Obituary for a Flea

Abstract

The Landsman–Reuvers proposal to solve the measurement problem from within quantum theory is extensively analysed. In favor of proposals of this kind, it is shown that the standard reasoning behind objections to solving the measurement problem from within quantum theory rely on counterfactual reasoning or mathematical idealisations. Subsequently, a list of objections/challenges to the proposal are made. Part of these objections are equally important for all attempts at solving the measurement problem, such as the problem of interpreting small numbers in the density matrix, the problem of reproducing the Born rule, the use of pure states as a tool to alleviate the interpretational issues of quantum states, and the necessity of introducing classical certainties which are not strictly present in quantum theory. The additional objections that are particular to the proposal, such as the physical interpretation/origin of the flea perturbation, the use of potentials to solve a dynamical problem, slow collapse times, the inability to handle unequal probabilities, and the dictatorial role of the flea perturbation, lead us to believe that the Landsman–Reuvers proposal is lacking in both physical grounding and theoretical promise. Finally, an overview is given of the challenges that were encountered in this attempt to solve the measurement problem from within quantum theory.

The research in this chapter is part of the project Experimental Tests of Quantum Reality, funded by the Templeton World Charity Foundation.

Appendices

5 Appendix 1: Convergence of States

For concreteness, let $\hbar$ take values in the unit interval [0, 1]. To each strictly positive $\hbar >0$, associate the non-commutative algebra $\mathfrak {A}_{\hbar }=\mathscr {K}(L^{2}(\mathbb {R}))$ of compact operators acting on the Hilbert space of square-integrable functions. To $\hbar =0$ associate the commutative algebra $\mathfrak {A}_{0}=C_{0}(\mathbb {R}^{2})$ of continuous real-valued functions on the phase space $\mathbb {R}^{2}$, which vanish at infinity. Through their disjoint union these algebras combine in a single algebra fibred over the unit interval, $\mathfrak {A}\rightarrow [0,1]$. Dual to this bundle there is the bundle of state spaces $\mathscr {S}\rightarrow [0,1]$ where , and $\mathscr {S}_{\hbar }$ denotes the state space of $\mathfrak {A}_{\hbar }$. For $\hbar >0$ the states are density operators acting on $L^{2}(\mathbb {R})$, and for $\hbar =0$ the states are probability measures on the phase space $\mathbb {R}^{2}$.

Next, we could consider the algebraic and topological aspects of these bundles. But since we are only concerned with the measurement problem, we refer the reader to [10] and proceed directly to the our main question; how is convergence of states defined in this scheme? More precise, when does a family of density operators $(\rho _{\hbar })_{\hbar \in (0,1]}$ converge to a classical state $\mu _{0}\in \mathscr {S}_{0}$? To define convergence, note that each density operator $\rho _{\hbar }$ defines a probability measure $\mu _{\hbar }$ on $\mathbb {R}^{2}$, through

$$\begin{aligned} \int _{\mathbb {R}^{2}}d\mu _{\hbar }f:=Tr\left( \rho _{\hbar }Q_{\hbar }(f)\right) ,\ \ \ \forall f\in C_{0}(\mathbb {R}^2) \end{aligned}$$

(20)

where $Q_{\hbar }(f)$ is the compact operator acting on $L^{2}(\mathbb {R})$, called the ‘Berezin quantisation’ of f. The Berezin quantisation map $Q_{\hbar }$ is defined as

$$\begin{aligned} Q_{\hbar }(f)=\int _{\mathbb {R}^{2}}\frac{dp dq}{2\pi \hbar }f(p,q)\vert \varPhi ^{(p,q)}_{\hbar }\rangle \langle \varPhi ^{(p,q)}_{\hbar }\vert , \end{aligned}$$

(21)

through the coherent states $\varPhi ^{(p,q)}_{\hbar }\in L^{2}(\mathbb {R})$;

$$\begin{aligned} \varPhi ^{(p,q)}_{\hbar }(x)=(\pi \hbar )^{-1/4}e^{-ipq/2\hbar }e^{ipx/\hbar }e^{-(x-q)^{2}/2\hbar }. \end{aligned}$$

(22)

The states $\rho _{\hbar }$ converge to the classical (possibly mixed) state $\mu _{0}$ iff the probability measures $\mu _{\hbar }$ converge weakly to $\mu _{0}$ in the sense

$$\begin{aligned} \lim _{\hbar \rightarrow 0}\int _{\mathbb {R}^{2}}d\mu _{\hbar }f=\int _{\mathbb {R}^{2}}d\mu _{0}f, \end{aligned}$$

for each $f\in C_{0}(\mathbb {R}^{2})$ with compact support.

For the double-well model, as $\hbar \rightarrow 0$, the ground state $\psi _{0}(x)$, or rather its associated density operator, converges to the classical mixed state (7), a convex combination of probability distributions with support in the two different wells. As emphasised in the main paper, this is a classical state which does not qualify as an outcome.

6 Appendix 2: Practical Necessities

The fundamental uncertainties of quantum theory prohibit the preparation of a pure state, let alone an eigenstate with respect to a fixed basis. For most purposes this is irrelevant since we can get close enough in terms of Born probabilities; but for the measurement problem, the distinction may matter. We illustrate the point using the simple example of preparing an initial state using a Stern–Gerlach experiment. The point will be that for any fully quantum mechanical treatment, the Born probabilities associated to both eigenvalues of the observable are always non-zero. If we started with an n-level system, the same would hold for all the eigenvalues of any observable.

Consider the Stern–Gerlach experiment where spin-1 / 2 particles are sent through an inhomogeneous magnetic field. The textbook view is that due to the spin-magnetic-field interaction the spin along the magnetic field gets correlated with the position of the particle. In this simple view of the device, the incoming particle is described by a wavepacket $\psi (\mathbf {x},t)$ and then after some interaction time the position of the wavepacket is correlated to the spin in the z-direction

$$\begin{aligned} \left| \varPsi (t=0)\right\rangle =&\left( \alpha \left| \uparrow \right\rangle +\beta \left| \downarrow \right\rangle \right) \int \psi (\mathbf {x},0)\left| \mathbf {x}\right\rangle \mathrm {d}\mathbf {x}\\ \rightarrow \left| \varPsi (t)\right\rangle =&\alpha \left| \uparrow \right\rangle \int \psi _{+}(\mathbf {x},t)\left| \mathbf {x}\right\rangle \mathrm {d}\mathbf {x}+\beta \left| \downarrow \right\rangle \int \psi _{-}(\mathbf {x},t)\left| \mathbf {x}\right\rangle \mathrm {d}\mathbf {x}, \end{aligned}$$

where $\psi _{\pm }(\mathbf {x},t)$ indicate the wavepackets as they leave the Stern–Gerlach apparatus. If the initial wavepacket is Gaussian, the wavepackets $\psi _{\pm }$ will also be approximately Gaussian but shifted upwards or downwards along the z-axis. For a derivation of the typical form of such wavepackets see [7, 16, 17].

For the Stern–Gerlach apparatus to serve its purpose, to distinguish spin states based on the position of the particle, the conditions of the experiment should be such that most particles will be detected at one of the two well-separated positions on a detector screen. Some of the dominant parameters, which determine the separation of the final wavepackets, are the initial wavepacket width, the strength and inhomogeneity of the magnetic field, and the time of flight during and after the interaction.

These parameters are varied by the experimenter when designing and testing the experiment until the overlap between the wavepackets $\int \psi _{+}^{*}(\mathbf {x},t)$ $\psi _{-}(\mathbf {x},t)\mathrm {d}\mathbf {x}$ becomes extremely small such that for all practical purposes the wavepackets seem to be separated in space. However, according to quantum theory the overlap will in general always be non-zero. In other words, spin is not perfectly correlated with position on the detector screen.

If a small slit is made in the detector in the region we identify with “spin-up”, the state immediately after the slit will be given by

$$ \alpha \left| \uparrow \right\rangle \int \psi _{+}(\mathbf {x},t)\left| \mathbf {x}\right\rangle \mathrm {d}\mathbf {x}+\beta \left| \downarrow \right\rangle \int \psi _{-}(\mathbf {x},t)\left| \mathbf {x}\right\rangle \mathrm {d}\mathbf {x}, $$

where now the integration is restricted to the small slit. If the experiment is well designed, one of the terms will be (exponentially) smaller than the other. In principle, we should also allow for an extremely small contribution where the particle tunnels through the detector screen.²

In practice, when the Stern–Gerlach apparatus is used as a preparation device, the smaller term will be discarded as the parameters of the setup were tuned specifically for reproducibility, i.e., it is tuned such that the smaller term is experimentally inaccessible to subsequent verification (using another device) due to the finite statistics and the resolution of any experiment. This leads to the erroneous conclusion that a pure state in spin-space can be obtained by application of the Stern–Gerlach apparatus. Theoretically, after the slit the following density matrix in the $\uparrow ,\downarrow$-basis is obtained

$$\begin{aligned} \rho _{s}=\int \left( \begin{array}{cc} \left| \alpha \psi _{+}\right| ^{2} &{} \alpha ^{*}\psi _{+}^{*}\beta \psi _{-}\\ \alpha \psi _{+}\beta ^{*}\psi _{-}^{*} &{} \left| \beta \psi _{-}\right| ^{2} \end{array}\right) \mathrm {d}\mathbf {x}. \end{aligned}$$

(23)

Experimentally, the factors in the density matrix can be tuned more-or-less continuously by the above-mentioned parameters, however, they will never be strictly equal to one or zero unless the exact initial spin-state was known, i.e., the exact value of $\alpha$ and $\beta$ is known beforehand.

A further fundamental complication is that the magnetic field must have zero divergence, which implies that it cannot have a gradient in the field in only one direction [16]. Therefore, as the wavepacket has a finite width in space, each part of it couples to its local direction of the magnetic field; and these local directions are not precisely aligned with the single z-axis that is considered theoretically. Thus particles with initially the same spin state along the quantization axis can, nevertheless, deviate according to that of the opposite spin.

Another important point is that the magnetic field and magnet were presumed to be classical. If the electromagnetic field is treated quantum mechanically as mediating interactions between the spin-1 / 2 test-particle and the particles in the Stern–Gerlach magnets, it would result in entanglement between the test-particle’s position and those of the charge carriers in the coils of the Stern–Gerlach magnets. Namely, the charge carriers in the magnet would undergo a momentum increase, and thereby change their position, along the z-direction depending on the spin-component of the test-particle’s wavefunction. After tracing out the states of the magnet, a density matrix is obtained similar to Eq. (23). Also, as spin exchange processes between the test-particle and the magnet’s particles are always possible, there are again contributions which cause incorrect deflections to occur. Such processes are easy to visualize in the path-integral picture, which sums the amplitudes over all possible paths and interactions, see Fig. 7. Agreed, the suppression of spin-flips can be argued to be to be very strong under typical circumstances due to Pauli blocking of transitions to already occupied electronic states in the magnet: whereby we normally assume classical properties to the magnet.

Fig. 7

A spin-flip process in the Stern–Gerlach experiment

Summarizing, the Stern–Gerlach experiment cannot be used as an ideal and reliable preparation device of spin states, even in principle, as there is no perfect one-to-one correspondence with position and spin. Note that the objections to this experiment creating pure states in spin are of a fundamental nature, namely they lie in the divergencelessness of the magnetic field, or the entanglement with the magnet with which it necessarily interacts, or the spatial extent of the wavefunctions.