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arXiv:2411.08532v1 [quant-ph] 13 Nov 2024
CONDITIONAL EXPECTATIONS IN QUANTUM MECHANICS AND CAUSAL
INTERPRETATIONS: THE BOHM MOMENTUM AS A BEST PREDICTOR
RAYMOND BRUMMELHUIS
Abstract. Given a normalized state-vector ψ, we define the conditional expectation Eψ(A|B)of a
Hermitian operator Awith respect to a strongly commuting family of self-adjoint operators Bas the
best approximation, in the operator mean square norm associated to ψ, of Aby a real-valued function
of B. A fundamental example is the conditional expectation of the momentum operator Pgiven the
position operator X, which is found to be the Bohm momentum. After developing the Bohm theory
from this point of view we treat conditional expectations with respect to general B, which we apply to
non-relativistic spin 1/2-particles modeled by the Pauli equation, conditioning with respect Xand to a
component of the spin operator. We derive the dynamics of the conditional expectations of momentum
and spin which can be interpreted in terms of a classical two-component fluid whose components carry
intrinsic angular momentum.
The joint spectrum of the conditioning operators Bcan be interpreted as a space of beables. To
arrive at a Bohm-style causal interpretation one additionally needs a particle dynamics on this beable
space which is compatible with the time evolution of the Born probability. This can be done for spin
1/2−particles by combining the de Broglie-Bohm guidance condition with J. Bell’s idea of using Markov
jump processes when the spectrum is discrete. A basic problem is that such auxiliary particle dynamics
are far from unique.
We finally examine the relation of Eψ(A|B)with conditional expectations of C∗-algebras when Bhas
discrete spectrum. As an application, we derive an evolution equation for Eψ(A|B)when ψsatisfies an
abstract Schrödinger equation idψ/dt =Hψ which takes a simple form when A=H. Two appendices
re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide
the details of the two-fluid interpretation.
1. Introduction
The Bohm interpretation [5], [6] associates to a solution ψ=ReiS of the non-relativistic time-
dependent Schrödinger equation i∂tψ= (−∆+V)ψon Rna particle (or a Gibbsian ensemble of particles)
moving on the configuration space Rnwith both a well-defined position and velocity or momentum. The
probability density for the particle’s position is given by the Born-rule, |ψ|2=R2, and its momentum
is given by the Bohm momentum pB:= ∇S= Im(∇ψ/ψ) = Im(ψ∇ψ/|ψ|2).Bohm’s motivation for
interpreting ∇Sas a momentum came from re-writing the Schrödinger equation as a coupled system of
evolution equations for R2and Sand observing that the equation for Shas the form of a Hamilton-Jacobi
equation for Swhile the equation for R2can be interpreted as the continuity equation for the density
of a particle flow with velocity field ∇S. Here, as elsewhere in this paper, we have set both the particle
mass and the reduced Planck constant equal to 1.
The starting point of the present paper is the observation that the Bohm momentum can be given
a natural interpretation as the conditional expectation of the momentum operator P=i−1∇given the
position operator X(= multiplication by the position variable x), with an appropriate definition condi-
tional expectation. In classical elementary probability theory, the definition of conditional probability is
based upon the notion of joint probability, which is notoriously controversial in a quantum mechanical
context. However, as is well-known, there exists an alternative definition of conditional expectation as an
orthogonal projection, or distance minimizing-map, and in this form it generalizes naturally within the
framework of quantum mechanics formulated in terms of operators on Hilbert spaces. Specifically, we can
regard Hermitian operators Aas the quantum analogues of classical real-valued random variables. The
wave function ψsupplies the probability measure, with the expectation of Agiven by Eψ(A) := (Aψ, ψ),
where we introduce the notation Eψ(·)to emphasize the analogy with classical probability. If we then
define the conditional expectation of Pgiven Xas the real-valued function f(X)of Xwhich minimizes
Eψ(P−f(X))2=||(P−f(X))ψ||2, then an elementary computation (for which see the proof of the-
orem 2.1 below) shows that the minimizing function fis precisely the Bohm momentum: f(x) = pB(x)
(assuming ψ(x)6= 0 a.e.). If we drop the requirement of fhaving to be real, then it turns out that the
solution of this minimization problem is given by the so-called weak value of Pin the state ψ,P(ψ)/ψ.
Key words and phrases. ABC .
1
2 RAYMOND BRUMMELHUIS
The same construction can be applied to define the conditional expectation Eψ(A|X)with respect to X
of an arbitrary Hermitian operator Ahaving ψin its domain: see theorem 2.1. In particular, if we take A
to be the kinetic energy operator 1
2P2, we find for example that Bohm’s quantum potential is equal to 1/2
times the conditional variance of Pwith respect to X,Eψ(P2|X)−(Eψ(P|X))2: see proposition 2.5. The
quantum potential associated to a wave-function ψcan therefore be seen as a measure of the fluctuation
of the momentum operator around its conditional expectation in the state ψ, giving it an operational
meaning within the framework of orthodox quantum mechanics and making it more natural and less ad
hoc than it is sometimes claimed to be (see for example [23]). See Hiley [26] for a related observation
based on the Wigner-Moyal theory, where it should be noted that conditional expectations based on the
Wigner-Moyal distribution will in general differ from our’s: see the next paragraph. Since the conditional
expectation is defined as a distance-minimizing ob ject, the corresponding minimum distance becomes an
interesting quantity to examine. One finds, for example, that for the Bohm momentum this minimum
distance squared, Eψ(P−pB(X))2, is equal to 1/4-th of the Fisher information of the probability
density |ψ|2, and is also equal to twice the expected value of the quantum potential: see section 2.1 and
2.2.
In classical probability theory conditional expectations and joint probabilities are related, and we can
use this to define a "joint probability density" of position and momentum operators (X, P ), starting
from the conditional expectations Eψ(f(P)|X)of real-valued functions of P, in particular, indicator
functions. This will not actually produce a classical probability density, but will give a quasi-probability
density, in the sense that negative probabilities are possible. A well-known example of such a quasi-
probability density is given by the Wigner distribution but the quasi-probability density we obtain from
our conditional expectations is different from Wigner’s. Conversely, starting with the Wigner distribution
one can define a conditional expectations of Pand of functions of Pby using the classical expression
of a conditional expectation in terms of a joint probability density. Such conditional expectations of
powers of the momentum operator were already considered by Moyal [33] under the name of space
conditional moments of the momentum: see also Hiley [25], [26]. For Pitself this will again give the
Bohm momentum, but already for the kinetic energy operator P2the answer is found to be different
from Eψ(P2|X).The advantage of the latter is that it can be generalized to general pairs of self-adjoint
operators on Hilbert spaces: see below.
The definition of the Bohm momentum as a conditional expectation is independent of any Schrödinger
dynamics and, in a sense, is purely kinematical. In case the wave function ψis a solution of the
non-relativistic time-dependent Schrödinger equation, we will show in section 4 that if x(t)satisfies
the de Broglie-Bohm guidance equation dx/dt =pB(x, t), then the pair (x(t), pB(x(t), t)) is a solution
of Hamilton’s equations with a Hamiltonian in which the quantum potential is added to the classical
Hamiltonian, establishing the connection with Bohm’s original approach which was based on interpreting
the PDE satisfied by the phase of the wave function as a Hamilton-Jacobi equation. We will more
generally verify this for the Schrödinger equation for a particle in an electromagnetic field, with an
appropriately modified guidance equation when a vector potential is present. The guidance equation
can also be considered as a constraint on the Hamiltonian system, which is satisfied for all times iff it is
satisfied at an initial time. As we note in Appendix A, as a set of constraints it makes the Hamiltonian
system in extended phase space completely integrable when restricted to the constraint surface. This is
natural, since we only have to integrate the guidance equation to determine the particle tra jectories.
Conditional expectations can be defined in the general axiomatic framework of orthodox Quantum
mechanics. In section 5 we replace the position operator Xby an arbitrary commuting set of self-
adjoint operators B, and use the spectral theorem to establish the existence and (essential) uniqueness
of the conditional expectation Eψ(A|B)of an Hermitian operator Awith respect to B, which is again
defined as the best approximation, in mean square sense, of Aby a real-valued function f(B)of B, for a
system in the state ψ. If B=P, we find for example that Eψ(X|P)is the particle position of Epstein’s
(incomplete) momentum-based variant of the Bohm interpretation (incomplete, since there is no natural
analogue of Bohm’s guidance equation, except in special cases, such as a Schrödinger operator with
quadratic potential). Another interesting example is that of spin, where we can consider the conditional
expectations of two of the spin operators respect to the third. In this paper we limit ourselves to the
spin 1/2case.
If σ1, σ2and σ3are the Pauli-matrices, then the conditional expectations of the spin operators, sj:=
Eψ(1
2σj|σ3)are diagonal matrices or, equivalently, pairs (sj,+, sj,−)or functions on spec(σ3) = {±}, the
spectrum of σ3.The latter would be the analogue of the configuration space Rn= spec(X)of the classical
Bohm theory. If we consider the elements of this spectrum as the beables (or, in older terminology, as
CONDITIONAL EXPECTATIONS AND CAUSAL INTERPRETATIONS 3
the hidden variables) of the model, then we can think of S±= (s1,±, s2,±, s3,±)as the value of the spin-
vector when the beable is ±, similar to pB(x)being the momentum when the particle is located at x.
Here s3,±=±1/2, the respective eigenvalue of 1
2σ3, so the third component identifies with the beable.
In section 6 we examine the dynamics of these conditional expectations sjfor a fixed (non-moving,
infinitely massive) spin-1/2particle in a possibly time-dependent magnetic field by deriving a system
of differential equations for S±:= (s1,±, s2,±, s3,±).The equations for S+and S−decouple, and the two
spin vectors evolve autonomously with the beable having at each instant t, a probability of 1
4||s+||−2
and 1
4||S−||−2of being +1
2respectively −1
2.In fact, S+determines S−and vice versa, and it suffices to
solve the ODE for either one (excepting possible singularities which may develop). There is at this point
no talk yet of a causal mechanism (deterministic or classically stochastic) for the change of S+to S−or
vice versa. In the Bohm interpretation, such a mechanism is provided by the guidance equation, which
introduces a mouvement on the beable space spec(X) = Rn, and which can be motivated by writing the
evolution equation for the probability density of position as the continuity equation for a particle flow.
This can no longer be done in the case of a spin 1/2particle but we can, following an idea of Bell [4],
introduce a continuous time Markov chain process on {±} which is compatible with the time-evolution
of the occupation probabilities of the ±-states as given by the Born rule. The conditional expectation
S:= (s1, s2, s3)of quantum mechanical spin is then found to follow a stochastic differential equation
driven by the Markov process. This necessitates the introduction of an additional force term which may
be thought of as an analogue of Bohm’s quantum force (the gradient of the quantum potential).
One problem with the Bell approach is that the Markov chain process is not unique, a problem shared
by Bohmian mechanics, which in essence retains the kinematics of the Bohm interpretation while putting
aside the dynamical aspects involving the quantum potential: there are infinitely many deterministic
(and also classical stochastic, cf. Bacciagaluppi [3]) guidance equations which are compatible with the
time evolution of the probability density |ψ(x, t)|2if ψis a solution of the non-relativistic Schrödinger
equation with Hamiltonian −1
2∆ + V, but the de Broglie-Bohm guidance equation is singled out by the
quantum Hamilton-Jacobi equation or, equivalently, by the dynamics of the Bohm momentum. See also
the discussion following theorem 4.3 in section 4 below, where it is pointed that changing the guidance
equation by adding a divergence-free vector field to the probability flux amounts to considering a particle
in a magnetic field, which is not the natural thing to do if the quantum Hamiltonian does not already
include a vector potential, thereby singling out the de Broglie-Bohm guidance equation in such cases. We
did not succeed in finding a similar physical criterion for privileging one Markov process over the others
in the spin-1/2model. Bell’s specific choice in [4] corresponds to the Markov process for which the sum
of the instantaneous transition probabilities from the +-state to the −state and from the −-state to the
+-state is as small as possible, which has the effect of, at each point in time, forbidding one of the two
transitions (such processes are called minimal jump processes in [16]). It is not clear, though, whether
this translates into a criterion based on physical notions such as action, energy, or entropy.
In section 7, we turn to non-relativistic spin 1/2particles with finite mass (normalized to 1), whose
wave function ψ= (ψ+, ψ−)satisfies the Pauli equation with an electro-magnetic field. We condition
with respect to the commuting set of operators (X , σ3), and derive two systems of equations for the
conditional expectations sj,±(x, t)of the spin operators (which are now also functions of x, not just of
t) and for the (analogue of the) Bohm momentum, Eψ(P|X, σ3): cf. theorem 7.1 and theorems 7.3,
7.4. The Bohm momentum, like the sj, is now a diagonal matrix-valued function or, equivalently, a
pair of functions (p+(x, t), p−(x, t)), which can be thought of1as the conditional expectation of the
momentum when σ3is in the up- or down state and the particle is located at xat time t. As we show in
Appendix B, the system of theorems 7.3 and 7.4 can be interpreted in terms of a two-component polar
fluid, generalizing Madelung’s fluid-dynamical interpretation [32] of the Schrödinger equation. Madelung
fluids differ from classical Eulerian compressible fluids in that their Cauchy stress tensor is non-isotropic
and furthermore depends on derivatives up to order two of the density, instead of just the density itself,
making them special cases of so-called Korteweg fluids. This feature persists for the stress tensors of the
two components of the two-fluid interpretation of the Pauli equation, to which are added couple stress
tensors which govern the evolution of the total angular momentum, plus additional terms for the mass-,
momentum- and angular momentum transfer between the components: details are given in Appendix B.
It is also possible to give a Bohm-Bell type interpretation in terms of a particle moving on the joint
spectrum R3× {±} of (X, σ3)with a velocity of p±−Aon R3× {±}, where Ais the vector potential,
1One should be slightly careful with such interpretations, which are based on a classical probabilistic intuition for
conditional expectation, when random variables are functions on a sample space. In Quantum mechanics there is no such
thing as a common sample space for all of the random variables, which are the self-adjoint operators. There is one, though,
for a maximal commuting family of self-adjoint operators, its joint spectrum, which here, for (X, σ3), is Rn× {±}.
4 RAYMOND BRUMMELHUIS
while being allowed to jump between R3× {+}and R3× {−} with transition probabilities which are
compatible with the time-evolution of the probability densities |ψ±|2of the spin up and -down states.
As for the pure-spin case of section 7, such an interpretation, while attractive for a purely kinematical
causal description of quantum mechanics (as in Bohmian mechanics), complicates matters considerably
as far as the dynamics of the spin-vector S±is concerned, since the introduction of the jump process has
to be compensated for by a stochastic force whose physical meaning is not transparent. This perhaps
makes such a Bohm-Bell interpretation less attractive, from a physical point of view, than the two-fluid
interpretation, at least at this point in time.
Section 8 returns to general considerations, and discusses the relation of our conditional expectations
with the conditional expectations of the theory of C∗-algebras, in particular the C∗-algebraic conditional
expectation onto the von Neumann algebra generated by the conditioning operator B. For technical
reasons (depending on the definition of C∗-algebraic conditional expectation, the latter may not always
exists if it is for example required to preserve a semi-finite state, or may not be unique if it isn’t) we
limit ourselves to finite quantum systems, for which the underlying Hilbert space is finite-dimensional,
or, slightly more generally, to operators Bwhose spectrum is discrete. As an application, we derive an
evolution equation for Eψ(A|B)when ψsatisfies an abstract Schrödinger equation idψ/dt =Hψ, which
takes a simple form when A=H.
The final section concludes with some general observations, open problems and potentially interesting
directions for future research.
We end this introduction by going into some more detail regarding our definition of conditional ex-
pectation of operators and its classical probabilistic motivation. Let (Ω,F,P)be a (classical) probability
space, with Faσ-algebra of subsets of Ωand Pa probability measure on F.If X∈L2(Ω,F,P)is a
square-integrable random variable (equivalently, a random variable having finite variance) and if G ⊂ F
is a sub-σ-algebra, then the conditional expectation E(X|G)can be defined as the orthogonal projection
of Xonto the subspace L2(Ω,G,P)of G-measurable L2-functions. It is therefore the unique element
of this subspace which minimises the L2-distance to X. If Yis another random variable, or vector of
random variables Y= (Y1, . . . , Yn), then the conditional expectation of Xgiven Yis, by definition the
conditional expectation of Xwith respect to the σ-algebra FYgenerated by Y:EP(X|Y) := EP(X|FY.
Now FY-measurable functions are all of the form f(Y)with fa Borel-measurable function, a result
which is sometimes known as the Doob-Dynkin lemma. It follows that we can define EP(X|Y)as the,
essentially unique, function f(Y)of Ywhich minimizes EP(X−f(Y))2over all Borel functions ffor
which f(Y)2has finite expectation. Using statistical terminology, the conditional expectation of Xgiven
Yis the best predictor, with respect to the mean square error norm, of Xby a function of Y, and in this
form it generalizes naturally to Quantum Mechanics.
In Quantum Mechanics, the rôle of the random variables is played by the quantum mechanical ob-
servables, which are modeled by possibly unbounded self-adjoint operators Aon some Hilbert space H,
and that of the probability measure by a normalized vector ψ∈ H (or more precisely, by an element of
the projective space P(H) = H/C∗). The expectation of Ain a state ψwhich is in the domain of Ais
defined as
(1) Eψ(A) = (Aψ, ψ),
where (·,·)is the inner product on H; we will sometimes also use the more condensed notation hAiψ.If B
is a self-adjoint operator, or n-tuple B= (B1,...,Bn)of strongly commuting self-adjoint operators, we
define the conditional expectation of a self-adjoint operator Agiven B, relative to the normalized state
ψ, as a function f(B)of Bwhich minimizes Eψ((A−f(B))2) = ||(A−f(B))ψ||2, where the minimization
is over real-valued Borel measurable functions f, so that f(B)is self-adjoint. Such a minimizer always
exists by theorem 5.1 below, and the conditional expectation is unique modulo functions of Bwhich
have ψin their kernel. The conditional expectation can be seen as the best predictor, as in non-linear
statistical regression, of a self-adjoint operator Aby a self-adjoint operator of the form f(B)when the
system is in the state ψ. The value of the minimum, which is mostly non-zero, becomes a potentially
interesting quantity in its own right.
It is clear that the definition also makes sense for non-self-adjoint A, but it is then no longer natural
to minimize over real-valued f’s only. If we allow complex-valued f’s, a minimizer still exists and is
related to a so-called weak value of A. The minimum can now always be 0, for example when Bis the
position operator on Rnor, more generally, if the (joint) spectrum of the Bj’s is multiplicity free.
We finally note that we can replace the pure state ψby a density operator ρ=Pνwν|ψνihψν|with
wν≥0and Pνwν= 1 and Eψ(A)by Eρ(A) := Tr(ρA), and define conditional expectations Eρ(A|B)
CONDITIONAL EXPECTATIONS AND CAUSAL INTERPRETATIONS 5
with respect to ρ. We will briefly examine this generalization in section 2.4 below, but will otherwise
restrict ourselves to conditional expectations with respect to pure states.
2. Conditional expectations with respect to the position operator
In this and the next two sections we will first study conditional expectations of operators on L2(Rn) =
L2(Rn, dx), with dx the Lebesgue measure, in the special case when we condition with respect to the
position operators X= (X1, . . . , XN), and discuss the relationship with the Bohm interpretation. The
general case, which uses the spectral theorem, will be treated in section 5.
Let ψ∈L2(Rn)be a normalized state, and let Abe a symmetric operator with ψ∈Dom(A), the
domain of A. We want to determine a real-valued measurable function fon Rnwhich best approximates
Ain the state ψin the sense that Eψ((A−f(X))2) = ||(A−f(X))ψ||2is as small as possible, while
ψ∈Dom(f(X)).An elementary argument shows that this minimisation problem has a solution, which
is unique modulo multiplication operators having ψin their kernel.
Theorem 2.1. If ψis a normalized state in the domain of a symmetric operator A, then the minimization
problem
(2) min {||(A−f(X))ψ||2:f:Rn→RBorel measurable, fψ ∈L2(Rn)}.
has as solution the function
(3) f(x) =
Re ψ(x)A(ψ)(x)
|ψ(x)|21{ψ(x)6=0}
and any two minimizers differ by a function which is a.e. to 0on {x:ψ(x)6= 0}.In particular, the
minimizer is unique if ψ(x)6= 0 a.e. The minimum (2) is equal to
(4) Z{ψ=0}|A(ψ)(x)|2dx +Z{ψ6=0}Im A(ψ)(x)
ψ(x)2
|ψ(x)|2dx.
Proof. Since fis real-valued we have
||(A−f(X))ψ||2− ||Aψ||2
=ZRnf(x)2|ψ(x)|2−2f(x) Re ψ(x)A(ψ)(x)1ψ6=0 dx
=ZRn f(x)−Re A(ψ)(x)ψ(x)
|ψ(x)|2!2
|ψ(x)|21ψ6=0 dx −ZRnRe A(ψ)(x))ψ(x)2
|ψ(x)|21ψ6=0 dx.
Clearly, f=|ψ|−2Re ψA(ψ)·1ψ6=0 minimises this expression, and the minimiser is unique a.e. with
respect to |ψ(x)|2dx. The minimum is equal to
ZRn|A(ψ)|2dx −Z{ψ6=0}ReA(ψ)
ψ2
|ψ|2dx
=Z{ψ=0}|A(ψ)|2dx +Z{ψ6=0}
A(ψ)
ψ
2
−ReA(ψ)
ψ2!|ψ|2dx,
which is (4).
Definition 2.2. We define the conditional expectation Eψ(A X)of a self-adjoint or symmetric operator
Ain the state ψ∈Dom(A)with respect to the position operator Xas the operator of multiplication
(5) Eψ(A|X) := f(X),
with fgiven by (3).
We will call (4) the (mean square) prediction error (of Aby a function of Xin the state ψ). The
terminology is motivated by non-linear statistical regression: if one wants to predict the values of a
(classical) random variable Y, considered as the dependent variable, on the basis of observed values of
another random variable Z, the optimal predictor, in mean-squared sense, is precisely the conditional
expectation EP(Y|Z), and the prediction error is EPY−EP(Y|Z))2=EP(Y2)−EPEP(Y|Z)2.Similarly,
the prediction error ((4) equals Eψ(A2)−EψEψ(A|X)2, by the first line of its proof above.
The conditional expectation of a self-adjoint operator is again a self-adjoint operator (with its natural,
maximal, domain as a multiplication operator), much as the conditional expectation of a random variable
6 RAYMOND BRUMMELHUIS
in classical probability is a random variable. Distinguishing between fand the multiplication operator
f(X)may seem a bit pedantic, but the operator point of view becomes important when conditioning with
respect to arbitrary Hermitian operators in section 5 below, for example when relating the conditional
expectations with respect to operators which are linked by a unitary transformation. This being said,
in this section and the next two, we will usually confound the two, when no confusion is possible, and
simply consider Eψ(A|X)to be the function defined by (3).
The function Eψ(A|X)is defined for any operator A, not necessarily symmetric, and will still minimize
||(A−f(X))ψ||2, which we can (subject to suitable domain conditions on ψand Aψ) interpret as
Eψ(|A−f(X)|2), where for any operator B,|B|2:= B∗B. We will however not interpret it as a conditional
expectation if Ais not Hermitian, since it then no longer is natural to restrict the minimization to real-
valued f’s . If we allow complex-valued f, the minimum of ||(A−f(X))ψ||2is 0, with minimizer
f(x) = A(ψ)(x)
ψ(x)1{ψ6=0}.
Using physicists’ bra-ket notation (which by convention is linear in the second argument),
(6) A(ψ)(x)
ψ(x)=hx|A|ψi
hx|ψi,
which is the so-called weak value of Ain the pre-selected state |ψiand post-selected state |xi=δx,
introduced by Aharonov, Albert and Vaidman [1]. When Ais Hermitian, its real part is Eψ(A|X).The
expectation of its imaginary part equals the prediction error (4). If Ais the momentum operator, it is
rminus Nelson’s osmotic velocity [34] and if Ais the free Schrödinger operator, it is related to the minus
divergence of the probability flux: see (15) and (22) below. For arbitrary Ait can be written as the
conditional expectation of i[A, Πψ], where Πψis the orthogonal projection onto (the subspace generated
by) ψ:
Proposition 2.3. If Ais Hermitian,
(7) Im A(ψ)
ψ=−Eψ(i[A, Πψ]|X)
Proof. Since ψis, by assumption, normalized, Πψ(χ) = (χ, ψ)ψ, and i[A, Πψ]sends χto i((χ, ψ)A(ψ)−(Aχ, ψ)ψ).
In particular, assuming that ψ(x)6= 0 a.e. (if not, multiply by the indicator function of {ψ6= 0}),
Eψ(i[A, Πψ]|X) = Re iAψ −i(Aψ, ψ )ψ
ψ= Re iAψ
ψ,
since (Aψ, ψ)∈R.
The linear functional A→Eψ(A|X)is R-linear and, when restricted to the real vector space of
symmetric operators having ψin their domain, satisfies the iterated expectations property
(8) Eψ(Eψ(A|X)) = Eψ(A).
It is possible to find a C-linear extension to the space of all operators which still satisfies (8): see
subsection 2.4 below.
Remarks 2.4. (i) Holland [27] calls Re(ψ(x)A(ψ)(x))/|ψ(x)|2the local expectation value of Ain the
state ψin the position representation, and observes that it equals the Bohm momentum if Ais the
momentum operator P, and gives rise to the quantum potential if A=1
2P2, the free (classical) energy
operator: see below. Theorem 2.1 provides an interpretation for this local expectation value in terms
of the standard formalism of Quantum mechanics, and suggests the value of the minimum (4) as a
potentially interesting new quantity to be computed and, if possible, physically interpreted. We will do
so below for the momentum and kinetic energy operators.
(ii) If we define the variance of a symmetric operator Bin the (normalized) state ψby Varψ(B) :=
Eψ(B2)−Eψ(B)2= (Bψ, B ψ)−(Bψ, ψ )2, then the minimum (4) can also be written as
||Aψ||2−EψEψ(A|X)2= Varψ(A)−VarψEψ(A|X),
we used (8). In particular, since the minimum is positive or zero, the map A→Eψ(A|X)is variance-
reducing. If Ais dispersion-free in the state ψ, in the sense that Eψ(A2) = Eψ(A)2, then ψwill then be
an eigenvector of A, by the case of equality of Cauchy-Schwarz, and Eψ(A|X) = λwill be a constant.
Another potentially interesting quantity to consider is the conditional variance,
Eψ(A2|X)−Eψ(A|X)2.
CONDITIONAL EXPECTATIONS AND CAUSAL INTERPRETATIONS 7
If A=Pthis turns out to be the Bohm-de Broglie quantum potential: see section 2.2 below.
(iii) The analogy with classical conditional expectations is not perfect: A≥0does not imply that
Eψ(A|X)≥0.Similarly, the analogue of EP(f(Z)Y|Z) = f(Z)EP(Y|Z)for ordinary random variables Y
and Zand bounded measurable f, is no longer true: while we do have that Eψ(f(X)A) = f·Eψ(A)for
bounded real-valued f, right-multiplication of Awith f(X)is more complicated: the best we can say is
that if f(X)ψ∈Dom(A)and f(x)6= 0 a.e. with respect to |ψ|2dx, then Eψ(Af (X)) = f·Ef(X)ψ(A).
In particular (remembering that we only speak of conditional expectations of Hermitian operators)
Eψ(f(X)Af(X)|X) = f(X)2Ef(X)ψ(A|X)
for such f.
2.1. The Bohm momentum. If Pj=1
i∇j, the j-th component of the momentum operator on Rn
with ∇j=∂/∂xj, and if we write the wave function in complex polar coordinates, ψ=ReiS , where we
will from now on assume for simplicity that ψ(x)6= 0 a.e., then
(9) Eψ(Pj)(x) = Re Pj(ψ)(x)
ψ(x)= Im ∇jψ(x)
ψ(x)=∇jS(x),
which is the j-th component of the Bohm momentum
pB(x) := pB,ψ (x) := ∇S(x),
introduced by Bohm [5, 6]. It follows that Eψ(P|X) = pB(X)(as vectors of multiplication operators),
and that pB,ψ (X)is the best approximation, in mean-square sense, of the momentum operator by a
multiplication operator for a system in a quantum state ψ. This gives a new perspective on the Bohm
momentum and on the causal interpretation of Quantum mechanics, one of whose main building blocks
is seen to have a natural interpretation within the standard formalism of Quantum mechanics.
The minimum (2) when A=Pjis given by
(10) ZRRe ∇jψ(x)
ψ(x)2
|ψ(x)|2dx =ZR
(∇jR(x))2dx,
where ψ=ReiS as before (note for example that |ψ|−2Reψ∇ψ=1
2|ψ|−2∇|ψ|2=R−1∇R). It may be
interesting to note that the right hand side is related to the Fisher information of the probability density
function
ρψ(x) := |ψ(x)|2=R(x)2,
which the Born interpretation associates to the wave function, assumed to be normalized. This is
clearest when n= 1: the Fisher information for the location parameter of a univariate probability
density function2ρ=ρ(x)is defined as
(11) IF(ρ) := ZRd
dx log ρ(x)2
ρ(x)dx.
Since IF(ρψ) = IF(R2) = 4 R(R′)2dx, we have
(12) min (P−f(X))2ψ, ψ:f:R→RBorel, ψ ∈D(f(X))=1
4IF(ρψ).
where P=i−1d/dx. If n > 1, then ρ=ρψis a multivariate probability density, whose Fisher information
is now defined as the variance-covariance matrix of ∇(log ρ)with respect to ρ,
(13) IF(ρ) = ZRn
(∇jlog ρ)(∇klog ρ)ρ dx1≤j,k≤n
,
noting that the expectation R∇jlog ρ)ρdx =R∇jρdx = 0 (assuming ρtends to 0 at infinity). It follows
that if A=Pjand ψ(x)6= 0 a.e., the minimum (2) is equal to
(14) min
f||(Pj−f(X))ψ||2=1
4IF(ρψ)jj .
We will see below that the Fisher information is also proportional to the expected value of the quantum
potential.
2In fact, the Fisher information is introduced in the context of parametrized statistics as the variance of the score
function of a parametrized pdf ρ(x, θ)with parameter θ:I(θ) = R(∂θlog ρ(x, θ))2ρ(x, θ)dx, the expected value of the score
as computed with the probability density ρ(x, θ)being 0.In case θis a location parameter, meaning that ρ(x, θ) = ρ(x−θ),
we find (11).
8 RAYMOND BRUMMELHUIS
Recall that the osmotic momentum of the wave-function ψis defined as
pO:= pO,ψ := ∇R/R;
cf. Nelson [34], Bohm and Hiley [8]. Since Im(P ψ /ψ) = −Re(∇ψ/ψ) = −1
2(ψ∇ψ+ψ∇ψ)/|ψ|2=
−1
2∇log |ψ|2, we see that the weak value (6) of Pis given by
(15) P(ψ)
ψ=pB,ψ −ipO,ψ ,
cf. Flack and Hiley [21]. Note that the minimum (10) can also be interpreted as Eψ(p2
O,j ).
2.2. Conditional expectation of kinetic energy and the quantum potential. The conditional
expectation Eψ(H0|X)of the free Hamiltonian H0=1
2PjP2
j=−1
2∆is given by multiplication by
(16) Eψ(H0)(x) = −Re ∆ψ
2ψ=1
2(X
j∇jS)2−∆R
2R,
where ψ=ReiS .We recognize the first term on the right as the square of the Bohm momentum,
p2
B:= pB·pB, the dot representing the euclidian inner product on Rn, while the final term is the de
Broglie-Bohm quantum potential:
(17) Q(x) := Qψ(x) := −∆R(x)
2R(x).
Recall that we are assuming that ψ(x)6= 0 a.e., otherwise we would multiply by the indicator function
of {x:ψ(x)6= 0}.With this notation, Eψ(H0|X), which we will call the Bohm energy, can be identified
with (the operator of multiplication by) 1
2pB(ψ)2+Qψ.This can also be stated as follows:
Proposition 2.5. The (operator of multiplication by the) quantum potential Qψis equal to 1/2 times
the conditional variance of Pgiven X:
(18) Qψ(X) = 1
2Eψ(P2|X)−Eψ(P|X)2,
where for any vector V,V2:= V·V, the Euclidean inner product of Vwith itself.
The quantum potential can therefore be interpreted as the conditional variance of the momentum oper-
ator with respect to the position operators.
We note the following relation between Fisher information and the expected value of Qψ(X), cf. [35].
Proposition 2.6.
(19) Eψ(Qψ(X)) = 1
8TrIF(ρψ)
Proof. Recalling that R=|ψ|, an integration by parts (justified if ψis in the domain of H0) shows that,
(Qψψ, ψ) = −1
2ZRn
R∆R dx =1
2ZRn
(∇R)2dx.
On the other hand,
IF(ψ)jj =ZRn∇jR2
R22
R2dx = 4 ZRn
(∇jR)2dx,
and summation over jfinishes the proof.
More generally, one can relate the conditional correlations of the components of Pto the matrix
elements of the Fisher information matrix (13): if we define the conditional covariance of Pjand Pkby
(20) covψ(Pj, Pk|X) := Eψ(PjPk)−Eψ(Pj|X)Eψ(Pk|X)
then we have:
Proposition 2.7.
(21) Eψ( covψ(Pj, Pk|X) ) = 1
4IF(ρψ)jk
CONDITIONAL EXPECTATIONS AND CAUSAL INTERPRETATIONS 9
Proof. By direct computation, Eψ(PjPk|X)equals (multiplication by) Re−∇2
jk ψ/ψ=−∇2
jk R/R −
∇jS∇kSwhich shows that covψ(Pj, Pk|X) = −∇2
jk R/R. Its expectation (in the state ψ) equals, after
integration by parts,
−ZRn
(∇2
jk R)Rdx =ZRn∇jR∇kRdx =1
4IF(ρψ)jk ,
on substituting ρψ=R2in (13).
We finally compute the weak value of H0and the value of the minimum (2) when A=H0.
Proposition 2.8. Let ψbe in the domain of H0with ψ(x)6= 0 a.e., and let pB:= pB,ψ , the Bohm-
momentum, pO:= pO,ψ , the osmotic momentum, and ρ=ρψ=|ψ|2, the probability density for position.
If
h0,B := 1
2p2
B+Qψ,
the Bohm energy of a free Schrödinger particle in the state ψ, then the weak value of H0equals
H0ψ
ψ=h0,B −i
2ρ−1∇ · (ρpB)(22)
=h0,B −i
2∇ · pB−ipB·pO
and the prediciton error
(23) min
freal
Eψ(H0−f(X))2=1
4ZRn
ρ−1(∇ · ρpB)2dx
Proof. Equation (22) follows from
ImH0ψ
ψ=−1
4i
ψ∆ψ−ψ∆ψ
|ψ|2=−1
2∇ · jψ
|ψ|2,
where jψ=1
2iψ∇ψ−ψ∇ψ= Im ψ∇ψ, the probability current, together with the relations pB(ψ) =
ρ−1
ψjψand ρ−1∇ρ= 2pO, and (23) is then a consequence of Theorem 2.1.
Remarks 2.9. (i) If the wave-function ψ=ψ(x, t)satisfies the time-dependend Schrödinger equation,
i∂tψ=H0ψ+V ψ,
with real-valued potential V, then ∂t|ψ|2+∇xjψ= 0, and the minimum (23) at time tcan also be written
as 1
4Z(∂tρψ)2ρ−1
ψdx,
where ρψ=|ψ|2as before. The integral is the expected value, with respect to ρψ, of (∂tlog ρψ(·,t))2which,
somewhat curiously, is the Fisher information of ρψ(x, t)when the latter is interpreted as a parametrized
family of probability densities with time tas the parameter. It is not clear to the author how to interpret
this in physical terms. See Frieden [22] for Fisher information as a unifying concept for physics, and
Lavis and Streatham [29] for a critical review.
(ii) Another way to derive (22) is to take the real and imaginary parts of the identity
∇ · ∇ψ
ψ=∆ψ
ψ−∇ψ
ψ·∇ψ
ψ,
and then to use (15). This gives an alternative expression for the quantum potential in terms of the
osmotic momentum pO:
(24) Qψ=−1
2∇ · pO+p2
O,
a formula going back to Wyatt and Bittner [46] if we substitute pO=∇log |ψ|; see also Sbitnev [36].
This can of course also be verified directly from the polar decomposition of the wave function. Another
expression for the quantum potential which is sometimes useful is
(25) Qψ=1
2∇R
R2
−1
4
∆ρ
ρ=1
8∇ρ
ρ2
−1
4
∆ρ
ρ,
as follows for example by calculating ∆ρ/ρ =∇ · ∇R2/R2.Note that the first term on the right corre-
sponds to the kinetic energy associated to the osmotic momentum.
10 RAYMOND BRUMMELHUIS
We next consider some natural extensions of the theory.
2.3. Vector-valued wave functions. One easily checks that the first formula of (3) for the conditional
expectation remains valid for operators Aacting vector-valued wave functions ψ= (ψ1,...,ψN)∈
L2(R,CN), with its standard inner product, if we replace ψA(ψ)by ψ∗A(ψ) = PνψνA(ψ)νand |ψ|2
by ψ∗ψ=Pνψνψν, where ψ∗=ψ(x)∗is the conjugate transpose of ψ(x)∈CN.Concretely, if Ais a
Hermitian operator on L2(Rn;CN), then Eψ(A|X)is the operator of multiplication by the function
(26) Eψ(A)(x) := Re ψ(x)∗A(ψ)(x)
ψ(x)∗ψ(x)·I
where Iis the N×Nidentity matrix. The minimum is now easily found to be
(27) min
f
Eψ((A−f(X))2) = ZRn(Im ψ∗A(ψ))2
ψ∗ψ+A(ψ)∗A(ψ)−|ψ∗A(ψ)|2
ψ∗ψdx,
(franging as before over real-valued Borel functions on Rn). Compared to the scalar case, there are two
additional terms in the integrand on the right ,whose net-contribution is point-wise positive because of
the Cauchy-Schwartz inequality on CN.We therefore have an inequality
(28) inf
f
Eψ(A−f(X))2≥ZRn
(Im ψ∗A(ψ))2
ψ∗ψdx,
where for N= 1 we had an equality. If A=Pj=i−1∇jon L2(R;CN)and ψ= (ψ1,...,ψN), then
(29) Eψ(Pj)(x) = Im Pνψν∇jψν
Pνψνψν
,
which for N= 2 is the familiar expression of the Bohm momentum in the causal interpretation of the
Pauli theory of a non-relativistic particle with spin [9]. Its motivation in that paper, as for the ordinary
Schrödinger equation [5], came from the continuity equation for the probability density (the right hand
side of (29) is the probability flux divided by the density) which is a consequence of quantum dynamics,
in the form of the time-dependent Pauli equation. Here the dynamics has not yet played a rôle and the
definition of Eψ(P)as the best approximation of Pby multiplication operators is, in a sense, purely
kinematical.
If A=Pj, then Im ψ∗Pjψ=−Re ψ∗∇jψ=−1
2∇jψ∗ψ, and the right hand side of (28) equals
1
4IF(ρψ)jj , where IF(ρψ)is the Fisher information matrix of ρψ=ψ∗ψ=P|ψν|2, but this is now only
a lower bound.
In the vector-valued case we can also enlarge the set of commuting operators with respect to which
we condition, for example by adding a number of commuting orthogonal projections Πνof the target
space CN.If we take for the Πν’s the orthogonal projections onto the vectors of the standard basis of
CN, then functions of (X, Π1,...,ΠN)are operators of multiplication by diagonal matrix functions
f1(x) 0 ··· 0
0f2(x)··· 0
.
.
..
.
....0
0 0 ··· fN(x)
with f1,...,fNBorel, and one finds that the best approximation Eψ(A|X, Π1, ...ΠN)of Aby such
multiplication operators in the state ψis given by multiplication by the diagonal matrix function
(30)
Re(A(ψ)1/ψ1) 0 ··· 0
0 Re(A(ψ)2/ψ2)··· 0
.
.
..
.
....0
0 0 ··· Re(A(ψ)N/ψN)
where A(ψ)j=A(ψ)j(x)are the components of A(ψ),j= 1,...,N. See section 5 below for such
conditional expectations in a more general framework and sections 6 and 7 for applications to spin and
to the Pauli operator.
CONDITIONAL EXPECTATIONS AND CAUSAL INTERPRETATIONS 11
2.4. Conditional expectation with respect to a mixed state. Let ρ=PνwνΠψνbe a mixed
state, with Πνthe orthogonal projection onto the (normalized) state ψνand wν∈[0,1] summing to 1
(ρ=Pνwν|ψνihψν|in physicist’s notation), and let
(31) Eρ(A) := X
ν
wν(A(ψν), ψν)
the associated expectation value. If, as before, we look for a function f(X)of the position variable X,
with freal-valued Borel-measurable, which minimizes Eρ((A−f(X))2), then one easily finds that the
minimum is attained for
(32) f(x) = PνwνRe ψν(x)A(ψν)(x)
Pνwν|ψν(x)|2,
an expression found by Maroney [32] for the case of the momentum operator A=Pusing Brown and
Hiley’s approach to the Bohm theory. (We again assume for simplicity that the denominator never
vanishes, if not (32) has to be multiplied by the indicator function of where the denominator is non-zero,
which then singles out a particular minimizer, as before.) The expression for the minimum is less clean
than for a pure state: it is equal to the integral over Rnof the function
X
ν
wν|A(ψν)|2−PνwνRe ψνA(ψν)2
Pνwν|ψν|2
=PνwνIm ψνA(ψν)2
Pνwν|ψν|2+X
ν
wν|A(ψν|2−PνwνψνA(ψν)2
Pνwν|ψν|2
Note that if ρ=PN
ν=1 wν|ψνihψν|is of finite rank, then these formulas are the same as for a vector-
valued wave-function (ψ1,...,ψN)∈L2(Rn,CN), with inner product (ψ, ϕ) = RPνwνψνψνdx and