbitrary C*-algebras [Landsman, 1998], and defines the precise connection between the Schrödinger picture and the Heisenberg picture that is relevant to decoherence. These pictures are equivalent, but in practice the reconstruction procedure may be difficult to carry through.

11.

For this reason it is of interest to have a direct description of decoherence in the Heisenberg picture. Such a description has been developed by Blanchard and Olkiewicz [2003], partly on the basis of earlier results by Olkiewicz [1999a, b; 2000]. Mathematically, their approach is more powerful than the Schrödinger picture on which most of the literature on decoherence is based. Let $\mathcal{A}$_S = $\mathcal{B}$($\mathcal{H}$_S) and $\mathcal{A}$_ε = $\mathcal{B}$($\mathcal{H}$_ε), and assume one has a total Hamiltonian H acting on $\mathcal{H}$_S ⊗ $\mathcal{H}$_ε as well as a fixed state of the environment, represented by a density matrix ρ_ε (often taken to be a thermal equilibrium state). If ρ_S is a density matrix on $\mathcal{H}$_S (so that the total state is ρ_S ⊗ ρ_ε), the Schrödinger picture approach to decoherence (and more generally to the quantum theory of open systems) is based on the time-evolution

(1)${\rho }_{\mathcal{S}}\left(t\right)={\text{Tr}}_{\mathcal{H}\varepsilon }\left(e^{-\frac{it}{\hslash }H}{\rho }_{\mathcal{S}}\otimes {\rho }_{\mathcal{E}}{e}^{\frac{it}{\hslash }H}\right)\cdot$

The Heisenberg picture, on the other hand, is based on the associated operator time-evolution for $\mathcal{A}$ ∈ $\mathcal{B}$($\mathcal{H}$_S) given by

(2)$A\left(t\right)={\text{Tr}}_{\mathcal{H}\varepsilon }\left({\rho }_{\mathcal{E}}{e}^{\frac{it}{\hslash }H}A\otimes 1e^{-\frac{it}{\hslash }H}\right),$

since this yields the equivalence of the Schrödinger and Heisenberg pictures expressed by

(3)${\text{Tr}}_{{\mathcal{H}}_{\mathcal{S}}}\left({\rho }_{\mathcal{S}}\left(t\right)A\right)={\text{Tr}}_{{\mathcal{H}}_{\mathcal{S}}}\left({\rho }_{\mathcal{S}}A\left(t\right)\right)\cdot$

More generally, let $\mathcal{A}$_S and $\mathcal{A}$_ε be unital C*-algebras with spatial tensor product $\mathcal{A}$_S ⊗ $\mathcal{A}$_ε, equipped with a time-evolution α_t and a fixed state ω_ε on $\mathcal{A}$_ε. This defines a conditional expectation $\mathcal{P}$_ε: $\mathcal{A}{\hspace{0.17em}}_{\mathcal{S}}$⊗$\mathcal{A}$_ε → $\mathcal{A}{\hspace{0.17em}}_{\mathcal{S}}$ by linear and continuous extension of $\mathcal{P}$_ε(A ⊗ B) = A_ωε(B), and consequently a reduced time-evolution A ↦ A (t) on $\mathcal{A}{\hspace{0.17em}}_{\mathcal{S}}$ via

(4)$A\left(t\right)=P_{\mathcal{E}}\left({\alpha }_t\left(A\otimes 1\right)\right)\cdot$

See, for example, Alicki and Lendi [1987]; in our context, this generality is crucial for the potential emergence of continuous classical phase spaces; see below.³¹⁹ Now the key point is that decoherence is described by a decomposition $\mathcal{A}$_S = $\mathcal{A}$_S⁽¹⁾ ⊗ $\mathcal{A}$⁽²⁾_S as a vector space (not as a C*-algebra), where $\mathcal{A}$⁽¹⁾_S is a C*-algebra, with the property that lim_t→∞ A (t) = 0 (weakly) for all A ∈ $\mathcal{A}$_S⁽²⁾, whereas A ↦ A (t) is an automorphism on $\mathcal{A}$_S⁽¹⁾ for each finite t. Consequently, $\mathcal{A}$⁽¹⁾_S is the effective algebra of observables after decoherence, and it is precisely the pure states on $\mathcal{A}$⁽¹⁾_S that are robust or einselected in the sense discussed before.

12.

For example, if $\mathcal{A}$_S = M₂(ℂ) and the states ↑ and ↓ are robust under decoherence, then $\mathcal{A}$_S = ℂ ⊕ ℂ and $\mathcal{A}$⁽²⁾_S consists of all 2 × 2 matrices with zeros on the diagonal. In this example $\mathcal{A}$⁽¹⁾_S is commutative hence classical, but this may not be the case in general. But if it is, the automorphic time-evolution on $\mathcal{A}$_S⁽¹⁾ induces a classical flow on its structure space, which should be shown to be Hamiltonian using the techniques of Section 6.³²⁰

In any case, there will be some sort of classical behaviour of the decohered system whenever $\mathcal{A}$_S⁽¹⁾ has a nontrivial center.³²¹ If this center is discrete, then the induced time-evolution on it is necessarily trivial, and one has the typical measurement situation where the center in question is generated by the projections on the eigenstates of a pointer observable with discrete spectrum. This is generic for the case where $\mathcal{A}$_S is a type i factor. However, type ii and iii factors may give rise to continuous classical systems with nontrivial time-evolution; see [Lugiewicz and Olkiewicz, 2002; 2003]. We cannot do justice here to the full technical details and complications involved here. But we would like to emphasize that further to quantum field theory and the theory of the thermodynamic limit, the present context of decoherence should provide important motivation for specialists in the foundations of quantum theory to learn the theory of operator algebras.³²²

1.

The starting point of the consistent histories formulation of quantum theory is conventional: one has a Hilbert space $\mathcal{H}$, a state ρ, taken to be the initial state of the total system under consideration (realized as a density matrix on $\mathcal{H}$) and a Hamiltonian $\mathcal{H}$ (defined as a self-adjoint operator on $\mathcal{H}$). What is unconventional is that this total system may well be the entire Universe. Each property α of the total system is mathematically represented by a projection $\mathcal{P}$_α on $\mathcal{H}$; for example, if α is the property that the energy takes some value ε, then the operator $\mathcal{P}$_α is the projection onto the associated eigenspace (assuming ε belongs to the discrete spectrum of H). In the Heisenberg picture, $\mathcal{P}$_α evolves in time as P_α(t) according to (12); note that P_α(t) is once again a projection.

A history ℍ_A is a chain of properties (or propositions) (α₁(t₁),…, α_n(t_n)) indexed by n different times t₁ < … < t_n; here A is a multi-label incorporating both the properties (α₁,…, α_n) and the times (t₁,…, t_n). Such a history indicates that each property α_i holds at time t_i, i = 1,…, n. Such a history may be taken to be a collection {α(t)}_t∈ℝ defined for all times, but for simplicity one usually assumes that α(t) ≠ 1 (where 1 is the trivial property that always holds) only for a finite set of times t; this set is precisely {t₁,…, t_n}. An example suggested by Heisenberg (1927) is to take α_i to be the property that a particle moving through a Wilson cloud chamber may be found in a cell Δ_i ⊂ ℝ⁶ of its phase space; the history (α₁(t₁),…, α_n(t_n)) then denotes the state of affairs in which the particle is in cell Δ₁ at time t₁, subsequently is in cell δ₂ at time t₂, etcetera. Nothing is stated about the particle's behaviour at intermediate times. Another example of a history is provided by the double slit experiment, where α₁ is the particle's launch at the source at t₁ (which is usually omitted from the description), α₂ is the particle passing through (e.g.) the upper slit at t₂, and α₃ is the detection of the particle at some location L at the screen at t₃. As we all know, there is a potential problem with this history, which will be clarified below in the present framework.

The fundamental claim of the consistent historians seems to be that quantum theory should do no more (or less) than making predictions about the probabilities that histories occur. What these probabilities actually mean remains obscure (except perhaps when they are close to zero or one, or when reference is made to some measurement context; see [Hartle, 2005]), but let us first see when and how one can define them. The only potentially meaningful mathematical expression (within quantum mechanics) for the probability of a history ℍ_A with respect to a state ρ is [Groenewold, 1952; Wigner, 1963]

(5)$p\left({ℍ}_A\right)=\text{Tr}\left(C_A\rho C_A^{*}\right),$

where

(6)$C_A=P_{{\alpha }_n}\left(t_n\right)\cdots P_{{\alpha }_1}\left(t_1\right)\cdot$

Note that C_A is generally not a projection (and hence a property) itself (unless all P_αi mutually commute). In particular, when ρ = [Ψ] is a pure state (defined by some unit vector Ψ ∈ $\mathcal{H}$), one simply has

(7)$p\left({ℍ}_A\right)={\Vert C_A\Psi \Vert }^{\text{2}}={\Vert P_{{\alpha }_n}\left(t_n\right)\cdots P_{{\alpha }_1}\left(t_1\right)\Psi \Vert }^2\cdot$

When n = 1 this just yields the Born rule. Conversely, see Isham (1994) for a derivation of (5) from the Born rule.³²⁴

2.

Whatever one might think about the metaphysics of quantum mechanics, a probability makes no sense whatsoever when it is only attributed to a single history (except when it is exactly zero or one). The least one should have is something like a sample space (or event space) of histories, each (measurable) subset of which is assigned some probability such that the usual (Kolmogorov) rules are satisfied. This is a (well-known) problem even for a single time t and a single projection P_α (i.e. n = 1). In that case, the problem is solved by finding a self-adjoint operator A of which P_α is a spectral projection, so that the sample space is taken to be the spectrum σ(A) of A, with α ⊂ σ(A). Given P_α, the choice of A is by no means unique, of course; different choices may lead to different and incompatible sample spaces. In practice, one usually starts from A and derives the P_α as its spectral projections P_α = ∫_α dP (λ), given that the spectral resolution of A is A = ∫_ℝ dP (λ) λ. Subsequently, one may then either coarse-grain or fine-grain this sample space. The former is done by finding a partition σ(A) = ∐_i α_i (disjoint union), and only admitting elements of the σ-algebra generated by the α_i as events (along with the associated spectral projection P_αi), instead of all (measurable) subsets of σ(A). To perform fine-graining, one supplements A by operators that commute with A as well as with each other, so that the new sample space is the joint spectrum of the ensuing family of mutually commuting operators.

In any case, in what follows it turns out to be convenient to work with the projections P_α instead of the subsets α of the sample space; the above discussion then amounts to extending the given projection on $\mathcal{H}$ to some Boolean sublattice of the lattice $\mathcal{P}$($\mathcal{H}$) of all projections on $\mathcal{H}$.³²⁵ Any state ρ then defines a probability measure on this sublattice in the usual way [Beltrametti and Cassinelli, 1984].

3.

Generalizing this to the multi-time case is not a trivial task, somewhat facilitated by the following device (Isham, 1994). Put $\mathcal{H}$^N = ⊗^N$\mathcal{H}$, where N is the cardinality of the set of all times t_i relevant to the histories in the given collection,³²⁶ and, for a given history ℍ_A, define

(8)${ℂ}_A=P_{{\alpha }_n}\left(t_n\right)\otimes \cdots \otimes P_{{\alpha }_1}\left(t_1\right)\cdot$

Here P_αi (t_i) acts on the copy of $\mathcal{H}$ in the tensor product $\mathcal{H}$^N labeled by t_i, so to speak. Note that ℂ_A is a projection on $\mathcal{H}$^N (whereas C_A in (6) is generally not a projection on $\mathcal{H}$). Furthermore, given a density matrix ρ on $\mathcal{H}$ as above, define the decoherence functional d as a map from pairs of histories into ℂ by

(9)$d\left({ℍ}_A,{ℍ}_B\right)=\text{Tr}\left(C_A\rho C_B^{*}\right)\cdot$

The main point of the consistent histories approach may now be summarized as follows: a collection {ℍA$\}{\hspace{0.17em}}_{\mathrm{A\in }\mathcal{A}}$ of histories can be regarded as a sample space on which a state ρ defines a probability measure via (5), which of course amounts to

(10)$p\left({ℍ}_A\right)=d\left({ℍ}_A,{ℍ}_A\right),$

provided that:

(a)

The operators {ℂ_A$\}{\hspace{0.17em}}_{\mathrm{A\in }\mathcal{A}}$ form a Boolean sublattice of the lattice $\mathcal{P}$($\mathcal{H}$^N) of all projections on $\mathcal{H}$^N;

(b)

The real part of d (ℍ_A, ℍ_B) vanishes whenever ℍ_A is disjoint from ℍ_B.³²⁷

In that case, the set {ℍ$\}{\hspace{0.17em}}_{\mathrm{A\in }\mathcal{A}}$ is called consistent. It is important to realize that the possible consistency of a given set of histories depends (trivially) not only on this set, but in addition on the dynamics and on the initial state.

Consistent sets of histories generalize families of commuting projections at a single time. There is no great loss in replacing the second condition by the vanishing of d (ℍ_A, ℍ_B) itself, in which case the histories ℍ_A and ℍ_B are said to decohere.³²⁸ For example, in the double slit experiment the pair of histories {ℍ_A, ℍ_B} where α₁ = β₁ is the particle's launch at the source at t₁, α₂ (β₂) is the particle passing through the upper (lower) slit at t₂, and α₃ = β₃ is the detection of the particle at some location L at the screen, is not consistent. It becomes consistent, however, when the particle's passage through either one of the slits is recorded (or measured) without the recording device being included in the histories (if it is, nothing would be gained). This is reminiscent of the von Neumann chain in quantum measurement theory, which indeed provides an abstract setting for decoherence (cf. item 1 in the preceding subsection). Alternatively, the set can be made consistent by omitting α₂ and β₂. See [Griffiths, 2002] for a more extensive discussion of the double slit experiment in the language of consistent histories.

More generally, coarse-graining by simply leaving out certain properties is often a promising attempt to make a given inconsistent set consistent; if the original history was already consistent, it can never become inconsistent by doing so. Fine-graining (by embedding into a larger set), on the other hand, is a dangerous act in that it may render a consistent set inconsistent.

4.

What does it all mean? Each choice of a consistent set defines a “universe of discourse” within which one can apply classical probability theory and classical logic [Omnès, 1992]. In this sense the consistent historians are quite faithful to the Copenhagen spirit (as most of them acknowledge): in order to understand it, the quantum world has to be looked at through classical glasses. In our opinion, no convincing case has ever been made for the absolute necessity of this Bohrian stance (cf. Subsection 3.1), but accepting it, the consistent histories approach is superior to Copenhagen in not relying on measurement as an a priori ingredient in the interpretation of quantum mechanics.³²⁹ It is also more powerful than the decoherence approach in turning the notion of a system into a dynamical variable: different consistent sets describe different systems (and hence different environments, defined as the rest of the Universe); cf. item 6 in the previous subsection.³³⁰ In other words, the choice of a consistent set boils down to a choice of “relevant variables” against “irrelevant” ones omitted from the description. As indeed stressed in the literature, the act of identification of a certain consistent set as a universe of discourse is itself nothing but a coarse-graining of the Universe as a whole.

5.

But these conceptual successes come with a price tag. Firstly, consistent sets turn out not to exist in realistic models (at least if the histories in the set carry more than one time variable). This has been recognized from the beginning of the program, the response being that one has to deal with approximately consistent sets for which (the real part of) d (ℍ_A, ℍ_B) is merely very small. Furthermore, even the definition of a history often cannot be given in terms of projections. For example, in Heisenberg's cloud chamber example (see item 1 above), because of his very own uncertainty principle it is impossible to write down the corresponding projections P_αi. A natural candidate would be $P_{\alpha }\hspace{0.17em}=\hspace{0.17em}{\mathcal{Q}}_{\hslash }^B(\chi \Delta )$, cf. (19) and (28), but in view of (21) this operator fails to satisfy P²_α = P_α, so that it is not a projection (although it does satisfy the second defining property of a projection P*_α = P_α). This merely reflects the usual property $\mathcal{Q}(f)2\hspace{0.17em}\ne \hspace{0.17em}\mathcal{Q}(f^2)$ of any quantization method, and necessitates the use of approximate projections [Omnès, 1997]. Indeed, this point calls for a reformulation of the entire consistent histories approach in terms of positive operators instead of projections [Rudolph, 1996a, b].

These are probably not serious problems; indeed, the recognition that classicality emerges from quantum theory only in an approximate sense (conceptually as well as mathematically) is a profound one (see the Introduction), and it rather should be counted among its blessings that the consistent histories program has so far confirmed it.

6.

What is potentially more troubling is that consistency by no means implies classicality beyond the ability (within a given consistent set) to assign classical probabilities and to use classical logic. Quite to the contrary, neither Schrödinger cat states nor histories that look classical at each time but follow utterly unclassical trajectories in time are forbidden by the consistency conditions alone [Dowker and Kent, 1996]. But is this a genuine problem, except to those who still believe that the earth is at the centre of the Universe and/or that humans are privileged observers? It just seems to be the case that — at least according to the consistent historians — the ontological landscape laid out by quantum theory is far more “inhuman” (or some would say “obscure”) than the one we inherited from Bohr, in the sense that most consistent sets bear no obvious relationship to the world that we observe. In attempting to make sense of these, no appeal to “complementarity” will do now: for one, the complementary pictures of the quantum world called for by Bohr were classical in a much stronger sense than generic consistent sets are, and on top of that Bohr asked us to only think about two such pictures, as opposed to the innumerable consistent sets offered to us. Our conclusion is that, much as decoherence does not solve the measurement problem but rather aggravates it (see item 2 in the preceding subsection), also consistent histories actually make the problem of interpreting quantum mechanics more difficult than it was thought to be before. In any case, it is beyond doubt that the consistent historians have significantly deepened our understanding of quantum theory — at the very least by providing a good bookkeeping device!

7.

Considerable progress has been made in the task of identifying at least some (approximately) consistent sets that display (approximate) classical behaviour in the full sense of the word [Gell-Mann and Hartle, 1993; Omnès, 1992; 1997; Halliwell, 1998; 2000; 2004; Brun and Hartle, 1999; Bosse and Hartle, 2005]. Indeed, in our opinion studies of this type form the main concrete outcome of the consistent histories program. The idea is to find a consistent set {ℍ$\}{\hspace{0.17em}}_{\mathrm{A\in }\mathcal{A}}$ with three decisive properties:

(a)

Its elements (i.e. histories) are strings of propositions with a classical interpretation;

(b)

Any history in the set that delineates a classical trajectory (i.e. a solution of appropriate classical equations of motion) has probability (10) close to unity, and any history following a classically impossible trajectory has probability close to zero;

(c)

The description is sufficiently coarse-grained to achieve consistency, but is sufficiently fine-grained to turn the deterministic equations of motion following from (b) into a closed system.

When these goals are met, it is in this sense (no more, no less) that the consistent histories program can claim with some justification that it has indicated (or even explained) ‘How the quantum Universe becomes classical’ [Halliwell, 2005].

Examples of propositions with a classical interpretation are quantized classical observables with a recognizable interpretation (such as the operators $\mathcal{B}$ħ (_χδ) mentioned in item 5), macroscopic observables of the kind studied in Subsection 6.1, and hydrodynamic variables (i.e. spatial integrals over conserved currents). These represent three different levels of classicality, which in principle are connected through mutual fine- or coarse-grainings.³³¹ The first are sufficiently coarse-grained to achieve consistency only in the limit ℏ → 0 (cf. Section 5), whereas the latter two are already coarse-grained by their very nature. Even so, also the initial state will have to be “classical” in some sense in order te achieve the three targets (a) – (c).

To a particle of the microworld, de Broglie ascribed wavefunction $\phi$. Realizing this determination in a literal sense, de Broglie introduced the relations

which connect energy $E$ and momentum $\mathbf{p}$ of a freely moving particle with frequency $\omega$ and wave vector $\mathbf{k}$ of a plane wave

in which $\omega =2\pi \nu =c|\mathbf{k}|\mathrm{and}\hslash =h/2\pi$. In other words, apart from the material, there is “something” that at each point $\mathbf{q}$ of space provides some information about the particle at the moment of time $t$. Born proposed the probability interpretation for function $\phi$. These waves are probability waves, for instance, the waves of probability where the particle is located at a concrete point of space or where the particle has a concrete value of energy. Despite his phenomenological explanation of the photoelectric effect, Einstein contested vigorously against the probability interpretation of a wavefunction: “God does not play dice.” Despite all its successes, Einstein, as is well known, treated quantum mechanics cautiously.

Depending on a representation, a wavefunction can be specified as the function of corresponding variables; for instance,

are images of a wavefunction in coordinate, momentum, and energy representations, respectively. To not indicate explicitly the chosen representation, it is convenient to express the states of quantum-mechanical systems through abstract vectors in a separable Hilbert space. According to Dirac’s notation, to a vector that characterizes some state $n$ we ascribe symbol $|n⟩$; furthermore,

is the vector that is the complex conjugate of $|n⟩$,

is the multiplication of vector by a complex number $c$,

is the scalar product of two vectors $|n⟩$ and $|m⟩$, and

is the matrix element of physical quantity $G$ between states $n$ and $m$. Vectors $|n⟩$ and $⟨n|$ are called ket and bra, respectively.

Any physical vector $|\mathrm{\Psi }⟩$ is expressible as a series expansion in terms of orthonormal states $|1⟩,|2⟩,\dots ,|\ell ⟩,\dots$, which in a set form a complete basis, and in a linear manner

in which $c_1,c_2,\dots ,c_{\ell },\dots$ are the pertinent coefficients; basis vectors $|\ell ⟩$ fail to be related to each other through any relation of linear type — they are linearly independent. The orthonormality condition means that

that is, the vectors are mutually orthogonal and each of them is normalized to unity. From a physical point of view, the state of a quantum-mechanical system is a superposition of all possible for this system state, and each of which gives its own contribution with a weight that is defined by corresponding coefficient $c_{\ell }$. Amplitude $c_{\ell }$ thus has a purely probability character, and quantity

can be interpreted as the probability of state $\ell$. Assuming that ket $|\mathrm{\Psi }⟩$ is normalized to unity, one might readily obtain this equality,

which proves that the total probability, as one should expect, equals unity; the event that the system occupies one available state, obviously, represents a certain event.

In common use, the coordinate representation is applied to the problems of quantum mechanics. In this case, the scalar product of wavefunctions is given by this expression

in which $M$ is the manifold, in which functions ${\phi }_n$ and ${\phi }_m$, corresponding to states $n$ and $m$, respectively, are defined and $\mathrm{d}\mathbf{q}$ is an element of volume. Quantity

according to Born’s probability interpretation, represents the probability that the values of coordinates of a system lie in an interval between $\mathbf{q}$ and $\mathbf{q}+\mathrm{d}\mathbf{q}$. Thus, diagonal matrix element $⟨n|G|n⟩,$ which is equal to

should be understood as the average value of physical quantity $G$ in state $n$; if $G$ is simply the coordinate function, then

The general formula for an arbitrary matrix element in the coordinate representation, by definition, has the form

Analogous considerations are applicable for the case of the momentum representation. For instance,

is the probability to detect the momentum of a system in the vicinity of point $\mathbf{p}$ in the element of volume $\mathrm{d}\mathbf{p}$ of the momentum space, and

represents the average value of quantity $G$ in the state with wavefunction ${\chi }_n$ that is defined in manifold $M\prime$.

Suppose that $q$ and $p$ are the canonically conjugate coordinate and momentum, $\phi (q)$ is the wavefunction in the coordinate representation, and $\chi (p)$ is the same function, but in the $p$-representation; quantities $\phi (q)$ and $\chi (p)$ are related between each other through the Fourier integral

The average value of momentum $p$, on the one side, is expressible as

and on the other side,

through which symbol “?” designates the operator of momentum in the coordinate representation. Our task,¹ taking into account that

is to determine the explicit form of “?.”

So, using the expansion of $\chi (p)$ in the Fourier integral, we have

Putting

and taking into account the boundary conditions $\phi (\pm \infty )=0$, we execute the integration by parts:

consequently,

As

then

Hence,

and, in the coordinate representation, the momentum is expressible as a linear differential operator:

With respect to coordinate, the analogous considerations are valid. In this case, we apply the inverse Fourier transformation

$p$ and $q$ are simply interchanged and $\mathrm{i}$ is converted into $-\mathrm{i}$. As a result,

represents the expression for the operator of coordinate in the momentum representation.

Having determined a commutator for two quantities $G$ and $F$ as follows,

we see that

hence,

and the conjugate coordinate and momentum in quantum mechanics fail to conform to the law of commutative multiplication. This condition constitutes properly the main distinction between quantum and classical theories. Classical physics operates with commuting to each other c-numbers, which are generally measurable in the experiment. The natural variables of quantum physics are opposite, unobservable q-numbers. The quantities of q-type, failing to commute generally with each other, act in a Hilbert space of abstract state vectors. The vectors are supposed to be transformed to each other through the action of q-numbers, which are in a sense the pertinent operators. Not all mathematical operators find their application in practice in physics. The principal requirements are linearity and hermitivity. Let

be arbitrary vectors, and let

in which $b$ and $c$ are numerical coefficients. By definition, $G$ is the linear operator if

$G$ is the Hermitian operator if

in which $G^{+}$ satisfies the relation

and represents an operator Hermitian conjugate to $G$. For instance, as is easily seen in the coordinate representation, the differential operator for momentum possesses the properties of both hermitivity and linearity.

However, there exist such states in which the dynamic variables of quantum origin have determinate values belonging to a class of observable c-numbers. We imply here the eigenvalues of physical operators. So, if q-number $G$ acts on state vector $|n⟩$ without altering its “direction,” an equation

is valid and $g_n$ is a number of c-type, then $|n⟩$ and $g_n$ are, respectively, eigenvector and eigenvalue of operator $G$. In a general case, $G$ has a set of eigenvectors, each of which corresponds to a determinate eigenvalue. We show that the eigenvalues of physical operators are strictly real numbers. So, on the one side,

on the other side, taking into account that $G=G^{+}$, we have

consequently,

completely proves our assertion.

The eigenvectors that belong to various eigenvalues are mutually orthogonal. Let $|n⟩$ and $|m⟩$ be eigenvectors of physical operator $G$; they correspond to the equations for eigenvalues, respectively, $g_n$ and $g_m$:

As $⟨m|G^{+}=g_m^{*}⟨m|$, $G^{+}=G$, and $g_m^{*}=g_m$, then

On comparing, we find

hence

because $g_n\ne g_m.$ The property of mutual orthogonality of eigenvectors plays an important role in the problems of quantum mechanics. Arbitrary vector $|\mathrm{\Psi }⟩$, for instance, can be represented in a form of expansion

in which $c_n$ are the amplitudes of states in terms of the complete system of eigenvectors $|n⟩$ of some operator $G$. If $g_n$ are the eigenvalues corresponding to states $n$, then the average value of $G$ in state $\mathrm{\Psi }$ is given by the expression

that, in probability theory, represents a formula for an expectation value of quantity $G$.

Finally, we discuss a useful result regarding the possibility of simultaneous measurement of two dynamical quantities. Suppose that

in which $|n⟩$ and $g_n$ represent, respectively, the eigenvectors and eigenvalues of physical operator $G$. On acting on this equation with some variable $F$ on the left, we have

If $[G,F]=0$, then

consequently, quantity $F|n⟩$ is the eigenvector of operator $G$. Hence,

in which $f_n$ are c-numbers. Commutative variables $G$ and $F$ thus have a common complete system of eigenstates. In this case, the dynamical quantities are simultaneously measurable.