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On classical systems and measurements in quantum mechanics


The recent rigorous derivation of the Born rule from the dynamical law of quantum mechanics Allahverdyan et al. (Phys Rep 525:1–166., 2013) is taken as incentive to reexamine whether quantum mechanics has to be an inherently probabilistic theory. It is shown, as an existence proof, that an alternative perspective on quantum mechanics is possible where the fundamental ontological element, the ket, is not probabilistic in nature and in which the Born rule can also be derived from the dynamics. The probabilistic phenomenology of quantum mechanics follows from a new definition of statistical state in the form of a probability measure on the Hilbert space of kets that is a replacement for the von Neumann statistical operator to address the lack of uniqueness in recovering the pure states included in mixed states, as was pointed out by Schrödinger. From the statistical state of a quantum system, classical variables are defined as collective variables with negligible dispersion. In this framework, classical variables can be chosen to define a derived classical system that obeys, by Ehrenfest’s theorem, the laws of classical mechanics and that describes the macroscopic behavior of the quantum system. The Born rule is derived from the dynamics of the statistical state of the quantum system composed of the observed system interacting with the measurement system and the role of the derived classical system in the process is exhibited. The approach suggests to formulate physical systems in second quantization in terms of local quantum fields to ensure conceptually equivalent treatment of space and time. A real double-slit experiment, as opposed to a thought experiment, is studied in detail to illustrate the measurement process.

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  1. 1.

    See Appendix A for a short formulation of QM.

  2. 2.

    See Appendix C.1 for details.

  3. 3.

    For ease of reference, the description of the measurement process derived by ABN is summarized in Appendix C.2.

  4. 4.

    See Eq. (B.8) in Appendix B.3 for the definition.

  5. 5.

    A brief summary of non-equilibrium statistical mechanics, both classical and quantum can be found in Appendix B.

  6. 6.

    The definition is given in Appendix C.1.

  7. 7.

    See Appendix C.3 for the definition.

  8. 8.

    A brief summary of the formulation of QM is given in Appendix A.

  9. 9.

    The composition of systems and the reverse, extracting subsystems from systems, are two operations that show the big difference between CM and QM, as discussed in Appendices A.2 and A.3.

  10. 10.

    See Theorem A.2 in Appendix A for the inspiration for the form of the statistical operator.

  11. 11.

    Given by Eq. (A.14) in Appendix A.

  12. 12.

    Definition B.1 in Appendix B.

  13. 13.

    One such exception the case of atomic nuclei in the realm of atomic, molecular, and materials physics: it is not necessary to treat the coordinates of the protons and neutrons, or quarks and gluons, inside the nucleus explicitly to obtain highly accurate results in atomic, molecular, and materials physics. The center of mass coordinate of the protons and neutrons decouples and can be used by itself to describe the nucleus in that realm.

  14. 14.

    In Appendix D, we summarize a few of the most relevant properties pertinent to the formulation of statistical QM.

  15. 15.

    See Theorem D.1 in Appendix D.

  16. 16.

    The definition of superposition and entanglement can be found in Appendix A.3.

  17. 17.

    See Appendix D for the definition.

  18. 18.

    The procedure is summarized in Appendix B.

  19. 19.

    The ABN description of the measurement process is summarized in Appendix C.2.

  20. 20.

    The definition is given and discussed in Appendix C.3.

  21. 21.

    Definition in Appendix C.3.

  22. 22.

    On spaces like \(\mathbb {R}^{2N}\) Borel sets are sets that can be created as unions and intersections of open hypercubes.


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My deepest appreciation goes to the anonymous reviewers for their thorough analysis of the paper and numerous substantive comments and suggestions that resulted in a very much improved end result. My thanks go to Henk Monkhorst and several anonymous referees for their valuable comments and suggestions on earlier expositions of the ideas presented in this paper.

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Brief formulation of quantum mechanics

For ease of reference and consistency of conventions, nomenclature, and notation, we give a brief overview of the formalism of QM.

In CM, a degree of freedom is described by a coordinate q and its velocity \(\dot{q}\) (Lagrangian formalism) or its conjugate momentum p (Hamiltonian formalism). With the set of degrees of freedom for a system specified, one selects the range of values, called the spectrum, of one coordinate in each pair, typically q, and assembles them into the configuration space. For example, a system consisting of N degrees of freedom, with each coordinate having the real numbers as spectrum, has the configuration space

$$\begin{aligned} \mathbb {F} = \mathbb {R}^N. \end{aligned}$$

The phase space in QM is then defined as the Hilbert space of complex-valued functions on the configuration space

$$\begin{aligned} \mathbb {H} = L^2(\mathbb {F},\lambda ,\mathbb {C}) = \otimes ^N L^2(\mathbb {R},\lambda ,\mathbb {C}). \end{aligned}$$

Here \(\lambda \) denotes the Lebesgue measure on \(\mathbb {F}\) and \(\mathbb {R}\), respectively. The second form of the phase space in Eq. (A.2) shows its structure as a composition, tensor product, of the N degrees of freedom [61], similar to the decomposition of the phase space in CM as

$$\begin{aligned} \mathbb {H}=\oplus ^N \mathbb {R}^2, \end{aligned}$$

where \(\mathbb {R}^2\) is a two-dimensional phase space of a single degree of freedom with the symplectic form

$$\begin{aligned} J=\left( \begin{array}{ll} 0 &{} 1 \\ -1 &{} 0 \end{array} \right) . \end{aligned}$$

This form of phase space is called the Schrödinger representation. The functions on configuration space that are the elements of phase space are called wave functions. The state of a system in QM is completely determined by a wave function, or an element of the phase space \(\mathbb {H}\) in any other representation. The abstract form of the state, introduced by Dirac [33], is the ket\(|{\varPsi }\rangle \). The two are related by

$$\begin{aligned} {\varPsi }(q_1,\ldots ,q_N) = \langle q_1,\ldots ,q_N | {\varPsi }\rangle . \end{aligned}$$

The multiplication operator \(\varvec{q}\)

$$\begin{aligned} \varvec{q} : \mathbb {H} \rightarrow \mathbb {H} : {\varPsi }\mapsto {\varPhi }= \varvec{q} {\varPsi }, \end{aligned}$$


$$\begin{aligned} {\varPhi }= \varvec{q} {\varPsi }: \mathbb {F} \rightarrow \mathbb {C} : q \mapsto {\varPhi }(q) = q {\varPsi }(q), \end{aligned}$$

is the observable associated with the coordinate q in the Schrödinger representation, which is the spectral representation of the observable as a Hermitian (technically self-adjoint) operator.

In the same representation the canonically conjugate coordinate is represented by the Hermitian operator that is the derivative operator \(\varvec{p}\)

$$\begin{aligned} \varvec{p} : \mathbb {H} \rightarrow \mathbb {H} : {\varPsi }\mapsto {\varPhi }= \varvec{p} {\varPsi }, \end{aligned}$$


$$\begin{aligned} {\varPhi }= \varvec{p} {\varPsi }: \mathbb {F} \rightarrow \mathbb {C} : q \mapsto {\varPhi }(q) = -i\hbar \frac{\mathrm {d} {\varPsi }(q)}{\mathrm {d} q}. \end{aligned}$$

The fact that the two coordinates are canonically conjugate is expressed by the commutation relation between the observables as operators

$$\begin{aligned} {[} \varvec{q}, \varvec{p} ] = i\hbar \varvec{I} \end{aligned}$$

with \(\varvec{I}\) the identity operator on \(\mathbb {H}\). A theorem by von Neumann establishes that the representation of conjugate coordinate by multiplication and derivative is a unique representation of a pair of operators satisfying the above commutation relation [61]. It is possible to represent the state of the system as a wave function on the spectrum of \(\varvec{p}\), which is called the momentum representation. In that representation the coordinate \(\varvec{q}\) is then represented as the derivative operator.

The two observables \(\varvec{q}\) and \(\dot{\varvec{q}}\), or \(\varvec{p}\), that make up a degree of freedom are inextricably intertwined in QM. This is the most significant difference with CM, where both coordinate and canonical momentum are separate variables that each take on a single numeric real value for any state of a classical system at all times during any dynamical process. The degree of freedom as part of the description of the state of any quantum system is represented by the wave function on the spectrum of one of the two observables as Hermitian operators. If the wave function is given as a function on the spectrum of the coordinate \(\varvec{q}\), then all information about the wave function as a function on the spectrum of the conjugate momentum \(\varvec{p}\) is already available and can be obtained by the spectral transform of the operator \(\varvec{p}\), which is the Fourier transform.

There is no way to split the information in the wave function into a part that has only the information about the coordinate \(\varvec{q}\) and another part that has only the information about the conjugate momentum \(\varvec{p}\).

It is useful to look at the Hilbert space of QM as a classical phase space with the real and imaginary parts of the wave function as conjugate variables and the complex structure of the Hilbert space as the symplectic structure [49]. However, there is no simple relation between the real or imaginary part by itself of the wave function in the position representation and any observable of a particle that could give it physical meaning.

The Wigner distribution function\(W_\psi (q,p)\) computed from a wave function \(\psi \) [8, p. 29] provides a way to visualize the position and momentum content of a wave function in one construct. The very fact that the Wigner distribution provides two views of the same wave function, smoothly connecting the wave function for position with its Fourier transform for momentum, illustrates the point that the position and momentum content are inextricably connected in the ket.


We formulate the basic assumptions of QM:

Assumption A.1

QM is a mechanics of kets, or wave functions, that represent physical systems in terms of their degrees of freedom. The kets are the basic elements of the mathematical formalism of QM.

The ket can be interpreted as the state of the physical system, but this requires an ontology and an interpretation to provide context, as is discussed in Appendix C.

Assumption A.2

In QM, the evolution in time of the ket is given by the solution of the first-order differential equation for kets \(|{\varPsi },t\rangle \) or wave functions \({\varPsi }(q_1,\ldots ,q_N,t)\) of the system

$$\begin{aligned} i \hbar \frac{\partial }{\partial t}{\varPsi }(q_1,\ldots ,q_N,t) = \varvec{H} {\varPsi }(q_1,\ldots ,q_N,t) \end{aligned}$$

called the Schrödinger equation (SE). All observable phenomena in physics can be derived from the evolution of kets of physical systems according to the SE.

The construction of the Hamiltonian operator \(\varvec{H}\) is an important step in the formulation of the quantum description of any system. The classical Hamiltonian function H(qp) often serves as a good guide.

Because the SE is a linear equation, its solution is explicitly known in general and it is given in terms of the one-parameter family of unitary operators

$$\begin{aligned} \varvec{U}(t) = \exp \left( -\frac{it}{\hbar } \varvec{H}\right) = \int _{E_0}^\infty \exp \left( -\frac{it}{\hbar }E\right) \varvec{M}(\mathrm {d}E), \end{aligned}$$

where \(\varvec{M}\) is a projection-operator-valued measure on the spectrum \([E_0,\infty [\) of \(\varvec{H}\) [66]. This family of operators forms an Abelian group: \(\varvec{U}(t_1+t_2) = \varvec{U}(t_1) \varvec{U}(t_2)\). The operators are defined with the spectral representation of the Hamiltonian operator \(\varvec{H}\), which is the unitarily equivalent form of the operator where it acts as a multiplicative operator [61, 66].

The relativistic description of a quantum system requires the extension of the one-parameter group of time translations Eq. (A.12) to a unitary representation \(\varvec{U}(\mathbf {r},t,{\varLambda })\) in \(\mathbb {H}\) of the 10-dimensional Poincaré Lie group with \((\mathbf {r},t)\) a translation in spacetime and \({\varLambda }\) an element of the 6-dimensional Lorentz Lie subgroup of rotations and Lorentz boosts [45, 83, 84, 92]. The transformation of an operator \(\varvec{A}\) under the full Poincaré group

$$\begin{aligned} \varvec{A}_{\mathbf {r},t} = \varvec{U}(\mathbf {r},t,I)^\dagger \varvec{A} \varvec{U}(\mathbf {r},t,I) \end{aligned}$$

is a natural relativistic generalization of the Heisenberg representation [33, p. 112].

An important phenomenological element of QM is Born’s rule [18, 19] which provides a connection between the formal description of a quantum system given by the ket and numbers obtained in experiments on the quantum system supposedly described by that ket. The rule is as follows.

For a quantum system in a pure state described by a ket \(|{\varPhi }\rangle \) the measurement of an observable represented by a self-adjoint operator \(\varvec{A}\) results in values a being obtained with a probability p(a) that is given by the modulus squared

$$\begin{aligned} p(a) = |{\varPhi }(a)|^2 \end{aligned}$$

of the wave function \({\varPhi }(a)\) representing the ket \(|{\varPhi }\rangle \) on the spectrum of \(\varvec{A}\).

The Born rule is variably used as a definition of the interpretation or the meaning of the wave function, and of the ket, or as an assumption in the formulation of QM to describe an alternative way the ket can change under the effect of interacting with other systems in addition to evolution under the SE as in Assumption A.2. Everybody uses the Born rule, but there is no universal consensus on what its true role is [23, 50, 60, 91]. In the work of ABN [3, 4] and in this paper, the Born rule is derived from from the LvNE Eq. (B.10) and the SE Eq. (A.11), respectively.

Composite systems

There are important differences between CM and QM in the way the theories handle composite systems. We exhibit the procedure in two theorems; the statement in CM is usually not discussed explicitly because it is considered self-evident.

Theorem A.1

Consider two systems A and B in CM with states

$$\begin{aligned} (q^A,p^A)=(q_1^A,\ldots ,q_M^A,p_1^A,\ldots ,p_M^A) \end{aligned}$$


$$\begin{aligned} (q^B,p^B)=(q_1^B,\ldots ,q_N^B,p_1^B,\ldots ,p_N^B). \end{aligned}$$

The state of the composite system \(C=A+B\) is then given by the vector \((q^A,q^B,p^A,p^B)\) in the direct sum \(\mathbb {R}^{2M}\oplus \mathbb {R}^{2N}\) of the phase spaces \(\mathbb {R}^{2M}\) and \(\mathbb {R}^{2N}\). This state is unique.

Conversely, any state \((q_1^C,\ldots ,q_K^C,p_1^C,\ldots ,p_K^C)\) of a system C can be decomposed into a state \((q^A,p^A)\) for subsystem A and a state \((q^B,p^B)\) for subsystem B with \(K=M+N\). This decomposition is unique.

The theorem is a direct consequence of the structure of classical phase space. As a consequence, every system can be uniquely characterized as a composition of its degrees of freedom as shown in Eq. (A.3).

The situation in QM is very different. The analysis is due to von Neumann [61, § VI.2]. We formulate his result in close parallel with the result in CM.

Theorem A.2

Consider two systems A and B in QM with respective kets \(|{\varPsi }^A\rangle \) and \(|{\varPsi }^B\rangle \). The state of the composite system C is given by \(|{\varPsi }^C\rangle = |{\varPsi }^A\rangle \otimes |{\varPsi }^B\rangle \) in the tensor product \(\mathbb {H}^A \otimes \mathbb {H}^B\) of the phase spaces \(\mathbb {H}^A\) and \(\mathbb {H}^B\). This ket is unique.

Conversely (von Neumann’s result), given a ket \(|{\varPsi }^C\rangle \) of a system C, with associated statistical operator \(\varvec{D}^C=|{\varPsi }^C\rangle \langle {\varPsi }^C|\), it is possible to find two statistical operators \(\varvec{D}^A\) and \(\varvec{D}^B\) for subsystems A and B such that

$$\begin{aligned} \varvec{D}^A= & {} \sum _{n=0}^\infty \lambda _n |{\varXi }^A,n\rangle \langle {\varXi }^A,n| \nonumber \\ \varvec{D}^B= & {} \sum _{n=0}^\infty \lambda _n |{\varUpsilon }^B,n\rangle \langle {\varUpsilon }^B,n| \nonumber \\ \varvec{D}^C= & {} \sum _{n=0}^\infty \lambda _n |{\varXi }^A,n\rangle |{\varUpsilon }^B,n\rangle \otimes \langle {\varXi }^A,n| \langle {\varUpsilon }^B,n| \end{aligned}$$

This decomposition is unique.

For a general ket \(|{\varPsi }^C\rangle \) more than one weight \(\lambda _n\) is different from zero, so that the separation of a system \(C=A+B\) does not lead to a unique ket for each of the two subsystems A and B; that only happens for product kets \(|{\varPsi }^C\rangle = |{\varPsi }^A\rangle \otimes |{\varPsi }^B\rangle \).

Given two statistical operators \(\varvec{D}^A\) and \(\varvec{D}^B\), the ket \(|{\varPsi }^C\rangle \) can be reconstructed from the sequences of kets \((|{\varXi }^A,n\rangle )_n\) and \((|{\varUpsilon }^B,n\rangle )_n\) together with the eigenvalues \((\lambda _n)_n\) and a sequence of phases \((\varphi _n)_n\)

$$\begin{aligned} |{\varPsi }^C\rangle = \sum _{n=0}^\infty \sqrt{\lambda _n} e^{i\varphi _n} |{\varXi }^A,n\rangle \otimes |{\varUpsilon }^B,n\rangle . \end{aligned}$$

The reconstruction of a general state \(|{\varPsi }^C\rangle \) from two statistical operators \(\varvec{D}^A\) and \(\varvec{D}^B\) cannot be completed without the phases \((\varphi _n)_n\).

The proof is well-known and can be found in many textbooks, in addition to von Neumann’s book, see for example [51, § 11-8]. The kets \((|{\varXi }^A,n\rangle )_n\) and \((|{\varUpsilon }^B,n\rangle )_n\) are obtained as the eigenvectors of the density operators, which are by definition real and positive, so that the eigenvectors are naturally chosen as real. The decomposition is a special case of the Schmidt decomposition [70, 71]. That decomposition is written without the phase factors because the functions \((|{\varXi }^A,n\rangle )_n\) and \((|{\varUpsilon }^B,n\rangle )_n\) do not have to be real in general.

These two theorems show that decomposition into subsystems is unambiguous in CM, but it is in general not possible in QM.

Superposition and entanglement

The phase space and the dynamical equation in QM are both linear: if \(|{\varPsi }\rangle \) and \(|{\varPhi }\rangle \) are valid kets for a physical system, then \(a|{\varPsi }\rangle + b|{\varPhi }\rangle \) is also a valid ket for any two complex numbers a and b; if \(|{\varPsi },t\rangle \) and \(|{\varPhi },t\rangle \) are valid evolutions for a physical system as solutions of the SE, then \(a|{\varPsi },t\rangle + b|{\varPhi },t\rangle \) is also a valid evolution and a solution of the SE for any two complex numbers a and b. Such a linear combination of states, kets, and wave functions is called a superposition.

Superpositions in spaces that are tensor products lead to a new feature that is very different from classical superpositions of waves in water or in the electromagnetic field. Consider a superposition \(a_n|\alpha ,n\rangle + b_n|\beta ,n\rangle \) for a single degree of freedom labeled n. A valid ket for the system with N degrees of freedom \(n=1,\ldots ,N\) is the tensor product ket

$$\begin{aligned} \bigotimes _{n=1}^N (c_{\alpha n}|\alpha ,n\rangle + c_{\beta n}|\beta ,n\rangle ). \end{aligned}$$

This ket is a superposition of the product kets

$$\begin{aligned} |p(1),1\rangle \otimes \ldots \otimes |p(N),N\rangle , \end{aligned}$$

where p is a map \(n \mapsto p(n) \in \{\alpha ,\beta \}\). There are \(2^N\) such kets. The superposition is a very special one in that it is itself still a tensor product of kets for each degree of freedom or each subsystem. As shown by von Neumann in Theorem A.2, this superposition is very much like a classical superposition of waves and fields.

But superpositions in QM are in no way restricted to such superposition; as a matter of fact, such superpositions are rare because the superposition coefficients must have the special product form \(c_{p(1) 1} \ldots c_{p(N) N}\). Because of interactions between degrees of freedom or subsystems, the SE quickly evolves tensor product states into states that are no longer tensor products. The kets and wave functions can still be written as superpositions of product kets like the ones in Eq. (A.20), but the expansion coefficients no longer have the product form. Such superpositions can be called entangled states, because there is no way to decompose the state of the system into states for the subsystems. The decomposition leads to von Neumann statistical operators for the description of the subsystem as given by Theorem A.2.

In a very interesting paper, Harshman and Ranade [46] illustrate the pervasive and relative nature of entanglement, by showing that any pure state can be written as an entangled state by constructing a set of tailored observables that give the Hilbert space in which the state is defined a tensor product structure in such a way that the pure state is not a product state. Their proof is valid for finite-dimensional Hilbert spaces because they rely on a matrix representation of the observables.

Non-equilibrium statistical mechanics

To set notations and clarify conventions, we give a short summary of statistical mechanics, both classical and quantum. We add the designation “non-equilibrium” to make clear that we are interested in the dynamics of systems, not in the equilibrium states introduced by Boltzmann and Gibbs [41] in what is now also known as statistical thermodynamics [77].

Classical mechanics

Consider a classical system with N degrees of freedom. The classical configuration space is the space \(\mathbb {R}^N\) of all vectors \(q=(q_1,\ldots ,q_N)\) and the classical phase space is \(\mathbb {H}=\mathbb {R}^{2N}\) with each point \((q,p)=(q_1,\ldots ,q_N,p_1,\ldots ,p_N)\) corresponding to a state of the classical system. Note that, in the state, the coordinate \(q_i\) and its canonically conjugate momentum \(p_i\) are specified by independent values. A single degree of freedom in mechanics is described by a coordinate \(q_i\) and its velocity \(\dot{q}_i\) or canonically conjugate momentum \(p_i\). It then makes sense to write the phase space of a system with N degrees of freedom as shown in Eq. (A.3).

Definition B.1

The statistical state of a classical system is a probability measure \(\sigma \) on phase space \(\mathbb {H}=\mathbb {R}^{2N}\) with associated probability distribution function (PDF) \(\rho (q,p)\) assigning a probability

$$\begin{aligned} \sigma (S) = \int _S \rho (q,p) \mathrm {d}^Nq\mathrm {d}^Np \end{aligned}$$

to any BorelFootnote 22 set \(S \subseteq \mathbb {H}\).

A probability measure is normalized so that the probability of the whole space is equal to 1

$$\begin{aligned} \sigma (\mathbb {H}) = \int _\mathbb {H} \rho (q,p) \mathrm {d}^Nq\mathrm {d}^Np = 1. \end{aligned}$$

The Hamilton flow of classical dynamics in phase space

$$\begin{aligned} (q_t,p_t)=F_t(q_0,p_0), \end{aligned}$$

i.e., the solution of Hamilton’s equations, then defines the evolution of the statistical state as [43, 53, 57, 97]

$$\begin{aligned} \rho _t(q,p) = \rho \big (F_{-t}(q,p)\big ). \end{aligned}$$

The evolution \(\sigma _t\) of the measure follows from inserting Eq. (B.4) into Eq. (B.1).

It follows from the above definition of the time evolution that the density \(\rho _t\) of the statistical state \(\sigma _t\) satisfies the differential equation

$$\begin{aligned} \frac{\partial \rho _t(q,p)}{\partial t} = \{ H(q,p) , \rho _t(q,p) \}, \end{aligned}$$

called the Liouville equation (LE) [97]. Here

$$\begin{aligned} \{f,g\} = \frac{\partial f}{\partial q} \frac{\partial g}{\partial p} - \frac{\partial g}{\partial q} \frac{\partial f}{\partial p} \end{aligned}$$

is the Poisson bracket.

The evolution of the statistical state under the LE Eq, (B.5) is completely equivalent to the states in phase space moving under the Hamiltonian flow Eq. (B.3) or to evolution under Hamilton’s equations.

The interpretation of the statistical state in the literature varies: Boltzmann [41, §3.2] conceives of a classical system, for example a gas, as a single system of a large number of particles, with the statistical state characterized by specifying that the state of the gas is confined to a given region, a subset \(S \subset \mathbb {H}\) of phase space. This makes the PDF the index function \(\iota _S\) of that set, equal to one for points in the set and zero otherwise.

Gibbs [41, §3.3], on the other hand, considers an ensemble of identical systems with coordinates chosen randomly with probability \(\rho (q,p) \mathrm {d}^Nq \mathrm {d}^Np\). This probability is interpreted as frequency over the ensemble or as an average over time [41, §3.3.4]. While the latter view is relevant for equilibrium states, it is not meaningful in the context of dynamically evolving non-equilibrium states.

As an example of a statistical state, the Boltzmann partition for a gas in a container with volume V can be defined as a finite measure by constructing the sets S(E) in phase space with total energy E. For an ideal gas, this energy is the sum of the kinetic energy of all atoms or molecules in the gas. Then the PDF is given by weighing the sets with the Boltzmann factor at temperature T

$$\begin{aligned} \rho _\mathrm {Boltz}(q,p) = C \exp \left( -\frac{E}{kT}\right) \iota _{S(E)}(q,p) \end{aligned}$$

with C the normalization constant to satisfy Eq. (B.2). Physically more accurate models of the gas add interactions between the atoms or molecules and between the atoms or molecules and the walls of the container, which results in modified sets \(S(E) \subseteq \mathbb {H}\).

Collective variables from the statistical state

We propose to view the probability measure \(\sigma \) Eq. (B.1) in a third way that is a blending of the two views of Boltzmann and Gibbs summarized in Section B.1. Consider the fact that the measurement of a macroscopic observable, like pressure, of a macroscopic system, like a gas, is never accomplished by explicitly and meticulously recording the position and momentum of every atom or molecule in the system and then computing the macroscopic variable. Rather, experiments measure macroscopic dynamical variables with macroscopic devices that contain macroscopic controls such as size, shape, position, pressure, volume, and temperature, and these controls then are associated with the value of observables of the system under observation.

While the atomic hypothesis is of great value in the theoretical description of observed phenomena and in deriving governing laws between observations, it is not directly relevant as part of the practice of making observations. The frequency interpretation of the statistical state as a probability measure \(\sigma \) is therefore somewhat artificial: the measure \(\sigma \) assigns a probability \(\rho (q,p)\mathrm {d}^Nq\mathrm {d}^Np\) for each microscopic state (qp), but in practice one never obtains these states as samples from a distribution as specified in the frequency interpretation. Therefore, it is possible to think of the probability measure \(\sigma \) itself as the mathematical description of the macroscopic state, i.e., we take the meaning of Definition B.1 to be:

  1. 1.

    In non-equilibrium statistical CM, the mathematical description of the state of a macroscopic physical system with N degrees of freedom is the statistical state of the system, i.e., the probability measure \(\sigma \) on the system’s phase space \(\mathbb {H}=\mathbb {R}^{2N}\).

  2. 2.

    The macroscopic observables (volume, pressure, temperature) are encoded in the probability measure \(\sigma \) and their evolution follows from that of \(\sigma _t\) as derived by Eq. (B.4) from the underlying flow \(F_t(p,q)\) in \(\mathbb {H}=\mathbb {R}^{2N}\), Eq. (B.3).

Quantum mechanics

The statistical state in quantum mechanics was introduced by von Neumann [61] to describe a state for a quantum system that is incompletely specified, i.e., as a mixture or Gemenge [23, p. 21]. It is defined as the positive symmetric operator

$$\begin{aligned} \varvec{D} = \sum _{n=0}^\infty p_n \varvec{P}_n = \sum _{n=0}^\infty p_n |n \rangle \langle n| \end{aligned}$$

specifying the system to be in one of a number of kets \(|n\rangle \) with positive weights \(p_n\). The normalization that the total probability equal 1 then requires the operator to have trace equal to 1

$$\begin{aligned} \mathrm {Tr} \varvec{D} = \sum _i p_i = 1. \end{aligned}$$

The operator \(\varvec{D}\) is called the statistical operator, density operator, or density matrix.

The dynamical evolution of the statistical state is given by the Liouville–von Neumann equation (LvNE)

$$\begin{aligned} i\hbar \frac{\mathrm {d}}{\mathrm {d}t}\varvec{D}_t = [\varvec{H}, \varvec{D}_t]. \end{aligned}$$

If the initial statistical operator is written with an orthonormal set of kets \((|n\rangle )_n\), then its evolution under the LvNE

$$\begin{aligned} \varvec{D}_t = \sum _{n=0}^\infty p_n |n,t \rangle \langle n,t| \end{aligned}$$

is equivalent to the evolution of the kets according to the SE and the numbers \(p_n\) do not change in time, as in the classical case.

As a positive symmetric operator, the statistical operator can always be diagonalized, resulting in a unique composition of the statistical operator as a mixture of orthogonal states with positive weights \(p_n\). But, as pointed first out by Schrödinger [23, 63, 75, 76], it is not possible to associate a unique set of pure states that are linearly independent, but not necessarily orthogonal, with a given statistical operator. Therefore, it is not possible to directly interpret the numbers \(p_n\) as probabilities with the meaning of frequency of occurrence in experiments on ensemble of systems in the statistical state described by \(\varvec{D}\). Hence there is a subtle difference between the evolution of the statistical operator governed by the LvNE and pure states governed by the SE. This situation is different from the situation in statistical CM and that is the core message of the statistical interpretation of QM [3, 4, 7]: it is not possible to clearly and uniquely distinguish statistical probabilities in mixtures from the innate probabilistic nature of pure states; there is only one single unavoidable probability notion in QM and it is not a purely statistical probability as it is in CM.

Empiricism and realism

We summarize the context of concepts relevant to the measurement problem in QM, see, for example, the book by de Muynck [60]. The analysis by ABN [3, 4] shows that the Born probability rule can be derived from the dynamics of QM provided the complexity necessarily associated with measurement instruments is given proper consideration. This approach is not new, for example d’Espagnat [27, §16.2] explores the options of a single principle, namely the SE, in QM for changing the state, represented by a ket or a wave function. The relative state formulation of Everett [39] is also based entirely on the SE as the only principle for changing states. The work by ABN appears to be the most exhaustive analysis to date that shows that evolution in QM under the LvNE leads to a unique outcome, which contradicts Everett’s claim [39, p. 457] that “It seems that nothing can ever be settled by such a measurement.”

Newton–Maxwell ontology

In Newtonian mechanics the ontology is clear: the world consists of “objects with substance” that are described by coordinates and their velocities subject to Newton’s law of force. The mass of the object gives a measure of the amount of substance. Newtonian mechanics is scale invariant in that the objects can be described at multiple levels, depending on desired accuracy, either as monolithic objects with substance, for example planets, or as composed of a number of moving parts with substance, such as cars, or, applying the atomic hypothesis, as assemblies of atoms with substance. The essence of the ontology is that the degrees of freedom are described by pairs of variables that at all times take on values that are real numbers, one degree of freedom being specified by two values. By using the ontology as a framework for thinking about the world and the physics processes taking place in it, the physicist can accurately and efficiently apply the formalism of Newtonian mechanics for computing descriptions of observed processes and make predictions of planned or expected future observations.

The same ontology exists for Maxwell’s theory of electromagnetism, but now applied to the ontological element of a “field of force” defined in all of space. The field is described by a dynamical coordinate and its associated velocity, both taking on a numeric value, defined in every point of space. Einstein’s theories of relativity are consistent with this ontology, both the special theory for particles and fields, and the general theory adding space to the list of ontological elements.

We refer to this ontology as the Newton–Maxwell ontology.

When Einstein does the statistical analysis [35] of light interacting with matter, he concludes that the observed behavior is consistent with that of a gas of photons. In Bohr’s development of a description for atomic spectra [14], a theory of the Periodic Table of Elements [15], and an explanation of the chemical bond [16], he uses electrons moving in classical circular and elliptic orbitals selected by the “quantum condition”. It is clear that Einstein and Bohr both work with the Newton–Maxwell ontology.

Einstein [36] summarizes his view by stating that there are two options:

(a) The (free) particle really has a definite position and a definite momentum, even if they cannot both be ascertained by measurement in the same individual case. According to this point of view, the \(\psi \)-function represents an incomplete description of the real state of affairs.


(b) In reality the particle has neither a definite momentum nor a definite position; the description by \(\psi \)-function is in principle a complete description. The sharply defined position of the particle, obtained by measuring the position, cannot be interpreted as the position of the particle prior to the measurement. The sharp localization which appears as a result of the measurement is brought about only as a result of the unavoidable (but not unimportant) operation of measurement.

Both Einstein [35] and Bohr [14] agreed on the appropriateness of the notion that physical systems have properties that are described mathematically by numbers, individual values that are captured during the measurement process [42, 50], in particular for the pair of canonically conjugate variables making up a degree of freedom. They were arguing about whether the systems, like particles, have these values before and after measurement, or only during the measurement process. This view was prevalent and very successful at the time and is beautifully described in the works by Sommerfeld [82] and Born [20]. While Bohr advocated a conceptual complementarity [17] to understand the principles of QM and the observations of quantum phenomena, Einstein wanted something more precise [37].

The ideas introduced by Einstein and Bohr remain valid, even though the precise calculations to quantify the photo-electric effect, the lines in atomic spectra, and the structure of molecules are now done with the formalism developed in 1926 instead of the application of the quantum condition. It is natural that Einstein and Bohr, and others, considered the 1926 formalism as a new and improved, and more complicated and highly unfamiliar, version of the quantum condition. They assumed that the prevailing Newton–Maxwell ontology was still applicable [17]. We argue in this paper that it is not applicable.

State–measurement connection

In CM it is an experiential fact that measurements exist that provide direct observation of the values taken by coordinates and velocities, at least for measurements on sufficiently simple systems. This leads to the positivist idea that physics theories can be and should be formulated such that experimental observations are the foundation for all concepts [29]. The historically most important example is the observation of the planetary positions (and velocities from multiple positions) by observing the Sun’s light reflected by the planets. The second prominent example is the definition of the electric and magnetic field strengths as the force felt by an infinitesimal test charge.

Note that this does not preclude that with complex systems, such as automobile engines, some variables may not be observable in a similarly direct way, but their values can be reconstructed, by using the law of force, from the values obtained for other dynamical variables that can be observed.

In their debate, Einstein and Bohr take the observation of localized flashes produced by photons or electrons to be direct information about the photons or electrons [42, 50]. The analysis by ABN [3, 4] conclusively shows that even the simplest experimental observation of a quantum mechanical system is a complex multi-step process. It is unwarranted to conclude that the state of a macroscopic device in an experiment provides a direct view on the microscopic quantum system that initiated the event.

On the basis of the Newton–Maxwell ontology and with the assumption that measurements in QM provide direct information about the state of quantum systems, Einstein and Bohr argued about how the observed facts could be derived from this ontology. In particular, they argued about whether the values for coordinates and velocities (or momenta) were present all the time or just during the measurement [42, 50]. They did not consider the possibility [36] that the ontology does not apply to QM and that all experimental observations in QM may have to be derived from the dynamics by complex computations of quantum systems as shown, for example, by ABN. Born’s probability rule [18, 19] provides an operational definition of the quantum mechanical state, ket, and wave function, supporting the Newton–Maxwell ontology, but it does not explain the connection.

The theory of measurement presented by ABN is rigorous and minimalist. No special assumptions are introduced, like hidden variables or spontaneous wave function collapse. QM is assumed to be irreducibly probabilistic [3, p. 6] and the analysis focuses on the ideal projection-operator measurement as defined by von Neumann [61]. Contrary to the Copenhagen interpretation, where two distinct processes are assumed, the SE for dynamical evolution and the Born rule for the measurement process, the measurement process is derived in full detail from the dynamical evolution of the statistical operator governed by the LvNE Eq. (B.10). That gives the evolution for both the observed quantum system and the macroscopic measurement apparatus.

We briefly summarize the stages in the dynamical process that are identified in the analysis by ABN. They identify three component systems in the analysis:

  1. 1.

    The microscopic quantum system S to be measured.

  2. 2.

    The macroscopic measurement apparatus and pointer system M that will register, after some time, an outcome correlated with the state of S.

  3. 3.

    A macroscopic system B that provides a heat bath to ensure that the measurement is irreversible as a dynamical process.

The system is described by a statistical operator \(\varvec{D}\), given by Eq. (B.8), that evolves under the LvNE Eq. (B.10). The dynamics is governed by the Hamiltonian with an important role played by the terms describing the interactions between the different systems. The different timescales in the process are carefully analyzed and described by ABN with the following summary of the process [3, Table 1, p. 103] [4]:

  1. 1.

    Preparation—create a metastable state in the apparatus or pointer M; the statistical operator is \(\varvec{D}=\varvec{D}^S \otimes \varvec{D}^M \otimes \varvec{D}^B\).

  2. 2.

    Initial truncation—decay off-diagonal blocks of statistical operator \(\varvec{D}^M\) of the pointer.

  3. 3.

    Irreversible truncation—further interaction between the pointer M and the bath B ensure that there is no recurrence of the off-diagonal blocks of \(\varvec{D}^M\).

  4. 4.

    Registration—\(S-M\) correlation is built up in diagonal blocks of the statistical operator \(\varvec{D}^M\) of the pointer to correlate with the diagonal elements of \(\varvec{D}^S\) in a process that is similar to a phase transition [3, §7] and involves a weak coupling to the bath B. This process ensures that the pointer M reflects the state of the system S as it was at the start of the measurement process: \(\varvec{D}^S_{aa} \leftrightarrow \varvec{D}^M_{aa}\).

  5. 5.

    Sub-ensemble relaxation—interaction terms in the Hamiltonian of the pointer ensure a consistent evolution in ensembles, so that sub-ensembles can be identified that ultimately lead to the outcome of a single run.

  6. 6.

    Reduction—gain of information in the pointer M about the initial state of S with probabilities in accordance with Born’s rule.

The majority of these steps will appear plausible to anyone with experience in studying the measurement problem, except possibly item 5. The need for the analysis of ensembles and sub-ensembles comes from Schrödinger’s observation [23, 63, 75, 76] the statistical operator cannot be uniquely decomposed in terms of pure states, as discussed in Section B.3. The reader is referred to ABN [4] for an in-depth and lucid discussion.

Heisenberg–Dirac ontology


Given Newton’s objects with substance governed by Newton’s law of force as the ontology of mechanics and the electromagnetic fields of force governed by Maxwell’s equations as the ontology for electromagnetism, it stands to reason to consider, similarly, the object governed by the Schrödinger equation as the basic element in the ontology for QM: that means taking the abstract ket introduced by Dirac [33], or the wave function in any representation, to be of the same quality of being as the objects with substance in Newtonian mechanics and the fields of force in Maxwell theory. We refer to this ontology as the Heisenberg–Dirac ontology.

d’Espagnat [29] discusses ontology of QM in depth, but his analysis remains within the Newton–Maxwell ontology. He does not follow Heisenberg, who, when he was thinking about tracks in the bubble chamber [42, 50], questioned the complexity of the measurement process and hoped to find understanding in the mathematical formalism as opposed to in experimental observations, as indicated by Rosenfeld’s remark quoted above.

What if everything is made of kets? What if there are no particles, no fields, no systems that can be described by degrees of freedom that consist of coordinate-momentum pairs that take on values that are real numbers?

This brings us to a point where we are led to accept as fundamental, realist postulates of QM the two assumptions in Appendix A.1 at face value:

  1. 1.

    The ontological building block of QM is the ket.

  2. 2.

    The ket is governed at all times by the SE.

A ket is not a number. Since experiments obtain numbers, it follows that measurements in QM cannot measure the ket of any quantum system; some assembly is required.

Finite measures on Hilbert spaces

The study of measures on infinite-dimensional spaces is the subject of stochastic analysis, a field in mathematics that was initiated by the work on the space of paths of Brownian motion by Norbert Wiener in 1923 [94].

The first counterintuitive property one encounters with measure on and volume in infinite-dimensional spaces is the fact that there is no translation-invariant measure like the Lebesgue measure on an infinite-dimensional space [52, p. 32]. For the mathematically inclined reader, the fundamental reason is that the unit ball in an infinite-dimensional space is not compact.

Theorem D.1

There does not exist a meaningful translation invariant measure on an infinite-dimensional Hilbert space.


Consider [24, p. 5] an orthonormal basis \((|{\varPhi }_n\rangle )_n\) in the Hilbert space and a ball \(B_n\) with radius 1 / 2 centered at \(|{\varPhi }_n\rangle \). Assume that a translation invariant measure \(\lambda \) exists. Then the measure of all these balls is the same \(V=\lambda (B_n)\) by translation invariance, because they are all just translations of the first one \(B_0\). It is easy to see that all these balls fit into a bigger ball B with radius 2 centered at the origin and that none of them overlap.

A proper measure must have the property that the volume of a set B is larger than the sum of the measures of any collection of non-overlapping sets \(B_n\) contained in B. Thus we must have

$$\begin{aligned} \sum _{n=0}^\infty \lambda (B_n) \le \lambda (B). \end{aligned}$$

It follows that \(\lambda (B)\) is infinite as the sum of an infinite number of equal terms.

This contradicts another requirement of reasonable measures, namely that the measure of a bounded set be finite. Thus the only measure satisfying all requirements is one that assigns the measure zero to all balls. It follows easily that that measure must be identically zero. \(\square \)

Note that the proof uses translation, but the problem really lies with rotations: it is possible to keep turning to new directions in an infinite-dimensional Hilbert space putting new balls \(B_{N+1}\) that do not overlap with any ball \(B_0,\ldots ,B_N\) we already have, while still remaining inside the big ball B.

Even though there is no \(\sigma \)-finite, translation invariant measure, it is possible to define finite measures on infinite-dimensional spaces, like a Hilbert space, with all the right properties [11, 80]. A finite measure is one that assigns to the total space a finite volume, which is chosen to be 1 by convention. A probability measure is an example of a finite measure.

The practical reader may suggest that there is no need to work with infinite-dimensional spaces, as any computations can be carried out in spaces with a finite number of dimensions to get approximations as close as needed. However, the issue is not confined to infinite-dimensional spaces: it does show up in spaces with a large number of dimensions and requires special attention to implement efficient and accurate algorithms [58, 87]. The methods developed to handle the infinite-dimensional case provide ways to build such efficient algorithms for the case with a very large number of dimensions.

In the literature on measure theory, with \(\lambda \) denoting the Lebesgue measure, the Lebesgue integral of a function f over a set \(S\in \mathbb {R}^n\) is written in various ways as [67, 81]

$$\begin{aligned} \int _S f \mathrm {d} \lambda = \int _S f(x) \mathrm {d} \lambda (x) = \int _S f(x) \lambda (\mathrm {d}x) = \int _S f(x) \mathrm {d}^n x, \end{aligned}$$

where the last form is identical to the way Riemann integrals are written. For general measures, any of these notations, except the last one, can be found in the literature.

The best-known measures on infinite-dimensional spaces are Gaussian measures \(\gamma \) [11, 80]. These are defined by specifying the mean \({\varTheta }\) as an element of \(\mathbb {H}\)

$$\begin{aligned} |{\varTheta }\rangle = \int _\mathbb {H} |{\varPsi }\rangle \gamma (\mathrm {d}|{\varPsi }\rangle ) \end{aligned}$$

and the covariance \({\varGamma }\) as an operator on \(\mathbb {H}\)

$$\begin{aligned} \langle {\varXi }| {\varGamma }| {\varUpsilon }\rangle = \int _\mathbb {H} (\langle {\varPsi }|-\langle {\varXi }|) (|{\varPsi }\rangle -|{\varUpsilon }\rangle ) \gamma (\mathrm {d}|{\varPsi }\rangle ). \end{aligned}$$

The covariance operator must be symmetric, positive definite, and trace class [11, 80], i.e.,

$$\begin{aligned} {\varGamma }> 0 \qquad \mathrm {Tr} {\varGamma }< \infty \end{aligned}$$

for the Gaussian measure to exist and have support in \(\mathbb {H}\) and not some larger space.

In finite dimensions, sets of measures zero can be safely ignored in physics. For example, the Lebesgue measure of individual points and curves in a two-dimensional space is zero: \(\lambda (\{(x_1,x_2)\})=0\) and \(\lambda (\{(x_1(t),x_2(t)) | t \in [0,1] \})=0\), respectively. A remarkable property of measures on infinite-dimensional spaces is that sets of measure zero can be big.

For example, the space of all kets \(|{\varPsi }\rangle \) in Hilbert space with finite expectation value \(\langle {\varPsi }| {\varGamma }^{-1} | {\varPsi }\rangle < \infty \) of the covariance operator \({\varGamma }\) is called the Cameron–Martin space\(\mathbb {E}_\gamma \) of the Gaussian measure \(\gamma \). It has measure zero, \(\gamma (\mathbb {E}_\gamma )=0\) [11, thm 2.4.7]. To illustrate the “size” of the Cameron–Martin space, consider the Wiener Bridge measure on the Hilbert space \(L^2([0,T])\) of square-integrable paths from \(x=0\) at time \(t=0\) to \(x=0\) at time \(t=T\) [11, 52]. This measure is Gaussian and the covariance operator \({\varGamma }\) has eigenvalues proportional to \(1/n^2\), so that it is trace class. The paths that count, i.e., that provide the weight of the measure, are paths that are Hölder continuous of order 1 / 2. These are paths that are continuous and almost nowhere differentiable. They are the paths of Brownian motion. They carry the full weight of the measure in that the measure of all such paths is 1. All paths that are in \(L^2([0,T])\) and that are not continuous or that are less smooth than Hölder continuous of order 1 / 2 form a set of measure zero; these rougher paths carry no weight for the Wiener Bridge measure. The subset of paths that are continuous and almost everywhere differentiable turn out to have finite expectation value for the covariance \({\varGamma }\) and thus form the Cameron–Martin space of the Wiener Bridge measure, which has measure zero. In other words, these smoother paths (differentiable) do not carry any weight in the Wiener Bridge measure either. That means they do not contribute to any integral.

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Deumens, E. On classical systems and measurements in quantum mechanics. Quantum Stud.: Math. Found. 6, 481–517 (2019).

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  • Quantum mechanics
  • Measurement theory
  • Quantum statistical mechanics

Mathematics Subject Classification

  • 81P15
  • 28C20
  • 82C10
  • 60G15