Lorentz-Invariant, Retrocausal, and Deterministic Hidden Variables

Abstract

We review several no-go theorems attributed to Gisin and Hardy, Conway and Kochen purporting the impossibility of Lorentz-invariant deterministic hidden-variable model for explaining quantum nonlocality. Those theorems claim that the only known solution to escape the conclusions is either to accept a preferred reference frame or to abandon the hidden-variable program altogether. Here we present a different alternative based on a foliation dependent framework adapted to deterministic hidden variables. We analyse the impact of such an approach on Bohmian mechanics and show that retrocausation (that is future influencing the past) necessarily comes out without time-loop paradox.

Appendix: A Foliation Dependent Bohmian Ontology for Bosonic Quantum Fields

We follow [71] and use the Schrödinger wave-functional picture.²⁰ For this purpose we consider in the Minkoswky flat space–time (as seen from a Lorentz frame with metric $\eta _{\mu \nu }$) the classical action $S=\int d^4x {\mathcal {L}}(\phi (x),\partial \phi (x),x)$ for a real scalar field $\phi (x)$. This action is equivalently analyzed using the curvilinear coordinate system $x':=[s,\xi ^i]$ ($i=1,2,3$) with the transformation $x^\mu =x^\mu (s,\xi ^i)$ defined such that s labels the leaves $\varSigma (s)\in {\mathcal {F}}_0$.

Following the ADM formalism [72] we define a ‘lapse’ function $N(x')$ and three tangential projections $p^\mu _i$ such that

$$\begin{aligned} dx^\mu =Nn^\mu ds+p^\mu _id\xi ^i \end{aligned}$$

(32)

and $p^\mu _in_\mu =0$ with $n^\mu$ the vector normal to the leaf $\varSigma (s)$ at point of coordinate $x^\mu$ (we use our freedom in the choice of coordinates to cancel the ‘shift’ function $N^i=0$ [71, 72]). With ADM notations we thus have $\frac{\partial x^\mu }{\partial s}=N n^\mu$, $\frac{\partial x^\mu }{\partial \xi ^i}=p^\mu _i$, $\frac{\partial s}{\partial x^\mu }=\frac{n_\mu }{N}$ characterizing the coordinate transformation. Writing $g'_{\mu \nu }$ the metric in the $x'$ coordinate system we deduce $g'_{00}=N^2$, $g'_{0i}=g'_{i0}=0$, $g'_{ij}=h_{ij}=\eta _{\mu \nu }p^\mu _i p^\nu _i$ and $\sqrt{-g'}=N\sqrt{-h}$ with $g'(x')$, and $h'(x')$ the determinant of $g'_{\mu \nu }$ and $h_{ij}$ respectively.

The action S for the field $\phi (x)=\phi '(s,\xi )$ reads now with ${\mathcal {L}}(\phi (x),\partial \phi (x),x)=\mathcal {L'}(\phi '(s,\xi ),\partial _s\phi '(s,\xi ),\nabla _i\phi '(s,\xi ),s,\xi )$:

$$\begin{aligned} S=\int dsd^3\xi N\sqrt{-h}\mathcal {L'}(\phi '(s,\xi ),\partial _s\phi '(s,\xi ),\nabla _i\phi '(s,\xi ),s,\xi ) \end{aligned}$$

(33)

with $\nabla _i$ a short hand notation for $\frac{\partial }{\partial \xi ^i}$, $\nabla _i\phi '=p^\mu _i\partial _\mu \phi$ and $\partial _s\phi '=N n^\mu \partial _\mu \phi$. Of course, Euler-Lagrange’s equation $\partial _s(\sqrt{-g'}\frac{\partial \mathcal {L'}}{\partial \partial _s\phi '})+\nabla _i(\sqrt{-g'} \frac{\partial \mathcal {L'}}{\partial \nabla _i\phi '})=\sqrt{-g'}\frac{\partial \mathcal {L'}}{\partial \phi '}$ deduced from Eq. 33 is rigorously equivalent to $\partial _\mu \frac{\partial {\mathcal {L}}}{\partial \partial _\mu \phi }=\frac{\partial {\mathcal {L}}}{\partial \phi }$ obtained in the x coordinate system in agreement with general relativistic covariance.

Writing $S=\int dsL_{\varSigma (s)}$ we use a Legendre transformation to define the Hamiltonian as $H_{\varSigma (s)}a=-L_{\varSigma (s)} +\int d^3\xi \varPi '\partial _s\phi '$ with $\varPi '=\frac{\delta L_\varSigma }{\delta \phi '}=\sqrt{-g'}\frac{\partial \mathcal {L'}}{\partial \partial _s\phi '}=\sqrt{-h}\frac{\partial {\mathcal {L}}}{\partial \partial _\mu \phi }n_\mu$ the canonical momentum conjugate to $\phi .$²¹ This entails

$$\begin{aligned} H_{\varSigma (s)}=\int d^3\xi N\sqrt{-h}(\frac{\partial \mathcal {L'}}{\partial \partial _s\phi '}\partial _s\phi '-\mathcal {L'})=\int _\varSigma d^3\sigma (x) N{\mathcal {H}}_\varSigma (x) \end{aligned}$$

(34)

with $d^3\sigma = d^3\xi \sqrt{-h}$ and ${\mathcal {H}}_\varSigma (x)=\mathcal {H'}_\varSigma (x')$ a foliation dependent scalar energy density such that ${\mathcal {H}}_\varSigma =T^{\mu \nu }n_\mu n_\nu$ with $T^{\mu \nu }=\frac{\partial {\mathcal {L}}}{\partial \partial _\mu \phi }\partial ^\nu \phi -\eta ^{\mu \nu }{\mathcal {L}}$ the full energy-momentum tensor.²² In the Hamilton formalism we have explicitly $\mathcal {H'}_\varSigma =\mathcal {H'}_\varSigma (\phi ',\pi '(s,\xi ),\nabla _i\phi ',x')$ where $\pi '_\varSigma =\frac{\varPi '}{\sqrt{-h}}=\frac{\partial {\mathcal {L}}}{\partial \partial _\mu \phi }n_\mu$ is introduced for further convenience.

In order to quantize this theory we introduce the equal-time commutation relations $[\hat{\phi '}(s,\xi ),\hat{\phi '}(s,\xi ')]=0=[\hat{\varPi '}(s,\xi ),\hat{\varPi '}(s,\xi ')]=0$ and

$$\begin{aligned} {[}\hat{\phi '}(s,\xi ),\hat{\varPi '}(s,\xi ')]=i\delta ^3(\xi -\xi ') \end{aligned}$$

(35)

written in the generalized Heisenberg picture adapted to the foliation where s plays the role of a time parameter.²³ The relation with the Schrödinger picture involves an unitary transformation such that for any local operator in the Heisenberg picture ${\hat{A}}^{(H)}(x):={\hat{A}}([\hat{\phi '}(s,\xi ),\hat{\varPi '}(s,\xi )],s)$ it exists a Schrödinger representation ${\hat{A}}^{(S)}(x):= {\hat{A}}([\hat{\phi '}(s_{in},\xi ),\hat{\varPi '}(s_{in},\xi )],s)$

$$\begin{aligned} {\hat{A}}^{(H)}(x)={\hat{U}}_{\varSigma (s),\varSigma _{in}(s_{in})}^{-1}{\hat{A}}^{(S)}(x){\hat{U}}_{\varSigma (s),\varSigma _{in}(s_{in})} \end{aligned}$$

(36)

where $s_{in}$ labels an initial leaf $\varSigma (s_{in})\in {\mathcal {F}}_0$.²⁴ The wave functional at time s (Schrödinger picture) is related to the one at time $s_{in}$ (Heisenberg picture) by $|\varPsi _{\varSigma (s)}\rangle ={\hat{U}}_{\varSigma (s),\varSigma _{in}(s_{in})}|\varPsi _{\varSigma _{in}(s_{in})}\rangle$ with the Schrödinger equation:

$$\begin{aligned} i\frac{d}{ds}|\varPsi _{\varSigma (s)}\rangle =\int _\varSigma d^3\sigma N\mathcal {{\hat{H}}'}_\varSigma (\hat{\phi '}(s_{in},\xi ),\nabla \hat{\phi '}(s_{in},\xi ),\frac{\hat{\varPi '}}{\sqrt{-h}}(s_{in},\xi ),s,\xi )|\varPsi _{\varSigma (s)}\rangle \nonumber \\ \end{aligned}$$

(37)

We emphasize that the Hamiltonian is here written in the Schrödinger picture.

To work with the Schrödinger functional representation we introduce the amplitude

$$\begin{aligned} \varPsi ([\phi '(\xi )],s)=\langle [\phi '(\xi )];s|\varPsi _{\varSigma _{in}(s_{in})}\rangle =\langle [\phi '(\xi )];s_{in}|\varPsi _{\varSigma (s)}\rangle \end{aligned}$$

(38)

with the eigenvectors condition $\hat{\phi '}(s,\xi )|[\phi '(\xi )];s\rangle =\phi '(\xi )|[\phi '(\xi )];s\rangle$ and the evolution $|[\phi '(\xi )];s\rangle ={\hat{U}}_{\varSigma (s),\varSigma _{in}(s_{in})}^{-1}|[\phi '(\xi )];s_{in}\rangle$. Furthermore, we have the representation

$$\begin{aligned} \langle [\phi '(\xi )];s_{in}|\hat{\varPi '}(\xi ,s_{in})|\varPsi (s)\rangle =-i\frac{\delta \varPsi ([\phi '(\xi )],s)}{\delta \phi '(\xi )} \end{aligned}$$

(39)

which yields

$$\begin{aligned} i\frac{\partial }{\partial s}\varPsi ([\phi '(\xi )],s)=\int _\varSigma d^3\sigma N\mathcal {\hat{H'}}_\varSigma (\phi '(\xi ),\nabla \phi '(\xi ),\frac{-i}{\sqrt{-h}}\frac{\delta }{\delta \phi '(\xi )},s,\xi )\varPsi ([\phi '(\xi )],s) \nonumber \\ \end{aligned}$$

(40)

Moreover, BM entails the introduction of field beables defined on $\varSigma (s)\in {\mathcal {F}}_0$. The foliation dependent formalism advocated here imposes thus the beables $\phi '(\xi ):=\phi '_{{\mathcal {F}}_0}(x')=\phi _{{\mathcal {F}}_0}(x)$ for points $x\in \varSigma (s)$ and allows us to write

$$\begin{aligned} \varPsi ([\phi '(\xi )],s):=\varPsi ([\phi _{{\mathcal {F}}_0}(x)]_{\varSigma (s)}) \end{aligned}$$

(41)

The fundamental Schrödinger wave-functional equation reads now

$$\begin{aligned} i\frac{\partial }{\partial s}\varPsi ([\phi _{{\mathcal {F}}_0}]_{\varSigma (s)})=\int _\varSigma d^3\sigma N\mathcal {\hat{H'}}_\varSigma (\phi _{{\mathcal {F}}_0},\nabla \phi _{{\mathcal {F}}_0},-i\frac{\delta }{\delta _\varSigma \phi _{{\mathcal {F}}_0}},x')\varPsi ([\phi _{{\mathcal {F}}_0}]_{\varSigma (s)}) \nonumber \\ \end{aligned}$$

(42)

(see footnotes 14 for notations). As an example we consider the field described classically by the Lagrangian ${\mathcal {L}}=\frac{1}{2}\partial _\mu \phi \partial ^\mu \phi - V(\phi )$ leading to the quantum Hamiltonian density (in the Schrödinger representation) $\mathcal {\hat{H'}}_\varSigma =\frac{\hat{\pi '}_\varSigma ^2}{2}-\frac{h^{ij}}{2}\nabla _i\hat{\phi '}\nabla _i\hat{\phi '}+V(\hat{\phi '})$. Eq. 42 entails

$$\begin{aligned} i\frac{\partial }{\partial s}\varPsi ([\phi _{{\mathcal {F}}_0}]_{\varSigma })=\int _\varSigma d^3\sigma N[\frac{-\delta ^2 }{2\delta _\varSigma \phi ^2_{{\mathcal {F}}_0}}-\frac{h^{ij}}{2}\nabla _i\phi _{{\mathcal {F}}_0}\nabla _i\phi _{{\mathcal {F}}_0}+V(\phi _{{\mathcal {F}}_0})]\varPsi ([\phi _{{\mathcal {F}}_0}]_{\varSigma }) \nonumber \\ \end{aligned}$$

(43)

The Madelung polar expansion $\varPsi ([\phi _{{\mathcal {F}}_0}]_{\varSigma (s)})=R([\phi _{{\mathcal {F}}_0}]_{\varSigma (s)})e^{iS([\phi _{{\mathcal {F}}_0}]_{\varSigma (s)})}$ leads to the Bohmian Hamilton–Jacobi equation

$$\begin{aligned} -\frac{\partial }{\partial s}S=\int _\varSigma d^3\sigma N[\frac{1}{2}\left( \frac{\delta S }{\delta _\varSigma \phi _{{\mathcal {F}}_0}}\right) ^2-\frac{h^{ij}}{2}\nabla _i\phi _{{\mathcal {F}}_0}\nabla _i\phi _{{\mathcal {F}}_0}+V(\phi _{{\mathcal {F}}_0})]+Q_\varSigma \nonumber \\ \end{aligned}$$

(44)

involving the quantum potential $Q_\varSigma =-\int _\varSigma d^3\sigma N\frac{1}{2R}\frac{\delta ^2 R }{\delta _\varSigma \phi ^2_{{\mathcal {F}}_0}}$, and the probability conservation

$$\begin{aligned} -\frac{\partial }{\partial s}R^2=\int _\varSigma d^3\sigma N \frac{\delta }{\delta _\varSigma \phi _{{\mathcal {F}}_0}}\left( R^2\frac{\delta S }{\delta _\varSigma \phi _{{\mathcal {F}}_0}}\right) \end{aligned}$$

(45)

from which we derive the probability conservation $\int {\mathcal {D}}\phi _{{\mathcal {F}}_0}R^2(s)=1$ (${\mathcal {D}}\phi _{{\mathcal {F}}_0}$ is a functional volume [73] defined in the configuration space at time s).

Most importantly, BM is driven by the guidance equation

$$\begin{aligned} \pi '_\varSigma =\frac{\delta S }{\delta _\varSigma \phi _{{\mathcal {F}}_0}(x)}=Im\left( \frac{1}{\varPsi }\frac{\delta \varPsi }{\delta _\varSigma \phi _{{\mathcal {F}}_0}(x)}\right) =n_{{\mathcal {F}}_0}^\mu (x)\partial _\mu \phi _{{\mathcal {F}}_0}(x)=\frac{\partial _s\phi _{{\mathcal {F}}_0}(x)}{N}\qquad \quad \end{aligned}$$

(46)

which is equivalent to Eq. 29 discussed in [25, 66, 67]. While BM is clearly a first-order dynamics we can yet deduce the Newton-like second-order differential equation by applying the functional derivative on both sides of Eq. 44. It yields:

$$\begin{aligned} \partial _\mu \partial ^\mu \phi _{{\mathcal {F}}_0}(x)= & {} \frac{1}{\sqrt{-g'}}\Bigg [\partial _s\Bigg (\frac{\sqrt{-h}}{N}\partial _s\phi '_{{\mathcal {F}}_0}(x')\Bigg )+\nabla _i(h^{ij}\nabla _j\phi '_{{\mathcal {F}}_0}(x'))\Bigg ]\nonumber \\= & {} -\frac{dV(\phi )}{d\phi }|_{\phi =\phi _{{\mathcal {F}}_0}(x)}-\frac{1}{N}\frac{\delta Q_\varSigma }{\delta _\varSigma \phi _{{\mathcal {F}}_0}(x)} \end{aligned}$$

(47)

which differs from the classical equation $\partial _\mu \partial ^\mu \phi _{{\mathcal {F}}_0}(x)=-\frac{dV(\phi )}{d\phi }|_{\phi =\phi _{{\mathcal {F}}_0}(x)}$ by the introduction of a nonlocal and foliation dependent quantum force responsible for the ‘super-implicate order’ advocated by Bohm and Hiley [3].

We emphasize that while we actually picked up a specific foliation ${\mathcal {F}}_0$ for representing the Schrödinger wave-functional problem, the full structure is still entirely relativistically covariant. To see this, we introduce the general transformation $|\varPsi _{\varSigma '}\rangle ={\hat{U}}_{\varSigma ',\varSigma }|\varPsi _{\varSigma }\rangle$ where $\varSigma , \varSigma '$ do not necessarily belong to ${\mathcal {F}}_0$. Let $x_\varSigma ^\mu$ be any point of $\varSigma (s)\in {\mathcal {F}}_0$. We then define an infinitesimal variation of the surface $\varSigma (s)\rightarrow \varSigma '$ by the transformation $x_\varSigma ^\mu \rightarrow x_\varSigma ^\mu +\epsilon n^\mu (x_\varSigma )$ where $\epsilon (\xi )$ is the infinitesimal and local amount of displacement normal to $\varSigma$. The unitary infinitesimal transformation relating $\varPsi ([\phi ]_{\varSigma '})$ and $\varPsi ([\phi ]_{\varSigma (s)})$ leads to

$$\begin{aligned} \varPsi ([\phi ]_{\varSigma '})-\varPsi ([\phi ]_{\varSigma (s)})\simeq & {} -i\int _\varSigma d^3\sigma \epsilon \mathcal {\hat{H'}}_\varSigma (\phi ,\nabla \phi ,-i\frac{\delta }{\delta _\varSigma \phi },x')\varPsi ([\phi ]_{\varSigma (s)}) \nonumber \\= & {} \int _\varSigma d^3\sigma \epsilon \frac{\delta }{\delta \varSigma (x)}\varPsi ([\phi ]_{\varSigma (s)}) \end{aligned}$$

(48)

where we introduced in the second line the definition of Schwinger’s functional derivative $\frac{\delta }{\delta \varSigma (x)}\varPsi ([\phi ]_{\varSigma (s)})$ [55, 74]. From this we deduce a multi-time Schwinger–Tomonaga equation [75] adapted to the Schrödinger–Heisenberg picture [76]

$$\begin{aligned} i\frac{\delta }{\delta \varSigma (x)}\varPsi ([\phi ]_{\varSigma })=\mathcal {\hat{H'}}_\varSigma (\phi ,\nabla \phi ,-i\frac{\delta }{\delta _\varSigma \phi },x')\varPsi ([\phi ]_{\varSigma }) \end{aligned}$$

(49)

which connects with the Bohmian description given in [25, 66, 67]. We emphasize that $[\mathcal {\hat{H'}}_\varSigma (x_1),\mathcal {\hat{H'}}_\varSigma (x_2)]=0$$\forall x_1,x_2\in \varSigma$ as it should be in this formalism [55, 75, 76].