##### Document Text Contents

Page 1

SpringerBriefs in Physics

For further volumes:

http://www.springer.com/series/8902

Editorial Board

Egor Babaev, University of Massachusetts, USA

Malcolm Bremer, University of Bristol, UK

Xavier Calmet, University of Sussex, UK

Francesca Di Lodovico, Queen Mary University of London, UK

Maarten Hoogerland, University of Auckland, New Zealand

Eric Le Ru, Victoria University of Wellington, New Zealand

James Overduin, Towson University, USA

Vesselin Petkov, Concordia University, Canada

Charles H.-T. Wang, The University of Aberdeen, UK

Andrew Whitaker, Queen’s University Belfast, UK

http://www.springer.com/series/8902

Page 54

fmb1.eps

42 3 The Einstein–Podolsky–Rosen Paradox, Bell’s Theorem and Nonlocality

Rosen paradox, what follows is a stronger conclusion. Not only do we have a conflict

of quantum mechanics with local realism, but Bell’s Theorem implies the incompat-

ibility of locality itself with quantum theory. I.e., one is left to conclude that quantum

theory is irreducibly nonlocal.

In this chapter, we review and elaborate on these arguments. We attempt to present

the issues in a very gradual manner, so that the essential nature of each step may

be easily grasped. We present first the Einstein–Podolsky–Rosen paradox and Bell’s

Theorem individually, and then consider the implications of their conjunction. It

is worth stressing here that the full ramifications of this chapter follow from this

combination which we present in the final section of the chapter.6

3.2 Review of the Einstein–Podolsky–Rosen Paradox

3.2.1 Rotational Invariance of the Spin Singlet State

and Perfect Correlations

The well-known work of Einstein, Podolsky, and Rosen, first published in 1935,

was not designed to address the possibility of nonlocality, as such. The title of the

paper was “Can Quantum Mechanical Description of Physical Reality be Considered

Complete?”, and the goal of these authors was essentially the opposite of authors

such as von Neumann: Einstein, Podolsky, and Rosen wished to demonstrate that the

addition of hidden variables to the description of state is necessary for a complete

description of a quantum system. According to these authors, the quantum mechan-

ical state description given by ψ is incomplete, i.e., it cannot account for all the

objective properties of the system. This conclusion is stated in the paper’s closing

remark: “While we have thus shown that the wave function does not provide a com-

plete description of the physical reality, we left open the question of whether such a

description exists. We believe, however, that such a theory is possible.”

Einstein, Podolsky, and Rosen arrived at this conclusion having shown that for

the system they considered, each of the particles must have position and momentum

as simultaneous “elements of reality”. Regarding the completeness of a physical

theory, the authors state: [1] (emphasis due to EPR) “Whatever the meaning assigned

to the term complete, the following requirement for a complete theory seems to be

a necessary one: every element of the physical reality must have a counterpart in

Footnote 5 (Continued)

a proof that realism is impossible in quantum physics. See Bethe [22], Gell-Mann [23], p. 172, and

Wigner [19], p. 291.

6 In particular, one cannot take the conclusion of the EPR paradox—the existence of noncontextual

hidden variables—as received and final. This conclusion is based not only on quantum mechanical

predictions, but also on the assumption of locality, which will ultimately be seen to fail. The status

of EPR becomes clearer when one recognizes that the analysis is in fact equivalent to a theorem, as

we demonstrate in Sect. 3.2.3.

Page 55

fmb1.eps

3.2 Review of the Einstein–Podolsky–Rosen Paradox 43

the physical theory”. This requirement leads them to conclude that the quantum

theory is incomplete, since it does not account for the possibility of position and

momentum as being simultaneous elements of reality. To develop this conclusion

for the position and momentum of a particle, the authors make use of the following

“sufficient condition” for a physical quantity to be considered as an element of

reality: “If without in any way disturbing a system, we can predict with certainty (i.e.

with probability equaling unity) the value of a physical quantity, then there exists an

element of physical reality corresponding to this quantity”.

What we shall present here7 is a form of the EPR paradox which was developed

in 1951, by David Bohm. Bohm’s EPR analysis involves the properties of the spin

singlet state of a pair of spin– 12 particles. Within his argument, Bohm shows that

various components of the spin of a pair of particles must be elements of reality

in the same sense as the position and momentum were for Einstein, Podolsky, and

Rosen. We begin with a discussion of the formal properties of the spin singlet state,

and then proceed with our presentation of the EPR incompleteness argument.

Because the spin and its components all commute with the observables associated

with the system’s spatial properties, one may analyze a particle with spin by separately

analyzing the spin observables and the spatial observables. The spin observables may

be analyzed in terms of a Hilbert space Hs (which is a two-dimensional space in the

case of a spin 12 particle) and the spatial observables in terms of L2(IR). The full

Hilbert space of the system is then given by tensor product Hs ⊗ L2(IR) of these

spaces. Hence, we proceed to discuss the spin observables only, without explicit

reference to the spatial observables of the system. We denote each direction in space

by its θ and φ coordinates in spherical polar coordinates, and (since we consider spin

1

2 particles) the symbol σ denotes the spin. To represent the eigenvectors of σθ, φ

corresponding to the eigenvalues + 12 and − 12 , we write | ↑ θ, φ〉, and | ↓ θ, φ〉,

respectively. For the eigenvectors of σz,we write simply |↑〉 and |↓〉.Often, vectors

and observables are expressed in terms of the basis formed by the eigenvectors of

σz . The vectors |↑ θ, φ〉, and |↓ θ, φ〉 when expressed in terms of these are

|↑ θ, φ〉 = cos(θ/2)| ↑〉 + sin(θ/2)eiφ | ↓〉

|↓ θ, φ〉 = sin(θ/2)e−iφ |↑〉 − cos(θ/2)|↓〉. (3.1)

For a system consisting of two spin 12 particles,

8 the states are often classified

in terms of the total spin S = σ (1) + σ (2) of the particles, where σ (1) is the spin

of particle 1 and σ (2) is the spin of particle 2. The spin singlet state, in which

we shall be interested, is characterized by S = 0. The name given to the state

reflects that it contains just one eigenvector of the z-component Sz of the total spin:

that corresponding to the eigenvalue 0. In fact, as we shall demonstrate, the spin

singlet state is an eigenvector of all components of the total spin with an eigenvalue

7 The Bohm spin singlet version and the original version of the EPR paradox differ essentially in

the states and observables with which they are concerned. We shall consider the original EPR state

more explicitly in Sect. 4.2.1.

8 See for example, Messiah [24], and Shankar [25].

http://dx.doi.org/10.1007/978-3-642-23468-2_4

Page 108

fmb1.eps

96 4 Schrödinger’s Paradox and Nonlocality

The linkage they make to human free will is quite superfluous, since the assumption

of free will can be dropped or even reversed in the argument without affecting its

conclusion. The argument’s assumptions SPIN, TWIN and FIN65 are just as reason-

able as the authors claim, but more powerful than they appear to have realized. By

the logical implications of the Schrödinger paradox (and Schrödinger’s Theorem)

these axioms are precisely what is needed to derive the inevitable existence of non-

contextual hidden variables on the appropriate components of S2

θ, φ

. The fact that the

Kochen and Specker theorem contradicts this leads finally to the conclusion that any

theory satisfying FIN must conflict with the quantum predictions.66 This is just the

same as the Schrödinger nonlocality argument we gave in Sect. 4.5.1, in which we

showed how a complex argument may be formed based on the Schrödinger paradox

and the Kochen and Specker theorem. What Conway and Kochen have developed is

a special case of the Schrödinger nonlocality proof.

All this might bring us to inquire how Schrödinger himself regarded his results,

and what further conclusions he drew within his remarkable paper. Clearly, it would

have been possible for him to argue for quantum nonlocality had he anticipated the

results of any of a wide variety of theorems including at least Gleason’s, Kochen

and Specker’s, or any other spectral incompatibility theorem. Instead, as we saw

in Chap. 1, Schrödinger essentially reproduced the von Neumann argument against

hidden variables, in his observation of a set of observables that obey a linear relation-

ship not satisfied by the set’s eigenvalues. What may be seen from von Neumann’s

result is just what Schrödinger noted—relationships constraining the observables do

not necessarily constrain their values. Schrödinger continued this line of thought by

speculating on the case for which no relation whatsoever constrained the values of the

various observables. In light of his generalization of the EPR paradox, these notions

led Schrödinger to the idea that the quantum system in question might possess an

infinite number of degrees of freedom, which concept is actually quite similar to that

of Bohmian mechanics.

If, on the other hand, Schrödinger had made von Neumann’s error, i.e., had con-

cluded the impossibility of a map from observables to values, this mistaken line of

reasoning would have allowed him to reach the conclusion of quantum nonlocal-

ity. Thus, insofar as one might regard the von Neumann proof as “almost” leading

to the type of conclusion that follows from Gleason’s theorem, one must consider

Schrödinger as having come precisely that close to a proof of quantum nonlocality.

It is quite interesting to see that many of the issues related to quantum mechanical

incompleteness and hidden variables were addressed in the 1935 work [8–10] of

Erwin Schrödinger. One could make the case that Schrödinger’s work is the most

far-reaching of the early analyses addressed to the subject. Schrödinger presented

here not only his famous “cat paradox,” but he developed results beyond those of the

Einstein, Podolsky, Rosen paper—an extension of their incompleteness argument,

65 As we noted in Sect. 4.5.2 there is nothing gained by appealing to the strong free will theorem

in which FIN is replaced by MIN.

66 Conflict with quantum mechanics cannot be laid at the feet of TWIN nor SPIN, since these are

taken from the theory itself.

http://dx.doi.org/10.1007/978-3-642-23468-2_1

Page 109

4.7 Summary and Conclusions 97

and an analysis of this result in terms closely related to von Neumann’s theorem.

It seems clear that the field of foundations of quantum mechanics might have been

greatly advanced had these features of Schrödinger’s paper been more widely appre-

ciated at the time it was first published.

References

1. Brown, H.R., Svetlichny, G.: Nonlocality and Gleason’s Lemma. Part I Deterministic Theories.

Found. Phys. 20, 1379 (1990)

2. Heywood, P., Redhead, M.L.G.: Nonlocality and the Kochen-Specker Paradox. Found. Phys.

13, 481 (1983)

3. Aravind, P.K.: Bell’s theorem without inequalities and only two distant observers. Found. Phys.

Lett. 15, 399–405 (2002)

4. Cabello, A.: Bell’s theorem without inequalities and only two distant observers. Phys. Rev.

Lett. 86, 1911–1914 (2001)

5. Bassi, A., Ghirardi, G.C.: The Conway-Kochen Argument and relativistic GRW Models. Found.

Phys. 37, 169 (2007). Physics archives: arXiv:quantph/610209

6. Tumulka, R.: Comment on ‘The Free Will Theorem’. Found. Phys. 37, 186–197 (2007). Physics

archives: arXic:quant-ph/0611283

7. Goldstein, S., Tausk, D., Tumulka, R., Zanghí, N.: What does the free will theorem actually

prove? Notices AMS 57(11):1451–1453 (2010) arXiv:0905.4641v1 [quant-ph]

8. Schrödinger, E.: Die gegenwA rtige Situation in der Quantenmechanik. Naturwissenschaften

23 807–812, 823–828, 844–849 (1935). The English translation of this work - ‘The present

situation in quantum mechanics’ - appears. In: Proceedings of the American Philosophical

Society 124, 323–338 (1980) (translated by J. Drimmer), and can also be found in [21, p. 152]

9. Schrödinger, E.: Discussion of probability relations between separated systems. Proc. Cam-

bridge Phil. Soc. 31, 555 (1935)

10. Schrödinger, E.: Probability relations between separated systems. Proc. Cambridge Phil. Soc.

32, 446 (1936)

11. Bohm, D.: Quantum Theory. Prentice Hall, Englewood Cliffs (1951)

12. Einstein, A., Podolsky, B., Rosen, N.: Con quantum mechanical description of physical reality

be considered complete? Phys. Rev. 47, 777 (1935). Reprinted in [21, p. 138]

13. Conway, J., Kochen, S.: “The Free Will Theorem” Found. Phys. 36(10); 1441–1473 (2006).

Physics archives arXiv:quant-ph/0604079

14. Conway, J., Kochen, S.: The strong Free Will Theorem. Notices AMS 56(2), 226–232 Feb

2009

15. Peres, A.: Quantum Theory: Concepts and Methods. Springer, Heidelberg (1995)

16. Bell, J.S.: Bertlemann’s socks and the nature of reality Journal de Physique. Colloque C2,

suppl. au numero 3, Tome 42 1981 pp. C2 41–61. Reprinted in [22, p. 139]

17. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Modern Phys. 38

447–452 (1966). Reprinted in [22, p. 1] and [21, p. 397]

18. Bell, J.S.: On the impossible pilot wave. Found. Phys. 12 989–999 (1982) Reprinted in

[22, p. 159]

19. Schilpp, P.A.: Albert Einstein: Philosopher-Scientist. Harper and Row, New York (1949)

20. Bell, J.S.: Quantum mechanics for cosmologists. p. 611 in: Quantum Gravity. vol. 2 Isham, C.,

Penrose, R., Sciama, D. (eds.) Clarendon Press, Oxford (1981). Reprinted in [22, p. 117]

SpringerBriefs in Physics

For further volumes:

http://www.springer.com/series/8902

Editorial Board

Egor Babaev, University of Massachusetts, USA

Malcolm Bremer, University of Bristol, UK

Xavier Calmet, University of Sussex, UK

Francesca Di Lodovico, Queen Mary University of London, UK

Maarten Hoogerland, University of Auckland, New Zealand

Eric Le Ru, Victoria University of Wellington, New Zealand

James Overduin, Towson University, USA

Vesselin Petkov, Concordia University, Canada

Charles H.-T. Wang, The University of Aberdeen, UK

Andrew Whitaker, Queen’s University Belfast, UK

http://www.springer.com/series/8902

Page 54

fmb1.eps

42 3 The Einstein–Podolsky–Rosen Paradox, Bell’s Theorem and Nonlocality

Rosen paradox, what follows is a stronger conclusion. Not only do we have a conflict

of quantum mechanics with local realism, but Bell’s Theorem implies the incompat-

ibility of locality itself with quantum theory. I.e., one is left to conclude that quantum

theory is irreducibly nonlocal.

In this chapter, we review and elaborate on these arguments. We attempt to present

the issues in a very gradual manner, so that the essential nature of each step may

be easily grasped. We present first the Einstein–Podolsky–Rosen paradox and Bell’s

Theorem individually, and then consider the implications of their conjunction. It

is worth stressing here that the full ramifications of this chapter follow from this

combination which we present in the final section of the chapter.6

3.2 Review of the Einstein–Podolsky–Rosen Paradox

3.2.1 Rotational Invariance of the Spin Singlet State

and Perfect Correlations

The well-known work of Einstein, Podolsky, and Rosen, first published in 1935,

was not designed to address the possibility of nonlocality, as such. The title of the

paper was “Can Quantum Mechanical Description of Physical Reality be Considered

Complete?”, and the goal of these authors was essentially the opposite of authors

such as von Neumann: Einstein, Podolsky, and Rosen wished to demonstrate that the

addition of hidden variables to the description of state is necessary for a complete

description of a quantum system. According to these authors, the quantum mechan-

ical state description given by ψ is incomplete, i.e., it cannot account for all the

objective properties of the system. This conclusion is stated in the paper’s closing

remark: “While we have thus shown that the wave function does not provide a com-

plete description of the physical reality, we left open the question of whether such a

description exists. We believe, however, that such a theory is possible.”

Einstein, Podolsky, and Rosen arrived at this conclusion having shown that for

the system they considered, each of the particles must have position and momentum

as simultaneous “elements of reality”. Regarding the completeness of a physical

theory, the authors state: [1] (emphasis due to EPR) “Whatever the meaning assigned

to the term complete, the following requirement for a complete theory seems to be

a necessary one: every element of the physical reality must have a counterpart in

Footnote 5 (Continued)

a proof that realism is impossible in quantum physics. See Bethe [22], Gell-Mann [23], p. 172, and

Wigner [19], p. 291.

6 In particular, one cannot take the conclusion of the EPR paradox—the existence of noncontextual

hidden variables—as received and final. This conclusion is based not only on quantum mechanical

predictions, but also on the assumption of locality, which will ultimately be seen to fail. The status

of EPR becomes clearer when one recognizes that the analysis is in fact equivalent to a theorem, as

we demonstrate in Sect. 3.2.3.

Page 55

fmb1.eps

3.2 Review of the Einstein–Podolsky–Rosen Paradox 43

the physical theory”. This requirement leads them to conclude that the quantum

theory is incomplete, since it does not account for the possibility of position and

momentum as being simultaneous elements of reality. To develop this conclusion

for the position and momentum of a particle, the authors make use of the following

“sufficient condition” for a physical quantity to be considered as an element of

reality: “If without in any way disturbing a system, we can predict with certainty (i.e.

with probability equaling unity) the value of a physical quantity, then there exists an

element of physical reality corresponding to this quantity”.

What we shall present here7 is a form of the EPR paradox which was developed

in 1951, by David Bohm. Bohm’s EPR analysis involves the properties of the spin

singlet state of a pair of spin– 12 particles. Within his argument, Bohm shows that

various components of the spin of a pair of particles must be elements of reality

in the same sense as the position and momentum were for Einstein, Podolsky, and

Rosen. We begin with a discussion of the formal properties of the spin singlet state,

and then proceed with our presentation of the EPR incompleteness argument.

Because the spin and its components all commute with the observables associated

with the system’s spatial properties, one may analyze a particle with spin by separately

analyzing the spin observables and the spatial observables. The spin observables may

be analyzed in terms of a Hilbert space Hs (which is a two-dimensional space in the

case of a spin 12 particle) and the spatial observables in terms of L2(IR). The full

Hilbert space of the system is then given by tensor product Hs ⊗ L2(IR) of these

spaces. Hence, we proceed to discuss the spin observables only, without explicit

reference to the spatial observables of the system. We denote each direction in space

by its θ and φ coordinates in spherical polar coordinates, and (since we consider spin

1

2 particles) the symbol σ denotes the spin. To represent the eigenvectors of σθ, φ

corresponding to the eigenvalues + 12 and − 12 , we write | ↑ θ, φ〉, and | ↓ θ, φ〉,

respectively. For the eigenvectors of σz,we write simply |↑〉 and |↓〉.Often, vectors

and observables are expressed in terms of the basis formed by the eigenvectors of

σz . The vectors |↑ θ, φ〉, and |↓ θ, φ〉 when expressed in terms of these are

|↑ θ, φ〉 = cos(θ/2)| ↑〉 + sin(θ/2)eiφ | ↓〉

|↓ θ, φ〉 = sin(θ/2)e−iφ |↑〉 − cos(θ/2)|↓〉. (3.1)

For a system consisting of two spin 12 particles,

8 the states are often classified

in terms of the total spin S = σ (1) + σ (2) of the particles, where σ (1) is the spin

of particle 1 and σ (2) is the spin of particle 2. The spin singlet state, in which

we shall be interested, is characterized by S = 0. The name given to the state

reflects that it contains just one eigenvector of the z-component Sz of the total spin:

that corresponding to the eigenvalue 0. In fact, as we shall demonstrate, the spin

singlet state is an eigenvector of all components of the total spin with an eigenvalue

7 The Bohm spin singlet version and the original version of the EPR paradox differ essentially in

the states and observables with which they are concerned. We shall consider the original EPR state

more explicitly in Sect. 4.2.1.

8 See for example, Messiah [24], and Shankar [25].

http://dx.doi.org/10.1007/978-3-642-23468-2_4

Page 108

fmb1.eps

96 4 Schrödinger’s Paradox and Nonlocality

The linkage they make to human free will is quite superfluous, since the assumption

of free will can be dropped or even reversed in the argument without affecting its

conclusion. The argument’s assumptions SPIN, TWIN and FIN65 are just as reason-

able as the authors claim, but more powerful than they appear to have realized. By

the logical implications of the Schrödinger paradox (and Schrödinger’s Theorem)

these axioms are precisely what is needed to derive the inevitable existence of non-

contextual hidden variables on the appropriate components of S2

θ, φ

. The fact that the

Kochen and Specker theorem contradicts this leads finally to the conclusion that any

theory satisfying FIN must conflict with the quantum predictions.66 This is just the

same as the Schrödinger nonlocality argument we gave in Sect. 4.5.1, in which we

showed how a complex argument may be formed based on the Schrödinger paradox

and the Kochen and Specker theorem. What Conway and Kochen have developed is

a special case of the Schrödinger nonlocality proof.

All this might bring us to inquire how Schrödinger himself regarded his results,

and what further conclusions he drew within his remarkable paper. Clearly, it would

have been possible for him to argue for quantum nonlocality had he anticipated the

results of any of a wide variety of theorems including at least Gleason’s, Kochen

and Specker’s, or any other spectral incompatibility theorem. Instead, as we saw

in Chap. 1, Schrödinger essentially reproduced the von Neumann argument against

hidden variables, in his observation of a set of observables that obey a linear relation-

ship not satisfied by the set’s eigenvalues. What may be seen from von Neumann’s

result is just what Schrödinger noted—relationships constraining the observables do

not necessarily constrain their values. Schrödinger continued this line of thought by

speculating on the case for which no relation whatsoever constrained the values of the

various observables. In light of his generalization of the EPR paradox, these notions

led Schrödinger to the idea that the quantum system in question might possess an

infinite number of degrees of freedom, which concept is actually quite similar to that

of Bohmian mechanics.

If, on the other hand, Schrödinger had made von Neumann’s error, i.e., had con-

cluded the impossibility of a map from observables to values, this mistaken line of

reasoning would have allowed him to reach the conclusion of quantum nonlocal-

ity. Thus, insofar as one might regard the von Neumann proof as “almost” leading

to the type of conclusion that follows from Gleason’s theorem, one must consider

Schrödinger as having come precisely that close to a proof of quantum nonlocality.

It is quite interesting to see that many of the issues related to quantum mechanical

incompleteness and hidden variables were addressed in the 1935 work [8–10] of

Erwin Schrödinger. One could make the case that Schrödinger’s work is the most

far-reaching of the early analyses addressed to the subject. Schrödinger presented

here not only his famous “cat paradox,” but he developed results beyond those of the

Einstein, Podolsky, Rosen paper—an extension of their incompleteness argument,

65 As we noted in Sect. 4.5.2 there is nothing gained by appealing to the strong free will theorem

in which FIN is replaced by MIN.

66 Conflict with quantum mechanics cannot be laid at the feet of TWIN nor SPIN, since these are

taken from the theory itself.

http://dx.doi.org/10.1007/978-3-642-23468-2_1

Page 109

4.7 Summary and Conclusions 97

and an analysis of this result in terms closely related to von Neumann’s theorem.

It seems clear that the field of foundations of quantum mechanics might have been

greatly advanced had these features of Schrödinger’s paper been more widely appre-

ciated at the time it was first published.

References

1. Brown, H.R., Svetlichny, G.: Nonlocality and Gleason’s Lemma. Part I Deterministic Theories.

Found. Phys. 20, 1379 (1990)

2. Heywood, P., Redhead, M.L.G.: Nonlocality and the Kochen-Specker Paradox. Found. Phys.

13, 481 (1983)

3. Aravind, P.K.: Bell’s theorem without inequalities and only two distant observers. Found. Phys.

Lett. 15, 399–405 (2002)

4. Cabello, A.: Bell’s theorem without inequalities and only two distant observers. Phys. Rev.

Lett. 86, 1911–1914 (2001)

5. Bassi, A., Ghirardi, G.C.: The Conway-Kochen Argument and relativistic GRW Models. Found.

Phys. 37, 169 (2007). Physics archives: arXiv:quantph/610209

6. Tumulka, R.: Comment on ‘The Free Will Theorem’. Found. Phys. 37, 186–197 (2007). Physics

archives: arXic:quant-ph/0611283

7. Goldstein, S., Tausk, D., Tumulka, R., Zanghí, N.: What does the free will theorem actually

prove? Notices AMS 57(11):1451–1453 (2010) arXiv:0905.4641v1 [quant-ph]

8. Schrödinger, E.: Die gegenwA rtige Situation in der Quantenmechanik. Naturwissenschaften

23 807–812, 823–828, 844–849 (1935). The English translation of this work - ‘The present

situation in quantum mechanics’ - appears. In: Proceedings of the American Philosophical

Society 124, 323–338 (1980) (translated by J. Drimmer), and can also be found in [21, p. 152]

9. Schrödinger, E.: Discussion of probability relations between separated systems. Proc. Cam-

bridge Phil. Soc. 31, 555 (1935)

10. Schrödinger, E.: Probability relations between separated systems. Proc. Cambridge Phil. Soc.

32, 446 (1936)

11. Bohm, D.: Quantum Theory. Prentice Hall, Englewood Cliffs (1951)

12. Einstein, A., Podolsky, B., Rosen, N.: Con quantum mechanical description of physical reality

be considered complete? Phys. Rev. 47, 777 (1935). Reprinted in [21, p. 138]

13. Conway, J., Kochen, S.: “The Free Will Theorem” Found. Phys. 36(10); 1441–1473 (2006).

Physics archives arXiv:quant-ph/0604079

14. Conway, J., Kochen, S.: The strong Free Will Theorem. Notices AMS 56(2), 226–232 Feb

2009

15. Peres, A.: Quantum Theory: Concepts and Methods. Springer, Heidelberg (1995)

16. Bell, J.S.: Bertlemann’s socks and the nature of reality Journal de Physique. Colloque C2,

suppl. au numero 3, Tome 42 1981 pp. C2 41–61. Reprinted in [22, p. 139]

17. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Modern Phys. 38

447–452 (1966). Reprinted in [22, p. 1] and [21, p. 397]

18. Bell, J.S.: On the impossible pilot wave. Found. Phys. 12 989–999 (1982) Reprinted in

[22, p. 159]

19. Schilpp, P.A.: Albert Einstein: Philosopher-Scientist. Harper and Row, New York (1949)

20. Bell, J.S.: Quantum mechanics for cosmologists. p. 611 in: Quantum Gravity. vol. 2 Isham, C.,

Penrose, R., Sciama, D. (eds.) Clarendon Press, Oxford (1981). Reprinted in [22, p. 117]