Measurement theory in classical mechanics

Measurement theory in classical mechanics is investigated in the formulation of classical mechanics by Koopman and von Neumann (KvN), using Hilbert space. It is shown that the classical and the quantum measurements give different"relative interpretations"of the measurement state and the recording state of the measurement device. The uncertainty relation in classical mechanics is also derived.


Introduction
To discuss the crucial differences between quantum mechanics and classical mechanics, it is essential to compare the two in the same formalism. Bohm rewrote quantum mechanics into classical mechanics [1], and he found that the presence or absence of quantum potential 2 leads to the difference between classical mechanics and quantum mechanics.
However, there is a clear difference between comparing them in the formalism of quantum mechanics and comparing them in the formalism of classical mechanics. This difference is non-commutativity of operators. Unlike classical mechanics, quantum mechanics is described by q-numbers, i.e., non-commutative physical quantities. This difference becomes apparent in the theory of measurement, such as Heisenberg's uncertainty principle [2].
In other words, quantum mechanics is the world of q-number. Correspondingly, classical mechanics is the world of c-number. That is the commutative world. However, such discussions are unclear because they are both in different formalisms.
To clarify this, it is important to rewrite classical mechanics into quantum mechanics and compare the two. Rewriting classical mechanics in the form of quantum mechanics is discussed in the formalism of KvN equations by Koopman and von Neumann in the early stages of quantum mechanics [3,4]. Later, Gozzi 1 So.Katagiri@gmail.com 2 Quantum potential is defined as follows. Rewriting ψ to ψ = R exp(iS), we obtain a Hamilton-Jacobi-like equation, where − 1 2m ∂ 2 ∂x 2 (log R) is called quantum potential. and Mauro examined the formalism in detail [5,6,7]. Here, the quantum description of classical mechanics related to the KvN equation is called the KvN formalism. Sudarshan discussed the KvN formalism as a model of quantumclassical interaction [8].
In this paper, we apply quantum measurement theory made in quantum mechanics to classical mechanics and investigate how we describe measurements in classical mechanics. Until now, measurement in classical mechanics has not been considered enough. Jens, Wilkens, and Lewenstein found that the formalism of quantum mechanics is useful to other than quantum mechanics [9]. It makes us expect the KvN formalism has a new meaning and application in classical mechanics.
The structure of this paper is as follows. First, we review the KvN formalism in Section 2. Next, we extend the KvN formalism to quantum mechanics and show that this is equivalent to quantum theory. After reviewing the observation problem using the von Neumann model in Section 4, we discuss in Section 5 the measurement theory in classical mechanics. In Section 6, we construct the classical mechanics' version of uncertainty relation. The last chapter will give a summary and discussion.
The appendices include the following. In Appendix A, we comment that the KvN formalism for free particles can be regarded as a von Neumann model. Next, in Appendix B, we discuss the von Neumann model in a formalism that extends the KvN formalism to quantum mechanics, which is discussed in Section 3. In Appendix C, we introduce the Kraus operator. In Appendix D, we discuss the case where the initial condition is the only known probability.

The KvN formalism
This section briefly introduces the KvN formalism. In quantum mechanics, the commutation relation between the position and momentum operator of a particle is given by, A state can be written |ψ as an expansion using position and momentum eigenvalue states, The time evolution of the state is described using the Hamiltonian operator H(x,p) as The wave function (probability amplitude) ψ(x) is a function of x only, and ψ(p) is its Fourier transform. So it is not a function on the phase space.
The essence of the KvN formalism is to introduce operatorsπ x ,π p in addition tox,p and require non-commutability betweenx,p andπ x ,π p whilex andp are made commutative.
Since position and momentum are commutative, state |ψ can be expanded by simultaneous eigenstates of position and momentum as |ψ = dxdp|x, p x, p|ψ . (2.5) That is, in the KvN formalism, the wave function (probability amplitude) ψ(x, p) = x, p|ψ is a complex function in phase space.
It should be noted that ψ(x, p) is not a pseudo-probability like a Wigner function [10] or a Husimi function [11], but a probability amplitude.
By Fourier transform, |ψ can be described as In addition, the Liouvillian will be introduced in correspondence with the Hamiltonian 3L The KvN equation corresponding to the Schrödinger equation is introduced as By applying x, p| from the left, is obtained. This form is the same as the Liouville equation, but in this case, there is a difference in that ψ is a complex function. In the KvN formalism, as with quantum mechanics, ψ(x, p) is regarded as the probability amplitude, and |ψ(x, p)| 2 is the probability density in the phase space.
As an example, let us consider a free particle H(x, p) = p 2 2m [5]. Using the LiouvillianL Now applying π x , p| from the left, we obtain Using proportionality factor A, the solution is given by (2.14) By Fourier transforming π x , we obtain Here, if the initial state is |x 0 , p 0 , it is A = 1 and This solution reproduces the linear orbit of a free particle in classical mechanics.

Relation to quantum mechanics
Now we consider the relationship between the KvN formalism and quantum mechanics.
In quantum mechanics, positionx q and momentump q satisfy the canonical commutation relation, The same algebra can be constructed usingx,π x ,p,π p . If we define operatorŝ x andp asx then we obtain A Similar algebra is discussed in [12] 4 . 4 These equations satisfy the Weyl relation, Then we obtain quantum wave functions in phase space, To recover the original KvN formalism, we expand the right hand side of (3.9) in power of , (3.11) Then, in the limit → 0 this equation returns to the KvN formalism.
We comment state |x q in quantum mechanics corrensponds to |x, π p , not |x, p . Therefore, |x, π p and |p, π x in the KvN formalism have a connection with quantum theory in spite of classical mechanics.
4 The von Neumann model as a Measurement theory of quantum mechanics

The von Neumann model
In this section, we introduce the von Neumann model as a simple example of the measurement model [13].
The system consists of a measurement target and a measurement device, and the corresponding physical quantities {x,p}, {X,P } satisfy the canonical commutation relations, and the other commutators ofx,p,X,P are 0.
As an interaction between the measurement target and the measurement device, we introduce a HamiltonianĤ Then, from the Stone-von Neumann theorem, There exists a unitary transformationÛ such thatÛ

7)
U e ip t/ = e ipq t/ Û . Then,x ,p describe quantum mechanics exactly. and the free Hamiltonian part is not considered for the sake of simplicity. Also, we take t = 1. Then, the time evolution operator is given bŷ Taking the initial state as where |φ is the initial state of the measurement target and |η is the initial state of the measurement device. The time evolution of state is expressed as We expand |η using |X to obtain Next, we perform a projective measurement on the measurement device. The probability that the measurement device (or needle) obtains x 0 is If the initial state of the measurement device is |0 X , this probability is This equation is consistent with the results of the projective measurement of the measurement target [14].

Relative state
Since the projection hypothesis is a problem inherent to quantum mechanics, it is conceptually difficult to think about the observation theory, including the projection hypothesis in classical mechanics. Therefore, this section introduces relative state 5 .
We use the notation |η[x] ≡ e −ixP |η .  |x ⊗ |η[x] is called relative state. In relative state interpretation, |x ⊗ |η[x] is interpreted as the measurement device observing its position as x 6 .
On the other hand, |ψ after can be expanded as follows, This is different from the previous one, and it can be interpreted that the measurement target observed the momentum of the measurement device as P .
Note that these two propositions do not hold in relative state at the same time.
In contrast, these two propositions will hold in relative state at the same time in measurement theory in classical mechanics.

Measurement theory in classical mechanics
In this section, we discuss measurement theory in classical mechanics using the KvN formalism and the von Neumann model.
As in the previous section, the system consists of a measurement target and a measurement device, and the corresponding physical quantities {x,p,π x ,π p }, {X,P ,π X ,π P } satisfy the canonical commutation relations, As with quantum mechanics, the time evolution of state is obtained as = dp dx dX dP φ(x, p)η(X, P )|x, The important difference from quantum mechanics is that in relative state interpretation, two propositions 1. the measurement device observed the position of the measurement target as x, 2. the measurement target observed the momentum of the measurement device as P , hold in relative state at the same time. 7 6 Uncertainty relations in classical mechanics

Uncertainty relation in classical mechanics withx and p
Here, we investigate the relationship between error and disturbance. The Ozawa's inequality is a relational expression for error and disturbance [17]. We discuss how we can obtain the Ozawa's inequality in classical mechanics. We introduce error operatorN (t) =X(t) −x and disturbance operator D(t) =p(t) −p.
Then, in this condition, we get If the unbiased condition is not satisfied, we get at t = 1, ǫ = X, η = P. (6.33) It represents the initial calibration of the device. In such a case, X = 0 or P = 0 is the condition of the equal sign of (6.15).
The case where the initial condition is the only known probability is discussed in Appendix D.

Uncertainty relationsin classical mechanics withπ x and π p
We clarify the role ofπ x andπ p in classical mechanics. Althoughπ x andπ p are hidden variables in classical mechanics, they can be expressed as physical quantities by combining quantum mechanics and classical mechanics. As confirmed in Section 3,π x andπ p are described bŷ π x = − 2 (p −p), (6.34) In these relational expressions, we can determineπ x andπ p by using both classical and quantum observables.
Note that these equations can also be regarded as differentiation by Planck's constant.
Therefore, in addition to the usual disturbanceD, we should consider another disturbance,D By a similar argument such as using Kennard-Robertson uncertainty relations, non-commutativity ofx andπ x gives an Ozawa-like inequality where η πx = D πx . Since the Planck constant does not appear in this inequality, it holds even in the classical mechanical limit.

Discussion
We constructed the measurement theory of classical mechanics.
In this KvN formalism extended to quantum mechanics, it is interesting to discuss the measurement theory with stepwise quantum corrections.
In contrast to quantum mechanics, we have found two propositions hold in relative state at the same time.
1. The measurement device observed the position of the measurement target as x.
2. The measurement target observed the momentum of the measurement device as P .
This difference in simultaneity corresponds to the result of the discussion of uncertainty relations, and Ozawa's inequality becomes trivial in classical mechanics. If the initial state is not well known, we can obtain a relational expression about the error and the disturbance in the von Neumann model. We extended the KvN formalism to quantum theory and determinedπ x and π p using both classical and quantum observables. Then, we also introduced another disturbance onπ x and obtained an Ozawa-like uncertainty relation. Since this relation is independent of Planck's constant, it holds in classical mechanics. This relation may be significant in the theory of intermediate scale between classical theory and quantum theory.
The application of these relational expressions to behavioral economics in recent years is astonishing. Through these applications, the role of phase in classical mechanics may be newly understood.
As an application of measurement theory in classical mechanics, it is possible to analytically formulate thought experiments in classical mechanics such as Maxwell's demon and Einstein's optical clock [20]. Although many have discussed these in the past, our study can contribute to the conceptual discussion of science. Further research will reveal them.

Appendix A. Measurement interpretation of classical mechanics
We comment that the KvN formalism for a free particle can be regarded as a von Neumann model.
In such a case, the Liouvillian is give bŷ If we regard |p, x as a composite of the measurement target |p and the measurement device |x , the time evolution of the free particle e −ip mπ x t |p |x = |p |x + p m t (7.2) can be regarded as the obseravation of position x by momentum p.
Note that the disturbance in this case isDπ p =π p (t) −π p .Dπ p is defined in Section 6.

Appendix B. Measurement theory in the extended KvN formalism
We discuss the von Neumann model in a formalism that extends the KvN formalism to quantum mechanics, which is discussed in Section 3.
The Hamiltonian of the von Neumann model iŝ We take |x, π p ⊗ |X, P as the initial condition. We assume in this initial condition that the measurement target is quantum, and the measurement device is classical.