Spin and Contextuality in Extended de Broglie-Bohm-Bell Quantum Mechanics

Abstract

This paper introduces an extension of the de Broglie-Bohm-Bell formulation of quantum mechanics, which includes intrinsic particle degrees of freedom, such as spin, as elements of reality. To evade constraints from the Kochen-Specker theorem the discrete spin values refer to a specific basis – i.e., a single spin vector orientation for each particle; these spin orientations are, however, not predetermined, but dynamic and guided by the (reduced, spin-only) wave function of the system, which is conditional on the realized location values of the particles. In this way, the unavoidable contextuality of spin is provided by the wave function and its realized particle configuration, whereas spin is still expressed as a local property of the individual particles. This formulation, which furthermore features a rigorous discrete-time stochastic dynamics, allows for numerical simulations of particle systems with entangled spin, such as Bohm’s version of the EPR experiment.

Appendices

Appendices

A No Surreal Trajectories with the eBBB Formulation

One of the non-classical features of quantum mechanics is the phenomenon that a particle can interfere “with itself”. In the causal Bohm interpretation this leads to trajectories with equally non-classical features. Consider a Gaussian wave function $\psi _p(x+x_0,t)$ for a particle in one dimension localized around $x=-x_0$ and moving in the positive direction with momentum p. The superposition

$$\begin{aligned} \psi (x,t_0) = (1/\sqrt{2})(\psi _p(x+x_0) + \psi _{-p}(x-x_0)) \end{aligned}$$

(A-1)

then represents a system in which the particle at $t=t_0$ has equal probability to be localized around $x=-x_0$ moving in the positive direction or be localized around $x=+x_0$ moving in the negative direction. After some time, when the wave packets are in the region around $x=0$, the particle (or rather the wave function) will “interfere with itself”, after which the two wave packets reemerge and continue on the paths they followed before entering the interference region.

Fig. 9

Trajectories for the system (A-2) describing a free particle with spin 1/2. The wave function consists of two wave packets with opposite spin, which move towards each other and are unperturbed when the packages cross around $x = 0$. Trajectories in the top figure (“surrealistically”) bounce back rather than cross, because they are guided by the marginal, location only, probability current; trajectories in the bottom figure cross $x=0$ without interference, just like the wave packets – because they are computed with the eBBB approach of the present paper

In the causal Bohm interpretation (as well as in the eBBB interpretation) the trajectories for such a particle will show the “bouncing” behavior discussed at length in the “surreal trajectories” papers [13,14,15,16,17,18]. In this situation, the observation that particle trajectories will not (cannot) cross $x=0$ is perhaps surprising at first glance, but this unexpected behavior can be justified or at least mollified by pointing to the distortions in the wave packets, due to this non-classical self-interference, which then apparently also impacts the particle trajectories.⁵ It is, however, less straightforward to argue that such a bouncing behavior of the causal Bohm trajectories is equally reasonable, when the particle (system) has an additional internal degree of freedom, for example spin [16,17,18] (cf. also ref. [11]). This is at the heart of the assertion in ref. [13, 14] that these Bohmian trajectories should be dubbed surreal.

The wave function (A-1) can be augmented with a binary degree of freedom. This could represent the spin up and down state of the particle, or the two states of a resonance cavity. Then the wave function

$$\begin{aligned} \psi (x,t_0) = (1/\sqrt{2})(\psi _p(x+x_0)\left( \begin{array}{c} 1 \\ 0 \end{array} \right) + \psi _{-p}(x-x_0))\left( \begin{array}{c} 0 \\ 1 \end{array} \right) \end{aligned}$$

(A-2)

can represent a state in which one of the wave packets has triggered a particle detection, which has flipped its spin, or flipped the binary state from $(1,0)^T$ to $(0,1)^T$ (as in the final stretch of the which-way experiment of ref. [13]). In the causal Bohm interpretation, also this wave function will generate particle trajectories that bounce away from $x_0$, even though the wave packets now are in orthogonal sectors of the Hilbert space and will not interfere with each other when crossing $x=0$. In the Bohm interpretation only particle locations are available as elements of reality - the binary state (be it spin or resonance cavity state), which allows the wave packets to cross in configuration space, is absent.

The easiest way to see the inevitability of the bouncing behavior of trajectories generated by the state (A-2), is to observe that in the Bohm approach the particle velocities $\dot{x}$ are obtained from the marginal probability current in which all internal particle degrees of freedom have been averaged over,

$$\begin{aligned} \dot{x}(t) = \overline{J}(x,t)/\overline{P}(x,t) \end{aligned}$$

(A-3)

with

$$\begin{aligned} \begin{array}{rcl} \overline{P}(x,t) &{} = &{} \sum _s \psi _s^*(x,t)\psi _s(x,t) \\ \overline{J(}x,t) &{} = &{} (1/2mi) \sum _s \left( \psi _s^*(x,t)\partial _x \psi _s(x,t) - \psi _s(x,t)\partial _x \psi _s^*(x,t) \right) . \end{array} \end{aligned}$$

(A-4)

For free moving particles with initial state (A-2) the marginal current is anti-symmetric in x and hence $\overline{J}=0$ at $x=0$. Together with the single valuedness of the particle velocities in (A-3) this implies that the trajectories cannot cross $x=0$ and will show the same bouncing behavior as they do when the guiding wave function does not have internal degrees of freedom. Since each of the two wave packets, $\psi _1(x,t)$ and $\psi _2(x,t)$, evolves as if the other component is absent, it is difficult to accept that the trajectories they guide along behave so differently, depending on the presence of absence of the other component.

In the eBBB formulation explored in the present paper, internal degrees of freedom (like spin) are elements of reality and the particle trajectories also carry this degree of freedom. Therefore, the (stochastic) eBBB-trajectories can (and will) cross each other at $x=0$, because the left and right moving particles have different spin (or cavity state) values. This is illustrated in Fig. 9, where the top figure shows a set of trajectories guided by the marginal current and the bottom figure trajectories (x-values only) guided by the transition probabilities computed using the eBBB approach. Here, the up- and down-moving trajectories have a different internal state index, and hence they can cross.

B Evolution Operators for the Simplified EPRB Experiment

This appendix provides further details of the simplified EPRB model used for the numerical simulation discussed in this paper. In particular, the construction of the two evolution operators, ${\hat{U}}^{(i)}$ and ${\hat{U}}^{(f)}$ will be described in more detail.

In the first stage of the experiment, the unitary evolution operator ${\hat{U}}^{(i)}$ must transform the location and magnet angle states from the initial state $|{\phi _0,x_r,0\pm }\rangle$, to states $|{ \alpha ,x_a,0\pm }\rangle$ or $|{\beta ,x_a,0\pm }\rangle$, at which point the magnets have assumed orientation $\alpha$ with probability $P_{ \alpha }$ or orientation $\beta$ with probability $P_{\beta }=1-P_{ \alpha }$. I.e., the magnet orientation subspace evolution operator is defined on the three-dimensional space spanned by $\{|{\phi _0}\rangle ,|{ \alpha }\rangle ,|{\beta }\rangle \}$ and acts on the ‘ready’ state $|{\phi _0}\rangle$ as

$$\begin{aligned} {\hat{U}}^{(i,\phi )}|{\phi _0}\rangle = \gamma _{ \alpha }|{ \alpha }\rangle + \gamma _{\beta }|{\beta }\rangle , \end{aligned}$$

(B-5)

which can be achieved, for example, by a unitary evolution operator of the form

$$\begin{aligned} {\hat{U}}^{(i,\phi )} = \gamma _{ \alpha }|{ \alpha }\rangle \langle {\phi _0}| + \gamma _{\beta }|{\beta }\rangle \langle {\phi _0}| - \gamma _{\beta }^*|{ \alpha }\rangle \langle { \alpha }| + \gamma _{ \alpha }^*|{\beta }\rangle \langle { \alpha }| + |{\phi _0}\rangle \langle {\beta }|. \end{aligned}$$

(B-6)

The complex-valued coefficients in Eq. (B-5) are such that $\vert \gamma _{ \alpha }\vert ^2 = P_{ \alpha }$ and $\vert \gamma _{\beta }\vert ^2 = 1-\vert \gamma _{ \alpha }\vert ^2=P_{\beta }$. The location subspace evolution operator for the initial stage acts as

$$\begin{aligned} {\hat{U}}^{(i,x)}|{x_r}\rangle = |{x_a}\rangle , \end{aligned}$$

(B-7)

which implies that it can, for example, be defined as

$$\begin{aligned} {\hat{U}}^{(i,x)} = \sum _{i\ne a,r} |{x_i}\rangle \langle {x_i}| + |{x_r}\rangle \langle {x_a}| + |{x_a}\rangle \langle {x_r}|. \end{aligned}$$

(B-8)

In this initial stage of the experiment the spin state does not change, hence the identity matrix can be used for its evolution:

$$\begin{aligned} {\hat{U}}^{(i,\sigma )} = |{0+}\rangle \langle {0+}| + |{0-}\rangle \langle {0-}|. \end{aligned}$$

(B-9)

Since the three substates evolve independently, the combined evolution operator for the first stage is the direct product

$$\begin{aligned} {\hat{U}}^{(i)} = {\hat{U}}^{(i,\phi )} \otimes {\hat{U}}^{(i,x)} \otimes {\hat{U}}^{(i,\sigma )}, \end{aligned}$$

(B-10)

and the ‘all-set’ state is generated from the ‘ready’ state as,

$$\begin{aligned} |{\psi _a}\rangle = {\hat{U}}^{(i)}|{\phi _0,x_r,0\pm }\rangle = \gamma _{ \alpha }|{ \alpha ,x_a,0\pm }\rangle + \gamma _{\beta }|{\beta ,x_a,0\pm }\rangle . \end{aligned}$$

(B-11)

In the second and final stage of the experiment, the interaction between Stern-Gerlach magnet and particle spin leads to a combined evolution in which the $|{ \alpha ,x_a,0\pm }\rangle$ state transforms into $|{ \alpha ,x_{ \alpha \pm }, \alpha \pm }\rangle$ and the $|{\beta ,x_a,0\pm }\rangle$ state transforms into $|{\beta ,x_{\beta \pm },\beta \pm }\rangle$. I.e., the $\alpha$ substate from the r.h.s. in Eq. (B-11) evolves into a state with spin in the $\alpha +$ or $\alpha -$ direction and corresponding location value $x_{ \alpha +}$ or $x_{ \alpha -}$ and similarly for the $\beta$ substate. In this stage, unlike in the first, there is no change in the device angles, hence the evolution operator can be trivial; likewise, there is no force acting on the particle spin, other than the impact of the Stern-Gerlach device, which forces the spin axis to be aligned with the magnet orientation. Therefore, in this second stage the realized macroscopic magnet orientation and changes in particle location and spin orientation become correlated. This evolution can no longer be described with a factorized operator of the form shown in Eq. (B-10). However, introducing projection operators allows employing suitable combinations of subspace evolution operators that implement the correlated state transitions.

Since there are four different components in the final state of each particle—reflecting the two values of the magnet angle and for each angle the two spin values—the combined evolution operator is composed of four terms:

$$\begin{gathered} \hat{U}^{{(f)}} = \hat{P}_{\alpha }^{{(\phi )}} \otimes \hat{U}_{{x_{{\alpha + }} }}^{{(x)}} \otimes \hat{P}_{{\alpha + }}^{{(\sigma )}} + \hat{P}_{\alpha }^{{(\phi )}} \otimes \hat{U}_{{x_{{\alpha - }} }}^{{(x)}} \otimes \hat{P}_{{\alpha - }}^{{(\sigma )}} \hfill \\ \quad \quad + \hat{P}_{\beta }^{{(\phi )}} \otimes \hat{U}_{{x_{{\beta + }} }}^{{(x)}} \otimes \hat{P}_{{\beta + }}^{{(\sigma )}} + \hat{P}_{\beta }^{{(\phi )}} \otimes \hat{U}_{{x_{{\beta - }} }}^{{(x)}} \otimes \hat{P}_{{\beta - }}^{{(\sigma )}} . \hfill \\ \end{gathered}$$

(B-12)

The projection operators on the magnet angle subspaces in this expression are defined as

$$\begin{aligned} {\hat{P}}^{(\phi )}_{ \alpha } = |{ \alpha }\rangle \langle { \alpha }|, \;\;\;\;{\hat{P}}^{(\phi )}_{\beta } = |{\beta }\rangle \langle {\beta }|, \end{aligned}$$

(B-13)

the x-space evolution operators map the $x_a$ state to the state indicated by their subscripts and can be defined as

$$\begin{aligned} {\hat{U}}^{(x)}_{ x_k} = \sum _{i\ne a,k} |{x_i}\rangle \langle {x_i}| + |{x_k}\rangle \langle {x_a}| + |{x_a}\rangle \langle {x_k}|,\;\;\;\;\mathrm{for}\; k= \alpha +, \alpha -,\beta +,\beta -, \end{aligned}$$

(B-14)

and the projectors on the (rotated) positive and negative spin states are

$$\begin{aligned} {\hat{P}}^{(\sigma )}_{ \alpha +} = |{ \alpha +}\rangle \langle { \alpha +}|, \;\;\;\;{\hat{P}}^{(\sigma )}_{ \alpha -} = |{ \alpha -}\rangle \langle { \alpha -}|. \end{aligned}$$

(B-15)

A similar definition applies to the projectors on spin states in the $\beta$ direction. Since the location subspace evolution matrices are unitary and the projectors are orthogonal, one can readily verify that also the combined evolution matrix ${\hat{U}}^{(f)}$ is unitary.

It is now straightforward to show that the combined evolution operator for the two particles

$$\begin{aligned} U^{(f)}U^{(i)}\otimes U^{(f)}U^{(i)}, \end{aligned}$$

(B-16)

transforms the initial spin-singlet ‘all-set’ state

$$\begin{aligned} |{\psi _r}\rangle = \left( |{\phi _0,x_r,0+}\rangle \otimes |{\phi _0,x_r,0-}\rangle - |{\phi _0,x_r,0-}\rangle \otimes |{\phi _0,x_r,0+}\rangle \right) /\sqrt{2} , \end{aligned}$$

(B-17)

to an end state with the expected correlations between measured values of the spins of the two particles. To simplify notations, from now on the $\otimes$ symbol will only be used to indicate direct products between the states or operators for particle one and two.

After applying the initial-stage evolution operator, the ‘ready’ state turns into the following ‘all-set’ state,

$$\begin{gathered} |\psi _{a} \rangle = U^{{(i)}} \otimes U^{{(i)}} |\psi _{r} \rangle = \hfill \\ \quad \quad \;\left( {(\gamma _{\alpha } |\alpha ,x_{a} ,0 + \rangle + \gamma _{\beta } |\beta ,x_{a} ,0 + \rangle ) \otimes (\gamma ^{\prime}_{\alpha } |\alpha ,x_{a} ,0 - \rangle + \gamma ^{\prime}_{\beta } |\beta ,x_{a} ,0 - \rangle )} \right. \hfill \\ \quad \quad \left. { - (\gamma _{\alpha } |\alpha ,x_{a} ,0 - \rangle + \gamma _{\beta } |\beta ,x_{a} ,0 - \rangle ) \otimes (\gamma ^{\prime}_{\alpha } |\alpha ,x_{a} ,0 + \rangle + \gamma ^{\prime}_{\beta } |\beta ,x_{a} ,0 + \rangle )} \right)/\sqrt 2 . \hfill \\ \end{gathered}$$

(B-18)

The same evolution operators are used for particle one and two; i.e., for both particles, the same two orientation angles $\alpha$ and $\beta$ for the Stern-Gerlach magnet can be assumed, only the coefficients $\gamma _{ \alpha ,\beta }$ that determine the relative probability to realize these orientations can be different, as is indicated by the prime on these coefficients in the ‘all-set’ state of particle two.

The second stage evolution operator will produce a ‘measured’ state which is a superposition of states for the four different combinations of the magnet orientations:

$$\begin{gathered} |\psi _{m} \rangle = U^{{(f)}} \otimes U^{{(f)}} |\psi _{a} \rangle = \hfill \\ \quad \quad \quad \gamma _{\alpha } \gamma ^{\prime}_{\alpha } (|\alpha \rangle \otimes |\alpha \rangle )|\psi _{{\alpha ,\alpha }} \rangle + \gamma _{\alpha } \gamma ^{\prime}_{\beta } (|\alpha \rangle \otimes |\beta \rangle )|\psi _{{\alpha ,\beta }} \rangle \hfill \\ \quad \quad \quad + \gamma _{\beta } \gamma ^{\prime}_{\alpha } (|\beta \rangle \otimes |\alpha \rangle )|\psi _{{\beta ,\alpha }} \rangle + \gamma _{\beta } \gamma ^{\prime}_{\beta } (|\beta \rangle \otimes |\beta \rangle )|\psi _{{\beta ,\beta }} \rangle . \hfill \\ \end{gathered}$$

(B-19)

This shows that the full state is a linear combination of four⁶ conditional states $|{\psi _{ \alpha ,\beta }}\rangle \equiv (U^{(f)}\otimes U^{(f)})(|{x_a, \alpha +}\rangle \otimes |{x_a,\beta -}\rangle - |{x_a, \alpha -}\rangle \otimes |{x_a,\beta +}\rangle )$, which can be computed as

$$\begin{gathered} |\psi _{{\alpha ,\beta }} \rangle = \left( {(c_{\alpha } |x_{{\alpha + }} ,\alpha + \rangle + s_{\alpha } |x_{{\alpha - }} ,\alpha - \rangle ) \otimes (s_{\beta } |x_{{\beta + }} ,\beta + \rangle - c_{\beta } |x_{{\beta - }} ,\beta - \rangle )} \right. \hfill \\ \quad \quad \quad - \left. {(s_{\alpha } |x_{{\alpha + }} ,\alpha + \rangle - c_{\alpha } |x_{{\alpha - }} ,\alpha - \rangle ) \otimes (c_{\beta } |x_{{\beta + }} ,\beta + \rangle + s_{\beta } |x_{{\beta - }} ,\beta - \rangle )} \right)/\sqrt 2 \hfill \\ \quad \quad \quad = \left( { - s_{{\alpha - \beta }} |x_{{\alpha + }} ,\alpha + \rangle \otimes |x_{{\beta + }} ,\beta + \rangle - c_{{\alpha - \beta }} |x_{{\alpha + }} ,\alpha + \rangle \otimes |x_{{\beta - }} ,\beta - \rangle } \right. \hfill \\ \quad \quad \quad \quad \left. { + c_{{\alpha - \beta }} |x_{{\alpha - }} ,\alpha - \rangle \otimes |x_{{\beta + }} ,\beta + \rangle - s_{{\alpha - \beta }} |x_{{\alpha - }} ,\alpha - \rangle \otimes |x_{{\beta - }} ,\beta - \rangle } \right)/\sqrt 2 . \hfill \\ \end{gathered}$$

(B-20)

To obtain these equalities, the following results were used for the projections of the $|{0\pm }\rangle$ states on the rotated $|{ \alpha \pm }\rangle$ states:

$$\begin{aligned} \langle { \alpha +}|0+\rangle = c_{ \alpha }, \;\;\;\;\langle { \alpha +}|0-\rangle = \langle { \alpha -}|0+\rangle = s_{ \alpha }, \;\;\;\;\langle { \alpha -}|0-\rangle = -c_{ \alpha }; \end{aligned}$$

(B-21)

with the following abbreviations for the sines and cosines in the expressions above:

$$\begin{aligned} c_{\phi } \equiv \cos (\phi /2), \;\;\;\;s_{\phi } \equiv \sin (\phi /2), \;\;\;\;\phi \in \{ \alpha , \beta , \alpha -\beta \}. \end{aligned}$$

(B-22)