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Introduction of a Classical Level in Quantum Theory

Continuous Monitoring


In an old paper of our group in Milano a formalism was introduced for the continuous monitoring of a system during a certain interval of time in the framework of a somewhat generalized approach to quantum mechanics (QM). The outcome was a distribution of probability on the space of all the possible continuous histories of a set of quantities to be considered as a kind of coarse grained approximation to some ordinary quantum observables commuting or not. In fact the main aim was the introduction of a classical level in the context of QM, treating formally a set of basic quantities, to be considered as beables in the sense of Bell, as continuously taken under observation. However the effect of such assumption was a permanent modification of the Liouville-von Neumann equation for the statistical operator by the introduction of a dissipative term which is in conflict with basic conservation rules in all reasonable models we had considered. Difficulties were even encountered for a relativistic extension of the formalism. In this paper I propose a modified version of the original formalism which seems to overcome both difficulties. First I study the simple models of an harmonic oscillator and a free scalar field in which a coarse grain position and a coarse grained field respectively are treated as beables. Then I consider the more realistic case of spinor electrodynamics in which only certain coarse grained electric and magnetic fields are introduced as classical variables and no matter related quantities.

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  1. As a rule, across the paper we shall denote by hatted letter \({\hat{A}},\,{\hat{B}},\ldots \) operators on the Hilbert space of the system; by \(A,\,B,\ldots \), the corresponding abstract observables, if they are self-adjoint; by the \(a,\,b\ldots \) the related macroscopic quantities, if the case.

  2. In references [4144], in the somewhat different perspective of a finite difference Liouville von Neumann equation, the energy is conserved but the second difficulty remains.

  3. A direct proof that \({\bar{K}}_{ij}\) is positive can be done under a somewhat more restrictive condition than \(T<T_p\). Note that, in order the matrix to be positive, it is necessary that \( \epsilon ^2 \omega ^2\) is smaller than \(\lambda _m\), the minimum eigenvalue of \({\bar{K}}_{ij}|_{\epsilon =0}\). Now such eigenvalues are given by the roots of a polynomial of the form \(\det {\bar{K}}_{ij}|_{\epsilon =0} - \lambda \delta _{ij}= A_0 - A_1 \lambda + A_2 \lambda ^2- \ldots +(-\lambda )^N \), where the coefficients \(A_0,\, A_1, \ldots \) are all positive and in, particular, \( A_0 =1\), \( A_1=N(N+1)/2\). Then we have \(\lambda _m > A_0 / A_1 = 2/N(N+1)\) and \(\epsilon ^2 \omega ^2 \equiv {\omega ^2 \,T^2 \over N^2} < {2 \over N(N+1)}\) if \(T<\sqrt{2} /\omega \).

  4. The situation is completely similar to that encountered in the calculation of the amplitude \(\langle Q_F,\,t_F|Q_0,\,t_0\rangle \) for the harmonic oscillator, that can be performed exactly (see [50]) both for a real or for an imaginary mass \(m=i\mu \) with \(\mu >0.\)

  5. Note that under the restriction of Q(t) to the subspace with Fourier components with \(|k|>\omega \) alone the classic action is actually minimal for \(Q(t)=Q_c(t)\).


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Warm thanks are due to my friend L. Lanz for many interesting discussion and critical remarks.

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Correspondence to G. M. Prosperi.


Appendix 1: A Brief Reference to Generalized Quantum Mechanics

Let us recall that in GQM a set of compatible observables \(A \equiv (A^1, A^2, \ldots A^p)\) is associated to a normalized effect or positive operator valued measure (p.o.m.) \({\hat{F}}_A(T)\) and the apparatus \(S_A\) for observing them to an instrument or operation valued measure (o.v.m.) \({\mathcal {F}}_{S_A}(T)\), T being a Borel subset of the real space \(\mathbb {R}^p\) of all possible values of A. That is, \({\hat{F}}_A(T)\) and \({\mathcal {F}}_{S_A}(T)\) are a positive operator on the Hilbert space \(\mathbb {H}\) associated to the system and a positive mapping of the set of the trace class operators in itself, respectively, satisfying the relations

$$\begin{aligned} {\hat{F}}\left( \cup _{j=1}^n T_j\right) =\sum _{j=1 ^n} {\hat{F}}(T_j) \qquad \ \mathrm{and} \qquad {\mathcal {F}}\left( \cup _{j=1}^n T_j\right) =\sum _{j=1 ^n} {\mathcal {F}}(T_j), \end{aligned}$$

if \( T_i\cap T_j = 0 \), and

$$\begin{aligned} {\hat{F}}_A(\mathbb R ^p) = {\hat{I}} \qquad \mathrm{Tr}\left[ {\mathcal {F}}_{S_A}(\mathbb R^p) {\hat{X}}\right] = \mathrm{Tr}{\hat{X}}. \end{aligned}$$

Further they must be related each other by the equation

$$\begin{aligned} {\hat{F}}_A(T)={\mathcal {F}}_{S_A}^\prime (T) {\hat{I}}, \end{aligned}$$

where by \({\mathcal {F}} ^\prime \) we denote the dual mapping of \({\mathcal {F}}\), defined by the equation

$$\begin{aligned} \mathrm{Tr}\left[ {\hat{B}} {\mathcal {F}} {\hat{X}} \right] = \mathrm{Tr}\left[ \left( {\mathcal {F}}^\prime {\hat{B}} \right) {\hat{X}} \right] , \end{aligned}$$

\({\hat{X}}\) being an arbitrary trace class operator and \({\hat{B}}\) an arbitrary bounded operator. So

$$\begin{aligned} \mathrm{Tr}[{\hat{F}}_{A}(T) {\hat{X}}]= \mathrm{Tr}\left[ {\mathcal {F}}_{S_A}(T){\hat{X}}\right] . \end{aligned}$$

As we told, we shall find convenient to work in Heisenberg picture. Then we have

$$\begin{aligned}&\qquad \quad {\hat{F}}_A(T,t)=e^{iHt}{\hat{F}}_{A}(T) e^{-iHt}\\ \nonumber&{\mathcal {F}}_{S_A}(T,t){\hat{X}} = e^{iHt}\left[ {\mathcal {F}}_{S_A}(T)(e^{-iHt}{\hat{Xe}}^{iHt})\right] e^{-iHt} \end{aligned}$$

and the probability of observing \(A \in T\) at the time t is

$$\begin{aligned} P(A \in T,t)=\mathrm{Tr}\left[ {\hat{F}}_A(T,t){\hat{\rho }}\right] =\mathrm{Tr}\left[ {\mathcal {F}}_{S_A}(T,t){\hat{\rho }}\right] , \end{aligned}$$

where \({\hat{\rho }}\) denotes the statistical operator representing the state of the system (a priori a mixture state.

The reduction of the state as consequence of having observed \(A \in T\) at the time \(t_0\) by the apparatus \(S_A\) must be written as

$$\begin{aligned} {\hat{\rho }} \rightarrow {\mathcal {F}}_{S_A}(T,t_0){\hat{\rho }} / \mathrm{Tr}\left[ {\mathcal {F}}_{S_A}(T,t_0){\hat{\rho }}\right] . \end{aligned}$$


$$\begin{aligned} \langle A^j \rangle = \mathrm{Tr}\left[ {\hat{A}}^j(t) {\hat{\rho }}\right] \end{aligned}$$


$$\begin{aligned} {\hat{A}}^j(t) = e^{iHt} {\hat{A}}^j e^{-iHt} \qquad \mathrm{and} \qquad {\hat{A}}^j = \int _{\mathfrak {R}^p} d {\hat{F}}(a) a^j. \end{aligned}$$

The operators \({\hat{A}}^j\) are Hermitian but generally they do not commute. Such a set of generalized compatible observables can be interpreted as corresponding to an approximate simultaneous measurement of possibly incompatible ordinary observables \({\hat{A}}_1,\,{\hat{A}}_2,\ldots \)

Now let us assume that we make repeated independent observations on A at subsequent times \(t_0, t_1, \ldots t_N\). Combining Eqs. (140) and (141) the Joint probability of observing a a sequence of results for A can be written

$$\begin{aligned}&P(A \in T_N, t_N; \ldots A \in T_1, t_1; A \in T_0, t_0) \qquad \nonumber \\&\quad =\mathrm{Tr}\left[ {\mathcal {F}}_{S_A}(T_N,t_N)\ldots {\mathcal {F}}_{S_A}(T_1,t_1) {\mathcal {F}}_{S_A}(T_0,t_0) {\hat{\rho }}\right] \end{aligned}$$

Notice that

$$\begin{aligned} {\mathcal {F}}( T_N, t_N; \ldots ; T_1, t_1; T_0, t_0) = {\mathcal {F}}_{S_A}(T_N,t_N)\ldots {\mathcal {F}}_{S_A}(T_1,t_1) {\mathcal {F}}_{S_A}(T_0,t_0))\quad \end{aligned}$$


$$\begin{aligned} {\hat{F}}( T_N, t_N; \ldots ; T_1, t_1; T_0, t_0) = {\mathcal {F}}_{S_A}^\prime (T_0,t_0) {\mathcal {F}}_{S_A}^\prime (T_1,t_1) \ldots {\mathcal {F}}_{S_A}^\prime (T_N,t_N)) {\hat{I}}\qquad \end{aligned}$$

define an instrument and a p.o.m. on a real space with \(p(N+1)\) dimensions \(\mathfrak {R}^{p(N+1)}\).


$$\begin{aligned}&P(A \in T_N, t_N; \ldots ; A \in T_0, t_0) \nonumber \\&\qquad = \mathrm{Tr}\left[ {\mathcal {F}}( T_N, t_N; \ldots ; T_0, t_0) {\hat{\rho }}\right] \nonumber \\&\qquad = \mathrm{Tr}\left[ {\hat{F}}( T_N, t_N; \ldots ; T_0, t_0) {\hat{\rho }}\right] . \end{aligned}$$

So in GQM the observation of a sequence of results at certain successive times can be put on the same foot as the observation of A at a single time.

Appendix 2: Recovery of Ordinary Quantum Mechanics for a Small System

In the perspective of the paper, as we stressed, any observation on a system has to be expressed in terms of the modification that the system induces on the classical e. m. field.

Let us then consider, e. g., a system of a small number of particles, characterized by a certain set of invariants (a total electric charge, baryon number, lepton number, etc), to which we shall refer as the object. Let us assume that such particles interact freely among themselves during a certain interval of time \((t_a,\,t_b)\). We can admit any kind of rearrangement inside the system, exchange of energy and momentum, production or destruction of particles, but no interaction with the external environment during such interval of time.

We assume that at the time \(t_b\) the system comes in contact with an apparatus, by which the specific type of final particles, their momenta, energies etc. can be detected. To be specific we may think of the apparatus as a set of counters, filling densely a certain region of the space possibly kept under the action of a magnetic field.

Both the object and the apparatus in their specific states must be thought as states of the same system of fields initially localized in different parts of the space. Such states can be expressed by appropriate composed creator operators applied to the vacuum state. Let us denote by \(|u_1(t)\rangle ,\,|u_2(t)\rangle ,\ldots \) and \(|U_1(t)\rangle ,\,|U_2(t)\rangle ,\ldots \) two orthogonal basis in the subspaces of the object system and of the apparatus at the time t and write

$$\begin{aligned} |u_j(t)\rangle = {\hat{a}}_j^\dagger (t) |0 \rangle \,\qquad |U_r(t)\rangle = {\hat{A}}_r^\dagger (t) |0 \rangle , \end{aligned}$$

\({\hat{a}}_j^\dagger (t)\) and \({\hat{A}}_j^\dagger (t)\) being ordinary Heisenberg picture operators which commute for \(t<t_b\).

Then let us assume the object described at the initial time \(t_a\) by the statistical operator

$$\begin{aligned}&{\hat{\rho }} ^\mathrm{O}(t_a)= \sum _{ij} |u_i(t_a) \rangle \rho _{ij}^\mathrm{O}(t_a) \langle u_j(t_a)| \nonumber \\&\qquad \qquad =\sum _{ij} {\hat{a}}^\dagger (t_a)|0 \rangle \rho _{ij}^\mathrm{O}(t_a) \langle 0| {\hat{a}}_i(t_a). \end{aligned}$$

The assumption that the number of particles is small implies that the classical fields \(\mathbf{E}_\mathrm{classic}(t,\,\mathbf{x})\) and \(\mathbf{B}_\mathrm{classic}(t,\,\mathbf{x}) \) remain negligible in the region occupied by the system until this does come in contact with the apparatus. Then, if \(L_0 \in \Sigma _{t_a}^{t_b}\) is the set of the histories of the field corresponding to such situation, the probability of occurrence of the complementary set must be null, \({\mathcal {F}}(L_0';\,t_b,\,t_a){\hat{\rho }}(t_a)=0 \). So at the time \(t_b\) we have

$$\begin{aligned}&{\hat{\rho }} ^\mathrm{O}(t_b)={\mathcal {F}}(L_0;\,t_b,\,t_a){\hat{\rho }} ^\mathrm{O}(t_a) = {\mathcal {G}}(\,t_b,\,t_a){\hat{\rho }} ^\mathrm{O}(t_a) \nonumber \\&\qquad \qquad = \sum _{ij} {\hat{a}}_i^\dagger (t_b)|0 \rangle \rho _{ij}^\mathrm{O}(t_b) \langle 0|{\hat{a}}_j(t_a), \end{aligned}$$


$$\begin{aligned} \rho _{ij}^\mathrm{O}(t_b)= \langle u_i(t_b)|{\mathcal {G}}(t_b,\,t_a) \left\{ \sum _{kl}{\hat{a}}_k^\dagger (t_a)|0 \rangle \rho _{kl}^\mathrm{O}(t_a) \langle 0|{\hat{a}}_l(t_a)\right\} |u_j(t_b) \rangle . \end{aligned}$$

Similarly let be

$$\begin{aligned} {\hat{\rho }}^\mathrm{A}(t_a) = \sum _{rs} {\hat{A}}_r^\dagger (t_a)|0 \rangle \rho _{rs}^\mathrm{A}(t_a) \langle 0|{\hat{A}}_s(t_a) \end{aligned}$$

the initial state of the apparatus. In this case we can assume that the counters remain in their charged states, corresponding to the classical e. m. field having certain specific stable values inside them, until any interaction with some external object occurs. Again this corresponds to the classical history of the e. m. field falling with certainty in an other set \(M_0 \in \Sigma _{t_a}^{t_b}\), being null the probability of occurrence of the complementary set. Then

$$\begin{aligned} {\hat{\rho }}^\mathrm{A}(t_b) = \sum _{rs} {\hat{A}}_r^\dagger (t_b)|0 \rangle \rho _{rs}^\mathrm{A}(t_b) \langle 0|{\hat{A}}_s(t_b) \end{aligned}$$


$$\begin{aligned} \rho _{rs}^\mathrm{A}(t_b)= \left\langle U_r(t_b)\left| {\mathcal {G}}(t_b,\,t_a) \left\{ \sum _{kl}{\hat{A}}_k^\dagger (t_a)|0 \rangle \rho _{kl}^\mathrm{A}(t_a) \langle 0|{\hat{A}}_l(t_a)\right\} \right| U_s(t_b )\right\rangle . \end{aligned}$$

Finally, since we have assumed that the object and the apparatus do not come in contact before \(t_b\), they must evolve independently during the interval \((t_a,\, t_b)\). Then, at the time \(t_b\) we have for their compound state

$$\begin{aligned} {\hat{\rho }}^\mathrm{T} (t_b)= & {} \sum _{ij}\sum _{rs}{\hat{a}}_i^\dagger (t_b) {\hat{A}}_r^\dagger (t_b) |0 \rangle \rho _{ij}^\mathrm{O}(t_b)\,\rho _{rs}^\mathrm{A}(t_b) \langle 0| {\hat{A}}_s (t_b) {\hat{a}}_j(t_b) \nonumber \\= & {} \sum _{ij}\sum _{rs}|u_i,\,U_r;\,t_b \rangle \rho _{ij}^\mathrm{O}(t_b)\,\rho _{rs}^\mathrm{A}(t_b) \langle u_j,\, U_s; t_b |. \end{aligned}$$

In a subsequent time interval \((t_b,\,t_c) \), as consequence of the interaction with the particles of the object, some of the counter shall discharge and every specific pattern of discharged counters is interpreted as corresponding to certain specific particles with specific energies and momenta present in the system at time \(t_b\). Then, if now we denote by \(N \in \Sigma _{t_b}^{t_c}\) the set of classical e. m. world histories corresponding to the parameters specifying the particles types, energies, momenta etc. falling in a certain set T, we have

$$\begin{aligned} p(T,\,t_b)= & {} P(t_c,\,t_b;\,N)\nonumber \\= & {} \mathrm{Tr} [{\mathcal {F}}(t_c,\,t_b;\,N)\,{\hat{\rho }} ^\mathrm{T}(t_b)]= \sum _{ij}F_{ij}(T,\,t_b)\, \rho _{ji}^\mathrm{O}(t_b), \end{aligned}$$

which is positive and where obviously

$$\begin{aligned}&F_{ij}(T,\,t_b)=\mathrm{Tr} \left[ {\mathcal {F}}(t_c,\,t_b;\,N) \{\sum _{rs} |u_i(t_b),\,U_r(t_b) \rangle \right. \nonumber \\&\quad \qquad \qquad \qquad \left. \rho _{rs}^\mathrm{A}(t_b) \langle u_j(t_b),\,U_s(t_b)|\} \right] . \end{aligned}$$

To be more explicit, let us assume that the vectors \(|u_j(t_b)\rangle \) already correspond to a specifications of the state of the particles at the time \(t_b\) and denote by \(N_j \in \Sigma _{t_b}^{t_c}\) the corresponding pattern of discharge of the counters. We can write

$$\begin{aligned}&{\mathcal {F}}(t_c,\,t_b;\,N_j)\,{\hat{\rho }}^\mathrm{T} (t_b) \nonumber \\&\qquad = \rho _{jj}^\mathrm{O} (t_b) \, {\mathcal {G}}(t_c,\,t_b) \{ \sum _{rs}|u_j,\,U_r;\,t_b \rangle \,\rho _{rs}^\mathrm{A}(t_b) \langle u_j,\, U_s; t_b |\rbrace . \end{aligned}$$

from which, since \({\mathcal {G}}(t_c,\,t_b) \) is trace-preserving, it follows

$$\begin{aligned} p_j(t_b)\equiv & {} P(t_c,\,t_b;\, N_j)= \mathrm{Tr} \{{\mathcal {F}}(t_c,\,t_b;\, N_j)\, {\hat{\rho }}^\mathrm{T}(t_b)\} \nonumber \\= & {} \rho _{jj}^\mathrm{O}(t_b)\mathrm{Tr} \left\{ \sum _{rs}|u_j,\,U_r;\,t_b \rangle \,\rho _{rs}^\mathrm{A}(t_b) \langle u_j,\, U_s; t_b |\right\} =\rho _{jj}^\mathrm{O}(t_b), \end{aligned}$$

that is the prescription of usual elementary Quantum Theory, up to the correction introduced in (152) by the action of the mapping \({\mathcal {G}}(t_b,\,t_a)\).

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Prosperi, G.M. Introduction of a Classical Level in Quantum Theory. Found Phys 46, 1426–1460 (2016).

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  • Continuous monitoring
  • Quantum mechanics
  • Classical variables
  • Spinor electrodynamics