Abstract
Based on his extension of the classical argument of Einstein, Podolsky and Rosen, Schrödinger observed that, in certain quantum states associated with pairs of particles that can be far away from one another, the result of the measurement of an observable associated with one particle is perfectly correlated with the result of the measurement of another observable associated with the other particle. Combining this with the assumption of locality and some “no hidden variables” theorems, we showed in a previous paper [11] that this yields a contradiction. This means that the assumption of locality is false, and thus provides us with another demonstration of quantum nonlocality that does not involve Bell’s (or any other) inequalities. In [11] we introduced only “spin-like” observables acting on finite dimensional Hilbert spaces. Here we will give a similar argument using the variables originally used by Einstein, Podolsky and Rosen, namely position and momentum.
We dedicate this paper to the memory of Giancarlo Ghirardi, who devoted his life to understanding quantum mechanics. He was a friend of John Bell, who was inspired by Giancarlo’s work. He was also a friend of two of us (J.B. and S.G.)
To appear in: Do wave functions jump? Perspectives on the work of G. C. Ghirardi. Editors: V. Allori, A. Bassi, D. Dürr, N. Zanghì.
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Notes
- 1.
In the literature on quantum mechanics, these theorems are often called “no hidden variables” theorems. But we prefer the expression “inexistence of a non-contextual value map” because, as we will discuss in Sect. 5, the expression “hidden variables” is really a misnomer.
- 2.
The correlations mentioned here are often called anti-correlations, for example when \(\tilde{O}=-O\), as in the example of the spin in remark 1 above.
- 3.
This is obvious for (2.3.2), a special case of (2.2.1). For (2.3.1) we observe that, since O is self-adjoint, we can write \(O = \sum _i \lambda _i P_{\lambda _i}\) where \(P_{\lambda _i}\) is the projector on the subspace of eigenvectors of eigenvalue \(\lambda _i\) of O and thus we have that \(f(O) = \sum _i f(\lambda _i) P_{\lambda _i}\). If we choose any f whose range is the set of eigenvalues of O and is such that \(f(\lambda _i)= \lambda _i\) \(\forall i\), we have that \(O=f(O)\) and, by (2.2.1), we obtain that \(v(O)=v(f(O))=f(v(O))\) and thus v(O) is an eigenvalue of O.
- 4.
This resembles a maximally entangled state, like (2.1.1), but it is not one because the sum in (3.1.1) extends to infinity and, for (3.1.1) to be a maximally entangled state, the set \(\{\psi _n\}_{n=1}^\infty \) should be orthonormal. But then the norm of (3.1.1) would be infinite and thus (3.1.1) would not belong the Hilbert space.
- 5.
This is a generalized wave function, which means that it is not an element of the Hilbert space \(L^2(\mathbb R^2)\), but rather a distribution, namely a linear function acting on a space of smooth functions that decay rapidly at infinity (see [33, Sect. 5.3] for a short introduction to distributions). We will not try to be rigorous about these generalized functions here, but we will give a regularized version of the same wave function in Sect. 3.6.
- 6.
This paper remained famous for his example of the cat that is “both dead and alive”, but that example will not concerned us here.
- 7.
We need two copies of the EPR state only in order to prove Theorem 4.1 below.
- 8.
Formally, since the state itself is not a vector in a finite dimensional Hilbert space. But, since we are not concerned here with mathematical rigor, we will put aside that issue.
- 9.
- 10.
We use lower case letters for the generic arguments of the wave function and upper case ones for the actual positions of the particles.
- 11.
The fact that the measurements of both the momentum and the position reveal the same intrinsic property may sound strange but that is just a peculiarity of the example considered here.
- 12.
This state ressembles a maximally entangled one, but it does not fit the definition of maximally entangled, since the Hilbert space here in infinite dimensional.
- 13.
That is because there is a unique solution of the first order equation (5.1) if the position is fixed at a given time.
- 14.
- 15.
In fact, \(X_m\) must lie between 0 and the nearest maximum of \(\cos ^2(kx + kY)\).
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Appendix 1: Proof of Clifton’s Theorem 4.1
Appendix 1: Proof of Clifton’s Theorem 4.1
The proof we give here is taken from a paper by Myrvold [29], which is a simplified version of the result of Clifton [14] and is similar to proofs of Mermin [28], and to Peres [31, 32] in the case of spins. We note that the same proof would apply to the regularized EPR state of Sect. 3.6.
Proof of Theorem 4.1
We will need the operators \(U_j (b)= \exp (-i b Q_j)\), \(V_j (c)= \exp (-i c P_j)\), \(j=1, 2\), with \(Q_j\), \(P_j\) defined by formulas (4.3), (4.4), but acting in \(L^2(\mathbb R^2)\) instead of \(L^2(\mathbb R^4)\), and \(b, c \in \mathbb R\). They act as
which follows trivially from (4.3), and
and similarly for \(V_2 (c)\). Equation (A.2) follows from (4.4) by expanding both sides in a Taylor series, for functions \(\Psi \) such that the series converges, and by extending the unitary operator \(V_2 (b)\) to more general functions \(\Psi \) (see, e.g., [33, Chap. 8] for an explanation of that extension).
We choose now the following functions of the operators \(Q_i\), \(P_i\,\):
where a is an arbitrary constant, and the functions are defined by (A.1), (A.2), and the Euler relations:
for \(j=1, 2\). Note that \(A_1,A_2, B_1,B_2\) are self-adjoint. By applying (4.8) several times to pairs of commuting operators made of products of such operators, we will derive a contradiction.
We have the relations
since the relevant operators act on different variables.
We can also prove:
To show (A.6), note that, from (A.1) and (A.2), one gets
for \(j=1,2\), which, for \(bc=\pm \pi \), means
Now use (A.4) to expand the product \(\cos (a Q_j) \cos (\pi P_j/ a)\), for \(j=1,2\), into a sum of four terms; each term will have the form of the left-hand side of (A.7) with \(b=\pm a\), \(c= \pm \pi / a\), whence \(bc=\pm \pi \). Then applying (A.8) to each term proves (A.6).
The relations (A.5) and (A.6) imply that
since two anticommutations (A.6) suffice to move the B’s to the left of the A’s. Similarly we have that
We also have, using (A.6) once, that
Thus, with \(C= (A_1 A_2)(B_2 B_1)\) and \(D= (A_1 B_2)(A_2 B_1)\), we have that
Now suppose there is a value map v as described in Theorem 4.1. Then, from (4.7) with \(f(x)=-x\), we have that
But by (A.5), (A.9) and (A.10), we also have, by (4.8), that
and
Thus \(v(C)= v(D)\). This is a contradiction unless \(v(C)=0\), i.e. unless at least one of \(v(A_i)\), \(v(B_i)\), \(i=1,2\) vanishes. But, by (4.7),
and
and thus a can be so chosen that \(v(A_i)\) and \(v(B_i)\) are all nonvanishing. \(\blacksquare \)
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Bricmont, J., Goldstein, S., Hemmick, D. (2021). EPR-Bell-Schrödinger Proof of Nonlocality Using Position and Momentum. In: Allori, V., Bassi, A., Dürr, D., Zanghi, N. (eds) Do Wave Functions Jump? . Fundamental Theories of Physics, vol 198. Springer, Cham. https://doi.org/10.1007/978-3-030-46777-7_2
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