We develop a new operator algebraic formulation of the Nakajima-Mori-Zwanzig (NMZ) method of projections. The new theory is built upon rigorous mathematical foundations, and it can be applied to both classical and quantum systems. We show that a duality principle between the NMZ formulation in the space of observables and in the state space can be established, analogous to the Heisenberg and Schrödinger pictures in quantum mechanics. Based on this duality we prove that under natural assumptions, the projection operators appearing in the NMZ equation must be conditional expectations. The proposed formulation is illustrated in various examples.

High-dimensional stochastic dynamical systems arise in many areas of mathematics, natural sciences, and engineering. Whether it is a physical system being studied in a lab or an equation being solved on a computer, the full microscopic state of the system as a point evolving in some phase space is often intractable to handle in all its complexity.

Instead, it is often desirable to attempt to reduce the complexity of the theoretical description by passing from a model of the dynamics of the full system to a model only of the observables of interest. Such observables may be chosen, for example, because they represent global macroscopic features of the bulk system, as in the derivation of the Boltzmann equation of nonequilibrium thermodynamics from microscopic descriptions1–3 or in the derivation of the dynamics of commutative subalgebras of observables in quantum mechanics.4 The observables may also represent features localized on a subsystem of interest, as in the Brownian motion of a particle in a liquid, where the master equation governing the position and momentum of the particle is derived from first principles (Hamiltonian equations of motion of the full system), by eliminating the degrees of freedom associated with the surrounding liquid.5,6 In the context of numerical approximation of stochastic partial differential equations (SPDEs), the observables may be chosen to define a finite-dimensional approximation of the phase space, for example, a finite set of Fourier-Galerkin coefficients.7–9 Whatever the reason behind this reduction of the set of observables, it is often desirable to then attempt to reduce the complexity of the theoretical description by passing from a model of the dynamics of the full system to a model only of the observables of interest. For example, we might have a high-dimensional dynamical system evolving as dx/dt = F(x), but we are only interested in a relatively small number of C-valued observable functions g1(x),,gm(x). The dynamics of this lower-dimensional set of observable quantities may be simpler than that of the entire system although the underlying law by which such quantities evolve in time is often quite complex. Nevertheless, approximation of such a law can in many cases allow us to avoid performing simulation of the full system and solve directly for the quantities of interest. If the resulting equation for {gi(x)} is low dimensional and computable, this provides a means of avoiding the curse of dimensionality.

In this paper, we study one family of techniques for performing such dimensional reduction, namely, the Nakajima-Mori-Zwanzig (NMZ) method of projections10–13 (see also Refs. 14–16). To this end, we place the NMZ formulation in the context of C*-algebras of observables, and in doing so, set rigorous foundations of this important and widely used technique. More importantly, the operator algebraic setting we propose unifies classical and quantum mechanical formulations. The method of projections derives its name from the use of a projection map from the algebra of observables of the full system to the subalgebra of interest. In this algebraic context, it will naturally emerge that the two common flavors of NMZ—for “phase space functions” and for probability density functions (PDFs)—are dual equations for observables and states, directly corresponding to the dual Schrödinger and Heisenberg pictures of quantum mechanics. Reasoning about information in these algebras and desiderata of the NMZ projection will reveal that the projection must be a conditional expectation in the operator algebraic sense.

The paper is organized as follows. We begin in Sec. II with a quick review of C*-algebras, their states and homomorphisms, and the relationship between topological spaces and algebras of functions. We then discuss the relationship between classical dynamical systems and observable algebras in Sec. III, deriving from the nonlinear dynamical system the equivalent linear dynamics on the observable algebra. In Sec. IV, the NMZ equation is introduced for the reduced dynamics on an observable algebra, along with the dual NMZ equation on the states of the algebra. We then look more closely at the NMZ projection operator in Sec. V, finding that under natural assumptions, the projection operator must be a conditional expectation. While the elements of C*-algebras are bounded observables, it is common to consider also unbounded observables (such as momentum); the incorporation of such affiliated observables into the NMZ framework is considered in Sec. V B. In Sec. VI, we consider the problem of “pushing” the dynamics from one space to another (typically lower dimensional) space using NMZ and discuss the application of NMZ to quantum open systems. In Sec. VII, two simple examples of the NMZ method are carried out analytically. Finally, the main results are summarized in Sec. VIII. We also include two brief  Appendices A and  B, in which we discuss technical questions related to non-degenerate homeomorphisms and state-preserving maps.

In this section, we provide a quick review of C*-algebras, their states and homomorphisms, and the relationship between topological spaces and algebras of functions. The material in this section is well-known and can be found throughout the literature on operator algebras and algebraic dynamics. Standard references for much of this material include, e.g., Refs. 17–20.

We are interested in developing the NMZ formalism simultaneously for dynamical systems

ẋ=F(x,ξ,t),
(1)

evolving on a (sufficiently) smooth manifold M, where ξ represents parameters drawn (perhaps randomly) from some parameter space Ξ, as well as for quantum mechanical systems. By “manifold” here we mean any of a large class of spaces on which (1) makes sense, including, at a minimum, finite-dimensional manifolds, Banach spaces, and more general Banach manifolds. In particular, the simple form of (1) can represent many different kinds of the initial value problem, including ordinary differential equations (ODEs), partial differential equations (PDEs), and functional differential equations.21 Thinking first of the classical dynamical system above and of M as a generalized phase space of the system, a classical observable will typically be a C-valued function on M. There are different possible choices for the set of such functions, but certain properties may be desirable.22,23 For example, if we can observe f and g, then we should be able to observe αf+βg for any α,βC. We should also expect to be able to observe the product fg. In this way, we should expect the space of observables to form an algebra of functions under pointwise addition and multiplication. More careful and detailed reasoning23 about arbitrary physical systems (be they classical or quantum) leads to the conclusion that the set of observables for any physical system can be represented as a C*-algebra. That algebra will generally be commutative in the classical case and noncommutative in the quantum case.

A Banach algebra is an algebra A over C with a norm making A a Banach space and such that xyxy for all x,yA. A C*-algebra is a Banach algebra A with an isometric involution xx* such that (xy)*=y*x* for all x,yA and such that x*x=x2 for all xA.

An important subclass of C*-algebras is formed by the von Neumann (i.e., W*-)algebras that are unital C*-algebras closed with respect to the ultraweak topology. They can be characterized as those C*-algebras that admit a Banach space predual,24 i.e., a Banach space whose dual space is isomorphic to the C*-algebra. Two key commutative algebras of functions to keep in mind are the following:

  1. A=C0(M). This is the algebra of continuous C-valued functions on M “vanishing at infinity.” This means that for any fA and any 𝜖>0, the set {xM:|f(x)|𝜖} is compact within M. The algebra C0(M) is endowed with the sup norm, i.e., f=supxMf(x). It is a unital algebra if and only if M is compact, in which case the identity is the function that is everywhere equal to one: 1(x)=1 for all xM. The dual space A* is isometrically isomorphic to the space M(M) of all complex-valued regular Borel measures (i.e., Radon measures) with finite norm
    ν=supfC0(M)Mf(x)dν(x)f.
    (2)
  2. A=L(M,μ). This is the algebra of (equivalence classes of) essentially bounded C-valued measurable functions on M, for some choice of localizable25 regular Borel measure (i.e., Radon measure) μ on M. The norm on this algebra is the essential supremum f=inf{r>0:|f(x)|rμ-a.e.}. The dual space A* is isometrically isomorphic to the space of finitely additive localizable complex-valued Borel measures absolutely continuous with respect to μ and with finite norm
    ν=supfL(M,μ)Mf(x)dν(x)f.
    (3)
    L(M,μ) is a von Neumann algebra, and, as such, admits a unique (up to isometric isomorphism) predual L(M,μ)*, which can be identified with L1(M,μ). Of course, L1(M,μ) can itself be identified with the space of localizable C-valued Borel measures absolutely continuous with respect to μ.

A function g in A=C0(M) or A=L(M,μ) is considered to be positive if g(x)0 everywhere (or almost everywhere, as appropriate). This will be denoted as g0. More abstractly, in a general C*-algebra A, G0 if there exist HA such that G=H*H. The positive elements of A form a closed convex cone. An element ϕ of the Banach dual space A* is positive if ϕ(G)0 for all G0. And ϕA* is a state if it is positive and ϕ=1. Note that in the case that A is unital, the normalization condition ϕ=1 is equivalent to ϕ(1)=1. We will denote the set of states as S(A)A*. The equivalence of A* with a Banach space of C-valued (at least finitely additive) measures on M implies that S(A) can be identified with the subset of probability measures on M. Thus for any ρS(A) and any GA, ρ(G) is the expectation value of G over the state (probability measure) ρ. In other words, if ρ^ is the measure associated with the state ρ, then ρ(G) can be interpreted as

ρ(G)=MG(x)dρ^(x).

When A is a von Neumann algebra such as L(M,μ), we identify the positive cone in the predual A* as follows: for any ϕA*, ϕ0 if G(ϕ)0 for all G0 in A. The set of normal statesSN(A) is the set of ρA* such that ρ0 and ρ=1. Because of the added complications that finitely additive measures impose, it may be advantageous to work with states in the predual when we take A=L(M,μ). In what follows, we will use the notation associated with duals (rather than preduals), but replacing this with the predual (and normal states) in the case of von Neumann algebras is straightforward.

Although ρ(G) is the expectation value of G, the state ρ carries more detailed statistical information about the result of measuring the observable G. Indeed, repeated measurements of a normal observable G (meaning G*G=GG*) against an ensemble of systems in identical states does not simply yield the expectation value but samples from a probability measure of values. Let A be any unital C*-algebra and ρS(A) be a state; if A is not unital, pass to the unitization à and extend ρ by ρ(1)=1 to a state of Ã. For any normal GA, let σ(G)C (topologized as a compact subset of C) be the spectrum of G and C(σ(G)) be the C*-algebra of continuous C-valued functions on σ(G). There is a continuous functional calculus, expressible as a unital *-morphism ΦG:C(σ(G))A, which is an isomorphism onto the unital subalgebra of A generated by G [Ref. 18, Proposition I.4.6]. Then ρG:=ΦG*ρ is a state of C(σ(G)) and therefore may be identified through the Riesz-Markov theorem with a Radon probability measure on σ(G) or equivalently with a Radon probability measure μ on C that is supported on σ(G). This measure describes the probability of observing value λC when measuring observable G on a system in state ρ.

Here we briefly review some categories of topological spaces, measure spaces, and algebras that will be relevant to the remainder of the paper.

  • LoCompHaus: locally compact Hausdorff topological spaces with proper continuous maps.

  • CommC*Alg: commutative (not necessarily unital) C* algebras with continuous (i.e., bounded) non-degenerate*-homomorphisms. Here f:ABnon-degenerate means SpanC{f(a)b:aA,bB} is dense in B.26 See also Lemma 1 in  Appendix A for an alternative characterization of non-degenerate morphisms.

  • LocMeas: localizable measure spaces25 with non-singular measurable functions (defined almost everywhere), i.e., f:(X,Σ,μ)(Y,Σ,ν) is such that μ(f1(A))=0 whenever ν(A)=0.

  • CommVNA: commutative (unital) von Neumann algebras with ultraweakly continuous unital*-homomorphisms.

  • C*Alg: (not necessarily unital) C* algebras with continuous (i.e., bounded) non-degenerate*-homomorphisms.

  • vNAlg: (unital) von Neumann algebras with ultraweakly continuous unital*-homomorphisms.

When describing the NMZ formalism for a dynamical system evolving on a manifold (perhaps infinite-dimensional), we will be working largely in the two categories of commutative algebras CommC*Alg and CommVNA. The more general setting—including the NMZ formalism for quantum mechanics—involves the categories of noncommutative algebras C*Alg and vNAlg.

Gelfand duality19,26 implies that LoCompHaus is equivalent to CommC*Algop (see Fig. 1). Thus, for example, any commutative C* algebra is realizable as the algebra C0(X) of continuous functions vanishing at infinity on an essentially unique locally compact Hausdorff (LCH) topological space X, and every nondegenerate C*-homomorphism between such algebras is realizable as an essentially unique proper map between LCH spaces. A similar Segal duality25 implies that LocMeas is equivalent to CommVNAop (see Fig. 2). These duality theories, which we discuss further in Subsection II D, allow for the transformation from nonlinear dynamical systems to completely equivalent linear dynamical systems in Banach spaces.

FIG. 1.

Sketch of typical spaces and maps associated with the Gelfand representation of locally compact Hausdorff spaces and proper maps.

FIG. 1.

Sketch of typical spaces and maps associated with the Gelfand representation of locally compact Hausdorff spaces and proper maps.

Close modal
FIG. 2.

Sketch of typical spaces and maps associated with the Segal representation of localizable measure spaces and measurable maps.

FIG. 2.

Sketch of typical spaces and maps associated with the Segal representation of localizable measure spaces and measurable maps.

Close modal

We will make extensive use of the composition operator (i.e., the Koopman operator27) and its adjoint, the transfer operator (i.e., the Perron-Frobenius operator).28,29 Let γ:MN be a continuous map. In keeping with our definitions of LoCompHaus and LocMeas, we require that γ is non-singular: in LoCompHaus, we require that γ is proper (the pre-image of a compact set is compact), and in LocMeas, we require that μ(γ1(A))=0 whenever ν(A)=0 (where ν is the Borel measure on N ). Then the composition operator (Koopman) Kγ is the C*-homomorphism from B=C0(N) to A=C0(M) (or from B=L(N,ν) to A=L(M,μ)), defined by

Kγg=gγ.
(4)

As it is a C*-homomorphism, Kγ is contractive, i.e., Kγ1. In fact, it is easy to see that Kγ=1 since one can construct a nontrivial gB supported in the range of γ, for which we clearly have Kγg=g (see Lemma 3 in  Appendix A).

The dual of the composition operator is the transfer operator (Perron-Frobenius)30Kγ*:A*B* that essentially “pushes” states forward along γ. It is straightforward to show that Kγ* is positive and contractive, i.e., Kγ*ϕϕ for all ϕA*. Moreover, for any ϕ0 in A*, Kγ*ϕ=ϕ (see Lemma 4 in  Appendix A) so that Kγ*S(A)S(B). When A=C0(M) and B=C0(N), then A*M(M) (the space of Radon measures on M), B*M(N), and for any measurable BN and μM(M), (Kγ*μ)(B)=μ(γ1(B)). In the von Neumann algebra setting, when A*L1(M,μ) and B*L1(N,ν), the transfer operator Kγ* may be regarded as a transformation between L1 functions, defined via the Radon-Nikodym derivative

Kγ*ρ(y)=dηρdν(y),
(5)

for all yN, where ηρ is the push-forward measure on N given by

ηρ(A):=γ1(A)ρ(x)dμ(x).

Given the C*-algebra of observables A on our system, we may choose to only observe the system through a subcollection CA. By only viewing the system through these observables in C, we obtain only partial information (relative to that obtained from using all of the observables in A). However, the set C may not fully represent the set of observables whose value we know if we observe using C. For example, if f,gC, then we know the value of αf+βgA for any a,bC. We also know the value of f*, g*, and, at least when f and g commute, we know the value of fg. So, to represent the set of observables whose value we know after measurement, we must expand any mutually commuting set of observables C at least to the *-algebra containing C. In this paper (and in keeping with much of the literature in quantum mechanics and quantum information), we generally view partial information only through the lens of C*-subalgebras of observables.

Consider a nonautonomous dynamical system in form (1) and assume that the flow Φ(t,t0) exists for all tt0. Let A be a commutative C*-algebra of C-valued “observable” functions on a manifold M, and let KΦ(t,s) be the Koopman operator associated with Φ(t,s). Clearly, KΦ(t,s) is an *-endomorphism acting on A.

Let Φ(t,t0) be the flow generated by the dynamical system (1). For any observable g(x), we have

g(x(t))=KΦ(t,t0)g(x0).
(6)

By differentiating this equation with respect to t, we obtain

ddtKΦ(t,t0)g(x0)=KΦ(t,t0)LFtξg(x0),
(7)

where Ftξ=F(,ξ,t) is the vector field from (1) at fixed t and ξ, and LFtξg=dg(Ftξ) is an ultraweakly densely defined, ultraweakly closed, generally unbounded linear operator on A. Therefore,

ddtKΦ(t,t0)=KΦ(t,t0)LFtξ,
(8)

which implies

KΦ(t,s)=TestLFτξdτ.
(9)

Here T is the time-ordering operator placing later operators to the right. It should be noted that for time-dependent Lt and irreversible flow Φ(t,t0), the observable g(t) = g(x(t)) does not generally obey a simple evolution equation of the form dg(t)/dt = Rtg(t). This is because there need not exist a time-dependent operator Rt such that

ddtg(t)=KΦ(t,0)Ltg0=RtKΦ(t,0)g0=Rtg(t).
(10)

Now, consider a fixed state ρ0S(A). By using (6) and (9), we have

ρ0(g(t))=ρ0Te0tLFτξdτg0.

Differentiation with respect to t yields

ddtρ0(g(t))=ρ0ddtg(t)=ρ0Te0tLFτξdτLFtξg0=LFtξ*Te0tLFτξ*dτρ0(g0)=LFtξ*ρ(t)(g0),
(11)

i.e.,

ddtρ(t)=LFtξ*ρ(t).
(12)

The formal solution to (12) is

ρ(t)=KΦ(t,s)*ρ(0),
(13)

where

KΦ(t,s)*=Te0tLFτξ*dτ
(14)

is the transfer operator (Perron-Frobenius) associated with the flow map Φ(t,s). The Gelfand and Segal dualities (see Figs. 1 and 2) imply that the linear dynamics of the composition operator in B(A) [or the linear dynamics of the transfer operator in B(A*)] is completely equivalent to the nonlinear dynamics generated by (1) on M.

So far we have framed the discussion around dynamical systems evolving on manifolds, leading to commutative observable algebras A. However, very little of what we will do will depend on the commutativity of A. By and large, if we have any C*-algebra A and a linear evolution operator that is an *-endomorphism on A that evolves as in (8), it is possible to apply the NMZ formulation to obtain a generalized Langevin equation for the reduced dynamics. In particular, this applies to quantum mechanics. In this section, we develop the formalism in this more general perspective.

Let A be a (not necessarily commutative) C*-algebra. We will typically let A* denote the Banach space dual, and S(A)A*, the closed convex set of positive norm one linear functionals on A, be the set of states on A. However, when A is a von Neumann algebra (i.e., it admits a Banach predual), we will abuse the notation to let A* denote the Banach space predual, and S(A)A*, the closed convex set of positive norm one elements of the predual, be the set of (normal) states on A. A (not necessarily bounded) linear operator L acting on A is called a *-derivation if for any f,gA, L(fg*)=(Lf)g*+f(Lg)*.

Suppose that we have available a time-dependent family of closed, densely defined linear *-derivations {Lt}0tT, along with a *-endomorphism E(t,s) on A, strongly continuous in both s and t, and satisfying for ts,

E(t,s)=idA+stE(τ,s)Lτdτ,
(15)

i.e., satisfying, in the sense of Carathéodory, the differential equation

ddtE(t,s)=E(t,s)Lt
(16)

such that E(s,s) is the identity morphism idA on A. In the case of von Neumann algebras, it suffices for {Lt} to be weak-*-densely defined and for E(t,s) to be weak-* continuous in s and t. Then,

E(t,s)=TestLτdτ
(17)

is contractive, i.e., E(t,s)XX for all ts and all XA, by virtue of being a *-endomorphism. This E(t,t0) serves as the evolution operator for observables in A, i.e., E(t,t0)(G0)=G(t). In the classical case described in Sec. III, E(t,t0)=KΦ(t,t0).

We now introduce a projection P on A and develop from it the equations that comprise the NMZ formalism. The nature and properties of P will be discussed in detail in Sec. V, but for now it will suffice to assume only that P is a bounded linear operator acting on A and that P2=P. The NMZ formalism describes the evolution of observables initially in the image of the P. Because the evolution of observables is governed by E(t,t0) [see Eq. (17)], we seek an evolution equation for E(t,t0)P. To this end, recall first the well-known Dyson identity: if

Y(t,s)=TestA(τ)dτandZ(t,s)=TestB(τ)dτ,
(18)

then

Y(t,t0)Z(t,t0)=t0tddsY(s,t0)Z(t,s)ds=t0tY(s,t0)(A(s)B(s))Z(t,s)ds.
(19)

Applying this to Y(t,t0)=E(t,t0) [Eq. (17)] and Z(t,t0)=Tet0tQLτdτ (here Q=1P denotes the complementary projection), we find

E(t,t0)=Tet0tQLτdτ+t0tE(s,t0)PLsTestQLτdτds.
(20)

A differentiation with respect to time and composition with P yields the following generalized Langevin equation for E(t,t0)P:

ddtE(t,t0)P=E(t,t0)PLtP+Tet0tQLτdτQLtP+t0tE(s,t0)PLsTestQLτdτQLtPds.
(21)

Letting f0=Pg0 represent an observable initially in the image of P, the NMZ equation describes its evolution as

ddtf(t)=E(t,t0)PLtf0+Tet0tQLτdτQLtf0+t0tE(s,t0)PLsTestQLτdτQLtf0ds.
(22)

The Banach dual of (21) yields the NMZ equation for states

ddtP*E(t,t0)*=P*Lt*P*E(t,t0)*+P*Lt*Tet0tQ*Lτ*dτQ*+P*Lt*t0tTestQ*Lτ*dτQ*Ls*P*E(s,t0)*ds.
(23)

In this case, the generalized Langevin equation describing the evolution of a projected state σ(t)=P*ρ(t)=P*E(t,t0)*ρ0 is

ddtσ(t)=P*Lt*σ(t)+P*Lt*Tet0tQ*Lτ*dτQ*ρ0+P*Lt*t0tTestQ*Lτ*dτQ*Ls*σ(s)ds.
(24)

The NMZ equations (22) and (24) describe the exact evolution of the reduced observable algebras and states. In the context of classical dynamical systems, Eqs. (22) and (24) describe, respectively, the evolution of a phase space function (observable) and the corresponding probability density function. We would like to emphasize that the duality we just established between the NMZ formulations (22) and (24) extends the well-known duality between Koopman and Perron-Frobenious operators to reduced observable algebras and states.

Unlike the NMZ evolution equation for states (24), the evolution equation for observables (22) involves the explicit application of the full evolution operator (17). In other words, while (21) is a generalized Langevin equation for the evolution operator E(t,t0)P, (22) is not explicitly in Langevin form. However, the equation may be put into a Langevin-type form by considering a basis for the image of P. To this end, suppose that

{gk}t[t0,T]D(Lt)t0stTDLsTestQLτdτQLt

is a basis for the image of P. Let {Ωij(t)}, {Ri(t)}, and {Kij(t, s)} be the unique functions such that

PLtgi=jΩij(t)gj,Ri(t)=Tet0tQLτdτQLtgi,PLsTestQLτdτQLtgi=jKij(t,s)gj,

i.e., Ω(t) and K(t, s) are the matrix representations of PLtP and PLsTestQLτdτQLtP, respectively. Then

dgdt(t)=Ω(t)g(t)+R(t)+t0tK(t,s)g(s)ds.
(25)

This is the generalized Langevin equation for the vector-valued observable g(t).

We now consider a special class of projections on A, i.e., the conditional expectations. After a brief introduction to these operators, we argue that the projection P used in the NMZ formalism should typically be a conditional expectation and discuss the problem of constructing these projections. In the C*-algebra literature, the notion of conditional expectation31 has been developed as a noncommutative generalization of the more traditional idea of conditional expectation known from probability theory. It is defined to be a positive contractive projection P on a C* algebra A with image equal to a C*-subalgebra BA such that P(bab)=bP(a)b for all aA and b,bB. This implies also that P is completely positive.32 

Next, we show that under natural assumptions, the NMZ projection P must be a conditional expectation. First, we require P to project onto a C*-subalgebra BA. This is because the projection represents a restriction to partial information about the system, represented as a subset of observables, and, as argued in Sec. II E, such partial information is embodied in the C*-subalgebra generated by the monitored observables. Second, keeping the dual pictures in mind, when we introduce the projection P on A, we want to ensure that P* preserves states, i.e., P* maps S(A) into itself so that the dual NMZ Langevin equation (24) describes the evolution of states associated with the reduced system. For any C*-algebra A of observables of a system and any state-preserving projection P onto a C*-subalgebra, P is a conditional expectation, as shown in  Appendix B.

It should be noted that since P is contractive (i.e., P=1), the complementary projection Q=1P is bounded: Q=1P1+P=2. Indeed, as we will see in Example 2 below, the norm of Q can achieve this bound, and therefore Q is not, in general, contractive.

We aim at constructing explicitly a projection operator representing an observable g(x) [see Eq. (6)]. In the language of operator algebras, this is equivalent to asking the following question: How do we find a conditional expectation projecting onto a chosen subalgebra BA? It should first be noted that for general BAC*-algebras, there need not exist a conditional expectation P:AB. In fact, such conditional expectations are rare.20,33 However, if A admits a faithful, tracial state τ and BA is a nondegenerate C*-subalgebra, then there exists a unique conditional expectationP:AB such that τ(AB)=τ(P(A)B) for each AA,BB (Ref. 34, Theorem 7). This conditional expectation is ultraweakly continuous if A and B are von Neumann algebras and τ is normal. Any faithful state ρ of A induces an inner product on A via x,y=ρ(x*y).35 Then there exists a unique orthogonal projection from A onto the closed subspace B. When τ is a faithful, tracial state, this projection is the unique “τ-preserving” conditional expectation P satisfying τ(AB)=τ(P(A)B).

On A=L(M,μ), the faithful, normal, tracial states are the probability measures ρ on M that are equivalent to μ, in the sense that μρ and ρμ. And on A=C0(M), the faithful, tracial states are the strictly positive Radon probability measures, i.e., the Radon probability measures μ for which μ(G)>0 for all nonempty open sets GM. Thus, when A is commutative, there always exists a conditional expectation onto BA. Moreover, for von Neumann A and commutative von Neumann BA, there exists a conditional expectation P:AB (Ref. 34, Proposition 6). Conditional expectations can also exist when A and B are both noncommutative. An example is considered in Sec. VI A. For now, let us provide simple examples involving commutative algebras of observables.

Example 1.
Consider a three-dimensional dynamical system such as the Kraichnan-Orszag system36 or the Lorenz system evolving on the manifold M=R3. Let A=L(R3,λ) where λ is the Lebesgue measure on R3, and let h:R3R (phase space function) be given by
h(x)=e(x12+x22),
(26)
where x1(t) and x2(t) are the first two phase variables of the system. Then, the subalgebra B generated by h is the set of all gA that factor over h, i.e., functions in the form g=fh for some f:RR. In other words, B is the subalgebra of A comprising those functions g that are constant on level sets of h. A conditional expectation projecting onto this subalgebra may be obtained by starting with the faithful, normal, tracial state τ on R3,
τ(f)=R3f(x)dμ(x),
given by the Gaussian measure
dμ(x)=1(2πσ2)32ex12+x22+x322σ2dλ(x).
(27)
The construction of the projection onto the subalgebra B generated by (26) then proceeds as follows. Since gB is constant on level sets of h, in order to ensure τ(fg)=τ(P(f)g) for all fA and all gB, we require that P(f)(x) is the mean value of f on the level set of h through x. In other words,
(Pf)(x)=Rey32/(2σ2)σ(2π)12f(0,0,y3)dy3,x1=x2=0,yC(x)ey32(2σ2)σ(2π)32x12+x22f(y)dy,
(28)
where C(x) is the cylinder {y:y12+y22=x12+x22}R3 (level set of h containing x). It is readily verified that this P is a projection from A to B and that τ(fg)=τ(P(f)g), as desired. Of course, this projection works for any dynamical system on R3 where we are measuring (26).

Example 2.
Let (S1,λ) be the circle with normalized Haar measure λ, represented, for example, as S1={e2iπs:s[0,1)} with λ Lebesgue measure on [0, 1). Let (M,μ)=T2 be the two-dimensional torus with normalized Haar measure, i.e., (M,μ)=(S1,λ)×(S1,λ). Let A=L(M,μ)=L(S1,λ)2. Now consider the observable function gA given by
ge2iπs1,e2iπs2=e2iπs1.
(29)
The von Neumann subalgebra BA generated by g is the subalgebra of functions that factor over g. In other words, B is the subalgebra of functions hA that are constant on level sets of g,
he2iπs1,e2iπs2=he2iπs1,1s2[0,1).
(30)
A conditional expectation P onto B is given by
Pf(e2iπs1,e2iπs2)=01fe2iπs1,e2iπrdr.
(31)
Next we show that the complementary projection Q=1P is not a contraction in this case. To this end, let
fn(e2iπs1,e2iπs1)=1+2en2(s21/2)22
(32)
so that fn=1, Pfnvn is a constant function with value
vn=1+201en2(r1/2)22dr1,
(33)
and Qfn=fnvn12 so that
Q=2.
(34)

By their nature, C*-algebras comprise bounded observables of the system. In many cases, however, one is interested in unbounded observables. For example, in studying physical particle systems, we are often interested in the positions and momenta of the particles. The theory of C*-algebras and von Neumann algebras have been extended (in two independent ways) to incorporate certain classes of well-behaved unbounded operators. In both cases, these operators are called affiliated operators, and the underlying idea is to identify those unbounded operators that may in some sense be approximated by the (bounded) elements of the algebra.

  1. von Neumann algebras. When A is a von Neumann algebra acting on a Hilbert space H, a closed, densely defined operator T is affiliated with A whenever U*TU=T for all unitary operators UB(H) that commute with A.17,37 For example, in the case AL(X,σ,μ) of a commutative von Neumann algebra isomorphic (via Segal duality) to the algebra of essentially bounded functions on a localizable measure space, the affiliated operators Aη are represented by the set of all measurable C-valued functions on (X,σ,μ). Given a von Neumann algebra A and a normal affiliated operator T, there is a minimal von Neumann subalgebra BA such that every T is affiliated with B. This B is the Abelian subalgebra generated by T (Ref. 17, Theorem 5.6.18). With respect to the polar decomposition T = U|T|, B contains U as well as all spectral projections of |T| (Ref. 19, Lemma 2.5.8).

  2. C*-algebras. When A is a C*-algebra, a densely defined operator T acting on A is affiliated with A when T admits a T* such that a*T(b)=[T*(a)]*b for all bD(T)A and all a in a dense subset of A and when idA+T*T has a dense range in A.38–40 For example, when AC0(X) for some locally compact Hausdorff space X, the affiliated operators are represented by the set C(X) of all continuous C-valued functions on X.38 If A is unital, then the set Aη of affiliated operators may be identified with A itself, which is analogous to the statement that every continuous function on a compact space is bounded.38 

As in the case of measuring a bounded normal operator (see Sec. II B), there is a continuous functional calculus for any normal affiliated operator TAη, i.e., an injective nondegenerate *-morphism ΦT:C0(σ(T))M(A), where M(A) is the multiplier algebra of A.38,39 Any state ρA extends uniquely to a state of M(A) and restricts to a state of C0(σ(T)), namely, ρT:=ΦT*ρ. As before, the Riesz-Markov theorem then yields a probability measure on C, supported on σ(T), which is interpreted as the probability of the possible outcomes of measurement of T on a system in state ρ. When A is unital, M(A)=A, and the image of ΦT is the subalgebra of A generated by T.

Example 3.
Consider a nonlinear dynamical system evolving on M=Rn, for example, the semi-discrete form of an initial/boundary value problem for a PDE. Let A=C0(Rn) and B be the subalgebra generated by the observable
h(x)=i=1naixi,
(35)
for some fixed a=(a1,,an)Cn. Note that h(x) may represent the series expansion of the solution to the aforementioned PDE. Although hA (it does not vanish at infinity), we can still represent the partial information embodied in h by the subalgebra BA of functions gA that factor over h, i.e., for which there exists r such that g=rh. Thus, B comprises functions in C0(M) that are constant on level sets of h. A conditional expectation onto this B is given by
(Pf)(x)=e|x,a|2a2C(x)ey22σ2(2πσ2)n12f(y)dy,
(36)
where C(x) = {y: h(y) = h(x)}.

In many cases, the reduction to a coarser algebra of observables involves pushing the problem from the original phase space M to a new (typically smaller, lower dimensional) space N via a continuous (and appropriately nonsingular) map γ:MN. In other words, rather than worrying about how a state ρ0S(A) evolves, we are interested only in how the push-forward state σ0:=Kγ*ρ0S(B) evolves, where Kγ:BA is the Koopman homomorphism from the appropriate observable algebra B on N to the original algebra A of observables on A and Kγ* is the corresponding transfer (i.e., Perron-Frobenius) operator pushing states forward from M to N. A similar situation can arise in the case of noncommutative algebras, for example, when K:BA is an embedding identifying the subalgebra of observables localized on a quantum subsystem of interest. To keep the discussion general, we will assume in this section that E(t,s) is a strongly continuous (or weak-* continuous, in the case of von Neumann algebras) family of *-endomorphisms generated by derivations Lt on A and that K:BA is a nondegenerate *-homomorphism. We then seek an appropriate evolution equation for the reduced state σ(t)=K*ρ(t). To this end, consider g0B. We have

[Kg](t)=E(t,0)Kg0=Te0tLτdτ(g0γ)
(37)

and

ddt[Kg](t)=Te0tLτdτLtKg0=Te0tLτdτLt(g0γ).
(38)

Then for any state ρ0S(A),

ρ0([Kg](t))=ρ0(etLKg0)=ρ(t)(Kg0)=K*ρ(t)(g0).
(39)

Therefore, as expected,

ddtK*ρ(t)=K*Lt*ρ(t).
(40)

This equation is still not a reduced-order equation since the right-hand side is not in terms of σt:=K*ρt. However, we can use the NMZ projection operator method to derive the reduced-order equation we are interested in. To this end, let us assume that K:BA is injective; if not, one can typically replace B by B/kerK and replace K by K̃:B/kerKA. In the case where K=Kγ is the Koopman morphism of a map γ:MN, this amounts to replacing N with Ñ=γ(M), i.e., the image of γ, and letting B be the appropriate algebra of observables on Ñ, say C0(Ñ). Since K is injective *-morphism, K:BA is an embedding of B into A. In other words, B^:=ImKA is isomorphic to B via K. Suppose that P:AA is a conditional expectation onto B^. Then P can be decomposed as the composition of two positive contractions: P=Kπ, where π:AB may be viewed as the projection P onto B^, followed by identification of B^ with B. Moreover, it is clear that πK is the identity map on B so that PK=K and πP=π. Using P, we get the standard NMZ evolution equation for P*ρ(t),

ddtP*ρ(t)=P*Lt*P*ρ(t)+P*Lt*Te0tQ*Lτ*dτQ*ρ0+P*Lt*0tTestQ*Lτ*dτQ*Ls*P*ρ(s)ds.
(41)

Now, replacing P* with P*=π*K*, acting on the left with K*, and using the fact that K*P*=K*, we get the desired Langevin equation for σ(t)=K*ρ(t),

ddtK*ρ(t)=K*Lt*π*K*ρ(t)+K*Lt*Te0tQ*Lτ*dτQ*ρ0K*Lt*0tTestQ*Lτ*dτQ*Ls*π*K*ρ(s)ds,
(42)
ddtσ(t)=K*Lt*π*σ(t)+K*Lt*Te0tQ*Lτ*dτQ*ρ0K*Lt*0tTestQ*Lτ*dτQ*Ls*π*σ(s)ds.
(43)

If ρ0S(A) is supported on B^, then ρ0=π*σ0 for some σ0S(B). Then, since πP=π, it follows that πQ=0 and Q*π*=0 so that Q*ρ0=Q*π*σ0=0 and the “random noise” term in the Langevin equation vanishes, leaving

ddtσ(t)=K*Lt*π*σ(t)+K*Lt*0tTestQ*Lτ*dτQ*Ls*π*σ(s)ds.
(44)

As we will see in the next example, this equation takes a particularly simple form if the dynamics is on a manifold M with a tensor product structure.

Example 4.
Let (N,ν) and (R,η) manifolds with Borel measures ν and η, respectively, and let (M,μ)=(N,ν)×(R,η), i.e., M=N×R and μ is the product measure ν×η. Then with A=L(M,μ), B=L(N,ν), and R=L(R,η), we have that A=BR. Let γ:MN be the map γ(x,y)=x for xN and yR. Then the composition (i.e., Koopman) operator Kγ:BA is given by
(Kγg)(x,y)=g(γ(x,y))=g(x)=(g1)(x,y),
for any gB. Let π:AB be given by
(πf)=(idBρR)(f),(πf)(x)=Rp(y)f(x,y)dη(y),
where ρRS(R) is a normal state and p is the corresponding probability density function on (R,η). Then P=Kγπ is a conditional expectation on A with image isomorphic to B, and πKγ is the identity morphism on B.
It is now straightforward to identify the predual operators (with the predual of L(M,μ) identified with L1(M,μ) and likewise for (N,ν) and (R,η))
(Kγ*ϕ)(x)=Rϕ(x,y)dη(y),(π*ψ)(x,y)=(ψp)(x,y)=ψ(x)p(y),(P*ϕ)(x,y)=p(y)Rϕ(x,y)dη(x),
for any ϕA*, ψB*. Then the NMZ equation (44) becomes
ddtσt(x)=RLt*(σtp)(x,y)dη(y)+RLt*0tTestQ*Lτ*dτQ*Ls*(σsp)ds(x,y)dη(y).
(45)

Remark 1.

The projection defined in Example 4 may be thought of as a generalization of Chorin’s conditional expectation14 when NRn and RRr, with ν and η the Lebesgue measures on these spaces. It is also a commutative (i.e., classical) example of the problem of reducing dynamics to a subsystem of interest that arises in the theory of open quantum systems and quantum information theory.

The steps necessary to undertake dimension reduction using the NMZ formalism are outlined in Algorithm 1. The last step obviously hides many important details involving approximation of memory integrals, noise terms, and implementation. These details are beyond the scope of the present paper and we refer to Refs. 14–16, 41, and 42 (see also Ref. 43). In general, solving the NMZ equations is a very challenging task that implicitly requires propagation of all information of the system. Only by suitable (typically problem-class-dependent) approximations and efficient numerical algorithms, can these equations be rendered tractable. Except under strong assumptions (e.g., scale-separation), these issues persist and present serious challenges to the development of efficient and accurate solution methods.

Algorithm 1.

NMZ algorithm.

1: Identify the system ẋ=F(x,ξ,t). 
2: Decide on the algebra of observables, typically either A=L(M,μ) or A=C0(M)
3: Identify the subalgebra of interest BA
4: Choose a faithful tracial state ρ of A. This is typically equivalent to choosing a strictly positive probability measure ρ^ 
on M
5: Extract a conditional expectation P as the unique projection P:ABA satisfying ρ(AB)=ρ(P(A)B) for all AA and 
BB. In the typical case, this is 
MA(x)B(x)dρ^(x)=MP(A)(x)B(x)dρ^(x). 
6: Set up the NMZ equations (22) and/or (24). 
7: Solve, by approximating the memory integral and/or noise term as needed. 
1: Identify the system ẋ=F(x,ξ,t). 
2: Decide on the algebra of observables, typically either A=L(M,μ) or A=C0(M)
3: Identify the subalgebra of interest BA
4: Choose a faithful tracial state ρ of A. This is typically equivalent to choosing a strictly positive probability measure ρ^ 
on M
5: Extract a conditional expectation P as the unique projection P:ABA satisfying ρ(AB)=ρ(P(A)B) for all AA and 
BB. In the typical case, this is 
MA(x)B(x)dρ^(x)=MP(A)(x)B(x)dρ^(x). 
6: Set up the NMZ equations (22) and/or (24). 
7: Solve, by approximating the memory integral and/or noise term as needed. 

In Sec. VI A we study an example of the NMZ formalism applied to quantum systems, in which the observable algebra is non-commutative.

Let HA and HB be Hilbert spaces for two quantum systems. Then the total Hilbert space for the two systems together is HAHB. We will look at the case where we know the Hamiltonian dynamics of the composite system but want to understand the dynamics of system A alone. This is a common starting point for the study of open quantum systems, where system B represents the (typically nuisance) environment from which we cannot entirely decouple our system A of interest. Let A=B(HAHB) and B=B(HA), where B(H) denotes the (noncommutative) von Neumann algebra of all bounded linear operators on H with the operator norm X=supv=1Xv. Note that the predual of B(H) is T(H), the Banach space of trace class bounded linear operators on H with norm A1=Tr|A|, where |A|=AA. As suggested by the definitions (and subscripts), B(H) and T(H) are in some ways analogous to L and L1 function spaces.

We consider the isometric *-homomorphism K:BA given by K(X)=X1. The predual of K is then K*(Y)=TrB(Y), the partial trace of Y over HB. We seek a pseudoinverse of K, i.e., a linear map π:AB such that KπK=K and πKπ=π. It is easily verified that for any fixed ρBS(B(HB)), π(X)=TrB(X1ρB) is such a pseudoinverse. Indeed, with this choice, πK=idB. This choice brings us in line with Ref. 44, Sec. 9.1. Then π*:A*B* is π*(A)=AρB. Let P be the conditional expectation Kπ on A so that P(X)=TrB(X1ρB)1 projects onto the subalgebra B1A and P*(A)=TrB(A)ρB projects onto B*ρBA*. Then, letting σ(t)=K*ρ(t), (43) yields

ddtσ(t)=TrBLFtξ*σ(t)ρB+TrBLFtξ*Te0tQ*LFτξ*dτQ*ρ0+TrBLFtξ*0tTestQ*LFτξ*dτQ*LFsξ*σ(s)ρBds.
(46)

This generalized Langevin equation should be compared with the classical subsystem reduction in (45). In many cases, further assumptions and approximations are made in order to eliminate the memory, yielding a Markovian master equation.45 

In this section, we study the dynamics of simple integrable systems on SU(2) and SO(3) and work through an analytic example in both the observable (Heisenberg) and state (Schrödinger) pictures.

Let M=SU(2) with normalized Haar measure μ and consider the observable g(U)=Tr(U)/2 in the Banach algebra A=L(SU(2),μ). Then g=g* (g is real-valued) and g=1. Consider the dynamical system

U̇=iHU,
(47)

where HC2×2 is Hermitian and Tr(H)=0 so that, by the Cayley-Hamilton theorem, H2=1detH. While (47) resembles a quantum mechanical model of a 2-state system in some respects, we will not think of it in those terms, but only as a simple ODE defined on a 3-dimensional compact manifold. In particular, the observable algebra we consider hereafter is a commutative algebra of C-valued functions on SU(2), rather than the non-commutative algebra B(C2). Then for any differentiable fL(SU(2),μ),

(Lf)(U)=df(iHU).
(48)

Recall that the class functions on SU(2) are those functions f:SU(2)C such that f(U)=f(ΩUΩ) for all ΩSU(2), i.e., the functions that are constant on conjugacy classes. Let BA be the von Neumann subalgebra of class functions within L(SU(2),μ). This subalgebra captures the observables on SU(2) that depend only on the spectra of the unitary operators. And, since in SU(2) the spectrum of U is determined by Tr(U), the class functions are exactly the functions that factor over g so that B is the von Neumann subalgebra of A generated by g. We take as the projection operator onto B the conditional expectation

(Pf)(U)=SU(2)f(ΩUΩ)dμ(Ω).
(49)

The NMZ equation for g is

ddtg(t)=etLPLg+R(t)+0te(ts)LPLR(s)ds,
(50)

where

R(t)=et(1P)L(1P)Lg.
(51)

To make this MZ equation more concrete, we first observe that

(Lg)(U)=i2Tr(HU),(L2g)(U)=12Tr(H2U)=det(H)g(U)

so that g and Lg span an invariant subspace of L. Moreover, g is a class function of SU(2), so Pg=g and

(PLg)(U)=SU(2)(Lg)(ΩUΩ)dμ(Ω)=i2SU(2)Tr(HΩUΩ)dμ(Ω)=i2TrSU(2)ΩHΩdμ(Ω)U=iTr(H)g(U)=0,
(52)

for all USU(2) since Tr(H)=0. Then Span{g,Lg} is also an invariant subspace of P (and therefore also of 1P). It follows that

(1P)Lkg=gk=0Lgk=10k2,
(53)

and therefore

R(t)=et(1P)L(1P)Lg=k=0tkk!(1P)Lk+1g=Lg.
(54)

This means that R(t)(U0)=R(U0)=iTr(HU0) is constant in time and that PLR=det(H)g. The NMZ equation then reduces to

ddtg(t)=R+det(H)0tg(ts)ds.
(55)

Now, since ODE (47) is linear, we can solve it exactly. This is particularly simple since, as we observed above, H2=det(H)1. We therefore immediately get

U(t)=eitHU0=cos(λt)U0isin(λt)λHU0,
(56)

where λ=detH so that σ(H)={±λ}. Thus,

g(t)=cos(λt)g+sin(λt)λLg.
(57)

It is easy to show by direct substitution that (91) is indeed the solution to the integro-differential equation (88), as desired.

We now turn to the problem of solving the predual NMZ equation for the evolution of a reduced normal state. Note that in the present example, this is a classical reduced-order probability distribution function on SU(2) and not a density matrix as would be typical in a quantum mechanical setting. We consider the subalgebra BA of bounded class functions, i.e., fA such that f(ΩUΩ)=f(U) for all U,ΩSU(2). The projector PB(A) given in (49) has predual P*B(A*) with the same form, i.e.,

(P*ρ)(U)=SU(2)ρ(ΩUΩ)dμ(Ω).
(58)

Here, and throughout this section, we will freely use the isomorphism L(M,μ)*L1(M,μ) to identify functionals in A* with μ-integrable functions. Likewise, the predual Liouvillian L* takes almost the same form as L, namely,

(L*ρ)(U)=dUρ(iHU).
(59)

Now, suppose we take as initial state the PDF ρ0(U)=Tr(U)2. This is positive valued on SU(2) because Tr is real-valued on SU(2), and it is normalized because

SU(2)Tr(U)2dμ(U)=TrSU(2)UUdμ(U)=Tr[12(1swap)]=1,
(60)

where swap is the operator on HH that permutes the two subsystems, i.e., swapψϕ=ϕψ. Because P*ρ0=ρ0, the NMZ equation (24) that we wish to solve reduces to

ddtP*ρ(t)=P*L*P*ρ(t)+P*L*0te(ts)Q*L*Q*L*P*ρ(s)ds.
(61)

Next, we look for suitable matrix representations of P* and L*. To this end, consider the linearly independent family of functions

{1,ρ0=4g2,4gL*g,4(L*g)2},
(62)

where the observable g(U)=Tr(U)/2 is as in Sec. VII A. It easy to show that the space spanned by these functions is invariant for L* and P*. It can also be verified that L* and P* have the following matrix representations relative to (62):

L*000000λ200202λ20010,
(63)
P*10043λ201013λ200000000.
(64)

Therefore,

P*L*0043λ200043λ2000000000,
(65)
P*L*e(ts)Q*L*Q*L*083λ2cos2λ(ts)3833λ3sin2λ(ts)383λ4cos2λ(ts)3083λ2cos2λ(ts)3833λ3sin2λ(ts)383λ4cos2λ(ts)300000000.
(66)

Since P*ρ(0)0100T, it follows that P*ρ(t)=a(t)1+b(t)ρ0, so we can reduce to the 2-dimensional invariant subspace spanned by 1 and ρ0, yielding

P*L*0000,
(67)
P*L*e(ts)Q*L*Q*L*083λ2cos2λ(ts)3083λ2cos2λ(ts)3.
(68)

The NMZ equation (61) then becomes

a(t)b(t)=83λ20tcos2λ(ts)30101a(s)b(s)ds.
(69)

Note that a(0) + b(0) = 1 [since a(0) = 0 and b(0) = 1] and a(t)+b(t)=0 so that a(t) + b(t) = 1 for all t0. Moreover, b(t) is described by the integro-differential equation

b(t)=83λ20tcos2λ(ts)3b(s)ds.
(70)

Differentiating this expression twice more, we find that

b(t)=83b(t)1633λ30tsin2λ(ts)3b(s)ds,b(t)=83b(t)329λ40tcos2λ(ts)3b(s)ds=4λ2b(t)
(71)

so that

b(t)=C+γsin(2λt)+κcos(2λt).
(72)

Using the initial conditions b(0) = 1, b(0)=0, b(0)=8λ2/3 (the last two are clear from the integro-differential equations for b and b above), we conclude that

b(t)=13(2cos(2λt)+1)
(73)

so that

P*ρ(t)=23(1cos(2λt))1+13(2cos(2λt)+1)ρ0=ρ0+23(1cos(2λt))(1ρ0).
(74)

This solution can also be obtained by exponentiating L* in the 4-dimensional invariant subspace, yielding

P*etL*143sin2(λt)23λsin(2λt)43λ2cos2(λt)01+2cos(2λt)323λsin(2λt)λ212cos(2λt)300000000.

Let M=SO(3) with normalized Haar measure μ and consider the observable g(O)=Tr(O)/3 in the Banach algebra A=L(SO(3),μ). Then g=g* (g is real-valued) and g=1. Consider the dynamical system

ddtO(t)=XO(t),
(75)

where XSO(3) is skew-symmetric of the form

X=0zyz0xyx0
(76)

with Tr(X)=0 and det(X)=0 so that, by the Cayley-Hamilton theorem, X3 = −r2X, where r2 = x2 + y2 + z2. The dynamical system (75) is then a simple ODE on a 3-dimensional compact manifold, which describes a one-parameter semigroup of orthogonal operators O(t) effecting a constant-rate rigid rotation about the axis along (x, y, z). Then for any differentiable fL(SO(3),μ),

(Lf)(O)=df(XO).
(77)

Recall that the class functions on SO(3) are those functions f:SO(3)C such that f(O)=f(ΩOΩT) for all ΩSO(3), i.e., the functions that are constant on conjugacy classes. Let BA be the von Neumann subalgebra of class functions within L(SO(3),μ). This subalgebra captures the observables on SO(3) that depend only on the spectra of the orthogonal operators. Moreover, since in SO(3) the spectrum of O is determined by Tr(O), the class functions are exactly the functions that factor over g so that B is the von Neumann subalgebra of A generated by g. We take as the projection operator onto B the conditional expectation

(Pf)(O)=SO(3)f(ΩOΩT)dμ(Ω).
(78)

The NMZ equation for g is

ddtg(t)=etLPLg+R(t)+0te(ts)LPLR(s)ds,
(79)

where

R(t)=et(1P)L(1P)Lg.
(80)

To make this NMZ equation more concrete, we first observe that

(Lg)(O)=Tr(XO)/3,(L2g)(O)=Tr(X2O)/3,(L3g)(O)=Tr(X3O)/3=r2Tr(XO)/3=r2(Lg)(O)
(81)

so that g, Lg, and L2g span an invariant subspace of L and

Lkg=g,k=0,(r2)k12Lg,kodd,k1,(r2)k22L2g,keven,k2.
(82)

Moreover, g is a class function of SO(3), and therefore we have Pg=g and

(PLkg)(O)=SO(3)(Lkg)(ΩOΩT)dμ(Ω)=13SO(3)Tr(XkΩOΩT)dμ(Ω)=13TrSO(3)ΩTXkΩdμ(Ω)O=Tr(Xk)3g(O),
(83)

for all OSO(3) and k1. Since Tr(X)=0 and Tr(X2)=2r2,

PLkg,=gk=0,0kodd,k1,23(r2)k2g,keven,k2.
(84)

Thus, Span{g,Lg,L2g} is also an invariant subspace of P (and therefore also of 1P). This implies that

[(1P)L]kg,=gk=0,(r2/3)k12Lg,kodd,k1,(r2/3)k22(L2g+(2/3)r2g),keven,k2,
(85)

and therefore

R(t)=et(1P)L(1P)Lg=k=0t2k(2k)![(1P)L]2k+1g+k=0t2k+1(2k+1)![(1P)L]2k+2g=k=0(1)k(rt/3)2k(2k)!Lg+3rk=0(1)k(rt/3)2k+1(2k+1)!L2g+23r2g=cosrt3Lg+3rsinrt3L2g+23r2g.
(86)

By applying PL to R(s), we obtain

PLR(s)=23r2cosrs3g.
(87)

Thus, the NMZ equation then reduces to

ddtg(t)=R(t)23r20tcosrs3g(ts)ds,
(88)

which may be solved (e.g., via Laplace transforms) to obtain

gt=g+sin(rt)rLg+1cos(rt)r2L2g.
(89)

Of course, since ODE (75) is linear, we can solve it exactly. This is particularly simple since, as we observed above, X3 = −r2X. We therefore find

O(t)=etXO0=k=0t2k(2k)!X2kO0+k=0t2k+1(2k+1)!X2k+1O0=O0+1rk=0(1)k(rt)2k+1(2k+1)!XO01r2k=1(1)k(rt)2k(2k)!X2O0=O0+sin(rt)rXO0+1cos(rt)r2X2O0.
(90)

Thus,

g(t)(O0)=Tr(O(t))/3=Tr(O0)/3+sin(rt)rTr(XO0)/3+1cos(rt)r2Tr(X2O0)/3=g(O0)+sin(rt)r(Lg)(O0)+1cos(rt)r2(L2g)(O0),
(91)

confirming the solution obtained through the NMZ formalism [compare (91) and (89)].

We now turn to the problem of solving the predual NMZ equation for the evolution of a reduced normal state. We consider the subalgebra BA of bounded class functions, i.e., fA such that f(ΩOΩT)=f(O) for all O,ΩSO(3). The projector PB(A) given in (78) has predual P*B(A*) with the same form, i.e.,

(P*ρ)(O)=SO(3)ρ(ΩTOΩ)dμ(Ω).
(92)

Here, and throughout this section, we will freely use the isomorphism L(M,μ)*L1(M,μ) to identify functionals in A* with μ-integrable functions. Likewise, the predual Liouvillian L* takes almost the same form as L, namely,

(L*ρ)(O)=dOρ(XO).
(93)

Now, suppose we take as initial state the PDF ρ0(O)=Tr(O)2. This is positive valued on SO(3) because the trace operator is real-valued on SO(3) and it is normalized,

SO(3)Tr(O)2dμ(O)=TrSO(3)OOdμ(O)=1.
(94)

This follows from the fact that

SO(3)OOdμ(O)
(95)

is the orthogonal projection onto the subspace spanned by

e1e1+e2e2+e3e3.
(96)

The NMZ equation (24) for the PDF ρ(t) takes the form

ddtP*ρ(t)=P*L*P*ρ(t)+P*L*0te(ts)Q*L*Q*L*P*ρ(s)ds.
(97)

Next, we look for suitable matrix representations of P* and L*. To this end, consider the 10-dimensional space spanned by the linearly independent functions

{1,g,g2,L(g),L2(g),gL*(g),gL*2(g),L*(g)2,L*(g)L*2(g),(L*2(g))2}.
(98)

It easy to show that the space spanned by these functions is invariant under L* and P*, and Q*=1P*. With respect to basis elements (98), these operators may be represented as

L*0000000000000000000000000000000100r2000000001000000002000r2000000001000000000100r200000001202r20000000010,
(99)
P*1000000r2015r4010023r2002r2025r400100023r23r20215r40000000000000000000000000000000000000000000000000000000000000000000000.
(100)

Next, we observe that P* restricted to the span of (98) has image the subspace spanned by {1, g, g2}. Using the fact that the spectrum (with multiplicity) of Q*L* on the span of (98) is

σ(Q*L*)={0,0,0,0,±ir/3,±r(α+iβ),±r(αiβ)},

where

α=712+6715,β=712+6715,
(101)

it can be verified that on the 3-dimensional space spanned by {1, g, g2},

P*L*000000000,
(102)
P*L*e(ts)Q*L*Q*L*002r2cosh(αrt)cos(βrt)+83r230αβsinh(αrt)sin(βrt)023r2cos(rt/3)1211842r2cosh(αrt)cos(βrt)+28319r230αβsinh(αrt)sin(βrt)+2cos(t/3)00223r2cosh(αrt)cos(βrt)+239r290αβsinh(αrt)sin(βrt).
(103)

With respect to {1, g, g2}, the NMZ equation (97) then becomes

dA(t)dt=0tP*L*e(τs)Q*L*Q*L*A(τ)dτ,
(104)

where A(t) = [a1(t), a2(t), a3(t)]T are the components of P*ρ(t) relative to {1, g, g2}, and P*L*e(τs)Q*L*Q*L* is the 3×3 matrix given explicitly in (103). The integro-differential equation (104) can then finally be solved via Langrange transforms to obtain

P*ρ(t)=185[2cos(rt)cos(2rt)]+49495[19179+61376cos(rt)42917cos(2rt)32241rtsin(rt)]g+115[67106cos(rt)+54cos(2rt)]ρ0,
(105)

where r2=x12+x22+x32 and g=Tr(O)/3.

We have developed a new formulation of the Nakajima-Mori-Zwanzig (NMZ) method of projections based on operator algebras of observables and associated states. The new theory does not depend on the commutativity of the observable algebra, and therefore it is equally applicable to both classical and quantum systems. We established a duality principle between the NMZ formulation in the space of observables and associated space of states, which extends the well-known duality between Koopman and Perron-Frobenious operators to reduced observable algebras and states. We also provided guidance on the selection of the projection operators appearing in NMZ by proving that the only projections onto C*-subalgebras that preserve all states are the conditional expectations—a special class of projections on C*-algebras. Such projections can be determined systematically for a broad class of bounded and unbounded observables. This allows us to derive formally exact NMZ equations for observables and states in high-dimensional classical and quantum systems. Computing the solution to such equations is usually a very challenging task that needs to address approximation of memory integrals and noise terms for which suitable (typically problem-class-dependent) algorithms are needed.

This work was supported by the Air Force Office of Scientific Research Grant No. FA9550-16-1-0092.

Definition 1

(Approximate Identity). Given a C*-algebra A, a net {Eα}A is an approximate identity for A if Eα0 and Eα1 for all α and if EαAA for all AA.

Lemma 1.

LetA, BbeC*-algebras andΨ:BAbe aC*-homomorphism. ThenΨis nondegenerate (i.e.,SpanC{Ψ(b)a:bB,aA}) if and only ifΨis approximately unital [i.e., for some (and therefore every) approximate identity22{Eβ}B, {Ψ(Eβ)}is an approximate identity forA].

Proof.
First, assume that Ψ is nondegenerate and let {Eβ}B be an approximate identity. For any bB, limβEβb=b, and by the continuity of Ψ, limβΨ(Eβb)=Ψ(b). Then for any aA, limβΨ(Eβb)a=Ψ(b)a. Therefore
limβΨ(Eβ)Ψ(b)aΨ(b)a=limβΨ(Eβb)aΨ(b)a=0,
(A1)
for any bB and aA. Since nondegeneracy of Ψ implies that SpanC{Ψ(b)a:bB,aA} is dense in A, we have found that Ψ(Eβ)aa for all a in a dense subspace of A. Since Eβ1 and therefore Ψ(Eβ)1, we conclude that limβΨ(Eβ)aa for all aA, i.e., {Ψ(Eβ)} is an approximate identity on A.

Now, suppose that Ψ is degenerate so that T:=SpanC{Ψ(b)a:bB,aA} is not dense in A. Then there exist 𝜖>0 and aA such that at>𝜖 for all tT. Then let {Eβ} be any approximate identity in B. Since Ψ(Eβ)aT for all β, Ψ(Eβ)aa>𝜖 for all β, and therefore Ψ(Eβ)aa so that {Ψ(Eβ)} is not an approximate identity. So, by contradiction, if {Ψ(Eβ)} is an approximate identity for some approximate identity {Eβ}, then Ψ must be nondegenerate.

It may be noted that if B is unital, then Ψ:BA is nondegenerate if and only if A is a unital algebra and Ψ is a unital C*-homomorphism. This follows from the simple fact that Eβ1 is the only possible constant approximate identity.

Lemma 2.

For any (contractive) approximate identityEαA, Eα1.

Proof.
For any nonzero aA,
lim infαEαlim infαEαaa=aa=1,
(A2)
and, since Eα1 for all α, lim supαEα1. Therefore limαEα=1.

Lemma 3.

LetA, BbeC*-algebras andΨ:BAbe a nondegenerateC*-homomorphism. ThenΨ=1.

Proof.
Since Ψ is a C*-homomorphism, Ψ1. Let {Eβ}B be an approximate identity. Then {Ψ(Eβ)} is also an approximate identity, and Eβ1 and Ψ(Eβ)1 so that
ΨlimβΨ(Eβ)Eβ=1,
(A3)
and therefore Ψ=1.

Lemma 4.

LetA, B be C*-algebras andΨ:BAbe a nondegenerateC*-homomorphism. Ψ*ϕ=ϕ, for anyϕ0, whereΨ*:A*B*is the adjoint operator.

Proof.
For any ϕA*, ϕ0, ϕ=limβ|ϕ(Fβ)|, where {Fβ}A is an approximate identity. Thus, for any approximate identity {Eβ}B,
Ψ*ϕ=limβ|(Ψ*ϕ)(Eβ)|=limβ|ϕ(Ψ(Eβ))|=ϕ
(A4)
since, by Lemma 1, {Ψ(Eβ)} is an approximate identity for A.

Theorem 1.

LetA,B be C*-algebras andΨ:ABbe a linear map satisfyingΨ*[S(B)]S(A). ThenΨis a positive contraction withΨ=1. IfAis unital, thenBis unital andΨ(1)=1.

Proof.
First note that Ψ*[S(B)]S(A) requires that Ψ* be positive and Ψ*ϕ=ϕ for all ϕ0 and by Ref. 17, Theorem 4.3.4, Ψ*0 if and only if Ψ0:
Ψ*ϕ0ϕ0Ψ*ϕ(g)0ϕ0,g0ϕ(Ψg)0ϕ0,g0Ψg0g0.
Now we pass to the second dual A** of A, which, via the Takeda-Sherman theorem,20,46 may be endowed with a multiplication that renders it a (unital) von Neumann algebra (the universal enveloping von Neumann algebra of A). We likewise endow B** with the structure of a (unital) von Neumann algebra. Then for each state ϕS(B), we have ϕ(1B**Ψ**(1A**))=ϕ(1B**)Ψ*ϕ(1A**)=ϕΨ*ϕ=0 by assumption about Ψ and Ref. 20, Proposition II.6.2.5. Since this holds for all states of B, which comprise all normal states of B**, and they separate points in B**, it follows that Ψ**(1A**)=1A**, i.e., Ψ** is a unital positive map, and therefore is a contraction47 with Ψ**=1. And because Ψ=Ψ*=Ψ**, Ψ is also a positive contraction with Ψ=1.

Corollary 1.

IfAis aC*-algebra, Pis a linear projection onAsatisfyingP*[S(A)]S(A), and the image ofPis aC*-subalgebraBA, thenPis a conditional expectation.

Proof.

By 1, P is a contractive projection onto a C*-subalgebra and therefore is a conditional expectation.20,48

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