A conjecture concerning determinism, reduction, and measurement in quantum mechanics

Abstract

It is shown that it is possible to introduce determinism into quantum mechanics by tracing the probabilities in the Born rules back to pseudorandomness in the absolute phase constants of the wave functions. Each wave function is conceived to contain an individual phase factor $\exp (\mathrm {i}\alpha )$. In an ensemble of systems, the phase constants $\alpha$ are taken to be pseudorandom numbers. A reduction process (collapse), independent of any measurement, is conceived to be a spatial contraction of two wavepackets when they meet and satisfy a certain criterion. The criterion depends on the phase constants of both wavepackets. The measurement apparatus fans out the incoming wavepacket into spatially separated eigenpackets of the observable and a reduction associates the point of contraction with an eigenvalue of the observable. The theory is nonlocal and contextual.

Appendices

Appendix 1: Valuation of the absolute phase

We here want to deal with the frequently encountered assertion that the absolute phase of a wave function is undetermined if the wave function represents a definite number of particles, in the same way as the position is undetermined if the wave function represents a particle of definite momentum [46–48]. This would contradict our approach, in which a definite phase is ascribed to every wave function.

We reject that assertion. It stems from the introduction of a phase operator, which does not commute with the particle-number operator. There are, however, serious difficulties with the construction of a phase operator. The original phase operator, introduced by Dirac [49], is not Hermitian and thus cannot be an observable. Subsequently there have been many attempts to construct a phase operator that is less deficient, but each proposal had its own difficulties and none has met with general approval. Details can be found in [50] and the literature cited there.

Moreover, in a plane wave $\exp (\mathrm {i}({\varvec{k}}\varvec{r}-\omega t+\alpha ))$ the phase constant appears on an equal footing with the time t. So, what holds for time should also hold for phase. Regarding time, Pauli [51] pointed out that it is generally not possible to satisfy the canonical commutation relations between the operators time and energy; he wrote:

We, therefore, conclude that the introduction of an operator t is basically forbidden and the time t must necessarily be considered as an ordinary number (‘c-number’) in Quantum Mechanics.

Indeed, there is a position operator but no time operator in non-relativistic quantum mechanics [21, p. 252], [51–53]. And in relativistic quantum field theory both position and time are parameters, not operators. We therefore take the stand that we do not need a phase operator for observing the phase, just as we do not need a time operator for observing the time. Dispensing with the phase operator frees us from the phase/number uncertainty relation and allows us to ascribe a definite phase to every wave function, though the actual values of the phases may be unknown to us.

Appendix 2: Phase-interval overlap

Here we estimate how much the probability $P_4= \rho \,{\text {d}}^3r\,(\alpha _{\text {S}}/2\pi ) =n\,(\alpha _{\text {S}}/2\pi )$ of (5.4) is changed if there is overlap of the phase-constant intervals $\alpha _{\text {S}}$ belonging to different grains, that is, if the phase constants of neighbouring grains are not always separated by more than $\alpha _{\text {S}}$.

$P_4$ is the ratio of the favorable interval to the total available interval $2\pi$. In the case of only one grain, $n=1$, the favorable interval is $\alpha _{\text {S}}$. In the case of n grains without overlap the favorable interval is $n\alpha _{\text {S}}$. In the (hypothetical) extreme opposite case of n grains with total overlap, that is, where all grains have the same phase constant, the favorable interval remains $\alpha _{\text {S}}$ as in the case $n=1$. Of course, owing to the quantum nature of the incoming wavepacket, only one of the n grains at a time can lead to a contraction.

In the case of partial overlap of the n intervals the value of the favorable interval will lie between $\alpha _{\text {S}}$ and $n\alpha _{\text {S}}$. We want to calculate the mean value of the favorable interval in many repetitions of the measurement. In order to get an estimate we divide the total interval of $2\pi$ into $2\pi /\alpha _{\text {S}}=861$ sections, each of length $\alpha _{\text {S}}$, and the number of favorable intervals is the number of intervals which are represented by at least one grain. We then substitute the overlap problem by an occupancy problem whose solution is known, namely by the birthday problem. In this problem the number 861 of sections corresponds to the 365 days of the year, and the number n of grains means the number of people in a group. The number of favorable intervals $\alpha _{\text {S}}$ corresponds to the number of birthdays, that is of days which are occupied by at least one person celebrating his or her birthday. This number in turn is the total number of 365 days minus the number $N_0$ of empty days (without any birthday). The mean value of the last number for a group of n persons is given in [16, Vol. I, p. 239, 493] as $\bar{N}_0= 364^n 365^{1-n}$. Thus the mean number $\bar{N}$ of birthdays is $365-\bar{N_0}$, and the mean fraction of birthdays is $P=(365-\bar{N}_0)/365$. Returning to the original problem we replace 365 by 861 and obtain

$$\begin{aligned} \bar{N}_0^{\prime }=861\times (860/861)^n=861(1-1/861)^n=861\exp (-c_1n) \end{aligned}$$

with $c_1=-\ln (1-1/861)=1/860.5$. With this the mean fraction of favorable intervals becomes

$$\begin{aligned} (\bar{N}_0^{\prime }-861)/861=1-\exp (-c_1n)=c_1n-\frac{1}{2}c_1^2n^2\pm \cdots , \end{aligned}$$

which goes to 1 for $n\rightarrow \infty$, as it should be. For small n it is approximately

$$\begin{aligned} \frac{1}{860.5}n\approx \frac{1}{861}n=\frac{\alpha _{\text {S}}}{2\pi }n= \frac{\alpha _{\text {S}}}{2\pi } \rho \Delta ^3r= P_4 \end{aligned}$$

with an error of ${\le }5\,\%$ for $n=\rho \,\mathrm {d}^3r\le 90$. As the chosen physical interval corresponding to the infinitesimal interval $\mathrm {d}^3r$ can be very small this is an acceptable approximation.

Appendix 3: Approximation in Eq. (5.6)

We here give an estimate of the degree of approximation in formula (5.6) under the conditions for a measurement of position. That is, we want to show that

$$\begin{aligned} \int _\mathrm{R^3} |\psi _{\text {{1}}}(\varvec{r})| \, |\psi _{\text {{2}}}(\varvec{r})| \, {\text {d}}^3r \approx \text {const}\; |\psi _{\text {{1}}}(\varvec{r}_0)|. \end{aligned}$$

To this end we choose the measured packet $|\psi _1|^2$ and the grain packet $|\psi _2|^2$ to be one-dimensional (normalized) Gaussian functions with full width $\sigma _1$ and $\sigma _2$, respectively. Thus

$$\begin{aligned} |\psi _1(x)|= & {} \left( \frac{2}{\pi \sigma _1^2}\right) ^{1/4}\, \exp \Big (-(x-[x_0-\delta ])^2/\sigma _1^2\Big ),\\ |\psi _2(x)|= & {} \left( \frac{2}{\pi \sigma _2^2}\right) ^{1/4}\, \exp \Big (-(x-x_0)^2/\sigma _2^2\Big ). \end{aligned}$$

The center of the grain function $|\psi _2|$ lies at the fixed position $x_0$, while the center of the measured function $|\psi _1|$ lies at the variable distance $\delta$ from $x_0$. The value of the integral

$$\begin{aligned}I:= \int _{-\infty }^{+\infty } |\psi _{\text {{1}}}(x)| \, |\psi _{\text {{2}}}(x)| \, {\text {d}}x \end{aligned}$$

can then be evaluated exactly, with the result

$$\begin{aligned} I(\delta )=\left[ \frac{2}{\sigma _1/\sigma _2+\sigma _2/\sigma _1}\right] ^{1/2}\,\exp \left( \frac{\delta ^2}{\sigma _1^2}\bigg [\frac{1}{1+(\sigma _1/\sigma _2)^2}-1\bigg ]\right) . \end{aligned}$$

(8.1)

This can be written as $I(\delta )=Q(\delta )\,\times |\psi _1(x_0)|, \text { with }$

$$\begin{aligned} Q(\delta )=\left[ \frac{2}{\sigma _1/\sigma _2+\sigma _2/\sigma _1}\right] ^{1/2}\, \bigg (\frac{\pi \sigma _1^2}{2}\bigg )^{1/4} \exp \left( \frac{\delta ^2}{\sigma _1^2}\bigg [\frac{1}{1+(\sigma _1/\sigma _2)^2}\bigg ]\right) . \end{aligned}$$

(8.2)

From (8.1) and (8.2) it is seen that $Q(\delta )$ varies only slowly where $I(\delta )$ has appreciable values. For example, with $\sigma _2=0.1\sigma _1$ (corresponding to the small $v_\mathrm{{gr}}$) and $\delta =2\sigma _1$, the value of Q has increased by only 4 % from its (maximum) value at $\delta =0$, whereas I thereby has decreased to 2 % of its value at $\delta =0$, and with this rarely satisfies the overlap condition (3.2b). We therefore conclude that (5.6) is an acceptable approximation.