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Bohm's 1957 paper “Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky” co-authored by Aharonov is about the nature of correlations that were the subject of the EPR paper.
They introduce a model to explain EPR correlations. According to that model the quantum mechanical (QM) description of many-body system breaks down when the systems are separated and the parties acquire a separable state which produces the same correlation on average whereas individual pairs are not always anti-correlated anymore. Then they remind an already done experiment that gives different observable results for this separable model and quantum entangled model. The experiment is in agreement with QM description.
Bohm uses an entanglement of discrete quantities, spin of spin-1/2 electrons or polarization of photons, to be able to have an experimental setup that is easier to implement than the original Gedanken experiment suggestion of EPR that involves entanglement of continuous DoF's such as position and momentum. Discrete version is mathematically simpler too.
Instead of
\[\Psi\left(x_{1},x_{2}\right)=\int_{-\infty}^{\infty}e^{i\left(x_{1}-x_{2}+x_{0}\right)p/\hbar}dp\]
Bohm proposes to use the singlet state
\[\psi=\frac{1}{\sqrt{2}}\left[\psi_{+}\left(1\right)\psi_{-}\left(2\right)-\psi_{-}\left(1\right)\psi_{+}\left(2\right)\right]\]
in our notation
\[\ket{\Psi}=\left(\ket{u+u-}-\ket{u-u+}\right)/2\]
where $u$ is an arbitrary direction. The reason of perfect correlation in momenta in the case of EPR state was the fact that it was a state of zero total momentum. In the singlet state the total spin angular momentum is zero. Hence the existence of correlation is independent of which direction is used for the spin measurement. As long as both parties are measured at the same direction the outcomes are perfectly anti-correlated, hence $u$ is arbitrary.
"Before the measurement has taken place (even while the atoms are still in flight) we are free to choose any direction as the one in which the spin of particle $A$ (and therefore of particle $B$) will become definite." This idea of changing the measurement direction on the flight is the basis of the Bell experiments that closed "communication loophole".
Bohm mentions and old and incorrect interpretation of the uncertainty principle as a representation of the disturbance effects of the measurement on incompatible quantities. “This point of view more generally implies that the definiteness of any desired component of the spin is (along with the indefiniteness of the other two components) a potentiality which can be realized with the aid of a suitably oriented spin-measuring apparatus” Bohm does not insist on using a concept such as element of reality (EoR) but simply uses the term “definiteness”. Definiteness is humbler than reality in scope and is a cleaner concept in definiton. According to this view wave-like and particle-like behaviors of an electron are different and mutually incompatible potentialities which are realized by applying some certain experimental setups.
Bohm says that this disturbance based explanation is not satisfactory in EPR experiment. Because of the lack of the direct physical interaction between the party-2 and the apparatus-1, why party-2 “realizes its potentiality for a definite spin is precisely the same direction as that of” party-1, cannot be explained.
At the time when the paper is written quantum optics and other means of manipulating individual quantum systems was not developed enough to directly implement the EPR Gedanken experiment or any other setup that will create EPR correlations. Hence Bohm feels free to develop alternative models to “yet experimentally unverified extrapolation of the many-body Schrödinger and Dirac equations to the case where the particle's wave functions do not overlap”. Einstein proposed the idea that “QM may break down when particles are far enough apart” in a private communication with Bohm.
All these detailed discussion and reasonable arguments against QM amazes me about QM's accuracy. The validity region of QM is so broad. Even though when it was put forward the experimental abilities were not enough to test all of its implications hence it basically consisted of speculations, those speculations turned out to be correct in the course of the history thanks to the deep intuitions of the founders.
The alternative model is proposed by Furry 1936. In that model, when particles “loose contact” wave function turns into a product state of zero total spin, as if entanglement is some kind of thread that snaps when pulled from both sides,
\[\psi=\psi_{+\theta,\varphi}\left(1\right)\psi_{-\theta,\varphi}\left(2\right)\]
\[\ket{\psi}=\ket{v+}\ket{v-}\]
where $v$ is an arbitrary unit vector. The down side of this state is that, now spin angular momentum is zero only for the $v$ direction. In other two orthogonal directions total spin fluctuates and is not zero for individual pairs.
\begin{eqnarray*}\sqrt{2}\sigma_{v}^{T}\ket{\psi} & = & \left(\sigma_{v}\otimes I+I\otimes\sigma_{v}\right)\ket{v+}\otimes\ket{v-}\\ & = & \sigma_{v}\ket{v+}\otimes I\ket{v-}+I\ket{v+}\otimes\sigma_{v}\ket{v-}\\ & = & +\ket{v+}\otimes I\ket{v-}-\ket{v+}\otimes\ket{v-}\\ & = & 0\ket{\psi}\end{eqnarray*}
\begin{eqnarray*}\sqrt{2}\sigma_{x}^{T}\ket{\psi} & = & \sigma_{x}\ket{z+}\otimes I\ket{z-}+I\ket{z+}\otimes\sigma_{x}\ket{z-}\\ & = & \ket{z-}\ket{z-}+\ket{z+}\ket{z+}\end{eqnarray*}
Half of the time they are both down and other half of the time both up so that on average total spin is zero too but not in individual cases. This is correct for only $x$ (or $y$) directions. For other directions the total spin will not be zero. So, the additional assumption is that, $v$ is a random direction with a uniform distribution in all solid angle.
In a future post I will try to clarify what the constraints of local reality imposes on the correlations between two spatially separated systems with two measurements both of which has two outcomes.
The only advantage of the this model is to avoid the so-called EPR paradox, the instantaneous interaction needed to explain the correlations.
But an experiment that refutes this model is shown hence the “paradox” cannot be resolved by breaking down QM for far away particles.
This is a moment of enlightenment about the nature of uncertainty principle: “we can no longer retain the notion that a definite value of a given variable is essentially realized through interaction with an appropriate apparatus, and that the indeterminacy principle represents only an uncontrollable fluctuation in complementary variables brought about by a disturbance originating in the apparatus.” Which means that the uncertainty principle is not an arbitrary lower limit on the amount of the disturbances due to making measurement, a limit on the means of gaining knowledge about systems, but it is an inherent constraint about the nature of the states that a physical system can have and it is independent of the measurement processes.
First view implies that the disturbance due to the mechanism of the apparatus a formerly definite quantity will have a value that is unknown until it is going to be measured, which will just give a probability distribution about the prior information about the system. This situation is different than a superposition, which is an inherent indefiniteness.
Bohm mentions Bohr's ideas to overcome the paradox. Once I tried to read Bohr's response to EPR paper, but it was so unintelligible. :-( What Bohr says, basically, according to Bohm, is that we are not allowed to ask questions that leads to the paradox, hence there is no paradox. Voila! “the observing apparatus plus what is observed form a single indivisible combined system not capable at the quantum-level of being analyzed correctly into separate and distinct parts” The apparatus and system at hand cannot be analyzed further, QM just lets us calculate the outcome probabilities using the “recipe”, and how those probabilities emerge is outside the realm of Bohr's QM and hence meaningless. It does not matter whether the system at hand is a single particle system or an entangled system of many particles or a system and an apparatus etc.
For sure, I do not like this approach. The combination of the system and the apparatus basically includes entanglement too, namely, the gauge of the apparatus is in one-to-one correlation with the quantity that is measured. And entanglement is basically the quantum version correlations. Though, the answer to the problem of how the superposition in that entangled combined system “collapses” to one of its terms, the infamous measurement problem, is still unknown.
Then Bohm mentions his own view that suggests a “deeper exploration of the quantum theory as a whole”. He agrees to treat the setup as a single combined system but says that single combined system may be subject to further analysis, according to his “casual interpretation of quantum theory” namely the Bohmian Mechanics (BM), where he introduces a hidden interaction via the so-called “quantum potential” that allows an interaction “even when their classical interaction potential is zero.” “It must be admitted, however, that this quantum potential seems rather artificial in form, besides being subject to the criticism ... that it implies instantaneous interactions between distant particles, so that it is not consistent with the theory of relativity”.
I am not knowledgeable in BM. But it looks like Bohm himself thinks that his framework is more like a mathematical data fitting than an actual physical theory because it is in conflict with special relativity. But I wonder whether it is really so. Does BM obeys no-signaling condition? If yes, then there is no contradiction.
Bohm also envisage a deeper theory underlying QM. “there has been developed a further new explanation of the quantum theory in terms of a deeper subquantum-mechanical level. The laws of this lower level are different from those of the quantum theory, but approach these latter laws as an approximation, much as the laws of atomic physics approach those of macroscopic physics when many atoms are involved”. Sound exciting!
Then Bohm describes the experiment that supports the existence of EPR-like correlations. Instead of working with spin-1/2 particles he proposes to use photons of which polarization DoF is effectively two dimensional too. The system at hand is “photons ... produced in the annihilation radiation of a positron-electron pair”. They have zero total momentum and their polarization is always orthogonal. He describes the math of bosons that uses creation and annihilations operators on the vacuum state to describe the state of the system that lives in Fock space, which is a bit more involved that the fermion version where number of particles is constant and the state lives in Hilbert space.
Most general state for a photon of wave number $k$ moving along $z$-axis is
\[\psi=rC_{k}^{x}\psi_{0}+sC_{k}^{y}\psi_{0}\]
\[\ket{\psi}=ra_{x}^{\dagger}\ket{\phi}+sa_{y}^{\dagger}\ket{\phi}\]
where $\ket{\phi}$ is the vacuum state with no photons, $a_{i}^{\dagger}$ are creation operators,$ $ $i$ indicates the direction of linear polarization. $r$ and $s$ are complex numbers. Then analogue of the singlet state will be
\begin{eqnarray*}\ket{\Psi} & = & \frac{1}{\sqrt{2}}\left(a_{x}^{\dagger}b_{y}^{\dagger}-a_{y}^{\dagger}b_{x}^{\dagger}\right)\ket{\phi}\\ & = & \frac{1}{\sqrt{2}}\left(a_{x'}^{\dagger}b_{y'}^{\dagger}-a_{y'}^{\dagger}b_{x'}^{\dagger}\right)\ket{\phi}\\ & \equiv & \frac{1}{\sqrt{2}}\left(\ket H\ket V-\ket V\ket H\right)\end{eqnarray*}
where $a$ is for left going photon and $b$ for right going. Primed coordinates are rotated coordinate around the $z$-axis. Last line shows the mathematical equivalency of this polarization state to the spin singlet state.
Because at that time it was experimentally not feasible to measure whether each and every pair of photons produced by the annihilation have orthogonal polarizations, Bohm applies to another experiment that has been done by Wu in 1950 that tests the existence of EPR-correlations in a more indirect way. ``This experiment consists in measuring the relative rate, $R$, of coincidences in the scattering of the two photons through some angle, $\theta$'' It is involved with the scattering theory of photons in a $e^{-}e^{+}\rightarrow\gamma\gamma$ scattering. I have shamefully no idea about the physics of this process. :-(
They calculate $R$ using 3 different assumptions:
A) QM is correct and particles are in the singlet state $\frac{1}{\sqrt{2}}\left(\ket H\ket V-\ket V\ket H\right)$
B1) Entanglement breaks down, particles are circularly polarized in opposite directions. $\left(\ket H+i\ket V\right)\left(\ket H-i\ket V\right)$.
B2) Entanglement breaks down, particles are linearly polarized in uniformly distribute random opposite directions $u$, $\ket{H_{u}}\ket{V_{u}}$.
B is the case where combined angular momentum is conserved only for the average of all pairs but not for individual pairs.
They call the plane that is spanned by the initial direction of motion of photon and direction after scattering $\pi$.
To do the calculations they use the scattering cross section of a single photon from an electron, according to the Klein-Nishina formula it is
\begin{eqnarray*}d\sigma_{1} & = & \frac{1}{2}r_{0}d\Omega\left(k^{2}/k_{0}^{2}\right)\left(\gamma-2\sin^{2}\theta\right)\\d\sigma_{2} & = & \frac{1}{2}r_{0}d\Omega\left(k^{2}/k_{0}^{2}\right)\gamma\end{eqnarray*}
where $\gamma=k_{0}/k+k/k_{0}$, $k_{0}$ is the wave number of the incident photon, $k$ that of the final photon, $r_{0}$ is the classical electronic radius, and $d\Omega$ is the element of solid angle. 1 is the case where initial direction of polarization is parallel to the plane $\pi$ and 2 is the case where it is perpendicular.
A general initial polarization state will be a linear combination of $\ket{\psi_{i}}=\left\{ \ket{HH},\ket{HV},\ket{VH},\ket{VV}\right\} $, $\ket{\Psi}=\sum_{i}c_{i}\ket{\psi_{i}}$. And in the most general case the calculation of cross-section contains terms $c_{i}c_{j}$ where $i\neq j$. But for the special case where $\pi_{1}$ and $\pi_{2}$ of two photons are parallel or perpendicular, those terms ``drop out from the expression for the probability of scattering''. So, if both $\pi_{i}$'s are placed on $zx$-plane, or, one of them on $zx$-and the other one on $zy$-plane, then the lack of crossing terms allows one to calculate the probabilities for individual $\ket{\psi_{i}}$ and then sum their contributions. And, ``for each case, $\ket{\psi_{i}}$, ... this probability reduces to just the product of the probabilities of scattering of the single photons''. Which simplifies the calculations!
Calculated value of $R$ (average for experimental solid angle) for the case A is $2.00$, case B1 $1.00$, B2 $1.5$. The experimental value is $2.04\pm0.08$.
This is a strong argument for the existence of singlet state for distant particles and against the “hypotheses implying a breakdown of the quantum theory that could avoid paradox of ERP”.
I think this article is a good example of clarifying philosophical concepts and carrying them into the realms of physics by quantifying them. It supports the existence of the controversial singlet state by finding experimental discrepancies between the observational fact and alternative models. It demonstrates a usage of scattering experiments in a “fundamentals experiment” which can be used to test the existence of entanglement in a two photon system. I wonder whether further applications can be found for the same physical setup.
One other unsolved mystery that was left is why Bohm changed the order of initials from EPR to ERP?
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