User:YohanN7/Bell's theorem

A hidden variable theory

The EPR paper^[1] suggests that quantum mechanics is incomplete in a particular sense. Bell's paper^[2] considers several more predictive example theories that do abide to local realism and causality.

The setting is that proposed by Bohm.^[3] Consider "real" unit spin vectors associated with each spin $.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄2$ -particle with real components (like classical angular momentum), but only one of which is measurable at any instant. The other two components are regarded as hidden variables.^[4] Denote this real spin vector by $s$ . The modeling of this necessarily is ad hoc.^[5] That is, it signifies a solution designed for a specific problem, less formally, the postulated behavior is made up to ensure compliance with experimental facts and reasonable expected behavior.

One-particle theory

Consider first a system consisting of one spin $1 ⁄ 2$ -particle.

The system is assumed to have a physically real spin vector $s$ associated to it. The following assumptions are made:

All components of $s$ exist and have definite real values. (element of physical reality)
Only one component can be determined by experiment at any time (to comply with quantum mechanics)
The result for that component must be $\pm1$ in units of $ℏ ⁄ 2$ . (experimentally true)

It is not required that the measured value is the same as the real value. This is generally impossible by the third bullet.

Assume that $s$ may lie anywhere on the unit sphere. Postulate that the measured value in an arbitrary direction $a$ is

s_{a,e}=\operatorname {sgn} \mathbf {a} \cdot \mathbf {s} ,

(H1)

where $sgn$ is the signum function. The extra subscript $e$ signifies an experimentally measured value.

Suppose now that the system has a definite spin polarization vector $ρ$ . This is to say that a measurement of the $ρ$ -component $s ρ$ will with certainty yield the value $+1$ . Such states can be prepared by simply measuring $s ρ$ by means of a Stern–Gerlach apparatus and choosing the appropriate output channel and filtering the output to the plus-channel. It is important to note that in general $ρ \neq s$ . All that is certain is that $sgn ρ \cdot s = 1$ by (H1), in other words, $S ρ, e = 1$ . In yet other words, the vector $s$ lies in the upper hemisphere defined by $ρ$ .

Now fix the arbitrary unit vector $a$ along which a measurement is to be made. Denote the angle between $ρ$ and $a$ with $θ ρ, a$ . By the above assumptions, the probabilities $P +$ of a positive result is proportional to overlapping area of the hemispheres defined by $ρ$ and $a$ , see spherical lune. The probabilities for a positive and negative result are^[6]

{\begin{aligned}P_{+}&=kA_{overlap}={\frac {1}{2\pi }}(2\pi -2\theta _{\rho ,a})=1-{\frac {\theta _{\rho ,a}}{\pi }},\\P_{-}&={\frac {\theta _{\rho ,a}}{\pi }},\quad 0\leq \theta _{\rho ,a}\leq \pi .\end{aligned}}

(P1)

At this point it is already clear that the theory fails to the experimental prediction of quantum theory, which is^[7]

{\begin{aligned}P_{+}&=\cos \theta _{\rho ,a},\\P_{-}&=1-\cos \theta _{\rho ,a}\quad 0\leq \theta _{\rho ,a}\leq 2\pi .\end{aligned}}

Modified one-particle theory

Now modify the failing theory as such: Postulate that the measured value in an arbitrary direction $a$ is

s_{a,e}=\operatorname {sgn} \mathbf {a} '\cdot \mathbf {s} ,\quad a'=\mathbf {a} '(\mathbf {a} ,\mathbf {s} )

(H2)

that is, the third vecor $a'$ may depend on both $a$ and $s$ . With this prescription one finds for the expectation value^[8]

E_{\rho ,a}=\iint _{{\mathcal {S}}^{2}}\operatorname {sgn}(\mathbf {a} '\cdot \mathbf {s} ){\frac {1}{2\pi }}\sin \theta d\theta d\phi =1-2\theta _{a',\rho }/\pi ,

according to (P1), where $θ a', ρ$ is the angle between $a'$ and $ρ$ .^{[nb 1]}

Define $a'$ by starting from $a$ and rotating towards $ρ$ until such that

E_{\rho ,a,H}=\cos \theta _{a,\rho }=1-2\theta _{a',\rho }/\pi

holds, where $θ a, ρ$ is the angle between $a$ and $ρ$ . With this definition

E_{\rho ,a,H}=E_{Q},

i.e. the hidden variable theory, quantum mechanics and experiment all agree.

Two particle theory

In a two particle theory one may define similarly a spin polarization vector $ρ 2 = - ρ 1$ by an experiment on particle 1. The formalism is entirely

{\begin{aligned}P_{++}&={\frac {\theta _{a}}{\pi }},\\P_{+-}&=1-{\frac {\theta _{a}}{\pi }},\\P_{-+}&=1-{\frac {\theta _{a}}{\pi }},\\P_{--}&={\frac {\theta _{a}}{\pi }},\quad 0\leq \theta _{a}\leq 1.\end{aligned}}

The expectation expectation value of measurement of $s c$ in the polarization state $ρ$ then becomes

E_{h1}=1-{\frac {2\theta _{a}}{\pi }},

where $θ c$ is the angle between $ρ$ and $a$ . This constitutes a simple hidden variable theory. The prediction of quantum mechanics is

E_{q}=\cos \theta _{a},

and, most emphatically,

E_{h1}\neq E_{q}.

This theory can be modified into one that does agree with quantum mechanics. Postulate instead that the measured value in an arbitrary direction $a$ is

s_{a,e}=\operatorname {sgn} \mathbf {s} \cdot {\boldsymbol {\alpha }}(\mathbf {s} ,\mathbf {a} )=\cos \theta _{\alpha },\quad {\text{(h2)}},

where $α$ may depend on $a$ and $s$ . Set as definition of $θ α$

E_{h2}=1-{\frac {2\theta _{\alpha }}{\pi }}=\cos \theta _{a}=E_{q}.

This is achieved if $α$ is obtained from $a$ by rotation towards $ρ$ . With this,

E_{h2}=E_{q},

and the two theories agree with each other and with experiment.

Now let the system be composed of two spin $1 ⁄ 2$ -particles in the singlet state. Additional assumption:

The spin vectors are equal and opposite. (classical mechanics)

The question is whether, if given a polarization $ρ 1$ of the first particle, it is possible to define hidden variables such that their predictions agree with quantum mechanics and experiment. Tentatively, set

s_{2a,e}=\operatorname {sgn} \mathbf {s} _{2}\cdot {\boldsymbol {\beta }}(\mathbf {s} _{2},\mathbf {a} )=\cos \theta _{\beta },\quad {\text{(h3)}}.

Remarks

^ The integral is easiest solved "by inspection" as the sum of $+1$ times overlap area of hemispheres defined by $ρ$ and $a'$ ( $1 - θ a', ρ ⁄ π$ ) and $-1$ times the non-overlapping area ( $θ a', ρ$ ⁄π). (The limits of the integral over $φ$ depends on $Θ$ and $θ a', ρ$ .)

Notes

^ Einstein, Podolsky & Rosen 1935
^ Bell 1964
^ Bohm 1952a, Bohm 1952b
^ Greiner 2001, Section 17.3
^ Greiner 2001, Chapter 17.3
^ Greiner, 2003 & Section 17.3
^ Greiner, 2003 & Section 17.3. See also calculation in Bell's theorem#Bell inequalities
^ Bell 1964

References

Bell, J. (1964). "On the Einstein Podolsky Rosen Paradox" (PDF). Physics. 1 (3): 195–200. doi:10.1103/PhysicsPhysiqueFizika.1.195.

Bohm, D. (1952a). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I". Physical Review. 85 (2). American Physical Society: 166–179. doi:10.1103/PhysRev.85.166. {{cite journal}}: Unknown parameter |subscription= ignored (|url-access= suggested) (help)

Bohm, D. (1952b). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. II". Physical Review. 85 (2). American Physical Society: 180–192. doi:10.1103/PhysRev.85.180. {{cite journal}}: Unknown parameter |subscription= ignored (|url-access= suggested) (help)

Einstein, A.; Podolsky, B.; Rosen, N. (1935). "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" (PDF). Physical Review. 47 (10): 777–780. Bibcode:1935PhRv...47..777E. doi:10.1103/PhysRev.47.777.

Griner, W. (2001) [1989]. Quantum Mechanics - An Introduction (4th ed.). Springer Verlag. ISBN 3-540-67458-6.

[9] The integral is easiest solved "by inspection" as the sum of $+1$ times overlap area of hemispheres defined by $ρ$ and $a'$ ( $1 - θ a', ρ ⁄ π$ ) and $-1$ times the non-overlapping area ( $θ a', ρ$ ⁄π). (The limits of the integral over $φ$ depends on $Θ$ and $θ a', ρ$ .)

[1] Einstein, Podolsky & Rosen 1935

[2] Bell 1964

[3] Bohm 1952a, Bohm 1952b

[4] Greiner 2001, Section 17.3

[5] Greiner 2001, Chapter 17.3

[6] Greiner, 2003 & Section 17.3

[7] Greiner, 2003 & Section 17.3. See also calculation in Bell's theorem#Bell inequalities

[8] Bell 1964

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